Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Das, Sourav
Doctor of Philosophy - PhD
University of British Columbia
This thesis presents the reliability analysis and probabilistic design of a one-storey moment-resisting frame coupled with a nonlinear energy sink (NES) with negative stiffness and sliding friction and a shape memory alloy (SMA)-based damped outrigger structure when the structures are subjected to earthquake excitation. Initially, the reliability-based design optimization (RBDO) of the above structures is carried out to estimate the tuning parameters of NES and SMA so that the proposed systems produce effective performances when exposed to uncertain environments. To do that, the probability of failure of the structure is estimated using an outcrossing rate approach based on a stationary assumption. To reduce the computational time, a surrogate model is used in the RBDO analysis. To overcome stationary assumption, the next work is devoted to formulate an efficient reliability analysis method using surrogate model-based probability density evolution method (PDEM). The PDEM is used to estimate the probability density function (PDF) of the structural responses. The partial differential equations (PDE) in PDEM are solved using the finite difference method coupled with total variation diminishing, in which a set of representative points of random parameters are generated using the generalized F-discrepancy scheme. To obtain satisfactory accuracy of the numerical solution, representative points are needed, which becomes computationally expensive for complex structures. To reduce the computation burden, the stochastic spectral embedding (SSE) surrogate model is used, which approximates the original response surface. The SSE is a class of supervised machine learning algorithm where it is trained by few observations and enables output prediction as spectral representation. In general, PDE in PDEM are solved using a finite difference scheme in which the accuracy of the numerical solution depends on the number of temporal and spatial discretizations, making them computationally inefficient for high-fidelity models. With this in view, the last work is devoted to forming a physics-informed neural network (PINN) based PDEM, for solving the PDE. This method does not need any interpolation or coordinate transformation, which is often seen in any numerical scheme, thus the computational budget is reduced. Lastly, time-dependent reliability analysis of the above structures is performed using the proposed PINN-based PDEM.
Surrogate Assisted ReliabilityAnalysis and Probabilistic Design ofStructures under UncertaintybySourav DasB.E., Jadavpur University, 2014M.Tech., Indian Institute of Technology, Guwahati, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)May 2023© Sourav Das, 2023The following individuals certify that they have read, and recommend to the College ofGraduate Studies for acceptance, a dissertation entitled:Surrogate Assisted Reliability Analysis and Probabilistic Design of Struc-tures under Uncertaintysubmitted by Sourav Das in partial fulfillment of the requirements for the degree of Doctorof PhilosophyDr. Solomon Tesfamariam, School of EngineeringSupervisorDr. Warren Hare, IKB Faculty of ScienceSupervisory Committee MemberDr. Abbas Milani, School of EngineeringSupervisory Committee MemberDr. Ahmad Al-Dabbagh, School of EngineeringUniversity ExaminerDr. Ertugrul Taciroglu, University of California - Los AngelesExternal ExamineriiAbstractThis thesis presents the reliability analysis and probabilistic design of a one-storeymoment-resisting frame coupled with a nonlinear energy sink (NES) with negative stiffnessand sliding friction and a shape memory alloy (SMA)-based damped outrigger structurewhen the structures are subjected to earthquake excitation.Initially, the reliability-based design optimization (RBDO) of the above structures iscarried out to estimate the tuning parameters of NES and SMA so that the proposed systemsproduce effective performances when exposed to uncertain environments. To do that, theprobability of failure of the structure is estimated using an outcrossing rate approach basedon a stationary assumption. To reduce the computational time, a surrogate model is usedin the RBDO analysis.To overcome stationary assumption, the next work is devoted to formulate an efficient re-liability analysis method using surrogate model-based probability density evolution method(PDEM). The PDEM is used to estimate the probability density function (PDF) of thestructural responses. The partial differential equations (PDE) in PDEM are solved usingthe finite difference method coupled with total variation diminishing, in which a set of rep-resentative points of random parameters are generated using the generalized F-discrepancyscheme. To obtain satisfactory accuracy of the numerical solution, representative pointsare needed, which becomes computationally expensive for complex structures. To reducethe computation burden, the stochastic spectral embedding (SSE) surrogate model is used,which approximates the original response surface. The SSE is a class of supervised machinelearning algorithm where it is trained by few observations and enables output prediction asspectral representation.In general, PDE in PDEM are solved using a finite difference scheme in which theaccuracy of the numerical solution depends on the number of temporal and spatial dis-cretizations, making them computationally inefficient for high-fidelity models. With this inview, the last work is devoted to forming a physics-informed neural network (PINN) basedPDEM, for solving the PDE. This method does not need any interpolation or coordinatetransformation, which is often seen in any numerical scheme, thus the computational budgetis reduced. Lastly, time-dependent reliability analysis of the above structures is performedusing the proposed PINN-based PDEM.iiiLay SummaryThe reliability-based design of structures is an emerging topic in which the structure isdesigned based on target probability of failure so that the structures produce reliable per-formance when they are subjected to uncertain dynamic loads such as winds, earthquakesetc. To do that, it is seen that the design methodology becomes computationally expensivewhen the structures are complex. Because the design needs multiple simulations of thecomplex structures. To reduce the computational cost, in this thesis, surrogate assistedreliability design of structures is used. Physics-informed neural network, a supervised ma-chine learning algorithm, is used in this thesis to solve the partial differential equation forestimating the probability density function of the performance function (such as maximumdisplacement, maximum inter-storey drift ratio of the structure). The probability of failureof structure is estimated from the obtained probability density function. The structures aredesigned by minimizing the probability of failure.ivPrefaceI, Sourav Das, prepared all the contents of this dissertation, including literature review,model development, analysis, interpretation of results, and writing of the manuscripts, underthe supervision of Dr. Solomon Tesfamariam. All of the contents of this dissertation arepublished or under review in the following journals.Journal Articles:1. S. Das, S. Tesfamariam, Y. Chen, Z. Qian, P. Tan, and F. Zhou. Reliability-based op-timization of nonlinear energy sink with negative stiffness and sliding friction. Journalof Sound and Vibration, 485:115560, 2020.– This paper includes Chapter 3 of this dissertation.2. S. Das and S. Tesfamariam. Optimization of SMA based damped outrigger structureunder uncertainty. Engineering Structures, 222:111074, 2020.– This paper includes Chapter 3 of this dissertation.3. S. Das and S. Tesfamariam. Reliability analysis of structures using probability densityevolution method and stochastic spectral embedding surrogate model. EarthquakeEngineering & Structural Dynamics, 1-18, 2023.– This paper includes Chapter 4 of this dissertation.4. S. Das and S. Tesfamariam. Multiobjective design optimization of multi-outriggertall-timber building: Using SMA-based damper and Lagrangian model. Journal ofBuilding Engineering, 51:104358, 2022.– This paper partially includes Chapter 4 of this dissertation.5. S. Das and S. Tesfamariam. Reliability assessment of stochastic dynamical systemsusing Physics Informed Neural Network based PDEM. Reliability Engineering & Sys-tem Safety. (Under Review)– This paper includes Chapter 5 of this dissertation.6. S. Das and S. Tesfamariam. State-of-the-Art review of Design of Experiments forPhysics Informed Neural Networks. Engineering Applications of Artificial Intelligence.(Under Review)v– This paper includes Chapter 2 of this dissertation.Book Chapter:1. S. Das and S. Tesfamariam. Reliability Based Design Optimization of Damped-Outrigger Timber Structure Using Stochastic Spectral Embedding Based ProbabilityDensity Evolution Method. In Seismic Isolation, Energy Dissipation and Active Vi-bration Control of Structures: 17th World Conference on Seismic Isolation (17WCSI),pp. 273-284. Cham: Springer International Publishing, 2023.– This paper includes Chapter 4 of this dissertation.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.1 Nonlinear Energy Sink . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Outrigger Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Research Gaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Research Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Chapter 2: Design of Experiments and Surrogate Models . . . . . . . . . . 102.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Overview of Design of Experiments . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Full and Fractional Factorial Design . . . . . . . . . . . . . . . . . . 132.2.2 Central Composite Design . . . . . . . . . . . . . . . . . . . . . . . . 14viiTABLE OF CONTENTS2.2.3 Box-Behnken Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.4 Doehlert Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.5 Optimal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.6 Orthogonal Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.7 Random Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.8 Quasi-Random Design . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.9 Full Grid Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.10 Sparse Grid Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.11 Rotational Sparse Grid Design . . . . . . . . . . . . . . . . . . . . . 272.3 Overview of Surrogate Models . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.1 Polynomial Chaos Expansion . . . . . . . . . . . . . . . . . . . . . . 282.3.2 Stochastic Spectral Embedding . . . . . . . . . . . . . . . . . . . . . 292.3.3 Universal Kriging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.3.4 Polynomial Chaos Based Kriging . . . . . . . . . . . . . . . . . . . . 322.3.5 Support Vector Regression . . . . . . . . . . . . . . . . . . . . . . . 322.3.6 Multilayer Perceptron . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3.7 Physics-Informed Neural Network . . . . . . . . . . . . . . . . . . . 352.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.1 Example 1: Viscous Burger’s Equation . . . . . . . . . . . . . . . . . 382.4.2 Example 2 : Shro¨dinger Equation . . . . . . . . . . . . . . . . . . . . 412.4.3 Example 3: 1D - Heat Equation . . . . . . . . . . . . . . . . . . . . 422.4.4 Example 4: Allen - Cahn Equation . . . . . . . . . . . . . . . . . . . 452.4.5 Example 5: Korteweg-de Vries Equation . . . . . . . . . . . . . . . . 472.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Chapter 3: Reliability Based Design of Structures using Level CrossingFormulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Level-Crossing Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Estimation of Probability of Failure . . . . . . . . . . . . . . . . . . . . . . 543.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.1 Example 1: Reliability-based Optimization of Nonlinear Energy Sinkwith Negative Stiffness and Sliding Friction . . . . . . . . . . . . . . 553.4.2 Example 2: Reliability-based Optimization of SMA-based DampedOutrigger Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84Chapter 4: Reliability Analysis of Structures using Probability DensityEvolution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Principle of Probability Conservation . . . . . . . . . . . . . . . . . . . . . . 874.2.1 Time Independent System . . . . . . . . . . . . . . . . . . . . . . . . 874.2.2 Time Dependent System . . . . . . . . . . . . . . . . . . . . . . . . . 88viiiTABLE OF CONTENTS4.3 Liouville Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4 Dostupov–Pugachev Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 894.5 Generalized Density Evolution Method . . . . . . . . . . . . . . . . . . . . . 904.6 Estimation of Reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.7 Finite Difference Method with TVD . . . . . . . . . . . . . . . . . . . . . . 934.8 Selection of Representative Points . . . . . . . . . . . . . . . . . . . . . . . 934.9 Numerical Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.10 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.10.1 Example 1: Four-branch Problem . . . . . . . . . . . . . . . . . . . . 974.10.2 Example 2: 10-storey Shear Building . . . . . . . . . . . . . . . . . . 974.10.3 Example 3: One-storey Frame with NES . . . . . . . . . . . . . . . . 1004.10.4 Example 4: SMA-based Damped Outrigger Timber Building . . . . 1014.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113Chapter 5: Reliability Assessment of Deteriorating Structures using PhysicsInformed Neural Network based PDEM . . . . . . . . . . . . . . 1155.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.2 Numerical Algorithm for PINN-based PDEM Framework . . . . . . . . . . 1165.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.1 Example 1: Van-der-Pol Oscillator . . . . . . . . . . . . . . . . . . . 1185.3.2 Example 2: One-storey Frame with NES . . . . . . . . . . . . . . . . 1225.3.3 Example 3: SMA-based Damped Outrigger Timber Building . . . . 1245.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127Chapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.1 Summary of the Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Scope of Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134ixList of TablesTable 2.1 A comparison of number of samples generated using factorial, centralcomposite, Box-Behnken and Doehlert design for different factors . . 16Table 2.2 Orthogonal array, OA(4, 3, 2, 2) . . . . . . . . . . . . . . . . . . . . . 17Table 3.1 Design parameter values for the present NES . . . . . . . . . . . . . 62Table 3.2 Statistical Properties of uncertain variables . . . . . . . . . . . . . . 63Table 3.3 Definitions of the evaluation criteria . . . . . . . . . . . . . . . . . . 64Table 3.4 Performance of proposed NES representing peak value of the floordisplacement under El Centro earthquake record . . . . . . . . . . . 65Table 3.5 Performance of proposed NES representing RMS value of the floordisplacement under El Centro earthquake record . . . . . . . . . . . 67Table 3.6 Performance of proposed NES representing peak value of the flooracceleration under El Centro earthquake record . . . . . . . . . . . . 67Table 3.7 Performance of proposed NES representing RMS value of the flooracceleration under El Centro earthquake record . . . . . . . . . . . . 67Table 3.8 Comparison of fundamental periods for Euler-Bernoulli and Timo-shenko beam models . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Table 3.9 Parameters of Kanai-Tajimi Spectrum . . . . . . . . . . . . . . . . . 79xList of FiguresFigure 1.1 Force displacement hysteresis diagrams for (a) Cubic NES and (b)NES with negative stiffness . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.2 SMA-based damped outrigger structure . . . . . . . . . . . . . . . . 6Figure 1.3 A schematic diagram shows objectives of this thesis . . . . . . . . . 8Figure 2.1 A schematic diagram shows the commonly used Design of Experi-ments and surrogate models . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.2 Commonly used DoE sampling strategies . . . . . . . . . . . . . . . 12Figure 2.3 2-levels (a) full and (b) fractional factorial design . . . . . . . . . . 13Figure 2.4 Central composite design; (a) Circumscribed, (b) Inscribed and (c)Faced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.5 Box-Behnken design . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.6 Doehlert design for (a) two and (b) three factors . . . . . . . . . . . 16Figure 2.7 (a) Orthogonal array and (b) Orthogonal array-based Latin hyper-cube design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Figure 2.8 (a) Gibbs sampling and (b) Metropolis-Hastings sampling . . . . . . 18Figure 2.9 Various sampling strategies for quasi-random design . . . . . . . . . 22Figure 2.10 Full tensor product grid for different levels for two and three factors 25Figure 2.11 Grid points generated using sparse grid scheme . . . . . . . . . . . . 27Figure 2.12 Support points generated using (a) sparse grid and (b) rotationalsparse grid scheme for two dimensional problem . . . . . . . . . . . 29Figure 2.13 Domain decomposition upto level of expansion M= 2 and Nm = 2m 30Figure 2.14 Schematic diagram of ANN . . . . . . . . . . . . . . . . . . . . . . . 34Figure 2.15 Schematic diagram of physics informed neural network . . . . . . . 36Figure 2.16 The MSE history for different levels of DoE for solving ViscousBurger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 2.17 Absolute error between original and predicted responses of viscousBurger’s equation considering different DoE strategies . . . . . . . . 39Figure 2.18 Viscous Burger’s equation: (a) Exact solution; (b) Solution usingPINN and (c) Absolute error between exact solution and solutionusing PINN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Figure 2.19 The MSE values for different number of DoE sets for different DoEtechniques for Shro¨dinger equation . . . . . . . . . . . . . . . . . . . 42xiLIST OF FIGURESFigure 2.20 Absolute error between original and predicted responses of Shro¨dingerequation considering different DoE strategies . . . . . . . . . . . . . 42Figure 2.21 Shro¨dinger equation: (a) Solution for u using PINN and (b) Absoluteerror between exact solution and solution using PINN . . . . . . . . 43Figure 2.22 Heat conduction in a rod . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 2.23 The MSE values for different number of DoE sets for different DoEtechniques for heat equation . . . . . . . . . . . . . . . . . . . . . . 44Figure 2.24 Absolute error between original and predicted responses of heat equa-tion considering different DoE strategies . . . . . . . . . . . . . . . 44Figure 2.25 (a) Solution for u using PINN and (b) Absolute error between exactsolution and solution using PINN for heat conduction equation . . . 45Figure 2.26 The MSE values for different number of DoE sets for different DoEtechniques for Allen-Cahn equation . . . . . . . . . . . . . . . . . . 46Figure 2.27 Absolute error between original and predicted responses of Allen-Cahn equation considering different DoE strategies . . . . . . . . . 47Figure 2.28 (a) Solution for u using PINN and (b) Absolute error between ex-act solution and solution using PINN for reaction - diffusion systemrepresented by Allen-Cahn equation . . . . . . . . . . . . . . . . . . 47Figure 2.29 The MSE values for different number of DoE sets for different DoEtechniques for KdV equation . . . . . . . . . . . . . . . . . . . . . . 48Figure 2.30 Absolute error between original and predicted responses of KdVequation considering different DoE strategies . . . . . . . . . . . . . 49Figure 2.31 (a) Solution for u using PINN and (b) Absolute error between exactsolution and solution using PINN for solitary waves represented byKdV equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Figure 3.1 Schematic diagram of level crossing . . . . . . . . . . . . . . . . . . 53Figure 3.2 (a) Schematic representation of moment resisting frame and (b) NESwith negative stiffness and sliding friction . . . . . . . . . . . . . . . 56Figure 3.3 Flowchart of the reliability-based design optimization . . . . . . . . 60Figure 3.4 The test structure installed on the shake table . . . . . . . . . . . . 61Figure 3.5 Prototype of negative stiffness device with sliding friction . . . . . . 61Figure 3.6 Comparison of experimental and analytical primary structure dis-placement time history under El Centro with PGA 0.15g; (a) Uncon-trolled and (b) Controlled responses . . . . . . . . . . . . . . . . . . 62Figure 3.7 Plot showing convergence of reliability index with different samplesize (nr) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 3.8 Nonlinear response surface for structure - NES with negative stiffnesssystem by minimizing top floor displacement of primary structureobtained using PCK . . . . . . . . . . . . . . . . . . . . . . . . . . . 64Figure 3.9 Primary structure response under El Centro earthquake record withPGA 0.15g (a) displacement and (b) acceleration time histories . . 