EFFICIENT FINITE ELEMENT RESPONSE SENSITIVITY ANALYSIS AND APPLICATIONS IN COMPOSITES MANUFACTURING by ARMIN BEBAMZADEH B.Sc. (Civil Engineering), Sharif University of Technology, 2000 M.Sc. (Civil Engineering/Structural Engineering), University of Tehran, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Civil Engineering) The University of British Columbia (Vancouver) April 2009 Armin Bebamzadeh, 2009 Abstract ii Abstract This thesis presents the development, implementation, and application of response sensitivities in numerical simulation of composite manufacturing. The sensitivity results include both first- and second- order derivatives. Such results are useful in many applications. In this thesis, they are applied in reliability analysis, optimization analysis, model validation, model calibration, as well as stand-alone measures of parameter importance to gain physical insight into the curing and stress development process. In addition to novel derivations and implementations, this thesis is intended to facilitate and foster increased use of response sensitivities in engineering analysis. The work presented in this thesis constitutes an extension of the direct differentiation method (DDM). This is a method that produces response sensitivities in an efficient and accurate manner, at the one-time cost of deriving and implementing sensitivity equations alongside the ordinary response algorithm. In this thesis, novel extensions of the methodology are presented for the composite manufacturing problem. The derivations include all material, geometry, and processing parameters in both the thermochemical and the stress development algorithms. A state-of-the-art simulation software is developed to perform first-order sensitivity analysis for composite manufacturing problems using the DDM. In this software, several novel techniques are employed to minimize the computational cost associated with the response sensitivity computations. This sensitivity-enabled software is also used in reliability, optimization, and model calibration applications. All these applications are facilitated by the availability of efficient and accurate response sensitivities. The derivation and implementation of second-order sensitivity equations is a particular novelty in this thesis. It is demonstrated that it is computationally feasible to obtain second-order sensitivities (the “Hessian matrix”) by the DDM for inelastic finite element problems. It is demonstrated that the direct differentiation approach to the computation of first- and second-order response sensitivities becomes increasingly efficient as the problem size increases, compared with the less accurate and less efficient finite different approach. Table of Contents iii Table of Contents Abstract ....................................................................................................................................................... ii Table of Contents....................................................................................................................................... iii List of Tables................................................................................................................................................x List of Figures ............................................................................................................................................ xi Acknowledgements .................................................................................................................................. xiii Dedication................................................................................................................................................. xiv Co-Authorship Statement .........................................................................................................................xv Chapter 1. Introduction .......................................................................................................................1 1.1. Background ..................................................................................................................................2 1.2. Research Objectives and Scope ...................................................................................................6 1.3. Organization of Thesis .................................................................................................................7 1.3.1. First-order Response Sensitivities (Chapter 2) ........................................................................7 1.3.2. Application of Response Sensitivities (Chapter 3) ..................................................................7 1.3.3. Second-order Response Sensitivities (Chapter 4)....................................................................8 1.3.4. Additional Derivations and Software User’s Guide (Appendices)..........................................8 1.4. References..................................................................................................................................10 Chapter 2. Response Sensitivity and Parameter Importance Studies in Composite Manufacturing ...........................................................................................................................................15 2.1. Introduction................................................................................................................................15 2.2. Governing Equations..................................................................................................................19 Table of Contents iv 2.2.1. Thermochemical Model.........................................................................................................19 2.2.2. Stress and Deformation Model ..............................................................................................22 2.3. Direct Differentiation Methodology ..........................................................................................25 2.3.1. Differentiation of the Thermochemical Model ......................................................................25 2.3.2. Differentiation of the Stress Development Model .................................................................27 2.4. Response Sensitivity Algorithms and Efficient Software Implementation................................29 2.5. Numerical Example....................................................................................................................31 2.6. Summary and Conclusions.........................................................................................................36 2.7. Tables .........................................................................................................................................37 2.8. Figures........................................................................................................................................38 2.9. References..................................................................................................................................42 Chapter 3. Application of Response Sensitivity in Composite Manufacturing.............................46 3.1. Introduction................................................................................................................................46 3.2. Real-time Validation of Analysis Results with Response Sensitivities .....................................48 3.3. Model Calibration with Gradient-based Algorithms..................................................................51 3.4. Reliability Analysis with Gradient-based FORM......................................................................54 3.5. Optimization with Gradient-based Algorithm and Reliability Constraint .................................59 3.6. Conclusions and Comparison with State-of-the-Art ..................................................................61 3.7. Tables .........................................................................................................................................64 3.8. Figures........................................................................................................................................67 3.9. References..................................................................................................................................75 Table of Contents v Chapter 4. Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation ...........................................................................................................................................83 4.1. Introduction................................................................................................................................83 4.2. Need for Second-order Response Sensitivities ..........................................................................84 4.3. Review of First-order Sensitivity Computations........................................................................87 4.4. Second Differentiation of Governing Response Equations........................................................88 4.5. Second Differentiation of Constitutive Equations .....................................................................90 4.6. Overview of the Computation Process.......................................................................................91 4.7. Efficient Computer Implementation ..........................................................................................94 4.8. Numerical Examples ..................................................................................................................97 4.8.1. Tower Truss with Uniaxial J2 Plasticity Material Model ......................................................97 4.8.2. 2D Cylinder Under Internal Pressure with Multiaxial J2 Plasticity Model............................99 4.9. Conclusions..............................................................................................................................101 4.10. Tables .......................................................................................................................................102 4.11. Figures......................................................................................................................................103 4.12. References................................................................................................................................110 Chapter 5. Conclusions and Future Work .....................................................................................112 5.1. Summary of Research and Contributions ................................................................................112 5.2. Future Research Directions ......................................................................................................114 5.2.1. Future Research Topics .......................................................................................................114 5.2.2. Future Software Implementations Directions ......................................................................115 5.3. References................................................................................................................................116 Table of Contents vi Appendix A. Thermochemical Model Sensitivity Equations ...........................................................117 A.1. Degree of Cure Sensitivity ............................................................................................................117 A.2. Heat Capacity Matrix Sensitivity ..................................................................................................118 A.3. Conductivity Matrix Sensitivity ....................................................................................................120 A.4. Boundary Convection Matrix Sensitivity......................................................................................122 A.5. Load Vectors Sensitivities.............................................................................................................122 A.6. References .....................................................................................................................................125 Appendix B. Stress Development Model Sensitivity Equations.......................................................126 B.1. 3D Ply Stiffness Matrix Sensitivity..........................................................................................126 B.2. 3D Ply Initial Strains Sensitivity..............................................................................................128 B.3. Plane Strain Ply Stiffness Matrix Sensitivity ...........................................................................129 B.4. Plane Strain Initial Strain Sensitivity .......................................................................................131 B.5. References................................................................................................................................134 Appendix C. Overall Direct Differentiation Software Flow.............................................................135 C.1. Top-level Algorithm ................................................................................................................135 C.2. Thermochemical Module Algorithm........................................................................................135 C.3. Stress Development Module Algorithm...................................................................................136 C.4. Thermochemical Sensitivity Module Algorithm......................................................................138 C.5. Stress Development Sensitivity Module Algorithm.................................................................138 Appendix D. Thermochemical and Stress Development Model Parameters..................................140 Appendix E. Temperature and Spring-in Sensitivity Results..........................................................143 Table of Contents vii Appendix F. Second-order Sensitivity Equations for Uniaxial J2 Plasticity ..................................146 F.1. Phase 0: Response..........................................................................................................................147 F.2. Phase 1: First-order Sensitivity Results .........................................................................................148 F.3. Phase 2: First-order Derivatives of the History Variables .............................................................149 F.4. Phase 3: Second-order Sensitivity Results.....................................................................................150 F.5. Phase 4: Second-order Derivatives of the History Variables.........................................................152 F.6. References......................................................................................................................................155 Appendix G. Second-order Sensitivity Equations for Multiaxial J2 Plasticity ..............................156 G.1. Phase 0: Response .........................................................................................................................159 G.2. Phase 1: First-order Sensitivity Results: .......................................................................................162 G.3. Phase 2: First-order Derivatives of the History Variables.............................................................164 G.4. Phase 3: Second-order Sensitivity Results ....................................................................................166 G.5. Phase 4: Second-order Derivatives of the History Variables ........................................................170 G.6. References .....................................................................................................................................174 Appendix H. Second-order Sensitivity Pseudo-code.........................................................................175 H.1. Top-level 1 st and 2 nd – order DDM algorithm ...............................................................................175 H.2. Response module algorithm at load step k (phase 0) ....................................................................176 H.3. Element module algorithm............................................................................................................176 H.4. Material module algorithm (2D Plain Strain)................................................................................177 H.5. 1 st -order sensitivity at time step k (phase 1) ..................................................................................178 H.6. Element 1 st -order conditional sensitivity module algorithm .........................................................179 Table of Contents viii H.7. Material 1 st -order sensitivity module algorithm (2D Plain Strain)................................................179 H.8. Update 1 st -order history variables sensitivity at time step k (phase 2) ..........................................182 H.9. Update element 1 st -order history variables sensitivity module algorithm.....................................182 H.10. 2 nd -order sensitivity at time step k (phase 3) ...............................................................................183 H.11. Material 1 st -order sensitivity module algorithm (2D Plain Strain)..............................................184 Appendix I. User’s Guide to Software Implementations ................................................................189 I.1. Composites Manufacturing Sensitivity, Reliability, and Design Optimization using Direct Differentiation.......................................................................................................................................189 I.1.1. First-order Sensitivity Analysis (SENCOM).......................................................................190 I.1.1.1. How to Run SENCOM ...............................................................................................190 I.1.1.2. How to Develop the Input File....................................................................................190 I.1.1.3. Outputs of SENCOM..................................................................................................195 I.1.1.4. SENCOM Subroutines Organizations.........................................................................197 I.1.2. Reliability Analysis (SENCOM-REL) ................................................................................203 I.1.2.1. FERUM Subroutines Organizations ...........................................................................203 I.1.2.2. How to Link FERUM and SENCOM (SENCOM-REL) ............................................205 I.1.2.3. How to Develop Input File for SENCOM-REL .........................................................207 I.1.2.4. Outputs of SENCOM-REL .........................................................................................209 I.1.3. Design Optimization............................................................................................................210 I.1.3.1. Deterministic Optimization by SENCOM-OPT .........................................................210 I.1.3.2. Reliability-based Design Optimization (RBDO) using SECOM-OPT-REL ..............213 I.2. Second-order Sensitivity Analysis of Inelastic Finite Element.......................................................214 Table of Contents ix I.2.1. How to Develop an Input File and Perform the Analysis....................................................215 I.2.2. Subroutines Organization ....................................................................................................218 I.3. References.......................................................................................................................................222 List of Tables x List of Tables Table 2.1: Comparison of run time used for direct differentiation method (DDM) and finite difference method (FDM) to calculate the sensitivity of response in combined thermochemical and stress development model.............................................................................................................................37 Table 3.1: Thermochemical model parameters ...........................................................................................64 Table 3.2: Boundary condition, cure cycle, and geometry parameters (see Figs. 3.8a and 3.8b for the definition of these parameters) ...........................................................................................................65 Table 3.3: Comparison of run time used for direct differentiation method (DDM) and finite difference method (FDM) to calculate the sensitivity of response in thermochemical model and run time used for reliability analysis using FORM by the DDM and FDM..............................................................65 Table 3.4: Optimized time of curing and design variables for different composites thicknesses and different constraints ............................................................................................................................66 Table 4.1: Comparison of computational cost associated with the DDM compared to the FDM to compute first- and second-order response sensitivities for the first example. The stand-alone finite element response analysis takes 0.9 seconds..................................................................................................102 Table 4.2: Comparison of computational cost associated with the DDM compared to the FDM to compute first- and second-order response sensitivities for the second example. The stand-alone finite element response analysis takes 12.0 seconds for the 40-element-mesh, and 674.2 seconds for the 250- element-mesh....................................................................................................................................102 Table D.1: Thermochemical model parameters ........................................................................................140 Table D.2: L-shaped geometric parameters (see Fig. 2.2 for the definition of these parameters).............141 Table D.3: Stress development model parameters ....................................................................................142 Table E.1: Sensitivity of maximum part temperature for the [0 o ]8 and [0 o ]32 lay-ups ...............................143 Table E.2: Sensitivity of the corner and warpage components of spring-in for the model parameters θ1 to θ42, for the [0o]8 lay-up ..................................................................................................................144 Table E.3: Sensitivity of the corner and warpage components of spring-in for the model parameters θ43 to θ83, for the [0o]8 lay-up..................................................................................................................145 List of Figures xi List of Figures Figure 2.1: Schematic of the boundary value problem under consideration ...............................................38 Figure 2.2: Finite element mesh of the L-shaped unidirectional composite part on a solid aluminium tool and the thermo-mechanical boundary conditions applied before tool removal ..................................38 Figure 2.3: Temperature and degree of cure evolution for the [0 o ]8 , [0 o ]32 , and [0 o ]128 lay-ups..................39 Figure 2.4: Schematic showing the definition of corner ∆θ1 and warpage ∆θ2 components of the total spring-in angle ∆θ of an L-shaped part...............................................................................................39 Figure 2.5: Comparison of maximum part temperature sensitivity by DDM and FDM for E∆=21θ , for the [0 o ]8 lay-up ....................................................................................................................................40 Figure 2.6: Change in maximum part temperature due to a 1% change in the parameters for the [0 o ]8, [0 o ]32 , and [0 o ]128 lay-ups ....................................................................................................................40 Figure 2.7: Change in the corner component of spring-in (nominal value = 1.3 o ) due to a 1% change in the parameters for the [0 o ]8 lay-up ............................................................................................................41 Figure 2.8: Change in the warpage component of spring-in (nominal value = 0.06 o ) due to a 1% change in the parameters for the [0 o ]8 lay-up ......................................................................................................41 Figure 3.1: (a) Finite element mesh of the flat unidirectional composite part on a solid aluminium tool and the thermo-mechanical boundary conditions applied before tool removal, (b) typical 2-hold temperature cure cycle, and (c) part after removal tool ......................................................................67 Figure 3.2: (a) Half of finite element mesh of the unidirectional cured composite part on a solid aluminium tool and the external thermal boundary conditions (b) autoclave temperature and pressure during the process ...............................................................................................................................68 Figure 3.3: (a) Temperature at the part’s top surface and (b) interface of the tool and part where hP=60 (W/m 2 K), hTT=60 (W/m 2 K), and hTB=140 (W/m 2 K)...........................................................................68 Figure 3.4: Change in temperature at the part’s top surface due to a 1% change in the parameters ...........69 Figure 3.5: Temperature at the part’s top surface and (b) interface of the tool and part where hP=55 (W/m 2 K), hTT=55 (W/m 2 K), and hTB=100 (W/m 2 K)...........................................................................69 Figure 3.6: (a) Convective heat transfer coefficient on the part’s top surface (b) and tool bottom using sensitivity analysis and Eq. (3.10) by Johnston (1997) ......................................................................70 List of Figures xii Figure 3.7: (a) Temperature at the part’s top surface and (b) interface of the tool and part using Eq. (3.10) by Johnston (1997) for convective heat transfer coefficients .............................................................70 Figure 3.8: Half of finite element mesh of the unidirectional composite part on a solid aluminium tool and the thermo-mechanical boundary conditions (b) cure cycle ...............................................................71 Figure 3.9: Variation of maximum and minimum peak temperature ( peakTmax , peakTmin ) and maximum and minimum degree of cure at the end of the process ( maxα , minα ) for the variation for the variation of (a) initial degree of cure, α0; (b) activation energy, ∆E; (c) autoclave second ramp, s2 .....................72 Figure 3.10: (a) Cumulative Probability Function (CDF) and (b) Probability Density Function (PDF) for the maximum and minimum peak temperature ( peakTmax , peakTmin ) (c) CDF and (d) PDF for maximum and minimum degree of cure at the end of the process.......................................................................73 Figure 3.11: Values of the importance vectors δ and η for three different limit-state functions: (a,b) peakTg max200 −= and (c,d) max825.0 α−=g ..................................................................................74 Figure 4.1: Extended objectives of a generic constitutive model enable second-order response sensitivity computations.....................................................................................................................................103 Figure 4.2: (a) Tower truss with uniaxial J2 material model and (b) time variation of the load at the top of the tower ...........................................................................................................................................104 Figure 4.3: (a) Force-displacement response at the top of truss and (b to e): first-order response sensitivities of the displacement response ........................................................................................105 Figure 4.4: Second-order displacement response sensitivity results .........................................................106 Figure 4.5: (a) Quarter model of a circular cylinder cross-section and (b) variation of the internal pressure, P ........................................................................................................................................107 Figure 4.6: (a) Pressure-displacement response and (b to e) first-order response sensitivities of the displacement response ......................................................................................................................108 Figure 4.7: Second-order displacement response sensitivity results .........................................................109 Acknowledgements xiii Acknowledgements I would like to acknowledge several individuals who truly made this work possible. First, I would like to thank my supervisors Dr. Terje Haukaas who has given me the chance to participate in several interesting research projects and submit various papers to the international conferences and journals. He has supported me with his encouragement and many fruitful discussions from the very beginning until the end, the huge amount of time which he always had for me, and all his productive tips and hints. I would also like to express my sincere thanks to my co-supervisors Dr. Reza Vaziri and Dr. Anoush Poursartip for their invaluable experiences and advices. I gratefully thank Dr. Göran Fernlund for his valuable comments on my papers. I would also thank Dr. Foschi for his great influence on my work. Financial support in the form of a University Graduate Fellowship from The University of British Columbia, for the last two years of my study is gratefully acknowledged. I also thank the funding provided by Dr. Terje Haukaas, Dr. Reza Vaziri, and Dr. Anoush Poursartip from the Natural Sciences and Engineering Research Council (NSERC) as well as Auto21 Network of Centers of Excellence of Canada. Many thanks go to my colleague, Smitha Kuduru and Mojtaba Mahsuli for scientific discussions and the pleasure of working together in the UBC Reliability Group. I also thank all the members of the UBC Composites Group for their friendship and help. Especially, I would like to thank Ali Rasekh, Ahamed Arafath, Alireza Forghani, Mehdi Haghshenas, and Navid Zobeiry for many fruitful discussions and their assistance on many occasions during the course of my studies. I truly value the friendship of several close friends, Anoush Hafezi, Hamid Karimian, Hamid Abdollahi, and Reza Molavi. I appreciate their enthusiasm, support, and encouragements during the highs and lows of my degree. Finally, my greatest thanks are due to my family. Their continuous love and patience are the keys to all my achievements. Dedication xiv Dedication This thesis is dedicated to my beloved parents. Verbal and written acknowledgments can never show a fraction of my love for them and their support for me. Co-Authorship Statement xv Co-Authorship Statement The main body of this thesis consists of two published journal papers (Chapter 2 and 4) and one submitted journal paper (Chapter 3). The paper in Chapter 2 presents my own derivations, implementation, and analyses that I carried out after being offered the topic of developing response sensitivity analysis in composite manufacturing simulation. I drafted the initial version of the paper and finalized it in an iterative process, with input from my co-supervisors Dr. Terje Haukaas and Dr. Reza Vaziri and subsequently from the other two supervisory committee members; Dr. Anoush Poursartip and Dr. Göran Fernlund. The paper in Chapter 3 is the result of suggestions from my supervisory committee to carry out applications of response sensitivity analysis. I carried out all the analyses with the Matlab®- based software that I created in the work related to Chapter 2. I provided the initial draft of the paper and finalized it in close collaboration with Dr. Terje Haukaas. Dr. Göran Fernlund provided substantial input for Section 3.2. The paper in Chapter 4 is based on an initial idea from Dr. Terje Haukaas, from which I carried out all the derivations and computer implementations. I drafted the initial version of this paper and it was finalized in collaboration with Dr. Terje Haukaas. The appendices of this thesis are written in their entirety by me. Chapter 1: Introduction 1 Chapter 1. INTRODUCTION A modern objective in many fields of engineering is the simulation of the behaviour of complex physical phenomena. Progress towards this goal has been made because of rapid increase in computer power and drastic improvements in predictive models. In particular, the finite element method has become a widely utilized tool for predicting the response of boundary value problems that model real-world mechanical systems. However, it is argued in this thesis that the prudent use of the finite element method must include sensitivity analysis in order to assess confidence in the results. Moreover, sensitivity analysis facilitates the treatment of uncertainties by means of reliability analysis, as well as gradient-based model calibration and optimization analysis. This motivates the work that is presented in this thesis. Essentially, sensitivity analysis is aimed at determining the model’s output variations resulting from changes to the input parameters. The simplest approach to compute the sensitivities is the finite difference approaches, such as the forward finite difference or central difference approaches. This approach is straightforward to implement and is based on perturbing one parameter at a time and repeatedly re- running the model. However, this approach suffers from accuracy concerns and requires a large number of re-runs of the finite element model when sensitivities are sought with respect to many parameters. Furthermore, calculating second-order sensitivities by the finite difference approach is increasingly cumbersome. This motivates the study of more accurate and efficient methods for computing the sensitivity of finite element responses. The direct differentiation method (DDM) offers an attractive alternative to the finite difference approach. In this approach, the response equations are analytically differentiated and implemented in the software alongside the ordinary response computations. DDM provides efficient and accurate response sensitivities at the one-time cost of deriving and implementing analytical sensitivity equations. Therefore, a large part of this study is devoted to the derivation and efficient implementation of the DDM to Chapter 1: Introduction 2 calculate first-order sensitivity of composites manufacturing problems, as well as the second-order sensitivities of inelastic finite element responses. In this thesis it is also demonstrated how the DDM facilitates gradient-based reliability analysis and optimization, as well as model calibration and validation in composites manufacturing. 1.1. BACKGROUND This section presents a brief overview of existing literature that is relevant to the overall research thrust of this thesis. In particular, various definitions and assumptions employed by other researchers are provided. In addition, an overview of alternative methods to perform sensitivity analysis, along with corresponding traditional applications, is discussed. Conversely, literature that pertains specifically to the journal papers presented in the subsequent three chapters is referenced in those chapters. Sensitivity analysis can be regarded as an analytical tool that may accompany any numerical modeling process (Kleiber 1997). Sensitivity analysis applications – and thus definitions and associated assumptions – vary significantly depending on the area of utilization. Sensitivity analysis tools are found in many disciplines, including advanced engineering, economics, physics, chemistry, computer sciences, statistics, social sciences, and medical decision making. Among the wide range of applications, two main definitions are found relevant to the contents of this study. Saltelli et al. (2000) define sensitivity analysis as the study of the output quantitative and/or qualitative variations as the result of changes to the model’s input parameters and assumptions. Similarly, Frey and Patil (2002) characterize sensitivity analysis as an assessment tool to analyze the impact of changes in input values on the model’s output. Interestingly, Saltelli et al. (2006) conduct a detailed review of previous research to explore recent articles containing sensitivity analysis as a keyword. Sensitivity analysis was originally developed to study the uncertainties associated with the model input variables parameters (Saltelli et al. 2000). Therefore, sensitivity analysis and uncertainty analysis are closely related and often utilized in combination. A key objective in uncertainty analysis is to identify Chapter 1: Introduction 3 output uncertainties resulting from uncertainties in the model’s input and assumptions. In this context, sensitivity analysis focuses on identifying influential parameters and assumptions that influences the model output (Saltelli 2000). Saltelli et al. (2000) categorize sensitivity analysis methods as “local” and “global.” Local sensitivity analysis is performed by partial derivative computation of the output functions with respect to the input parameters. For nonlinear models, local sensitivities are accurate only in the vicinity of the realization of the parameters at which the sensitivities are computed. This is the case for the sensitivities that are computed in this thesis. Turanyi (1990) and Turanyi and Rabitz (2000) classify the numerical methods for computing local sensitivities in two groups: 1) The finite difference method, which is the most straightforward method to calculate the local sensitivities. It is based on perturbation of one parameter at a time, while keeping all other fixed. 2) The DDM, in which the response equations are analytically differentiated and implemented on the computer alongside the ordinary response computations. This method provides efficient and accurate response sensitivities at the one-time cost of differentiating and implementing analytical sensitivity equations. Based on its accuracy and efficiency properties, the DDM approach is the best method for the numerical calculation of local sensitivities, as confirmed by Turanyi and Rabitz (2000). Many researchers have extended the DDM in the context of structural finite element analysis, including Choi and Santos (1987), Tsay and Arora (1990), Liu and Der Kiureghian (1991), Zhang and Der Kiureghian (1993), Kleiber et al. (1997), Roth and Grigoriu (2001), Cont et al. (2003), Scott et al. (2003), and Haukaas and Der Kiureghian (2005). Developments are also found in the context of composite manufacturing, including Hou and Sheen (1987), Li et al. (2001), Li and Tucker (2002), and Zhu and Geubelle (2002). The Green Function (Hwang et al. 1978) and Polynomial Approximation methods (Hwang 1983) are also discussed by Turanyi and Rabitz (2000) for local sensitivity computations, but these are not widely used. Chapter 1: Introduction 4 While a local first-order sensitivity analysis produces the mathematical gradient vector of partial derivatives of a response function with respect to its input parameters, a sensitivity analysis is considered to be global if all the parameters are changed simultaneously and/or the impact of the range of the probability distribution for each input parameter are incorporated into the analysis. This approach is outside the scope of this study, and the interested reader may use the literature in Saltelli et al. (2000) and Saltelli et al. (2008) as a starting point for further studies in that direction. Local sensitivity analysis, for brevity called sensitivity analysis in this thesis, is associated with a wide range of applications. Frey and Patil (2002) and Frey et al. (2004) indicate that the sensitivity analysis is crucial to a model’s verification and validation in the development and refinement processes (see also Kleijnen 1995 and Fraedrich and Goldberg 2000). The main objective of the sensitivity analysis in model verification is to ensure that the sensitivity results correspond to the expected sensitivity of the model response. If a model responds inappropriately to one or more input changes then the source of the discrepancy should be investigated and resolved. Frey et al. (2004) also indicate that sensitivity analysis is helpful in the validation process. Significant contradictions in the relative impact of input changes and real world experiences would indicate potential flaws in the model. Frey et al (2004) also state that the identification of “insignificant” input parameters should lead to the elimination of corresponding input or components, and thus simplification of the model. This is an integrated part of “mechanism reduction” and “model lumping,” where a simpler model can be built or extracted from a more complex one (see also Saltelli and Scott 1997, Saltelli et al. 1999, Turanyi 1990, Turanyi and Rabitiz 2000, and Saltelli et al. 2006). Similarly, Saltelli and Scott (1997), Rabitz (2001), Saltelli (2002), Saltelli et al. (2000), and Saltelli et al. (2004) utilize sensitivity analysis to identify a set of significant parameters for model calibration. This idea is adopted in Chapter 4 of this thesis. Local sensitivities are also useful in certain design optimization procedures. For example, gradient-based optimization algorithms require the derivatives of the objective function and constraint function(s). This necessitates the local derivation of the model’s response with respect to the design variables Chapter 1: Introduction 5 (Schittkowski 1986 and Polak 1997). Local sensitivity analysis is also an important ingredient in the first- and second-order reliability methods (FORM and SORM). These analyses provide a means of approximately obtaining the probability of response events in the presence of input parameter uncertainty (Der Kiureghian and Taylor 1983; Ditlevsen and Madsen 1996). In this methodology, optimization algorithms are required to identify the most likely realization of the random variables that will cause the response event. An important class of optimization algorithms to solve this problem requires the derivatives of the response with respect to each of the random variables (Hasofer and Lind 1974; Schittkowski 1986; Liu and Der Kiureghian 1991; Zhang and Der Kiureghian 1997; and Polak 1997). In the field of sensitivity analysis, the “adjoint method” appears in the literature (Haug and Rousselet 1980 a and b, Cea 1981, Zolesio 1981, Dems and Mroz 1983, Dems and Mroz 1984, Choi 1985, Haug et al. 1986, Haug and Choi 1986, Haber 1987, Dems and Haftka 1989, Dems and Mroz 1993, and Kleiber 1997). The primary objective of this methodology is to avoid the computation all response sensitivities. Rather, in the adjoint method, the objective function and constraint sensitivities are computed by solving the “adjoint problem.” In general optimization problems, this is preferred compared when the number of the responses involved in the objective function and constraints is small compared to the number of input parameters. However, the adjoint method is not suitable for path-dependent problems, in which the derivatives of the history variables are required at each analysis step (Kleiber 1997, Cho and Choi 2000). Thus, the adjoint method is not appropriate for the path-dependent problems considered in this thesis. Certain reliability and optimization algorithms also require second-order sensitivities. Examples include SORM (Ditlevsen and Madsen 1996) and the sequential quadratic programming algorithm (Luenberger 1984 and Polak 1997). Saltelli et al. (2000) show that these higher order sensitivities are also useful to study the interaction between parameters. The computation of second-order sensitivities by the DDM is a particular novelty in this thesis. Chapter 1: Introduction 6 Frey and Patil (2002) mention additional applications of sensitivity analysis, including risk analysis aimed at identifying the most significant “exposure or risk factors” and prioritizing the risk mitigation sequence (see also Baker et al. 1999 and Jones 2000). Saltelli et al. (2000) and Frey and Patil (2002) also apply sensitivity analysis to examine the modeling assumptions to identify the “robustness” of the model outputs. 1.2. RESEARCH OBJECTIVES AND SCOPE The research presented in this thesis extends and applies the DDM as an accurate and efficient tool for sensitivity analysis. The main objective is to facilitate and foster sensitivity analysis in practical applications. This is achieved by analytical derivations and implementation of efficient DDM algorithms to compute both first-order and second-order sensitivity results, with particular emphasis on the numerical simulation of composites manufacturing. The sensitivity analysis results are subsequently utilized to demonstrate several applications, including reliability analysis, design optimization, and parameter-value- calibration for the composites manufacturing problem. In addition, novel verification and validation approaches are presented, utilizing the DDM. The scope of the research is not limited to composite manufacturing problems, although the applications are limited to that field. For example, the development of second-order sensitivities by the DDM is applicable to the broad range of applications of the finite element method, including other structural mechanics problems. The DDM is comprehensively implemented to obtain the response sensitivities with respect to all attributes involved in composites processing, including material, geometric, and processing parameters. The novel second-order sensitivity equations are developed for general inelastic finite element problems, but in this thesis the implementations are limited to the uniaxial and multiaxial J2 plasticity inelastic material model. Several novel implementation strategies are investigated to improve the computational efficiency of the DDM for both first- and second- order sensitivity computations. These implementation strategies can be utilized in any finite element based implementation of the DDM as well. Chapter 1: Introduction 7 1.3. ORGANIZATION OF THESIS This thesis is “manuscript-based”. Consequently, each of the main chapters (excluding the Introduction, and the Conclusions) are produced as stand-alone manuscripts for journal publication. Each chapter’s content is assembled to be independent of the other chapters. The study commences with the response sensitivity and parameter importance studies in composites manufacturing described in Chapter 2. Next, Chapter 3 presents practical applications of DDM in composites manufacturing problems. Chapter 4 present novel developments to obtain second-order sensitivities of inelastic finite element response by the DDM. Chapter 5 presents the concluding remarks. A synopsis of each chapter is as follows. 1.3.1. First-order Response Sensitivities (Chapter 2) In this chapter, the derivation, implementation, and verification of equations to compute the sensitivity of responses from numerical simulation of composites manufacturing are presented. The responses considered are part temperature and degree of cure, as well as process-induced deformation of the cured part. The direct differentiation method is used, which entails a one-time investment of effort to differentiate the governing response equations analytically. The implementation of the derivative equations facilitates efficient and accurate computation of response sensitivities in all subsequent analyses. This paper extends the direct differentiation methodology developed earlier for mechanical problems. Novel shape sensitivity equations and efficient implementation techniques are also included. In order to verify the implementations, the model predictions are compared with those obtained from the less efficient finite difference approach. A comprehensive example is presented where the usefulness and interpretation of response sensitivities are emphasized. It is observed that the responses are particularly sensitive to certain model parameters, for which further data gathering and model improvement efforts should be focused. 1.3.2. Application of Response Sensitivities (Chapter 3) This chapter demonstrates the use of response sensitivities in numerical simulation of composite manufacturing. A state-of-the-art simulation software with response sensitivity capabilities is utilized to Chapter 1: Introduction 8 perform real-time result validation, model calibration, reliability analysis, and optimization. The applications are facilitated by the efficient and accurate direct differentiation method. Several comprehensive numerical examples are presented and analyzed. 1.3.3. Second-order Response Sensitivities (Chapter 4) In this chapter analytical equations are developed and implemented to obtain second-order derivatives of finite element responses with respect to input parameters. The work extends previous work on first-order response sensitivity analysis. Of particular interest is the computational feasibility of obtaining second- order response sensitivities. In the past, the straightforward finite difference approach has been available, but this approach suffers from serious efficiency and accuracy concerns. In this study it is demonstrated that analytical differentiation of the response algorithm and subsequent implementation on the computer provides second-order sensitivities at a significantly reduced cost. The sensitivity results are consistent with and have the same numerical precision as the ordinary response. The computational cost advantage of the direct differentiation approach increases as the problem size increases. Several novel implementation techniques are developed in this paper to optimize the computational efficiency. The derivations and implementations are demonstrated and verified with two finite element analysis examples. 1.3.4. Additional Derivations and Software User’s Guide (Appendices) Appendices A to E are related to the first journal paper in Chapter 2. Detailed sensitivity equations of thermochemical and stress development model involved composite manufacturing simulation are presented in Appendices A and B. The details of software implementation utilized in this manuscript is presented in Appendix C. Appendix D shows the thermochemical and stress development model parameters involved in this manuscript. Appendix E summarizes the sensitivity results with respect to the model parameters in Appendix D. Appendices F and G are related to the journal paper that is presented in Chapter 4 (on second- order response sensitivities). Due to space limitations in the journal paper, only certain aspects of the Chapter 1: Introduction 9 second-order sensitivity derivations were presented in the manuscript`s appendices. In fact, the paper references this thesis for further details. Hence, Appendices F and G in this thesis presents the detailed and comprehensive equations for the uniaxial and multiaxial J2 material models, both for first- and second-order sensitivities. Appendix H shows the step-by-step pseudo code that is necessary to implement second-order response sensitivity capabilities in any finite element software. Appendix I presents a user’s guide to the new Matlab based software developed in this thesis. The derivations and implementations presented have resulted in a comprehensive software for simulation of composite manufacturing, with emphasis on first-order sensitivity results. 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Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 15 Chapter 2. RESPONSE SENSITIVITY AND PARAMETER IMPORTANCE STUDIES IN COMPOSITE MANUFACTURING 1 2.1. INTRODUCTION The primary objective in this chapter is to develop and implement response sensitivity equations for the purposes of parameter importance studies. The problem under consideration is the numerical simulation of composites manufacturing. Previously developed numerical models for prediction of temperature, degree of cure, and residual stresses are differentiated analytically. Next, the derivative equations are implemented into a comprehensive in-house finite element code for analysis of composites processing problems. Subsequently, sensitivity and parameter importance measures are utilized to provide physical insight, and to offer understanding of the robustness of the process. It is noted that response sensitivities are useful also in other applications, including reliability and optimization analysis. The long-term goal of this study is the development of robust numerical tools to simulate composites manufacturing. This will enable engineers to predict the outcome of a manufacturing process and in doing so reduce the reliance on time consuming and costly trial-and-error based experiments. The added benefit of quantifying the response sensitivities of model predictions is that the efforts required in material characterization and model enhancements can be directed towards a subset of parameters and submodels. Through efficient techniques response sensitivities can readily be quantified within the same computational run used for standard process simulation. 1 A version of this chapter has been published. Bebamzadeh, A., Haukaas, T., Vaziri, R., Poursartip, A., and Fernlund, G., “Response sensitivity and parameter importance studies in composite manufacturing.” Journal of Composites Materials, 43(6), 621-569. Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 16 Several strategies are available to improve the numerical models and to gain physical insight into composites manufacturing. Of particular interest in this study is the use of response sensitivities. By obtaining information about the change in the response (temperature, degree of cure, and residual stresses) due to changes in the individual model parameters it is possible to focus attention on the most significant parameters, as well as assessing the confidence in the predicted response. For example, if the response is sensitive to a parameter for which the actual value is uncertain then a high confidence in the result is unreasonable. Three short-term objectives are addressed in this chapter: 1) development of exact response sensitivity equations, 2) software implementation of response sensitivity equations to obtain response sensitivities in an efficient and accurate manner, and 3) investigation of the relative significance of each intervening parameter. Two approaches are available to obtain response sensitivities: the finite difference method (FDM) and the direct differentiation method (DDM). The finite difference approach is simply involves re-runs of the response analysis with perturbed parameter values to estimate the response sensitivity. Consequently, it is a computationally inefficient approach. Moreover, the FDM approach suffers from accuracy concerns. It is not a trivial task to select the value of the parameter perturbation for nonlinear problems. If the perturbation is too small then round-off errors may be dominant. On the other hand, if the perturbation is too large then nonlinearities in the parameter-response relationship may lead to inaccurate sensitivity estimates. The DDM represents an attractive alternative to the FDM whereby efficient and accurate response sensitivities are obtained at the one-time cost of deriving and implementing analytical sensitivity equations. It is emphasized that no finite difference computations take place within the DDM. Instead, the response equations are analytically differentiated and implemented in the software alongside the ordinary response computations. A number of researchers have contributed to the development of the DDM. Efforts by some investigators have primarily focused on obtaining sensitivities of displacement and force responses from structural (finite element) analysis, including Choi and Santos (1987), Tsay and Arora Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 17 (1990), Liu and Der Kiureghian (1991), Zhang and Der Kiureghian (1993), Kleiber et al. (1997), Roth and Grigoriu (2001), Cont et al. (2003), Scott et al. (2003), and Haukaas and Der Kiureghian (2005). Static and dynamic problems have been considered; with and without material inelasticity and geometric nonlinearities. In the field of composite manufacturing, a few studies, including Hou and Sheen (1987), Rai and Pitchumani (1997), Li et al. (2001), Li and Tucker (2002), Zhu and Geubelle (2002), and Li et al. (2002) employ response sensitivity analysis to obtain the response gradient required in optimization analysis. Rai and Pitchumani (1997) and Li et al. (2002) use the FDM by perturbing the values of the decision variables to calculate the gradient of the objective function and the constraints. Conversely, Hou and Sheen (1987), Li et al. (2001), Li and Tucker (2002), and Zhu and Geubelle (2002) use the DDM by analytically differentiating the response equations with respect to the design variables. Hou and Sheen (1987) and Li et al. (2001) differentiate coupled thermal and cure equations. However, only cure cycle parameters including autoclave temperature and their durations are considered as design variables. Consequently, only analytical differentiation of the ambient temperature at convective boundaries is introduced. That is, the complexity of differentiating equations with respect to material and geometry variables is not addressed in these papers. Li and Tucker (2002) couple the thermochemical model with the consolidation model and perform the analytical optimization to determine the time-optimal temperature and pressure cycles. Again, the thermochemical and consolidation models are analytically differentiated with respect to the autoclave parameters. Material and geometry sensitivities are not considered in that study and only one dimensional problem is studied. Zhu and Geubelle (2002) perform the tool shape optimization in order to minimize the final displacements. A thermochemical model is combined with a viscolelastic model to calculate the residual stress during the curing process and final displacements after tool removal. Only tool shape parameters, such as model curvature and angle are considered as design variables. Since these parameters do not affect the thermochemical model, the sensitivity of the thermochemical model is neglected. Therefore, only the viscolelastic model is analytically differentiated with respect to model shape parameters. Consequently, the sensitivity of parameters that dramatically affect the thermochemical model, such as tool thickness, tool and part Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 18 thermal properties, cannot be studied. Moreover, mechanical variables, such as part and tool coefficients of thermal expansion, are not included in the differentiation of the viscolelastic model, which affect both stiffness and strains. However, in the present chapter, comprehensive sensitivity equations of thermochemical and stress development models are formulated and solved in an efficient manner (solving only one system of equations per analysis step) with respect to all variables, including thermal and mechanical material, geometry, and processing parameters (over 80 parameters). The link between the thermochemical sensitivity and stress development sensitivity is included. This enables the study of the sensitivity of parameters that affect the temperature and degree of cure, and subsequently the residual stresses and final displacements. A number of software packages are available to numerically simulate composites manufacturing. Examples include ABAQUS (Hibbitt et al. 2008) and COMPRO (Johnston et al. 2001). The latter has been developed by the Composites Group at The University of British Columbia. In this study a comprehensive software is developed in-house using Matlab® to demonstrate the implementation and use of the derived equations. Coding in Matlab is convenient because of its user-friendly programming environment and the extensive, yet straightforward, debugging features. These are important issues when embarking on the task of implementing the composite manufacturing response algorithm and augmenting it with sensitivity equations. The resulting software to the best of our knowledge constitutes the first comprehensive 2D composite processing software with built-in response sensitivity capabilities. In the following, the governing response equations for the composite manufacturing process are first formulated to facilitate the subsequent sensitivity developments. In particular, we utilize previously developed models for thermochemical and stress analyses that predict the degree of cure, temperature, and residual stress development during the curing process. Thereafter, the DDM is presented in which the response equations are analytically differentiated to obtain the sought sensitivity equations. The overall methodology is presented in the body of this chapter, while the details are provided in separate appendices. One appendix is particularly devoted to the detailed algorithms that are used to implement the Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 19 sensitivity equations in the software. Finally, the implementations are verified by finite difference computations for a 2D L-shaped composite part. The results are also interpreted and discussed to indicate the usefulness of response sensitivity results. 2.2. GOVERNING EQUATIONS The modelling of composite processing has an extensive history, spanning more than three decades. A comprehensive review of all the available models is beyond the scope of this study. It suffices to say that the utilized model is divided into two sub-models as shown in Fig. 2.1: the prediction of heat transfer and resin cure (thermochemical model) and the prediction of residual stress and deformation (stress development model) (Johnston et al. 2001). In passing, it is noted that some models presented by Hubert et al. (1999) include a “flow of resin” model that is not considered in this study. In addition, the material viscoelasticity is neglected due to the small effect on the residual stresses and the final deformations (Zobeiry et al. 2006). In principle, the DDM can be applied to any response computation procedure. Essentially, the methodology requires the consideration of all model equations and their analytical differentiation. For this purpose, the thermochemical and stress development equations are reviewed in the following. They are subsequently differentiated to obtain the sought response sensitivity equations. It is emphasized that it is the discretized response algorithm that must be differentiated; that is, the theoretical models must first be discretized in space and time before performing the differentiation (Conte et al. 2003). 2.2.1. Thermochemical Model Temperature and resin degree of cure are the two state variables in modeling of the thermoset matrix, and in turn drive the evolution of the constitutive behaviour of the composite material during manufacturing. As outlined in Fig. 2.1, the temperature and degree of cure are computed by what is termed the thermochemical model, and they serve as input to the model that predicts the development of residual stresses and deformations. The thermochemical model consists of a combination of two sub-models; one Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 20 for heat transfer and one for resin reaction kinetics. These sub-models are coupled for two reasons. First, the degree of cure is dependent on the resin temperature. Second, the heat generated by the resin is a function of the cure rate. Consequently, the sub-models are treated as coupled and repeated runs of each model are performed to achieve convergence at the individual time steps of the solution procedure (Johnston 1997 ). The spatially and temporally discretized form of the equilibrium equation for the heat transfer using the finite element approach is (Johnston 1997 and Reddy 1993) kkk }{}{][ TT FTK = (2.1) in which the implicit (backward) Euler technique is adopted for the temporal discretization, and where k}{T is the nodal temperatures at the end of time step k and [KT]k is the global effective temperature stiffness matrix given by kkkk t ][][][ 1 ][ PT HΚCK ++∆ = κ (2.2) in which ∆t is the time step size . k][ PC , k][ κΚ , and k][H are known as the global heat capacity matrix , the global conductivity matrix and the global boundary convection matrix, respectively at the end of time step k, which are given by ∑ ∫ = Ω Ω= nele e P T dC e 1 TTP ][][][ NNC ρ (2.3) ∑ ∫ = Ω Ω= nele e T e d 1 TT ]][[][][ BκBΚκ (2.4) ∑ ∫ = Γ Γ= nele e T e dh 1 TT ][][][ NNH (2.5) Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 21 where nele is the total number of elements in the discretized domain, ][ TN is the matrix of element (temperature) shape functions, ρ is the mass density of the element volume eΩ , and Cp is the specific heat capacity of the element, ][ TB is the derivative of the element (temperature) shape function matrix, ][κ is the thermal conductivity matrix in the global axes (x-y-z), and h is the convective heat transfer coefficient of the element traction boundary eΓ . Moreover, the right-hand side of Eq. (2.1) recognized as {FT}k is the global temperature load vector defined by 1PTQThTqT }{][ 1 }{}{}{}{ −∆ +++= kkkkkk t TCFFFF (2.6) where k}{ TqF , k}{ ThF , and k}{ TQF are known as the global heat flux, the global boundary convection, and the global heat generation vectors, respectively, at the end of time step k. These global vectors are expressed as ∑ ∫ = Γ Γ= nele e b T e dq 1 TTq ][}{ NF (2.7) [ ]∑ ∫ = Γ ∞ Γ= nele e T e dhT 1 TTh}{ NF (2.8) ∑ ∫ = Ω Ω= nele e T e dQ 1 TTQ ][}{ &NF (2.9) where qb is the externally imposed surface heat flux (e.g., inductive heating), T∞ is the temperature of the boundary fluid, and Q& is the rate of internal heat generation, which is a key quantity in the coupling between the temperature and the degree of cure calculated as (Johnston 1997) dt d HVQ Rrf α ρ)1( −=& (2.10) Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 22 where Vf is the fibre volume fraction, ρr is the resin density, HR is the heat of reaction, and α is the degree of cure. A number of different models are available in the literature for calculation of the cure rate, including Lee et al. (1982), Scott (1991), Bogetti and Gillespie (1992), Lee et al. (1992), Kenny (1992), White and Hahn (1992a and b), and Hubert et al. (1995), which can be represented as ),,( αθ α Tf t = ∂ ∂ (2.11) where θ are the model parameters. Using the backward-Euler integration technique, one obtains the degree of cure at time step k as 1),,( −+∆= kkk Tft ααθα (2.12) The forward-Euler integration technique could also be used; in which case no iterations would be required. However, due to the highly non-linear form of the cure rate, that technique would entail extremely small time steps to ensure stability. The equation for temperature (Eq. 2.1) and the resin degree of cure (Eq. 2.12) are coupled, that is, the value of the parameters in one equation affects the other. Therefore, at each time step k an iterative procedure is required. For this purpose, the resin degree of cure is first calculated by using Eq. (2.12); utilizing the previous iteration values for temperature and degree of cure. Then, the temperature is computed using Eq. (2.1). The iteration is performed until the norms of the change in the trial solution between iteration steps for both temperature and degree of cure are sufficiently small (Johnston 1997). It should be noted that the heat transfer in the autoclave is predominantly of a convection nature and that the effect of the autoclave radiation is ignored in Eq. (2.1). 2.2.2. Stress and Deformation Model The stress model predicts the process-induced stress and deformation, with the temperature and resin degree of cure computed by the thermochemical model as input. During autoclave processing, a number of different processes lead to the development of residual stress and deformation. From the mechanisms Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 23 for the development of process-induced stress and deformation described in the literature, five main sources are identified by Johnston et al. (2001): 1) thermal strains, 2) resin cure shrinkage strains, 3) gradients in component temperature and resin degree of cure, 4) resin pressure gradients (resulting in resin flow), and 5) tooling mechanical constraints. A comprehensive treatment of all these sources is outside the scope of this study; in this study we consider the most researched sources. The solution technique used by the stress model is an incremental instantaneously linear elastic analysis, as presented by Johnston et al. (2001). Using this approach, an elastic analysis is carried out at each time step to determine the change in response within that step. Subsequently, this incremental result is added to the response from the previous step to determine the total solution at the end of the step. In this approach, all mechanical properties are re-calculated at the start of each time step and assumed to remain constant during the time step. For these calculations, the temperature and resin degree of cure calculated from the thermochemical model are employed at each Gauss integration point to compute material properties at these points. Thus, for example, the total nodal displacement at the end of time step k is determined by the summation of the calculated displacements at all previous steps, i.e. kkk }{}{}{ 1 δδδ ∆+= − (2.13) The change in nodal displacements, k}{ δ∆ during a time step is determined using kkk }{][}{ 1 FKδ ∆=∆ − (2.14) where ][K is the global stiffness matrix during the step calculated from the material elastic constants by (Cook et al. 1989 and Johnston et al. 2001) Ω= ∑ ∫ = Ω d nele e T e ][][][][ 1 δδ BCBK (2.15) where ][ δB is the matrix of (displacement) shape function derivatives, ][C is the instantaneous material stiffness matrix. According to the micromechanical models by Bogetti and Gillespie (1992), the elastic Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 24 constants of the transversely isotropic ply material – which is the combination of fibres and resin – are determined by employing the fibre and resin elastic properties, as well as the instantaneous fibre volume fraction Vf. The fibre mechanical properties are assumed to be transversely isotropic and are linear functions of the temperature at any time during processing. The resin is isotropic and is modelled as a “cure-hardening instantaneously linear elastic material” (CHILE material). This description specifies that the modulus of the instantaneously linear elastic resin develops monotonically with the progression of cure (Bogetti and Gillespie 1992; Johnston et al. 2001). }{ F∆ in Eq. (2.14) is the change in the global force vector during the step. In the current setup of the analysis, no external loads are applied. Thus, the change in the global force vector for each step is composed entirely of contributions from the thermal and cure shrinkage strains of elements, i.e. ∑ ∫ = Ω Ω∆=∆ nele e T d e 1 0}{][][}{ εCBF δ (2.16) where for the composite plies, the change in ply initial strains, }{ 0ε∆ is computed from the fibre and resin mechanical properties and strains including the fibre and resin thermal strains as well as resin cure shrinkage strains (Bogetti and Gillespie 1992). It should be noted that for multilayer composite elements, the numerical integration of Eqs. (2.15) and (2.16) within each element is evaluated using (Arafath 2007) ( ) ( )∑ ∑∫ = =Ω ≈ nply l l npoint m lmm t t fdf e 1 , 1 , ξxJξxx ω (2.17) where nply is the number of layers, npoint is the number of integration points, lm,ξ is the thm integration point of the thl layer, mω are the corresponding weights, t is the total thickness and lt is the thl layer thickness, (.)f is the integrand, and ξxJ , is the determinant of the Jacobian matrix of transformation between the x and ξ co-ordinates. Finally, the stress at the element level is computed as ( )}{}]{[][}{ 00 εδBCσ ∆−∆=∆ δk (2.18) Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 25 and the total stress at each time step is updated kkk }{}{}{ 1 σσσ ∆+= − (2.19) 2.3. DIRECT DIFFERENTIATION METHODOLOGY The DDM methodology consists of differentiating the governing response equations. In the following, the thermochemical model is differentiated to obtain the sensitivity of the temperature and the degree of cure. Subsequently, the stress development model is differentiated. Since most of the mechanical properties of the composite material, such as thermal strains and resin shrinkage, are dependent on the temperature and the degree of cure, the stress sensitivity computations require the aforementioned sensitivity of the temperature and the degree of cure. 2.3.1. Differentiation of the Thermochemical Model Consider the governing equation for heat transfer at the end of time step k in Eq. (2.1). Response sensitivities are obtained by differentiating this equation with respect to a generic model parameter θ, which may represent any of the input parameters provided by the user. In general, all quantities in Eq. (2.1) may depend on θ. Differentiation yields θθθ ∂ ∂ = ∂ ∂ + ∂ ∂ k k kk k }{ }{ ][}{ ][ TTT F T KT K (2.20) where [KT]k is the global temperature stiffness matrix, defined in Eq. (2.2), the derivative of which is given by θθθθ κ ∂ ∂ + ∂ ∂ + ∂ ∂ ∆ = ∂ ∂ kkkk t ][][][1][ PT HΚCK (2.21) and {FT}k is the global temperature load vector, defined in Eq. (2.6) for which the derivative reads θθθθθθ ∂ ∂ ∆ + ∂ ∂ ∆ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ − − 1 P1 PTQThTqT }{][ 1 }{ ][1}{}{}{}{ k kk kkkkk tt T CT CFFFF (2.22) Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 26 The complete details of each term in Eqs. (2.21) and (2.22) are presented in App. A. App. A-1 shows that the sensitivity of the degree of cure is a linear function of the sensitivity of the temperature. Consequently, no iterations are necessary to compute the sensitivities, as was the case for the response. App. A-2, A-3, and A-4 describe how each term in Eq. (2.21) is calculated. The load term sensitivity in Eq. (2.22) is written out in App. A-5. Substituting Eqs. (A.13), (A.17), (A.24), (A.25), (A.26), and (A.30) into Eqs. (2.21) and (2.22) and solving for θ∂∂T from Eq. (2.20), one obtains the linear thermochemical sensitivity equation }{][ }{ 1 BA T −= ∂ ∂ θ k (2.23) where ][A is defined by { } ( ) ( ) ( ) { } { } ′′− ∂ ∂ + ∂ ∂ + ∆ − ∂ ∂ + ∂ ∂ + ++= − = = = Γ == = ∑ ∑ ∑ ∑∑ ∑ ][][][ ][][ ][ ][][][ ][][][][][][][][ TTTT T 1 TT 1 , 1 1 ,TT 11 TTTT, 1 NaNa TT t a C T C hC TT kkPPT nele e m npoint m nele e T m npoint m nele e T P T m npoint m B κκ B N TT NNJ JNNBκBNNJA ξx ξxξx L K α ρρ ω ωρω (2.24) where the symbols are defined in App. A. It is noted that the first line in Eq. (2.24) is the global effective temperature stiffness matrix obtained by Eq. (2.2). Hence, for efficiency, this expression is not re- computed; the stiffness matrix from the response computations is utilized. The additional terms in Eq. (2.24) stem from the fact that the temperature sensitivity also appears in Eqs. (2.21) and (22). }{B in Eq. (2.23) is calculated as Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 27 { } ( ) ( ) ( ) ( ) ( ) ( )( ) Γ − ∞ ∞ − Γ == −− − −− == ∂ ∂ ++ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∆ ++′′ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∆ − ∂ ∂ + ∂ ∂ + ∂ ∂ = ∑∑ ∑∑ ξ N JNN TN ξ N JNNNNJ T NNNJNN ξ N TN ξ N JBκB B κB Bκ B B κ BB κ B TT N ξ N JNN NNNNJB ξx ξxξx ξx ξx ξx ξx θ θ θ θ α θ α θθ θ ω θ ρ θ θαθ ρ α ρ θ ρ ω T b TbT TTT m npoint m nele e k P TTTT T T TT T TTT kkT P T PTPT m npoint m nele e hTq hTq h h C t Qb b t C b CC ,TT T,TTTT, 11 1 TTT,TT T,T T T TTTT fixed fixed T T 1 T,TT TTT fixed fixed T T, 11 ][ )( ][ }{][][][][][ }{ ][][][ 1 ][][ }{][][][][ ][ ][][ ][][ ][ ][ ][ ][][ ][ ][ }{}{ ][][][ ][ )( ][][ )( ][ K K& K K K K (2.25) It is noted that the terms in Eq. (2.25) that include ξN ∂∂ θ account for “shape sensitivities;” the inclusion of the case in which θ is a geometric parameter is a particular novelty in this study (Haukaas and Der Kiureghian 2005). Next, the sensitivity of the temperature field, θ∂∂T , which is obtained by multiplying the nodal temperature sensitivities from Eq. (2.23) by the shape functions, is utilized in Eq. (A.3) to compute the sensitivity of the degree of cure, θα ∂∂ . The sensitivities of the temperature and degree of cure are in turn employed to calculate the sensitivity of the mechanical properties of the composite material, the thermal strain, and the resin shrinkage, as shown below. 2.3.2. Differentiation of the Stress Development Model Consider the increment of nodal displacements,{ }δ∆ , from Eq. (2.14). Response sensitivities are obtained by differentiating Eq. (2.14) with respect to the generic model parameter θ. It is noted that all quantities in Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 28 Eq. (2.14) may depend on θ. Differentiating Eq. (2.14) and solving for the response sensitivity, θ∂∆∂ k}{ δ , yields ∆ ∂ ∂ − ∂ ∆∂ = ∂ ∆∂ − k kk k k }{ ][}{ ][ }{ 1 δ KF K δ θθθ (2.26) where k][K is the global stiffness matrix during the step, defined by Eq. (2.15). The derivative of the global stiffness matrix in the last term of Eq. (2.26) is first addressed. By noting that Eq. (2.17) is used to numerically evaluate the integral in Eq. (2.15), the derivative of the Jacobian determinant will appear in this expression. This derivative is given in Eq. (A.15) according to Haukaas and Der Kiureghian (2005). Hence ( ) ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ − = = = ∑∑ ∑ ξ N JBCB B CBB C BBC B J K ξx ξx θ δδ δ δδδδ δ θθθ ω θ TT nele e TT Tnply l l npoint m m t t , 1 1 , 1 ]][[][ ][ ][][][ ][ ][]][[ ][][ K (2.27) where θN represents the shape functions in N corresponding to the degree of freedom matching the nodal coordinate represented by θ . Next, { }F∆ in Eq. (2.26) is the global load vector, described by Eq. (2.16). Its derivative is calculated by using the same technique as was applied to obtain Eq. (2.27). The result is { } { } { } { }( ) ∂ ∂ ∆+ ∂ ∆∂ +∆ ∂ ∂ +∆ ∂ ∂ = ∂ ∆∂ − = = = ∑∑ ∑ ξ N JεCB ε CBεC B ε C BJ F ξx ξx θ δ δ δ δ θθθ ω θ TT nele e nply l npoint m T T Tl m t t ,0 1 1 1 0 00, ][][ ][][][ ][][ ][ }{ K (2.28) The sensitivity of the total displacement in Eq. (2.13) now reads θθθ ∂ ∆∂ + ∂ ∂ = ∂ ∂ − kkk }{}{}{ 1 δδδ (2.29) Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 29 From Eqs. (2.27) and (2.28) it is apparent that the derivatives of the ply stiffness matrix [C] and the change in the ply initial strain { }0ε∆ are needed. The non-differentiated expressions for these quantities are provided by Johnston (2001). Their differentiations are presented in Appendix B. App. B-1 and B-3 describe the sensitivity of the ply stiffness matrix for 3D and plane strain problems, respectively. The sensitivity of the change in the ply initial strain for 3D and plane strain cases are outlined in App. B-2 and B-4. Finally, after calculating the sensitivity of the displacement, the sensitivity of the stress is computed by differentiating Eq. (2.18) as follows ( ) ∂ ∆∂ − ∂ ∆∂ +∆ ∂ ∂ +∆−∆ ∂ ∂ = ∂ ∆∂ θθθθθ δ δ δ }{}{ ][}{ ][ ][}{}]{[ ][}{ 00 000 εδ Bδ B CεδB Cσ k (2.30) Consequently, the total stress sensitivity reads θθθ ∂ ∆∂ + ∂ ∂ = ∂ ∂ − kkk }{}{}{ 1 σσσ (2.31) 2.4. RESPONSE SENSITIVITY ALGORITHMS AND EFFICIENT SOFTWARE IMPLEMENTATION The software developed in this study constitutes a collection of Matlab subroutines to predict the temperature, degree of cure, and stress development of 2D composite manufacturing problems. Importantly, the software includes response sensitivity capabilities according to the equations developed above. In the DDM, the analytically differentiated response equations are implemented alongside the ordinary response computations to produce response sensitivities. This section provides an overview of the overall analysis procedure. For brevity, the details of the implementations are presented in App. C. Consider first App. C-1, which shows the overall solution algorithm including the response and the response sensitivities for both the thermochemical and the stress development parts of the problem. It is noted that the response Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 30 sensitivity computations in Items 2c and 2d of App. C-1 are executed at every time step subsequent to the computation of the response in Items 2a and 2b. Each of the Items 2a to 2d contains a reference to subsequent algorithms provided in App. C-2 to App. C-4. App. C-2 shows the algorithm for the calculation of the temperature and degree of cure within a time step. References are made to the relevant equations. Next, in App. C-3 the temperature and degree of cure are utilized to determine the mechanical properties of the composite, as well as the incremental thermal strains, resin shrinkage and residual stresses. App. C-4 shows how the thermochemical sensitivity equations are implemented to compute the sensitivity of the temperature and the degree of cure. It is noted that the loop to compute the derivative of the stiffness matrix and the load vector is carried out over all the elements. This is done because, in the present study, the parameter θ is common to all elements. Also, similar to the DDM developed for inelastic mechanical boundary value problems, there are additional reasons to loop over all elements to gather contributions to the right-hand side of Eq. (2.23) (see Haukaas and Der Kiureghian 2005). The response sensitivities obtained by App. C-4 finally enters the algorithm in App. C-5, which outlines the calculation of response sensitivities of the stresses and displacements. A note is made regarding the efficiency of the sensitivity implementations. Instead of looping over all θ parameters, for which response sensitivities are sought, an expanded version of Eqs. (2.23 and 2.26) are solved once by collecting the sensitivity vectors in matrices. Specifically, a unique feature of the presented implementations is that the left-hand side of Eqs. (2.23 and 2.26) is considered as an n×m matrix, where n here is the number of degrees-of-freedom and m is the number of parameters for which sensitivities are sought. That is, this matrix contains the vectors pertaining to each of the parameters for which sensitivities are sought. Furthermore, the sensitivity results from Eqs. (2.23 and 2.26) is a product of the n×n tangent stiffness matrix by the aforementioned n×m matrix. Consequently, the response sensitivity vectors are obtained by inverting the n×n matrix and multiplying it by the n×m matrix. In the stress development model this inverse matrix is already factorized during the calculation of the ordinary Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 31 response and there is no need for re-calculation. However, in the thermochemical model, where the secant stiffness matrix is employed (see Eq. 20), this matrix is established in the sensitivity computations. 2.5. NUMERICAL EXAMPLE In this section an example is presented to verify the sensitivity results provided by the DDM equations developed and implemented above. For this purpose, the sensitivity of the temperature development and the deformations of an L-shaped composite part laid on a tool are computed during the curing process. At the end of the process, when the tool is removed, the sensitivity of the part “spring-in” is studied. The composite part is made of unidirectional layers of Hercules AS4/8552 (Johnston 2001) and the tool is made of aluminium. Fig. 2.2 shows the tool part assembly as well as the mechanical and thermal boundary conditions. A cure cycle, as illustrated in Fig. 2.3, is applied on all external boundaries. The part is removed form the tool at the end of the cure cycle and the spring-in angle is calculated (Albert and Fernlund 2002). As shown in Fig. 2.4 the spring-in angle consists of two components, a corner and a warpage component (Albert and Fernlund 2002). The corner component is mainly due to anisotropy in shrinkage, with a larger shrinkage through-thickness than in-plane during processing. This differential shrinkage will cause “spring-in” of enclosed angles. The warpage component is due to mechanical interaction of the part with the tool, causing a residual bending moment to form in the part. When the part is removed from the tool, the residual bending moment will cause the part to warp away from the tool. The thermophysical parameters employed in this example for both the carbon fibre/epoxy resin composite part and the aluminium tool are shown in Table D.1 in App. D. In addition, the parameters of the resin cure kinetics model selected from Hubert et al. (1995) are provided in Table D.1. All part geometry parameters are shown in Fig. 2.2 and listed in Table D.2. Fig. 2.3 also shows the predicted time history of the temperature and degree of cure at the point where the maximum temperature occurs for the three part lay-ups studied: [0 o ]8, [0 o ]32 , and [0 o ]128. Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 32 In the following, sensitivities of the temperature response are examined. Sensitivity results are shown in Fig. 2.5 for the parameter 21θ , which represents the activation energy E∆ in the cure kinetics equation. The result is compared with sensitivities obtained by the FDM. It is observed that the FDM results converge to the DDM results as the parameter perturbation value in the FDM is reduced. This confirms the accuracy of the DDM results. Excellent agreement is observed between the DDM and FDM results when the perturbation is 0.001 times the parameter value. This has also been verified for the other sensitivity results. Table E.1 in App. E presents the sensitivity of the temperature response at the time when the maximum temperature occurs for the 8 and 32 ply parts with respect to the 36 material parameters listed in Table D.1 and 6 geometric parameters listed in Table D.2. It is observed that the sensitivity results for the [0 o ]32 lay-up (column 6) are larger than the [0 o ]8 lay-up (column 3). This is reasonable because a larger exotherm occurs in the [0 o ]32 part than in the [0 o ]8 part. Therefore, the maximum temperature is greater and the temperature is more sensitive to the model parameters in the [0 o ]32 lay-up. It is noted that the sensitivity results in columns 3 and 6 in Table E.1 alone are not sufficient to rank the parameters according to their relative importance because they have inhomogeneous units. Importance rankings may be obtained from first-order reliability analysis (Hohenbichler and Rackwitz 1986; Bjerager and Krenk 1989; and Haukaas and Der Kiureghian 2005); however, this is a computationally costly approach that is outside the scope and objective of this study. Instead, a straightforward approach is adopted whereby the DDM results are utilized to predict the variation in the response due to a 1% variation in the individual parameters. This is intended to provide a practical feel for the sensitivity of the response with respect to the input parameters, given that there is limited knowledge of the actual variability in both the individual parameters and the response parameters of composite manufacturing processes. In fact, very few of the parameters studied here are routinely measured and tracked in industry. Thus the main objective of this numerical example is to identify what parameters we should measure and track, and to get a general understanding of how the variability in the individual input Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 33 parameters affects the response of the process. Note that several of the parameters examined are derived or calculated properties, especially the material related parameters. Variability in these parameters is due to both physical variability of the material as well as variability in materials characterization and subsequent data reduction. Columns 4 and 7 in Table E.1 show the estimated percentage change in response as a result of a 1% change in the parameters for the [0 o ]8 and [0 o ]32 lay-ups. Based on these results the parameters are ranked in columns 5 and 8. Zero-valued parameters are neglected since 1% change of these parameters does not have any meaning. A summarized view of the ranking of the thermochemical model parameters for the [0 o ]8 , [0 o ]32, and [0 o ]128 lay-ups is provided in Fig. 2.6. It is observed that for all three cases the activation energy E∆=21θ in the cure kinetics model is the most important parameter. Furthermore, fibre volume fraction, fV=28θ , heat of reaction RH=27θ , resin density )0(5 rρθ = and cure kinetics parameters m=22θ and n=23θ rank next. This is not surprising as the maximum temperature is mainly driven by the cure rate of the resin and the heat given off during the exothermic reaction. Therefore, parameters which are related to the cure and the consequent heat generation, have higher sensitivities. The other composite material parameters, such as initial degree of cure, fibre density, resin and fibre heat capacity, and heat transfer coefficient do not play a significant role in the heat generation. Therefore, they have lower sensitivities. As shown in Fig. 2.6, the tool parameters such as the tool density, heat capacity, and thickness have low sensitivities for the three different lay-ups. This is reasonable since the thickness of the tool is small. In fact, the effect of the tool properties is similar to the convective heat transfer coefficient between the autoclave and the part. Typically, a composite part is allowed to exotherm 5-10 o C above the hold temperature, in this case 180 o C. Therefore the maximum permissible temperature is typically at most 190 o C. As some of this temperature increase is already consumed by the baseline response of the [0 o ]128 and [0 o ]32 laminates, a change of even 2 o C in the exotherm, that is 1% change in temperature, may be of concern. Inspection of Fig. 2.6 in this light suggests that the fibre volume fraction of the material is a concern for the [0 o ]128 Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 34 laminate and that the cure kinetics of the resin is important for all thicknesses. Conversely, part and tool thickness are less important than might be thought. One important purpose of the numerical modelling of composites processing is to obtain the residual stresses in the part at the end of the cycle and hence the spring-in that occurs when the tool is removed. As shown in Fig. 2.4, the total spring-in as measured away from the corner includes a corner and a warpage component. In this section, an analysis is carried out to obtain the sensitivity of the spring- in angle at the end of the curing process. For this purpose the DDM equations derived in this study are implemented for the 2D plane strain case. A total of 83 parameters are present, including the thermochemical parameters, (Table D.1), the L-shaped geometry parameters (Table D.2) and the stress development parameters (Table D.3). To predict the warpage component of the spring-in angle a thin elastic shear layer is placed between the tool and the part as shown in Fig. 2.2 to simulate the effect of friction at the tool-part interface. The results of the DDM sensitivity analysis were again compared to the FDM results to verify the DDM implementation. Tables E.2 and E.3 in App. E show the sensitivities of the spring-in angle for the part with [0 o ]8 lay-up with respect to the 36 parameters of the thermochemical model, 6 parameters of the L-shaped geometry and the 41 parameters of the stress development model listed in Tables D.1, D.2 and D.3 in App. D. The nominal corner component of the spring-in angle equals 1.3 o , and the corresponding sensitivity results are presented in column 3 of Tables E.2 and E.3. Furthermore, the nominal warpage component of the spring-in angle is 0.06 o , with corresponding sensitivity results given in column 6 of the same tables. As in the case of temperature sensitivity, these results have inhomogeneous units and are therefore not employed to rank the parameters. Instead, physical insight is gained by observing the results in columns 4 and 7. These columns provide the percent of change in the corner and warpage components of the spring-in angle due to a 1% change in the non-zero parameters. The ranked parameters for both the corner and warpage components are presented in columns 5 and 8. A summary of the 11 most important parameters is shown in Figs. 2.7 and 2.8. It is observed that the activation energy, ∆E, is by far the most Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 35 influential parameter on the warpage component of the spring-in. This can be explained by the fact that ∆E affects the timing of the curve, which in turn determines the modulus development in the part, this being a critical parameter in determining warpage [31]. Conversely, the corner spring-in is influenced by several parameters, including the activation energy, coefficient of thermal expansion for the resin, the fibre-volume fraction, and the parameters αcr and αc0, which are related to vitrification of the polymer matrix. All these parameters influence the contribution of thermal and cure shrinkage stresses. In conclusion, Figs 7 and 8 show that the corner component of the spring-in is sensitive to the parameters that affect the differential shrinkage through-thickness and in-plane, whereas the warpage component is more sensitive to the timing of the hardening behaviour via the cure kinetics parameters. Although the warpage component is smaller than the corner component, it is more sensitive to changes in the process conditions. It is of significant interest to study the computational cost required to obtain the above sensitivity results. For this purpose, Table 2.1 compares the computational cost associated with obtaining sensitivity results by the DDM and the FDM. (The computations in this thesis are carried out on a desktop computer with a Pentium® 4 processor with clock frequency 2.80GHz and 512MB internal memory.) The finite element analysis including DDM sensitivities takes 323 minutes for the problem at hand. The FDM requires 83 re-runs of the finite element analysis (because there are 83 parameters), each with perturbed value of one of the parameters. It is therefore observed that the DDM is highly efficient compared with the FDM, and that the FDM takes almost 11 times longer than the DDM. This serves as an indication that the one-time endeavour of deriving and implementing DDM sensitivity equations often is well worth the effort. For completeness, the labour cost associated with implementing the DDM sensitivity equations in the software should be included. Although we argue that the relative cost of this one-time effort will decrease over time – because computational savings are earned in every subsequent sensitivity analysis – the required computer programming is a significant task. The research presented in this study to derive and implement DDM sensitivity equations is estimated at 6 months for a full-time PhD student, including Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 36 software debugging and testing. The time required to implement these equations in another software, with formulas provided by this study, is estimated at 1 month’s full time work, assuming that the software is well known to the programmer. 2.6. SUMMARY AND CONCLUSIONS Response sensitivity equations for composites manufacturing are developed and implemented by utilizing the direct differentiation approach. This provides a valuable tool for studying the relative importance of model parameters in composites manufacturing. It also renders available gradient results required in reliability and optimization applications. The methodology significantly reduces the cost of obtaining sensitivity results compared to the finite different approach. Detailed derivations are presented to obtain sensitivities with respect to material and geometry parameters. Differentiation of the thermochemical and stress development equations is carried out in order to compute the sensitivities. The resulting algorithms are implemented in a comprehensive software developed in-house. The utility of the software is demonstrated by means of a 2D composite processing problem. The problem is described by 83 parameters, including 36 thermochemical model parameters, 6 geometric parameters, and 41 stress development parameters. The example is utilized to study the effect of the parameters on the resulting maximum temperature during cure and the spring-in distortion when the cured part is removed from the tool. An overview of the computational cost to obtain the sensitivity results is also provided. It is concluded that the DDM provides response sensitivities in an accurate and efficient manner, at the one- time cost of deriving and implementing analytical sensitivity equations, as is accomplished in this chapter. Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 37 2.7. TABLES Table 2.1: Comparison of run time used for direct differentiation method (DDM) and finite difference method (FDM) to calculate the sensitivity of response in combined thermochemical and stress development model Case Analysis time (Sec.) Ratio 1 run of the FE analysis 2580 1 DDM sensitivity analysis 19380 7.5 FDM sensitivity analysis 216720 84 Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 38 2.8. FIGURES Figure 2.1: Schematic of the boundary value problem under consideration Figure 2.2: Finite element mesh of the L-shaped unidirectional composite part on a solid aluminium tool and the thermo-mechanical boundary conditions applied before tool removal thP = 1.6 mm Convective Heat Transfer LT = 17 mm Lp= 50 mm Tool Part Shear Layer x z thT = 4.5 mm thsh = 0.1 mm RT= 5 mm Applied temperature Applied temperature Composite material Physical model Numerical model Temperature Degree of cure Stress development model Thermochemical model Residual stresses Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 39 0 50 100 150 200 250 300 0 50 100 150 200 T em p e ra tu re ( o C ) 320 0 0.2 0.4 0.6 0.8 1 time (min) D e g re e o f cu re Autoclave [0 o ] 8 [0 o ] 32 [0 o ] 128 Figure 2.3: Temperature and degree of cure evolution for the [0o]8 , [0 o ]32 , and [0 o ]128 lay-ups Figure 2.4: Schematic showing the definition of corner ∆θ1 and warpage ∆θ2 components of the total spring-in angle ∆θ of an L-shaped part Warpage Component, ∆θ2/2 Total spring-in angle: ∆θ = ∆θ1 + ∆θ2 Original Shape Corner Component, ∆θ1/2 θ Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 40 0 50 100 150 200 250 300 320 -6 -5 -4 -3 -2 -1 0 1 2 3 4 x 10 -4 time (min) T em p e ra tu re S e n si ti v it y f o r θ 2 1 = ∆ E ( o C /J /g m o l) DDM FDM, ∆θ 21 =0.01θ 21 FDM, ∆θ 21 =0.001θ 21 Figure 2.5: Comparison of maximum part temperature sensitivity by DDM and FDM for E∆=21θ , for the [0 o ]8 lay-up Figure 2.6: Change in maximum part temperature due to a 1% change in the parameters for the [0 o ]8, [0 o ]32 , and [0 o ]128 lay-ups -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 T em p ra tu re c h an g e (% ) 32]0[ 8]0[ 128]0[ T em p er at u re c h an g e (% ) Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 41 Figure 2.7: Change in the corner component of spring-in (nominal value = 1.3o) due to a 1% change in the parameters for the [0 o ]8 lay-up Figure 2.8: Change in the warpage component of spring-in (nominal value = 0.06o) due to a 1% change in the parameters for the [0 o ]8 lay-up -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 C o rn er S p ri n g -i n c h an g e (% ) -5 0 5 10 15 20 25 30 35 W ar p ag e ch an g e (% ) Chapter 2: Response Sensitivities and Parameter Importance Studies in Composite Manufacturing 42 2.9. 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Nonlinear structural design sensitivity analysis for path dependent problems, part1: general theory.” Computer Methods in Applied Mechanics and Engineering, 81, 183- 208. White, S.R. and Hahn, H.T. (1992a). “Process modeling of composite materials: residual stress development during cure, part I: model formulation.” Journal of Composite Materials, 26 (16), 2402- 2422. White, S.R. and Hahn, H.T. (1992b). “Process modeling of composite materials: residual stress development during cure, part II: experimental validation.” Journal of Composite Materials, 26 (16), 2423-2453. Zhang, Y. and Der Kiureghian, A. (1993). “Dynamic response sensitivity of inelastic structures.” Computer Methods in Applied Science and Engineering, 108, 23-36. Zhu, Q. and Geubelle, P.H. (2002). “Dimensional accuracy of thermoset composites: shape optimization.” Journal of Composite Material, 36(6), 647-672. Chapter 3: Application of Response Sensitivity in Composite Manufacturing 46 Chapter 3. APPLICATION OF RESPONSE SENSITIVITY IN COMPOSITE MANUFACTURING 2 3.1. INTRODUCTION The primary objective in this paper is to demonstrate and expand the use of response sensitivities in numerical simulation of composite manufacturing. The premise for this work is recent developments in numerical simulation of the composite manufacturing process. Worldwide, several software applications are now available to simulate the curing and stress development in composite material parts (e.g., Johnston et al. 2001). Such parts are frequently used to assemble large and complicated structures. Consequently, an important challenge in composite manufacturing is to ensure dimensional consistency. Potential shape imperfections are influenced by a number of parameters that are generally categorized into three groups: material, geometric, and processing parameters. The material parameters include resin chemistry, fibre volume fraction, and fibre properties. The geometric parameters involve the form of the fibre material (textile, unidirectional, short fibre, random fibre, etc.), part thickness, and tool surface conditions. The processing parameters include tool temperature, pressure, and cure time. Given values for all these input parameters, a numerical simulation of the composite manufacturing process is aimed at forecasting the curing, stress development, and resulting shape imperfection. In this paper, a new Matlab®-based software for simulation of composite manufacturing is employed. This software was created and extended with response sensitivity capabilities by Bebamzadeh et al. (2009). In this software the sensitivity of the curing and stress development response with respect to 2 A version of this chapter is being submitted for publication as a journal paper. Bebamzadeh, A., Haukaas, T., Vaziri, R., Poursartip, A., Fernlund, G., “Application of DDM response sensitivities in numerical simulation of composite manufacturing.” Chapter 3: Application of Response Sensitivity in Composite Manufacturing 47 all input parameters are available, including all material, geometric, and processing parameters. The efficient and accurate direct differentiation method (DDM) is utilized. In this approach, all response equations, including the thermochemical and stress development models, are analytically differentiated and implemented in the software alongside the ordinary response computations. The relative cost of this one-time effort will decrease over time, because computational savings are earned in every subsequent sensitivity analysis. Consequently, the DDM represents an attractive alternative to the less accurate and less efficient finite difference methods (FDMs). Although, FDMs are straightforward to implement, they suffer from accuracy concerns due to the difficulty in selecting the value of the parameter perturbation. Also FDMs are computationally inefficient. They require repeated re-runs of the finite element model with perturbed parameter values to estimate the response sensitivity. Many researchers have extended the DDM in the context of structural finite element analysis, including Choi and Santos (1987), Tsay and Arora (1990), Liu and Der Kiureghian (1991), Zhang and Der Kiureghian (1993), Kleiber et al. (1997), Roth and Grigoriu (2001), Cont et al. (2003), Scott et al. (2003), and Haukaas and Der Kiureghian (2005). Developments are also found in the context of composite manufacturing, including Hou and Sheen (1987), Li et al. (2001), Li and Tucker (2002), and Zhu and Geubelle (2002). Sensitivity analysis is associated with a wide range of applications. Kleijnen (1995), Saltelli et al. (2000), and Frey et al. (2004) suggest the sensitivity analysis for model validation and verification. For this purpose, the model sensitivity must reflect the modeled system characteristics. Any dependency on presumable non-influential parameters and assumptions necessitates the model’s modification. Saltelli (2000) et al. and Frey et al. (2004) also recommend the sensitivity analysis for model reduction by neglecting insignificant parameters. They also suggest sensitivity analysis for model calibration by providing a set of significant parameters to calibrate the model. Other applications associated with the sensitivity analysis include reliability analysis and system optimization where the gradient of response is required (Hasofer and Lind 1974; Schittkowski 1986; Liu and Der Kiureghian 1991; Zhang and Der Chapter 3: Application of Response Sensitivity in Composite Manufacturing 48 Kiureghian 1997; and Polak 1997). In addition, certain reliability and optimization algorithms also require second-order sensitivities (Ditlevsen and Madsen 1996 and Polak 1997). Saltelli et al. (2000) show that these higher order sensitivities are also useful to study the interaction between parameters. In this paper, the efficient and accurate computation of response sensitivities is utilized to explore applications and innovations in four directions: 1) A novel alert system that uses response sensitivity results to notify the user if the numerical results deviate from expected “rules” or scaling laws; 2) Model calibration with gradient-based algorithm to tune the simulation model with observed information; 3) Gradient-based reliability analysis to obtain the probability of response events when input parameters are characterized as random variables; and 4) Efficient optimization analysis to guide the user in decisions related to minimization of processing time and associated manufacturing cost. Although the last three analysis types are well-known in engineering analysis, the efficient and accurate response sensitivities with respect to all input parameters facilitates novel applications and results. The first topic; the extension of engineering simulation software with error alerts based on response sensitivity results is an interesting idea that is explored in this paper. Comparison with state-of-the-art literature and software in each of the four areas is provided in the final section of this paper. 3.2. REAL-TIME VALIDATION OF ANALYSIS RESULTS WITH RESPONSE SENSITIVITIES The first application of response sensitivities does not involve additional algorithms, as is necessary in reliability and optimization applications. Instead, the basic idea in this section is to monitor the sensitivity results and raise warning flags if they do not conform to fundamental rules and scaling laws. The general idea of using sensitivity results for verification and validation purposes is not new. The approach has been explored earlier by Kleijnen (1995), Fraedrich and Goldberg (2000), Saltelli et al. (2000), and Frey et al. (2004). The main objective of the sensitivity analysis in model verification is to ensure that the sensitivity results correspond to the expected sensitivity of the model response. If a model responds inappropriately to one or more input changes then the source of the discrepancy should be investigated and resolved. Frey Chapter 3: Application of Response Sensitivity in Composite Manufacturing 49 et al. (2004) also indicate that sensitivity analysis is helpful in the validation process. Significant contradictions in the relative impact of input changes and real world experiences would indicate potential flaws in the model. The basis for the proposed methodology is that the simulation of complex physical phenomena – such as composite manufacturing – must conform to certain general behaviour characteristics. In other words, the response that is the result of complex computer algorithms can be roughly checked against fundamental physical laws, experimental observations, or expert judgment. To this end, suppose y is the response of a complex simulation model, and that it is a function of n variables xi ),...,( 21 nxxxfy = (3.1) Further, as an example, assume that y is known to approximately vary over a certain domain as n m n mm xxxy L21 21∝ (3.2) where mi are integer coefficients. Now observe that the partial derivative of y with respect to one of the variables xi is ni m n m ii m i xxmx x y LL 11 1 −∝ ∂ ∂ (3.3) Multiplying Eq. (3.3) by xi and combining Eqs. (3.2) and (3.3) yields y x x y m i i i ⋅∂ ∂ = (3.4) The coefficient mi in Eq. (3.4) is computed in “real-time” during the simulation analysis – in which response sensitivities ixy ∂∂ are available – and compared with the corresponding coefficient mi in Eq. (3.2), which is based on empirical data or other information. If the values differ considerably, then warning flags are raised to prompt further investigation or debugging of the simulation algorithms. Chapter 3: Application of Response Sensitivity in Composite Manufacturing 50 A simple conceptual example is first presented to illustrate the methodology. Laboratory tests in composite manufacturing show that the out-of-plane displacement δ due to the tool-part interaction for a laminate of length L and thickness t approximately follows the law 23 −⋅∝ tLδ (3.5) Hence, according to Eq. (3.4) it should be checked during the analysis that the response sensitivities satisfy the equations δ δ L L m ⋅ ∂ ∂ =1 and δ δ t t m ⋅ ∂ ∂ =2 (3.6) The software simulation example that is utilized to demonstrate the methodology put forward in this section involves a flat composite part cured on a tool. When the tool is removed, the sensitivity of the warpage is computed. The composite part is made of eight layers of unidirectional Hercules AS4/8552 (Johnstion 2001) and the tool is made of aluminium. Fig. 3.1(a) shows the tool and part assembly as well as the mechanical and thermal boundary conditions. The curing cycle illustrated in Fig. 3.1(b) is applied to all external boundaries. The composite part is removed form the tool at the end of curing cycle and the warpage is calculated as shown in Fig. 3.1(c). The predicted maximum warpage, wmax, is 1.55 mm and the sensitivity of the warpage with respect to the length is Lw ∂∂ max = 3.75×10 -2 . Given the length L=150 mm Eq. (3.4) yields mL = 3.63. In other words, the analysis result indicates that the warpage response is proportional to L 3.63 . Conversely, empirical results by Twigg et al. (2004) show that the warpage is proportional to L 3 . This discrepancy, albeit not dramatic, signals potential inaccuracies in the model. A potential reason may be in the shear layer properties assumptions. The finite element model assumes that the shear layer elastic modulus (G) is constant. However, Twigg et al. (2004) show that the shear layer modulus varies with respect to the length (L) and thickness (t) as Chapter 3: Application of Response Sensitivity in Composite Manufacturing 51 s s s t t L L GG = (3.7) where Gs is the calibrated shear modulus for a particular too-part interface condition with length, Ls and thickness, ts. Using the chain rule of differentiation the sensitivity of warpage with respect to length is GL w L G G w L w ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ (3.8) The sensitivity analyses show the values of G Lw ∂∂ max and Gw ∂∂ max are 3.75×10 -2 and 5.32×10 -8 , respectively. In addition, the value of LG ∂∂ is obtained by differentiation from Eq. (3.7). Using Eq. (3.8) and Eq. (3.4), mL is computed to be 2.74, which shows the warpage varies approximately as L 2.74 . This value is somewhat closer to the empirical results showing the shear layer modulus dependencies on its own dimensions and demonstrates how the methodology can be employed to improve the model. 3.3. MODEL CALIBRATION WITH GRADIENT-BASED ALGORITHMS An extension of the model validation efforts in the previous section is to actually calibrate the model with the experimental observations, sometimes referred to as system identification. Essentially, this is an optimization problem in which the objective is to minimize the difference between the simulated response and experimental results. Response sensitivities enter as a central ingredient in this methodology. As a starting point, consider the objective function and constraints ( )∑ = −= M m mm uuE 1 2* )()( θθ (3.9) where M is the number of empirically observed responses, * mu are the observed responses, )(θmu are the corresponding model responses, and θ are the model parameters that will be varied to improve the model prediction. The calibrated model is obtained by determining the value of θ that minimizes )(θE , Chapter 3: Application of Response Sensitivity in Composite Manufacturing 52 potentially subject to constraints on θ . Efficient optimization algorithms to minimize E utilizes response sensitivities θ∂∂ mu , where θ is a parameter in θ . The sensitivity-enabled software (Bebamzadeh et al. 2009) is linked with the optimization algorithm to obtain the calibrated parameters in composites manufacturing problems. The model depicted in Fig. 3.2(a) is considered for calibration of the convective heat transfer coefficients. It is a composite part composed of cured unidirectional layers of Hercules AS4/8552 (Johnston et al. 2001) with 38.1mm thickness laid on a 25.4mm aluminum tool. Fig. 3.2(b) shows the convective air temperature applied on the boundaries. Fig. 3.3(a) shows the predicted temperature at the top surface of the composite part utilizing the software, as well as the empirical temperature using measurements. The convective heat transfer coefficient used on the part’s top surface and bottom of the tool are 60 and 140 (W/m 2 K), respectively. Fig. 3.3(b) shows the temperatures at the interface of the tool and the part. Figs. 3.3(a) and (b) show the discrepancy between the predicted and the measured temperature. The sensitivity analysis results, shown in Fig. 3.4, indicate that the temperature at the top surface of the part is influenced by the convective heat transfer coefficient. Combined with engineering judgment, this parameter is selected as the variable in the model calibration exercise. In other words, the objective is to minimize the gap between the model and empirical results by finding the appropriate value of the convective heat transfer coefficients at the part’s top surface and at the tool. The optimization is carried out with a gradient-based algorithm that employs the DDM response sensitivity results. The resulting calibrated coefficients are 55 and 100 (W/m 2 K) at the part’s top surface and the bottom of the tool, respectively. Figs. 3.5(a) and (b) show the temperature at the part’s top surface and the interface of the tool and part with the calibrated convective heat transfer coefficients. A closer match between the model and empirical results is evident, compared with the original results in Fig. 3.3. However, some difference still exists. To address the persisting difference between the model and empirical observations, the optimization is carried out at each time instant instead of one optimization over the entire event. This Chapter 3: Application of Response Sensitivity in Composite Manufacturing 53 turns out to be a successful approach that yields interesting results. Fig. 3.6 presents the final convective heat transfer coefficients and their “optimal” values along the time axis. The fluctuation over time contradicts the fundamental model assumption that the convective heat transfer coefficient is constant during the curing process. Hence, from a philosophical viewpoint this analysis demonstrates that the calibration approach is useful not only to calibrate the input parameters but also to address fundamental model assumptions. There are two key parameters that fluctuate during the experiments: the autoclave pressure and the temperature. It is therefore reasonable to assume that these two parameters impact the observed time- varying behaviour of the convective heat transfer coefficient. This agrees with Johnston (1997), who showed that the convective heat transfer coefficient is significantly influenced by the pressure and temperature. Johnston (1997) proposes that the convective heat transfer coefficient changes by pressure and temperature as 5 4 = ∗T P ah (3.10) where P and T * are the autoclave pressure and temperature ( o K), respectively and a is a variable that depends on different parameters such as air velocity, conductivity, and air dynamic viscosity. In this context it is noted that air parameters may be very different for different autoclaves. Next, the model is revised by using Eq. (3.10) for the convective heat transfer coefficients of the part’s top surface and the tool bottom. The only input parameter that is added to the model is the constant value, a, in Eq. (3.10), which varies for different autoclave conditions. Eq. (3.10) is used to calculate the values at the part’s top surface (aP) and tool bottom (aTB). Figs. 3.7(a) and 3.7(b) compare the empirical results to the revised model. The results are now identical. For completeness, the optimization is then performed to revise the values of a at each individual time instant instead of one value over the entire test time. However, this does not contribute to a better correlation between the model and empirical results. This implies that the a value does not vary during the process. It is concluded that Eq. (3.10), namely the Chapter 3: Application of Response Sensitivity in Composite Manufacturing 54 pressure dependency in the convective heat transfer coefficient, is a reasonable model of the observed time-variation of the convective heat transfer coefficient. 3.4. RELIABILITY ANALYSIS WITH GRADIENT-BASED FORM There are many factors that affect the simulated predictions in composites manufacturing simulation. These factors can generally be divided into three categories: material parameters, geometric parameters, and processing parameters. Unavoidable uncertainties are present in each category. Uncertain errors are also present in the modeling and analysis procedures themselves. Therefore, it can be argued that predictions of composites manufacturing should only be made in a probabilistic manner. A coupling of finite element analysis with reliability analysis algorithms addresses this issue. This approach facilitates probabilistic predictions of responses, such as temperature, degree of cure, stresses, and deformations in composites manufacturing simulations. The primary concern in reliability analysis is to estimate probability of failure to achieve a predefined limit-state, e.g, the response exceeding a user-defined threshold. This is calculated as the multiple integral of the joint probability density function of random variables over the failure domain. Methods to address this integral include sampling and the first- and second-order reliability methods (FORM and SORM) (Ditlevsen and Madsen 1996). In FORM, the reliability problem is solved by approximating the integration boundary (the “limit-state surface”) by a hyper-plane that is tangential to the limit-state surface at the point on the surface with maximum probability density, called the most probable point (MPP). This approximation is made in the transformed space of uncorrelated standard normal random variables. Finding the MPP requires solving a constrained optimization problem. Several algorithms are available, including those in Hasofer and Lind (1974), Rackwitz and Fiessler (1978), Schittkowski (1986), Liu and Der Kiureghian (1991), Polak (1997), and Zhang and Der Kiureghian (1997). These algorithms share the need for the gradient of the limit-state function, and thus the sensitivity of the responses that enter the limit-state function. This is seen by applying the chain rule of Chapter 3: Application of Response Sensitivity in Composite Manufacturing 55 differentiation to obtain the needed gradient of the limit-state function, G, in the standard normal space (the limit-state function is denoted g when expressed in terms of the original random variables): y x x u uy ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ =∇ gG G (3.11) where x and y are the vector of random variables in original space and in uncorrelated standard normal space. yx ∂∂ is the Jacobian matrix, Jx,y, of the x–y transformation. u∂∂g is straightforwardly obtained because g is typically an algebraic function of the response quantity u. xu ∂∂ is the response sensitivities that form the topic of this paper. A particularly interesting aspect of FORM is the importance vectors that are available as a by- product of the analysis. As pointed out in Bebamzadeh et al. (2009), the ranking of the input parameters with only stand-alone sensitivity results is a challenging task because of the differing units of the different response sensitivities. This is solved by carrying out FORM analysis; see Haukaas and Der Kiureghian (2005) for examples in the context of finite element reliability analysis. Additionally, FORM is advantageous from a computational cost viewpoint when efficient DDM response sensitivities are available. Unless convergence difficulties are encountered, FORM typically requires 5 to 10 evaluations of the limit-state function to obtain the probability estimate. On the downside, non-smoothness limit-state function (discontinuities in the response sensitivities in the space of random variables) renders FORM analysis by gradient-based methods infeasible. This may be a significant impediment in finite element based reliability analysis (Haukaas and Der Kiureghian 2006). In this paper, the sensitivity-enabled software presented in Bebamzadeh et al. (2009) is coupled with a similar Matlab-based “toolbox” for reliability analysis named FERUM (Finite Element Reliability Using Matlab), which was developed at the University of California, Berkeley in the late 1990’s (Der Kiureghian et al. 2003). In the following, a FORM reliability analysis is carried out with this merged software to obtain probabilistic estimates for temperature and the degree of cure responses, as well as to investigate the relative importance of different material, geometry, and processing parameters. The Chapter 3: Application of Response Sensitivity in Composite Manufacturing 56 example consists of a flat composite sample placed on an aluminium tool. The composite part is composed of unidirectional layers of Hercules AS4/8552 (Johnston et al. 2001). Fig. 3.8(a) shows the geometry and boundary condition parameters. A two-hold curing cycle with eight (8) parameters is applied to all boundaries as presented in Fig. 3.8(b). In this example, geometry and processing parameters, boundary conditions, and material properties including resin thermochemical, fibre and tool thermal are considered as random variables. The probability distributions are listed in Tables 3.1 and 3.2. In total, there are 48 random variables, which are assumed to be uncorrelated. The limit-state functions are defined in terms of maximum and minimum peak temperatures ( peakTmax , peakTmin ) and the maximum and minimum degrees of cure at the end of the process. In passing, it is noted for the reliability-oriented audience that these are acceptable in FORM although “maximum responses” are used, because there is only one peak during the entire process. Otherwise, this would be a more challenging time-variant (first-excursion) reliability problem. Limit-state functions are defined so that a negative outcome defines the event for which the probability is sought. Hence: peak allowable TTg min−= , allowable peak TTg ′−= min (3.12,3.13) allowableg αα −= max , allowableg αα −= min (3.14,3.15) where allowableT , allowableT ′ and allowableα are the manufacturing recommended maximum and minimum peak temperatures and the degree of cure limits, respectively. To investigate the feasibility of FORM, the potential presence of gradient discontinuities in the space of random variables is investigated. Figs. 3.9(a) to (c) present variation of responses including the maximum and minimum peak temperature and maximum and minimum degree of cure at the end of the process due to variation in the cure α0, activation energy ∆E, and the cure cycle second ramp slope s2. The results indicate that the sensitivities are continuous with respect to variation in these parameters. In fact, Chapter 3: Application of Response Sensitivity in Composite Manufacturing 57 the only piece-wise linear model that is utilized in the thermochemical analysis is the two-hold cure cycle. This motivates the study in Figs. 3.9(a) to (c), which show that the piece-wise cure cycle does not result in non-smooth responses. It is concluded that there is no gradient discontinuity involved in the thermochemical analysis with the utilization of limit-state function containing maximum temperature and piece-wise cure cycle. Therefore, the study proceeds to carrying out the FORM reliability analysis. The reliability analysis of the limit-state functions in Eqs. (3.12-3.15) is carried out with several threshold values. Consequently, probability distributions of the response are obtained. Figs. 3.10(a-d) present the resulting cumulative distribution function (CDF) and the corresponding probability density function (PDF). Fig. 3.10(a) shows that the probability of the maximum temperature exceeding 185 o C (manufacturer’s recommended temperature), is 0.85. This high probability is the result of the composite sample’s thickness (25.4mm). A reduced probability of exceeding the manufacturer’s maximum temperature is achieved in the optimization analysis in the next section by modifying the cure cycle parameters. Fig. 3.10(c) shows the probability of the minimum degree of cure at the end of process to be less than 0.7 and 0.8 are 0.15 and 0.35 respectively. Figs. 3.11(a-d) graphically shows the importance measures δ and η from FORM analysis (Haukaas and Der Kiureghian 2005) for the limit-state functions pealTg max200 −= and max825.0 α−=g . Essentially, δ is the vector of normalized sensitivities of the reliability index with respect to the means of the random variables. Similarly, η is the vector of normalized sensitivities of the reliability index with respect to the standard deviations of the random variables. Each plot shows the value of the importance vector for the 10 most important random variables. It is noted that the parameter ranking facilitated in reliability analysis is an important extension of the parameter sensitivity considerations in Bebamzadeh et al. (2009), in which only stand-alone response sensitivity results were available. Fig. 3.11(a) shows that the mean of the cure cycle parameters 1s , 1t , 2s , and 3t (Fig. 3.8b) and activation energy E∆ have the highest influence on the reliability related to the maximum peak Chapter 3: Application of Response Sensitivity in Composite Manufacturing 58 temperature. Conversely, Fig. 3.11(b) shows that the standard deviation of the degree of cure 0α and reaction order n are most influential for this reliability. It is interesting to observe that the initial cure kinetics parameters show up as most important in the η-ranking. This means that uncertainty in the early curing stages propagates through the analysis and has significant effect on the final reliability. Conversely, it is observed that the δ-ranking (sensitivity of the mean) is most significant in the later stages of the curing process. The amplification of variation in the standard deviation in the initial stages is directly associated with the nature of the resin cure kinetics model shown in Table 3.1. The temperature reliability is sensitive to the uncertainty (standard deviation) associated with the slope and duration of the first ramp of the cure cycle, compared to the second ramp shown in Fig. 3.11(b). In other words, the uncertainty in the first ramp propagates through the entire analysis and has more significant impact on the final results than the second ramp ones. This is opposite for the mean values, as shown in Fig. 3.11(a). Figs. 3.11 (c) and (d) shows the importance ordering for the maximum degree of the cure at the end of the process. Fig. 3.11(c) indicates that the reliability associated with this response is sensitive to the cure kinetics parameters, including CTαθ =26 , 025 Cαθ = , and E∆=1̀2θ , namely the cure kinetics parameters from the later stages of the curing process. Fig. 3.11(d) shows that the standard deviation of the activation energy is most influential on this reliability result. Table 3.3 compares the computational cost associated with the FORM reliability analysis using DDM and the finite difference method (FDM) to obtain response sensitivities. The finite element analysis including DDM sensitivities takes 574 seconds for the previous problem, compared to 7973 seconds taken with the FDM. In comparison, the stand-alone finite element analysis takes 163 seconds. The high computational cost associated with the FDM is due to 48 repeated runs of the finite element response analysis with perturbed parameter values. For the limit-state function peakTg max200 −= , FORM analysis requires two iterations to obtain the design point and calculate the probability of failure. At each iteration response and its gradients with respect to all 48 parameters are required. Consequently, FORM analysis Chapter 3: Application of Response Sensitivity in Composite Manufacturing 59 utilizing DDM takes 1386 seconds, compared to 16270 seconds for the FORM using the FDM. In conclusion, the reliability analysis using FORM by the FDM takes almost 12 times longer than with response sensitivities computed by the DDM. 3.5. OPTIMIZATION WITH GRADIENT-BASED ALGORITHM AND RELIABILITY CONSTRAINT A key objective in composites manufacturing is to achieve minimum “cost,” where the cost includes manufacturing costs, testing costs, and costs related to under-performance of the resulting composite parts. For example, the cost must include ‘shim-to-fit’ in the assembly phase. Clearly, the assessment of the total cost is associated with uncertainty, as demonstrated in the previous section. Traditionally, this cost-benefit optimization has been addressed through experience and trial and error. In this paper, a link between the new software and optimization algorithms is established to address this problem. In the literature, this is often referred to as reliability-based design optimization (RBDO). This study adopts the most basic RBDO approach; the nested bi-level approach proposed by Enevoldsen and Sorensen (1994). This approach involves a standard nonlinear optimization algorithm and the failure probabilities being evaluated whenever the values of the objective function and the reliability constraints are required. The decoupled nature of this approach is not ideal from a computation cost viewpoint but is straightforward to implement on the computer. The example studied in the previous section shows that probability of the maximum temperature exceeding 185 o C, which is the manufacturer’s recommendation, is 85%. To decrease this probability, the cure cycle parameters should be varied such that the curing time is minimized. This example is extended in this section to demonstrate an RBDO application. The objective is to minimize the cost of the curing time subjected to the manufacturing constraints under material, geometric and processing uncertainties. The design variables are the processing parameters. The cost, C, is a function of the time of achieving required degree of cure ( ct *α ), which is in turn a function of design variables. The total cost is then written Chapter 3: Application of Response Sensitivity in Composite Manufacturing 60 ccc total dttptCC *** )()( ααα∫= (3.16) where )( * ctp α is the time of reaching required degree of cure probability. Assuming that the cost is a linear function of curing time as ctCC *0 α= , where C0 is a constant value, one obtains )()( **** 00 cccc total tECdttptCC αααα == ∫ (3.17) Here, )( * ctE α is the mean of the time of reaching the required degree of cure. Eq. (3.17) describes that in order to minimize the cost, the mean of achieving the required degree of cure should be minimized. Li et al. (2001) state that the time of reaching a certain degree of cure, ct *α is calculated as ttt nn n nc ∆ − − += − − 1 * 1 * αα αα α (3.18) where 1−nα and nα are the degrees of cure at the time step before and after the node passes *α , 1−nt and nt are the corresponding origin and end of study times , and 1−−=∆ nn ttt is the time step. The derivative of Eq. (3.18), which is required for the reliability analysis, is then computed using the chain rule of differentiation θ α αθ α αθ ααα ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ − − n n cn n cc ttt *** 1 1 (3.19) where 1 * −∂∂ nct α α and α α ∂∂ ct * are simply calculated by differentiation of Eq.