65xiiLIST OF FIGURESFigure 3.10 NES Response under El Centro earthquake record with PGA 0.15g(a) Relative drift time history of NES w.r.t. primary structure; (b)Control force - NES displacement hysteresis . . . . . . . . . . . . . 66Figure 3.11 Fourier amplitude spectrum for (a) Primary structure displacementand (b) Relative drift of NES w.r.t. primary structure under ElCentro earthquake record with PGA 0.15g . . . . . . . . . . . . . . 66Figure 3.12 Energy dissipation for different stiffness ratio of primary structureunder different PGA values of El Centro earthquake record for (a)0.05g to 0.10g and (b) 0.15g to 0.20g . . . . . . . . . . . . . . . . . 68Figure 3.13 Peak displacement reduction of primary structure for different stiff-ness ratio of primary structure under different PGA values of ElCentro earthquake record (a) 0.05g to 0.10g and (b) 0.15g to 0.20g 68Figure 3.14 RMS displacement reduction of primary structure for different stiff-ness ratio of primary structure under different PGA values of ElCentro earthquake record for (a) 0.05g to 0.10g and (b) 0.15g to 0.20g 69Figure 3.15 Schematic representation of SMA based damped outrigger structure 70Figure 3.16 Typical stress-strain-temperature hysteresis behavior of SMA . . . . 73Figure 3.17 Simplified model for super-elastic SMA proposed by Yan and Nie,compared with Greasser -Cozzarelli model . . . . . . . . . . . . . . 74Figure 3.18 Force displacement hysteresis for SMA; (a) Classical flag shaped and(b) After stochastic equivalent linearization . . . . . . . . . . . . . . 75Figure 3.19 Flowchart for the efficient global optimization . . . . . . . . . . . . 78Figure 3.20 Plan view of 40-storey building . . . . . . . . . . . . . . . . . . . . . 79Figure 3.21 Modal frequencies of undamped outrigger structure for different lo-cations of outrigger . . . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 3.22 (a) Inter-storey drift and (b) inter-storey drift angle for Euler-Bernoulliand Timoshenko beam model . . . . . . . . . . . . . . . . . . . . . . 80Figure 3.23 (a) Nonlinear response surface obtained from Kriging ; (b) Compar-ison of failure probability between efficient global optimization anddeterministic optimization . . . . . . . . . . . . . . . . . . . . . . . 82Figure 3.24 (a) PSDF and (b) Time histories of top floor displacement for thebare structure, undamped outrigger and SMA based outrigger systemat the optimal design values . . . . . . . . . . . . . . . . . . . . . . 83Figure 3.25 Comparison of ; (a) Interstorey drift and (b) Interstorey drift anglefor bare structure, traditional undamped and SMA based outrigger 83Figure 3.26 Comparison of ; (a) Storey shear force and (b) Overturning momentfor bare structure, traditional undamped and SMA based outrigger 84Figure 3.27 Probability of failure for different (a) threshold level and superstruc-ture damping and (b) seismic intensity with different level of uncer-tainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84xiiiLIST OF FIGURESFigure 4.1 GF-discrepancy-based representative points for (a) standard uniformand (b) standard normal distribution . . . . . . . . . . . . . . . . . 95Figure 4.2 Flowchart for reliability analysis using PDEM . . . . . . . . . . . . 96Figure 4.3 (a) Contour plot of four failure domains with the training samples,(b) Subdomains of the training samples in SSE and (b) Degree ofaccuracy corresponding to increasing number of subdomain in SSE 98Figure 4.4 PDF of (a) virtual stochastic process,V(τ) and (b) equivalent extremeevent, YEEV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 4.5 (a) PDF of equivalent-extreme event, YEEV; (b) Failure probabilityfor different threshold values . . . . . . . . . . . . . . . . . . . . . . 100Figure 4.6 (a) PDF of equivalent-extreme event, YEEV; (b) Failure probabilityresponses corresponding to design variables . . . . . . . . . . . . . . 102Figure 4.7 (a) Displacement time history of the primary structure and (b) Amountof energy primary structure dissipates before and after optimization 103Figure 4.8 Schematic representation of SMA-based damped outrigger structure 103Figure 4.9 Wide-column effect of core and outrigger . . . . . . . . . . . . . . . 108Figure 4.10 Nonlinear hysteresis of SMA . . . . . . . . . . . . . . . . . . . . . . 110Figure 4.11 Response spectra of selected ground motions and target responsespectra for 2% probability of exceedance in 50 years . . . . . . . . 111Figure 4.12 (a) PDF of equivalent extreme event and (b) Probability of failuresurface corresponding to design parameters . . . . . . . . . . . . . . 114Figure 5.1 Flowchart for reliability analysis using PINN-based PDEM . . . . . 118Figure 5.2 Propagation of joint PDF between displacement and velocity of os-cillator estimated from FPK Equation;(a) t = 0 s, (b) t = 20 s, (c)t = 40 s and (d) t = 60 s . . . . . . . . . . . . . . . . . . . . . . . . 121Figure 5.3 Stationary solution of marginal PDF of (a) displacement and (b)velocity estimated from FPK Equation and exact solution . . . . . 122Figure 5.4 Comparison between the solution using PINN-based PDEM, FPKequation and MCS for the marginal PDF of the displacement re-sponse of the oscillator at time instants: (a) t = 0 s, (b) t = 20 s, (c)t = 40 s and (d) t = 60 s . . . . . . . . . . . . . . . . . . . . . . . . 123Figure 5.5 (a) PDF of the virtual stochastic process, V(τ); (b) Convergencehistory of PINN; (c) PDF of equivalent extreme event, YEEV whenage of the structure is 0, 10 and 20 years and (d) Time dependentprobability of failure of the structure . . . . . . . . . . . . . . . . . 125Figure 5.6 (a) PDF of equivalent extreme event, YEEV and (b) Time dependentprobability of failure of the outrigger structure . . . . . . . . . . . . 127xivList of AbbreviationsBBD Box-Behnken designBLUP Best Linear Unbiased PredictionCCD Central composite designCDF Cumulative distribution functionCLT Cross laminated timberCVT Centroidal voronoi tesselationDoE Design of ExperimentEEV Equivalent extreme-eventFORM First order reliability methodFPK Fokker-Planck-KolmogorovGDEE Generalized density evolution methodGF-discrepancy Generalized F-discrepancyLHS Latin hypercube samplingMCS Monte Carlo simulationMLP Multilayer perceptronMPP Most probable pointMSE Mean squared errorNES Nonlinear energy sinkPCE Polynomial chaos expansionPCK Polynomial chaos based KrigingPDE Partial Differential EquationPDEM Probability density evolution methodPDF Probability density functionPGA Peak ground accelerationPINN Physics-informed neural networkRBDO Reliability based design optimizationRMS Root mean squareSGD Stochastic gradient descentSMA Shape memory alloySORM Second order reliability methodSSE Stochastic spectral embeddingSVR Support vector regressionTVD Total variation diminishingUK Universal KrigingxvList of Symbols⌈·⌉ Ceil of a functionE[·] Expectation operator∇ Gradient operatorΨ(·) Basis functionϕ(·) Activation functionδ(·) Dirac-delta functionα Threshold value of a performance functionn Number of random variablesσx Root mean square value of displacementN Total number of samples generatedT Duration of a random processM Mass of the structureC Damping of the structureξ Damping ratio of the structureK Stiffness of the structured Design variablesNX Number of crossing a random process above threshold valuepXY (x, y) Joint probability density function between X and YPf Probability of failureβˆ Reliability Indexη Learning rate∆ DeflectionΘ Random variablessgn(·) Signum functionκ(·) Kernel functionL(·) Loss functionH(·) Heaviside functionPr(·) Probability operatorSgg Power spectral density functionS0 Spectral intensity of bedrock motionsωg Natural frequency of soil layerηg Critical damping ratio of soil layerx¨g Ground accelerationxviAcknowledgmentsI take this opportunity to express my sincere thanks and gratitude to my thesis supervi-sor,Dr. Solomon Tesfamariam, for his invaluable guidance, support, and encouragementduring this research. His willingness to guide and his flexible timing have been of great helpto me in the completion of this thesis. His suggestions and encouragement have provided agood basis for the present thesis. I would also like to thank Natural Sciences and Engineer-ing Research Council of Canada under the Discovery Grant programs (RGPIN-2019-05584)and the BC Forestry Innovation Investment’s (FII) Wood First Program for providing thefunds to conduct this study.Sincere thanks to the members of the Doctoral Committee, Dr. Warren Hare, Dr. AbbasMilani and my supervisor for their critical reviews, suggestions and encouragement whichdirected me in making this work a worthy effort.I would like to thank Dr. Yangyang Chen, Dr. Zhichao Qian, Dr. Ping Tan and Dr.Fulin Zhou for providing me the experimental results which are used in this thesis.I would also like to thank School of Engineering, The University of British Columbia(Okanagan) for providing me the opportunity and facilities for the study.I would also like to thank my seniors and friends, especially Dr. Souvik Chakraborty,Dr. Kostas Skandalos, and Dr. Edel Martinez, for their help. Finally, I would like to thankmy parents and other family members for their support and encouragement.xviiI dedicate this thesis to my parents,who wished the Dr. title for me when I was a high school student.xviiiChapter 1Introduction1.1 GeneralDue to the increase in complexity of engineered structures, reliability-based structuraldesign becomes an essential tool to satisfy the desired level of reliability, including the safetyof the system when the structure is exposed to an uncertain environment. These uncertain-ties may occur due to the variability of the components of the structure, the uncertaintyin loading (wind, earthquake, etc.), the assumptions considered in the numerical modeling,etc. Broadly, it is classified into two different categories, which are aleatory and epistemicuncertainty. The aleatory uncertainty cannot be predicted as these are random events. Onthe other hand, epistemic uncertainty is seen due to a lack of information. Different struc-tural design schemes are established, i.e., working stress design method, limit state method,plastic design method, etc., in which the different safety factors are formulated consider-ing aleatory and epistemic uncertainty. The structural design is broadly classified into twocategories: deterministic design and probabilistic design. In deterministic design, the de-sign variables are fixed-valued variables, whereas in probabilistic design, those variables areassumed to be random variables.In probabilistic design, the safety of the structure is evaluated by performing an op-timization of a performance function based on the probability of failure of the structuralsystem. For those, the design variables involved in the design process are considered to beuncertain. The propagation of uncertainty through the design variables are the intensity ofloads acting on the structure, the geometrical properties, the material strength, etc. Basedon the design’s requirements, such as accuracy, importance, etc., different levels of relia-bility assessment are performed, such as Level 1, Level 2, and Level 3 reliability methods.Level 1 reliability method computes the reliability at the structural element level. Basedon that, different partial safety factors are evaluated that are used in the limit state designspecified in Canadian design codes. The Level 2 reliability method evaluates the safety ofthe structure based on performance function at a specific point of interest. In this method,the failure probability of the system or element is computed at that point. Lastly, theLevel 3 reliability method computes the reliability of the complete system. Besides thiscategorization, the reliability analysis can be categorized as time-invariant and time-variantreliability analysis, which are the basic themes of this thesis.11.2. BACKGROUND1.2 BackgroundNumerous reliability methods have been proposed to compute the failure probability,which are categorized into four sub-groups. These are analytical approximation method, nu-merical sampling-based method, surrogate-based method, and numerical integration method.The analytical approximation method includes the first- and second-order reliability meth-ods (FORM and SORM), which approximate the performance function about the mostprobable point (MPP) by using first- and second-order Taylor expansions [HM00]. How-ever, this method produces erroneous results when multiple MPPs are present or the per-formance function is nonlinear. Monte Carlo simulation [Shi72], subset simulation [AB01],importance sampling [ER93], line sampling [PSKC07], etc. are common and robust nu-merical sampling-based methods compared to FORM and SORM. These methods are com-putationally expensive for high-fidelity models or when estimating low failure probabilitiesbecause they require a large number of model evaluations. To reduce the computationbudget, surrogate-based methods have been attractive in recent decades. Surrogate modelsreplace the original performance function using a few observations. There are many surro-gate models available in the literature. Among them, Kriging [GTS14], polynomial chaosexpansion [MS18], radial basis function [MA10], support vector machine [BDL11], artifi-cial neural network [CTN+15] are popular. The numerical integration method estimatesthe failure probability by integrating the probability density function (PDF) of the perfor-mance function over the failure domain. The PDF is estimated under statistical moments,which include central moments [ZL07], fractional moments [IT03], point estimation method[LHX+16], univariate dimension reduction method [RX04], multiplicative-dimension reduc-tion method [ZP13], cubature formulation method [XL17], etc.Numerous numerical integration methods have been developed to estimate the response’sPDF for the stochastic system. For instance, the probability transformation method wasproposed by Falsone and Settineri [FS13] to estimate the PDF of the responses of linearstructures. They also extended their work to include random parameters and non-Gaussianexcitation [SF14, FS14]. Frank et al. [Fra05] studied the transition PDF of structuralresponses of a nonlinear system subjected to a Gaussian white noise process using theFokker–Planck–Kolmogorov (FPK) equation. However, due to the complexity of the FPKequation, i.e., the FPK equation is coupled with the state-space form of the system, theexact stationary solution for most of the nonlinear structural systems with a large numberof degrees of freedom is unavailable and limited to some specific systems [To00, Zhu06].The Monte Carlo simulation is one of the oldest methods for nonlinear systems, but it iscomputationally expensive and hence unfeasible for high-fidelity models.To address the issue related to the complexity of the FPK equation, Li and Chen [LC08]proposed the probability density evolution method (PDEM), in which a generalized densityevolution equation (GDEE) is formulated based on the principle of probability conservation.This method is useful for estimating the PDF of nonlinear multi-degrees of freedom systemsand their reliability analysis [LC04, Li16]. The advantage of the GDEE compared to theFPK equation is that the evolution equation is completely decoupled from the physical21.3. LITERATURE REVIEWsystem. Xu and Feng [XF19] investigated the stochastic response analysis of a 15-storeynonlinear shear building frame under non-stationary ground motion using PDEM. Thequantification of the uncertainties of the random parameters in a structure was studiedby Chen and Wan [CW19] using PDEM. Wan et al. [WCLA20] developed an efficientPDEM algorithm based on change of probability measure for the time-variant reliabilityassessment of deteriorating structures. Wang et al. [WSL21] proposed a reproducing kernelparticle method based PDEM for stochastic analysis of dynamic systems. The reliabilityanalysis of the shell structure coupled with a viscoelastic damper was studied by Xu etal. [XXY20] where the probabilistic response of the structure was estimated using PDEM.Xian et al. [XSG21] showed the seismic reliability analysis of the structure coupled withbuckling restrained braces using a hybrid method of PDEM and an explicit time-domainmethod. Feng et al. [FXXQ20] investigated the robustness of the reinforced concretestructure subjected to progressive collapse using PDEM. Seismic liquefaction analysis ofearth dams was shown by Huang and Xiong [HX17] using PDEM. Ang et al. [ADLF21]investigated the life-cycle cost-based optimal design of the structural system using PDEM.In the above literature related to PDEM, GDEE is solved using the finite differencemethod with total variation diminishing (TVD), proposed by Li and Chen [LC09] wherea proper strategy is required to generate the representative points of the random variablespresent in the system. The purpose of using a strategy for generating points is that the repre-sentative points should be covered over the entire probability space, which helps PDEM esti-mate PDF accurately within the distribution domain. There are numerous sample strategiesavailable, which include the number-theoretical method [LC07], the tangent sphere method[CL08], the quasi-symmetrical point method [XCL12], the GF discrepancy-based method[CYL16], and the partially stratified sampling-based method [XF19]. Among them, theGF discrepancy-based method is the most common and popular for estimating PDF usingGDEE. Though GDEE estimates PDF accuracy, it requires a large number of representa-tive points, which becomes inefficient for time-variant reliability analysis [WC16, JWWH18]and reliability-based design optimization [KHM17, HMW+19] of high-fidelity models. Toreduce the computational burden, different researchers have proposed the efficient PDEMcoupled with surrogate models. For example, Zhou and Peng [ZP22a, ZP22c] proposed anactive learning function for an adaptive polynomial chaos kriging-based PDEM. A Gaussianprocess regression-based PDEM was studied by Peng et al. [PZL21] for high-dimensionalstructural reliability analysis. Reliability analysis of the nonlinear structures using PDEMcoupled with a surrogate model can be found in [NPS16, ZP21, ZP22d, ZP22b].1.3 Literature ReviewIn this thesis, the reliability analysis is performed on two different structures: a nonlinearenergy sink (NES) with negative stiffness and sliding friction equipped at top of a steelmoment resisting frame and a shape memory alloy (SMA) based damped outrigger structure.The literature reviews of these structures are presented in the following subsections.31.3. LITERATURE REVIEW1.3.1 Nonlinear Energy SinkNonlinear Energy Sink (NES), a passive vibration absorber, consists of a mass attachedto a nonlinear spring [Gen01, GLP07, MGM+03, Vak01]. NES absorbs energy over a broadfrequency range and is less prone to detuning due to stiffness degradation of the primarystructure. A tuned mass damper (TMD), on the other hand, is tuned close to the natu-ral frequency of the structure and is prone to detuning [RC17]. Various NES types, withdifferent nonlinear restoring forces, have been proposed. Type I NES [LWH+14] utilizes amass coupled with the primary structure and dissipates the energy through linear viscousdamping and a nonlinear spring. The nonlinear spring property was employed by introduc-ing cubic stiffness [MKK+05], Fig. 1.1. The hysteretic behavior of the cubic NES is realizedby using a pair of elastic springs, vertically aligned with the moving mass by the hinges.In this position, the springs are neither elongated nor compressed. Thus, when the massmoves in the horizontal direction, the restoring force in the NES follows the cubic function[VGB+08]. Type II NES [WWSJL14, WWSJL15] utilizes nonlinear viscous damping anda nonlinear spring that is attached to the primary structure. Type III NES is two TypeI NES connected in series [SQVB12]. Bumper NES, also known as Vibro-Impact NES, isemployed for large-scale structures where the nonlinear restoring force is considered a non-smooth zig-zag function [Ibr09]. Al-Shudeifat et al. [ASWQ+13] investigated effectivenessof NES to reduce vibration on the large-scale structures by using elastomeric bumpers. Inthe above-mentioned NES, only cubic stiffness was employed.The configuration of Type I NES is modified by introducing negative linear and nonlinearstiffness (Fig. 1.1). Al-Shudeifat et al. [AS14] proposed modified Type I NES by introduc-ing negative linear stiffness considering geometric nonlinearities. The elastic springs aredesigned to be compressed when they are aligned in the vertical position. Due to this ar-rangement, the NES assumes bi-stable equilibrium positions where the springs are neitherstretched nor compressed, i.e., at u = ± uc as shown in Fig. 1.1. However, with this arrange-ment, instabilities such as distortions of hinged springs in the negative stiffness region arepossible [CQC+19]. Chen et al. [CQC+19] proposed an NES device with negative stiffnesscoupled with sliding friction. The sliding friction is provided by introducing sliding tracksand guiders, which are designed to carry the NES mass.1.3.2 