(3.18) with respect to 1−nα and nα . The sensitivities θα ∂∂ −1n and θα ∂∂ n are computed by the DDM in the utilized software. The constraints are applied to the probabilities of maximum and minimum peak temperatures exceeding certain thresholds. In summary, the RBDO problem reads } )( and )( )({argmin minmax * * pTTppTTp tE allowable peak allowable peakc )) ≤′<≤>∈ α x (3.20) Chapter 3: Application of Response Sensitivity in Composite Manufacturing 61 Here, 80% is set as the required degree of cure and maximum and minimum allowable peak temperatures are selected as 185 o C and 175 o C, respectively. Eq. (3.20) is then summarized as } )175( and )185( )({argmin minmax8.0 * pTppTp tE peakpeakc )) ≤<≤>∈x (3.21) Table 3.4 represents the results of Eq. (3.21) for thicknesses 5, 12.7, and 25.4 mm. The first column shows the applied constraints for different p ) values, ranging from 20% to 50%. Columns 3 to 7 show the optimum design variable values x * . The minimized time of reaching 80% degree of cure is shown in the last column. It is observed for 25.4 mm thickness, which is studied in previous example that changing the cure cycle parameters results in reducing the probabilities of exceeding 185 o C form 85% to 40%. Table 3.4 shows that for some p ) values and thicknesses the solution is not available. For example, for 5 and 12.7 mm thicknesses and =p) 20%, Eq. (3.21) does not have a solution. This is because within the constraints on the reliability and on the parameter values, the probability of exceeding the allowable peak temperature cannot be reduced to less than 20%. 3.6. CONCLUSIONS AND COMPARISON WITH STATE-OF-THE-ART Several authors have conducted research to address the uncertainties in composites manufacturing simulations. Fernlund et al. (1999) present parametric studies on the input variables utilizing deterministic analysis to identify the variability in the process-induced deformations. Padmanabhan and Pitchumani (1999a) utilize a stochastic model exploring the impacts of process and material uncertainties on the variability of fill time and degree of cure in the Resin Transfer Molding (RTM). Padmanabhan and Pitchumani (1999b) use the same route to investigate the variability of the nonisothermal filling process resulting from the uncertainties associated with the material and process. Mawardi and Pitchumani (2004) and Acquah et al. (2006) investigate a new approach to identify optimal cure temperature cycles under material and process uncertainties for polymer-matrix composites fabrication. Li et al. (2002) utilize a probabilistic-based approach to analyze the variability of process-induced deformations due to material Chapter 3: Application of Response Sensitivity in Composite Manufacturing 62 uncertainties in the RTM assignments. Those references assume that the finite element analysis is independent of the probabilistic analysis. In addition, they have all utilize sampling-based approaches. In this paper it is demonstrated that FORM is an efficient alternative, which yields importance vectors to rank the input parameters. In the context of finite-element based reliability analysis, much work has been carried out in structural engineering applications. Der Kiureghian and Taylor (1983) initiated the link between finite element analysis and first-order reliability analysis. Gutierre et al. (1994), Liu and Der Kiureghian (1991), Zhang and Der Kiureghian (1997), Der Kiureghian and Zhang (1999), Sudret and Der Kiureghian (2000), Imai and Frangopol (2000), Haldar and Mahadevan (2000), Frier and Sorensen (2003), and Haukaas and Der Kiureghian (2007) further develop this subject. Liu and Mahadevan (2000) utilize FORM in composites manufacturing using the ANSYS software, with a finite difference approach to obtain response sensitivities. Li et al. (2002) also carries out reliability analysis in the context of composites manufacturing, but with a response surface approach and only a few random variables. Shiaoa and Chamis (1999), Arafath et al. (2001), and Chamis (2004) utilize the same approach for probabilistic analysis of other composite structures. Relative to the optimization analysis, Hou and Sheen (1987), Rai and Pitchumani (1997), Li et al. (2001), and Li and Tucker (2002) apply nonlinear optimization programming to deterministically develop the optimized cure cycle under temperature constrains. Rai and Pitchumani (1997) utilize the finite difference method to calculate the needed response sensitivities. Hou and Sheen (1987), Li et al. (2001), and Li and Tucker (2002) use the DDM for this purspose. Zhu and Geubelle (2002) perform optimization to minimize the final displacements using DDM sensitivities in the optimization algorithm.. Rai and Pitchumani (1996) employ the neural network approach in optimization analysis. All those references neglect uncertainties. Mawardi and Pitchumani (2004) and Acquah et al. (2006) incorporate uncertainties into the optimal cure cycle design with a sampling-based approach, but without addressing the cost- benefit problem directly. Li et al. (2002) address this issue by merging reliability analysis with a simple Chapter 3: Application of Response Sensitivity in Composite Manufacturing 63 cost model associated with shimming and composites part failures. The response surface method is utilized in the reliability analysis, with only a few random variables. The reliability-based optimization carried out in this paper with dozens of random variables is believed to be state-of-the-art in composite manufacturing analysis. However, more advanced RBDO algorithms are available in other fields. Madsen and Hansen (1992), Enevoldsen and Sorensen (1994), Chen et al. (1997), Der Kiureghian and Polak (1998), Gasser and Schueller (1998), Kirjner-Neto et al. (1998), Torczon and Trosset (1998), Kuschel and Rackwitz (2000), Nakamura et al. (2000), Du and Chen (2002), Eldred et al. (2002), Wang and Kodiyalam (2002), Agarwal et al. (2003), Igusa and Wan (2003), Agarwal and Renaud (2004), Royset et al. (2006), and Liang et al. (2007) develop and utilize RBDO in various fields. Chapter 3: Application of Response Sensitivity in Composite Manufacturing 64 3.7. TABLES Table 3.1: Thermochemical model parameters Description Parameter Definition Mean Value Coefficient of Variation Distribution 01 T=θ C20 o 5% Lognormal Initial Values 02 αθ = 2103 −× 40% Uniform )0(3 fρθ = 33 kg/m 10790.1 × 2% Lognormal faρθ =4 Ckg/m 0 3 o 2% Lognormal )0(5 rρθ = 33 kg/m 10300.1 × 5% Lognormal raρθ =6 Ckg/m 0 3 o 5% Lognormal Fibre Density: )( 0)0( TTa fff −+= ρρρ Resin Density: )()( 00)0( ααρρ ρρ −+−+= rrrr bTTa rbρθ =7 3kg/m 0 5% Lognormal )0(8 fP C=θ KJ/kg 10913.7 2× 2% Lognormal fCp a=9θ 2J/kgK 050.2 2% Lognormal )0(10 rP C=θ J/kgK 10005.1 3× 5% Lognormal rCp a=11θ 2J/kgK 740.3 5% Lognormal Fibre Specific Heat Capacity: )( 0)0( TTaCC fff CpPP −+= Resin Specific Heat Capacity: )()( 00)0( αα −+−+= rrrr CpCpPP bTTaCC rCp b=12θ J/kgK 0 5% Lognormal )0(13 lfκθ = W/mK002.8 2% Lognormal lfaκθ =14 22 W/mK10560.1 −× 2% Lognormal )0(15 tfκθ = W/mK501.2 2% Lognormal tfaκθ =16 23 W/mK10070.5 −× 2% Lognormal )0(17 rκθ = W/mK158.0 5% Lognormal raκθ =18 24 W/mK10430.3 −× 5% Lognormal Fibre Thermal Conductivity: )(:alLongitudin 0)0( TTa lflflf −+= κκκ )(:Transverse 0)0( TTa tftftf −+= κκκ Resin Thermal Conductivity: )()( 00)0( αακκ κκ −+−+= rrrr bTTa rbκθ =19 W/mK10070.6 2−× 5% Lognormal A=20θ /s10528.1 5× 5% Lognormal E∆=21θ molJ/g 10650.6 4× 5% Lognormal m=22θ 813.0 5% Lognormal n=23θ 736.2 5% Lognormal C=24θ 090.43 5% Lognormal 025 Cαθ = 684.1− 5% Lognormal Resin Cure Kinetics: ( ){ }TC nm CTCe K dt d ααα ααα +−+ − = 01 )1( where RTEAeK /∆−= TCαθ =26 C /10475.5 3 o−× 5% Lognormal RH=27θ J/kg 10400.5 5× 5% Lognormal dt d HVQ Rrf α ρ)1( :GenerationHeat −=& fV=28θ 573.0 2% Lognormal )0(29 Tρθ = 33 kg/m 10707.2 × 1% Lognormal Tool Density: )( 0)0( TTa TTT −+= ρρρ Taρθ =30 Ckg/m 0 3 o 1% Lognormal )0(31 TP C=θ J/kgK 10960.8 2× 1% Lognormal Tool Specific Heat Capacity: )( 0)0( TTaCC TTT CpPP −+= TCp a=32θ 2J/kgK 0 1% Lognormal )0(33 Tκθ = W/mK10670.1 2× 1% Lognormal Tool Conductivity: )( 0)0( TTa TTT −+= κκκ Taκθ =34 2 W/mK0 1% Lognormal Chapter 3: Application of Response Sensitivity in Composite Manufacturing 65 Table 3.2: Boundary condition, cure cycle, and geometry parameters (see Figs. 3.8a and 3.8b for the definition of these parameters) Description Parameter Definition Parameter Value Coefficient of Variation Distribution Ph=35θ KW/m50 2 20% Uniform TBh=36θ KW/m110 2 20% Uniform Convective Heat Transfer: )( TThq −= ∞ TTh=37θ KW/m50 2 20% Uniform 138 s=θ min/C2 o 5% Lognormal 139 t=θ min 45 5% Lognormal 240 t=θ min 60 5% Lognormal 241 s=θ min/C2 o 5% Lognormal 342 t=θ min 35 5% Lognormal 443 t=θ min 120 5% Lognormal Cure Cycle Parameters 344 s=θ min/C3 o 5% Lognormal PL=45θ mm 2.76 0.1% Lognormal Tth=46θ mm 4.25 0.1% Lognormal PTL −=47θ mm 2.76 0.1% Lognormal Geometric Parameters Pth=48θ mm 5.42 0.1% Lognormal Table 3.3: Comparison of run time used for direct differentiation method (DDM) and finite difference method (FDM) to calculate the sensitivity of response in thermochemical model and run time used for reliability analysis using FORM by the DDM and FDM Case Analysis time (Sec.) 1 run of the FE analysis 163 DDM sensitivity analysis 574 FDM sensitivity analysis 7973 FORM analysis using DDM 1386 FORM analysis using FDM 16270 Chapter 3: Application of Response Sensitivity in Composite Manufacturing 66 Table 3.4: Optimized time of curing and design variables for different composites thicknesses and different constraints Constraints thp )(mm s1 min)/( C o t1 (min) t2 (min) s2 min)/( C o t3 (min) ct 8.0 (min) %50 )175( %50 )185( min max ≤< ≤> peak peak Tp Tp 5 2.80 46.15 50.00 1.20 30.00 171.68 %40 )175( %40 )185( min max ≤< ≤> peak peak Tp Tp 5 2.80 45.29 50.00 1.20 30.00 179.10 %30 )175( %30 )185( min max ≤< ≤> peak peak Tp Tp 5 2.80 44.42 50.00 1.20 30.00 189.60 %20 )175( %20 )185( min max ≤< ≤> peak peak Tp Tp 5 N/A N/A N/A N/A N/A N/A %50 )175( %50 )185( min max ≤< ≤> peak peak Tp Tp 12.7 2.80 45.78 50.00 1.20 30.00 173.06 %40 )175( %40 )185( min max ≤< ≤> peak peak Tp Tp 12.7 2.80 44.91 50.00 1.20 30.00 181.16 %30 )175( %30 )185( min max ≤< ≤> peak peak Tp Tp 12.7 2.73 44.96 88.34 1.23 30.04 222.74 %20 )175( %20 )185( min max ≤< ≤> peak peak Tp Tp 12.7 N/A N/A N/A N/A N/A N/A %50 )175( %50 )185( min max ≤< ≤> peak peak Tp Tp 25.4 2.80 44.49 51.62 1.20 30.00 182.24 %40 )175( %40 )185( min max ≤< ≤> peak peak Tp Tp 25.4 2.80 43.17 56.32 1.20 30.00 208.04 %30 )175( %30 )185( min max ≤< ≤> peak peak Tp Tp 25.4 N/A N/A N/A N/A N/A N/A %20 )175( %20 )185( min max ≤< ≤> peak peak Tp Tp 25.4 N/A N/A N/A N/A N/A N/A Chapter 3: Application of Response Sensitivity in Composite Manufacturing 67 3.8. FIGURES Figure 3.1: (a) Finite element mesh of the flat unidirectional composite part on a solid aluminium tool and the thermo-mechanical boundary conditions applied before tool removal, (b) typical 2-hold temperature cure cycle, and (c) part after removal tool (b) L=150 mm Hp = 1.6 mm (Part) HSH = 0.1 mm (shear layer) HT = 5 mm (Tool) (a) (c) wmax Convective heat transfer 0 40 80 120 160 200 240 280 320 20 50 80 110 140 170 200 time (min) T em p er at u re ( o C ) Chapter 3: Application of Response Sensitivity in Composite Manufacturing 68 Figure 3.2: (a) Half of finite element mesh of the unidirectional cured composite part on a solid aluminium tool and the external thermal boundary conditions (b) autoclave temperature and pressure during the process Figure 3.3: (a) Temperature at the part’s top surface and (b) interface of the tool and part where hP=60 (W/m 2 K), hTT=60 (W/m 2 K), and hTB=140 (W/m 2 K) 0 400 800 1200 1600 2000 20 40 60 80 100 time (sec) T em p er at u re ( o C ) Experiment Finite Element (a) (b) 0 400 800 1200 1600 2000 20 40 60 80 100 time (sec) T em p er at u re ( o C ) Experiment Finite element 0 400 800 1200 1600 2000 0 20 40 60 80 100 120 time (sec) T em p e ra tu re ( o C ) 0 100 200 300 400 500 600 P re ss u re ( K P a) Convective heat transfer, hP Tool Part hTT hTB thT = 25.4 mm thP = 38.1 mm LP = 76.2 mm LT-P =76.2 mm (a) (b) Chapter 3: Application of Response Sensitivity in Composite Manufacturing 69 Figure 3.4: Change in temperature at the part’s top surface due to a 1% change in the parameters Figure 3.5: Temperature at the part’s top surface and (b) interface of the tool and part where hP=55 (W/m 2 K), hTT=55 (W/m 2 K), and hTB=100 (W/m 2 K) -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 1 T em p er at u re c h an g e (% ) (a) (b) 0 400 800 1200 1600 2000 20 40 60 80 100 time (sec) T em p er at u re ( o C ) Experiment Finite Element 0 400 800 1200 1600 2000 20 40 60 80 100 time (sec) T em p er at u re ( o C ) Experiment Finite element Chapter 3: Application of Response Sensitivity in Composite Manufacturing 70 Figure 3.6: (a) Convective heat transfer coefficient on the part’s top surface (b) and tool bottom using sensitivity analysis and Eq. (3.10) by Johnston (1997) Figure 3.7: (a) Temperature at the part’s top surface and (b) interface of the tool and part using Eq. (3.10) by Johnston (1997) for convective heat transfer coefficients (a) (b) 0 400 800 1200 1600 2000 20 40 60 80 100 time (sec) T em p er at u re ( o C ) Experiment Finite element, a TB =0.25 Finite element, a TB =0.33 0 400 800 1200 1600 2000 20 40 60 80 100 time (sec) T em p er at u re ( o C ) Experiment Finite element, a P =0.15 Finite element, a P =0.17 0 400 800 1200 1600 2000 0 20 40 60 80 time (sec) h P ( W /m 2 K ) Estimated value using DDM 15.0 5 4 = ∗T P hP 0 400 800 1200 1600 2000 0 30 60 90 120 time (sec) h T B ( W /m 2 K ) Estimated value using DDM 5 4 25.0 = ∗T P hTB (a) (b) , , Chapter 3: Application of Response Sensitivity in Composite Manufacturing 71 Figure 3.8: Half of finite element mesh of the unidirectional composite part on a solid aluminium tool and the thermo-mechanical boundary conditions (b) cure cycle 0 40 80 120 160 200 240 280 320 20 60 100 140 180 220 time (min) T em p er at u re ( o C ) 1 1 s 2 t 5 t 4 t 3 t 2 t 1 s 3 1s1Tool Part hTT hTB Convective heat transfer, hP thT = 25.4 mm thP = 25.4 mm LP = 76.2 mm LT-P =76.2 mm (a) (b) Chapter 3: Application of Response Sensitivity in Composite Manufacturing 72 Figure 3.9: Variation of maximum and minimum peak temperature ( peakTmax , peakTmin ) and maximum and minimum degree of cure at the end of the process ( maxα , minα ) for the variation for the variation of (a) initial degree of cure, α0; (b) activation energy, ∆E; (c) autoclave second ramp, s2 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 180 186 192 198 204 210 T em p er at u re ( o C ) 0.82 0.824 0.828 0.832 0.836 0.84 α 0 D eg re e o f cu re α max α min T peak max T peak min 6 6.2 6.4 6.6 6.8 7 7.2 7.4 x 10 4 180 184 188 192 196 200 ∆ E (J/gmol) T em p er at u re ( o C ) 0.45 0.54 0.63 0.72 0.81 0.9 D eg re e o f cu re α min T peak min T peak max α max 1.8 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 170 178 186 194 202 210 S 2 ( o C/min) T em p er at u re ( o C ) 1.85 1.9 1.95 2 2.05 2.1 2.15 2.2 5 2.3 0.75 0.78 0.81 0.84 0.87 0.9 D eg re e o f cu re ( o / in) T peak max T peak min α max α min (a) (b) (c) Chapter 3: Application of Response Sensitivity in Composite Manufacturing 73 Figure 3.10: (a) Cumulative Probability Function (CDF) and (b) Probability Density Function (PDF) for the maximum and minimum peak temperature ( peakTmax , peakTmin ) (c) CDF and (d) PDF for maximum and minimum degree of cure at the end of the process (c) (d) (a) (b) 0.0 0.2 0.4 0.6 0.8 1.0 165 175 185 195 205 215 225 Temperature ( o C) C D F peakTmax peakTmin 0.000 0.012 0.024 0.036 0.048 0.060 165 175 185 195 205 215 225 Temperature ( o C) P D F peakTmax peakTmin 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Degree of cure C D F maxα minα 0.0 1.6 3.2 4.8 6.4 8.0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Degree of cure P D F maxα minα Chapter 3: Application of Response Sensitivity in Composite Manufacturing 74 Figure 3.11: Values of the importance vectors δ and η for three different limit-state functions: (a,b) peakTg max200 −= and (c,d) max825.0 α−=g (a) (b) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 1 2 3 4 5 6 7 8 9 10 d δ 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 1 2 3 4 5 6 7 8 9 10 h η (c) (d) 0.000 0.003 0.006 0.009 0.012 0.015 0.018 0.021 0.024 1 2 3 4 5 6 7 8 9 10 h η 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 1 2 3 4 5 6 7 8 9 10 d δ Chapter 3: Application of Response Sensitivity in Composite Manufacturing 75 3.9. 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Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 83 Chapter 4. SECOND-ORDER SENSITIVITIES OF INELASTIC FINITE ELEMENT RESPONSE BY DIRECT DIFFERENTIATION 3 4.1. INTRODUCTION The primary objective in this chapter is to explore the feasibility of computing second-order derivatives of finite element response with respect to input parameters. Such results are desirable in a number of applications, including reliability and optimization analysis. Examples include the second-order reliability method (SORM) (Ditlevsen and Madsen 1996) and optimization algorithms that employ the “Hessian matrix;” see, e.g., Luenberger (1984) and Polak (1997). However, second-order sensitivities of finite element response have previously been considered unattainable due to high computational cost. Instead, previous work in the field of response sensitivity analysis has focused on first-order derivatives. Significant progress has been made in the last two decades to compute first-order sensitivities of finite element response in an efficient and accurate manner, including inelastic and dynamic problems. This has been achieved by means of the direct differentiation method (DDM) (Choi and Santos 1987; Tsay and Arora 1990; Liu and Der Kiureghian 1991; Zhang and Der Kiureghian 1993; Kleiber et al. 1997; Roth and Grigoriu 2001; Conte et al. 2003; Scott et al. 2003; Haukaas and Der Kiureghian 2005). In essence, the DDM consists of differentiating the response algorithm analytically and implementing the resulting equations alongside the ordinary response algorithm. Notably, the DDM does not require repeated runs of the finite element analysis, as is the case with the less efficient finite difference approach. Moreover, the difficulties associated with selecting appropriate perturbation values in the finite difference approach is 3 A version of this chapter has been published. Bebamzadeh, A. and Haukaas, T. (2008), “Second-order sensitivities analysis of inelastic finite element response by direct differentiation.” ASCE Journal of Engineering Mechanics, 134(10), 867-880. Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 84 circumvented by the DDM. In the present study, the DDM is extended to offer second-order response sensitivities. Previous work to obtain second-order response sensitivities in an analytical manner has focused on linear, problems limited to a few degrees-of-freedom, or specialized problems (Lee and Lim 1998; Park et al. 1999; Chen and Choi 1994; Hwang et al. 1997; Kita et al. 1997). Additionally, algorithms have been developed to automatically differentiate computer code, e.g., Ozaki et al. (1995). However, the automatic differentiation strategy does not provide the physical insight and opportunity for manually optimizing the implementations for computational efficiency that the direct differentiation approach offers. This is particularly critical for second-order derivatives, which is the focus in this study. This chapter represents the first complete presentation and verification of second-order analytically differentiated equations for inelastic multi-degree-of-freedom finite element problems. Detailed equations for the extensively utilized J2 plasticity material model — both the uniaxial and the multiaxial version — are presented. The presentation starts by reviewing the background for the need for second-order sensitivities, as well as the fundamentals of first-order sensitivity analysis. Next, the governing response equations for inelastic finite element problems are differentiated analytically once more to obtain second- order response sensitivities. Subsequently, it is described how this necessitates additional results from the constitutive (material) model. Algorithms and detailed equations are presented for the aforementioned J2 plasticity models. The derivations are verified by finite difference results by applying the new implementations to two numerical examples. This chapter concludes with novel studies of the computational cost to provide insight into the feasibility of computing second-order finite element response sensitivities in practical applications. 4.2. NEED FOR SECOND-ORDER RESPONSE SENSITIVITIES In gradient-based optimization analysis the derivatives of the objective and constraint functions with respect to intervening parameters (e.g., material, geometry, and load parameters) are required. Similarly, Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 85 in reliability analysis by the first- and second-order reliability methods (FORM and SORM) the derivatives of the limit-state function with respect to the transformed random variables are sought. These circumstances necessitate the computation of finite element response sensitivities in cases where finite element response quantities enter the functions. To illustrate this, consider g to be the function for which derivatives are sought and let u, x, and y be the vectors containing the response quantities, the intervening physical parameters, and the transformed parameters, respectively. Generally, the function reads )))((()))((( pim yxuggg == yxu (4.1) where g is explicitly shown to be a function of the responses u, which are dependent on the physical model parameters x, which in turn are dependent on the variables y, which constitute the space in which the gradient-based search for a “design point” is carried out. As an example, u may represent displacement responses from a finite element analysis, while x may represent material parameters in that finite element model. The transformed variables y are employed within the reliability analysis. Index notation is introduced in the last part of Eq. (4.1) to facilitate the subsequent derivations in this study. While the vector notation (boldface symbols) is cumbersome in this study the index notation is appealing because the order of differentiation and multiplication is unambiguous. Moreover, index notation facilitates the direct implementation of the derived equations on the computer; simply by looping over indices. In Eq. (4.1), Mm ,,2,1 K= , Ii ,,2,1 K= , and Pp ,,2,1 K= , where M is the number of response quantities, and I and P both represent the number of intervening parameters. The customary summation over repeated indices is implied unless otherwise noted, e.g., xixi = x1x1 + x2x2 + x3x3 +… To obtain the first-order sensitivity of g with respect to y the chain rule of differentiation is invoked to obtain p i i m mp y x x u u g y g ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ (4.2) Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 86 Here, mug ∂∂ is straightforwardly obtained because g is generally an algebraic function of the response quantities um. Furthermore, pi yx ∂∂ simply represents the Jacobian matrix, Jx,y, of the x–y transformation. The factor for which the main effort must be devoted is im xu ∂∂ , namely the derivative of the finite element response with respect to the original input parameters, which is addressed by the aforementioned first-order response sensitivity analysis. Attention is now turned to the second-derivatives required to establish the Hessian matrix (curvatures) of g with respect to the transformed parameters. Eq. (4.2) is differentiated by invoking the product rule and chain rule of differentiation to obtain ∂∂ ∂ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ = ∂∂ ∂ 4342132143421 C qp i i m mp i q j B ji m mp i i m q j j n A nmqp yy x x u u g y x y x xx u u g y x x u y x x u uu g yy g 2222 (4.3) The observation is here made that the first-order derivatives that are present in Eq. (4.2) are also required to obtain qp yyg ∂∂∂ 2 . Additionally, the second-order derivatives identified by A, B, and C are required. The factor A is readily obtained because g is an algebraic function of the response quantities. The factor C is obtained by differentiating the parameter transformation twice, which poses no conceptual problems. Again, the main effort must be devoted to the response sensitivities, here in the form of the second-order response sensitivities identified as factor B. The same requirement is present in optimization algorithms that utilizes the second-derivative of the objection function and/or constraints. This motivates the developments in this study. In passing, it is noted that for cases where transformation of the intervening parameters is not performed, such as deterministic optimization applications, then Eqs. (4.2) and (4.3) simplify. Effectively, the last term in Eq. (4.3) vanishes and the derivatives with respect to y in other terms are set equal to one. Nevertheless, the need for response sensitivities remains. Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 87 4.3. REVIEW OF FIRST-ORDER SENSITIVITY COMPUTATIONS To benefit subsequent developments, the DDM to obtain first-order sensitivities is briefly reviewed for the inelastic static case. The linear static case is a trivial simplification of this case and the inclusion of dynamic effects poses no conceptual difficulties – because the key issue is the derivative of the static internal forces in the structure – but introduces more lengthy equations. The fundamental undertaking in the DDM is to analytically differentiate the governing response equations. To this end, consider the governing response equations for inelastic problems that express equilibrium between the internal forces of the structure, here denoted Pn, and the external load vector Fn, written as ( ) )(),( iniimn xFxxuP = (4.4) where the dependence of the internal force vector Pn on the is the displacement vector um is shown, as well as the implicit and explicit dependence upon the parameter xi. This is the parameter with respect to which sensitivity results are sought; that is, we typically seek the sensitivity result im xu ∂∂ . The parameter xi typically represents a material parameter, but as shown in Eq. (4.4) it could also represent a parameter that influences the external load vector Fn. Differentiation of Eq. (4.4) with respect to xi yields (Zhang and Der Kiureghian 1993; Kleiber et al. 1997) n iui n i m m n x F x P x u u P m ∂ ∂ = ∂ ∂ + ∂ ∂ ∂ ∂ (4.5a) ∂ ∂ − ∂ ∂ = ∂ ∂ ⇒ − mu i n i n nm i m x P x F K x u 1 (4.5b) where nmmn KuP =∂∂ is the tangent stiffness matrix and the vertical bar denotes differentiation for fixed displacements to account for the explicit dependence on xi. It is noted from Eq. (4.5b) that although inelastic problems are considered the response sensitivities are obtained from a linear equation of the Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 88 same form as that of the linearized incremental response equation that appears in the Newton-Raphson algorithm to solve Eq. (4.4). Consequently, upon attainment of equilibrium according to Eq. (4.5b), no further iterations are needed to determine first-order response sensitivities; the sensitivities are obtained by solving Eq. (4.5b) for the parameters of interest. This is a cornerstone in the efficiency of the DDM, as pointed out by Zhang and Der Kiureghian (1993), who also noted that the evaluation of the conditional derivative mu in xP ∂∂ requires two calls to the constitutive model for inelastic problems. First, the conditional derivative is obtained, and second, unconditional derivatives of the history variables are computed and stored for the next analysis step. This issue will be revisited in the developments of second- order sensitivities in the present study. Further details on derivation, implementation, and computation of first-order sensitivities of finite element response, including a wide array of material models and geometry parameters, are found in Zhang and Der Kiureghian 1993, Kleiber et al. 1997, Conte et al. 1999, Roth and Grigoriu 2001, Scott et al. 2003, Haukaas and Der Kiureghian 2005 and elsewhere. In summary, response sensitivities by the DDM for static and dynamic inelastic finite element problems with respect to material, geometry, and load parameter are currently available. 4.4. SECOND DIFFERENTIATION OF GOVERNING RESPONSE EQUATIONS In order to obtain second-order response sensitivity results by the DDM in this chapter, the governing response equations are differentiated twice. The second differentiation may obviously be performed with respect to a different parameter than the first differentiation. Consequently, the sought sensitivity is generically written as jim xxu ∂∂∂ 2 , where xi and xj may be the same or different parameters of the finite element model. When differentiating Eq. (4.5b) once more it is noted that because the internal forces and the current tangent stiffness matrix may depend on the parameter both explicitly and implicitly through the displacement response, we have the partial results Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 89 ou j nm j o o nm j nm m n j x K x u u K x K u P x ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ = ∂ ∂ ∂ ∂ (4.6) and omomm u ji n j o ui no uji n j o ui n oui n j xx P x u x K xx P x u x P ux P x ∂∂ ∂ + ∂ ∂ ∂ ∂ = ∂∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ 22 (4.7) The last equality in Eq. (4.7) is obtained by employing the principle of invariance to the order of differentiation. The complete differentiation of Eq. (4.5b) with respect to xj and substitution of Eqs. (4.6) and (4.7) yields jiuji n j o ui no ji m m n i m uj nm i m j o o nm xx F xx P x u x K xx u u P x u x K x u x u u K omo ∂∂ ∂ = ∂∂ ∂ + ∂ ∂ ∂ ∂ + ∂∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∂ ∂ n 222 (4.8a) ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ ∂ ∂ − ∂∂ ∂ = ∂∂ ∂ ⇒ − omo u ji n j o ui no i m uj nm i m j o o nm ji nr ji r xx P x u x K x u x K x u x u u K xx F K xx u 2n 2 1 2 (4.8b) The dummy index m in the third term of Eq. (4.8a) is replaced with r in Eq. (4.8b) to avoid violation of index notation rules in Eq. (4.8b). Several observations are made with regard to Eq. (4.8b). First, the second-order sensitivity is obtained from a linear system of equations with the same stiffness matrix as that of both the incremental response equation and the first-order response sensitivity equation in Eq. (4.5b). That is, only the right-hand side of the system of equations in Eq. (4.8b) need to be reassembled to obtain second-order sensitivities. It is strongly emphasized, however, that the methodology addresses inelastic problems. Furthermore, it is stressed that no approximations are introduced in Eq. (4.8). Specifically, no Taylor expansion is involved in the derivation of Eq. (4.8). Next, it is observed that the quantity onm uK ∂∂ is required to determine the second-order response sensitivities. This derivative of the stiffness matrix is effectively the second-derivative of the internal force vector with respect to the displacement vector. By counting indices in this quantity it is clear that Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 90 this derivative is a third order tensor. As shown in the subsequent derivations for the J2 plasticity material, it is obtained in a straightforward manner, albeit at additional computational cost compared to the first- order sensitivity computations. Third, it is observed that the first-order response sensitivities are required to obtain the second-order response sensitivities. Fourth, the matrix ou jnm xK ∂∂ is required to determine the second-order response sensitivities. Because the stiffness matrix at the element, section, and material levels is ordinarily available in an algorithmic form amenable to differentiation, also this quantity will subsequently be obtained in a straightforward manner. In passing, it is also noted that for nonlinear problems the current tangent stiffness matrix varies with the displacements. Consequently, the first-order response sensitivities will generally appear in the computation of ou jnm xK ∂∂ . Finally, it is observed in Eq. (4.8b) that ou jin xxP ∂∂∂ 2 is required to determine the second-order response sensitivities. Conceptually, the computation of this quantity does not differ significantly from the computation of mu in xP ∂∂ in first-order sensitivity analysis, as will be demonstrated in the following. 4.5. SECOND DIFFERENTIATION OF CONSTITUTIVE EQUATIONS Compared to first-order sensitivity computations, the derivations above show that the quantities onm uK ∂∂ , ou jnm xK ∂∂ , and ou jin xxP ∂∂∂ 2 are required to obtain second-order response sensitivities by the DDM for inelastic problems. The stiffness matrix is the derivative of the internal force vector Pn with respect to the displacement vector um and the internal force vector reads U structure element elrnrn dVBP = ∫ σ (4.9) Eq. (4.9) denotes the assembly over all elements of the element integral of the strain-displacement matrix Bnr and the stress vector σr, where r counts over the number of stress (and strain) components of the material model. The case in which the parameters xi and xj represent material parameters will be Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 91 considered in the following. This implies that the boundaries of the integral in Eq. (4.9), as well as Bnr, are independent of the parameters for which the differentiation is carried out. Cases in which xi and xj represent load parameters are conceptually simpler, while cases in which xi and xj represent geometry parameters require extensions analogue to those carried out for first-order sensitivities in Haukaas and Der Kiureghian (2005). Consequently, by inspection of Eq. (4.9), the above-stated need for the three derivatives onm uK ∂∂ , ou jnm xK ∂∂ , and ou jin xxP ∂∂∂ 2 directly translates into the need for the derivatives trsk ε∂∂ , t jrs xk ε ∂∂ , and t jir xx ε σ ∂∂∂2 from the material model. Two constitutive models are considered in this study: the uniaxial and the multiaxial J2 plasticity models (Simo & Hughes 1998). Due to space constraints, all the derivative equations cannot be displayed in this paper. Appxs. F and G provide equations to demonstrate the differentiation technique. In this study, focus is on the presentation of the overall differentiation and implementation process, followed by computational cost studies. To this end, the following section provides a step-by-step outline of the process. 4.6. OVERVIEW OF THE COMPUTATION PROCESS In this study, all equations pertaining to first- and second-order response sensitivities are implemented on the computer and verified through examples in the following sections. It is emphasized that although the second-order sensitivity equations, such as those presented in Appxs. F and G are more comprehensive than those previously published for first-order sensitivity analysis, no conceptual differences or difficulties are present. The process remains straightforward; namely, to differentiate algebraic expressions and implementing the results on the computer. Although this may seem a cumbersome task it is a one-time investment that provides significant computational savings in all subsequent use of the finite element code. In fact, the alternative finite difference approach has opposite characteristics: it is straightforward to implement but the sensitivity results come at a significant computational cost and loss of accuracy in all subsequent use of the code. Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 92 To make the entire process clear to the reader, the step-by-step procedure that must be implemented in the finite element code is offered below. These operations are executed at each analysis increment. That is, after convergence to equilibrium for the ordinary finite element response, the following sensitivity computations are executed: 1. Compute first-order sensitivity results: a. Compute the first-order derivative of the internal force vector conditioned upon fixed displacement. This quantity is appears in Eq. (4.5). This task is identical to what is done in ordinary first-order sensitivity analysis. b. Solve for first-order sensitivity results; that is, solve Eq. (4.5b). 2. Compute and store first-order derivatives of the history variables for inelastic materials. This is done without the condition of fixed displacements. As with the previous tasks, this step is identical to what is done in ordinary first-order sensitivity analysis. 3. Compute second-order sensitivity results: a. Compute the following three quantities that appear in Eq. (4.8): The second-order derivative of the internal force vector conditioned upon fixed displacements; the first- order derivative of the current tangent stiffness matrix with respect to the sensitivity parameter, and the first-order derivative of the current tangent stiffness matrix with respect to the displacement vector. b. Solve for second-order sensitivity results; that is, solve Eq. (4.8b). 