Outrigger StructureOutrigger beams, which connect a building’s core(s) with perimeter columns (Fig. 1.2),contribute to an increase in the lateral stiffness of the structural system [SS81]. To reduceexcessive force demand on the perimeter columns and outrigger beams, Smith and Wil-ford [SW07] utilized a viscous damper. Deng et al. [DPLX14] investigated effectivenessof hysteresis-based damped outrigger system. Ding et al. [DWW18] investigated seismicenergy dissipation mechanism, optimal position of the outrigger, and the viscous damper’sparameters. Beltran et al. [MBTYP18] carried out parametric study on outrigger loca-tion, damping ratio, and rigidity ratios of core-to-outrigger and core-to-column. Jiang etal. [JLZ17] used buckling restraining brace (BRB)-based energy dissipation outrigger sys-41.3. LITERATURE REVIEWFigure 1.1: Force displacement hysteresis diagrams for (a) Cubic NES and (b) NES withnegative stiffnesstem. Lin et al. [LTM18a], for an outrigger beam incorporating BRB, investigated optimaloutrigger location and the relationship between outrigger truss flexural stiffness and BRB’saxial stiffness with yield roof drift ratio. Mostly conventional bracing, energy dissipators,and BRB have been reported in the literature. Under severe earthquake, however, dueto the hardening of the BRB outrigger, local buckling induced deterioration and damagesare observed [LLC+19]. To overcome this problem, Lu et al. [LLC+19] proposed a novelsacrificial energy dissipation outrigger system for enhancing seismic performance. Besidesthe passive damped outrigger system, innovative control strategies and new materials havegained popularity. For example, Asai et al. [AW17] proposed an outrigger-tuned inertialmass electromagnetic transducer system for tall buildings under long-period earthquake ex-citation. Liu et al. [LTM+18b] proposed a rotational inertia damper for seismic mitigationof outrigger structures with good displacement amplification. In a resiliency-based designframework, the functionality of the structure in the post-earthquake is of importance. Withexcellent recovery potential, a SMA-based damper outrigger structure is considered.SMA has the ability to recover its initial state while it undergoes a large strain (up to51.3. LITERATURE REVIEWαHHCore StructurePerimeter ColumnOutriggerẍgSMA M kcForceksመ𝐤sƸ𝐜sDisplacementEcAc EcAcEIe eFigure 1.2: SMA-based damped outrigger structure10%) either by changing the temperature or by inducing stress. The process associated withthe change in temperature to capture the self-centering (i.e., residual strain becomes zero)property of SMA is known as the shape memory effect [TNT95]. On the other hand, theprocess related to the stress-strain loading-unloading phenomenon is termed superelasticityor pseudoelasticity [DMSW90]. This behavior is widely used in passive vibration controlapplications [ASM+16, QZ17]. De Domenico et al. [DGQ20] proposed an SMA-based gapdamper for an isolation system that was placed in parallel with a classical pendulum system.This configuration introduces additional stiffening and energy dissipation to the isolationsystem only when the displacement exceeds a certain threshold, or gap displacement, whilenot being engaged for lower displacements, thus avoiding residual displacements for low-to-moderate earthquakes. Mulay and Shmerling [MS21] investigated the optimal location ofSMA-based damper for the seismic retrofit of unsymmetrical-plan yielding shear-frames andconcluded the effectiveness of SMA dampers at the storey level experiencing the maximuminter-storey drift. The above-mentioned literature motivates the study of the enhancedperformance of outrigger systems using SMA springs, which are installed at the junction ofthe outrigger and the column. It is seen that in general, the timber structures experience alarge residual inter-storey drift [LDWH17, LWH+19, DHCL21]. To minimize the residualdeformation of the timber structure and to improve energy dissipation, an SMA spring isused in this study, which may help to minimize the repair costs at post-earthquake. Thereare various types of SMA springs available for different applications, such as ring springs[FYLZ15, FWR+18], coil springs [HKM19], washer springs [FZO+16, FLZ+20], etc. SMA-61.4. RESEARCH GAPSbased coil springs are widely used [TT91, LR97, FCAL12, Koh18]. The other uses of SMAfor mitigating dynamic force-induced vibration may be found in [DCM00, LMO08, LWA21].For efficient and reliable design, optimization of the outrigger system (location of out-rigger(s) and damper properties) is needed. Chen and Wang [CW09] exploited the naturalfrequencies and damping of a damped outrigger structure using the assumed mode method.Chen et al. [CMW+10] proposed, for a single damper, a characteristic equation for optimallocation and size of the damper. Tan et al. [FTCZ15] proposed a general solution for thedynamic characteristics and structural responses of tall buildings equipped with multipledamped and undamped outriggers by combining general rotational stiffness and dynamicstiffness methods. Hoenderkamp et al. [Hoe08] investigated the optimum location of thesecond outrigger by fixing the first outrigger with the objective of the top floor maximumdrift reduction. Zeidabadi et al. [ZMM04] studied the optimal location of outriggers usedto stiffen coupled shear walls by minimizing the top floor drift of the structure. Zhou etal. [ZZL16] studied the optimal location of outriggers, considering up to three outriggersystems, by minimizing inter-storey drift under wind and earthquakes. Fang et al. [Fan12]studied the dynamic reliability analysis for the damped outriggers with viscous dampers orBRB installed in high-rise buildings.1.4 Research GapsResearch Gaps, to which this thesis is called to provide answers are:− The effectiveness of reliability based estimation of the optimal design variables of theNES with negative stiffness for the steel portal frame and the SMA-based outriggerstructure subjected to stationary Gaussian white noise excitation on the structuralperformance.− The National Building Code (NBC), 2015 in Canada, provides design guidelines forup to six-storey timber buildings. This thesis is aimed to provide an alternative designstrategy of high-rise timber building by incorporating SMA-based outrigger structure.− This thesis is aimed to provide a numerical framework which describes how to estimatethe PDF of the structural responses for the non-stationary earthquake excitation toevaluate the probability of failure from limit state function.1.5 Research ContributionThe principal goal of this study is to address the probabilistic design of the NES withnegative stiffness for the moment-resisting frame and SMA-based outrigger structure. Todo that, reliability analyses of these structures are performed. The traditional reliabilityanalysis is time-consuming for nonlinear structures. To reduce the computational burden,a surrogate model is used. In this context, the following contributions are made:71.6. ORGANIZATION❏ The first contribution is the development of a closed-form solution to the probabilityof failure using the level-crossing formulation for the stationary process. The optimaldesign parameters are obtained by minimizing the closed-form solution. Also, a gen-eral numerical model of the SMA-based timber outrigger building is formulated usinga Lagrangian formulation.❏ The second contribution is the estimation of the PDF of the structural responsesusing the probability density evolution method (PDEM). This is a general methodthat overcomes the drawbacks considered in the first contribution.❏ The third contribution is the estimation of the failure probability of deterioratingstructures in which the differential equations involved in PDEM are solved by machinelearning techniques (physics-informed neural networks).Figure 1.3: A schematic diagram shows objectives of this thesis1.6 OrganizationThis thesis shall be presented in a number of chapters, starting with an introductionand ending with conclusions and future works. Each of the chapters will have a precisetitle reflecting the contents of the chapter. A chapter can be subdivided into sections andsubsections so as to present the content discretely and with due emphasis. The chaptersand their content are divided as follows:❏ Chapter 1 contains introduction of the problem followed by literature review, theresearch contributions and organization of the report.81.6. ORGANIZATION❏ Chapter 2 contains a comprehensive review of the design of experiments. Differenttypes of surrogate models are described briefly in this chapter with only the relevantequations. This chapter also includes the performance of physics-informed neuralnetworks for different design of experiment schemes.❏ Chapter 3 contains details formulation of estimation of probability of failure of struc-tures using level-crossing formulation. The closed-form solution is derived based ona stationary assumption. The reliability-based design optimization coupled with thesurrogate model is carried out for two structures: a one-storey frame coupled with anonlinear energy sink and sliding friction and an SMA-based damped outrigger struc-ture. The optimization is performed under ground motion and the system’s parameteruncertainties. This chapter also includes details on the formulations of the governingequations of the motions of those structures.❏ Chapter 4 describes the complete overview of the reliability analysis of the stochasticsystem, in which a description of PDEM and its derivation is presented. This chapteralso includes a description of the estimation of first-passage reliability using PDEM,the selection of representative points, and a numerical method for solving PDEMcoupled with surrogate model. In this chapter, the reliability analysis of the abovestructures is performed using surrogate model-based PDEM.❏ Chapter 5 provides a brief description of a physics-informed neural network and thedetails of the proposed framework of PINN-based PDEM. In this chapter, the struc-tures mentioned in previous chapters are assumed to be a deteriorating structureswhose material properties deteriorate with time, which turns the problem into a time-variant reliability analysis.❏ Chapter 6 concludes the thesis. The thesis ends with Future works where areas ofpossible research and investigation are identified.9Chapter 2Design of Experiments andSurrogate ModelsQuantifying uncertainties in a physical system is an active area of research in the sci-ence and engineering fields. In design optimization under uncertainty, Monte Carlo simu-lation (MCS) is the oldest and most widely used technique. The MCS, however, requiresa large number of model evaluations and is computationally costly for high-fidelity mod-els. Surrogate-assisted models, developed using Design of Experiment (DoE) (Fig. 2.1),are attractive options. Once the training data set through the DoE is generated, differentmethodologies are used to build the surrogate model. The surrogate models minimize thenoise between actual and predicted responses using fitting tools, based on the knowledge ofDoE samples. The purpose of using DoE is that the samples should cover the entire designspace; otherwise, predicted responses will be erroneous.2.1 Objectives∗ This chapter provides a comprehensive review of the design of experiments and sur-rogate models.∗ In particular, this chapter demonstrates the necessity of the design of experimentschemes for the Physics-Informed Neural Network (PINN), which belongs to the su-pervised learning class.∗ To establish the importance of choice of DoE scheme, five different PDEs are used fornumerical purposes, which are viscous Burger’s equation, Shro¨dinger equation, heatequation, Allen-Cahn equation, and Korteweg-de Vries equation.2.2 Overview of Design of ExperimentsExperiments used to be performed by trial-and-error to find sample (or training sam-ples or support) points [Mon17]. Fisher [Fis26] introduced the concept of a DoE strategyfor investigating the probabilistic behavior of agricultural crop systems in the early 20thcentury. The DoEs are used in the engineering field [KO96, SWNW03] and are gainingpopularity due to the reduction in computational costs. The DoE can be grouped intotwo categories, such as Classical DoE [AM74, LA93, MGH03, Ant14, Mon17] and Mod-ern DoE [SSW89, CTBA02, Cra02, SWNW03]. The DoE samples are generated using two102.2. OVERVIEW OF DESIGN OF EXPERIMENTSSurrogate ModelMethodologyDoE Fitting ToolEntropySequential ModellingBackpropagationBLUPLog-likelihoodRegressionMulti-point approx.Space FillingRandomClassicalQuasi-RandomEnsemble SurrogateANNRadial BasisSVRMARSDecision TreePCEPCE-KrigingHDMRKrigingResponse SurfaceFigure 2.1: A schematic diagram shows the commonly used Design of Experiments andsurrogate modelsapproaches. Those are domain-based (or non-adaptive or model-free) and response-based(or adaptive, or model-based, or sequential) approaches. In the domain-based approach,the DoE samples are generated based on the information obtained from the design space,whereas in the response-based approach, the samples are chosen from the information ob-tained from the surrogate models. In this chapter, extensive reviews on generation of DoEsample strategies are presented. The study is mainly focused on four different samplingschemes: deterministic sampling (classical DoE), space-filling sampling, random sampling,and quasi-random sampling. The DoE schemes are shown in Fig. 2.2 and further discussedbelow.112.2. OVERVIEW OF DESIGN OF EXPERIMENTSDoEQuasi-Random SamplingFaure SamplingHalton SamplingSobolSamplingHammersley SamplingCVTMaximin Latin HypercubeLatin HypercubeOrthogonal ArrayUniform DesignRandom SamplingMCMC SamplingGibbs SamplingMetropolis Hastings SamplingClassicalSamplingDeterministic SamplingSpace-filling SamplingSimple GridsOptimal DesignCentral Composite DesignDoehlertDesignBoxBehnken DesignFactorial DesignFullFactorial DesignFractional Factorial DesignFigure 2.2: Commonly used DoE sampling strategiesIn the deterministic scheme, the sample points are generated uniformly over the designspace, and the number of sample points is also fixed with the aim of minimizing the randomerror of the simulation. Therefore, the samples are generated in such a manner that most ofthe samples are located at the boundary of the design space, with few samples in the interiorspace. There are various types of classical DoE schemes available. Among them, factorialdesign [Mon17], central composite design [BW51], Box-Behnken design [FBF+07], optimaldesign [KW59], etc. are common. Space-filling sampling also lies within the classical DoEapproach. This sampling technique is mainly used to enhance the accuracy of the surrogatemodels. Initially, a surrogate model is built with a few samples, and that model is updatediteratively by adding a sample to the previous DoE set used until satisfactory accuracy isachieved. In this category, simple grids, Latin hypercube [HD03], orthogonal array [HSS99],and uniform design [Li02] are the most popular DoE generation schemes.Random sampling and quasi-random sampling lie within the modern DoE approach. Inrandom sampling, Markov chain Monte Carlo (MCMC) sampling [KTB13], Gibbs sampling[Gel00], and Metropolis-Hastings sampling [CG95] are popular. The samples are generatedwith an equally likely probability of occurrence within the design space. This techniqueis suitable only when the proper knowledge of design space is available because clusteringor poor coverage regions of samples have been observed for high-dimensional problems.In quasi-random sampling, the popular schemes are centroidal Voronoi tessellation (CVT)[DFG99], maximin Latin hypercube [JMY90], Sobol sampling [Sob67], Halton sampling[Hal60], Hammersley sampling [WLH97], Faure sampling [Fau82], etc.122.2. OVERVIEW OF DESIGN OF EXPERIMENTS2.2.1 Full and Fractional Factorial DesignFactorial design is one of the oldest techniques for the design of experiment samples[Mon17]. Every input variable is discretized into discrete levels. Fig. 2.3(a) shows a two-level full factorial design. Every input variable is bound by lower and upper limits, whichare the minimum and maximum values of the variable, expressed by (-1) and (+1). For athree-level factorial design, a sample point is taken at the center of the variable, which is themean value of the input variable. More concisely, a two-level design takes sample points atthe lower and upper limits of the variable, whereas a three-level design takes sample pointsat the lower, upper, and center. In general, the number of runs or simulations required isexpressed as 2n or 3n for two- and three-level designs, respectively, where n is the numberof input variables.(a) (b)Figure 2.3: 2-levels (a) full and (b) fractional factorial designThough full factorial design is easy to use, it has some drawbacks, as the number of runsor simulations depends on the number of input variables considered. For high-dimensionalproblems, the number of runs increases with an increasing number of variables, whichbecomes computationally expensive for high-fidelity models. The fractional factorial designlowers the computational cost because fewer sampling points are generated than in the fullfactorial design. For fractional factorial design, the number of samples is expressed as 2n−kand 3n−k, for two- and three-level design, respectively, where k represents the fraction ofthe full factorial design. Fig. 2.3 depicts a 23 design, i.e. two-level and three factors for fulland fractional factorial design. Though factorial design is a commonly used DoE scheme,it fails to capture the nonlinearity of the system responses as it considers only the lower,center, and upper values of the variables.132.2. OVERVIEW OF DESIGN OF EXPERIMENTS2.2.2 Central Composite DesignThe central composite design (CCD), proposed by Box and Wilson [BW51], is a variantof factorial design where additional sampling points are considered at the center and axialdirections. Fig. 2.4 shows the two-level CCD for three factors. The extremity samplesare located at the lower and upper limits of each variable, which is the same as in a fullfactorial design. Additionally, two axial points for each variable and the center point ofthe hypercube are considered. For a two-level central composite design, the number ofsimulations required for a given number of input variables is 2n + 2n + 1. The CCD are(a) (b)(c)Figure 2.4: Central composite design; (a) Circumscribed, (b) Inscribed and (c) Faced142.2. OVERVIEW OF DESIGN OF EXPERIMENTScategorized into three types: circumscribed (CCC), inscribed (CCI), and faced (CCF). Inthe CCC design, the axial points are located at the outside of the cube, where the cubepoints take (-1) and (+1) values. The CCI design is the same as the CCC design, but scaledso that the axial points are taken (-1) and (+1) values and the cube points are located insidethe cube. Unlike in CCC design, where axial points are located on the outside of the cube,in CCF design, the axial points are placed on the face of the cube. Fig. 2.4 shows all threetypes of CCD for three factors.2.2.3 Box-Behnken DesignThe Box-Behnken design (BBD) [FBF+07] is similar to CCD, where the sample pointsare located at the midpoint of each edge and one at the center of the hypercube. It does nothave any corners or extreme points. Fig. 2.5 illustrates the Box-Behnken design for threefactors. A comparison of DoE samples obtained from all types of CCD and BBD is shown inTable 2.1. It is seen that BBD requires a smaller number of simulations compared to otherCCDs. Like factorial design and CCD, it also suffers from the “curse of dimensionality”for high-dimensional problems as the number of simulations increases with the increasingnumber of input variables, which affects the computational cost.Figure 2.5: Box-Behnken design2.2.4 Doehlert DesignDoehlert [Doe70] proposed another efficient design of experiment scheme that takesfewer simulations than CCD and BBD. This strategy has been proven more economicaland highly efficient than other deterministic DoE sample strategies [Hib12]. The maindifference between this design and CCD and BBD is that the samples generated from theDoehlert design are not rotatable due to the number of estimations for different inputvariables or factors. At the same time, this design is efficient in filling the space uniformly.For two factors, samples are generated by forming a circular domain. For three factors,152.2. OVERVIEW OF DESIGN OF EXPERIMENTSit becomes a spherical domain, and for four or more factors, it becomes a hypersphericaldomain [BSO+08]. This design allows the total number of simulation as n2+n+1, where nis the number of factors. Fig. 2.6 illustrates the Doehlert design for two and three factors.Table 2.1 shows the comparison of the number of samples generated using factorial, centralcomposite, Box-Behnken, and Doehlert designs for different factors.-1 -0.5 0 0.5 1-1-0.500.51(a) (b)Figure 2.6: Doehlert design for (a) two and (b) three factorsTable 2.1: A comparison of number of samples generated using factorial, central composite,Box-Behnken and Doehlert design for different factorsNo. of FactorsNo. of Runs/ SimulationsFactorial CCD BBD Doehlert2 9 9 - 73 27 15 13 134 81 25 25 215 243 43 41 316 729 77 61 437 2187 143 85 578 6561 273 113 732.2.5 Optimal DesignThe optimal design, proposed by Kiefer and Wolfowitz [KW59], is one of the commonclassical DoE strategies in which the optimality of any design is achieved by minimizing thestatistical parameters, such as the variance of the predicted responses. Later, a number of162.2. OVERVIEW OF DESIGN OF EXPERIMENTSoptimal criteria were developed by Elfving [Elf52], Kiefer [Kie61], Bondar [Bon83], namedas ϕ-optimality criteria. The advantages of optimal design are that it requires a smallernumber of samples than other classical DoE approaches because it involves optimization ofthe predicted responses. The applications of this strategy can be found in [Bre91, Puk06,PSP93, JCS05].2.2.6 Orthogonal ArraysThe orthogonal array, also known as the Taguchi method [Tag59, KLF91, Owe92,HSS99], is one of the most popular classical design of experiment schemes. An orthog-onal array is expressed as OA (N, sm11 , sm22 , . . . , smpp , β) of strength β and dimension N ×mwhere 0 ≤ t ≤ m and m = m1+m2+ · · ·+mp. Every column (i.e., mi) has si ≥ 2 elementssuch that for any strength β, all possible combinations of the elements appeared equally inthe matrix. For example, OA(4,3,2,2) (sometimes expressed as OA4(23)) is the 4/23 = 1/2fraction of a 23 full factorial design. Table 2.2 displays OA(4,3,2,2) with the elements (0,0),(0,1), (1,0) and (1,1) appearing in the matrix only once. The orthogonal array for threefactors is shown in Fig. 2.7.Table 2.2: Orthogonal array, OA(4, 3, 2, 2)Row No.Column No.1 2 31 0 0 02 0 1 13 1 0 14 1 1 02.2.7 Random DesignRandom design, where the probability of occurrence of all samples is equal, is one of thepopular methods for generating samples. Popular methods under random design are simplerandom sampling [Gen03, FLS05], Markov Chain Monte Carlo sampling, Gibbs sampling,Metropolis-Hastings sampling, etc. [Liu01, RC04, KTB13]. The main drawback of thismethod is that complete knowledge of the design space i.e., the distribution of randomvariables is needed. Also, this method shows poor coverage regions or clustering for high-dimension problems. Fig. 2.8 illustrates the Gibbs sampling and the Metropolis-Hastingsampling for two factors.2.2.8 Quasi-Random DesignThe quasi-random design has been widely used as a DoE sampling strategy. The sam-ples generated using this scheme are neither fully random nor regular (like in classicalDoE schemes). The advantages of a quasi-random design are that it uses the flexibility172.2. OVERVIEW OF DESIGN OF EXPERIMENTS(a) (b)Figure 2.7: (a) Orthogonal array and (b) Orthogonal array-based Latin hypercube design-4 -2 0 2 4-4-2024(a)-4 -2 0 2 4-4-2024(b)Figure 2.8: (a) Gibbs sampling and (b) Metropolis-Hastings samplingof random sampling techniques as well as the advantages of a grid scheme (as used inclassical DoE designs). The samples using a quasi-random design are uniform for high-dimensional problems, and they are also statistically dependent. For quasi-random design,a various methods are available, such as Latinized Centroidal Voronoi Tesselation sampling[DFG99, RBGP06], maximin Latin Hypercube sampling (LHS) [Ste87, CHHW13, Via16],Sobol sampling [BF88, Owe98, SAKK11], Halton sampling [CMW05, WCC16], Hammer-182.2. OVERVIEW OF DESIGN OF EXPERIMENTSsley sampling [HA03, DW09], Faure sampling [Fau82], which are briefly discussed in thefollowing subsections.Centroidal Voronoi Tesselation SamplingCentroidal Voronoi tesselation (CVT) is commonly used as a space-filling DoE samplingtechnique in applied mathematics, engineering, and computer science field [DFG99]. Inthis technique, the generators, i.e., a set of points and a distance function, are considered.A Voronoi tesselation is a subset of the points which are closer to one of the generatorsthan other generators [JDG02]. CVT is the tesselation when the generators coincide withthe center of mass of each subset. In general, the generators become the centroids of theVoronoi cell, in which the region of the cell is defined by the subset. The algorithm ofgenerating sampling using CVT begins with selecting location of the generators chosenrandomly. The actual location of the generators is estimated iteratively and summarized inAlgorithm 1. The convergence criteria of this iteration are usually taken as the differencebetween the centroids obtained in two successive iterations. The samples generated usingCVT are shown in Fig. 2.9(a).Minimax Latin Hypercube SamplingLHS is a popular DoE sampling strategy [Via16] for building surrogate models. Togenerate N samples using LHS, all n dimensions are divided into N equally spaced intervals,and a coordinate value is randomly selected from each interval along a dimension, yieldingNcoordinate values in all n dimensions. Later, a sample point consisting of n coordinate valuesis built by randomly selecting one coordinate from each dimension. Thus, a two-dimensionalregion called the Latin square is formed for every sample. Details of the conventional LHSstrategy are shown in Algorithm 2. The main drawback of LHS is that there is no guaranteethat the samples generated cover the entire design space uniformly. For this reason, minimaxcriteria [JMY90], which minimum distance between two samples is maximized, is usedduring LHS sample generation. Algorithm 3 illustrates the pseudo-code for the minimaxLHS strategy. The samples generated using minimax criteria are shown in Fig. 2.9(b).Sobol SamplingThe Sobol sequence is commonly used quasi-random design, proposed by Sobol [Sob67].The samples are generated using Sobol sequence are drawn from a special binary fraction(vji ) of length w bits where i = 1, 2, . . ., w and j = 1, 2, . . ., n. n is the dimension of theproblem. The numbers vji are known as direction numbers. To generate direction numbersof dimension j, a primitive polynomial over the field F2 with elements {0,1} is considered,which is expressed aspj(x) = xq + b1xq−1 + · · ·+ bq−1x+ 1 (2.1)192.2. OVERVIEW OF DESIGN OF EXPERIMENTSAlgorithm 1 Pseudo-code for generation of CVT Sampling [JDG02]1: Aim: Generate N samples over the n-dimensional space which are located at centroidsof the centroidal Voronoi tesselation.2: Define the number of iterations(NIter)and the number of random points generated ineach iteration (m).3: Set the parameters α1, α2, β1 and β2 which define how the centroids are updated inevery iteration. Also, α2, β2 > 0, α1 + α2 = 1 and β1 + β2 = 1.4: Choose an initial set of random samples, xi, i = 1, 2, . . . , N which are assumed centroidsover the domain.5: Set pi = 1, i = 1, 2, . . . , N , which represents the number of times every centroid isupdated.6: Begin7: for Iteration = 1 to NIter do8: Generate the random samples yk, k = 1, 2, . . . ,m according to pre-defined probabil-ity density function, ρ(x)9: for i = 1 to n do10: Collect the set Wi of all sample yk which are closest to xi and estimate theirmean ui.11: if Wi = ϕ (i.e. null set) then12: Do nothing13: else14: Update the centroid, xi as,xi =(α1pi + β1)xi + (α2pi + β2)uipi + 1, pi = pi + 115: end if16: end for17: If the convergence criteria is satisfied, i.e., the difference of centroids obtained in twosuccessive iterations is less than tolerance, terminate.18: end for202.2. OVERVIEW OF DESIGN OF EXPERIMENTSAlgorithm 2 Pseudo-code for generation of basic Latin Hypercube Sampling1: Aim: Generate N samples over the n-dimensional space.2: Let xij is the j-th coordinate of the i-th sample, xi.3: Begin4: for j = 1 to n do5: Generate Pj , a random permutation of the set {1, . . ., N}.6: end for7: for j = 1 to n do8: for i = 1 to N do9: Generate xij =Pj(i)−Uj(i)n , where Uj is the uniform random number in between[0,1].10: end for11: end forAlgorithm 3 Pseudo-code for generation of Minimax Latin Hypercube Sampling [Kam21]1: Aim: Generate N samples over the n-dimensional space in which the minimum distancebetween samples is maximized.2: Define the number of iterations(NIter)and the number of interchanges for every iter-ation (m).3: Begin4: for Iteration = 1 to NIter do5: Generate N samples using basic Latin hypercube sampling, shown in Algorithm 2.6: Assume p1 and p2 are the indices of samples with minimum distance.7: for i = 1 to m do8: Select index random either p1 or p2.9: For a random integer, rcol in [1, . . ., n], and a random row, rrow in [1, . . ., N ],interchange the values in column rcol, row index, and rrow.10: If the minimum distance of samples is increased, accept the interchanges andupdate the values of p1 and p2.11: end for12: If the minimum distance between samples is smaller than the previous iteration,terminate.13: end for212.2. OVERVIEW OF DESIGN OF EXPERIMENTS-1 -0.5 0 0.5 1-1-0.500.51(a) Centroidal Voronoi Tesselation-1 -0.5 0 0.5 1-1-0.500.51(b) Minimax Latin Hypercube-1 -0.5 0 0.5 1-1-0.500.51(c) Sobol Samples-1 -0.5 0 0.5 1-1-0.500.51(d) Halton Samples-1 -0.5 0 0.5 1-1-0.500.51(e) Hammersley Samples-1 -0.5 0 0.5 1-1-0.500.51(f) Faure SamplesFigure 2.9: Various sampling strategies for quasi-random design222.2. OVERVIEW OF DESIGN OF EXPERIMENTSThe direction numbers, vji in the dimension j are generated using the following recurrencerelation, is given byvji = b1vji−1 ⊕ b2vji−2 ⊕ · · · ⊕ bq−1vji−(q−1) ⊕ vji−q ⊕(vji−q/2q); i > q (2.2)In the above equation, the notation ⊕ represents the bitwise XOR. Finally, in the dimensionj, the Sobol sequence is written asxjN = a1vj1 ⊕ a2vj2 ⊕ · · · ⊕ awvjw (2.3)where N =∑wi=0 ai2i. The coefficient ai is the random number in between [0,1]. Fig. 2.9(c)illustrates the samples generated using the Sobol sequence.Halton SamplingThe Halton sampling, proposed by Halton [Hal60], is a quasi-random design. Consider,first n prime numbers as m1, m2, . . ., mn. The Halton sequence for n-dimensional spacecan be expressed asxN =(Φm1(N), . . . ,Φmi(N), . . . ,Φmn(N))(2.4)where Φmi(N) is the i-th radical inverse function, which is expressed in the mathematicalform asΦmi(N) =l(i)∑j=0aj(i,N)m−(j+1)i (2.5)In the above equation, aj is the integer coefficient, and it is given as aj(i,N) ∈ [0,mi − 1].Also, the integer terms N and l are given byN =l(i)∑j=0aj(i,N)mji , l(i) =⌈logmi N⌉(2.6)The samples generated using the Halton sequence are only bounded in between [0,1]n whichare spread more evenly within the design space compared to random sampling. Fig. 2.9(d)shows the samples generated using the Halton sequence. It is noted that the samples inFig. 2.9(d) are scaled considering the bound [-1,1].Hammersley SamplingThe Hammersley sampling is a popular quasi-random design, similar to the Halton se-quence. The Hammersley sequence differs from the Halton sequence in that the Hammersleysequence employs the (n-1), Φ sequence. The Hammersley sequence for n-dimensional spaceis expressed asxN =(Nn,Φm1(N), . . . ,Φmi(N), . . . ,Φmn−1(N))(2.7)232.2. OVERVIEW OF DESIGN OF EXPERIMENTSThe primary distinction between the Halton and Hammersley sequences is that theHalton sequence can generate an infinite number of samples (i.e., infinite N), whereas theHammersley sequence requires an upper bound on the number of samples. Fig. 2.9(e) showsthe samples generated using the Hammersley sequence.Faure SamplingThe Faure sequence, which is similar to the Halton sequence and was proposed by Faure[Fau82], is a well-known quasi-random sample generation technique. Consider, m is the firstprime number such that m ≥ N , where N is the total number of samples. Also, the upperbound of the sample size is mp. This is the main difference between the Faure and theHalton sequences. For example, if N = 50 i.e., 50 samples are required to generate, the lastHalton sequence (i.e., in dimension 50) uses the 50-th prime number, which is 229, whereasthe Faure sequence uses the first prime number after 50. The main advantage of usingFaure sampling over Halton sampling is that the Faure sequence is faster to fill the gapsfor high-dimensional problems. Also, it prevents correlation problems in high dimensions,which are seen in the Halton sequence.Let cij =(ij)(mod m) where 0 ≤ j ≤ i ≤ p. This implies {cij − (ij)} is multiple of m.The base m representation of n is expressed asn =p−1∑i=0ai(n)mi (2.8)where the coefficients ai ∈ [0,m) take integer values. The samples produced by the Fauresequence are defined byxN =p−1∑j=0aj(N)m−(j+1) (2.9)where the coefficient aj =∑p−1l=j cljal(N) (mod p), j ∈ {0, 1, . . . , p − 1}. Fig. 2.9(f) showsthe samples generated using the Faure sequence.2.2.9 Full Grid DesignFull grid design, also known as the collocation technique, is a well established scheme forgenerating sample points [Tat96, MHZ05, XH05]. In this study, a Lagrangian polynomialis used for the interpolation. Let, the support nodes are denoted by xji which are expressedas xji ∈ xj = {xj1, xj2, . . . , xjN} ∈ [-1, 1] be a sequence of abscissas for Lagrange interpola-tion on [-1, 1]. The collocation method is based on one-dimensional Lagrange polynomialinterpolation at p Gauss quadrature points and is expressed asU j(y) =N∑i=1y(xji)· Lji (x) (2.10)242.2. OVERVIEW OF DESIGN OF EXPERIMENTSwhere N is the total number of grid points. Also, in the above equation, L is the Lagrangepolynomial, given byLji (x) =N∏k=1,k ̸=i(x− xjkxji − xjk)(2.11)Eq. 2.10 defines the one-dimensional problem. For multi-variate cases, the full tensor prod-uct interpolation is written asInU(y) =(U j1 ⊗ U j2 ⊗ · · · ⊗ U jn)(y) =N1∑i1=1n2∑N2=1· · ·Nn∑in=1y(xj1i1 ,xj2i2, . . . ,xjnin)·(Lj1i1 (x)⊗ Lj2i2(x)⊗ · · · ⊗ Ljnin (x)) (2.12)where n is the dimension of the problem. Fig. 2.10 illustrates the grid points generated-1 -0.5 0 0.5 1-1-0.500.51(a) µ = 1, n = 2-1 -0.5 0 0.5 1-1-0.500.51(b) µ = 2, n = 2-1 -0.5 0 0.5 1-1-0.500.51(c) µ = 3, n = 2(d) µ = 1, n = 3 (e) µ = 2, n = 3 (f) µ = 3, n = 3Figure 2.10: Full tensor product grid for different levels for two and three factorsusing the collocation technique for various levels (µ) of sparse grid for two and three randomvariables.252.2. OVERVIEW OF DESIGN OF EXPERIMENTS2.2.10 Sparse Grid DesignThe sparse grid scheme has been proven an efficient scheme for generating supportpoints [Hol10]. In this study, Smolyak’s algorithm [KW05] is utilized to generate samplingbased on the hierarchical sampling scheme [BG04]. The samplings generated from previousinterpolation depths are used in the subsequent interpolation to produce the new samplingset. The sparse grid (SG) can be formulated asSGµ =∑∑ni=1 ji≤µ+n−1(∆j1 ⊗∆j2 ⊗ · · · ⊗∆jn)(y)(2.13)where µ is the level of the sparse grid design and n denotes the dimension of the problem.In Eq. 2.13, the difference function, ∆j equals to (U j − U j−1) where U is the interpolationfunction for the univariate case. The unidimensional interpolation function can be expressedusing quadrature form, is given byU j(y) =Nj∑i=1y(xji)· A˘ji (x) (2.14)In the above equation, Nj denotes the number of grid points for j-th random variable. Also,A˘ji is the basis function, which is A˘ji ∈ C([−1, 1]), A˘ji (xjk) = δik. Here, xji are the supportnodes, which are expressed as xji ∈ xj = {xj1, xj2, . . . , xjNj} and xjk ∈ [−1, 1], 1 ≤ k ≤ Nj .The interpolation function for the univariate case, defined in Eq. 2.14, can be extended tothe multivariate case as:(U j1 ⊗ U j2 ⊗ · · · ⊗ U jn)(y) =N1∑i1=1N2∑i2=1· · ·Nn∑iβ=1y(xj1i1 ,xj2i2, . . . ,xjnin)·(A˘j1i1 (x)⊗ A˘j2i2(x)⊗ · · · ⊗ A˘jnin (x)) (2.15)The sparse grid interpolation presented above uses two types of basis functions, i.e., piece-wise linear and polynomial basis functions. In this study, the Chebyshev-Gauss-Lobatto grid[BNR00] is used, which is one of the most commonly used polynomial grids (i.e., higher or-der basis functions are considered) and provides higher accuracy. In the Chebyshev-Gauss-Lobatto grid scheme, the total number of grid points for each random variable and gridpoints are given as:Nj ={1, if j = 12j−1 + 1, if j > 1(2.16a)xji = − cosπ(i− 1)Nj − 1 , i = 1, 2, . . . , Nj (2.16b)262.2. OVERVIEW OF DESIGN OF EXPERIMENTSAlso, in Eq. 2.15, the basis function, A˘ji is given byA˘ji = 2Nj−1[1− cos(π(i−1))Nj(Nj−2) − 2∑(Nj−3)/2k=114k2−1 · cos 2πk(i−1)Nj−1], for i = 2, . . . , (Nj − 1)1Nj(Nj−2) , for i = 1&Nj(2.17)Fig. 2.11 shows the grid points generated using a sparse grid scheme for various levels (µ)of sparse grid for two and three random variables.-1 -0.5 0 0.5 1-1-0.500.51(a) µ = 1, n = 2-1 -0.5 0 0.5 1-1-0.500.51(b) µ = 2, n = 2-1 -0.5 0 0.5 1-1-0.500.51(c) µ = 3, n = 2(d) µ = 1, n = 3 (e) µ = 2, n = 3 (f) µ = 3, n = 3Figure 2.11: Grid points generated using sparse grid scheme2.2.11 Rotational Sparse Grid DesignRecently, rotational sparse grid design has been developed as one of the efficient design ofexperiment schemes, as proposed byWu et al. [WZJ+21]. In this method, the initial supportpoints are generated through a sparse grid scheme, as mentioned in Section 2.2.10. To obtainmaximum information from those support points, an optimization has been performed byrotating all support points with a fixed angle of rotation. The rotational matrix is given by272.3. OVERVIEW OF SURROGATE MODELSfor the two-dimensional problem as:R(θ) =[cos θ − sin θsin θ cos θ](2.18)where θ is the angle of rotation. Similarly, for three-dimensional problems, the rotationalmatrix is written as:Rx(θ) =1 0 00 cos θ − sin θ0 sin θ cos θ Ry(θ) = cos θ 0 sin θ0 1 0− sin θ 0 cos θ Rz(θ) =cos θ − sin θ 0sin θ cos θ 00 0 1(2.19)In the above equation, Rx(θ) is the rotational matrix where the rotation is performed withrespect to the x-axis. Similarly, Ry(θ) and Rz(θ) are the rotational matrix for rotationsabout the y- and z-axes, respectively. For a n-dimensional problem, the rotational matrixis given byR(θ) =1 · · · 0 · · · 0 · · · 0.... . ..... . ..... . ....0 · · · cos θ · · · − sin θ · · · 0.... . ..... . ..... . ....0 · · · sin θ · · · cos θ · · · 0.... . ..... . ..... . ....0 · · · 0 · · · 0 · · · 0i -th j -th k -thi -thj -thk -th(2.20)The support points on j-th and k-th axes are rotated with respect to the i-th axis in thiscase. The grid points using a sparse grid scheme are shown in Fig. 2.12 for a two-dimensionalproblem. The angle of rotation (θ) is estimated by performing the optimization problemwhile considering some objective functions from the designer’s perspective.2.3 Overview of Surrogate ModelsIn this section, a brief description of different surrogate models is presented.2.3.1 Polynomial Chaos ExpansionPolynomial chaos expansion (PCE), also known as spectral expansion, is one of the mostpopular surrogate models [Nor38]. In this method, a series of multivariate polynomials areused to approximate the original performance function. The output of PCE is written as[XK02]Y =P∑|i=0|αiΨi(X) (2.21)282.3. OVERVIEW OF SURROGATE MODELS-1 -0.5 0 0.5 1-1-0.500.51(a)-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5(b)Figure 2.12: Support points generated using (a) sparse grid and (b) rotational sparse gridscheme for two dimensional problem.where α is the coefficient of the orthogonal basis function Ψi(X) which is selected basedon the distribution of the input random variable X. Though it provides acceptable outputdata, it is seen that it suffers from the curse-of-dimensionality, as its coefficients growfactorially with the number of random variables. To address this issue, adaptive PCE hasbeen proposed [BS10, BS11], where only the important coefficients are retained while othersare neglected. Least-angle regression and the Bayesian framework with sparsity-promotingpriors are one of the methods used in adaptive PCE.2.3.2 Stochastic Spectral EmbeddingStochastic spectral embedding is a novel surrogate model proposed by Marelli et al.