4. Compute and store second-order derivatives of the history variables. As in Item 2 above, this is done without the assumption of fixed displacements. The order in which the above operations are performed is significant. Specifically, sensitivity results must be obtained before the derivatives of the history variables can be computed and stored. In this study, the Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 93 items listed above are identified as “phases” of the sensitivity computations. These phases are reflected at the constitutive level as well, for which Fig. 4.1 provides an overview. For brevity, Fig. 4.1 shows quantities that are involved in the uniaxial case. However, the concepts and the information flow are valid also for the multi-axial case. The input and output quantities are then vectors, matrices, and tensors, instead of scalars. To provide further insight into the sensitivity computation process, consider first this uniaxial case. The traditional objective of the material model is to compute the stress σn+1and the stiffness kn+1 at analysis step n+1 for a given input strain εn+1, where subscript indices are omitted in the uniaxial case because these are scalar quantities. This original objective is shown at the top of Fig. 4.1. To compute second-order sensitivities, the four sensitivity computation phases are added to the deliverables required by the implementation of the material model. The flow of information is indicated by arrows in Fig. 4.1. In paticular, it is noted that Phase 1 takes the current strain and the history variables as input to produce the conditional stress derivative that is needed to form the conditional derivative of the internal forces needed to compute first-order sensitivities by Eq. (4.5b). Phase 2 serves the purpose of storing unconditional derivatives of the history variables for subsequent analysis increments and also for the second-order sensitivity computations. Next it is noted that Phase 3, analogue to Phase 1, takes the current strain sensitivity and the first-order derivative of the history variables as input to produce the derivatives that are needed to solve for second-order response sensitivities in Eq. (8b). Finally, Phase 4 is analogue to Phase 2 in that it stores unconditional second-order derivatives of the history variables for use in subsequent analysis increments. The reason why the unconditional derivatives of the history variables could not be computed within Phases 1 and 3 is subtle but important: the conditional derivative output from Phases 1 and 3 are conditioned upon fixed current strain. That is, they are not conditioned upon fixed strain in previous analysis steps. Consequently, the derivative of the history variables should not be conditioned upon fixed strain. To obtain these unconditional derivatives the response sensitivities ix∂∂ε (and subsequently ji xx ∂∂∂ ε 2 ) must be known before the derivatives of the history variables are Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 94 computed. This is why Phase 1 is first carried out to obtain the first-order response sensitivities, followed by the computation of unconditional first-order derivatives of the history variables in Phase 2. Similarly, Phase 3 is carried out to obtain the second-order response sensitivities, followed by the computation of unconditional second-order derivatives of the history variables in Phase 4. A concluding remark about the history variables (sometimes referred to as path-dependent internal variables) is in order. It is known from first-order sensitivity analysis that the derivative of these variables must be computed and stored (Zhang and Der Kiureghian 1993). This is identified in Phase 2 in the process discussed above. Of particular significance in this study is the treatment of the history variables in second-order sensitivity analysis. Analogue to Phase 2, Phase 4 appears in the second-order sensitivity analysis. It is stressed that although Phase 4 necessitates additional computational effort, there are no conceptual difficulties associated with the path-dependent internal variables in second-order sensitivity analysis. 4.7. EFFICIENT COMPUTER IMPLEMENTATION The derived equations, of which a selection is presented in Appxs. F and G, are implemented in an in- house Matlab-based toolbox for nonlinear finite element analysis. An important purpose of the implementations – in addition to the verification carried out in the next section – is to investigate the computational cost, and thus feasibility, of computing second-order response sensitivities. To this end, several novel implementation strategies are explored in this study. First, for the first-order sensitivity computations, instead of looping over all parameters for which response sensitivities are sought, an expanded version of Eq. (4.5b) is solved only once. This is achieved by collecting the different right-hand sides of Eq. (4.5b) – one for each parameter – in one matrix. This produces a significant saving of computer time when solving the system of equations in Eq. (4.5b). Consequently, the left-hand side of Eq. (4.5b) is considered as an M×I matrix, where M is the number of degrees-of-freedom and I is the number Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 95 of parameters for which sensitivities are sought. That is, the left-hand side matrix in Eq. (4.5b) contains the vectors pertaining to each of the parameters for which sensitivities are sought. Similar to the above considerations for first-order sensitivity computations, in the second-order sensitivity computations the left-hand side of Eq. (4.8b) is a third order tensor (M×I×I); that is, for each of the M degrees-of-freedom a matrix of second-order sensitivities is computed. In other words, the left- hand side is the Hessian matrix for all degrees-of-freedom. In this case, the right-hand side of Eq. (4.8b) consists of the triple product of the M×M×M tensor onm uK ∂∂ , the M×I matrix jo xu ∂∂ and the M×I matrix im xu ∂∂ , as well as the product of the M×M×I tensor ou jnm xK ∂∂ and the M×I matrix im xu ∂∂ . A novel approach is presented in this study to reduce the order of the tensors; transforming them to matrices, and solving Eq. (4.8b) only once instead of looping over all parameters. As an illustrative example, consider the second term in the right-hand side of Eq. (4.8b); the triple product mentioned above. Instead of differentiating the global stiffness matrix with respect to the displacement vector followed by the multiplication with jo xu ∂∂ and im xu ∂∂ this product is evaluated in a novel and efficient manner at the element level as follows: UU UU ∫∫ ∫∫ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ ∂ ∂ el el i s j t t rs nr el el i m j o smto t rs nr el el i m j o sm o t t rs nr i m j o el elsm o rs nr i m j o o nm dV xx k BdV x u x u BB k B dV x u x u B u k B x u x u dVB u k B x u x u u K εε εε ε ε (4.10) The quantity after the last equality in Eq. (4.10) is an M×I×I tensor. This tensor is calculated at each element and then assembled to the global tensor required in Eq. (4.8b). This novel technique significantly reduces the computational cost, particularly for large number of the degrees-of-freedom. In particular, it prevents the computation and storage of the M×M×M tensor onm uK ∂∂ at the global level by converting Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 96 it into an M×I×I tensor at the element level. A similar approach is adopted for the product of ou jnm xK ∂∂ and im xu ∂∂ : UUU ∫∫∫ ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ el el i s j rs nr el el i m sm j rs nr i m el elsm j rs nr i m uj nm dV xx k BdV x u B x k B x u dVB x k B x u x K ttto ε εεε (4.11) Eq. (4.11) demonstrates that this product is also reduced to an M×I×I tensor (after the last equality) instead of saving the M×M×I tensor ou jnm xK ∂∂ at the global level. In addition, the last term in Eq. (4.8b), ou ji xxP ∂∂∂ 2 , is an M×I×I tensor. Consequently, all the terms in Eq. (4.8b) are converted to M×I×I tensors. In addition to the above developments to save computational cost, the tensors are saved as matrices M×I 2 . Specifically, the tensors treated above consist of a vector with dimension M for the parameters xi and xj. Due to symmetry, i.e., ijji xxxx ∂∂∂=∂∂∂ 22 , the size of the M×I 2 matrices are reduced to ( )2/)1( +× IIM by considering only the upper triangle of the symmetric matrices. Thus, instead of looping over all parameters xi and xj for which second-order response sensitivities are sought, an expanded version of Eq. (4.8b) is solved only once by collecting all second-order sensitivity tensors (left- hand sides of Eq. 4.8b) in matrices. Accordingly, the second-order sensitivities are obtained by inverting the M×M tangent stiffness matrix, which is already factorized in the last iteration of the Newton-Rophson algorithm employed to solve for the response, applied to the right-hand side of Eq. (4.8b), which now is a ( )2/)1( +× IIM matrix. The significant benefits of these computational cost saving strategies are explored in the following section. Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 97 4.8. NUMERICAL EXAMPLES In this section, two examples are considered to demonstrate and verify the computation of second-order response sensitivities. The results are accompanied by computational cost studies to reveal the practicability of computing second-order response sensitivity in inelastic finite element applications. Two examples are considered: a truss structure and a 2D problem that utilizes four-node quadrilateral finite elements. The uniaxial model is employed in conjunction with the truss elements, while the multiaxial model is employed with the 2D elements. It is emphasized, however, that the uniaxial material model has a significantly wider application than the truss elements employed in this study. For instance, the fiber- discretized cross-sections employed in nonlinear beam-column elements – which are prevalent in modern nonlinear structural analysis – utilize uniaxial material models. Other applications of the two material models differentiated herein are found aplenty in nonlinear structural analysis. In the following, comparisons are made between the efficient and accurate results from the DDM algorithm implemented in this study and first- and second-order sensitivities obtained by the less efficient and generally less accurate finite difference approach. 4.8.1. Tower Truss with Uniaxial J2 Plasticity Material Model Fig. 4.2 shows the geometrical dimensions of the tower truss that is considered for sensitivity analysis in this study. The cross section area of all members is 2.5 cm 2 . The cyclic load F is applied at the top of the tower with load variation over time shown in Fig. 4.2(b). The tower is modeled with truss elements with the uniaxial J2 plasticity model. The four material parameters are the Young’s modulus E=207 GPa, the yield stress σy=212 MPa, the isotropic hardening modulus Hiso=15.0 GPa, and the kinematic hardening modulus Hkin=1.0 GPa. The load-displacement response at the top of the truss is shown in Fig. 4.3(a). The first-order sensitivity results obtained by the DDM implementations are shown in Fig. 4.3(b) to Fig. 4.3(e) for the four parameters, which here are considered to be the same in all members of the truss. The results are compared with sensitivities obtained by the finite difference method (FDM). Excellent agreement is Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 98 observed between the DDM and FDM results when the perturbation in the FDM is 0.005 times the parameter value. Of particular significance to this study, Fig. 4.4 shows the second-order sensitivity results based on the DDM equations developed above and presented in detail in Appx. F. The effort devoted to implementation and debugging is rewarded by agreement between the DDM and FDM results. However, further comments are required because differences are observed between the DDM and FDM results for yEu σ∂∂∂ / 2 , isoHEu ∂∂∂ / 2 , kinHEu ∂∂∂ / 2 , and 22 / yu σ∂∂ . First, observe that the ordinate axis in these plots are in the order of 10 -14 to 10 -18 , that is, equal to zero from a computer precision standpoint. Next, observe that the FDM results exhibits a numerical noise behavior. It is readily concluded that the FDM results, although being practically zero as the DDM results correctly are, exhibits round-off errors that make them slightly different from the DDM results. This demonstrates the accuracy of the DDM results and the errors in the FDM results due to the difficulty in selecting parameter perturbation value. Fig. 4.4 effectively displays the evolution of the upper triangle of the Hessian matrix (matrix of second- derivatives) for this structural problem. The lower triangle is not shown because ji xxu ∂∂∂ / 2 is equal to ij xxu ∂∂∂ / 2 . Although it is outside the scope of this study, it is noted that for the truss problem several of the second-derivatives are equal to zero throughout the analysis. This is due to the bi-linear material behaviour for which several first-order sensitivities are constant, and hence, several second-order sensitivities are zero. Such information is valuable in reliability and optimization applications to judge the significance of curvatures in the parameter space. Table 4.1 provides an overview of the computational cost associated with computing the aforementioned sensitivity results. The first-order response sensitivities with respect to the 4 parameters are obtained in 1.6 seconds by the DDM, compared to 4.7 seconds by the FDM. The FDM cost is readily computed by adding the cost of 4 stand-alone response analyses – one for each parameter – to the initial response analysis. As indicated in the right-most column in Table 4.1, the DDM results are obtained with Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 99 a computational cost that is a factor 2.9 less than the finite difference results. This is in part due to the linear nature of Eq. (4.5b), but also the novel implementation approach described earlier. The supremacy of the DDM approach is expected to be further pronounced when the number of variables for which sensitivity results are sought increases. Table 4.1 shows, remarkably, that the second-order sensitivity results are obtained by the DDM in 2.7 seconds, while the FDM requires 13.9 seconds. That is, for second-order sensitivities the DDM approach is 5.1 times more efficient than the FDM. These results include the consideration given to the FDM approach that the matrix of second-order sensitivity results presented in Fig. 4.4 is symmetric. Consequently, only 10 additional re-runs of the stand-alone finite element analysis is counted in the FDM cost. These results clearly demonstrate the advantage of the DDM approach in the computation of second- order derivatives, and indicate that the computation of second-order derivatives of finite element responses is indeed feasible. This supposition is further examined in the next example. 4.8.2. 2D Cylinder Under Internal Pressure with Multiaxial J2 Plasticity Model In this section the implementations for the multiaxial J2 plasticity model are demonstrated and verified. Again the results from the DDM implementations and the less efficient FDM approach are compared. For this purpose, consider the 2D plane strain cylinder in Fig. 4.5(a), which models a quarter of the cylinder modelled with four-node quadrilateral 2D elements. The inner radius of the cylinder is 5cm and the outer radius is 6cm. A cyclic uniform pressure is applied at the internal radius of the cylinder. As shown in Fig. 4.5(b), the cyclic pressure is increased linearly from zero to 50MPa, followed by a decrease to -55 MPa, and followed by an increase to zero. The cylinder is made of steel, which is idealized as a multiaxial J2 plasticity material model with five parameters: Young’s modulus E=207GPa, Poisson’s ratio ν=0.3, yield stress σy=212MPa, isotropic hardening Hiso=15GPa, and kinematic hardening Hkin=1.0GPa. The prediction of the pressure- displacement response at the inner radius of the cylinder is shown in Fig. 4.6(a). The first-order sensitivity results are shown in Fig. 4.6(b) to Fig. 4.6(e) for five parameters; E, ν , σy, Hi, and Hk. Again, excellent Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 100 agreement is observed between the DDM and FDM results when the perturbation is 0.005 times the parameter value. This verifies the accuracy of the first-order response sensitivity computations by the DDM. The second-order sensitivity results obtained for the cylinder problem are shown in Fig. 4.7. Again, the effort to implement and debug the code is rewarded with excellent agreement between the DDM and FDM results. Similar to the first example, only the upper triangle of the Hessian matrix is shown due its symmetry. Application of the results in Fig. 4.7 is outside the scope of this study; however, it is observed that contrary to the truss problem all second-order response sensitivities are non-zero. Consequently, the results indicate that none of the elements of the Hessian matrix for this type of problems can be neglected. Table 4.2 compares the difference in the computational cost associated with obtaining the DDM and FDM sensitivity results for this example. Consider first the computation of first-order response sensitivities. For a mesh with 40 elements the DDM results are obtained in 24.7 seconds, while the FDM results are obtained in 72.1 seconds (6 times the cost associated with one stand-alone finite element analysis, because there are 5 parameters for which sensitivities are sought with respect to). That is, the DDM results are obtained with a factor 2.9 less cost than the FDM results. Interestingly, for a mesh with 250 elements (for the same physical problem) the efficiency ratio between the DDM and the FDM increases to 4.3. This indicates that the FDM approach, including the novel implementation techniques outlined above, becomes increasingly efficient as the stand-alone finite element cost increases. Next, consider the computational cost associated with the computation of second-order sensitivity results. For the 40-element mesh the DDM results are obtained in 56.9 second versus 252.4 seconds for the FDM results. That is, the computational cost ratio is 4.4, which is comparable to the results observed in the first example. However, notice the dramatic increase in the computational cost ratio for the 250- element mesh. In this case, the DDM cost is 1366.8 seconds while the FDM results are obtained in 14,158.2 seconds. Remarkably, the cost ratio is now 10.4. This clearly indicates that the DDM approach Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 101 to compute second-order response sensitivities becomes increasingly efficient as the problem size increases compared with the generally less accurate finite different approach. 4.9. CONCLUSIONS It is concluded in this study that the computation of second-order derivatives of finite element response has become practicable owing to the direct differentiation method. Significant improvement of the computational efficiency is observed compared to the alternative “brute-force” finite difference approach. The direct differentiation approach also circumvents the difficult problem of selecting parameter perturbation value in finite difference computations. Numerical examples are provided in this chapter to demonstrate and verify the implementations, with particular focus on computational cost. Detailed equations are also provided to enable the extension of any finite element code with response sensitivity capabilities. An important consequence of the developments in this paper – in addition to stand-alone response sensitivity studies – is that it is now feasible to employ second-order derivatives – that is, the Hessian matrix – in finite element-based reliability and optimization applications. Notably, these second- order derivatives have three important characteristics: they are computed efficiently; they are consistent with the approximations in the ordinary response; and have the same precision as the other analysis results. Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 102 4.10. TABLES Table 4.1: Comparison of computational cost associated with the DDM compared to the FDM to compute first- and second-order response sensitivities for the first example. The stand- alone finite element response analysis takes 0.9 seconds Time DDM (Sec.) Time for FDM (Sec.) Cost ratio 1 st -order sensitivity 1.6 4.7 2.9 2 nd -order sensitivity 2.7 13.9 5.1 Table 4.2: Comparison of computational cost associated with the DDM compared to the FDM to compute first- and second-order response sensitivities for the second example. The stand-alone finite element response analysis takes 12.0 seconds for the 40-element- mesh, and 674.2 seconds for the 250-element-mesh Mesh Time for DDM (Sec.) Time for FDM (Sec.) Cost ratio 40 24.7 72.1 2.9 1 st -order sensitivity 250 939.5 4045.2 4.3 40 56.9 252.4 4.4 2 nd -order sensitivity 250 1366.8 14158.2 10.4 Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 103 4.11. FIGURES Figure 4.1: Extended objectives of a generic constitutive model enable second-order response sensitivity computations • Compute trial elastic stress state • Determine if the step is elastic or plastic • Compute stress σ • Compute tangent stiffness k Compute 1 st -order conditional derivative of the stress εn+1 Phase 0 (original objective): Phase 1: Phase 2: σn+1 History variables 1 1 +∂ ∂ + ni n x ε σ Compute and store 1 st -order unconditional derivative of the history variables i n x∂ ∂ +1ε • Compute 2nd-order conditional derivative of the stress • Compute 1st-order conditional derivative of the stiffness wrt. the parameter • Compute derivative of the stiffness wrt. the strain Phase 3: 1 12 + ∂∂∂ + nji n xx ε σ 1 1 + ∂∂ + nj n xk ε 11 ++ ∂∂ nnk ε Derivative of history variables Phase 4: Compute and store 2 nd -order unconditional derivative of the history variables ji n xx ∂∂ ∂ +12ε kn+1 Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 104 Figure 4.2: (a) Tower truss with uniaxial J2 material model and (b) time variation of the load at the top of the tower Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 105 -0.05 0 0.05 0.1 0.15 -50 0 50 u (m) (a) F o rc e ( k N ) -0.05 0 0.05 0.1 0.15 -2 0 2 x 10 -7 u (m) (b) ∂ u / ∂ E -0.05 0 0.05 0.1 0.15 -1 -0.5 0 x 10 -3 u (m) (c) ∂u / ∂σ y -0.05 0 0.05 0.1 0.15 -1 -0.5 0 x 10 -5 u (m) (d) ∂u / ∂H is o -0.05 0 0.05 0.1 0.15 -2 -1 0 x 10 -5 u (m) (e) ∂ u / ∂ H k in 1 st -order DDM 1 st -order FDM, perturbation size = 0.005 Figure 4.3: (a) Force-displacement response at the top of truss and (b to e): first-order response sensitivities of the displacement response Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 106 0 0.05 0.1 -2 0 2 x 10 -12 u (m) ∂2 u / ∂ E 2 0 0.05 0.1 -1 0 1 x 10 -17 u (m) ∂2 u / ∂ E ∂σ y 0 0.05 0.1 -5 0 5 x 10 -18 u (m) ∂2 u / ∂ E ∂ H is o 0 0.05 0.1 -5 0 5 x 10 -17 u (m) ∂2 u / ∂ E ∂ H k in 0 0.05 0.1 -2 0 2 x 10 -14 u (m) ∂2 u / ∂σ y2 0 0.05 0.1 0 0.5 1 x 10 -7 u (m) ∂2 u / ∂σ y ∂ H is o 0 0.05 0.1 0 1 2 x 10 -7 u (m) ∂2 u / ∂σ y ∂ H k in 0 0.05 0.1 0 0.5 1 x 10 -9 u (m) ∂2 u / ∂ H is o 2 0 0.05 0.1 0 1 2 x 10 -9 u (m) ∂2 u / ∂ H is o ∂ H k in 0 0.05 0.1 0 2 4 x 10 -9 u (m) ∂2 u / ∂ H k in 22 nd -order DDM 2 nd -order FDM, perturbation size = 0.005 Figure 4.4: Second-order displacement response sensitivity results Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 107 Figure 4.5: (a) Quarter model of a circular cylinder cross-section and (b) variation of the internal pressure, P Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 108 Figure 4.6: (a) Pressure-displacement response and (b to e) first-order response sensitivities of the displacement response -1 0 1 2 3 x 10 -4 -5 0 5 x 10 -10 u (m) (b) ∂ u / ∂ E 1 st -order DDM 1 st -order FDM , perturbation size = 0.005 -1 -0.5 0 0.5 1 1.5 2 x 10 -4 -50 0 50 u (m) (a) P re ss u re ( M P a ) -1 -0.5 0 0.5 1 1.5 2 x 10 -4 -5 0 5 x 10 -10 u (m) (b) ∂ u / ∂ E -1 -0.5 0 0.5 1 1.5 2 x 10 -4 -2 0 2 x 10 -4 u (m) (c) ∂ u / ∂ν -1 -0.5 0 0.5 1 1.5 2 x 10 -4 -4 -2 0 x 10 -6 u (m) (d) ∂ u / ∂σ y -1 -0.5 0 0.5 1 1.5 2 x 10 -4 -1 0 1 x 10 -8 u (m) (e) ∂ u / ∂ H is o -1 -0.5 0 0.5 1 1.5 2 x 10 -4 -2 -1 0 x 10 -8 u (m) (e) ∂ u / ∂ H k in Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 109 -2 0 2 x 10 -4 -5 0 5 x 10 -15 u (m) ∂2 u / ∂ E 2 -2 0 2 x 10 -4 -5 0 5 x 10 -10 u (m) ∂2 u / ∂ E ∂ν -2 0 2 x 10 -4 -2 0 2 x 10 -12 u (m) ∂2 u / ∂E ∂σ y -2 0 2 x 10 -4 -5 0 5 x 10 -15 u (m) ∂2 u / ∂ E ∂ H is o -2 0 2 x 10 -4 0 0.5 x 10 -14 u (m) ∂2 u / ∂ E ∂ H k in -2 0 2 x 10 -4 -5 0 5 x 10 -4 u (m) ∂2 u / ∂ν 2 -2 0 2 x 10 -4 -5 0 5 x 10 -6 u (m) ∂2 u / ∂ν ∂σ y -2 0 2 x 10 -4 -5 0 5 x 10 -9 u (m) ∂2 u / ∂ν ∂ H is o -2 0 2 x 10 -4 -1 0 1 x 10 -8 u (m) ∂2 u / ∂ν ∂ H k in -2 0 2 x 10 -4 -5 0 5 x 10 -8 u (m) ∂2 u / ∂σ y2 -2 0 2 x 10 -4 0 1 2 x 10 -10 u (m) ∂2 u / ∂σ y ∂ H is o -2 0 2 x 10 -4 0 5 x 10 -10 u (m) ∂2 u / ∂σ y ∂ H k in -2 0 2 x 10 -4 -1 0 1 x 10 -12 u (m) ∂2 u / ∂ H is o 2 -2 0 2 x 10 -4 0 1 2 x 10 -12 u (m) ∂2 u / ∂ h is o ∂ H k in -2 0 2 x 10 -4 0 1 2 x 10 -12 u (m) ∂2 u / ∂ H k in 2 2 nd -order DDM 2 nd -order FDM, perturbation size = 0.005 Figure 4.7: Second-order displacement response sensitivity results Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 110 4.12. REFERENCES Chen, C.J., and Choi, K.K. (1994). “Continuum approach for second-order shape design sensitivity of three-dimensional elastic solids.” AIAA Journal, 32(10), 2099-2107. Choi, K.K. and Santos, J.L.T. (1987). “Design sensitivity analysis of nonlinear structural systems, part 1: theory.” International Journal for Numerical Methods in Engineering, 24, 2039-2055. Conte, J.P, Vijalapura, P.K. and Meghella, M. (2003). “Consistent finite-element response sensitivity analysis.” ASCE Journal of Engineering Mechanics, 129(12), 1380-1393. Ditlevsen, O. and Madsen, H.O. (1996). Structural Reliability Methods. Wiley, Chichester, New York, NY. Haukaas, T. and Der Kiureghian, A. (2005). “Parameter sensitivity and importance measures in nonlinear finite element reliability analysis.” ASCE Journal of Engineering Mechanics, 131(10), 1013-1026. Hwang, H.Y., Choi, K.K., and Chang, K.H. (1997). “Second-order shape design sensitivity using P- version finite element analysis.” Structural and Multidisciplinary Optimization, 14(2-3), 91-99. Kita, E., Kataoka, Y., and Kamiya, N. (1997). “Application of element-free Trefftz method to second- order design sensitivity analysis of two-dimensional elastic problem.” JSME International Journal, series A, Solid Mechanics and Material Engineering, 40(4), 375-381. Kleiber, M., Antunez, H., Hien, T. and Kowalczyk, P. (1997). Parameter Sensitivity in Nonlinear Mechanics, John Wiley and Sons Ltd., West Sussex, U.K. Lee, B.W., and Lim, O.K. (1998). “Application of stochastic finite element method to optimal design of structures.” Computers and Structures, 68 (5), 491-497. Liu, P-L. and Der Kiureghian, A. (1991). “Optimization algorithms for structural reliability.” Structural Safety, 9(3), 161-178. Chapter 4: Second-order Sensitivities of Inelastic Finite Element Response by Direct Differentiation 111 Luenberger, D.G. (1984). Linear and Nonlinear Programming. 2 nd edition, Addison-Wesley, Reading, M.A. Ozaki, I., Kimura, F., and Berz, M. (1995). “Higher-order sensitivity analysis of finite element method by automatic differentiation.” Computational Mechanics, 16(4), 223-234. Park, S., Kapania, R.K., and Kim, S.J. (1999). “Nonlinear transient response and second-order sensitivity using time finite element method.” AIAA Journal, 37 (5), 613-622. Polak, E. (1997). Optimization: Algorithms and Consistent Approximations. Applied Mathematical Sciences, Vol. 124, Springer-Verlag, NY. Roth, C. and Grigoriu, M. (2001). Sensitivity Analysis of Dynamic Systems Subjected to Seismic Loads, Report No MCEER-01-0003, Multidisciplinary Center for earthquake Engineering Research, State University of New York, Buffalo, NY. Scott, M.H., Franchin, P., Fenves, G.L. and Filippou, F.C. (2003). “Response Sensitivity for Nonlinear Beam-Column Elements.” ASCE Journal of Structural Engineering, 130(9), 1281-1288. Simo, J.C., and Hughes, T.J.R. (1998). Computational Inelasticity. Interdisciplinary applied mathematics, Springer-Verlag, NY. Tsay, J.J. and Arora, J.S. (1990). Nonlinear structural design sensitivity analysis for path dependent problems, part1: general theory.” Computer Methods in Applied Mechanics and Engineering, 81, 183- 208. Zhang, Y. and Der Kiureghian, A. (1993). “Dynamic response sensitivity of inelastic structures.” Computer Methods in Applied Science and Engineering, 108, 23-36. Chapter 5: Conclusions and Future Work 112 Chapter 5. CONCLUSIONS AND FUTURE WORK 5.1. SUMMARY OF RESEARCH AND CONTRIBUTIONS The contributions in this thesis are viewed in the context of numerical simulation of the composite manufacturing process. Although the novel second-order response sensitivities developed in this thesis are applicable to a wider range of mechanical problems, it is the composite manufacturing problem that is in focus throughout this thesis. For this problem, a number of steps are taken: Analytical sensitivity equations are derived, efficient computer implementations are carried out, sensitivity results are demonstrated and verified, and the sensitivity-enabled software is applied for a number of practical problems. An itemized summary of the contributions are as follows: • The direct differentiation method is applied to both the thermochemical and the stress development models, and analytical differentiation is carried out with respect to all input parameters. • The link between the thermochemical sensitivity and stress development sensitivity is included in the response sensitivity derivations. This enables the study of the sensitivity of parameters that affect the temperature and degree of cure, and subsequently the residual stresses and final displacements. This link is neglected by Zhu and Geubelle (2002). • Novel shape sensitivity equations are included to produce response sensitivities with respect to geometry parameters. • A comprehensive Matlab®-based software is developed to demonstrate the implementation and use of the derived equations. In this software, several novel techniques are employed to minimize the computational cost associated with the response sensitivity computations. Furthermore, the Chapter 5: Conclusions and Future Work 113 response sensitivity results provided by the software are verified by the less efficient finite difference approach. • The developed software provides response sensitivities of the temperature, degree of cure, stresses, and displacements before and after tool removal with respect to material, geometric, and processing parameters (Bebamzadeh et al. 2009). • By employing the sensitivity-enabled software, a number of observations are made to gain physical insight into the problem and to guide further model refinement and resource allocation efforts. Other applications of the software include a new alert system that uses response sensitivity results to notify the user if the numerical results deviate from expected “rules” and scaling laws. • The sensitivity-enabled software is also used in reliability, optimization, and model calibration applications. All these applications are facilitated by the availability of efficient and accurate response sensitivities. • The derivation and implementation of second-order sensitivity equations is a particular novelty in this thesis. It is demonstrated that it is computationally feasible to obtain second-order sensitivities (the “Hessian matrix”) by the direct differentiation method for inelastic finite element problems (Bebamzadeh and Haukaas 2008). • In addition to software implementations, a step-by-step pseudo code is provided to facilitate the extension of any finite element code with second-order sensitivity capabilities. Several novel implementations techniques are developed to minimize the computational cost. • Two examples are considered to demonstrate and verify the computation of second-order response sensitivities: a truss structure and a 2D problem that utilizes four-node quadrilateral finite elements. The uniaxial J2 plasticity model is employed in conjunction with the truss elements, while the multiaxial J2 plasticity model is employed with the 2D elements. Chapter 5: Conclusions and Future Work 114 • It is demonstrated that the direct differentiation approach to the computation of second-order response sensitivities becomes increasingly efficient as the problem size increases, compared with the less accurate and less efficient finite different approach (Bebamzadeh and Haukaas 2008). 5.2. FUTURE RESEARCH DIRECTIONS Several issues that are outside the scope of this thesis work, but that require further research, are identified in the course of this study. These research directions are categorized in two groups: Development of knowledge and development of software. They are listed in the following. 5.2.1. Future Research Topics • There is limited knowledge of the actual variability in both the physical parameters and the fitting parameters of the composite manufacturing processes. Few of the parameters studied in this thesis are routinely measured and tracked in industry. Further research is encouraged to ascertain the variations of composites manufacturing parameters. Interestingly, the response sensitivity results presented in this thesis will provide guidance to identify which parameters should be prioritized to measure and track. • It is suggested that sensitivity results be utilized to improve prediction models (perhaps simplify those for which the end result is not sensitive) and to boost the development of simplified scaling laws that yield synthesized engineering insight. • Further studies are suggested to investigate the feasibility of gradient-based reliability and optimization algorithms. Specifically, these applications require the response sensitivities to be “smooth” in the parameter space. Research into the development of material models, etc. is suggested to ensure continuous gradients. • Research into the classification and characterization of sources of uncertainty in composites manufacturing is suggested. This would include the identification of epistemic (model) Chapter 5: Conclusions and Future Work 115 uncertainty, which would be useful to identify the models with most significant model uncertainty, which in turn would be useful for directed model improvement efforts. • The new second-order sensitivity results can be implemented in second-order reliability analysis (SORM) (Ditlevsen and Madsen 1996) and optimization applications that make use of the “Hessian matrix” (the matrix of second-order function derivatives) (Luenberger 1984 and Polak 1997). 5.2.2. Future Software Implementations Directions • In the software developed in this thesis, the response sensitivities with respect to all input variables are computed at once. Although this is done in an efficient manner, the user may not be interested in all these sensitivity results. Additional programming work is suggested to allow the user to manually select the parameters with respect to which the response sensitivities are calculated. • The composites manufacturing simulation studied in this thesis include the thermochemical and stress development models. Advanced models including resin flow and tool-part contact models are not included, and may be investigated as part of an extended study. This would entail analytical differentiation and implementation efforts for which the work carried out in this thesis would provide significant guidance. • The software created in this thesis, which is developed for first- and second-order sensitivity analysis with inelastic finite elements, include uniaxial and multiaxial J2 plasticity models in conjunction with 1-D truss and 2-D plain strain plane quadrilateral elements, respectively. Further implementations are proposed to include other elements and non-linear material models. The well-organized architecture of the software enables the user to add these new elements and material models in a straightforward manner. Chapter 5: Conclusions and Future Work 116 5.3. REFERENCES Bebamzadeh, A. and Haukaas, T. (2008). “Second-order sensitivities analysis of inelastic finite element response by direct differentiation.” ASCE Journal of Engineering Mechanics, 134(10), 867-880. Bebamzadeh, A., Haukaas, T., Vaziri, R., Poursartip, A., and Fernlund, G. (2009). “Response sensitivity and parameter importance studies in composite manufacturing.” Journal of Composites Materials. (To appear) Ditlevsen, O. and Madsen, H.O. (1996). Structural Reliability Methods. Wiley, Chichester, New York, NY. Luenberger, D.G. (1984). Linear and Nonlinear Programming. 2 nd edition, Addison-Wesley, Reading, M.A. Polak, E. (1997). Optimization: Algorithms and Consistent Approximations. Applied Mathematical Sciences, Vol. 124, Springer-Verlag, NY. Appendix A: Thermochemical Model Sensitivity Equations 117 Appendix A. THERMOCHEMICAL MODEL SENSITIVITY EQUATIONS A.1. DEGREE OF CURE SENSITIVITY Differentiation of Eq. (2.12) using the chain rule of differentiation yields θ α θ α αθθθ α α ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∆= ∂ ∂ −1 fixed fixed T k k k fT T ff t (A.1) Rearranging and solving for θα ∂∂ gives ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∆ ∂ ∂ ∆− = ∂ ∂ − θ α θθ α θ α α 1 fixed fixed T 1 1 k k k k T T ff t f t (A.2) It is observed that the sensitivity of the degree of cure, θα ∂∂ k , is a linear function of θ∂∂ kT : b T a kk + ∂ ∂ = ∂ ∂ θθ α (A.3) where k k f t T f t a ∂ ∂ ∆− ∂ ∂ ∆ = α 1 (A.4) and k k k f t f t b ∂ ∂ ∆− ∂ ∂ + ∂ ∂ ∆ = − α θ α θ α 1 1 fixed fixed T (A.5) Appendix A: Thermochemical Model Sensitivity Equations 118 Finally, the sensitivity of the rate of cure is calculated as ttt kk ∆ ∂ −∂ − ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ θ α θ α θ αα θ 1 (A.6) Substituting for θα ∂∂ k from Eq. (A.2) into Eq. (A.6), one obtains ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ∆− = ∂ ∂ ∂ ∂ − θ α αθθ α α θ α 1 fixed fixed T 1 1 k k k k fT T ff f t t (A.7) Similar to θα ∂∂ k , the sensitivity of the rate of cure, θα ∂∂ k& , is a linear function of θ∂∂ kT : b T a kk ′+ ∂ ∂ ′= ∂ ∂ θθ α& (A.8) where k k f t T f a ∂ ∂ ∆− ∂ ∂ =′ α 1 (A.9) and k k k k f t ff b ∂ ∂ ∆− ∂ ∂ ∂ ∂ + ∂ ∂ =′ − α θ α αθ α 1 1 fixed fixed T (A.10) A.2. HEAT CAPACITY MATRIX SENSITIVITY The first term in right hand side of Eq. (2.21), which is the derivative of the heat capacity matrix is calculated by differentiation of Eq. (2.3) with respect to θ. Applying the chain rule of differentiation Appendix A: Thermochemical Model Sensitivity Equations 119 knowing that the ρ and PC are the only variables which are the functions of temperature and the degree of cure results in, Ω ∂ ∂ ∂ ∂ +Ω ∂ ∂ ∂ ∂ + Ω ∂ ∂ = ∂ ∂ ∫∫ ∫ ΩΩ Ω d C d T T C dC ee e PTPT P Te ][ )( ][][ )( ][ ][][ ][ TTTT fixed fixed T TT P NNNN NN C θ α α ρ θ ρ ρ θθ α K (A.11) It was stated in Eq. (A.3) that the derivative of the degree of cure is a linear function of the derivative of the temperature, hence we substitute Eq. (A.3) into Eq. (A.11) to obtain Ω + ∂ ∂ ∂ ∂ +Ω ∂ ∂ ∂ ∂ + Ω ∂ ∂ = ∂ ∂ ∫∫ ∫ ΩΩ Ω db T a C d T T C dC ee e PTPT P Te ][ )( ][][ )( ][ ][][ ][ TTTT fixed fixed T TT P NNNN NN C θα ρ θ ρ ρ θθ α K (A.12) Multiplying by the temperature vector }{T and replacing θ∂∂T by θ∂∂ }{][ T TN , and entering the vector }{T into the integration for the terms where appear, we find θα ρρ α ρ ρ θθ α ∂ ∂ Ω ∂ ∂ +Ω ∂ ∂ + Ω ∂ ∂ + Ω ∂ ∂ = ∂ ∂ ∫∫ ∫∫ ΩΩ ΩΩ }{ ][}{][ )( ][][}{][ )( ][ }{][ )( ][][][}{ ][ TTTTTT TT fixed fixed T TT P T NTNNNTNN TNNNNT C da C d T C db C dC ee ee PTPT PT P T e e L (A.13) Eq. (A.13) shows that the differentiation of the heat capacity matrix times the temperature vector {T} is a linear function of the response sensitivity vector θ∂∂ }{T . All the integrals in the Eq. (A.13) are evaluated by using isoparametric formulation as ( ) ( ) ξxJξxx , 1 ∑∫ =Ω ≈ npoint m mm fdf el ω (A.14) Appendix A: Thermochemical Model Sensitivity Equations 120 where mξ are the quadrature points, mω are the corresponding weights, npoint is the number of integration points, ( ).f is the integrand, and ξxJ , is the determinant of the Jacobian matrix of the coordinate transformation between x and ζ . Since θ in the first term of Eq. (A.13) may represent geometry parameters, the derivative of the determinant of the Jacobian matrix is required. According to Haukaas and Der Kiureghian (2005), this derivative is computed as ( ) ξ N JJ J ξxξx ξx ∂ ∂ = ∂ ∂ − θ θ T ,, , (A.15) where θN represent the shape functions in N corresponding to the degree of freedom matching the nodal coordinate represented by θ . Using Eq. (A.14) and (A.15) the first term of Eq. (A.13) is written as ( ) ∂ ∂ + ∂ ∂ ≈ Ω ∂ ∂ − =Ω ∑∫ ξ N JJNN JNNNN ξxξx ξx θ α α ρ θ ρ ωρ θ T P T PT m npoint m P T C C dC e ,,TT ,T fixed fixed T T 1 fixed fixed T TT ][][ ][ )( ][][][ K (A.16) where the first term indicates the material sensitivity and the second term stands for the “shape sensitivities.”