[MWLS21] that is a combination of spectral representation and adaptive domain decompo-sition. Consider a computational model M(θ) which contains a set of random parameters,θ = [θ1, θ2, . . . , θn]T with joint probability density function θ ∼ fΘ(θ). Therefore, the out-put response of the model M(θ) can be expressed in the form of spectral representation,which is given byY =M(θ) =∞∑i=1aiΨi(θ) (2.22)where ai is the coefficient of the orthonormal basis function with the following propertyE[Ψi(θ)Ψj(θ)]= δij (2.23)292.3. OVERVIEW OF SURROGATE MODELSDue to the orthogonal property of the basis function, the second moment of the outputbecomes finite, which is one of the essential properties of spectral decomposition. It iswritten asE[M(θ)2] = ∞∑i=1a2i < +∞ (2.24)The output response in Eq. 2.22 is truncated, which is expressed asY =M(θ) ≈ Mˆ(θ) =P∑i=1aiΨi(θ) (2.25)where P is the truncation order of the polynomial expansion. The residual is given byR(θ) =M(θ)− Mˆ(θ) (2.26)The basis function Ψ has support on the entire domain ΩΘ in polynomial chaos expansion(PCE). Due to that, sometimes PCE is unable to capture the nonlinearity present in themodel. In stochastic spectral embedding, the entire domain is divided into smaller subdo-mains (i.e., Ωm,nΘ ⊆ ΩΘ, where m is the level of expansion and n is the subdomain indexwithin the level). Fig. 2.13 shows a domain decomposition up to a level of expansion 2.When m = 0 i.e., a single domain is considered, Ω0,1Θ = ΩΘ (Fig. 2.13(a)), then residual isthe same as for PCE, in Eq. 2.26. For multi-level of expansion, Eq. 2.25 is re-written asθ1θ2𝛀𝚯0,1𝛀𝚯1,1𝛀𝚯1,2𝛀𝚯2,1𝛀𝚯2,2𝛀𝚯2,3𝛀𝚯2,4θ1θ2θ2θ1(a) m= 0 (b) m= 1 (c) m= 2Figure 2.13: Domain decomposition upto level of expansion M= 2 and Nm = 2mMˆ(θ) =M∑m=0Nm∑n=1IΩm,nΘ(θ)Rm,n(θ) (2.27)where the total number of expansion levels and number of subdomains at level m ∈[0, . . . ,M ] are denoted by M and Nm, respectively. In Eq. 2.27, I represents the indi-cator function of subdomain Ωm,nΘ . Rm,n(θ) is the truncated expansion of residual up to302.3. OVERVIEW OF SURROGATE MODELSlevel (m-1) which is expressed in following formRm(θ) =M(θ)−m−1∑l=0Nl∑n=1IΩl,nΘ(θ)Rl,n(θ) (2.28)where each residual term in the above equation is expanded asRl,n(θ) =∑ial,ni Ψl,ni (θ) (2.29)Readers may refer [MWLS21] for a complete description of stochastic spectral embedding.2.3.3 Universal KrigingKriging, also known as the Gaussian process, is an interpolation method where theoutput responses are produced by minimizing the mean square error of the training dataset, assuming correlated errors. The output of Kriging is written as [SSW89]Y =m∑i=1γiψi(X) + ϵ(X) = ψ(X)Tγ + ϵ(X) (2.30)where ψ(X) is the vector of m numbers of known basis function and γ is the regressioncoefficients. Also, in Eq. 2.30, ϵ is assumed to be zero mean stationary Gaussian processwith autocovariance function asE[ϵ(X(i))ϵ(X(j))]= σ2R[ϵ(X(i))ϵ(X(j))](2.31)where σ is the standard deviation of the Gaussian process and R is the autocorrelationfunction, which is expressed as a product, given byR[ϵ(X(i))ϵ(X(j))]=n∏k=1Rk[ϵ(X(i)k)− ϵ(X(j)k )] (2.32)where n is the number of the input random variables. In this study, the Gaussian correlationfunction is used, which is asR[θ, ϵ(X(i))ϵ(X(j))]=n∏k=1exp(− θk∣∣ϵ(X(i)k )− ϵ(X(j)k )∣∣2) (2.33)where θ is the hyperparameters in Kriging that are estimated by maximizing the likelihoodof the data. By solving the regression problem F γ ≃ Ys using least square method, thebest linear unbiased prediction of the response is written as [Kay05]Y = ψ(X)γˆ + r(X)TR−1(Ys − Fγˆ)(2.34)where Ys is the model response corresponding to the design of experiment points and r(X)is the cross-correlation between the design of experiment points and a new point of interest.312.3. OVERVIEW OF SURROGATE MODELS2.3.4 Polynomial Chaos Based KrigingPC-Kriging is the novel surrogate model, which is the combination of PCE and Kriging.The prediction is almost the same as Kriging, whereas the trend function is constructedusing orthonormal polynomials used in PCE and is written as [SSW15, SSM16]Y =P∑|i=0|αiΨi(X)︸ ︷︷ ︸PCE+ ϵ(X)︸ ︷︷ ︸Kriging(2.35)where αi and Ψi denote the unknown coefficients and orthonormal polynomial of ith degree,respectively, same as in Eq. 2.21. In Eq. 2.35, ϵ(X) is zero mean Gaussian process withvariance σ2, same as in Eq. 2.30. Although PCK has advantages over PCE and Kriging, itstill suffers from the curse-of-dimensionality when the number of input variables increasesand the unknown coefficients in the PCE polynomial increase factorially. To resolve thisissue, optimal PCE-Kriging is used where least-angle regression (LAR) is used. LAR isbasically a model selection technique where major components in PCE polynomials areretained. For further details, readers may refer [SSW15, SSM16].2.3.5 Support Vector RegressionSupport vector regression is a class of support vector machine, which minimizes the per-turbation using statistical learning theory [Vap98]. The output from SVR can be expressedasY = ⟨W ·X⟩+ b0 (2.36)where W and b0 are the two parameters of the canonical hyperplane, denote weight vectorand bias, respectively. In Eq. 2.36, ⟨·⟩ represents the inner product. The optimal regressionfunction is estimated by solving a minimization problem and is given bymin12||W ||2s.t.{Yi − ⟨W ·Xi⟩ − b0 ≤ ϵ⟨W ·Xi⟩+ b0 − Yi ≤ ϵ(2.37)where {Xi, Yi} are the training data set, and ϵ is the non-negative precision tolerance. Inreality, there may not be a function that satisfies the condition of Eq. 2.37. For this issue, theabove formulation is modified by introducing two slack variables (ξi, ξ∗i ≥ 0), is expressedasmin12||W ||2 + CN∑i=1(ξi + ξ∗i )s.t.{Yi − ⟨W ·Xi⟩ − b0 ≤ ϵ+ ξi⟨W ·Xi⟩+ b0 − Yi ≤ ϵ+ ξ∗i(2.38)322.3. OVERVIEW OF SURROGATE MODELSwhere C is the regularization constant that regulates between the flatness of Y and fittingerror of Y when exceeds ϵ. The optimization problem in Eq. 2.38 is solved by Lagrangeduality, where the corresponding primal Lagrangian is expressed asLPrim(W, b0, ξ, ξ∗, α, α∗, η, η∗) = 12||W ||2 + CN∑i=1(ξi + ξ∗i )−N∑i=1αi(ϵ+ ξi − Yi + ⟨W ·Xi⟩+ b0)−N∑i=1α∗i(ϵ+ ξ∗i + Yi − ⟨W ·Xi⟩ − b0)−N∑i=1(ηiξi + η∗i ξ∗i )(2.39)where α, α∗, η and η∗ are the Lagrangian multipliers. Using Karush-Kuhn-Tucker (KKT)conditions, the above primal Lagrangian LPrim is converted into dual Lagrangian formula-tion which is given bymin12N∑i=1N∑j=1(αi − α∗i )(αj − α∗j )⟨Xi ·Xj⟩+ ϵN∑i=1(αi + α∗i )−N∑i=1Yi(αi − α∗i )s.t.N∑i=1Yi(αi − α∗i ) = 0, αi, α∗i ∈ [0, C](2.40)Also, the weight vector can be computed using KKT conditions as W =∑Ni=1(αi − α∗i )Xi.Replacing this weight vector into Eq. 2.36 yields,Y =N∑i=1(αi − α∗i )⟨X ·Xi⟩+ b0 (2.41)This above formulation holds for linear regression. To convert it into nonlinear regression,the kernel function is used, as followsY =N∑i=1(αi − α∗i )κ(X, Xi) + b0 (2.42)where κ denotes the kernel function. Here, the Gaussian kernel function is considered, whichis given byκ(Xi, Xj) = exp(− ||Xi −Xj ||22σ2)(2.43)where σ denotes the hyper-parameter of the kernel function.332.3. OVERVIEW OF SURROGATE MODELS2.3.6 Multilayer PerceptronThe multilayer perceptron is a class of direct feedforward artificial neural network(ANN). Generally, it consists of at least three layers, which are the input layer, the hiddenlayer, and the output layer. When multiple hidden layers are used, the network is called“deep”. In each hidden layer, one or more neurons are used. For MLP, it is necessary tofind out the number of hidden layers and number of neurons in each hidden layer dependingupon the complexity and nonlinearity of the observed system. The output of the MLPnetwork is generally considered a linear combination of the neurons, which is related tothe weight of the connection (called synaptic weight) and a transfer function (called theactivation function).Figure 2.14: Schematic diagram of ANNA typical structure of MLP is shown in Fig. 2.14 where x1, x2, . . ., xn are the n-inputvariables and the output is denoted by Yˆ . Consider, there are m hidden layers and eachhidden layer is consisting n neurons. Each connection has a synaptic weight, say wkij wherek represents kth hidden layer (k = 1, . . . ,m); i and j denote ith node of (k − 1)th layerand jth node of kth layer (i, j = 1, . . . , n). The bias, output, and activation function of thejth neuron in the kth layer are denoted by θkj , Yˆkj and ϕkj , respectively. The output of thecorresponding neuron in that layer is written asYˆ kj = ϕkj( n∑i=1(wkij Yˆk−1i + θk−1i))(2.44)In the above equation, ϕ is the activation function. Common activation functions usedin ANN are linear, tan hyperbolic, sigmoid, rectified linear unit function, etc. When onehidden layer with a Gaussian-type activation function is used, the neural network is calleda radial basis neural network. Currently, wavelet neural networks have gained popularitywhere wavelet functions are used as an activation function. Common wavelet functions areMorlet, Maxican hat, Shannon, Meyer wavelet function, etc.342.3. OVERVIEW OF SURROGATE MODELSFor better prediction of the output response (Yˆ ), an updation of each synaptic weightin each layer are needed. In that case, a backpropagation learning algorithm is used. Inthis procedure, an optimization is carried out by minimizing mean square error betweenoutput obtained from network (Yˆ ) and desired output Y for all training points which canbe written asE =12N∑l=1(Yˆl − Yl)2 (2.45)where N is the total number of training points. The synaptic weight of each connectionin each layer is updated by minimizing error function as defined in Eq. 2.45 applying thegradient descent algorithm, which is given by[wkij ](o) = [wkij ](o−1) − η ∂E∂wkij(2.46)where o is current iteration and η denotes the learning rate. It is noted that no synapticweight is directly related to error, E, except preceding layer from the output layer. In thatcase, the chain rule is used to compute the gradient. Readers may refer [DS19] for furtherdetails on this topic.2.3.7 Physics-Informed Neural NetworkA physics-informed neural network (PINN) is a special class of feed-forward neuralnetwork where physical conditions are imposed on the neural network as a loss function.The data-driven neural network, as discussed in Section 2.3.6, is often cost-effective for acomplex physical system where a large amount of solution data is required. PINN is usedto solve problems where only a few data points are available, such as noisy data from anexperiment. To estimate the accurate solution of the physical system, the physical laws ofthe governing system are imposed on the network. A schematic diagram of PINN is shownin Fig. 2.15. To illustrate the algorithm of PINN, a nonlinear partial differential equationis considered, which is expressed in the following form∂u(x, t)∂t+N [u(x, t);λ] = 0; x ∈ Ω, t ∈ Tu(x, t) = G(x, t); x ∈ Ω, t ∈ Tu(x, 0) = H(x); x ∈ Ω(2.47)The latent solution in the above equation is u(x, t), which is a function of time t ∈ [0, T ]and a spatial variable x ∈ Ω, where Ω represents a space in Rn. Also, in Eq. 2.47, N [u;λ]denotes a non-linear differential operator with coefficients λ. The initial and boundaryconditions are represented by H(x) and G(x, t), respectively. Like in feed-forward neuralnetworks, the weights and bias in PINN are estimated by minimizing a cost function, whichis given asL(θ) = Lu(θ) + β1Lb(θ) + β2L0(θ) (2.48)352.3. OVERVIEW OF SURROGATE MODELStxuˆtx22x22u u 0.01 uf : ut x x   = + −      ( ) ( )u fN N2 2i i i i iu u f fi 1 i 1u f1 1MSE : u t ,x u f t ,xN N= == − + Training DataValidationPhysics-driven Trainingargminw,bInput Layer Hidden Layers Output Layer NN Auto-differentiationFigure 2.15: Schematic diagram of physics informed neural networkwhere Lu, Lb and L0 denote the residual of a PDE and the residual corresponding to bound-ary and initial conditions, respectively. The coefficients β1 and β2 are the weight parametersfor adjustment of relative importance of each residual term. Also, θ = (θ1, θ2, . . . , θn) arethe parameters of neural networks, i.e., weights and bias.The algorithm of PINN starts with generating the training samples. The samples{xiu, tiu}Nui=1are generated from the domain x ∈ Ω, t ∈ T . The corresponding boundarycondition samples and initial condition samples are generated as{xib, tib,Gib = G(xib, tib)}Nbi=1and{xi0,Hi0 = H(xi0)}N0i=1, respectively, from the given boundary and initial conditions. Thenumbers of samples for every residual are denoted by Nu, Nb, and N0, respectively. Basedon these samples, the residuals in Eq. 2.48 are constructed, which are given byLu(θ) = 1NuNu∑i=1∣∣∣∣∣∂u(xiu, tiu; θ)∂t +N [u(xiu, tiu);λ]∣∣∣∣∣2; Lb(θ) = 1NbNb∑i=1∣∣∣u(xib, tib; θ)− G(xib, tib)∣∣∣2L0(θ) = 1N0N0∑i=1∣∣∣u(xi0, 0; θ)−H(xib, 0)∣∣∣2(2.49)Once the residual functions are formulated, the parameters θ of the neural network areestimated through optimization. Most of the studies used the mini-batch stochastic gradi-ent descent (SGD) algorithm for solving an unconstrained optimization problem, which isexpressed asθ∗ = argminθLu(θ) + β1Lb(θ) + β2L0(θ)︸ ︷︷ ︸L(θ)(2.50)362.4. NUMERICAL EXAMPLESThe SGD is an iterative solver in which the gradient of the loss function, L(θ) is approx-imated in every iteration through backpropagation, and as a result, the neural networkparameters (θ) are updated. The updated model parameters are expressed as followsθi+1 = θi − η∇θL(θ) = θi − η(∇θLu(θ) + β1∇θLb(θ) + β2∇θL0(θ)) (2.51)where the learning rate and the gradient operator are denoted by η and ∇, respectively.The current iteration step is represented by i. It is shown from Eq. 2.51 that the efficiencyand accuracy of PINN are dependent on the choice of the weight parameters, i.e., β1 andβ2. Few studies have been found where the estimation of weight parameters is discussed.For instance, soft attention mechanism (SAM) is proposed by Levi and Ulisses [MBN20] toupdate the weight parameters. Wang et al. [WYP22] proposed neural tangent kernel bywhich the weight parameters are updated. Algorithm 4 depicts the pseudo-code for PINN.Algorithm 4 Pseudo-code for physics-informed neural networks1: Generate the training samples i.e.,{xiu, tiu}Nui=1,{xib, tib,Gib = G(xib, tib)}Nbi=1and{xi0,Hi0 =H(xi0)}N0i=1from the domain, boundary and initial conditions.2: Specify the architecture of the neural networks i.e., number of hidden layers and numberof neurons in each hidden layer.3: Initialize the model parameters, θ04: Begin5: while L(θi) > ϵ do6: θi+1 = θi − η∇θL(θ)7: end whilePINN is used for either data-driven interference or data-driven identification of differen-tial equations. Data-driven interference is a forward problem in which λ of the differentialequation (Eq. 2.47) is known and u(x, t) is solved based on initial and boundary conditions,whereas data-driven identification identifies coefficients λ of the differential equation basedon scattered data of the solution u(x, t).2.4 Numerical ExamplesIn this section, numerical examples are presented to illustrate the performance of a PINNconsidering various design experiment schemes to generate the training data set. For thispurpose, five different nonlinear differential equations are considered, i.e., Viscous Burger’sequation, Shro¨dinger equation, one-dimensional heat equation, Allen-Cahn equation andKorteweg-de Vries equation. The neural network is designed to solve those PDEs, whichapproximates the true solution u(t, x), and the approximated solution is denoted by uˆ(t, x).The number of neurons in the hidden layers is constant for a given problem. The totalnumber of hidden layers and the number of neurons in each hidden layer are obtained bycomparing the prediction responses using a neural network with the actual solution multiple372.4. NUMERICAL EXAMPLEStimes. The hyperbolic tangent tanh is used as an activation function for every neuron, andthis is same for all problems. To train the neural network, different DoE schemes areused, which are factorial design (FD), central composite design (CCD), centroidal Voronoitesselation (CVT), minimax Latin hypercube sampling (MLH), Sobol sampling, Haltonsampling, Hammersley sampling, Faure sampling, full grid design (FGD), and sparse griddesign (SGD). The accuracy of the predicted responses using a neural network depends onthe number of DoE samples in the training set. Therefore, a preliminary study is devotedto its accuracy by carrying out the simulation while considering increasing the size of thetraining set (i.e., from 50 to 500 with a step size of 50). The accuracy is evaluated usingthe mean squared error (MSE) between actual and predicted responses. For the simulationpurposes, python using TensorFlow [ABC+16] and SciANN [HJ21], MATLAB softwares areused.2.4.1 Example 1: Viscous Burger’s EquationAs the first example, viscous Burger’s equation is illustrated, whose applications areseen in the field of fluid mechanics, acoustics, and traffic flow [Daf05]. The governing PDEequation for viscous Burger’s equation is given by [Rai18]∂u∂t+ u∂u∂x− ν ∂2u∂x2= 0 (2.52)where (x, t) ∈ Ξ = [-1 , 1] × [0 , 1]; ν is the viscosity of the system, which is taken as(0.01/π). The initial and boundary conditions of the above equation are given byu(t = 0, x) = −sin(πx); u(t, x = ±1) = 0 (2.53)With the above equation, a function f(t, x) is constructed which provides the physicalinformation of the neural network. It is expressed as follows:f(t, x) =∂u∂t+ u∂u∂x− ν ∂2u∂x2(2.54)The optimal parameters of the neural network are estimated by minimizing the loss function,which is given byL = Lu + Lf (2.55)where Lu denotes mean square error loss, which is calculated using the data correspondingto the initial and boundary conditions, whereas Lf denotes the loss corresponding to thefunction f(t, x), defined in Eq. 2.54. These two loss functions are expressed in the followingformsLu = 1NuNu∑i=1∣∣∣u(tiu, xiu)− ui∣∣∣2 (2.56a)Lf = 1NfNf∑i=1∣∣∣f(tif , xif )∣∣∣2 (2.56b)382.4. NUMERICAL EXAMPLESIn Eq. 2.56, {tiu, xiu, ui} denotes the initial and boundary training data of the function u(t, x)where Nu is the total number of training data. Similarly, {tif , xif} is the training data ofthe spatio-temporal domain where Nf is the total number of training data to train f(t, x).In this problem, eight hidden layers, each with 20 neurons are used construct the neuralnetwork. The MSE values for various sizes of DoE sets are shown in Fig. 2.16. FromFig. 2.16, it is clear that 400 samples are needed to stabilize the MSE values.0 100 200 300 400 50000.20.40.60.8MLHSobolHaltonHammersleyFaureFigure 2.16: The MSE history for different levels of DoE for solving Viscous Burger’sequationFD CCD CVT MLH Sobol Halton Hammersley Faure FGD SGD00.511.52Figure 2.17: Absolute error between original and predicted responses of viscous Burger’sequation considering different DoE strategiesWith this, a comparative study is performed with all DoE schemes, and the absoluteerrors between actual and predicted responses are shown using a Box and Whisker plot asshown in Fig. 2.17. In Fig. 2.17, the red line represents the median value of the errors. By392.4. NUMERICAL EXAMPLES(a) (b)(c)Figure 2.18: Viscous Burger’s equation: (a) Exact solution; (b) Solution using PINN and(c) Absolute error between exact solution and solution using PINNcomparing the median values, it can be concluded that FD, CCD, and CVT perform theworst over-fitting. Among those, Hammersley performs best due to a lower median value.SGD and Sobol sampling yield the next-best results. Once the accuracy is established,the predicted responses are evaluated considering the 400 DoE samples generated usingHammersley sampling. The contour plots of the actual solution (u(t, x)) and predictedsolution (uˆ(t, x)) of the viscous Burger’s equation are shown in Fig. 2.18(a) and Fig. 2.18(b).The error between these two solutions is shown in Fig. 2.18(c).402.4. NUMERICAL EXAMPLES2.4.2 Example 2 : Shro¨dinger EquationHere, the Shro¨dinger equation is considered as our next example, which is a classical fieldequation, applied in quantum mechanical systems, nonlinear wave propagation in opticalfibres, etc. The nonlinear Shro¨dinger equation subjected to periodic boundary conditionsis written as [RPK19]:i∂u∂t+12∂2u∂x2+ |u|2u = 0u(0, x) = 2sech(x); u(t,−5) = u(t, 5)∂u∂x(t,−5) = ∂u∂x(t, 5)(2.57)where (x, t) ∈ Ξ = [-5, 5] × [0, π/2]. In the above equation, u(x, t) is a complex-valued func-tion. A loss function is considered, which has to be minimized to estimate the parametersrelated to neural networks. It is given byL = L0 + Lb + Lf (2.58)where L0, Lb, and Lf are the residual loss functions on the initial data, the periodic bound-ary condition and Shro¨dinger equation, are written asL0 = 1N0N0∑j=1∣∣∣u(0, xj0)− uj0∣∣∣2 (2.59a)Lb = 1NbNb∑j=1{∣∣∣uj(tjb,−5)− uj(tjb, 5)∣∣∣2+∣∣∣ujx(tjb,−5)− ujx(tjb, 5)∣∣∣2} (2.59b)Lf = 1NfNf∑j=1∣∣∣f(tjf , xjf)∣∣∣2 (2.59c)In Eq. 2.59b, ux denotes the first partial derivative of u with respect to x, i.e., ∂u/∂x.Also, in Eq. 2.59c, f is the function representing Shro¨dinger equation, as in Eq. 2.57 i.e.,f = iut + 0.5uxx + |u|2u.Fig. 2.19 depicts the MSE values for various DoE set sizes. It is seen that 400 samplesare needed to stabilize the MSE values. With this, a comparative study is performedwith all DoE schemes, and the absolute errors between actual and predicted responses areshown using a Box and Whisker plot as shown in Fig. 2.20. By comparing the medianvalues, it is seen that FD, CCD, and CVT perform the worst. Apart from FGD, the otherDoE strategies such as MLH, SGD, Sobol, Halton, Hammersley, and Faure sampling almostperform the same. Among those, Hammersley sampling slightly performs better. Oncethe accuracy is established, the predicted responses are evaluated considering the 400 DoEsamples generated using Hammersley sampling. Fig. 2.21(a) and Fig. 2.21(b) show thecontour plot of the predicted solution of the Shro¨dinger equation and the error between theexact and predicted solutions, respectively.412.4. NUMERICAL EXAMPLES0 100 200 300 400 50000.10.20.30.4MLHSobolHaltonHammersleyFaureFigure 2.19: The MSE values for different number of DoE sets for different DoE techniquesfor Shro¨dinger equationFD CCD CVT MLH Sobol Halton Hammersley Faure FGD SGD00.511.5Figure 2.20: Absolute error between original and predicted responses of Shro¨dinger equationconsidering different DoE strategies2.4.3 Example 3: 1D - Heat EquationAs an example, consider the one-dimensional heat equation, which is a second-orderPDE that describes heat diffusion through a material and is written as∂u∂t=∂2u∂x2(2.60)where (x, t) ∈ Ξ = [0 , L] × [0 ,1]. Initially, the temperature is a nonzero constant.Therefore, the initial condition is given byu(x, 0) = T0 (2.61)422.4. NUMERICAL EXAMPLES(a) (b)Figure 2.21: Shro¨dinger equation: (a) Solution for u using PINN and (b) Absolute errorbetween exact solution and solution using PINNAlso, the boundary conditions imposed on the system, which are zero at the left boundaryFigure 2.22: Heat conduction in a rodand nonzero at the right boundary, are shown in Fig. 2.22. The boundary conditions areexpressed asu(0, t) = 0 u(L, t) = 1 (2.62)In this study, the parameters L and T0 are taken as 1 and 0.5, respectively. The optimalneural network parameters are estimated by minimizing the loss function L = Lu+Lf . Thetwo components of the loss function are given byLu = 1NuNu∑i=1∣∣∣u(tiu, xiu)− ui∣∣∣2 (2.63a)Lf = 1NfNf∑i=1∣∣∣f(tif , xif )∣∣∣2 (2.63b)432.4. NUMERICAL EXAMPLESThe loss function corresponding to the initial and boundary conditions is denoted by Lu, andthe loss function corresponding to the function f(t, x), which is defined as f = (ut−uxx), isdenoted by Lf . The network contains eight hidden layers, each with 20 neurons. The MSEvalues for various sizes of DoE sets are shown in Fig. 2.23. It is seen that 400 samples areneeded to stabilize the MSE values. With this, a comparative study is performed with all0 100 200 300 400 50000.050.10.150.2MLHSobolHaltonHammersleyFaureFigure 2.23: The MSE values for different number of DoE sets for different DoE techniquesfor heat equationFD CCD CVT MLH Sobol Halton Hammersley Faure FGD SGD00.10.20.30.40.50.6Figure 2.24: Absolute error between original and predicted responses of heat equationconsidering different DoE strategiesDoE schemes, and the absolute errors between actual and predicted responses are shownusing a Box and Whisker plot as shown in Fig. 2.24. By comparing the median values, it isseen that FD and CVT perform the worst. Among others, Hammersley sampling performsbetter as its median value is the lowest. Once the accuracy is established, the predicted442.4. NUMERICAL EXAMPLESresponses are evaluated considering the 400 DoE samples generated using Hammersleysampling. Fig. 2.25(a) and Fig. 2.25(b) show the contour plot of the predicted solution ofthe one-dimensional heat conduction equation and the error between exact and predictedsolutions, respectively.(a) (b)Figure 2.25: (a) Solution for u using PINN and (b) Absolute error between exact solutionand solution using PINN for heat conduction equation2.4.4 Example 4: Allen - Cahn EquationAllen - Cahn equation, a second - order PDE, is used to represent the physical problemssuch as crystal growth [SSB16], image segmentation [BCM04], motion by mean curvatureflows [SSQB18] etc. Mainly, this equation is used to study phase transitions and interfacialdynamics in material science [SY11]. The governing differential equation is given as∂u∂t− 0.0001∂2u∂x2+ 5u3 − 5u = 0 (2.64)where (x, t) ∈ Ξ = [-1 , 1] × [0 ,1]. The initial and boundary conditions of the aboveequation are given byu(t = 0, x) = x3cos(πx)u(t, x = −1) = u(t, x = 1)∂u∂x(t, x = −1) = ∂u∂x(t, x = 1)(2.65)The parameters related to the neural networks are estimated by minimizing a loss function,which is expressed in the following formL = L0 + Lb + Lf (2.66)452.4. NUMERICAL EXAMPLESwhere L0, Lb, Lf are the residual loss functions on the initial data, the periodic boundarycondition and Allen-Cahn equation are as followsL0 = 1N0N0∑j=1∣∣∣u(0, xj0)− uj0∣∣∣2 (2.67a)Lb = 1NbNb∑j=1{∣∣∣uj(tjb,−1)− uj(tjb, 1)∣∣∣2+∣∣∣ujx(tjb,−1)− ujx(tjb, 1)∣∣∣2} (2.67b)Lf = 1NfNf∑j=1∣∣∣f(tjf , xjf)∣∣∣2 (2.67c)In Eq. 2.67c, f represents the Allen-Cahn equation, which is defined in Eq. 2.64, i.e.,f = ut − 0.0001uxx + 5u3 − 5u. To solve the second-order PDE representing the Allen-Cahn equation, a neural network is designed that contains four hidden layers, each with200 neurons. The MSE values for various sizes of DoE sets are shown in Fig. 2.26. It isseen that 400 samples are needed to stabilize the MSE values. Once the size of the DoE0 100 200 300 400 50000.10.20.30.4MLHSobolHaltonHammersleyFaureFigure 2.26: The MSE values for different number of DoE sets for different DoE techniquesfor Allen-Cahn equationset is fixed for quasi-random sampling schemes, a comparative study is performed with allDoE schemes, and the absolute errors between actual and predicted responses are shownusing the Box and Whisker plot as shown in Fig. 2.27. It is seen that Hammersley samplingperforms better as its maximum value of absolute error is the lowest compared to others.Once the accuracy is established, the predicted responses are evaluated considering the 400DoE samples generated using Hammersley sampling. Fig. 2.28(a) and Fig. 2.28(b) showthe contour plot of the predicted solution of the Allen-Cahn equation and the error betweenexact and predicted solutions, respectively.462.4. NUMERICAL EXAMPLESFD CCD CVT MLH Sobol Halton Hammersley Faure FGD SGD00.10.20.30.40.5Figure 2.27: Absolute error between original and predicted responses of Allen-Cahn equationconsidering different DoE strategies(a) (b)Figure 2.28: (a) Solution for u using PINN and (b) Absolute error between exact solutionand solution using PINN for reaction - diffusion system represented by Allen-Cahn equation2.4.5 Example 5: Korteweg-de Vries EquationAs the final example of this study, the Korteweg-de Vries (KdV) equation is considered,which is a nonlinear PDE comprising the higher-order derivatives. This equation is usedto describe the small amplitude shallow-water waves, ion-phonon waves, and fluctuationphenomena in biological systems, etc. [WG17, KST18]. The governing KdV equation for472.4. NUMERICAL EXAMPLESshallow-water waves is expressed as∂u∂t+ u∂u∂x+∂3u∂x3= 0 (2.68)where (x, t) ∈ Ξ = [0 , 2π] × [0 , 1]. With the above equation, periodic boundary conditions,i.e., u(t, x = 0) = u(t, x = 2π) are considered. The initial condition of the KdV equation isgiven byu(t = 0, x) = Asech2(√A12(x− π))(2.69)The optimal neural network parameters are determined by minimizing the loss functions(L = Lu + Lf ), which are written as followsLu = 1NuNu∑i=1∣∣∣u(tiu, xiu)− ui∣∣∣2 (2.70a)Lf = 1NfNf∑i=1∣∣∣f(tif , xif )∣∣∣2 (2.70b)where Lu and Lf are the loss functions corresponding to the initial and boundary conditions,and the function f(t, x), defined in Eq. 2.68. The function f in Eq. 2.70 represents the KdVequation, defined in Eq. 2.68, i.e., f = ut + uux + uxxx. To solve the higher-order PDErepresenting the KdV equation, a neural network is designed that contains seven hiddenlayers, each with 120 neurons. The MSE values for various sizes of DoE sets are shown inFig. 2.29. It is seen that 350 samples are needed to stabilize the MSE values.0 100 200 300 400 500024681012MLHSobolHaltonHammersleyFaureFigure 2.29: The MSE values for different number of DoE sets for different DoE techniquesfor KdV equation482.4. NUMERICAL EXAMPLESFD CCD CVT MLH Sobol Halton Hammersley Faure FGD SGD0102030Figure 2.30: Absolute error between original and predicted responses of KdV equationconsidering different DoE strategiesOnce the size of the DoE set is fixed for quasi-random sampling schemes, a comparativestudy is performed with all DoE schemes, and the absolute errors between actual and pre-dicted responses are shown using the Box and Whisker plot as shown in Fig. 2.30. It is seenthat Hammersley sampling performs better. Once the accuracy is established, the predictedresponses are evaluated considering the 350 DoE samples generated using Hammersley sam-pling. Fig. 2.31(a) and Fig. 2.31(b) show the contour plot of the predicted solution of theKdV equation and the error between exact and predicted solutions, respectively.(a) (b)Figure 2.31: (a) Solution for u using PINN and (b) Absolute error between exact solutionand solution using PINN for solitary waves represented by KdV equation492.5. SUMMARY2.5 SummaryThe selection of the design of experiment strategy plays an important role in predictingthe output using surrogate models. The surrogate model predicts the output based on thetraining data by minimizing the residuals between the output corresponding to the trainingdata and predicted responses. An extensive survey of the different DoE strategies anddifferent surrogate models is presented in this chapter. A review of classical and modernexperimental strategy design is also provided.The importance of the choosing a DoE scheme for physics-informed neural network isestablished by considering five different partial differential equations, which are ViscousBurger’s equation, Shro¨dinger equation, one-dimensional heat equation, Allen-Cahn equa-tion and Korteweg-de Vries equation. It is seen that quasi-random sampling techniques,especially the Hammersley sampling scheme, yield better results compared to classical DoEschemes. This argument is established based on comparing the absolute error between theactual and predicted solutions. This reflects the fact that the Hammersley sampling coversthe whole design space better than other DoE strategies.The size of the DoE set affects the accuracy of the surrogate model. For high-fidelitymodels where the computational budget is expensive, the smallest size of the training setis needed to reduce the computational budget as well as the accuracy of the prediction.In this chapter, it is seen that at least 350 to 400 DoE samples are taken to get accuratepredicted responses.50Chapter 3Reliability Based Design ofStructures using Level CrossingFormulationThe reliability of a system is defined as the probability that the structural responses willnot exceed a specified threshold value in its design lifetime. In other words, reliability mea-sures the probability of no-crossing an acceptable threshold at any instant. The thresholdvalue can be the maximum top floor displacement, inter-storey drift, or any critical measure.In this chapter, the reliability of a system is estimated using a level-crossing formulation,which is described in the following section.3.1 Objectives∗ This chapter provides details on the formulation used for the estimation of the prob-ability of failure of a structure using a level-crossing formulation when the structureis subjected to stationary earthquake motions.∗ This chapter develops a reliability-based optimization scheme to estimate the tuningparameters of NES to provide maximum control to structures during any uncertainfuture earthquake. In this context, the uncertainties considered are parameters as-sociated with the base excitation, like local soil characteristics (such as the naturalfrequency, damping, and intensity of ground motions) and system parameters (suchas the mass, stiffness, and damping ratio of the primary structure). The outcrossingrate approach is employed to evaluate the failure probability of the structure. To re-duce the computational cost, surrogate models are used for optimization. The geneticalgorithm (GA) is used as an optimization tool.∗ This chapter develops an efficient strategy for vibration control by introducing addi-tional stiffness into the outrigger structure. For this purpose, SMA is introduced atthe junction of the outrigger structure and the peripheral column. The main aim isto improve the structural performance of the outrigger structure under earthquakes.A reduced-order model of the high-rise building is formulated using the Timoshenkobeam model. A surrogate-assisted optimization scheme is used to find the optimaldesign variables, such as tuning parameters of the SMA and location of the outrig-513.2. LEVEL-CROSSING FORMULATIONger, so that the proposed damped outrigger system can offer maximum control of thestructure during an uncertain future seismic event.∗ Sensitivity analyses are carried out to establish the effectiveness of the proposed struc-tures for mitigating structural vibration due to earthquakes.3.2 Level-Crossing FormulationLetX(t) be a stochastic process which is the structural responses, as shown in Fig. 3.1(a).Also, it is assumed that α is the threshold limit. The level-crossing problem is defined asthe occurrence of events that describes the crossing of X(t) above α within the time interval[0,T ] i.e., NX(α, 0, T ). It is clear that as X(t) is a stochastic process, thus NX(α, 0, T ) isalso a stochastic process. To estimate the expected value of NX(α, 0, T ), a random processZ(t) is assumed which is shown in Fig. 3.1, defined asZ(t) = U[X(t)− α] (3.1)where U is the step function. More clearly, the above equation is expressed asZ(t) = 0,when X(t) < α= 1,when X(t) > α(3.2)To count the crossing when X(t) crosses α, the derivative of Z(t) is estimated which isexpressed asZ˙(t) = δ[X(t)− α]X˙(t) (3.3)where δ(·) represents the Dirac-delta function. It is seen from Fig. 3.1 that if the directionof Z˙(t) counts, it nullifies as it consist a up-crossing and a down-crossing. For this reason,the absolute value of the Z˙(t) is considered, as shown in Fig. 3.1. The total number of timesthat X(t) crosses the threshold limit α is given byNX(α, 0, T ) =∫ T0|Z˙(t)|dt =∫ T0|X˙(t)|δ[X(t)− α]dt (3.4)The expected value of the NX(α, 0, T ) is estimated asE[NX(α, 0, T )] = E[ ∫ T0|X˙(t)|δ[X(t)− α]dt]=∫ T0E[|X˙(t)|δ[X(t)− α]]dt=∫ T0[ ∫ ∞−∞∫ ∞−∞|x˙|δ(x− α)pXX˙(x, x˙; t)dxdx˙]dt=∫ T0[ ∫ ∞−∞|x˙|pXX˙(α, x˙; t)dx˙]dt(3.5)where pXX˙(α, x˙; t) is the joint probability density function between the response x and itstime derivative x˙. In this chapter, it is assumed that the structural responses are a zero-mean523.2. LEVEL-CROSSING FORMULATIONX(t)αZ(t)ሶZ(t)ሶZ(t)NX (t)(a)(b)(c)(d)(e)Figure 3.1: Schematic diagram of level crossingstationary Gaussian process. Due to stationary assumption, the joint probability densityfunction pXX˙(α, x˙; t) is independent of time t and the response x and its time derivative x˙are uncorrelated. The joint probability density function is given bypXX˙(α, x˙) = pX(α) · pX˙(x˙) =12πσxσx˙exp{− 12[(ασx)2+(x˙σx˙)2]}(3.6)Substituting Eq. 3.6 into Eq. 3.5 yieldsE[NX(α, 0, T )] =∫ T0[ ∫ ∞−∞|x˙|pXX˙(α, x˙; t)dx˙]dt=∫ T0[ ∫ ∞−∞|x˙| 12πσxσx˙exp{− 12[(ασx)2+(x˙σx˙)2]}dx˙]dt=12πσxσx˙exp{− 12α2σ2x}∫ T0[ ∫ ∞−∞|x˙| exp{− 12x˙2σ2x˙}dx˙]dt=22πσxσx˙exp{− 12α2σ2x}∫ T0[ ∫ ∞0x˙ exp{− 12x˙2σ2x˙}dx˙]dt=σx˙πσxexp{− 12α2σ2x}∫ T0dt=σx˙Tπσxexp{− 12α2σ2x}(3.7)533.3. ESTIMATION OF PROBABILITY OF FAILUREIn the above equation, σx and σx˙ are the root mean squares of the displacement andvelocity-time history responses, respectively.3.3 Estimation of Probability of FailureIn the engineering problem, we are most often interested on the extreme value of thesystem’s response. In this context, the probability of failure is defined when the maximumvalue of the stochastic process X(t) crosses the threshold value α. The extreme value ofthe stochastic process is given byXˆ = max0≤t≤TX(t) (3.8)The probability of failure is expressed asPf = P[Xˆ > α] = 1− P[Xˆ ≤ α] (3.9)In the previous section, the expected value of the crossing is estimated, as in Eq. 3.7. Thecrossings are assumed to be Poisson processes. The probability of N crossings within thetime interval [0, T ] is written as followsP[N, t] =λNN !exp (−λ) (3.10)where λ is the Poisson distribution’s parameter. For no-crossing i.e., N = 0, Eq. 3.10 isre-written asP[0, t] = exp (−λ) (3.11)For the Poisson distribution, the mean is also λ which is equals to E[NX(α, 0, T )], definedin Eq. 3.7. Therefore, the probability of failure, Pf is written asPf = 1− P[Xˆ ≤ α]= 1− exp (−λ) = 1− exp[− σx˙Tπσxexp{− 12α2σ2x}](3.12)In the case where the system consists of design variables, d and random parameters, Z, theconditional probability of failure of an event Fe for a given duration T can be calculated byusing Rice’s formula, defined in Eq. 3.12,P[Fe|d,Z, T ] = 1− exp[− σx˙(d,Z)Tπσx(d,Z)exp{− 12α2(σx(d,Z))2}](3.13)Also, the arrival of uncertain external excitation is modeled as Poisson’s process [CLE+19,CLMT19]. Thus, the probability of failure for a design lifetime period of TD and the ngnumber of seismic events can be written asP[Fe|d,Z, TD]=∞∑j=1P[Fe|d,Z, t, ng = j]P[ng = j|TD](3.14)543.4. NUMERICAL EXAMPLESwhereP[Fe|d,Z, t, ng = j] = 1−(1− P[Fe|d,Z, t])j (3.15a)P[ng = j|TD] =(νTD)jexp(− νTD)j!(3.15b)where ν is the arrival rate of events. In Eq. 3.14, the probability of failure is dependent onthe random parameters Z, which are characterized by joint probability density function fZ.Using the total probability theorem, the resulting probability of failure is [CLE+19]Pf (d) = EZ(P[Fe|d,Z, TD])=∫ZP[Fe|d,Z, TD]fZ(z)dz(3.16)For a given sample size, the probability of failure can be written as [CLE+19]Pˆf (d) =1nrnr∑i=1P[F |d,Z(i), TD](3.17)The reliability index for the system can be estimated byβˆ(d) = −Φ−1(Pˆf (d)) (3.18)3.4 Numerical ExamplesIn this section, two numerical examples are presented to illustrate the reliability-baseddesign optimization of structures using level-crossing formulation. The first example is asteel moment resisting frame in which a nonlinear energy sink with negative stiffness andsliding friction are equipped at top of the frame. The second example is a shape memoryalloy-based damped outrigger structure.3.4.1 Example 1: Reliability-based Optimization of Nonlinear EnergySink with Negative Stiffness and Sliding FrictionAs the first example, a nonlinear energy sink (NES) with negative stiffness and slidingfriction equipped at the top of a frame is considered. In this example, the tuning parametersof the NES are evaluated using a reliability-based optimization scheme so that the dampercan provide maximum control to the structure during any uncertain future earthquake.The ground motions are modeled as a stationary filtered Gaussian white noise process ex-pressed by the Kanai - Tajimi spectrum. In this context, the uncertainties considered areparameters associated with the base excitation, like local soil characteristics (such as nat-ural frequency, damping, and intensity of ground motions involved in the Kanai-Tajimispectrum model) and system parameters (such as mass, stiffness, and damping ratio of the553.4. NUMERICAL EXAMPLESprimary structure). As the ground motions are stationary zero-mean Gaussian processesand there is a linear transformation between the external excitation and structural response,the structural responses become zero-mean stationary Gaussian processes, and hence theoutcrossing rate approach can be employed to evaluate the failure probability of the struc-ture, as described in Section 3.2 and 3.3. The details of the governing equations of motionof the structure, the algorithm for RBDO, and the numerical results are presented below.Dynamics of Primary Structure - NES SystemHere, a one-storey moment-resistant frame incorporating NES with negative stiffnessand sliding friction is considered (Fig. 3.2). The governing equations of motion of thecombined structural system under the earthquake loading can be formulated asMx¨+ Cx˙+Kx− F = −Mx¨g (3.19a)m(x¨+ u¨) + F = −mx¨g (3.19b)where M , K and C are mass, stiffness and damping of the primary structure, respectively;CK/2 K/2FSFkNESmM xx+uẍg(a)kNESkNESL0mLNegative Stiffness IntervalFNES (u)ucFSFuFu0(b)Figure 3.2: (a) Schematic representation of moment resisting frame and (b) NES withnegative stiffness and sliding frictionx is the storey drift of the primary structure and u is the relative drift of the NES mass;563.4. NUMERICAL EXAMPLESm is the mass of the NES, and F is the control force by the NES. From Fig. 3.2(b), forrelative drift of NES u, the deflection (∆) in each spring is given as∆ =√L20 + (u− uc)2 − L (3.20)where L and L0 are the spring physical unloaded length and spring compressed length,respectively. uc is written asuc =√L2 − L20 (3.21)The control force by combined NES FNES and sliding friction FSF can be written asF = FNES + FSF= 2kNES(u− uc)[ ∆√L20 +(u− uc)2]+ FSF= 2kNES(u− uc)[1− L(L20 +(u− uc)2)−1/2]+ FSF(3.22)where kNES is the spring geometric nonlinear stiffness. The FSF is computed asFSF =fv.sgn(u˙) |u˙| > 0fs.sgn(FNES) |u˙| = 0, and |FNES | > |fs|FNES |u˙| = 0, and |FNES | ≤ |fs|(3.23)where sgn(·) is the signum function, fv is the sliding friction force and fs is the maximumstatic friction force beyond which sliding can happen. The fv and fs can be expressed asfv = µvmg (3.24a)fs = µsmg (3.24b)where µv and µs are the sliding and static friction factors, respectively.Kanai-Tijimi Spectrum ModelThe ground acceleration x¨g in Eq. 3.19 is modeled as a stationary filtered Gaussianwhite noise process expressed by Kanai - Tajimi spectrum [Kan57, Taj60]. According tothis model, the soil layer at a particular site is represented by a SDOF system, and theseismic motion at the bedrock level is modeled as zero-mean stationary Gaussian whitenoise. The power spectral density function is expressed asSgg(ω) = S0ω4g + 4η2gω2gω2(ω2 − ω2g)2+ 4η2gω2gω2(3.25)573.4. NUMERICAL EXAMPLESwhere ωg and ηg, respectively, are the natural frequency and the critical damping ratio ofthe soil layer at the site, and S0 is the spectral intensity of bedrock motions. The groundacceleration time history can be simulated using the linear superposition of an infinitenumber of sinusoids, expressed asx¨g =∞∑i=1√2Sgg(ωi)∆ωi. cos (ωit+ ϕi) (3.26)where ϕi is the uniformly distributed phase angle between 0 to 2π.Optimal Design of NES Under UncertaintyThe problem defined in Section 3.4.1 has two design variables, namely kNES and L0.The uncertain parameters considered in this study are classified into three groups:− Parameters associated with the primary structure, Zp = [M,C,K].− Parameters associated with the NES, ZNES = [kNES , L0].