. A.3. CONDUCTIVITY MATRIX SENSITIVITY By differentiating Eq. (2.4), one obtains the second term in Eq. (2.21), which is the derivative of the conductivity matrix. Similar to the differentiation of the heat capacity matrix in previous section, the differentiation of the conductivity matrix multiplied by the temperature vector is written as Appendix A: Thermochemical Model Sensitivity Equations 121 θα αθθ α κ ∂ ∂ Ω ∂ ∂ +Ω ∂ ∂ + Ω ∂ ∂ + Ω ∂ ∂ = ∂ ∂ ∫∫ ∫∫ ΩΩ ΩΩ }{ ][}{][ ][ ][][}{][ ][ ][ }{][ ][ ][][][][}{ ][ TTTTTT TT fixed fixed T TT T NTB κ BNTB κ B TB κ BBκBT k dad T dbd ee ee TT TT e e K (A.17) which is also is a linear function of the response sensitivity vector θ∂∂ }{T . In conclusion to include the geometry variations, the first term in the parenthesis of the right hand side is defined by exploiting Eqs. (A.14), and (A.15) as ( ) ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = Ω ∂ ∂ − =Ω ∑∫ ξ N JJBκB J B κBJBκ B JB κ BBκB ξxξx ξxξx ξx θ α α θθ θ ω θ TT T T T m npoint m T e d ,,TT , T T,T T ,T fixed fixed T T 1 fixed fixed T TT ]][[][ ][ ][][][][ ][ ][ ][ ][][][][ K K (A.18) The second and third terms in Eq. (A.18) require θ∂∂ ][ TB . Since ][ TB is dependent on the inverse of the Jacobian matrix, the derivative of the inverse of the Jacobian matrix is needed. This term is obtained as 1 , ,1 , 1 , −− − ∂ ∂ −= ∂ ∂ ξx ξx ξx ξx J J J J θθ (A.19) where ξ NJ ξx ∂ ∂ = ∂ ∂ θ θ , (A.20) For further details, see Haukaas and Der Kiureghian (2005). In addition, ][κ in Eq. (A.18) is the thermal conductivity matrix in the global axes which for composite plies is defined as ]][][[][][][ * βφφβ TTκTTκ TT= (A.21) Appendix A: Thermochemical Model Sensitivity Equations 122 where ][κ is the element conductivity matrix in the ply principal axes. ][ φT and ][ βT are the transformation matrices from the ply principal axes (1-2-3) to the local element axes (x’-y’-z’) and from the local element axes (x’-y’-z’) to the global axes (x-y-z) respectively. Therefore the derivative of the conductivity matrix is obtained by ][ ][ ][][][]][][[ ][ ][ ][ ]][[][][]][][[][ ][ ]][[ ][ ][][ ][ ** ** fixed fixed T * fixed fixed T β φ φββφ φ β β φφββφφ β βφ α φβ α θθ θθ θθ T T κTTTTκ T T T TκTTTTκT T TT κ TT κ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ TT T TTT T TT L K (A.22) A.4. BOUNDARY CONVECTION MATRIX SENSITIVITY The derivative of the boundary convective matrix in Eq. (2.21), is calculated by differentiation of Eq. (2.5) with respect to θ, as Γ ∂ ∂ = ∂ ∂ ∫ Γe dhTe ][][ ][ TT NN h θθ (A.23) Using Eqs. (A.14) and (A.15) yields ( ) Γ − = ∂ ∂ + ∂ ∂ ≈ ∂ ∂ ∑ ξ N JJNNJNN h ξxξxξx θ θ ω θ TTT m npoint m e h h ,,TT,TT 1 ][][][][ ][ (A.24) The subscript Γ in above equation denotes that the shape functions and corresponding Jacobian matrix should be computed on the element traction boundary. A.5. LOAD VECTORS SENSITIVITIES The derivatives of the load vectors in Eq. (2.22) are obtained by differentiation of Eqs. (2.7), (2.8), and (2.9) with respect to θ. Consequently, sensitivities of the heat flux vector and the boundary convection vector utilizing Eqs. (A.14) and (A.15), are expressed as Appendix A: Thermochemical Model Sensitivity Equations 123 ( ) Γ − = ∂ ∂ + ∂ ∂ ≈ ∂ ∂ ∑ ξ N JJNJN f ξxξxξx θ θ ω θ T b TbT m npoint m e q q ,,T,T 1 Tq ][][ }{ (A.25) ( ) Γ − ∞ ∞ = ∂ ∂ + ∂ ∂ ≈ ∂ ∂ ∑ ξ N JJNJN f ξxξxξx θ θ ω θ TTT m npoint m e hT hT ,,T,T 1 Th ][ )( ][ }{ (A.26) where, θ∂∂h , θ∂∂ ∞T , and θ∂∂ bq are zero unless h, ∞T , or bq are dependent upon θ. Again the subscript Γ in above equations denotes summation is performed on the element traction boundary. Furthermore, the derivatives of the heat generation vector is written as ( ) ∂ ∂ + ∂ ∂ ≈ ∂ ∂ − = ∑ ξ N JJNJN f ξxξxξx θ θ ω θ TTT m npoint m e Q Q ,,T,T 1 TQ ][][ }{ & & (A.27) where the derivative of Q& is given by differentiation of Eq. (2.10) as ( ) ( ) ( ) θ ρ α θ ρα ρ α θθ ∂ ∂ −+ ∂ ∂ −+− ∂ ∂ = ∂ ∂ R rfR r fRrf H V dt d HV dt d HV dt dQ 111 & (A.28) Earlier it was established that θα ∂∂ & and θρ ∂∂ R are linear functions of θ∂∂T . Furthermore, Vf and HR are not functions of either the temperature or the degree of cure. Thus, the sensitivity of Q& is a linear function of θ∂∂T and Eq. (A.28) can be written in compact form as b T a Q ′′+ ∂ ∂ ′′= ∂ ∂ θθ & (A.29) where the expressions for a ′′ and b ′′ are trivially obtained by collecting terms in Eq. (A.28). By substituting Eq. (A.29) into Eq. (A.27) and replacing θ∂∂T by θ∂∂ }{][ T TN , we re-write Eq. (A.27) as Appendix A: Thermochemical Model Sensitivity Equations 124 ( ) θ ω ω θ θ ∂ ∂ ′′+ +′′ ∂ ∂ ≈ ∂ ∂ ∑ ∑ = − = }{ ][][ ][][ }{ ,TT npoint 1 ,,T,T 1 TQ T JNN JJNJN ξ Nf ξx ξxξxξx a Qb T m m TTT m npoint m e K& (A.30) Appendix A: Thermochemical Model Sensitivity Equations 125 A.6. REFERENCES Haukaas, T. and Der Kiureghian, A. (2005). “Parameter sensitivity and importance measures in nonlinear finite element reliability analysis.” ASCE Journal of Engineering Mechanics, 131(10), 1013-1026. Appendix B: Stress Development Model Sensitivity Equations 126 Appendix B. STRESS DEVELOPMENT MODEL SENSITIVITY EQUATIONS B.1. 3D PLY STIFFNESS MATRIX SENSITIVITY In this section, the formulations to calculate the sensitivity of 3D ply stiffness matrix used by Eqs. (2.27) and (2.28) are derived. The ply stiffness matrix, ][ *C in the ply principle axes (1-2-3) is defined by inversing the ply compliance matrix, ][ *S , which is given by (Johnston et al. 2001) − −− =∗ c c c c c c c c c c c c G G Sym E G EE EEE 23 13 33 12 22 23 22 11 13 11 12 11 1 0 1 00 1 000 1 000 1 000 1 ][ ν νν S (B.1) where the compliance matrix components are computed from ply properties using micromechanics models (Bogetti and Gillespie 1992). Then, the ply stiffness matrix in the global axes (x-y-z) is expressed as ]][[][][][][ 1* βφφβ TTSTTC −= TT (B.2) where ][ φT and ][ βT are the transformation matrices from the ply principal axes (1-2-3) to the local element axes (x’-y’-z’) and from the local element axes (x’-y’-z’) to the global axes (x-y-z) respectively. The derivative of ][C is calculated by differentiating the expression in Eq. (B.2) with respect to θ, such that Appendix B: Stress Development Model Sensitivity Equations 127 ][ ][ ][][][]][[][ ][ ][ ][ ][][][][]][[][][ ][ ]][[ ][ ][][ ][ 1*1* 1*1* 1* β φ φββφ φ β β φφββφφ β βφφβ θθ θθ θθ T T STTTTS T T T TSTTTTST T TT S TT C ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ −− −− − TT T T TTT T TT L L (B.3) This task involves the differentiation of the inverse compliance matrix; namely, θ∂∂ −1* ][S . To this end, the property ISS =−1** ][][ , where I is the identity matrix, is invoked to obtain 1* * 1* 1* ][ ][ ][ ][ −− − ∂ ∂ −= ∂ ∂ S S S S θθ (B.4) where the derivative of the compliance matrix, θ∂∂ ][ *S , is calculated by differentiating Eq. ( B-1) using the chain rule as ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ θθθ θ ν νθ ν νθ ν ν θθθθ c c c c c c c c c c c c c c c c c c G G G G G G E E E E E E 23 23 * 13 13 * 12 12 * 23 23 * 13 13 * 12 12 * 33 33 * 22 22 * 11 11 ** ][][][ ][][][ ][][][][ SSS SSS SSSS L L (B.5) where the terms outside the parentheses are the derivatives of the compliance matrix with respect to its components. The terms inside the parentheses in Eq. (B.5) are the derivatives of the ply mechanical properties given by Bogetti and Gillespie (1992) with respect to θ. For example, the derivative of E11c is calculated using the chain rule of differentiation to obtain ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ = ∂ ∂ θθθ ν ν θ ν νθθθ ν νθθ f f cf f cf f c f f cf f cf f cr r cr r cc V V EG G EE EE E EE E EEE E EE 1113 13 1123 23 11 13 13 1122 22 1111 11 11111111 (B.6) Appendix B: Stress Development Model Sensitivity Equations 128 where the ply micromechanical models are differentiated to obtain the terms outside the parentheses. The terms inside the parentheses are calculated by differentiating the resin and fibre mechanical properties models. B.2. 3D PLY INITIAL STRAINS SENSITIVITY The change in the ply initial strain in the ply principle axes (1-2-3) is expressed as (Johnston et al. 2001) { }000}{ 321*0 ccc εεε ∆∆∆=∆ε (B.7) where the ply strain components, c1ε∆ , c2ε∆ , and c3ε∆ are calculated from the fibre and resin thermal strains and resin shrinkage strains using micromechanical models (Bogetti and Gillespie 1992). Then, the change in ply initial strain in the global axes using transformation matrices is written by }{][][}{ *0 11 0 εTTε ∆=∆ −− φβ (B.8) Therefore the derivative of the ply strains in the global axes (x-y-z) is calculated by differentiating the above equation as { } { }*0111 * 0 111 * 0110 ][ ][ ][][ ][][ ][ ][ }{ ][][ }{ εT T TT εTT T T ε TT ε ∆ ∂ ∂ − ∆ ∂ ∂ − ∂ ∆∂ = ∂ ∆∂ −−− −−−−− φ φ φβ φβ β βφβ θ θθθ L (B.9) where θ∂∆∂ }{ *0ε is the derivative of the change of the initial ply strain vector in the principal directions (1-2-3) obtained by the differentiation of Eq. (B.7) as ∂ ∆∂ ∂ ∆∂ ∂ ∆∂ = ∂ ∆∂ 000 321 * 0 θ ε θ ε θ ε θ cccε (B.10) in which the local strain sensitivity components θε ∂∆∂ c1 , θε ∂∆∂ c2 , and θε ∂∆∂ c3 are calculated by differentiating the ply micromechanical strains models . For example, by employing the chain rule of differentiation, the derivative of c1ε∆ is calculated by Appendix B: Stress Development Model Sensitivity Equations 129 ∂ ∂ ∂ ∆∂ + ∂ ∆∂ ∆∂ ∆∂ + ∂ ∂ ∂ ∆∂ + ∂ ∂ ∂ ∆∂ + ∂ ∆∂ ∆∂ ∆∂ = ∂ ∆∂ θ ε θ ε ε ε θ ε θ ε θ ε ε ε θ ε r r cr r c f f cf f cf f cc E E V V E E 11 111 11 11 1 11 K (B.11) The task in above equation includes the differentiation of the change in the resin strain. The change in resin strain arises from the change in resin thermal and shrinkage strains. Therefore, the sensitivity of the change in resin strain is written as θ ε θ ε θ ε ∂ ∆∂ − ∂ ∆∂ = ∂ ∆∂ sr T rr (B.12) where θε ∂∆∂ Tr and θε ∂∆∂ s r are the derivatives of the resin thermal expansion strain and resin cure shrinkage strain models, respectively. For example, the resin thermal strain sensitivity is expressed as θθθ ε ∂ ∆∂ +∆ ∂ ∂ = ∂ ∆∂ T CTET CTE r r T r (B.13) where rCTE is the resin coefficient of the thermal expansion which is a linear function of resin temperature and degree of cure. θ∂∂ rCTE is computed by the differentiation of the resin thermal expansion coefficient as ( ) ( ) ∂ ∂ − ∂ ∂ +− ∂ ∂ + ∂ ∂ − ∂ ∂ +− ∂ ∂ + ∂ ∂ = ∂ ∂ θ α θ α αα θθθθθθ 0 0 0 0 )0( CTEr CTEr CTEr CTErrr b BTT aTT aCTECTE (B.14) B.3. PLANE STRAIN PLY STIFFNESS MATRIX SENSITIVITY The in-plane compliance matrix in the local element axes (x’-z’) is calculated by constraining all out-of- plane strains as (Johnston et al. 2001) ][]][[][][ 2 1 143 SSSSS ′′′−′= − (B.15) where Appendix B: Stress Development Model Sensitivity Equations 130 [ ] ′ ′′ ′′ =′ 66 3323 2322 1 00 0 0 S SS SS S ; [ ] ′ ′′ ′′ =′ 56 3413 2412 2 00 0 0 S SS SS S [ ] ′ ′′ ′′ =′ 55 4414 1411 3 00 0 0 S SS SS S ; [ ] ′ ′′ ′′ =′ 56 3424 1312 4 00 0 0 S SS SS S (B.16) ijS ′ in above equation are the terms of the transformed 3D compliance matrix in the element local axes, which is obtained by ( ) 1*1 ][][][][ −−=′ Tφφ TSTS (B.17) where ][ *S is the 3D compliance matrix in the ply principle axes (1-2-3) given by Eq. (B.1) and ][ φT is the 3D transformation matrix from the ply principle axes (1-2-3) to local element axes (x’-y’-z’). The in- plane stiffness matrix is calculated by inversing the in-plane compliance matrix given by Eq. (B.15) as 1][][ −= SC (B.18) Therefore, the in-plane ply stiffness matrix in the global axes (x-z) is written as ]][[][][ ββ TCTC T= (B.19) where ][ βT is the 2D transformation matrix from the local element axes (x’-z’) to the global axes (x-z). The derivative of ][C is obtained by differentiating Eq. (B.19) as θθθθ β ββ β ββ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ][ ][][]][[ ][ ][ ][ ][ ][ T CTTC T T C T C T T T (B.20) Computing the above equation involves the differentiation of ][C . Using Eq. (B.18) and the property that ISS =−1]][[ yields Appendix B: Stress Development Model Sensitivity Equations 131 11 ][ ][ ][ ][ −− ∂ ∂ −= ∂ ∂ S S S C θθ (B.21) where the derivative of the in-plane compliance matrix, θ∂∂ ][S is computed by differentiation of Eq. (B.15) as θθθθθ ∂ ′∂ ′′−′′ ∂ ′∂ ′′+′′ ∂ ′∂ − ∂ ′∂ = ∂ ∂ −−−− ][]][[][][ ][ ]][[][][ ][][][ 21 142 1 1 11 142 1 1 43 SSSSS S SSSS SSS (B.22) where θ∂′∂ ][ 1S , θ∂′∂ ][ 2S , θ∂′∂ ][ 3S , and θ∂′∂ ][ 4S are calculated by differentiating the matrices in Eq. (B.16). For example θ∂′∂ ][ 1S is written as θθθθθ ∂ ′∂ ′∂ ′∂ + ∂ ′∂ ′∂ ′∂ + ∂ ′∂ ′∂ ′∂ + ∂ ′∂ ′∂ ′∂ = ∂ ′∂ 66 66 133 33 123 23 122 22 11 ][][][][][ S S S S S S S S SSSSS (B.23) where θ∂′∂ ijS are the components of the derivative of ][S′ , which is obtained by the differentiation of Eq. (B.17) as ( ) ( ) ( ) ( ) 11*11*111 * 1 ][ ][ ][][][][][][ ][ ][][ ][ ][ ][ −−−−−−−− ∂ ∂ − ∂ ∂ − ∂ ∂ = ∂ ′∂ T T TTT φ φ φφφφ φ φφφ θθθθ T T TSTTST T TT S T S (B.24) where the sensitivity of 3D ply compliance matrix, θ∂∂ ][ *S is calculated as explained in App. B.1 by Eq. (B.5). B.4. PLANE STRAIN INITIAL STRAIN SENSITIVITY Plane strain initial strain vector in the element axes (x’-z’) is calculated by forcing all out-of-plane strains, },,{}{ 000)(0 zyyxy T op ′′′′′ ′′′=′ γγεε to zero. The change in in-plane strains, },,{}{ )( zx0z0x0 T ip0 ′′′′ ′′′=′ γεεε induced by these forces is written as (Johnston et al. 2001) }]{[][}{ )(02 1 1)(0 opip εCCε ′′′=′∆ − (B.25) Appendix B: Stress Development Model Sensitivity Equations 132 where ′ ′′ ′′ =′ 55 4414 1411 1 00 0 0 ][ C CC CC C ; ′ ′′ ′′ =′ 66 3424 1312 2 C00 0CC 0CC ][C (B.26) and ijC′ are the components of the transformed 3D stiffness matrix defined by ][][][][][ 1*1 φφ TSTSC −− =′=′ T (B.27) Therefore, the total in-plane strain is the summation of the initial in-plane strains and the changes in these strains induced by the out of plane constraints. It is written as }{}{}{ )(0)(00 ipip εεε ′∆+′= (B.28) Finally the strain vector in global axis is obtained by }{][}{ 0 1 0 εTε −= β (B.29) The task to calculate the derivative of }{ 0ε involves the differentiation of }{ )(0 ipε′∆ . Differentiating Eq. (B.25) yields { } { } { } { })(0211)(0211111)(0211)(0 ][][][][][][][][ ipipipip εCCεCCCC ε CC ε ′ ∂ ′∂ ′+′′′ ∂ ′∂ ′− ∂ ′∂ ′′= ∂ ′∆∂ −−−− θθθθ (B.30) where θ∂′∂ ][ 1C and θ∂′∂ ][ 2C are computed by the differentiation of the matrices in Eq. (B.26). For instance, the derivative of ][ 1C′ is written as θθθθθ ∂ ′∂ ′∂ ′∂ + ∂ ′∂ ′∂ ′∂ + ∂ ′∂ ′∂ ′∂ + ∂ ′∂ ′∂ ′∂ = ∂ ′∂ 55 55 144 44 114 14 111 11 11 ][][][][][ C C C C C C C C CCCCC (B.31) where θ∂′∂ ][ ijC are the components of the derivative of ][C′ , which is obtained by differentiating Eq. (B.27) as θθθθ φ φφφφ φ ∂ ∂ + ∂ ∂ − ∂ ∂ = ∂ ′∂ −−−− ][][][][][ ][ ][][][][ ][][ 1*1* * 1*1* T STTS S STTS TC TT T (B.32) Appendix B: Stress Development Model Sensitivity Equations 133 where the sensitivity of the ply stiffness matrix in the principle axes, θ∂∂ ][ *S is calculated as discussed in App. B.1 by Eq. (B.5). Then, the total in-plane strain sensitivity is obtained by the summation of initial in-plane strain sensitivity and changes in in-plane strain sensitivities due to constraining out-of-plane strains as θθθ ∂ ′∆∂ + ∂ ′∂ = ∂ ∂ }{}{}{ )(0)(00 ipip εεε (B.33) At last, the derivative of in-plane strain vector in the global axes (x-z) is computed by the differentiation of Eq. (B.29) as }{][ ][ ][ }{ ][ }{ 0 11010 εT T T ε T ε −−− ∂ ∂ − ∂ ∂ = ∂ ∂ β β ββ θθθ (B.34) where ][ βT is the 2D transformation matrix from the local element axes (x’-z’) to the global axes (x-z). Appendix B: Stress Development Model Sensitivity Equations 134 B.5. REFERENCES Bogetti, T.A. and Gillespie, Jr,.J.W. (1992). “Processed-induced stress and deformation in thick-section thermoset composite laminate.” Journal of Composite Materials, 26(5), 626-660. Johnston, A.A., Vaziri, R. and Poursartip, A. (2001). “A plane strain model for process-induced deformation of laminated composite structures.” Journal of Composite Materials, 35(16), 1435-1469. Appendix C: Overall Direct Differentiation Software Flow 135 Appendix C. OVERALL DIRECT DIFFERENTIATION SOFTWARE FLOW This appendix outlines the overall structure of the in-house software for the response sensitivity study of the composite processing. C.1. TOP-LEVEL ALGORITHM The overall flow of the software is as follow: 1. Read input file and set initial values 2. Loop over all time steps a. Calculate k}{T and k}{α for time step kt using thermochemical module (App. C-2) b. Calculate k}{δ and k}{σ for time step kt using stress development module (App. C-3) c. Calculate θ∂∂ k}{T and θ∂∂ k}{α for time step kt using thermochemical sensitivity module (App. C-4) d. Calculate θ∂∂ k}{δ and θ∂∂ k}{σ for time step kt using stress development sensitivity module (App. C-5) 3. End C.2. THERMOCHEMICAL MODULE ALGORITHM 1. Begin iteration loop (iteration number i) a. Loop over all model elements i. Calculate i kα& and update i kα using Eq. (2.12) Appendix C: Overall Direct Differentiation Software Flow 136 ii. Update element thermal properties such as ρ , pC and ][κ using material properties models iii. Calculate ee ][ and ,][ P κkC using Eqs. (2.3), and (2.4) at the element level and Assemble them into global matrices i k][ TK , Eq. (2.2) iv. Calculate e}{ TQf using Eq. (2.9) at the element level and assemble it into global load vector i k}{ TF , Eq. (2.6) b. End c. Add contribution from i k][H , Eq. (2.5) for boundary elements to the global i k][ TK , Eq. (2.2) d. Add contribution from external load vectors i k i k }{ and }{ TQTh FF by Eqs. (2.8) and (2.9) for boundary elements to the total thermal load vector i k}{ TF , Eq. (2.6) e. Solve for ( ) ikikik }{][}{ T1T FKT −= using Eq. (2.1) f. Check convergence by max 1 }{}{}{ TTT ∆≤− −ik i k and max 1 }{}{}{ ααα ∆≤− −ik i k 2. If not CONVERGED GOTO 1 C.3. STRESS DEVELOPMENT MODULE ALGORITHM 1. Loop over all model elements a. loop over all plies in the element i. Calculate the resin and mechanical properties such as rE , rν , fE11 , and fG13 using material models Appendix C: Overall Direct Differentiation Software Flow 137 ii. Calculate the ply mechanical properties using micromechanical models by iii. Calculate the ply stiffness matrix in the global axes ][C iv. Calculate the fibre and resin thermal and shrinkage strains v. Calculate the change in ply initial strain }{ 0ε∆ vi. Calculate the ply stiffness matrix el][k and load vector el}{ f∆ vii. Add contribution of el][k and el}{ f∆ into element stiffness matrix and load vector, e][k and e}{ f∆ b. End c. Assemble the element stiffness matrix e][k , and load vector e}{ f∆ into global stiffness matrix k][K , and global load vector k}{ F∆ using Eqs. (2.15) and (2.18) 2. End 3. Solve for k}{ δ∆ , Eq. (2.14) 4. Update displacement, k}{δ , Eq. (2.13) 5. Loop over all model elements a. loop over all plies in the element i. Calculate change in stress, k}{ σ∆ , Eq. (2.18) ii. Update stress, k}{σ , Eq. (2.19) b. End 6. End Appendix C: Overall Direct Differentiation Software Flow 138 C.4. THERMOCHEMICAL SENSITIVITY MODULE ALGORITHM 1. Loop over all model elements a. Calculate kk }{][ and}{][ P TkTC θθ κ ∂∂∂∂ , Eqs. (A.13), and (A.17) and Assemble them into global sensitivity matrices kk }{][ T TK θ∂∂ , using. Eq. (2.21) b. Calculate θθθ ∂∂∂∂∂∂ −− 1P1PTQ }{][ and ,}{][ ,}{ kk TCTCf , Eqs. (A.13), and (A.30) and Assemble them into global sensitivity load vector θ∂∂ k}{ TF using Eq. (2.22) 2. End 3. Add contribution from k}{][ TH θ∂∂ using Eq. (A.24) for boundary elements to the global matrix kk }{][ T TK θ∂∂ , Eq. (2.21) 4. Add contribution from external load vectors θθ ∂∂∂∂ kk }{ and }{ ThTq FF , using Eqs. (A.25) and (A.26) for boundary elements to the total thermal load vector θ∂∂ k}{ TF , Eq. (2.22) 5. Solve for θ∂∂ k}{T using Eq. (2.23) 6. Loop over all element a. Update θα ∂∂ k at each Gauss point using θ∂∂ kT and Eq. (A.3) 7. End C.5. STRESS DEVELOPMENT SENSITIVITY MODULE ALGORITHM 1. Loop over all model elements a. Calculate the sensitivity of the element stiffness matrix θ∂∂ e][k using Eq. (2.27) at the element level Appendix C: Overall Direct Differentiation Software Flow 139 b. Calculate the sensitivity of the element load vector θ∂∂ e}{f Eq. (2.28) c. Assemble the sensitivity of the element stiffness matrix θ∂∂ e][k and the sensitivity of the element load vector θ∂∂ e}{f into the global stiffness matrix ][K , the global sensitivity of the stiffness matrix θ∂∂ ][K , and the global sensitivity of load vector θ∂∂ }{F 2. End 3. Solve for θ∂∆∂ k}{ δ using Eq. (2.26). 4. Update displacement sensitivity θ∂∂ k}{δ using Eq. (2.29). 5. Loop over all model elements a. Calculate the sensitivity of change in stress θ∂∆∂ k}{ σ , Eq. (2.30) b. Update stress sensitivity θ∂∂ k}{σ , Eq. (2.31) 6. End Appendix D: Thermochemical and Stress Development Model Parameters 140 Appendix D. THERMOCHEMICAL AND STRESS DEVELOPMENT MODEL PARAMETERS Table D.1: Thermochemical model parameters Description Parameter Definition Parameter Value 01 T=θ C20 o Initial Values 02 αθ = 2105 −× )0(3 fρθ = 33 kg/m 10790.1 × faρθ =4 Ckg/m 0 3 o )0(5 rρθ = 33 kg/m 10300.1 × raρθ =6 Ckg/m 0 3 o Fibre Density: )( 0)0( TTa fff −+= ρρρ Resin Density: )()( 00)0( ααρρ ρρ −+−+= rrrr bTTa rbρθ =7 3kg/m 0 )0(8 fP C=θ KJ/kg 10913.7 2× fCp a=9θ 2J/kgK 050.2 )0(10 rP C=θ J/kgK 10005.1 3× rCp a=11θ 2J/kgK 740.3 Fibre Specific Heat Capacity: )( 0)0( TTaCC fff CpPP −+= Resin Specific Heat Capacity: )()( 00)0( αα −+−+= rrrr CpCpPP bTTaCC rCp b=12θ J/kgK 0 )0(13 lfκθ = W/mK002.8 lfaκθ =14 22 W/mK10560.1 −× )0(15 tfκθ = W/mK501.2 tfaκθ =16 23 W/mK10070.5 −× )0(17 rκθ = W/mK158.0 raκθ =18 24 W/mK10430.3 −× Fibre Thermal Conductivity: )(:alLongitudin 0)0( TTa lflflf −+= κκκ )(:Transverse 0)0( TTa tftftf −+= κκκ Resin Thermal Conductivity: )()( 00)0( αακκ κκ −+−+= rrrr bTTa rbκθ =19 W/mK10070.6 2−× A=20θ /s10528.1 5× E∆=21θ molJ/g 10650.6 4× m=22θ 813.0 n=23θ 736.2 C=24θ 090.43 025 Cαθ = 684.1− Resin cure kinetics: ( ){ }TC nm CTCe K dt d ααα ααα +−+ − = 01 )1( where RTEAeK /∆−= TCαθ =26 C /10475.5 3 o−× RH=27θ J/kg 10400.5 5× dt d HVQ Rrf α ρ)1( :GenerationHeat −=& fV=28θ 573.0 )0(29 Tρθ = 33 kg/m 10707.2 × Tool Density: )( 0)0( TTa TTT −+= ρρρ Taρθ =30 Ckg/m 0 3 o )0(31 TP C=θ J/kgK 10960.8 2× Tool Specific Heat Capacity: )( 0)0( TTaCC TTT CpPP −+= TCp a=32θ 2J/kgK 0 )0(33 Tκθ = W/mK10670.1 2× Tool conductivity: )( 0)0( TTa TTT −+= κκκ Taκθ =34 2 W/mK0 )( :TransferHeat Convective TThq −= ∞ h=35θ KW/m20 2 )1( :re temperatuAutoclave TautoclaveTT ∆+=∞ T∆=36θ 0 Appendix D: Thermochemical and Stress Development Model Parameters 141 Table D.2: L-shaped geometric parameters (see Fig. 2.2 for the definition of these parameters) TL=37θ m 10700.1 2−× Tth=38θ m 105.4 3−× PL=39θ m 105 2−× R=40θ m 105 3−× shth=41θ m 101 4−× Geometric Parameters Pth=42θ 8 o3 ][0for m 106.1 −× 32 o3 ][0for m 104.6 −× 128 o3 ][0for m 106.25 −× Appendix D: Thermochemical and Stress Development Model Parameters 142 Table D.3: Stress development model parameters Description Parameter Definition Parameter Value 0 43 rE=θ Pa1067.4 4× ∞= rE44θ GPa 67.4 gaT=45θ K 268 gbT=46θ K 220 * 147 aCT=θ K 161.44 * 148 bCT=θ K 10874.19 2−×− * 249 CT=θ K 12− Resin Modulus: ( ) ( )[ ] ( ) TTTT TTTT TTaEE TTEE TTTEE TT TT EE TTEE bCaCC gbga Errr Crr CCrr CC C rr Crr ×+= −×+= −+′= >=′ <<− − − +=′ <=′ ∞ ∞ * 1 * 1 * 1 * 0 * 2 * * 2 ** 1 0 * 1 * 2 * 1 * 0 * 1 *0 :where 1 α Era=50θ CPa/0 o )0(51 rνθ = 37.0 raνθ =52 0 Resin Poisson’s Ratio: )()( 00)0( αανν νν −+−+= rrrr bTTa rbνθ =53 0 ∞= SrV54θ 099.0 155 Cαθ = 055.0 256 Cαθ = 670.0 Resin Cure Shrinkage: ( ) 12 1 2 21 2 1 where 0 CC C SC S r S r CCS S rS S r C S r VV AVAV V αα αα ααα ααααα αα − − =>= ≤≤−+= <= ∞ ∞ A=57θ 173.0 )0(58 rCTE=θ Cµ/ 144 o rCTEa=59θ Cµ/ 0 o Resin Coefficient of Thermal Expansion: )()( 00)0( αα −+−+= rCTErCTErr bTTaCTECTE rCTEb=60θ Cµ/ 72 o− )0(1161 fE=θ GPa 102 fEa 1162 =θ CPa/ 0 o )0(2263 fE=θ GPa 7.241 fEa 2264 =θ CPa/ 0 o )0(1265 fG=θ GPa 7.62 fGa 1266 =θ CPa/ 0 o )0(1267 fνθ = 2.0 fa 1268 νθ = 0 )0(2369 fνθ = 25.0 Fibre Elastic Properties: )( 011)0(1111 TTaEE fEff −+= )( 022)0(2222 TTaEE fEff −+= )( 012)0(1212 TTaGG fGff −+= )( 012)0(1212 TTa fff −+= ννν )( 023)0(2323 TTa fff −+= ννν fa 2370 νθ = 0 )0(71 lfCTE=θ Cµ/ .030 o lfCTEa=72θ Cµ/ 0 o )0(73 tfCTE=θ Cµ/ .27 o Fibre Coefficient of Thermal Expansion: )( :alLongitudin 0)0( TTaCTECTE lfCTElflf −+= )( :Transverse 0)0( TTaCTECTE tfCTEtftf −+= tfCTEa=74θ Cµ/ 0 o )0(75 TE=θ GPa 69 Tool Elastic Modulus: )( 0)0( TTaEE TETT −+= TEa=76θ CPa/ 0 o )0(77 Tνθ = 327.0 Tool Poisson’s Ratio: )( 0)0( TTa TTT −+= ννν Taνθ =78 0 )0(79 TCTE=θ Cµ/ 3.62 o Tool Coefficient of Thermal Expansion: )( 0)0( TTaCTECTE TCTETT −+= TCTEa=80θ Cµ/ 0 o 1181 E=θ Pa 109.6 4× 3382 E=θ GPa 69 Shear Layer Properties: 1383 G=θ Pa 106.2 4× Appendix E: Temperature and Spring-in Sensitivity Results 143 Appendix E. TEMPERATURE AND SPRING-IN SENSITIVITY RESULTS Table E.1: Sensitivity of maximum part temperature for the [0 o ]8 and [0 o ]32 lay-ups [0 o ]8 : time=158 min temperature=181.48 ( o C) [0 o ]32 : time=155 min temperature=187.70 ( o C) Parameter Parameter value Sensitivity Percent change (%) Ranking Sensitivity Percent change (%) Ranking θ1 20 4.757×10 -4 5.242×10-5 24 7.444×10-3 7.932×10-4 22 θ2 5×10 -2 -7.719 -2.127×10-3 13 -43.854 -1.168×10-2 12 θ3 1.790×10 3 -7.383×10-5 -7.283×10-4 16 -1.071×10-3 -1.022×10-2 13 θ4 0 -1.210×10 -2 --- --- -0.179 --- --- θ5 1.300×10 3 1.491×10-2 1.068×10-2 4 7.951×10-3 5.507×10-2 4 θ6 0 0.239 --- --- 1.299 --- --- θ7 0 7.426×10 -4 --- --- 3.598×10-3 --- --- θ8 7.913×10 2 -1.170×10-4 -5.102×10-4 17 -1.686×10-3 -7.106×10-3 16 θ9 2.050 -1.930×10 -2 -2.180×10-4 19 -0.2847 -3.109×10-3 19 θ10 1.005×10 3 -6.334×10-5 -3.507×10-4 18 -9.124×10-4 -4.885×10-3 18 θ11 3.740 -1.045×10 -2 -2.153×10-4 20 -1.541E-01 -3.070×10-3 20 θ12 0 -2.797×10 -5 --- --- -3.739×10-4 --- --- θ13 8.002 -6.017×10 -5 -2.653×10-6 32 -3.363×10-3 -1.434×10-4 31 θ14 1.560×10 -2 -9.692×10-3 -8.331×10-7 34 -0.560 -4.653×10-5 33 θ15 2.501 -3.555×10 -4 -4.900×10-6 30 -2.743×10-2 -3.656×10-4 26 θ16 5.070×10 -3 -6.850×10-2 -1.914×10-6 33 -4.696 -1.268×10-4 32 θ17 0.158 -1.080×10 -2 -9.396×10-6 26 -1.474 -1.240×10-3 21 θ18 3.430×10 -4 -2.459 -4.648×10-6 31 -2.535×102 -4.632×10-4 25 θ19 6.070×10 -2 -2.568×10-2 -8.588×10-6 27 -0.919 -2.970×10-4 30 θ20 1.528×10 5 -3.838×10-6 -3.232×10-3 12 -8.483×10-6 -6.906×10-3 17 θ21 6.650×10 4 1.882×10-4 6.895×10-2 1 5.316×10-4 0.188 1 θ22 0.813 2.070 9.273×10 -3 6 9.164 3.969×10-2 7 θ23 2.736 -0.458 -6.910×10 -3 7 -2.960 -4.315×10-2 6 θ24 43.090 2.991×10 -5 7.102×10-6 29 1.529×10-6 3.509×10-7 34 θ25 -1.684 8.643×10 -4 -8.020×10-6 28 -3.511×10-2 3.150×10-4 28 θ26 5.475×10 -3 1.277 3.854×10-5 25 -10.67 -3.112×10-4 29 θ27 5.400×10 5 3.780×10-6 1.125×10-2 3 2.191×10-5 6.302×10-2 3 θ28 0.573 -4.785 -1.511×10 -2 2 -28.866 -8.812×10-2 2 θ29 2.707×10 3 -3.922×10-4 -5.851×10-3 8 -1.395×10-3 -2.012×10-2 9 θ30 0 -6.437×10 -2 --- --- -0.231 --- --- θ31 8.960×10 2 -1.185×10-3 -5.851×10-3 9 -4.215×10-3 -2.012×10-2 10 θ32 0 -0.1945 --- --- -0.698 --- --- θ33 1.670×10 2 -1.554×10-4 -1.430×10-4 22 -6.619×10-4 -5.889×10-4 23 θ34 0 -2.497×10 -2 --- --- -0.108 --- --- θ35 20 -3.563×10 -2 -3.927×10-3 11 -0.201 -2.146×10-2 8 θ36 0 1.759×10 2 --- --- 1.689×102 --- --- θ37 1.700×10 -2 -22.549 -2.112×10-3 14 -85.509 -7.745×10-3 15 θ38 4.5×10 -3 -2.309×102 -5.726×10-3 10 -8.254×102 -1.979×10-2 11 θ39 5×10 -2 7.394 2.037×10-3 15 29.187 7.775×10-3 14 θ40 5×10 -3 5.652 1.557×10-4 21 21.966 5.851×10-4 24 θ41 1×10 -4 -1.680×102 -9.259×10-5 23 -6.025×102 -3.210×10-4 27 1.600×10-3 1.136×103 1.002×10-2 5 θ42 6.400×10-3 1.393×103 4.749×10-2 5 Appendix E: Temperature and Spring-in Sensitivity Results 144 Table E.2: Sensitivity of the corner and warpage components of spring-in for the model parameters θ1 to θ42, for the [0o]8 lay-up Corner Spring-in (1.3 o ) Warpage (0.06 o ) Parameter Parameter value Sensitivity Percent change (%) Ranking Sensitivity Percent change (%) Ranking θ1 20 -9.305×10-4 -1.429×10-2 32 -3.087×10-4 -0.106 29 θ2 5×10 -2 0.651 2.499×10-2 30 -0.189 -0.162 27 θ3 1.790×10 3 1.735×10-7 2.385×10-4 50 1.234×10-6 3.792×10-2 34 θ4 0 1.258×10-4 --- --- 1.672×10-4 --- --- θ5 1.300×10 3 1.916×10-6 1.913×10-3 40 -2.189×10-6 -4.883×10-2 32 θ6 0 4.462×10-4 --- --- -3.369×10-4 --- --- θ7 0 1.240×10-6 --- --- -7.350×10-7 --- --- θ8 7.913×10 2 -1.432×10-7 -8.705×10-5 52 2.070×10-6 2.810×10-2 35 θ9 2.050 2.068×10-4 3.256×10-4 48 2.791×10-4 9.818×10-3 41 θ10 1.005×10 3 -7.753×10-8 -5.984×10-5 53 1.120×10-6 1.932×10-2 38 θ11 3.740 1.119×10-4 3.215×10-4 49 1.510×10-4 9.694×10-3 42 θ12 0 -6.802×10-7 --- --- 1.963×10-7 --- --- θ13 8.002 1.697×10-6 1.043×10-5 58 -2.117×10-7 -2.907×10-5 59 θ14 1.560×10 -2 2.583×10-4 3.095×10-6 60 -2.832×10-5 -7.583×10-6 60 θ15 2.501 2.527×10-5 4.854×10-5 54 1.245×10-5 5.346×10-4 53 θ16 5.070×10 -3 1.439×10-3 5.603×10-6 59 7.378×10-4 6.419×10-5 58 θ17 0.158 1.300×10-3 1.577×10-4 51 5.872×10-4 1.591×10-3 50 θ18 3.430×10 -4 8.677×10-2 2.286×10-5 57 3.895×10-2 2.293×10-4 55 θ19 6.070×10 -2 7.695×10-4 3.587×10-5 56 4.598×10-4 4.789×10-4 54 θ20 1.528×10 5 -7.064×10-7 -8.290×10-2 20 -6.141×10-7 -1.610 6 θ21 6.650×10 4 2.951×10-5 1.507 1 2.571×10-5 29.341 1 θ22 0.813 0.104 6.506×10-2 22 0.100 1.400 8 θ23 2.736 7.920×10-2 0.166 14 5.301×10-2 2.489 3 θ24 43.090 1.519×10-4 5.027×10-3 36 -3.323×10-6 -2.457×10-3 47 θ25 -1.684 -.504 0.652 5 -1.663×10-3 4.805×10-2 33 θ26 5.475×10 -3 -2.284×102 -0.961 4 -0.732 -6.881×10-2 31 θ27 5.400×10 5 3.981×10-9 1.651×10-3 43 -8.400×10-9 -7.785×10-2 30 θ28 0.573 -2.602 -1.145 3 -8.983×10-2 -0.883 11 θ29 2.707×10 3 -8.384×10-7 -1.743×10-3 42 6.333×10-6 0.294 19 θ30 0 6.142×10-4 --- --- 8.589×10-4 --- --- θ31 8.960×10 2 -2.533×10-6 -1.743×10-3 41 1.913×10-5 0.294 20 θ32 0 1.856×10-3 --- --- 2.595×10-3 --- --- θ33 1.670×10 2 -3.037×10-7 -3.896×10-5 55 -2.437×10-8 -6.985×10-5 57 θ34 0 -6.798×10-5 --- --- 1.263×10-6 --- --- θ35 20 -4.253×10-5 -6.533×10-4 45 -8.335×10-4 -0.286 21 θ36 0 -0.4084 --- --- -0.480 --- --- θ37 1.700×10 -2 -5.017×10-2 -6.551×10-4 44 -4.335×10-2 -1.265×10-2 40 θ38 4.5×10 -3 9.220 3.186×10-2 27 1.724 0.133 28 θ39 5×10 -2 0.751 2.884×10-2 28 1.860 1.596 7 θ40 5×10 -3 47.612 0.183 13 2.529 0.217 23 θ41 1.000×10 -4 -6.927×102 -5.320×10-2 23 -4.185×102 -0.718 16 θ42 1.600×10 -3 -1.542×102 -0.190 12 -34.043 -0.935 9 Appendix E: Temperature and Spring-in Sensitivity Results 145 Table E.3: Sensitivity of the corner and warpage components of spring-in for the model parameters θ43 to θ83, for the [0o]8 lay-up Corner Spring-in (1.3 o ) Warpage (0.06 o ) Parameter Parameter value Sensitivity Percent change (%) Ranking Sensitivity Percent change (%) Ranking θ43 4.670×10 4 9.782×10-9 3.508×10-4 47 -1.855×10-9 -1.487×10-3 51 θ44 4.670×10 9 -1.415×10-11 -5.076×10-2 24 -3.118×10-12 -0.250 22 θ45 268 3.121×10-3 0.642 6 -6.838×10-4 -3.145 2 θ46 220 2.041×10-3 0.345 10 -4.521×10-4 -1.707 5 θ47 44.161 -2.798×10-3 -9.489×10-2 19 5.815×10-4 0.441 17 θ48 19.874×10 -2 -1.272 0.194 11 0.264 -0.902 10 θ49 -12 -3.233×10-4 2.980×10-3 37 1.023×10-4 -2.107×10-2 37 θ50 0 17.155 --- --- -7.695 --- --- θ51 0.370 1.988 0.565 7 3.200×10-2 0.203 25 θ52 0 1.532 --- --- 2.092×10-2 --- --- θ53 0 1.670×102 --- --- 5.781 --- --- θ54 0.099 1.656 0.126 15 0.518 0.881 12 θ55 0.055 4.532×10-2 1.914×10-3 39 1.491×10-2 1.407×10-2 39 θ56 0.670 0.895 0.461 9 0.209 2.398 4 θ57 0.173 -0.802 -0.107 17 -0.244 -0.726 14 θ58 1.440×10 -4 1.183×104 1.308 2 0.833 2.057×10-3 48 θ59 0 9.262×105 --- --- 5.211×102 --- --- θ60 -72×10 -6 9.187×103 -0.508 8 3.883 -4.798×10-3 43 θ61 2.100×10 11 -1.690×10-13 -2.726×10-2 29 -1.120×10-13 -0.404 18 θ62 0 3.176×10-10 --- --- 6.279×10-11 --- --- θ63 17.240×10 9 7.794×10-13 1.032×10-2 35 -5.499×10-15 -1.627×10-3 49 θ64 0 2.091×10-10 --- --- -2.730×10-11 --- --- θ65 27.600×10 9 -1.286×10-13 -2.726×10-3 38 -4.343×10-13 -0.206 24 θ66 0 5.274×10-10 --- --- -2.283×10-10 --- --- θ67 0.2 -9.202×10-2 -1.413×10-2 33 -7.726×10-4 -2.652×10-3 46 θ68 0 -5.160 --- --- -0.186 --- --- θ69 0.25 0.517 9.917×10-2 18 8.599×10-4 3.689×10-3 44 θ70 0 46.159 --- --- -1.081 --- --- θ71 3×10 -8 -2.219×104 -5.112×10-4 46 -1.643×103 -8.458×10-4 52 θ72 0 -1.754×106 --- --- -1.329×105 --- --- θ73 7.200×10 -6 1.191×104 6.584×10-2 21 1.787 2.208×10-4 56 θ74 0 9.320×105 --- --- 5.443×102 --- --- θ75 6.900×10 10 2.860×10-13 1.516×10-2 31 -1.845×10-14 -2.185×10-2 36 θ76 0 2.284×10-11 --- --- -1.320×10-12 --- --- θ77 0.327 0.138 3.466×10-2 26 3.427×10-2 0.192 26 θ78 0 11.275 --- --- 2.833 --- --- θ79 2.360×10 -5 6.342×103 0.115 16 2.094×103 0.848 13 θ80 0 5.201×105 --- --- 1.722×105 --- --- θ81 6.900×10 4 -1.357×10-12 -7.192×10-8 61 -3.218×10-12 -3.811×10-6 61 θ82 6.900×10 10 8.145×10-13 4.316×10-2 25 -2.825×10-15 -3.346×10-3 45 θ83 2.600×10 4 5.246×10-7 1.048×10-2 34 1.614×10-6 0.720 15 Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 146 Appendix F. SECOND-ORDER SENSITIVITY EQUATIONS FOR UNIAXIAL J2 PLASTICITY The algorithm outlined in Fig. 4.1 is here presented in detail for the uniaxial J2 plasticity material model. The reader is referred to, e.g. Simo and Hughes (1998) for background and interpretation of this widely utilized bilinear model. The primary objective of this exposition is to present the implementation-ready algorithm that produces all quantities required to obtain second-order response sensitivities. After introducing the material parameters and the history variables, the presentation below provides the original objective of the material model (producing a stress and a tangent stiffness for a given strain) followed by the equations for Phases 1 through 4 as outlined in Fig. 4.1. Superscript is employed to denote the analysis step number. It is noted that derivatives of material parameters, such as ixE ∂∂ , are either one or zero; depending on which parameter that xi or xj represents. Summary of material parameters (Response sensitivities are sought with respect to any of these.) • σy is the yield stress • E is the Young’s modulus • Hiso is the isotropic hardening modulus • Hkin is the kinematic hardening modulus Summary of history variables that are available for Phase 0 (These are re-computed and stored at each analysis step. Their initial value is zero.) • ε~ is the plastic strain • q is the back stress • α is the internal hardening variable Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 147 F.1. PHASE 0: RESPONSE 1) Compute the trial elastic stress: ( )nnn E εεσ ~ˆ 11 −= ++ (F.1) 2) Evaluate the yield function: ( )nisoynn Hqf ασσ +−−= +1ˆ (F.2) 3) If 0≤f the step is “elastic” compute: a) Final stress: 11 ˆ ++ = nn σσ b) Tangent stiffness: Ek n =+1 c) History variables: nn εε ~~ 1 =+ ; nn qq =+1 ; nn αα =+1 4) If 0>f the step is “plastic” compute: a) Plastic consistency parameter: kiniso HHE f ++ =∆γ (F.3) b) Final stress: ( )nnnn qE −⋅⋅∆−= +++ 111 ˆsignˆ σγσσ (F.4) c) Tangent stiffness: kiniso kinison HHE HHE k ++ + =+ )(1 (F.5) d) History variables: Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 148 ( ) ( ) γαα σγ σγεε ∆+= −∆+= −∆+= + ++ ++ nn nn kin nn nnnn qHqq q 1 11 11 ˆsign ˆsign~~ (F.6) Summary of history variables that are available for Phase 1 (These are re-computed and stored at each analysis step. Their initial value is zero.) • i i i x xq x ∂∂ ∂∂ ∂∂ α ε~ F.2. PHASE 1: FIRST-ORDER SENSITIVITY RESULTS 5) If the step is elastic ( 0≤f ) compute: a) Conditional first-order sensitivity of final stress: ( ) i n nn ii n x E x E x n ∂ ∂ −− ∂ ∂ = ∂ ∂ + + + ε εε σ ε ~ ~1 1 1 (F.7) 6) If the step is plastic ( 0>f ) compute: a) Conditional first-order sensitivity of trial stress: ( ) i n nn ii n x E x E x n ∂ ∂ −− ∂ ∂ = ∂ ∂ + + + ε εε σ ε ~ ~ˆ 1 1 1 (F.8) b) Conditional first-order sensitivity of yield function: ( ) i n iso n i iso i ynn i n i n i x H x H x q x q xx f nn ∂ ∂ − ∂ ∂ − ∂ ∂ −− ∂ ∂ − ∂ ∂ = ∂ ∂ + + ++ α α σ σ σ εε 1 1 ˆsign ˆ 11 (F.9) c) Conditional first-order sensitivity of plastic consistency parameter: Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 149 ( ) ( ) ( )2kiniso i kiniso kiniso i i HHE x HHE fHHE x f x 1n 1n ++ ∂ ++∂ −++ ∂ ∂ = ∂ ∆∂ + + ε ε γ (F.10) d) Conditional first-order sensitivity of final stress: ( ) ( )nn i nn ii n i n q x E qE xxx nnn − ∂ ∂ ∆−− ∂ ∆∂ − ∂ ∂ = ∂ ∂ ++ ++ +++ 11 11 ˆsignˆsign ˆ 111 σγσ γσσ εεε (F.11) F.3. PHASE 2: FIRST-ORDER DERIVATIVES OF THE HISTORY VARIABLES 7) If the step is elastic ( 0≤f ) compute: a) History variables: i n i n i n i n i n i n xx x q x q xx ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ + + + αα εε 1 1 1 ~~ (F.12) 8) If the step is plastic ( 0>f ) compute: a) First-order sensitivity of trial stress: ( ) ∂ ∂ − ∂ ∂ +− ∂ ∂ = ∂ ∂ ++ + i n i n nn ii n xx E x E x εε εε σ ~~ˆ 1 1 1 (F.13) b) First-order sensitivity of yield function: ( ) i n iso n i iso i ynn i n i n i x H x H x q x q xx f ∂ ∂ − ∂ ∂ − ∂ ∂ −− ∂ ∂ − ∂ ∂ = ∂ ∂ + + α α σ σ σ 1 1 ˆsign ˆ (F.14) c) First-order sensitivity of plastic consistency parameter: Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 150 ( ) ( ) ( )2kiniso i kiniso kiniso i i HHE x HHE fHHE x f x ++ ∂ ++∂ −++ ∂ ∂ = ∂ ∆∂ γ (F.15) d) History variables: ( ) ( ) ( ) ii n i n nn i kinnn kin ii n i n nn ii n i n xxx q x H qH xx q x q q xxx ∂ ∆∂ + ∂ ∂ = ∂ ∂ − ∂ ∂ ∆+− ∂ ∆∂ + ∂ ∂ = ∂ ∂ − ∂ ∆∂ + ∂ ∂ = ∂ ∂ + ++ + + + γαα σγσ γ σ γεε 1 11 1 1 1 ˆsignˆsign ˆsign ~~ (F.16) (Note that the strain sensitivity is generally non-zero also when the parameter is not in this material; which is why all materials in the domain must always be called after the first step.) Summary of history variables that are available for Phase 3 (These are re-computed and stored at each analysis step. Their initial value is zero.) • ji ji ji xx xxq xx ∂∂∂ ∂∂∂ ∂∂∂ α ε 2 2 2~ F.4. PHASE 3: SECOND-ORDER SENSITIVITY RESULTS 9) If the step is elastic ( 0≤f ) compute: a) Conditional Second-order sensitivity of final stress: ( ) ji n i n jj n i nn jiji n xx E xx E xx E xx E xx n ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ −− ∂∂ ∂ = ∂∂ ∂ + + + εεε εε σ ε ~~~ ~ 2 1 212 1 (F.17) b) Conditional first-order sensitivity of tangent stiffness: Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 151 jj n x E x k n ∂ ∂ = ∂ ∂ + + 1 1 ε (F.18) c) Gradient of tangent stiffness respect to strain: 0 1 1 = ∂ ∂ + + n nk ε (F.19) 10) If the step is plastic ( 0>f ) compute: a) Conditional second-order sensitivity of trial stress: ( ) ji n i n jj n i nn jiji n xx E xx E xx E xx E xx n ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ −− ∂∂ ∂ = ∂∂ ∂ + + + εεε εε σ ε ~~~ ~ˆ 2 1 212 1 (F.20) b) Conditional second-order sensitivity of yield function: ( ) ji n iso i n j iso j n i iso n ji iso ji ynn ji n ji n ji xx H xx H xx H xx H xx q xx q xxxx f nn ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂∂ ∂ − ∂∂ ∂ −− ∂∂ ∂ − ∂∂ ∂ = ∂∂ ∂ + + ++ ααα α σ σ σ εε 2 22 1 2122 ˆsign ˆ 11 K (F.21) c) Conditional second-order sensitivity of plastic consistency parameter: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 2 2 2 2 1 1 11 1 2 kiniso j kiniso i kiniso kiniso i kiniso ji kiniso i kiniso j kiniso j kiniso i kiniso ji ji HHE x HHE x HHE fHHE x f HHE xx HHE f x HHE x f HHE x HHE x f HHE xx f xx n n nn n ++ ∂ ++∂ ∂ ++∂ −++ ∂ ∂ − ++ ∂∂ ++∂ + ∂ ++∂ ∂ ∂ − ++ ∂ ++∂ ∂ ∂ +++ ∂∂ ∂ = ∂∂ ∆∂ + + ++ + ε ε εε ε γ L L (F.22) Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 152 where ji xxE ∂∂∂ 2 , jiiso xxH ∂∂∂ 2 , and jikin xxH ∂∂∂ 2 are zero. d) Conditional second-order sensitivity of final stress: ( ) ( ) ( ) ( )nn iji nn ij nn ji nn jiji n ji n q xx E q x E x q x E x qE xxxxxx nn nnn − ∂∂ ∂ ∆− − ∂ ∂ ∂ ∆∂ −− ∂ ∂ ∂ ∆∂ − − ∂∂ ∆∂ − ∂∂ ∂ = ∂∂ ∂ + ++ + ++ ++ +++ 1 2 11 1 21212 ˆsign ˆsignˆsign ˆsign ˆ 11 111 σγ σ γ σ γ σ γσσ εε εεε K K (F.23) e) Conditional first-order sensitivity of tangent stiffness: ( ) ( ) ( ) ( )2 2 1 )( )( )( 1 kiniso j kiniso kiniso kiniso kiniso j kiniso kiniso j j n HHE x HHE HHE HHE HHE x HH EHH x E x k n ++ ∂ ++∂ + − ++ ++ ∂ +∂ ++ ∂ ∂ = ∂ ∂ + + L ε (F.24) f) Gradient of tangent stiffness respect to strain : 0 1 1 = ∂ ∂ + + n nk ε (F.25) F.5. PHASE 4: SECOND-ORDER DERIVATIVES OF THE HISTORY VARIABLES 11) If the step is elastic ( 0≤f ) compute: a) History variables: Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 153 ji n ji n ji n ji n ji n ji n xxxx xx q xx q xxxx ∂∂ ∂ = ∂∂ ∂ ∂∂ ∂ = ∂∂ ∂ ∂∂ ∂ = ∂∂ ∂ + + + αα εε 212 212 212 ~~ (F.26) 12) If the step is plastic ( 0>f ) compute: a) Second-order sensitivity of trial stress: ( ) ∂∂ ∂ − ∂∂ ∂ + ∂ ∂ − ∂ ∂ ∂ ∂ + ∂ ∂ − ∂ ∂ ∂ ∂ +− ∂∂ ∂ = ∂∂ ∂ + ++ + + ji n ji n i n i n jj n j n i nn jiji n xxxx E xxx E xxx E xx E xx εε εεεε εε σ ~ ~~ ~ˆ 212 11 1 212 K (F.27) b) Second-order sensitivity of yield function: ( ) ji n iso i n j iso j n i ison ji iso ji ynn ji n ji n ji xx H xx H xx H xx H xx q xx q xxxx f ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂∂ ∂ − ∂∂ ∂ −− ∂∂ ∂ − ∂∂ ∂ = ∂∂ ∂ + + ααα α σ σ σ 22 2 1 2122 ˆsign ˆ K (F.28) c) Second-order sensitivity of plastic consistency parameter: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )3 2 2 2 2 2 kiniso j kiniso i kiniso kiniso i kiniso ji kiniso i kiniso j kiniso j kiniso i kiniso ji ji HHE x HHE x HHE fHHE x f HHE xx HHE f x HHE x f HHE x HHE x f HHE xx f xx ++ ∂ ++∂ ∂ ++∂ −++ ∂ ∂ − ++ ∂∂ ++∂ + ∂ ++∂ ∂ ∂ − ++ ∂ ++∂ ∂ ∂ +++ ∂∂ ∂ = ∂∂ ∆∂ K L γ (F.29) Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 154 where ji xxE ∂∂∂ 2 , jiiso xxH ∂∂∂ 2 , and jikin xxH ∂∂∂ 2 are zero. d) History variables: ( ) ( ) ( ) ( ) ( ) jiji n ji n nn ji kinnn i kin j nn j kin i nn kin jiji n ji n nn jiji n ji n xxxxxx q xx H q x H x q x H x qH xxxx q xx q q xxxxxx ∂∂ ∆∂ + ∂∂ ∂ = ∂∂ ∂ − ∂∂ ∂ ∆+− ∂ ∂ ∂ ∆∂ + − ∂ ∂ ∂ ∆∂ +− ∂∂ ∆∂ + ∂∂ ∂ = ∂∂ ∂ − ∂∂ ∆∂ + ∂∂ ∂ = ∂∂ ∂ + ++ ++ + + + γαα σγσ γ σ γ σ γ σ γεε 2212 1 2 1 11 2212 1 2212 ˆsignˆsign ˆsignˆsign ˆsign ~~ L (F.30) Appendix F: Second-order Sensitivity Equations for Uniaxial J2 Plasticity 155 F.6. REFERENCES Simo, J.C., and Hughes, T.J.R. (1998). Computational Inelasticity. Interdisciplinary applied mathematics, Springer-Verlag, NY. Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 156 Appendix G. SECOND-ORDER SENSITIVITY EQUATIONS FOR MULTIAXIAL J2 PLASTICITY The algorithm outlined in Fig. 4.1 is here presented in detail for the multiaxial J2 plasticity material model for 2D and 3D plain strain problems. The reader is referred to, e.g. Simo and Hughes (1998) for background and interpretation of this widely utilized multiaxial plasticity model. The primary objective of this exposition is to present the implementation-ready algorithm that produces all quantities required to obtain second-order response sensitivities. After introducing the material parameters and the history variables, the equations for multiaxial J2 plasticity model are followed by the equations for Phases 1 through 4 as outlined in Fig. 4.1. Superscript is employed to denote the analysis step number. Also subscripts indicate the vector or matrix components. Summary of material parameters (Response sensitivities are sought with respect to any of these.) • σy is the yield stress • υ is the Poisson’s ratio • E is the Young’s modulus • Hiso is the isotropic hardening modulus • Hkin is the kinematic hardening modulus Summary of history variables that are available for Phase 0 (These are re-computed and stored at each analysis step. Their initial value is zero.) • e~ is the vector of plastic strains • q is the vector of back stresses • α is the vector of internal hardening variables Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 157 Predefined matrices (For convenience, the implementation of this material model also makes use of the vector m and the matrices Io and Idev defined as below) • Trace projector matrix [ ] 6:1000111 === kmkTm (G.1) • Special identity matrix 6:1, 5.000000 05.00000 005.0000 000100 000010 000001 == = mkI kmooI (G.2) • Deviatoric matrix 6:1, 2/100000 02/10000 002/1000 0003/23/13/1 0003/13/23/1 0003/13/13/2 == −− −− −− = mkI kmdevdevI (G.3) Elastic constants and their derivatives (These constant and their derivations are used along the algorithm) • Bulk modulus κ )21(3 ν κ − = E (G.4) o First derivatives Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 158 ( )2)21(3 6)21(3 ν ν ν κ − ∂ ∂ + ∂ ∂ − = ∂ ∂ ii i x E x E x (G.5) o Second derivatives ( ) ( )3 2 22 2 )21(3 6)21(312 )21(3 666)21(3 ν νν ν ν ννν ν κ − ∂ ∂ ∂ ∂ + ∂ ∂ − + − ∂∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ − ∂∂ ∂ − = ∂∂ ∂ jii jiijijji ji xx E x E xx E xx E x E xxx E xx L (G.6) where ji xxE ∂∂∂ 2 and ji xx ∂∂∂ ν 2 are zero. • Lame constant µ )1(2 ν µ + = E (G.7) o First derivatives ( )2)1(2 2)1(2 ν ν ν µ + ∂ ∂ − ∂ ∂ + = ∂ ∂ ii i x E x E x (G.8) o Second derivatives ( ) ( )3 2 22 2 )1(2 2)1(24 )1(2 222)1(2 ν νν ν ν ννν ν µ + ∂ ∂ ∂ ∂ − ∂ ∂ + − + ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ + ∂∂ ∂ + = ∂∂ ∂ jii jiijijji ji xx E x E xx E xx E x E xxx E xx L (G.9) Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 159 where ji xxE ∂∂∂ 2 and ji xx ∂∂∂ ν 2 are zero. • Lame constant )21()1( νν ν λ −+ = E (G.10) o First derivatives ( )2)21()1( )41()21()1( νν ν ννννν ν λ −+ ∂ ∂ ++−+ ∂ ∂ + ∂ ∂ = ∂ ∂ iii i x E x E E x x (G.11) o Second derivatives ( ) ( ) ( )3 2 2 2 22 2 )21()1( )41()41()21()1( 2 )21()1( )41(4)41()41( )21()1( )41()21()1( νν ν ν ν ννννν ν νν ν νν νν ν ν νν νν ν νν ν νν ν ννν ννν λ −+ ∂ ∂ + ∂ ∂ ++−+ ∂ ∂ + ∂ ∂ + −+ ∂∂ ∂ ++ ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ ++ ∂ ∂ ∂ ∂ + + −+ ∂ ∂ + ∂ ∂ + ∂ ∂ −−+ ∂∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂∂ ∂ = ∂∂ ∂ jiii jijijiji jiijiijjiji ji xx E x E E x xx E xx E x E xxx E xx E E xxx E x E xx E x E xx xx L L (G.12) where ji xxE ∂∂∂ 2 and ji xx ∂∂∂ ν 2 are zero. G.1. PHASE 0: RESPONSE The J2 plasticity model for plane strain 2D problems and 3D problems takes as input the strain tensor and provides the corresponding stress tensor as output. In Voight notation these tensors are written as vectors: [ ]132312332211 ,,,,, εεεεεε=ε and [ ]132312332211 ,,,,, σσσσσσ=σ . 