− Parameters associated with the seismic excitation, Zgm = [S0, ωg, ηg].With this, the minimization problem can be expressed as:d∗ = argmind[− βˆ (d) ],s.t. dll ≤ d ≤ dul. (3.27)where d = [µkNES , µL0 ] represents the mean value of the design variables; dll and dul are thelower and upper bounds of the design variables, respectively. It is noted that the solutionof Eq. 3.27 involves a nested loop. The outer loop deals with the optimization problemwhereas the inner loop is for solving the uncertainty quantification problem. This kindof optimization problem is computationally expensive. One way to address this issue isto use surrogate models to approximate the computationally expensive dynamic problem.In this study, the Polynomial-Chaos-based Kriging is used as a surrogate model. In thiswork, we have used a coupled framework for solving the optimization problem, i.e., thesurrogate model and optimization are coupled so that the response surface is obtained in anadaptive manner. Within this framework, first the domain of design variables d (i.e., ωd)and the statistical properties (i.e., mean and standard deviation) of random variables Z aredefined. An initial design point d∗ ∈ ωd is assumed, ns training samples of d are generatedusing Design of Experiment scheme. In this study, the statistical properties of Z are fixed.Now another set of support points (nr in Eq. 3.17) of Z are generated at each trainingpoint of d. The reliability indices are calculated as defined in Eq. 3.18 corresponds to eachof the training points d(i). With the outputs of training samples, the surrogate model istrained. Once the response surface is generated, the optimization problem in Eq. 3.27 issolved using GA to evaluate the new design point d∗. For convergence, a parametric indexis used i.e. absolute value of difference between optimal solution obtained from successive583.4. NUMERICAL EXAMPLESiteration (ϵ = |dj+1 − dj |, where j is iteration number). The iteration stops when ϵ < ϵopt,where ϵopt is assumed as 1E-05. It is noted that if the convergence criteria is not satisfiedwith the optimal solution, d∗ obtained from the present iteration, then d∗ is added to theset of training samples used in the present iteration. The new set of training samples isused to build the response surface in the next iteration. These steps are continued for moreaccuracy towards an optimal solution until convergence is achieved. The overall frameworkis shown in the form of a flowchart in Fig. 3.3.Experimental Test, Numerical Results & DiscussionDescription of Negative Stiffness Device & Experimental Setup The experimentwas conducted in the laboratory of Earthquake Engineering Research & Test Centre atGuangzhou University, China [CQC+19]. The primary structure consists of a steel momentresisting frame with a 1000 mm × 1000 mm × 16 mm floor as shown in Fig. 3.4. Thecross-section of each column is 100 mm × 12 mm, with a clear height of 2000 mm. Thetotal weight of the primary structures, including all connection accessories, is 241.9 kg. Toensure high-storey drift capacity without yielding, the columns were designed with a steelyield strength of 690 MPa. The direction of weak bending stiffness of the columns wasadopted as the operational direction of the test.The equivalent mass, stiffness, and damping ratio of the SDOF system representation ofthe moment-resisting frame were estimated with fixed joint boundary conditions. The valueswere subsequently obtained by a trial-and-error procedure to match the natural vibrationtest. The final lumped mass (M), stiffness (K), natural frequency, and modal dampingratio of the equivalent SDOF system were determined to be 189.4 kg, 26.2 Nmm−1, 1.87Hz, and 0.35%, respectively.The NES device is shown in Fig. 3.5. The device consists of a stack of steel plates as theNES mass, a pair of tube sliders on top of which the movable mass can slide, and a pair ofcoil springs. The total mass of the NES includes the mass of the sliders. By means of linearbearings, the sliders are set on two straight, parallel tubes grounded on the top of the mainstructure, which provides reasonable friction in the operational directions. Each springis connected to a movable mass by hinged connections at one end and a steel pedestalat the other end. The springs are fixed at an inclination which depends on the springphysical unloaded length (L) and spring compressed length (L0). To avoid eccentricity, asthe assembled NES mass slides, its mass centroid and the axes of the springs are on thesame level. To prevent buckling of the hinged coil springs, each spring is coupled with alow-friction driven sleeve.Validation of Numerical Model For the experiment, the 1940 El Centro ground accel-eration is used to validate the numerical simulation. The peak ground acceleration (PGA)of the earthquake motions for the shake table is modulated to 0.05 g, 0.1 g, 0.15 g, and 0.2g. The experiment is performed with the NES design values summarized in Table 3.1.The floor displacement time histories for the experimental and numerical simulationsunder the El Centro earthquake record (for PGA 0.15 g) are shown in Fig. 3.6. Maximum593.4. NUMERICAL EXAMPLESStartDefine design variables d with bounds (di,ll , di,ul), ∀iProvide tolerance εoptSelect initial value of dGenerate support points (d(i)) for dEvaluate β(i) for each support points d(i) as in Eq.14Train surrogate modelEvaluate response statistics by using MCS on trained modelOptimizationNew design variables, di,jConvergence?j j+1 StopNoYesAdaptive LoopYesNoSet j = 1j < jmaxOptimization LoopGenerate nr sample points for d(i)Figure 3.3: Flowchart of the reliability-based design optimization603.4. NUMERICAL EXAMPLESFigure 3.4: The test structure installed on the shake tableFigure 3.5: Prototype of negative stiffness device with sliding frictionpeak floor displacements for uncontrolled responses from experimental and numerical simu-lations are found to be 44.85 mm and 43.56 mm, respectively. The error between these twosimulations is 2.87%. Similarly, for controlled responses, the maximum peak displacementsare found to be 27.54 mm and 26.73 mm, respectively. The error between these two sim-613.4. NUMERICAL EXAMPLESTable 3.1: Design parameter values for the present NESParameter kNES (Nm−1) L0 (m) L (m) m (kg) µv µsTheoretical design value 5850 0.140 0.150 9.18 0.033 0.0350 10 20 30 40 50 60Time (s)-40-30-20-10010203040Displacement (mm)AnalyticalExperimental(a)0 10 20 30 40 50 60Time (s)-30-20-100102030Displacement (mm)AnalyticalExperimental(b)Figure 3.6: Comparison of experimental and analytical primary structure displacement timehistory under El Centro with PGA 0.15g; (a) Uncontrolled and (b) Controlled responsesulations is 2.9%. Small inconsistencies in response are observed between experimental andnumerical simulations due to the effects of structure-table interaction, nonlinear frictionproperties, etc. that are not considered in the numerical models.Optimization Results In this study, all random variables are assumed to be uniformlydistributed within a given bound, tabulated in Table 3.2. For the optimization, the designvectors with lower and upper bounds are considered as kNES = [4000, 7000] Nm−1 and L0= [0.07, 0.15] m of the present NES. Other parameters such as design lifetime period TD andearthquake occurrence rate ν are considered as 50 years and 0.1 (1 event in every 10 years)[NLC+18], respectively. For the optimization, the genetic algorithm is considered with amaximum generation of 50, a population size of 100 in each generation, and a crossoverfraction of 0.9 [MG15].With the above parameters, a PCK surrogate-based design optimization procedure iscarried out in this example. The support points are generated using the Sobol sequence[WDG+14]. In this simulation, 75 numbers of support points of design variables are con-623.4. NUMERICAL EXAMPLESTable 3.2: Statistical Properties of uncertain variablesVariable Description Unit Distribution & BoundM Mass of primary structure kg U (173,206)ξ Damping ratio of primary structure % U (0.26,0.44)K Stiffness of primary structure Nmm−1 U (19,33)ωg Natural frequency of soil layer rads−1 U (7.9,16.2)ηg Critical damping ratio of soil layer % U (0.26,0.53)S0 Spectral intensity of bedrock motion m2s−3 U (0.016,0.033)sidered, and the objective function is the minimization of the probability of failure of theprimary structure under uncertainty. In Eq. 3.7, the barrier α = 40 mm is chosen as thetop floor displacement [CR11]. Once the surrogate model is trained using support points,10,000 sample points are generated, and the surrogate model is used to compute the non-linear response surface. The estimation of the accurate value of the probability of failureis dependent on the sample size, as depicted in Eq. 3.17. Fig. 3.7 shows the convergenceof the reliability index with increasing sample size. The samples, Z(i) are generated usingMATLAB considering the same seed of random number generator. It is seen from Fig. 3.7that at least 350 samples are required to stabilize the optimum value of the reliability index.With these samples, the optimization is carried out, and the nonlinear response surface isdepicted in Fig. 3.8, which corresponds to 350 samples. The optimal design vectors kNESand L0 are found to be 5656.4 Nm−1 and 0.13 m, respectively. The main purpose of thecontroller is to reduce the vibration level, which reduces the probability of failure of thestructure. It is seen that at the optimal design vector, the reliability index is obtained as3.3.100 200 300 400 500 600 700-3.6-3.4-3.2-3-2.8-2.6- Figure 3.7: Plot showing convergence of reliability index with different sample size (nr)633.4. NUMERICAL EXAMPLESFigure 3.8: Nonlinear response surface for structure - NES with negative stiffness systemby minimizing top floor displacement of primary structure obtained using PCKComparison between proposed NES and cubic NES A comparative study is carriedout to show the enhanced performance of the present NES compared to the cubic NES. Thefloor displacement and acceleration time history are shown in Fig. 3.9(a) and Fig. 3.9(b),respectively. Peak and root mean square (rms) displacements of the primary structure arereduced by 33% and 48% using cubic NES, whereas in the case of present NES, the sameare obtained as 38% and 64%, respectively. Similarly, the peak and rms accelerations of theprimary structure are reduced by 34% and 48% using the cubic NES, whereas the same areobtained as 38% and 65%, respectively, for the present NES. It is seen that the transientinternal resonance captured by present NES is more than cubic NES due to combining effectsof negative stiffness and friction, which results in the more RMS reduction by present NES.In this study, sensitivity analysis is carried out by considering performance indices, whichare tabulated in Table 3.3. Table 3.4, 3.5, 3.6 and 3.7 summarize the performance of theTable 3.3: Definitions of the evaluation criteriaEvaluation Criterion Related Structural Response Variable ExpressionJ1 Energy Dissipation J1 = Ec/EucJ2 Maximum Absolute Floor Displacement J2 = Xp,c/Xp,ucJ3 RMS Floor Displacement J3 = Xrms,c/Xrms,ucJ4 Maximum Absolute Floor Acceleration J4 = X¨p,c/X¨p,ucJ5 RMS Floor Acceleration J5 = X¨rms,c/X¨rms,ucAbbreviation: uc : Uncontrolled ; c : Controlled ; p : peak ; rms : root mean squareproposed NES and compared it with the performance of the cubic NES for different ground643.4. NUMERICAL EXAMPLESaccelerations with different PGA values. It is observed that the promising performanceof the present NES is achieved in the context of peak and RMS reduction. The NESdisplacement time history and force-displacement hysteresis behavior for the cubic NESand the present NES are shown in Fig. 3.10(a) and Fig. 3.10(b), respectively. Fig. 3.11(a)and Fig. 3.11(b) show the Fourier spectrum for primary structure floor displacement andNES displacement, respectively.0 10 20 30 40 50 60Time (s)-40-30-20-10010203040 Displacement (mm)UncontrolledWith Cubic NESWith Present NES(a)0 10 20 30 40 50 60Time (s)-600-400-2000200400600 UncontrolledWith Cubic NESWith Present NES(b)Figure 3.9: Primary structure response under El Centro earthquake record with PGA 0.15g(a) displacement and (b) acceleration time historiesTable 3.4: Performance of proposed NES representing peak value of the floor displacementunder El Centro earthquake recordPGAPeak Storey Drift (mm) 100 - J2 (%)xuc xC,NES xP,NES C,NES P,NES0.05g 15.74 12.92 10.27 17.92 34.750.10g 31.98 19.87 19.73 37.86 38.310.15g 45.82 30.34 28.23 33.78 38.390.20g 54.04 36.35 34.52 32.73 36.12Abbreviation: C,NES : Cubic NES ; P,NES : Present NESTo investigate the performance of present NES over the cubic stiffness, the stiffnessof the primary structure is varied from the 30% to 150% of the design value. Fig. 3.12shows the energy dissipated of primary structure by present NES and cubic NES underdifferent PGA values of El Centro earthquake record. Similarly Fig. 3.13 and Fig. 3.14653.4. NUMERICAL EXAMPLES0 10 20 30 40 50 60Time (s)-50050100150200 Relative drift (mm)With Cubic NESWith Present NES(a)-50 0 50 100 150Relative drift (mm)-300-200-1000100200300Control force (N)With Cubic NESWith Present NES0 50 100-20-1001020(b)Figure 3.10: NES Response under El Centro earthquake record with PGA 0.15g (a) Relativedrift time history of NES w.r.t. primary structure; (b) Control force - NES displacementhysteresis0 1 2 3 4Frequency (Hz)10-610-410-2 Fourier amplitudeUncontrolledWith Cubic NESWith Present NES1.4 1.6 1.8 2 2.210-410-310-2(a)0 1 2 3 4Frequency (Hz)10-510-410-310-210-1 Fourier amplitudeWith Cubic NESWith Present NES(b)Figure 3.11: Fourier amplitude spectrum for (a) Primary structure displacement and (b)Relative drift of NES w.r.t. primary structure under El Centro earthquake record withPGA 0.15gshow the peak and RMS reduction of the floor displacement under the same PGA values663.4. NUMERICAL EXAMPLESTable 3.5: Performance of proposed NES representing RMS value of the floor displacementunder El Centro earthquake recordPGARMS Storey Drift (mm) 100 - J3 (%)xuc xC,NES xP,NES C,NES P,NES0.05g 5.02 3.06 1.56 39.04 68.920.10g 10.46 7.73 3.76 26.10 64.050.15g 14.85 7.70 5.21 48.15 64.920.20g 17.63 15.45 7.01 12.36 60.24Table 3.6: Performance of proposed NES representing peak value of the floor accelerationunder El Centro earthquake recordPGAPeak Acceleration(cms−2)100 - J4 (%)xUC xC,NES xP,NES C,NES P,NES0.05g 217.78 178.77 142.16 17.91 34.720.10g 442.43 274.93 272.94 37.86 38.310.15g 633.87 419.66 390.55 33.79 38.390.20g 747.62 502.91 480.31 32.73 35.75Table 3.7: Performance of proposed NES representing RMS value of the floor accelerationunder El Centro earthquake recordPGARMS Acceleration(cms−2)100 - J5 (%)xUC xC,NES xP,NES C,NES P,NES0.05g 69.47 42.43 21.68 38.92 68.790.10g 144.73 106.96 52.08 26.10 64.020.15g 205.48 106.55 72.10 48.15 64.910.20g 243.95 213.76 98.30 12.37 59.71of El Centro earthquake record, respectively. It is evident that the present NES is morerobust in nature compared to the cubic NES when the natural frequency of the primarystructure is shifted from the design value.3.4.2 Example 2: Reliability-based Optimization of SMA-basedDamped Outrigger StructureStructural model descriptionA reduced-order model of a high-rise building, which is idealized as a cantilever beam(Fig. 3.15a), is considered. The structural deformations are formulated by the behavior ofthe core. The SMA-based damper is used for the damped outrigger system, which is located673.4. NUMERICAL EXAMPLES0.3 0.6 0.9 1.2 1.5Stiffness ratio0 20 40 60 80 100100 - J1 (%)Cubic NES (0.05g)Present NES(0.05g)Cubic NES (0.10g)Present NES(0.10g)(a)0.3 0.6 0.9 1.2 1.5Stiffness ratio0 20 40 60 80 100100 - J1 (%)Cubic NES (0.15g)Present NES(0.15g)Cubic NES (0.20g)Present NES(0.20g)(b)Figure 3.12: Energy dissipation for different stiffness ratio of primary structure under dif-ferent PGA values of El Centro earthquake record for (a) 0.05g to 0.10g and (b) 0.15g to0.20g0.3 0.6 0.9 1.2 1.5Stiffness ratio01020304050607080100 - J2 (%)Cubic NES (0.05g)Present NES(0.05g)Cubic NES (0.10g)Present NES(0.10g)(a)0.3 0.6 0.9 1.2 1.5Stiffness ratio01020304050607080100 - J2 (%)Cubic NES (0.15g)Present NES(0.15g)Cubic NES (0.20g)Present NES(0.20g)(b)Figure 3.13: Peak displacement reduction of primary structure for different stiffness ratioof primary structure under different PGA values of El Centro earthquake record (a) 0.05gto 0.10g and (b) 0.15g to 0.20gin between the outrigger and the peripheral column, as shown in Fig. 3.15. A simplified683.4. NUMERICAL EXAMPLES0.3 0.6 0.9 1.2 1.5Stiffness ratio0 20 40 60 80 100100 - J3 (%)Cubic NES (0.05g)Present NES(0.05g)Cubic NES (0.10g)Present NES(0.10g)(a)0.3 0.6 0.9 1.2 1.5Stiffness ratio0 20 40 60 80 100100 - J3 (%)Cubic NES (0.15g)Present NES(0.15g)Cubic NES (0.20g)Present NES(0.20g)(b)Figure 3.14: RMS displacement reduction of primary structure for different stiffness ratio ofprimary structure under different PGA values of El Centro earthquake record for (a) 0.05gto 0.10g and (b) 0.15g to 0.20gmodel is considered where the damped outrigger is represented by an equivalent rotationalspring at the junction of the core and outrigger Fig. 3.15.The perimeter columns are assumed to be axially stiff and the outrigger is considered as arigid body [SW07]. The numerical model (shown in Fig. 3.15) consists of three components:the central core, peripheral columns, and SMA damper. The governing equation of motionfor the coupled system can be written as:Mx¨(t) +Cx˙(t) +[K+ Λ(ne2keq)]x(t) = Fg(t) (3.28)whereM, C, andK are the global mass, damping, and stiffness matrices of the central core,respectively; keq is the additional rotational stiffness to the core at the outrigger level due tothe perimeter column and SMA; Fg is the force exerted into the system due to ground accel-eration; n is the total number of SMA dampers used in between the outrigger and perimetercolumn; and e is the distance from the control device to the centre of the core, as depictedin Fig. 3.15. Λ represents the location of the outrigger with respect to the specific rotationaldegree of freedom, and x is the vector representing the translational and rotational degreesof freedom (DOFs) of the floors, which is written as x =[x1 θ1 x2 θ2 · · · xN θN]T,where N is the total number of floors. As the perimeter column and SMA are connected inseries, the equivalent stiffness (keq) is equal to (kˆs + iωcˆs)/(kc + kˆs + iωcˆs) [Gam08] wherei =√−1. kc = EcAc/αH is the axial stiffness of each perimeter column, where EcAc isthe axial stiffness of the column; H is the total height of the core and αH is the locationof the SMA outrigger. kˆs and cˆs are stochastically equivalent stiffness and damping offered693.4. NUMERICAL EXAMPLESαHHCore StructurePerimeter ColumnOutriggerẍgSMA M kcForceksመ𝐤sƸ𝐜sDisplacementEcAc EcAcEIe eFigure 3.15: Schematic representation of SMA based damped outrigger structureby SMA, respectively, which are discussed in Section 3.4.2.The high-rise building can be modeled by either an Euler-Bernoulli beam or Timoshenko-beam element. In the Euler-Bernoulli beam model, the cross-section of the element is as-sumed to be perpendicular to the bending line. This model does not capture the exactdynamics of the high-rise building [FSJX+19]. The Timoshenko-beam model allows ro-tation between the cross-section and the bending line. This rotation arises due to sheardeformation, which is not considered in the Euler-Bernoulli beam. A beam element is con-sidered to have four DOFs, i.e., two translational (xk , xk+1) and two rotational (θk , θk+1)at two nodes. For the Timoshenko-beam model, the mass matrix [TWW73] for one elementcan be written asMe =m¯L420(1 + ϕ2)·(156 + 294ϕ+ 140ϕ2) (22 + 772 ϕ+352 ϕ2)L(54 + 126ϕ+ 70ϕ2)−(13 + 632 ϕ+352 ϕ2)L(22 + 772 ϕ+352 ϕ2)L(4 + 7ϕ+ 72ϕ2)L2(13 + 632 ϕ+352 ϕ2)L −(3 + 7ϕ+ 72ϕ2)L2(54 + 126ϕ+ 70ϕ2) (13 + 632 ϕ+352 ϕ2)L(156 + 294ϕ+ 140ϕ2)−(22 + 772 ϕ+352 ϕ2)L−(13 + 632 ϕ+352 ϕ2)L −(3 + 7ϕ+ 72ϕ2)L2 −(22 + 772 ϕ+352 ϕ2)L(4 + 7ϕ+ 72ϕ2)L2(3.29)703.4. NUMERICAL EXAMPLESwhere m¯ is the distributed mass along with the height of the core, and L is the height ofeach storey. The stiffness matrix for one beam element is written asKe =EIL3(1 + ϕ)12 6L −12 6L6L (4 + ϕ)L2 −6L (2− ϕ)L2−12 −6L 12 −6L6L (2− ϕ)L2 −6L (4 + ϕ)L2 (3.30)where EI is the bending stiffness of the core. In the above equations, ϕ = 12EIGkAL2is aconstant, where G is the shear modulus of the beam element, A is the cross-sectional areaof the element, and k is the shear coefficient [Cow66]. The Euler-Bernoulli beam, the massand stiffness matrices for one element are formulated by using ϕ = 0 in Eq. 3.29 and 3.30.Once the global mass (M) and stiffness (K) matrices are formulated, using Me and Ke,global damping matrix is written asC = β1M+ β2K (3.31)where the coefficients β1 and β2 are computed as[β1β2]= 2[ 1ω1ω11ω2ω2]−1 [ξ1ξ2](3.32)where ω1 and ω2 are the first and second modal frequencies of the bare structure, anddamping ratios corresponding to the modes are represented by ξ1 and ξ2, respectively. Theinertial force caused by an earthquake (x¨g) on one element is given byFg,e = −mLx¨g12[6 L 6 −L]T (3.33)Frequency Domain AnalysisFor random earthquake excitation, the power spectral density is written asSgg(ω) =1T[Xg(ω)X∗g (ω)] (3.34)where Xg(ω) is the FFT of xg(t); X∗g (ω) is the conjugate of Xg(ω); T is the strong motionduration of excitation. For the frequency-domain solution, taking the Fourier transforma-tion on Eq. 3.28, the transfer function is given byH(jω) =1(K− ω2M) + jωC (3.35)The power spectral matrix of displacement responses of the coupled system is obtained asfollowsSxx(ω) =[H(jω)MI]Sgg(ω)[ITMTHT (−jω)] (3.36)713.4. NUMERICAL EXAMPLESwhere I is the influence vector. As Sgg(ω) is considered as a zero-mean process as in Eq. 3.25,it is obvious that the structural responses should be zero mean processes. The mean squareof structural displacement responses isσ2x = E[x2] =∫ ∞−∞Sxx(ω)dω (3.37)Similarly, mean square response for velocity and acceleration responses of the coupled sys-tem areσ2x˙ = E[x˙2] =∫ ∞−∞ω2Sxx(ω)dω σ2x¨ = E[x¨2] =∫ ∞−∞ω4Sxx(ω)dω (3.38)Material Model of Shape Memory AlloyShape memory alloy (SMA) is a smart material that transforms between two stablephases (i.e., austenite and martensite) and can recover the original, undeformed shapeby changing temperature (i.e., shape memory effect) or removal of stresses (i.e., super-elasticity effect) [FYLX14, SHC10]. Fig. 3.16 shows the typical stress - strain - temperaturehysteresis of SMA where the path A-B-C-D-E-A shows the shape memory effect and super-elasticity effected is showed by path G-H-I-J-K-L-G. Different types of SMA are available inthe market, such as nickel-titanium (Ni-Ti), copper, and ferrous-based SMA. Among those,Nitinol (i.e., Ni-Ti alloy) is popular for its excellent thermo-mechanical and thermo-electricalproperties. SMA offers excellent fatigue and corrosion resistance [DC01a, DC01b]. Due tothese properties, SMAs are utilized for different vibration control applications. Wilde et al.