1) Import strain Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 160 [ ] kT εεεεεεε == 231312332211ε (G.13) where 2/1212 γε = , 2/1313 γε = , 2/2323 γε = , and 6:1=k 2) Decompose the strain into a) Volumetric strain 321 1 εεεε ++=+nv (G.14) b) Deviatoric strain k n v k n k m ε εe 3 1 1 + + −= (G.15) where 6:1=k 3) Evaluate trial deviatoric stress ( )nknknk eeµs ~2ˆ 11 −= ++ (G.16) 4) Evaluate the yield function: ( )nynn ασf +−−= + 3 2 ˆ 1 qs (G.17) where ( ) ( ) ( ) ( ) ( ) ( )2616251524142313221221111 ˆ2ˆ2ˆ2ˆˆˆˆ nnnnnnnnnnnnnn qsqsqsqsqsqs −+−+−+−+−+−≡− +++++++ qs (G.18) 5) If 0≤f the step is “elastic” compute: a) Back deviatoric stress 11 ˆ ++ = nk n k ss (G.19) b) Final stress: Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 161 111 ˆ +++ += nkk n v n k smκεσ (G.20) c) Tangent stiffness: mkkmo n km mmλIµk += + 21 (G.21) d) History variables: 11 ~~ ++ = nk n k ee ; 11 ++ = nk n k qq ; nn αα =+1 (G.22) 6) If 0>f the step is “plastic” compute: a) Plastic consistency parameter ( )isokin HH f ++ =∆ 3 2 2µ γ (G.23) b) Yield surface normal vector nn n k n kn k qs n qs − − = + + + 1 1 1 ˆ ˆ (G.24) c) Back deviatoric stress 111 2ˆ +++ −= nk n k n k n∆µss γ (G.25) d) Final stress 111 +++ += nkk n v n k smεκσ (G.26) e) Deviatoric tangent stiffness ( )[ ]11111 2 +++++ ⋅−−⋅−= nmnkdevnmnkdevnkmdev nnIBnnAIµk (G.27) where Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 162 ( )kiniso HHµ µ A ++ = 3 1 (G.28) nn ∆γµ B qs − = +1ˆ 2 (G.29) f) Tangent stiffness km n kmdev n km mmkk κ+= ++ 11 (G.30) g) History variables: iso nn n kkin n k n k n k n k n k H∆γαα nH∆γqq n∆γee 3 2 3 2 ~~ 1 11 11 += += += + ++ ++ (G.31) Summary of history variables that are available for Phase 1 (These are re-computed and stored at each analysis step. Their initial value is zero.) • i ik ik xα/ x/q x/e ∂∂ ∂∂ ∂∂~ G.2. PHASE 1: FIRST-ORDER SENSITIVITY RESULTS: 7) Import the strain kε and decompose it into volumetric and deviatoric part: 321 1 εεεε ++=+nv (G.32) k n v k n k m ε εe 3 1 1 + + −= (G.33) Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 163 where 6:1=k 8) If the step is elastic ( 0≤f ) compute: a) Conditional first-order sensitivity of deviatoric stress ( ) i n kn k n k iεi n k x e µee x µ x s n ∂ ∂ −− ∂ ∂ = ∂ ∂ + + + ~ 2~2 1 1 1 (G.34) b) Conditional first-order sensitivity of final stress 11 1 1 1 ++ ∂ ∂ + ∂ ∂ = ∂ ∂ ++ + n o n o ε i n k k n v iεi n k x s mε x κ x σ (G.35) 9) If the step is plastic ( 0>f ) compute: a) Conditional first-order sensitivity of trial deviatoric stress ( ) i n kn k n k iεi n k x e µee x µ x s n o ∂ ∂ −− ∂ ∂ = ∂ ∂ + + + ~ 2~2 ˆ 1 1 1 (G.36) b) Conditional first-order sensitivity of yield function ∂ ∂ + ∂ ∂ − ∂ −∂ = ∂ ∂ + + + i n i y i nn i x α x σ xx f n o n o 3 2ˆ 1 1 1 εε qs (G.37) c) Conditional first-order sensitivity of plastic consistency parameter ( ) ( ) 2 3 2 2 3 2 2 3 2 2 1 1 ++ ∂ ∂ + ∂ ∂ + ∂ ∂ − ++ ∂ ∂ = ∂ ∆∂ + + isokin i iso i kin i isokin i i HH x H x H x fHH x f x n o n o µ µ µ γ ε ε (G.38) d) Conditional first-order sensitivity of yield surface normal vector Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 164 2 1 11 1 1 1 ˆ ˆˆ ˆ ˆ 1 1 1 nn n k n k i nn nn i n k i n k i n k qs x x q x s x n n o n o n o qs qs qs − − ∂ −∂ − − ∂ ∂ − ∂ ∂ = ∂ ∂ + ++ + + + + + + ε ε ε (G.39) e) Conditional first-order sensitivity of back deviatoric stress 222 ˆ 111 1 11 11 +++ ∂ ∂ − ∂ ∂ − ∂ ∂ − ∂ ∂ = ∂ ∂ +++ ++ n o n o n o i n kn k i n k ii n k i n k x n ∆γµn x ∆γ µn∆γ x µ x s x s εεε (G.40) f) Conditional first-order sensitivity of final stress 11 1 1 1 ++ ∂ ∂ + ∂ ∂ = ∂ ∂ ++ + n o n o ε i n k k n v iεi n k x s mε x κ x σ (G.41) G.3. PHASE 2: FIRST-ORDER DERIVATIVES OF THE HISTORY VARIABLES 10) Import the strain ik x∂∂ε and decompose it into volumetric and deviatoric part: iiii n v xxxx ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + 321 1 εεεε (G.42) k i n v i k i n k m x ε x ε x e ∂ ∂ − ∂ ∂ = ∂ ∂ ++ 11 3 1 (G.43) 11) If the step is elastic ( 0≤f ) compute: a) History variables i n i n i n k i n k i n k i n k x α x α x q x q x e x e ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ + + + 1 1 1 ~~ (G.44) Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 165 where 6:1=k . 12) If the step is plastic ( 0>f ) compute: a) First-order sensitivity of Trial deviatoric stress ( ) ∂ ∂ − ∂ ∂ +− ∂ ∂ = ∂ ∂ ++ + i n k i n kn k n k ii n k x e x e µee x µ x s ~ 2~2 ˆ 1 1 1 (G.45) b) First-order yield function ∂ ∂ + ∂ ∂ − ∂ ∂ − ∂ ∂ = ∂ ∂ + + i n i yn k i n k i n k i x α x σ n x q x s x f 3 2ˆ 1 1 (G.46) c) First-order plastic consistency parameter ( ) ( ) 2 3 2 2 3 2 2 3 2 2 ++ ∂ ∂ + ∂ ∂ + ∂ ∂ − ++ ∂ ∂ = ∂ ∆∂ isokin i iso i kin i isokin i i HH x H x H x fHH x f x µ µ µ γ (G.47) d) First-order yield surface normal vector 2 1 11 1 1 1 ˆ ˆˆ ˆ ˆ nn n k n k i nn nn i n k i n k i n k qs x x q x s x n qs qs qs − − ∂ −∂ − − ∂ ∂ − ∂ ∂ = ∂ ∂ + ++ + + + (G.48) e) History variables i iso iso ii n i n i n k kin n k i kinn kkin ii n k i n k i n kn k ii n k i n k x H ∆γH x ∆γ x α x α x n H∆γn x H ∆γnH x ∆γ x q x q x n ∆γn x ∆γ x e x e ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + + ++ + + + + 3 2 3 2 3 2 3 2 3 2 ~~ 1 1 11 1 1 1 1 (G.49) Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 166 Summary of history variables that are available for Phase 3 (These are re-computed and stored at each analysis step. Their initial value is zero.) • ji jik jik xxα xxq xxe ∂∂∂ ∂∂∂ ∂∂∂ 2 2 2~ G.4. PHASE 3: SECOND-ORDER SENSITIVITY RESULTS 13) Import the strain kε and decompose into volumetric and deviatoric part: 321 1 εεεε ++=+nv (G.50) k n v k n k m ε εe 3 1 1 + + −= (G.51) where 6:1=k . 14) If the step is elastic ( 0≤f ) compute: a) Conditional second-order sensitivity of deviatoric stress ( ) ji n k i n k jj n k i n k n k jiεji n k xx e µ x e x µ x e x µ ee xx µ xx s n o ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ −− ∂∂ ∂ = ∂∂ ∂ + + + ~ 2 ~ 2 ~ 2~2 2 1 212 1 (G.52) b) Conditional second-order sensitivity of final stress 11 12 1 212 ++ ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ ++ + nn o ε ji n k k n v jiεji n k xx s mε xx κ xx σ (G.53) c) Conditional first-order sensitivity of tangent stiffness mk i kmo iεi n km mm x λ I x µ x k n o ∂ ∂ + ∂ ∂ = ∂ ∂ + + 2 1 1 (G.54) d) Gradient of tangent stiffness respect to strain Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 167 0 1 1 = ∂ ∂ + + n o n km ε k (G.55) 15) If the step is plastic: ( 0>f ) a) Conditional second-order sensitivity of deviatoric stress ( ) ji n i n k jj n k i n k n k jiεji n k xx e µ x e x µ x e x µ ee xx µ xx s n ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ −− ∂∂ ∂ = ∂∂ ∂ + + + ~ 2 ~ 2 ~ 2~2 ˆ 21 212 1 (G.56) b) Conditional second-order sensitivity of yield surface ∂∂ ∂ + ∂∂ ∂ − ∂∂ −∂ = ∂∂ ∂ ++ + ji n ji y ji nn ji yxxxxxxx f n o n o ασ εε 22122 3 2ˆ 11 qs (G.57) c) Conditional second -order sensitivity of plastic consistency parameter ( ) ( ) ( ) ( ) ( ) 3 2 222 2 2 2 3 2 2 3 2 2 3 2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 1 1 11 1 ++ ∂ ∂ + ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ + ∂ ∂ − ++ ∂ ∂ − ++ ∂∂ ∂ + ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ − ++ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ + ++ ∂∂ ∂ = ∂∂ ∆∂ + + ++ + isokin j iso j kin ji iso i kin i isokin i isokin ji iso ji kin jii iso i kin ij isokin j iso j kin ji isokin ji ji HH x H x H xx H x H x fHH x f HH xx H xx H xx f x H x H xx f HH x H x H xx f HH xx f xx n o n o n o n o n o µ µµ µ µ µµ µ µ µ γ ε ε εε ε L L (G.58) where jiiso xxH ∂∂∂ 2 and jikin xxH ∂∂∂ 2 are zero. d) Conditional second -order sensitivity of yield surface normal vector Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 168 ( ) ( ) ˆ ˆˆ ˆ2 ˆ ˆ ˆ ˆ ˆˆ ˆ ˆˆ ˆ ˆ 3 1 11 1 2 1 12 1 2 1 11 2 1 11 1 212 12 11 111 1 1 1 1 nn j nn i nn n k n k nn ji nn n k n k nn i nn j n k j n k nn j nn i n k i n k nn ji n k ji n k ji n k n o n o n o n o n o n o n o n o n o xx qs xx qs xx q x s xx q x s xx q xx s xx n qs qsqs qs qs qs qs qs qs qs − ∂ −∂ ∂ −∂ − + − ∂∂ −∂ − − − ∂ −∂ ∂ ∂ − ∂ ∂ − − ∂ −∂ ∂ ∂ − ∂ ∂ − − ∂∂ ∂ − ∂∂ ∂ = ∂∂ ∂ + ++ + + + + + ++ + ++ + + + ++ +++ +++ + εε εεε εεε ε K L (G.59) e) Conditional second-order sensitivity of back deviatoric stress 222 222 222 ˆ 1111 1111 1111 1211 1 1 2 1 1 11 21212 ++++ ++++ ++++ ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂∂ ∂ − ∂∂ ∂ = ∂∂ ∂ +++ + ++ + ++ ++ n o n o n o n o n o n o n o n o n o n o n o n o εji n k εi n k jεi n k j εj n k i n k ji n k ij εj n k i n k ji n k jiεji n k εji n k xx n ∆γµ x n x ∆γ µ x n ∆γ x µ x n x ∆γ µn xx ∆γ µn x ∆γ x µ x n ∆γ x µ n x ∆γ x µ n∆γ xx µ xx s xx s ε εεε ε L L (G.60) f) Conditional second-order sensitivity of final stress 11 12 1 212 ++ ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ ++ + n o n o ε ji n kn v jiεji n k xx s mε xx κ xx σ (G.61) g) Conditional first-order sensitivity of deviatoric tangent stiffness Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 169 ( )[ ] ( ) ⋅− ∂ ∂ +⋅ ∂ ∂ + ∂ ∂ ⋅−+⋅ ∂ ∂ −− ⋅−−⋅− ∂ ∂ = ∂ ∂ ++++ + ++ + ++++ + +++ + 1111 1 11 1 1111 1 111 1 )()(2 2 n m n kkmdev i n m n k ii n mn k n m i n k n m n kkmdev n m n kkmdev ii n kmdev nnI x B nn x A x n nBAn x n BAµ nnIBnnAI x µ x k n o n o n o n o εεε ε K (G.62) where ( ) ( ) 2 3 1 3 1 3 1 1 ++ ∂ ∂ + ∂ ∂ + ∂ ∂ − ++ ∂ ∂ = ∂ ∂ + kiniso i kin i iso i kiniso i i HH x H x H x HH x x A n o µ µ µµ µ ε (G.63) and 2 1 1 1 ˆ ˆ 2 ˆ 2 11 1 nn i nn nn ii i n o n o n o x ∆γµ x ∆γ µ∆γ x µ x B qs qs qs − ∂ −∂ − − ∂ ∂ + ∂ ∂ = ∂ ∂ + + + ++ + εε ε (G.64) h) Conditional first-order sensitivity of tangent stiffness km ii n kmdev i n km mm xx k x k n o ∂ ∂ + ∂ ∂ = ∂ ∂ ++ + κ ε 11 1 (G.65) i) Gradient of tangent stiffness respect to strain 1 1 1 1 + + + + ∂ ∂ = ∂ ∂ n o n kmdev n o n km kk εε (G.66) since ( ) 0 1 = ∂ +n o kmmm ε κ (G.67) j) Gradient of deviatoric tangent stiffness respect to strain Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 170 qodevn q n kmdev n o n q n q n kmdev n o n kmdev I e ke e kk 1 1 1 1 1 1 1 1 + + + + + + + + ∂ ∂ = ∂ ∂ ∂ = ∂ εε (G.68) since 0 1 1 = ∂ ∂ + + n v n kmdev k ε (G.69) k) Gradient of deviatoric tangent stiffness respect to deviatoric strain ( ) ⋅− ∂ ∂ + ∂ ∂ ⋅−+⋅ ∂ ∂ −−= ∂ ∂ ++ ++ + ++ + + + + 11 11 1 11 1 1 1 1 )()(2 nm n kdevn q n q n mn k n mn q n k n q n kmdev nnI e B e n nBAn e n BAµ e k (G.70) since 0= ∂ ∂ n qe A and where ( ) ∂ −∂ − − = − ∂ −∂ − − − ∂ ∂ = ∂ ∂ + + + ++ + + + + + + + + 1 1 1 121 1 1 1 1 1 1 1 1 ˆ ˆ 1 ˆ ˆ ˆ ˆ ˆ n kn q nn kqnnnn n q nn n k n k nn n q n k n q n k n e e qs e s e n qs qsqs qs qs δ (G.71) 2 1 1 1 1 1 1 ˆ ˆ 2 ˆ 2 nn n q nn nn n q n q e ∆γµ e ∆γ µ e B qs qs qs − ∂ −∂ − − ∂ ∂ = ∂ ∂ + + + + + + (G.72) ( ) ( )isokin n q nn isokin n q n q HH e HH e f e ++ ∂ −∂ = ++ ∂ ∂ = ∂ ∆∂ + + + + 3 2 2 ˆ 3 2 2 1 1 1 1 µµ γ qs (G.73) G.5. PHASE 4: SECOND-ORDER DERIVATIVES OF THE HISTORY VARIABLES 16) Import the first- and second-order strain sensitivities ik x∂∂ε and jik xx ∂∂∂ ε 2 and decompose them into volumetric and deviatoric parts: Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 171 iiii n v xxxx ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + 321 1 εεεε , jijijiji n v xxxxxxxx ∂∂ ∂ + ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ + 3 2 2 2 1 212 εεεε (G.74) k i n v i k i n k m x ε x ε x e ∂ ∂ − ∂ ∂ = ∂ ∂ ++ 11 3 1 , k ji v ji k ji n k m xx ε xx ε xx e ∂∂ ∂ − ∂∂ ∂ = ∂∂ ∂ + 2212 3 1 (G.75) 17) If the step is elastic ( 0≤f ) compute: a) History variables ji n ji n ji n k ji n k ji n k ji n k xx α xx α xx q xx q xx e xx e ∂∂ ∂ = ∂∂ ∂ ∂∂ ∂ = ∂∂ ∂ ∂∂ ∂ = ∂∂ ∂ + + + 212 212 212 ~~ (G.76) 18) If the step is plastic ( 0>f ) compute: a) Second-order sensitivity of deviatoric stress ( ) ∂∂ ∂ − ∂∂ ∂ + ∂ ∂ − ∂ ∂ ∂ ∂ + ∂ ∂ − ∂ ∂ ∂ ∂ +− ∂∂ ∂ = ∂∂ ∂ ++ + + + ji n k ji n k i n k i n k j j n k j n k i n k n k jiji n k xx e xx e µ x e x e x µ x e x e x µ ee xx µ xx s ~ 2 ~ 2 ~ 2~2 ˆ 2121 1 1 212 K (G.77) b) Second-order sensitivity of yield surface ∂∂ ∂ + ∂∂ ∂ − ∂∂ −∂ = ∂∂ ∂ + ji n ji y ji nn ji yxxxxxxx f ασ 2 2122 3 2ˆ qs (G.78) c) Second-order sensitivity of plastic consistency parameter Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 172 ( ) ( ) ( ) ( ) ( ) 3 2 222 2 2 2 3 2 2 3 2 2 3 2 2 3 2 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 3 2 2 −+ ∂ ∂ − ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ + ∂ ∂ − −+ ∂ ∂ − −+ ∂∂ ∂ − ∂∂ ∂ + ∂∂ ∂ + ∂ ∂ − ∂ ∂ + ∂ ∂ ∂ ∂ − −+ ∂ ∂ − ∂ ∂ + ∂ ∂ ∂ ∂ + −+ ∂∂ ∂ = ∂∂ ∆∂ isokin j iso j kin ji iso i kin i isokin i isokin ji iso ji kin jii iso i kin ij isokin j iso j kin ji isokin ji ji HH x H x H xx H x H x fHH x f HH xx H xx H xx f x H x H xx f HH x H x H xx f HH xx f xx µ µµ µ µ µµ µ µ µ γ L L (G.79) d) Second-order sensitivity of yield surface normal vector ( ) ( ) ˆ x ˆ x ˆ qŝ2 ˆ xx ˆ qŝ ˆ x ˆ x q x ŝ ˆ x ˆ x q x ŝ ˆ xx q xx ŝ xx n 3 n1n j n1n i n1n n k 1n k 2 n1n ji n1n2 n k 1n k 2 n1n i n1n j n k j 1n k 2 n1n j n1n i n k i 1n k n1n ji n k 2 ji 1n k 2 ji 1n k 2 qs qsqs qs qs qs qs qs qs qs − ∂ −∂ ∂ −∂ − + − ∂∂ −∂ − − − ∂ −∂ ∂ ∂ − ∂ ∂ − − ∂ −∂ ∂ ∂ − ∂ ∂ − − ∂∂ ∂ − ∂∂ ∂ = ∂∂ ∂ + ++ + + + + + ++ + ++ + + + K L (G.80) Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 173 e) History variables ji iso i iso jj iso i iso jiji n ji n ji n k kin i n k j kin i n k kin j j n k i kinn k ji kinn k i kin j j n k kin i n k j kin i n kkin jiji n k ji n ji n k i n k jj n k i n jiji n k ji n k xx H ∆γ x H x ∆γ x H x ∆γ H xx ∆γ xx α xx α xx n H∆γ x n x H ∆γ x n H x ∆γ x n x H ∆γn xx H ∆γn x H x ∆γ x n H x ∆γ n x H x ∆γ nH xx ∆γ xx q xx q xx n ∆γ x n x ∆γ x n x ∆γ n xx ∆γ xx e xx e ∂∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ ∂∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ ∂∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂∂ ∂ + ∂∂ ∂ = ∂∂ ∂ + +++ + ++ + ++ + +++ + + 22212 1211 1 1 2 1 1 11 2212 1211 1 2212 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 ~~ K K (G.81) Appendix G: Second-order Sensitivity Equations for Multiaxial J2 Plasticity 174 G.6. REFERENCES Simo, J.C., and Hughes, T.J.R. (1998). Computational Inelasticity. Interdisciplinary applied mathematics, Springer-Verlag, NY. Appendix H: Second-order Sensitivity Pseudo-code 175 Appendix H. SECOND-ORDER SENSITIVITY PSEUDO-CODE The advantage of the DDM when compared to the FDM is the efficiency of this methodology. Special concern should be devoted to the implementation of this method. Existing codes for DDM are not employed in an efficient manner especially when the number of parameters or the number of elements is large. This chapter aims to present the novelties implemented in the 1 st and 2 nd – order DDM algorithms to make this mythology highly efficient and valuable. This chapter comprises the step by step description of the DDM algorithms followed by the novel implementation strategies at each step. H.1. TOP-LEVEL 1 ST AND 2 ND – ORDER DDM ALGORITHM 1. Input data defining geometry, loading, boundary conditions, material properties and parameters for Gauss-integration points, load step size, and Newtown-Raphson tolerance. 2. Initialize some needed quantities 3. Loop over all load steps (k =1:n) a. Calculate response by attaining equilibrium using Newton-Raphson Iteration, phase 0 (H.2),{ }kU b. Calculate 1 st -order response sensitivity, phase 1 (H.5), k ∂ ∂ x U c. Update unconditional 1 st -order derivatives of the ‘future’ history variables, phase 2 (H.8) d. Calculate 2 nd -order response sensitivity, phase 3 (H.10), k 2 xx U ∂∂ ∂ e. Update unconditional 2 nd -order derivatives of the ‘future’ history variables, phase 4 (H.10) Appendix H: Second-order Sensitivity Pseudo-code 176 f. Set ‘future’ history variables as history (n+1) =history (n) 4. Return H.2. RESPONSE MODULE ALGORITHM AT LOAD STEP K (PHASE 0) 1. Begin iteration loop (iteration number i) a. Loop over all model elements i. Calculate the element stiffness matrix , [ ]ek and internal force vector { }ep using element module (H.3) ii. Assemble the element stiffness matrix [ ]ek , and internal force vector { }ep into global stiffness matrix [ ]K , and global internal force vector{ }P b. Return c. Calculate the residual as the difference of the external and internal load vectors using { } { } { }PFP −=∆ d. Compute incremental displacement by{ } [ ] { }PKU ∆=∆ −1i e. Update displacement by{ } { } { }iii UUU ∆+=+1 f. Check convergence, { } tolerance<∆P 2. If not CONVERGED GOTO 1 H.3. ELEMENT MODULE ALGORITHM 1. Initialize some declarations, { } ep and [ ]ek 2. Call Gaussian rule (Gauss points coordinates and weights),{ }GX and { }Gw 3. Loop over all Gauss points (Gauss number i =1:G) a. Get shape functions, derivatives and Jacobian matrix at Gauss point, iii j,B,N and Appendix H: Second-order Sensitivity Pseudo-code 177 b. Compute strains at Gauss point using { } [ ] { }iii uBε = c. Get material state including stress and stiffness tangent at Gauss point using material module (H.4): { }iσ and [ ]ik d. Calculate volume change by twjv iii =∆ ( where t is the element thickness) e. Calculate element internal force for residual computation by{ } { } [ ] { } iiTiee v∆+= σBpp f. Calculate element tangent stiffness using { } { } [ ] [ ] [ ] iTiiTiee v∆+= BkBkk 4. End H.4. MATERIAL MODULE ALGORITHM (2D PLAIN STRAIN) 1. Set some predefined vectors and matrices, { }m , [ ]oI , and [ ]devI 2. Set elastic constants,κ , µ , and λ 3. Convert strains to use mechanical strains by{ } { } 6:46:4 5.0 εε = 4. Calculate volumetric and deviatoric part of strains, vε and { }e 5. Calculate trial deviatoric stress: { }ŝ 6. Calculate norm of trial deviatoric stress, qs −ˆ 7. Evaluate yield function, f 8. If the step is elastic, 0≤f a. Compute stress,{ }σ b. Compute tangent stiffness, [ ]k Appendix H: Second-order Sensitivity Pseudo-code 178 c. Update ‘future’ history variables, { } 1n~ +e ,{ } 1n+q , and 1n+α 9. Else if the step is plastic, 0f > a. Compute consistency parameter, γ∆ b. Compute yield surface normal, { }n c. Compute back deviatoric stress, { }s d. Compute stress, { }σ e. Compute tangent stiffness, [ ]k f. Update ‘future’ history variables, { } 1n~ +e ,{ } 1n+q , and 1n+α End H.5. 1 ST -ORDER SENSITIVITY AT TIME STEP K (PHASE 1) 1. Loop over all model elements a. Calculate element displacement using global displacement obtained from phase 0, { }eu b. Calculate conditional 1 st -order sensitivity of element internal force matrix, e e u x p ∂ ∂ using element 1 st -order conditional sensitivity module (H.6) c. Assemble the e e u x p ∂ ∂ into conditional 1 st -order sensitivity of global internal force matrix Ux P ∂ ∂ 2. Return Appendix H: Second-order Sensitivity Pseudo-code 179 3. Compute 1 st -order displacement sensitivity using [ ] Ux P K x U ∂ ∂ = ∂ ∂ −1 H.6. ELEMENT 1 ST -ORDER CONDITIONAL SENSITIVITY MODULE ALGORITHM 1. Initialize some declarations, e e u x p ∂ ∂ 2. Call Gaussian rule (Gauss points coordinates and weights) , { }GX and { }Gw 3. Loop over all Gauss points (Gauss number i =1:G) a. Get shape functions, derivatives and Jacobian matrix at Gauss point, iii j,B,N and b. Compute strains at Gauss point by{ } [ ] { }eii uBε = c. Get conditional 1 st –order stress sensitivity matrix at Gauss point using (H.7), ε x σ i ∂ ∂ d. Calculate volume change, twjv iii =∆ ( where t is the element thickness) e. Calculate element 1st-order internal force sensitivity matrix using [ ] i i T i ee v∆ ∂ ∂ + ∂ ∂ = ∂ ∂ x σ B x p x p 4. End H.7. MATERIAL 1 ST -ORDER SENSITIVITY MODULE ALGORITHM (2D PLAIN STRAIN) 1. Set some predefined vectors and matrices, { }m , [ ]oI , and [ ]devI 2. Set elastic constants,κ , µ , and λ 3. Convert strains to use mechanical strains using{ } { } 6:46:4 5.0 εε = Appendix H: Second-order Sensitivity Pseudo-code 180 4. Calculate volumetric and deviatoric part of strains, , vε and { }e 5. If phase 1 a. Set deviatoric strain as zero, 0 x ε = ∂ ∂ 6. Else if phase 2 a. Convert 1 st –order sensitivity of strains to use mechanical strains sensitivity, :,6:4:,6:4 5.0 ∂ ∂ = ∂ ∂ x ε x ε b. Calculate volumetric and deviatoric part of 1 st –order strains sensitivity, ∂ ∂ x vε and ∂ ∂ x e 7. End 8. Extract current plastic strain, { }n~e 9. Call unconditional 1 st –order sensitivity of the current history variables and Collect them in matrices or vectors, n ∂ ∂ x e~ , n ∂ ∂ x α , and n ∂ ∂ x α 10. Calculate 1 st –order sensitivity of the elastic constants and Collect them in vectors, ∂ ∂ x κ , ∂ ∂ x µ , and ∂ ∂ x λ 11. If the step is elastic, 0≤f a. Compute conditional first-order sensitivity of deviatoric stress, εx s ∂ ∂ (phase 1) Appendix H: Second-order Sensitivity Pseudo-code 181 b. Compute conditional first-order sensitivity of final stress, εx σ ∂ ∂ (phase 1) It should be noted if the step is elastic and phase is 2, nothing is calculated in here. 12. Else if the step is plastic a. Recover some needed quantities such as i. Consistency parameter, γ∆ ii. Yield surface normal, { }n iii. Yield function, f b. Compute conditional or unconditional first-order sensitivity of trial deviatoric stress, εx s ∂ ∂ˆ (phase 1) and ∂ ∂ x ŝ (phase 2) c. Compute conditional or unconditional first-order sensitivity of yield function, εx ∂ ∂f (phase 1) and ∂ ∂ x f (phase 2) d. Compute conditional or unconditional first-order sensitivity of plastic consistency parameter, εx ∂ ∆∂ γ (phase 1) and ∂ ∆∂ x γ (phase 2) e. Compute conditional or unconditional first-order sensitivity of yield surface normal vector, εx n ∂ ∂ (phase 1) and ∂ ∂ x n (phase 2) f. Compute conditional first-order sensitivity of back deviatoric stress, εx s ∂ ∂ (phase 1) Appendix H: Second-order Sensitivity Pseudo-code 182 g. Compute conditional first-order sensitivity of final stress, εx σ ∂ ∂ (phase 1) 13. End 14. If phase 2 a. Update ‘future’ history variables 1 st - order sensitivities 15. End H.8. UPDATE 1 ST -ORDER HISTORY VARIABLES SENSITIVITY AT TIME STEP K (PHASE 2) 1. Loop over all model elements a. Calculate element displacement using global displacement obtained from phase 0, { }eu b. Calculate element 1 st - order displacement sensitivity using global 1 st - order displacement sensitivity obtained from phase 1, e ∂ ∂ x u c. Update element 1 st -order unconditional derivative of the history variables using e ∂ ∂ x u 2. Return H.9. UPDATE ELEMENT 1 ST -ORDER HISTORY VARIABLES SENSITIVITY MODULE ALGORITHM 1. Call Gaussian rule (Gauss points coordinates and weights) , { }GX and { }Gw 2. Loop over all Gauss points (Gauss number i =1:G) a. Get shape functions, derivatives and Jacobian matrix at Gauss point, iii j,B,N and b. Compute strains at Gauss point using{ } [ ] { }eii uBε = Appendix H: Second-order Sensitivity Pseudo-code 183 c. Compute strains 1 st -orderr sensitivity at Gauss point by [ ] e i i ∂ ∂ = ∂ ∂ x u B x ε d. Update history variable 1 st –order sensitivity at Gauss point using (H.7) by setting phase=2 3. End H.10. 2 ND -ORDER SENSITIVITY AT TIME STEP K (PHASE 3) 1. Loop over all model elements a. Calculate element displacement using global displacement obtained from phase 0, { }eu b. Calculate element 1 st -order displacement sensitivity using global displacement obtained from phase 1, e ∂ ∂ x u a. Calculate the conditional 2 nd -order sensitivity of element internal force matrix, e e 2 u xx p ∂∂ ∂ and assemble into global matrix. (H.11) b. Calculate ee e ∂ ∂ ∂ ∂ x u x k u , ee ∂ ∂ ∂ ∂ ∂ ∂ x u x u u k and assemble into global matrices (H.11) 2. Return 3. Compute 2 nd -order displacement sensitivity, xx U ∂∂ ∂ 2 Appendix H: Second-order Sensitivity Pseudo-code 184 H.11. MATERIAL 1 ST -ORDER SENSITIVITY MODULE ALGORITHM (2D PLAIN STRAIN) 1. Set some predefined vectors and matrices, { }m , [ ]oI , and 2. Set elastic constants,κ , µ , and λ 3. Convert strains to use mechanical strains { } { } 6:46:4 5.0 εε = 4. Calculate volumetric and deviatoric part of strains, vε and { }e 5. If phase 3 a. Set 1-order sensitivity of deviatoric strain as zero, 0 x ε = ∂ ∂ b. Set 2-order sensitivity of deviatoric strain as zero, 0 xx ε = ∂∂ ∂ 2 6. Else if phase 4 a. Convert 1 st and 2 nd -order sensitivity of strains to use mechanical strains sensitivity, i. :,6:4:,6:4 5.0 ∂ ∂ = ∂ ∂ x ε x ε ii. :,6:4 2 :,6:4 2 5.0 xx ε xx ε ∂∂ ∂ = ∂∂ ∂ b. Calculate volumetric and deviatoric part of 1 st and 2 nd -order strains sensitivity, i. ∂ ∂ x vε and ∂ ∂ x e ii. 2 xx∂∂ ∂ vε and xx e ∂∂ ∂ 2 Appendix H: Second-order Sensitivity Pseudo-code 185 7. End 8. Extract current plastic strain, { }n~e 9. Call unconditional 1 st and 2 nd -order sensitivity of the current history variables and Collect them in matrices or vectors, a. nn ∂ ∂ x e~ , n ∂ ∂ x q , and n ∂ ∂ x α b. nn xx e ∂∂ ∂ ~2 , n xx q ∂∂ ∂ 2 , and n xx∂∂ ∂ α2 10. Calculate 1 st and 2 nd –order sensitivity of the elastic constants and Collect them in vectors, a. ∂ ∂ x κ , ∂ ∂ x µ , and ∂ ∂ x λ b. xx∂∂ ∂ κ2 , xx∂∂ ∂ µ2 , and xx∂∂ ∂ λ2 11. Recover some needed quantities (Step 11-13 H.7) i. γ∆ , { }n , and f ii. εx s ∂ ∂ˆ , εx ∂ ∂f , εx ∂ ∆∂ γ , and εx n ∂ ∂ (phase 1) iii. ∂ ∂ x ŝ , ∂ ∂ x f , ∂ ∆∂ x γ , and ∂ ∂ x n (phase 2) 12. If phase 3 a. Loop over all parameters (i =1: I) Appendix H: Second-order Sensitivity Pseudo-code 186 i. Calculate ∂ ∂ ε k and then ∂ ∂ ∂ ∂ ix ε ε k ii. Calculate ∂ ∂ ix k b. End 13. Else if 14. End 15. Loop over all parameters (i =1: I) (to save 2-order sensitivity tensors in a matrix) a. Loop over all parameters (j =1:J) i. Calculate matrix seating parameter, 2/)1()1( −−+−= iijIik (using this variable the 2-order sensitivity tensors are stored in a matrix) ii. If the step is elastic 0≤f 1. Compute conditional 2 nd -order sensitivity of deviatoric and final stress, ε xx s k 2 ∂∂ ∂ and ε xx σ k 2 ∂∂ ∂ (phase 3) 2. Compute k ∂ ∂ ∂ ∂ ∂ ∂ x ε x ε ε k 3. Compute k ∂ ∂ ∂ ∂ x ε x k iii. Else if the step is plastic 1. Compute conditional or unconditional 2 nd -order sensitivity of trial Appendix H: Second-order Sensitivity Pseudo-code 187 deviatoric stress, ε xx s k 2ˆ ∂∂ ∂ (phase 3) and k 2ˆ xx s ∂∂ ∂ (phase 4) 2. Compute conditional or unconditional 2 nd -order sensitivity of yield function, ε xx k 2 f ∂∂ ∂ (phase 3) and k 2 f xx∂∂ ∂ (phase 4) 3. Compute conditional or unconditional 2 nd -order sensitivity of plastic consistency parameter, ε xx k 2 ∂∂ ∆∂ γ (phase 3) and k 2 xx∂∂ ∆∂ γ (phase 4) 4. Compute conditional or unconditional 2 nd -order sensitivity of yield surface normal vector, ε xx n k 2 ∂∂ ∂ (phase 3) and k 2 xx n ∂∂ ∂ (phase 4) 5. Compute conditional first-order sensitivity of back deviatoric stress, ε xx s k 2 ∂∂ ∂ (phase 3) 6. Compute conditional 2 nd -order sensitivity of final stress, ε xx σ k 2 ∂∂ ∂ (phase 3) 7. Compute k ∂ ∂ ∂ ∂ ∂ ∂ x ε x ε ε k (phase 3) 8. Compute k ∂ ∂ ∂ ∂ x ε x k (phase 3) iv. End b. End Appendix H: Second-order Sensitivity Pseudo-code 188 16. End 17. Else if phase 4 a. Update ‘future’ history variables 2 nd - order sensitivities 18. End Appendix I: User’s Guide Software Implementations 189 Appendix I. USER’S GUIDE TO SOFTWARE IMPLEMENTATIONS In this study, a software is developed to perform first-order sensitivity analysis for composite manufacturing problems and the second-order sensitivity analysis of inelastic finite element problem using the direct differentiation method (DDM). The software is developed in Matlab®, which is selected for several reasons: Software development in Matlab is convenient due to its user-friendly programming environment. Matlab also includes extensive, yet straightforward, debugging features. These are important issues when embarking on the comprehensive task of extending complex response algorithms with sensitivity equations. In addition, comprehensive optimization and reliability analysis are available in Matlab. Composite sensitivity analysis in conjunction with these toolboxes provides a powerful and efficient tool for reliability analysis and optimal design of composites manufacturing problems. In this section the software is discussed in detail. In addition, the link between the composites sensitivity analysis software and optimization and reliability analysis toolboxes is explained. I.1. COMPOSITES MANUFACTURING SENSITIVITY, RELIABILITY, AND DESIGN OPTIMIZATION USING DIRECT DIFFERENTIATION The Matlab-based software developed in this thesis is named “SENCOM” (SENsitivity analysis for COMposite problems). SENCOM is developed in this study to perform first-order sensitivity analysis using DDM for composite manufacturing problems. SENCOM is linked with a reliability analysis toolbox developed in Matlab to perform the reliability analysis for composites processing (SENCOM-REL). SENCOM-OPT is a software from merging SENCOM and Matlab optimization toolbox to perform optimal design for composites manufacturing problems. Matlab optimization toolbox is also linked with SENCOM-REL to perform reliability-based design optimization (SENCOM-REL-OPT). In the following section, the details of execution and the subroutines organization of the SENCOM are discussed. Next, the extension of SENCOM to carry out reliability and optimization analysis is explained. Appendix I: User’s Guide Software Implementations 190 I.1.1. First-order Sensitivity Analysis (SENCOM) SENCOM is a collection of subroutines for MATLAB to perform first-order sensitivity analysis. I.1.1.1. How to Run SENCOM After opening MATLAB, the user should change the current working directory to the directory where the SENCOM subroutines are available. This can be done at the command prompt as >> cd('D:\My work\4noded_Themral_sensitivity_dec2508') or by using the current directory field in the desktop toolbar. Before running the analysis, all the model and analysis parameters must be available in the MATLAB workspace. Therefore, an input file must be created and loaded into the workspace by giving the file name at the command prompt. (The syntax of this input file will be covered shortly.) For example: >> Input_example_Temp_stress After loading the input file, the analysis to calculate the model responses including temperature, degree of cure, displacements, and stresses as well as their sensitivities, is carried out with the command >> SENCOM I.1.1.2. How to Develop the Input File All model and analysis parameters are defined in the input file. In addition, from within the input file a call is made to a subroutine that generates the finite element mesh, using user-defined geometry parameters. The model parameters include material, cure cycle, and geometry parameters as well as convective heat transfer coefficients. In this study, the composites manufacturing simulation includes two models: thermochemical and stress development models. The thermochemical model requires thermochemical parameters of composites part as well as tool thermal parameters for temperature analysis and degree of cure development during the cure processing. The stress development model require composites part, shear layer, and tool material parameters to calculate the displacement and stresses Appendix I: User’s Guide Software Implementations 191 during the cure processing. In the thermochemical analysis it is assumed that the shear layer specification is the same as the tool. The model material parameters are utilized by the following variables: Tinit defines the initial temperature of the composites part and tool. Alphaint indicates the initial degree of cure of the composites part. Thermal_mat composites part thermochemical and tool thermal parameters are defined as following: Thermal_mat(1,:) includes all the composites part’s thermochemical parameters including fibre and resin density, heat capacity, conductivity, resin curing model parameters, and fibre volume fraction. Available composite thermochemical model includes 26 parameters (See θ3 to θ28) in Table D.1 of Appendix D. As an example: Thermal_mat(1,:)= [Ro0F aroF Ro0R aroR broR Cp0F aCpF Cp0R aCpR bCpR kk0Fl akkFl kk0Ft akkFt kk0R akkR bkkR A DE M N C C0 CT Hr vf] Thermal_mat(2,:): This variable includes tool thermal parameters including density, heat capacity, and conductivity (See θ29 to θ34 in Table D.1 of Appendix D). As an example: Thermal_mat(2,:)=[ Ro0T aroT 0 Cp0T aCpT 0 kk0T akkT zeros(1,18)] Convective heat transfer coefficients of the surrounding tool and composites part are identified by the variable “heat_tran”. For example for a flat composites part laid on a flat tool, this variable includes three (3) parameters as follows. heat_tran = [hPT hTB hTT]; where hPT, hTB, hTT are the convective heat transfer coefficients at the top of composites part, and the bottom and top of the tool, respectively. Appendix I: User’s Guide Software Implementations 192 Autoclave cure cycle parameters are defined by the variable “autoParams”. This variable defines the cure cycle’s ramps and their durations. For example a two-hold cure cycle is described by seven (7) parameters as follows. autoParams = [s1 ts1 th1 s2 ts2 th2 s3]; where s1, s2, s3 are ramp slopes; ts1 , and ts2 are the first and second ramp time durations; and th1 and th2 are the hold time durations. Subsequent to the thermochemical parameters definitions development, the stress development material parameters including composites part, tool, and shear layer are identified using the variable “Stress_mat”. As an example: Stress_mat(1,:) involves composites part stress development parameters including the modulus, Poission’s ratio, coefficients of thermal expansion of resin and fibre, and the resin cure shrinkage parameters. The available composites stress development model includes 32 parameters (See θ43 to θ74 in Table D.3 of Appendix D). As an example: Stress_mat(1,:)= [ER0 ERI TGA TGB TC1A TC1B TC2 AER NUR0 ANUR BNUR VLSI ALC1 ALC2 ASH CTER0 ACTER BCTER E1F AE1 E2F AE2 G12F AG12 NU12F ANU12 NU23F ANU23 CTE1F0 ACTE1F CTE2F0 ACTE2F] Stress_mat(2,:) involves the tool modulus, the Poisson’s ratio, and the coefficients of thermal expansion (See θ75 to θ80 in Table D.3 of Appendix D). As an example: Stress_mat(2,:)= [ET0 AET NUT0 ANUT CTET0 ACTET zeros(1,26)] Stress_mat(3,:) involves the shear layer elastic modulus parameters (See θ81 to θ83 in Table D.3 of Appendix D). As an example: Stress_mat(3,:)= [ESH1 ESH3 GSH zeros(1,29)] Appendix I: User’s Guide Software Implementations 193 Once the model’s thermochemical and stress development parameters are defined, the model’s geometry must be developed. The model geometric parameters are defined using the variable “GeoParams”. This variable includes geometric specifications of the tool and the part, which are used to generate the model geometry. For example a flat part laid on a flat tool (see example 1 in Chapter 3) is identified by four (4) geometric parameters as follows. GeoParams = [LP HT LT HP]; where LP is the length of the part; HT is the height of the tool; LT is the difference between length of the tool and part; and HP is the height of the composites part. Subsequently, the input file makes a call to a subroutine that generates the finite element mesh, utilizing the user-defined geometry parameters. The subroutine generates the model nodal coordinates and elements automatically with the geometry parameters. The thermal and mechanical boundary conditions are also assigned to the model in this subroutine. In addition, the derivatives of the nodal coordinates with respect to the geometric parameters are identified in this subroutine if geometric sensitivity analysis is required. This is facilitated by the parametric definitions of the nodal coordinates in this subroutine. Two geometry subroutines are currently available in SENCOM including the flat and L-shaped composites parts laid on the tool. The inputs to this subroutine include the geometric parameters and convective heat transfer coefficients. For example for a flat part this subroutine function is developed as follows. function [node, der_node, el,surf,nodPart,elenumT,elenumP ,fixed_dof, fixed_dofSHPA,fixed_dofPA, nelSH1, nelSH2, nelPA1, nelPA2]= Flat_Tool_Part(GeoParams, heat_tran); where the output of this subroutine includes: node is the (x-y) nodal coordinates. For example for the node number nodnum , it reads: node(nodnum,:)=[ X Y ]; X and Y coordinates are calculated based on the geometry and the meshing parameters. Appendix I: User’s Guide Software Implementations 194 der_node is the derivative of nodal coordinates with respect to the geometry parameters. For example for the node number nodnum this derivative with respect to the first geometry variable reads: der_node(nodnum,:,1)=[der_X_thet1 der_Y_theta1]; el is the element variable defining the nodes included in the element, material, and the thickness assigned to the element. For the element number elemnum, it reads: el(elenum,:)=[nodnum1 nodnum2 nodnum4 nodnum3 MatID Thick LayFor]; where nodnum1 to nodnum4 are the nodes number included in the element elenum. MatID identifies the type of material assigned to the elenum. MatID is “1” for the composites, “2” for the tool, and “3” for the shear layer elements. thick is the element thickness. LayFor defines the angles of lamina included in the composites element. It is zero for non-composites elements. surf defines the surfaces, on which the autoclave cure cycle are assigned. As an example: surf(1,:) = [nodnum1 nodnum2 heat_tran(1) 1 thick]; where nodnum1 and nodnum2 are two points, where the surface in between is assigned to the autoclave cure cycle with convective heat transfer coefficient heat_tran(1). thick is the element thickness. nodPart is a vector of the composites part node numbers. There are other variables required for the stress development analysis, including: fixed_dof is the vector of constrained degrees of freedom during the curing. fixed_dofSHPA is the vector of all tool and shear layer degrees of freedom. Appendix I: User’s Guide Software Implementations 195 fixed_dofPA is the vector of constrained composites part degrees of freedom after the tool removal. nelSH1 and nelSH2: are the beginning and ending shear layer elements numbers, with the element in between are in contact with the composites part. nelPA1 and nelPA2: are the beginning and ending elements of composites part. The analysis variables are required to be defined in the input file or the analysis will fail. The user must therefore define the following parameters. Totaltime is the total time of the analysis. numIncr is the number of the time increments. Tol is the analysis iteration tolerance. Order is the Gaussian integration order used in thermochemical analysis. It should be noted that the element integration is performed at each layer of lamina included in the composite element in the stress development analysis. Disp_flag is “1” to include stress development analysis and displacement calculation, otherwise “0”. stress_flag is “1” to include the stress calculation during the process, otherwise “0”. tool_rem_flag is “1” to remove the tool at the end of the analysis, otherwise is “0”. grad_flag is “1” to include sensitivity analysis using direct differentiation method, otherwise “0”. I.1.1.3. Outputs of SENCOM Thermochemcial analysis outputs include: Temperature is the matrix of the nodal temperature during the curing process. Sen_Temperature is a third-order tensor of the derivatives of temperature with respect to the thermochemical parameters during the curing process. Appendix I: User’s Guide Software Implementations 196 history is a collection of history variables to obtain degree of cure and its sensitivities. As an example: history(i).element(j).quadpoint(k).alpha: is the degree of cure in the kth Guess point of element j th at the time increment i th . As proved in section A.1 of Appendix A, the derivatives of the degree of cure is a linear function of the temperature sensitivity as b T a kk + ∂ ∂ = ∂ ∂ θθ α (A.3) history(i).element(j).quadpoint(k).d_alpha_Temp: is referred to “a” in the above equation and history(1).element(i).quadpoint(j).d_alpha_theta refers to “b”. Some of stress development outputs include: Displacement is the matrix of nodal displacement during the curing process. Sen_Displacement is the tensor of nodal displacement with respect to all thermochemical and stress development parameters during the curing process. Stress_part is the tensor of stresses of the composites part elements in the integration points during the curing process. Sen_Stress_part is the tensor of sensitivities of stresses of the composites part elements in the integration points during the curing process. Node_Disp_rem is the vector of total composites part displacements after the tool removal. Sen_Node_Disp_rem is the matrix of sensitivities of the total composites part displacements after the tool removal. Appendix I: User’s Guide Software Implementations 197 I.1.1.4. SENCOM Subroutines Organizations During each time increment, an iteration procedure is performed to calculate the temperature and the degree of cure in the thermochemical analysis. When the convergence is obtained, the displacements are calculated using the stress development model. At the end of each time increment, the first-order sensitivity analysis of the thermochemical model is performed to obtain the temperature and the degree of cure sensitivities. These variables are populated into the stress development model sensitivity analysis to compute the first-order sensitivity analysis of displacements developed during the curing process. At the end of the analysis, the tool is removed. The displacements and their sensitivities are subsequently computed. This process is summarized in the following six stages. Stage 1- The thermochemical analysis is performed by the following functions. create_conn_Thermal_quad4.m creates the thermal connectivity matrix for the quad elements to connect the local thermal stiffness matrices and the thermal forces to the global ones. quad4_STF_RHS_part.m calculates the thermal stiffness matrix and load vector at each of the quadrilateral composite elements. It includes the following sub-functions. gaussianQuadrature.m calculates the Gauss points and their integration weights. shapeFunctions4node.m calculates the shape functions and Jacobian matrix in each integration point. Alphafunc.m calculates the degree of cure using the curing model at the current temperature and Guess point. composite_density_heatcap.m calculates the composite density, heat capacity using the resin and fibre thermal properties, and volume fraction at the current temperature and degree of cure. Appendix I: User’s Guide Software Implementations 198 composite_conductivity.m calculates the composite conductivity matrix using the resin, longitudinal and transverse fibre conductivities. quad4_STF_RHS_tool.m calculates the local thermal stiffness matrix and the load vector for the quadrilateral tool and shear layer elements. surface_STF_RHS.m calculates the boundary convective matrix and vector using the following function. ftime13.m calculates the applied autoclave temperature at each time increment. Stage 2- The stress development analysis is performed using the thermochemical responses with the following functions. create_bound.m establishes the matrix for the boundary condition introduction. create_conn_Stress_quad4.m creates the mechanical connectivity matrix for quad elements to connect the local mechanical stiffness matrices and forces to the global ones. It should be noted that the mechanical connectivity matrix is different than the thermal one because the mechanical quadrilateral elements includes eight (8) degrees of freedom while the thermal elements includes four (4) degrees of freedom. quad4_Stress_STF_RHS_part.m calculates the local mechanical stiffness matrix and the load vector of the quadrilateral composites element in the global directions. It should be noted that the element integration is performed at each layer of lamina. . The main functions involved in this subroutine are: fibre_Resin_prop.m calculates the resin and fibre mechanical properties including modulus and Poisson’s ratio, resin and fibre coefficients of thermal expansions, resin incremental strain including thermal the shrinkage strains, and the fibre incremental thermal strain. Appendix I: User’s Guide Software Implementations 199 local_stifness_part.m evaluates the composites laminate local stiffness matrix using the resin and fibre mechanical properties and micromechanical formula in the laminate’s local direction. local_strains_part.m evaluates the laminate local strains vector using the resin and fibre incremental strains and micromechanical formula in the element’s local directions. quad4_Stress_STF_RHS_tool_shear.m calculates the local mechanical stiffness matrix and the load vector of the quadrilateral tool and the shear layer element. This subroutine includes the following functions. tool_prop.m calculates the tool mechanical properties including the modulus, the Poisson’s ratio, the tool coefficient of thermal expansion, and the tool incremental thermal strain. local_stifness_tool.m evaluates the tool’s local stiffness matrix in the Gauss integration. local_strains_tool.m evaluates the tool’s local strains vector at the Gauss integration point. local_stifness_shear.m evaluates the shear layer local stiffness matrix in the Gauss integration point quad4_forces_part_shear.m calculates the incremental force vector at the shear layer elements, which are in contact with the composite part. This load vector is used to calculate the total force applied to the composites part after the tool removal. quad4_Stress_STF_RHS_part_CAL calculates the incremental stresses in the composites part elements. quad4_Stress_STF_RHS_part_CAL calculates the incremental stresses in the tool and shear layer elements. Appendix I: User’s Guide Software Implementations 200 Stage 3- The thermochemical sensitivity is computed. In Chapter 2, it is proved that the sensitivity equations are linear equations and are calculated at the end of each time increment when the convergence is obtained. The temperature sensitivity is then computed by }{][ }{ 1 BA T −= ∂ ∂ θ k (2.23) Direct differentiation algorithm in SENCOM consists of the following subroutine to compute the temperature sensitivity matrix, which contains the nodal temperature sensitivities with respect to the thermochemical model parameters including: sen_quad4_STF_RHS_part.m calculates the ][A matrix and }{B vector in Eq. (2.23) at quadrilateral composite elements by the following functions: sen_shapeFunctions4node.m calculates the derivatives of strain-displacement matrix (B) with respect to the element nodal coordinates and the derivative of the Jacobian matrix with respect to the geometry parameters. This subroutine is developed for the geometry sensitivity analysis. sen_alphafunc.m calculates the derivative of curing modal and update the sensitivity history including “a” and “b” of Eq. (A.3) to calculate the degree of cure sensitivities as: b T a kk + ∂ ∂ = ∂ ∂ θθ α (A.3) sen_composite_density_heatcap.m calculates the conditional derivatives of the composites density and the heat capacity with respect to the thermochemical parameters with constant temperature and the degree of cure. sen_composite_conductivity.m calculates the conditional derivatives of conductivity with respect to the thermochemical parameters with constant temperature and the degree of cure. Appendix I: User’s Guide Software Implementations 201 sen_quad4_STF_RHS_tool.m calculates the contribution of tool elements in sensitivity Eq. (2.23) by computing tool elements ][A matrix and }{B vector. sen_surface_STF_RHS.m calculates the contribution of boundary elements in sensitivity equation by computing the boundary elements ][A matrix and }{B vector in Eq. (2.23) including: der_ftime13 calculates the derivatives of applied autoclave temperature with respect to the autoclave parameters. Stage 4 - The stress development sensitivity is calculated. The displacement sensitivities of the stress development model are obtained by Eq. (2-26) ∆ ∂ ∂ − ∂ ∆∂ = ∂ ∆∂ k kkk k }{ ][}{}{ ][ δ KFδ K θθθ (2.26) The algorithm to compute the right hand side of above equation consists of following subroutines: sen_quad4_Stress_STF_RHS_part.m calculates the contribution of each of the composite elements in the right hand side of Eq. (2.26) by using the following functions. sen_fibre_Resin_prop.m calculates the derivatives of the resin and fibre mechanical properties; the resin and fibre coefficients of thermal expansions, the resin incremental strain including the thermal and shrinkage strains, and the fibre incremental thermal strain with respect to the thermochemical and stress development models’ parameters. sen_stiffness_disp_part.m calculates the contribution of each of the laminate in the right hand side of Eq. (2.26) using the micromechanical formula with the fibre and resin properties, and their sensitivities. sen_quad4_Stress_STF_RHS_tool_shear.m calculates the contribution of tool and shear layer element in the right hand side of Eq. (2.26). The two main functions included in this subroutine that calculate the Eq.(2.26) in each Gauss integration point for the tool and shear layer are Appendix I: User’s Guide Software Implementations 202 sen_stiffness_disp_tool.m and sen_stiffness_disp_shear.m.These subroutines utilize the tool and shear layer stiffness matrices and their sensitivity as well as strains. sen_quad4_forces_part_shear.m calculates the sensitivities of incremental force vector in the shear layer elements, which are in contact with the composite part. These sensitivities are used to obtain the total force sensitivities that are applied to the composite part after the tool removal by the following functions. sen_stiffness_force_shear.m calculates the derivatives of shear layer incremental forces in each integration point using strains sensitivities calculated in the previous step. sen_quad4_Stress_STF_RHS_part_CAL.m calculates the sensitivity of the incremental stress in the composite part elements (Eq. 2.30). sen_quad4_Stress_STF_RHS_tool_shear_CAL.m calculates the sensitivity of the incremental stresses in the tool and shear layer elements. Stage 5 - The composite elements displacements are calculated after the tool removal. This stage contains the calculation of the composite part displacements }{U after the tool removal by solving }{][}{ 1 FKU −= . Where ][K is the composite part global stiffness matrix and }{F is the total forces vector in the shear layer degrees of freedom, which are in contact with composite part. No extra subroutines are required at this stage since ][K and }{F are calculated in the final step before the tool removal. Stage 6 - The sensitivities of the composites displacement with respect to the model parameters after the tool removal is computed. These sensitivities are obtained by solving ( ){ }}{][}{][}{ 1 UKFKU θθθ ∂∂−∂∂=∂∂ − . Where θ∂∂ }{F is the derivative of the shear layer forces at the degrees of freedom, which are in contact with Appendix I: User’s Guide Software Implementations 203 the composite part and calculated in the final step prior to the tool removal. The functions included in this stage are as follows. sen_quad4_Stress_STF_RHS_part_rem.m calculates the ( ) }{][ UK θ∂∂ at the composites part elements. This subroutine uses the element displacement calculated at Stage 5 and the composites element stiffness matrix and their derivatives in each element lamina utilizing the subroutine sen_stiffness_disp_part_rem.m. I.1.2. Reliability Analysis (SENCOM-REL) To perform finite element reliability analysis for the curing process, the reliability analysis software is linked to SENCOM to develop the SENCOM-REL. The chosen reliability software is FERUM (Finite Element Reliability Using Matlab®), which was developed at the University of California, Berkeley in the late 1990’s (Der Kiureghian et al. 2003). FERUM is a collection of functions in Matlab®. The FERUM, currently contains first- and second-order reliability method (FORM and SORM) as well as the Monte Carlo and importance sampling algorithms. In this study, the FERUM and SENCOM are linked to allow the input parameters of the SENCOM being characterized as random variables. SENCOM produces the derivatives of curing process responses using direct differentiation method (DDM). As shown in Chapter 3, this advantage increases the efficiency of the gradient-based reliability analysis such as FORM significantly. In order to describe the link between FORM and SENCOM, the subroutines used in FERUM are explained as follows. I.1.2.1. FERUM Subroutines Organizations Each subroutine/function in the FERUM is a new Matlab command. The following are the main subroutines included in FERUM to perform the FORM analysis. ferum.m is a “shell” function to run. This function is not performing any computation as it solely organizes the analysis and user can select the type of analysis (i.e. FORM, SORM, importance sampling Appendix I: User’s Guide Software Implementations 204 simulation, system analysis or inverse FORM analysis). In this study FORM is chosen and the related subroutines are discussed as follows. form.m contains the iHLRF algorithm to find the design point in addition to the post-processing to compute the first-order approximation to the probability of failure (Liu et al. 1989; Zhang and Der Kiureghian 1997). Linear transformation is used to convert correlated random variable “x” to the uncorrelated standard normal space “y”. For this purpose two subroutines, mod_corr.m and x_to_u.m, are utilized . mod_corr.m modifies the correlation matrix R according to the Nataf transformation to the R0. x_to_u.m uses L0, which is the Choleskly decomposition of R0 to transform the user-given starting point in the original space into the standard normal space. Utilizing iHLRF to find design point the iterative loop is launched: u_to_x.m Back-transforms vector “yi” from the standard normal space to “xi” in the original space. This is because the evaluation of limit-state function is only possible in the original space. jacobian.m calculates the Jacobian of the probability transformation, xyJ , . gfun.m evaluates the limit-state function, )(xg and its gradient, )(xg∇ . This is the most important subroutine in finite element reliability analysis, where the interaction between finite element and reliability software is introduced. The details are discussed later in this document. The Jacobioan matrix obtained in previous step is utilized to transform the limit-state function gradient into the normal space by T xyJxgyg ,).()( ∇=∇ .The next step is to check convergence criteria to verify if the point founded is close enough to the design point. The two convergence criteria are as below: Appendix I: User’s Guide Software Implementations 205 ;)2 ;)1 2 1 0 eyy e G G T <− < αα Where G G ∇ ∇ −=α When convergence is not achieved, a further step must be taken: search_dir.m selects search direction vector (λ ) in the search for the design point. step_size.m chooses step size (d ) in search for the design point and subsequently the new “y” is updated as, dyy mm .1 λ+=+ . I.1.2.2. How to Link FERUM and SENCOM (SENCOM-REL) The most important ingredient in a gradient-based reliability analysis such as FORM is the evaluation of the limit-state function and its gradient. There are two approaches to define the limit-state function in FERUM. Simple limit-state functions are defined in the input file. However, in case of complex limit- state functions, where for example the limit-state is a function of a finite element response, the evaluation of the limit-state is perfumed by a user defined subroutine. This subroutine defines the link between reliability analysis and finite element. Finite element software is utilized to evaluate the limit-state and its derivatives at iteration toward the design point in the reliability analysis within the defined subroutine. In addition, there are two methods available in FERUM to calculate the gradient of the limit-state function: direct differentiation method (DDM) and finite difference method (FDM). To perform FROM utilizing the DDM, the user must define the gradient of limit-state function by either an expression in the input file or in the user defined subroutine. Subsequently, in order to link the SENCOM to FERUM to perform reliability analysis for composites manufacturing problems, a user defined subroutine must be defined as user_lsf.m. The first interaction between SENCOM and FERUM is introduced in gfun.m. Therefore, following changes are required in gfun.m. Appendix I: User’s Guide Software Implementations 206 case 'matlabfile' %The limit-state function is evaluated by a user defined subroutine if grad_flag == 'no ' grad_g = 0; flag=0; % [G,dummy] = user_lsf(x,flag); elseif grad_flag == 'ffd' %to calculate the gradient of limit-state using finite difference method. flag=0; [G,dummy] = user_lsf(x,flag); parameter = probdata.parameter; original_x = x; for j = 1 : length(x) h = parameter(j,2)/2; x = original_x; x(j) = x(j) + h; [G_a_step_ahead,dummy] = user_lsf(x,flag); grad_g(j) = (G_a_step_ahead - G)/h; end elseif grad_flag == 'ddm' flag=1; % [G, grad_g] = user_lsf(x,flag); %to calculate the gradient of limit-state using direct differentiation method. else disp('ERROR: Invalid method for gradient computations'); end end A user defined subroutine should then be developed as function [g,der_g]= user_lsf(x,grad_flag); where grad_flag as explained in Section 5.2.1 is the analysis parameter indicating the DDM is performed by the SENCOM. x is vector of random variables resulting from the reliability analysis. The user_lsf is initiated by defining the SENCOM input parameters in the same order as explained in Section 5.21. If any of the material, geometry, and processing parameters of SENCOM is characterized as a random variable, the value of the parameter must be replaced by the random variable vector x. In case all the thermochemical parameters are random variables: Tinit = x(1); alphaint = x(2); Thermal_mat(1,:)= [x(3:28)]; Appendix I: User’s Guide Software Implementations 207 Thermal_mat(2,:)=[x(29) x(30) 0 x(31) x(32) 0 x(33) x(34) zeros(1,18)]; heat_tran= [x(35:37)]; % For flat composites with 3 heat transfer coefficients autoParams=[x(38:42)]; GeoParams= [x(43:346)]; % For flat composites part laid on tool This indicates that all composites and tool material parameters, convective heat transfer coefficients, processing and geometry parameters are characterized as random variables. The geometry subroutine and SENCOM analysis parameters are then called similarly as explained in Section 5.2.1. Subsequently, SENCOM.m is called to obtain the responses and their derivatives. Finally the limit-state function and its gradient are expressed at the end of user_lsf. Limit-states are algebric functions of responses. Consequently, their derivatives with respect to the responses and then the random variables are easy to calculate using the sensitivity results obtained by SENCOM. I.1.2.3. How to Develop Input File for SENCOM-REL FERUM assumes all necessary data are available in the MATLAB workspace. Then, an input file is required to be created and loaded into the workspace. The main data, which must be defined in the input file, are as followings: 1) Random variables parameters: probdata.marg is a marginal distribution for each random variable as follows. probdata.marg(1,:) = [(type) (mean) (std.dev.) (startpoint) (p1) (p2) (p3) (p4) (Input_type)]; where type indicates the probability distribution of the random variable. type is “1” for Normal distribution, “2” for Lognormal distribution, and “3” for Gamma distribution. Other types of distributions are also available in FERUM, which are not considered in this study. p1 to p4 are the probability Appendix I: User’s Guide Software Implementations 208 distribution parameters. Input_type is “0” when the distribution defined using the mean and std.dev and “1” when distribution defined using the distribution parameters pi. For example, the initial temperature is defined as a random variable with lognormal distribution with the mean, Tinit and the standard deviation, 0.05*Tinit as: probdata.marg(1,:) = [ 2 Tinit 0.05*Tinit Tinit 0 0 0 0 0]; probdata.correlation is the correlation matrix (square matrix with dimensions equal to the number of random variables) 2) Search algorithm parameters: analysisopt.ig_max is the number of global iterations allowed in the search algorithm. analysisopt.il_max is the maximum number of line iterations allowed in the search algorithm. analysisopt.e1 is the tolerance on how close design point is to limit-state surface. analysisopt.e2 is the tolerance on how accurately the gradient points towards the origin. analysisopt.step_code is “0” for step size by Armijo rule, otherwise, given value (0 < s <= 1) is the step size. analysisopt.grad_flag: is 'DDM' for direct differentiation and 'FFD' for forward finite difference to calculate the limit-state function derivative. 3) Parameters to define the evaluation of the limit-state function and its derivatives. gfundata(1).evaluator is 'basic' in this case. Other alternatives are 'FERUMlinearfecode', 'FERUMnonlinearfecode',and 'fedeas', which are not used in this study. Appendix I: User’s Guide Software Implementations 209 gfundata(1).type is 'matlabfile' to indicate that the limit-state function is evaluated by the user defined MATLAB file. The other option is 'expression' for simpler limit-state functions, which is not used in this study. gfundata(1).parameter is 'no' when the limit-state function is calculated by the user defined subroutine, or 'yes' if the limit-state function is defined by an expression and the corresponding parameters. I.1.2.4. Outputs of SENCOM-REL Following outputs variables are available in the MATLAB workspace as the results of FORM analysis: formresults.iter is the number of iterations. formresults.beta1 is the reliability index beta from FORM analysis. formresults.pf1 is the failure probability pf1. formresults.dsptu is the design point in normal space. formresults.dsptx is the design point in original space. formresults.alpha is the alpha vector. formresults.imptg is the importance vector gamma. formresults.gfcn is the recorded values of the limit-state function during search. formresults.stpsz is the a recorded step size value during search. formresults.beta_sensi_thetaf is the index beta sensitivities with respect to the distribution parameters. formresults.pf_sensi_thetaf is the probability of failure sensitivities with respect to the distribution parameters. This variable is used in the reliability-based design optimization, where the derivatives of the objective and constraints functions include the derivatives of the failure probabilities. Appendix I: User’s Guide Software Implementations 210 I.1.3. Design Optimization SENCOM, which is empowered with DDM, provides an efficient tool for gradient-based optimal design for the composites manufacturing problems. SENCOM can be linked directly to the gradient-based optimization algorithms to perform deterministic optimal design. In addition, Gradient-based optimization algorithms can also be linked to the finite element reliability analysis (FERA) to perform bi-level “reliability-based design optimization (RBDO)” as discussed in Chapter 3. As previously shown, FERA, which utilizes FROM in conjunction with SENCOM (SENCOM-REL), provides an efficient tool to calculate the composites manufacturing failure probabilities and their derivatives with respect to the random variables. These probabilities and their derivatives are in turn used in the optimization algorithm to perform RBDO. This section explains the links between the selected nonlinear optimization programming to convert the SENCOM and SENCOM-REL to SENCOM-OPT and SENCOM-REL-OPT. The preferred optimization algorithm is sequential quadratic programming (SQP) using MATLAB. I.1.3.1. Deterministic Optimization by SENCOM-OPT This section discusses the link between SENCOM and optimization toolbox in MATLAB. MATLAB toolbox provides a wide range of optimization algorithms. In this study, constrained nonlinear optimization algorithm using SQP is selected. The link between SENCOM and these algorithms contains three (3) steps as follows. Step 1- Write a M-file for the objective function and gradient: function [f,G] = objfungrad(x); where x is the vector of design variables. f is objective function. G is the derivative of objective function. Appendix I: User’s Guide Software Implementations 211 The objective function (f) may contain the composites manufacturing simulation responses. For example, one may be interested in minimizing the time of reaching to a certain degree of cure. A user defined subroutine should be developed similar to Section 5.1.2.3 to evaluate the response and its gradients. For example to minimize the time of reaching to a certain degree of cure, curcrit: [tcrit,Sen_tcrit]= user_lsf_obj(x,grad_flag,curcrit); where grad_flag is the analysis parameters, which is explained earlier. grad_flag is “1” to perform DDM. user_lsf_obj: provides the input parameters of the SENCOM and calls it similar to the reliability analysis using SENCOM in Section 5.1.2.2. The parameters, which are characterized as design variables, should be replaced by the values resulting from optimization analysis, x. In cases where the first and second ramps as well as first hold of cure cycle are design variables, it reads: autoParams = [x(1) x(2) x(3) x(4) x(5) th2 s3]; Then SENCOM is called by defining SENCOM.m in user_lsf_obj. At the end of this subroutine, an objective function is created, which is in turn a function of SENCOM responses. For example, tcrit is the time of reaching a certain degree of cure (curcrit) and is defined as the objective function (f). tcrit is in turn a function of degrees of cure obtained from SENCOM. Finally the derivatives of the objective function, which are also a function of sensitivity results provided by SENCOM, are defined. For example, Sen_tcri is the derivative of the time reaching to a certain degree of cure with respect to the all parameters. Finally, the derivatives of objective function with respect to design variables are assigned to G. As an example: G = [Sen_tcrit(38:42)]; where in this example, design variables x(1:5)corresponds to the 38 th to 42 nd parameters of SENCOM. Appendix I: User’s Guide Software Implementations 212 Step 2- Write a M-file for the nonlinear constraints and the gradients of the nonlinear constraints. function [c,ceq,DC,DCeq] = confungrad(x) where c is the inequality constraints. ceq is the nonlinear equality constraint. DC is the gradient of the inequality constraints. DCeq is the nonlinear equality constraint. If the constraints include the composites manufacturing responses, SENCOM should be called to calculate the responses and their derivatives. This can be performed by developing a user defined subroutine similar to the objective function. Step 3-Invoke the constrained optimization routine. This step includes defining initial estimate of the design variables as x0 = [2 45 60 2 35]; and defining the optimization options to indicate the types of utilized algorithms by options = optimset('LargeScale','off'); options = optimset(options,'GradObj','on','GradConstr','on'); Then, the lower and upper bounds of design variables must be defined as lb = [1.2 30 40 1.2 20]; ub = [2.8 70 80 2.8 60]; Finally optimization function is called as: [x,fval]=fmincon(@objfungrad,x0,[],[],[],[],lb,ub,@confungrad,options) the constraints are subsequently checked at x by: Appendix I: User’s Guide Software Implementations 213 [c,ceq] = confungrad(x) The reader is referred to MATLAB Optimization Toolbox manual for further details on the optimization functions including optimset.m and fmincon.m. I.1.3.2. Reliability-based Design Optimization (RBDO) using SECOM-OPT-REL This section discusses the link between the MATLAB optimization tool box and finite element reliability analysis using SENCOM. This link enables performing of bi-level reliability-based design optimization (RBDO). The optimization steps are similar to the deterministic optimization algorithm as explained in the previous Section. However, the objective function and constraints may include probability of failures. FORM analysis in conjunction with SENCOM is called at each iteration of optimization algorithm to calculate the objective function, constrains, and their gradients with respect to the design variables. This includes calling subroutine form.m to evaluate the failure probabilities either in the evaluation of objective function (objfungrad.m) or constraints (confungrad), or both. An input file must be created to define the reliability analysis input parameters where form.m is required. This is performed by: [probdata,analysisopt,gfundata,femodel,randomfield]= Rel_opt_thermal(x); Rel_opt_thermal is created similar to the reliability analysis input file as explained in Section 5.1.2.3. However the parameters, which are characterized as design variables must be replaced by x. As described in Section 5.1.2, the evaluation of the limit-stat function and its derivative is performed by the user defined subroutine, user_lsf.m. However, the objective function and constraints may include failure probabilities of different limit-state functions. This necessitates changing user_lsf.m automatically in objfungrad.m and confungrad to specify the limit-state function that should be evaluated. This is facilitated by introducing an additional variable (limit_flag) in form.m as: function formresults = form(lsf,probdata,analysisopt,gfundata,femodel, randomfield,limit_flag) Appendix I: User’s Guide Software Implementations 214 Different limit-state functions are developed in the user_lsf.m. However, limit_flag defines which one should be evaluated at each reliability analysis. Currently three (3) different limit-state functions are developed in the user_lsf.m. and they are studied in the RBDO. limit_flag is 1: where the limit state function is a function of maximum peak temperature in the composite part during the process as g = 185 - temp_max; 2: where limit state function is a function of the minimum peak temperature in the composite part as g = temp_min - 175; 3: where the limit state is a function of the time of reaching to a certain degree of cure g = 2O0-tcrit; It should be noted temp_max, temp_min, and tcrit are functions of thermochemical model responses obtained by SENCOM in the user_lsf.m. limit_flag must be introduced in any reliability analysis subroutines where the evaluation of the limit-state function and its derivatives are required. This includes subroutines such as form.m, gfun.m, and step_size. FORM analysis provides the derivatives of the failure probabilities with respect to probabilities parameters including the mean and standard deviations by the output variable, formresults.pf_sensi_thetaf. This is an important component to calculate the objective and constraints derivatives in objfungrad.m and confungrad.m. I.2. SECOND-ORDER SENSITIVITY ANALYSIS OF INELASTIC FINITE ELEMENT In this study nonlinear finite element using MATLAB is extended to calculate the first- and second-order sensitivity analysis for inelastic problems. Sensitivity equations for 1-D truss and 2-D quadrilateral plain strain elements are efficiently implemented. The uniaxial and multiaxial J2 plasticity models are Appendix I: User’s Guide Software Implementations 215 employed in conjunction with the truss and 2D four node quadrilateral elements, respectively. In this section, the developed software to compute the second-order sensitivity analysis is explained. Following steps are required to perform fist- and second-order sensitivity analysis using the nonlinear finite element software developed for this study: 1) Creating and loading an input file at the command prompt. For example for a 2-D cylinder in put file: >> cylinder 2) Performing analysis. There are two types of analysis subroutines that are available in the developed software: truss_plasticity.m performs the first- and second order sensitivity analysis for nonlinear 1-D truss elements. quad4_plasticity.m. performs the first- and second order sensitivity analysis for nonlinear 2-D quadrilateral plain strain elements. For example for the developed 2-D cylinder input file the analysis command is: >> quad4_plasticity In the following section, the requirements for developing an input file, the analysis subroutine organization are described. I.2.1. How to Develop an Input File and Perform the Analysis The developed software assumes that all the model and analysis parameters are available in MATLAB workspace. Therefore, it is required to create and load an input file before executing the analysis subroutine. The following variables are required in the input file for both truss and quadrilateral elements: node is the model nodal information. For example for node number i it reads: Appendix I: User’s Guide Software Implementations 216 node(i,:) = [ (x-coord) (y-coord) ]; el is the model elements information. For element number i, it reads : el(i,:) = [ (node1) (node2) (node3) .. (parameter1) (parameter2) .. (type)]. Available element types and their corresponding node and parameters list are: J2 plastic quad4: (node1) (node2) (node3) (node4) (E) (nu) (t) (sy) (Hi) (Hk) (PS) (4) J2 plastic truss: (node1) (node2) (E) (A) (nu) (sy) (Hi) (Hk) (5) Where: Nodei is the global node number coinciding element node i. E is the elastic modulus. A is the truss element cross area. Nu is the Poisson's ratio. T is the quadrilateral element thickness. Sy is the yield stress. Hi is the isotropic hardening modulus. Hk is the kinematics hardening modulus. PS is the “1” for plane-strain analysis and “2” for plane stress analysis. Only PS=1 option available for J2 plastic quad4 at this time. Loading is the nodal load for linear and nonlinear finite element analysis. For linear case applied for load number i is: femodel.loading(i,:) = [(node number) (magnitude) (direction) (factor)]; and for the nonlinear finite element for load number i is: Appendix I: User’s Guide Software Implementations 217 femodel.loading(i,:) = [ (node number) (magnitude) (direction) (factor) (time) (loadfactor) (time) (load factor) .. ]; where direction is: 1: x-direction 2: y-direction 3: clockwise moment A load in negative direction is indicated by a minus sign on the direction specification. nodal_spring applies the concentrated nodal spring stiffness. It is “0” if no springs are included in the model. Otherwise it is: nodal_spring(i,:) = [ (node number) (magnitude) (direction) ]; fixed_dof constraints the degrees of freedom. For example to fix nodes number 1, 2,3 , and 4: fixed_dof = [ 1 2 3 4]; id identifies where the r.v.'s enter into the finite element model: id(i,:) = [ (r.v.number) (phys.meaning) (load/node/el number) ]; where Phys.meaning is “1” for nodal load, “2” for Young's modulus (E), “3” for moment of inertia (I), “4” for Cross sectional area (A), “5” for nodal spring stiffness, “6” for Poisson's ratio (nu), “7” for thickness of 2D element, “8” for yield stress (s0), “9” for isotropic hardening (Hi), and “10” for kinematics hardening (Hk). Analysis parameters are then identified after defining the model input parameters such as nodes, elements, loadings, constraints, and etc. These parameters are identified as: numIncr is the number analysis increments. dt is time increment size. Appendix I: User’s Guide Software Implementations 218 rf is “0” since no random field is involved in the analysis. grad_flag1 is “1” to include first-order sensitivity analysis otherwise is “0”. grad_flag2 is “1” to include second-order sensitivity analysis otherwise is “0”. I.2.2. Subroutines Organization As mentioned earlier there are two types of analysis subroutines available: truss_plasticity.m for 1-D truss element and quad4_plasticity.m for quadrilateral plain strain elements. Each includes a collection of functions to perform the first- and second order sensitivity analysis for nonlinear finite element problems. This section discusses the functions utilized in the quad4_plasticity.m subroutine as an example. This subroutine starts with functions: Position.m positions the random variables. create_bound.m establishes the matrix for introduction of boundary condition. The analysis is then launched. As explained in Chapter 4, the second-order sensitivity analysis contains four stages. The functions used in each stage are described as follows. Stage Zero: Newton-Raphson iteration scheme is performed at each analysis increment to achieve convergence to equilibrium. During each iteration, the global stiffness matrix and the internal force vector are calculated by looping over all elements. The nodal displacement and resulting residual forces are then computed. If the convergence is not achieved, iteration must be repeated. The main functions in this stage are: create_conn_quad4.m creates the connectivity matrix, which is used to assemble element stiffness matrix to structure stiffness matrix by the operation: K = conn' * k * conn. tangres_quad4_plasticity.m creates plasticity stiffness matrix, internal force vector, and update the history variables for quad element. This function includes: gaussianQuadrature.m calculates the Gauss integration points and their weights. Appendix I: User’s Guide Software Implementations 219 shapeFunctions4node.m calculates the shape function, their derivatives and the Jacobinan matrix. Plastmatresponse.m calculates the stresses, stiffness matrix, and update history variables at integration points. Stage 1: If the convergence is obtained, the Stage 1 is performed at the end of the analysis increment to compute the first-order sensitivities. This includes the looping over of all elements by: fstord1_quad4_plasticity.m calculates the first-order conditional derivative of internal force for quad element (Eq. 4.5b). The main function in this subroutine is: plasmatfstorder.m calculates the first-order conditional derivative of stresses at the element integration points. Stage 2: After computing the first-order sensitivities, history variables are updated and stored. This is done by looping over all elements without the condition of fixed displacements to calculate the second- order sensitivities in the next stage by: fstord2_quad4_plasticity.m calculates and stores the first-order unconditional derivative of history variables using. It includes: plasmatfstorder.m calculates the first-order unconditional derivative of history variables at element integration points without fixing strains. Stage 3: Second-order sensitivities are then computed after calculating first-order sensitivities and updating history variables in previous stages by looping over all elements using: secord1_quad4_plasticity.m computes the contribution of each quad element to the right hand side component of Eq. (4.8b). It should be noted that efficient implementation schemes are presented to compute these components in Section 4.7. These include computing the: Appendix I: User’s Guide Software Implementations 220 1) M×I×I tensor of second-order conditional derivative of internal force ou ji xxP ∂∂∂ 2 at the element level and store it in a matrix ( )2/)1( +× IIM . 2) Triple product of the M×M×M tensor onm uK ∂∂ , the M×I matrix jo xu ∂∂ and the M×I matrix im xu ∂∂ at the element level (Eq. 4.10) and save it in a ( )2/)1( +× IIM matrix. 3) Compute the product of the M×M×I tensor ou jnm xK ∂∂ and the M×I matrix im xu ∂∂ at the element level (Eq. 4.11) and store it as a ( )2/)1( +× IIM matrix. The main function to compute above components is: Plasmatsecorder.m. calculates the following components in the integration point. 1) Second-order conditional derivative of stresses t jir xx ε σ ∂∂∂2 2) Triple product of the unconditional derivative of the stiffness matrix with respect to strains ( trsk ε∂∂ ), the derivative of strains with respect to the variables ( jt x∂∂ε ), and again the derivative of strains with respect to the variables ( is x∂∂ε ). 3) Product of the first-order conditional derivative of stiffness matrix with respect to the variables ( t jrs xk ε ∂∂ ) and the first-order derivative of the strains with respect to the variables ( is x∂∂ε |) Stage 4: When the computation of the second-order sensitivities of nodal displacements are completed, the second-order sensitivities of history variables are updated and stored without the assumption of fixed displacements for next steps by looping over all elements and using: secord2_quad4_plasticity:.m calculates and stores the second-order unconditional derivative of history variables, which include the following function. Appendix I: User’s Guide Software Implementations 221 Plasmatsecorder.m calculates the second-order unconditional derivatives of history variables in the element integration points without the assumption of fixed strains. The output of this analysis includes the nodal displacements, the stresses in the elements Gauss integration points and their first- and second order sensitivities during each analysis increment. It should be noted that the subroutine organization in the truss_plasticity.m for the1-D truss element is similar to quad4_plasticity.m for quadrilateral element. Appendix I: User’s Guide Software Implementations 222 I.3. REFERENCES Der Kiureghian, A., Haukaas, T., Hahnel, A., Sudret, B., Song, J., and Franchin, P. (2003). http://www.ce.berkeley.edu/~FERUM. Department and Environmental Engineering, University of California, Berkeley, CA. Liu, P.-L., Lin, H.-Z., and Der Kiureghian, A. (1989). CalREL User Manual. Report No UCB/SEMM- 89/18, Department of Civil and Environmental Engineering, University of California, Berkeley, CA. Zhang, Y. and Der Kiureghian, A. (1997). Finite Element Reliability Methods for Inelastic Structures. Report No. UCB/SEMM-97/05, Department of Civil and Environmental Engineering, University of California, Berkeley, CA.
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Efficient finite element response sensitivity analysis and applications in composites manufacturing Bebamzadeh, Armin 2009
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Title | Efficient finite element response sensitivity analysis and applications in composites manufacturing |
Creator |
Bebamzadeh, Armin |
Publisher | University of British Columbia |
Date Issued | 2009 |
Description | This thesis presents the development, implementation, and application of response sensitivities in numerical simulation of composite manufacturing. The sensitivity results include both first- and second-order derivatives. Such results are useful in many applications. In this thesis, they are applied in reliability analysis, optimization analysis, model validation, model calibration, as well as stand-alone measures of parameter importance to gain physical insight into the curing and stress development process. In addition to novel derivations and implementations, this thesis is intended to facilitate and foster increased use of response sensitivities in engineering analysis. The work presented in this thesis constitutes an extension of the direct differentiation method (DDM). This is a method that produces response sensitivities in an efficient and accurate manner, at the one-time cost of deriving and implementing sensitivity equations alongside the ordinary response algorithm. In this thesis, novel extensions of the methodology are presented for the composite manufacturing problem. The derivations include all material, geometry, and processing parameters in both the thermochemical and the stress development algorithms. A state-of-the-art simulation software is developed to perform first-order sensitivity analysis for composite manufacturing problems using the DDM. In this software, several novel techniques are employed to minimize the computational cost associated with the response sensitivity computations. This sensitivity-enabled software is also used in reliability, optimization, and model calibration applications. All these applications are facilitated by the availability of efficient and accurate response sensitivities. The derivation and implementation of second-order sensitivity equations is a particular novelty in this thesis. It is demonstrated that it is computationally feasible to obtain second-order sensitivities (the “Hessian matrix”) by the DDM for inelastic finite element problems. It is demonstrated that the direct differentiation approach to the computation of first- and second-order response sensitivities becomes increasingly efficient as the problem size increases, compared with the less accurate and less efficient finite different approach. |
Extent | 3829930 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-05-01 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0063127 |
URI | http://hdl.handle.net/2429/7801 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2009-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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