[WZGF98] proposed a device to control the vibration incorporating the bending of Nitinolbars. Aiken et al. [ANWK93] showed that by using SMA in the bracing element, thestructural damping is increased from 0.5% to 3% and reliably mitigate structural responses.Inaudi and Kelly [IK94] showed an effective nonlinear damping mechanism by using SMAin an orthogonal direction to the motion of a tuned mass damper. Savi and Mamiya [SM95]investigated the super-elasticity behavior of SMA and proposed that it can be effective tomitigate the vibration near-resonant conditions.Constitutive Model of SMA The different constitutive models proposed representedthe nonlinear behavior of SMA. Out of these, Brinson model [Bri93] has illustrated theSME and SE behavior simultaneously under all temperatures. Practically, this model is toocomplicated to use. To overcome this problem, the Graesser-Cozzarelli model [GC91, GC94]is frequently used for passive vibration control problems. This model has illustrated the SEbehavior above austenite finish temperature in which the model is based on classical Bouc-Wen model [Wen80] representing one-dimensional force-deformation nonlinear hysteresisof SMA. Yan and Nie [YN00] proposed an improved force-displacement hysteresis modelto facilitate the process of stochastic linearization. This model is widely used in SMA-based nonlinear vibration control problems [MGC13]. In this literature, Yan and Nie model723.4. NUMERICAL EXAMPLESFigure 3.16: Typical stress-strain-temperature hysteresis behavior of SMA[YN00] is used. According to this model, the restoring force offered by superelastic SMA isexpressed asFs(xs, x˙s) = αsksxs + (1− αs)kszs (3.39)where xs is the elastic part and zs is the hysteretic part of the displacement in SMA. αsrepresents the stiffness ratio of post- to pre-yielding and ks is the initial stiffness of SMA inthe austenite phase. The hysteretic part of displacement in SMA, zs is expressed aszs = [1− sgn{sgn(|xs| − ua) + 1}]xs + sgn(|xs| − ua) + 12[(ub − ua)sgn(xs) + sgn(x˙s)2+ua sgn(xs)](3.40)where ua and ub represent maximum displacement in the austenite phase and displacementtriggering martensite transformation, respectively, as shown in Fig. 3.17. In Eq. 3.40, x˙s isthe velocity in the SMA, and sgn(xs) is the signum function, which is written assgn(xs) =1, if xs > 00, if xs = 0−1, if xs < 0(3.41)Stochastic Equivalent Linearization of Nonlinear Hysteresis Model of SMA Theforce-displacement hysteresis behavior of SMA is highly nonlinear. For frequency domainanalysis, it is more convenient to use a simplified hysteresis model instead of the highly non-linear model. For this purpose, the stochastic equivalent linearization [LS04] is a promisingtool to simplify the nonlinear model. In the stochastic equivalent linearization method, the733.4. NUMERICAL EXAMPLESGreasser-CozzarelliYan-NieDisplacement (xs)Force(F s)ua ubFigure 3.17: Simplified model for super-elastic SMA proposed by Yan and Nie, comparedwith Greasser -Cozzarelli modelnonlinear equation is replaced by equivalent linearized terms by minimizing the residualerrors between the actual nonlinear model and the equivalent linear terms. However, eventhough the governing equations of motion become a linearized model, the stochastic equiv-alent parameters such as stiffness and damping remain functions of the structural responsesthat maintain the nonlinearity of the system. In Eq. 3.39, the hysteretic part zs is replacedby linearizing terms, can be expressed aszˆs = kesxs + cesx˙s (3.42)where kes and ces are the stochastically equivalent stiffness and damping, respectively. Asmentioned, the error between nonlinear terms and their equivalent linearized version is givenbyE(e2) = E[(zˆs − zs)2] = E[(kesxs + cesx˙s − zs)2] (3.43)To find the unknown equivalent parameters kes and ces, the least square method is employedwhich is expressed as∂E(e2)∂kes= 0∂E(e2)∂ces= 0 (3.44)By solving Eq. 3.44, the closed-form solution for the equivalent linearized stiffness (kes) anddamping (ces) are given bykes =E[xs · zs]E[x2s]ces =E[x˙s · zs]E[x˙2s](3.45)743.4. NUMERICAL EXAMPLESSubstituting Eq. 3.40 into Eq.3.45, yields,kes =ub + ua√2πσxsexp(− u2a2σ2xs)ces =ub − ua√2πσx˙s[1− erf(ua√2σxs)](3.46)where σxs and σx˙s are the mean square values of displacement (xs) and velocity (x˙s) ofSMA, respectively. Since the values of σxs and σx˙s are unknown a priori, an iterativecomputation is needed to obtain the structural responses. In Eq. 3.46, erf(·) is the errorfunction is written aserf (xs) =2√π∫ xs0e−η2dη (3.47)Substituting Eq. 3.42 into Eq. 3.39 yields,Fs = αsksxs + (1− αs)ks(kesxs + cesx˙s)= kˆsxs + cˆsx˙s(3.48)where the equivalent stiffness and damping from Eq. 3.48 are askˆs = αsks + (1− αs)kskes = αsks + (1− αs)ksub+ua√2πσxsexp(− u2a2σ2xs)cˆs = (1− αs)ksces = (1− αs)ks ub−ua√2πσx˙s[1− erf(ua√2σxs)] (3.49)-300 -200 -100 0 100 200 300xs (mm)-10-50510F s (MN)(a)-300 -200 -100 0 100 200 300xs (mm)-8-6-4-202468F s (MN)(b)Figure 3.18: Force displacement hysteresis for SMA; (a) Classical flag shaped and (b) Afterstochastic equivalent linearization753.4. NUMERICAL EXAMPLESOptimization of Damped Outrigger under UncertaintyEvaluation of Reliability The optimization is utilized to design the SMA-based dampedoutrigger system with the objective of minimizing the probability of failure of the structureunder earthquakes. The optimization problem is formulated asd∗ = argmind[Pf (d)],s.t. dll ≤ d ≤ dul. (3.50)where d represents the design vector with bounded by lower and upper values, denoted bydll and dul, respectively. Pf is the probability of failure of the structure, which is estimatedusing a level-crossing formulation. In this example, maximum top floor displacement istaken as a threshold limit.Efficient Global Optimization In this study, efficient global optimization method isadopted, proposed by Jones et al. [JSW98]. In this method, the set of training samples areformed in an adaptive manner using the combination of Kriging prediction Yˆ (x) and meansquare error s2(x). The details of Kriging model is presented in Section 2.3.3. ExpectedImprovement is one of the most effective Kriging-based adaptive sampling criteria. Thealgorithm starts with initial training points x =[x1 x2 . . . xN]Tand correspondingresponses are Y =[Y1 Y2 . . . YN]T. Here, the training points x are the samples ofdesign vector, d, generated using DoE strategy, mentioned in Chapter 2. Once an initialKriging model Yˆ (x) is built using DACE toolbox [LNS02]. With the approximate modelresponse and mean square error s2(x), the new training samples are formulated using theimprovement measure I(x) = (Ymin− Yˆ (x)), where Ymin = min(Y ). An alternative methodof adaptive sampling is to estimate the expected value of improvement I(x), which is provedan efficient method [JSW98]. The expected value can be estimated as followsE[I(x)] =(Ymin − Yˆ (x))Φ(Ymin − Yˆ (x)s)+ sϕ(Ymin − Yˆ (x)s)(3.51)where Φ(·) and ϕ(·) are the normal cumulative distribution function and normal probabilitydensity function, respectively. A new training sample is found by maximizing the expectedimprovement, as in Eq. 3.51,xnew = argmax{E[I(x)]}, ∀x (3.52)Then, the training data is updated by including new sample, x = x ∪ {xnew}. Withthe augmented set of training samples, the Kriging model is reformulated. The wholeprocess is continued until convergence is achieved. For more clarity, the overall frameworkis represented by a flowchart, as depicted in Fig. 3.19.763.4. NUMERICAL EXAMPLESAlgorithm 5 Kriging based Efficient Global Optimization1: Start the iteration i = 1.2: Generate initial training samples, x using the Sobol sequence.3: Evaluate the responses Y corresponding to each training sample using Eq. 3.17.4: Train Kriging model(Yˆ)using x and Y .5: Find minimum value of Y , Ymin = min(Y ).6: while i = 2, 3, . . ., ≤ imax or∣∣∣Yˆ iopt − Yˆ i−1opt ∣∣∣ > ϵtol do7: Compute expected value of improvement using Eq. 3.51.8: Evaluate new point, xnew using Eq. 3.52.9: Include new point to training samples set, used in previous iteration, x = x∪{xnew}.10: Train Kriging model using new set of training samples.11: Estimate optimal value of probability of failure using brute-force MCS method.12: end while13: Outcome: x∗ = argminx[Yˆ (x)].Numerical Results & DiscussionHere, a 40-storey building coupled with SMA-based damped outriggers is considered fordesign optimization purpose. The building has 30 m × 30 m plan dimension (Fig. 3.20) andeach storey height is 4 m. The concrete core of the building has a plan dimension of 14 m× 14 m with a 0.4 m thickness. The building consists of eight perimeter columns made ofsteel with a cross-sectional area of 0.15 m2. The structural parameters considered in thisstudy are: the net area of the core of the building is 21.76 m2, the moment of inertia ofthe core is 671.37 m4, the moduli of elasticity for the concrete core and steel columns are24.82 GPa and 200 GPa, respectively. The distributed mass along the height of the buildingis 115584 kgm−1. The structural damping is assumed to be 1% for each mode. The spanof the outrigger (e) is taken as 8 m. The radius of the core (R) is estimated by the squareroot of the moment of inertia over the net area of the core, which is found to be 5.55 m.The slenderness ratio (i.e., ratio of the total height of the building to the radius of the core)is calculated as 28.8. Readers may refer Gamaliel et al. [Gam08] for more details of thebuilding.Fundamental time periods of the bare structure using results from Gamaliel et al.[Gam08] (where Euler - Bernoulli beam model is considered) and Timoshenko beam modelsare summarized in Table 3.8. It is found that the relative error of the fundamental timeperiod between the above-mentioned models is 0.8%, whereas this error becomes muchlarger in higher modes. For an undamped outrigger structure, the first and second modalfrequencies for different locations of outrigger along the height of the structure are shownin Fig. 3.21. Fig. 3.22 shows inter-storey drift and inter-storey drift angle for an undampedoutrigger system obtained using Euler-Bernoulli and Timoshenko beam models. It is seenfrom Fig. 3.22 that with the Euler-Bernoulli beam model, responses of the outrigger struc-ture are under-estimated.773.4. NUMERICAL EXAMPLESStartDefine design variables d with bounds (dj,ll , dj,ul), ∀jProvide tolerance εtolGenerate initial training samples x for dEvaluate model response for each support points using Eq.28Train surrogate modelEvaluate response statistics by using MCS on trained modelOptimizationConvergence?StopNoYesYesNoSet i = 1i < imaxEstimate new training sample (xnew) by maximizing expected value of improvement measure using Eq.30 Update the training data set as x = x ⋃ {xnew}Optimization LoopFigure 3.19: Flowchart for the efficient global optimization783.4. NUMERICAL EXAMPLES8m14m8m8m 8m14mCoreOutriggerPerimeterColumnFigure 3.20: Plan view of 40-storey buildingTable 3.8: Comparison of fundamental periods for Euler-Bernoulli and Timoshenko beammodelsPeriods Euler-Bernoulli Model Timoshenko Model Error (%)(s) 1O 2O {( 2O- 1O)/ 1O}× 1001st 3.81 3.84 0.802nd 0.61 0.64 5.203rd 0.22 0.24 11.164th 0.11 0.13 18.185th 0.07 0.09 28.57Table 3.9: Parameters of Kanai-Tajimi SpectrumRandom Variables Units Mean Value COV Distributionωg rad/s 12.100 20% Uniformηg - 0.400 20% UniformS0 m2/s3 0.025 20% UniformHere, SMA-based damped outrigger system is proposed, whose design parameters areoptimized to ensure reliable performance when exposed to uncertain environment. The seis-793.4. NUMERICAL EXAMPLES0 10 20 30 40Location of Outrigger (Story)0.260.280.30.320.340.361st Mode Natural Frequency (Hz)Euler-BernoulliTimoshenko(a)0 10 20 30 40Location of Outrigger (Story)1.551.61.651.71.751.82nd Mode Natural Frequency (Hz)Euler-BernoulliTimoshenko(b)Figure 3.21: Modal frequencies of undamped outrigger structure for different locations ofoutrigger0 20 40 60 80Interstorey Drift (mm)0510152025303540StoryTimoshenkoEuler-Bernoulli(a)-5 0 5 10 15 20Interstorey Drift Angle (rad) 10-40510152025303540StoryTimoshenkoEuler-Bernoulli(b)Figure 3.22: (a) Inter-storey drift and (b) inter-storey drift angle for Euler-Bernoulli andTimoshenko beam modelmic excitation is assumed to be a zero-mean Gaussian process, which is characterized by theKanai-Tajimi model. The parameters related to this model, such as the spectral intensity ofbedrock motion, natural frequency, and critical damping ratio, are considered random. The803.4. NUMERICAL EXAMPLESdistributions of these random variables are assumed to be uniform distribution, as describedin Table 3.9. The probability of failure of the system is estimated using the first-passageformulation as described in Section 3.4.2, which is considered as an objective function foroptimization. The threshold level is considered the maximum top floor displacement. Thelimit of this threshold is taken as 0.0025H [Gam08] where H is the total height of the build-ing. Also, duration of seismic ground motion is taken as 30 s [MRC13]. For the designoptimization, the design parameters are considered as the location of the outrigger, α asshown in Fig. 3.15, which ranges from 0 to 1 and normalized transformation strength ofSMA F0 = [0.05 , 0.5], where F0 = ksub/(m¯Lg). The other parameters associated withSMA are assumed to be ua = 0.005 m, ub = 0.02 m, and αs = 0.001 [MGC13].For optimization, the Kriging meta-model is used to approximate the original responsesurface. Since the accuracy of the surrogate-assisted optimization is highly dependent onthe meta-model, the adaptive sampling technique is used to generate the support points fortraining the Kriging model. For this purpose, an efficient global optimization technique isused, which helps to build the set of support points in an adaptive manner by maximizingthe expected improvement performance metric. The details are described in Section 3.4.2.Initially, 75 support points are generated using the Sobol sequence to train the surrogate.The objective function is the minimization of the probability of failure of the structureunder stationary ground excitation, as characterized by the Kanai-Tajimi model. Once thesurrogate model is trained using support points, 50,000 prediction points are simulated us-ing the Monte-Carlo method and passed through the trained surrogate model to form thenonlinear response surface. The new support point is estimated using the response of eachprediction point obtained from Kriging by maximizing the expected value of the improve-ment performance metric. This new support point is added to the set of support pointsused in previous iteration, and this new set of support points is used to train surrogate inthe next iteration. This entire procedure continues until convergence is achieved. Once thefinal response surface is obtained, the optimal design parameters are estimated correspond-ing to the minimum probability of failure of the structure. The nonlinear response surfaceis shown in Fig. 3.23(a). The optimal location of the outrigger is found as at 29-th storeyand the optimal value of F0 is obtained as 0.49. The probability of failure at the designpoints is found to be 2.5E-4, i.e., reliability index (βˆ, in Eq. 3.18) = 3.48. Fig. 3.23(b)shows enhancement of the performance level after optimization under uncertainty. Here,deterministic design is referred to as response-based optimal design with no uncertainty.The probability of failure in this case is found to be 4.1E-3. Therefore, the proposed op-timization algorithm improves the performance of the proposed damped outrigger over thedeterministic design by a factor of 17.With the optimized design variables, the performance of SMA based outrigger is shownin Fig. 3.24. In Fig. 3.24(a), the power spectral density function of top floor displacement forthe bare structure, undamped outrigger, and SMA-based outrigger structure is shown. Theroot mean square (RMS) value of the top floor displacement for the SMA-based outriggeris reduced by 41% compared to the undamped outrigger structure. The corresponding timehistories for top floor displacement are shown in Fig. 3.24(b). The classical flag shape and,813.4. NUMERICAL EXAMPLES(a)EGO Deterministic012345P f10-3(b)Figure 3.23: (a) Nonlinear response surface obtained from Kriging ; (b) Comparison offailure probability between efficient global optimization and deterministic optimizationafter stochastic linearization, force-displacement hysteresis for SMA are shown in Fig. 3.18.It is evident that the re-centering property of SMA has been captured. With the optimizeddesign variables, the inter-storey drift and inter-storey drift angle are calculated for all thefloors for the undamped and SMA-based outrigger systems, as shown in Fig. 3.25. It is seenthat inter-storey drift is reduced using SMA outriggers for all floors, which help to reducebase shear and overturning moments at the base of the structure. The storey shear forceand overturning moment are shown in Fig. 3.26.Sensitivity Analysis Finally, an effort is made to investigate the first-passage probabil-ity of failure of the proposed damped outrigger system for different levels of design criteria.To investigate it, the threshold levels of maximum top floor displacement, superstructuredamping ratio, and seismic intensity are varied, and the optimal probability of failure isplotted in Fig. 3.27. In Fig. 3.27(a), the threshold limit is ranged from 0.002H to 0.01Hand the superstructure damping ratio is ranged from 1% to 3%, keeping the same valuesof ground excitation parameters as tabulated in Table 3.9. It is seen that with increasingthreshold levels, the first-passage probability of failure decreases. Also, it is evident fromFig. 3.27(a) that no significant improvement is observed while varying the superstructuredamping ratio. In Fig. 3.27(b), seismic intensity is varying from 0.025 m2s−3 to 0.075 m2s−3with uncertainty level of 10, 20 and 30%, respectively, keeping threshold level and super-structure damping as 0.0025H and 1%, respectively. It is seen that the probability of failureincreases with increasing seismic intensity.823.4. NUMERICAL EXAMPLES0 1 2 3 4 5Frequency (rad/s) 10-410-310-210-1100101PSDF of top floor displacement Bare StructureUndamped OutriggerSMA Based Outrigger(a)0 50 100 150 200t (s)-2000-1000010002000u (mm)Bare StructureUndamped OutriggerSMA Based Outrigger(b)Figure 3.24: (a) PSDF and (b) Time histories of top floor displacement for the bare struc-ture, undamped outrigger and SMA based outrigger system at the optimal design values0 20 40 60 80 100 120Interstory Drift (mm)Base10203040StoryBare StructureUndamped OutriggerSMA Based Outrigger(a)0 1 2 3 4Interstory Drift Angle (rad) 10-3Base10203040Story(b)Figure 3.25: Comparison of ; (a) Interstorey drift and (b) Interstorey drift angle for barestructure, traditional undamped and SMA based outrigger833.5. SUMMARY0 50 100 150 200 250 300Story Shear Force (GN)Base10203040StoryBare StructureUndamped OutriggerSMA Based Outrigger(a)0 0.5 1 1.5 2 2.5 3Overturning Moment (GN-m) 104Base10203040Story(b)Figure 3.26: Comparison of ; (a) Storey shear force and (b) Overturning moment for barestructure, traditional undamped and SMA based outriggerThreshold Level (m) P f(a)0.02 0.04 0.06 0.08S0 (m2/s3) 10-410-310-2P fCOV = 10%COV = 20%COV = 30%(b)Figure 3.27: Probability of failure for different (a) threshold level and superstructure damp-ing and (b) seismic intensity with different level of uncertainty3.5 SummaryIn this chapter, the effectiveness of the nonlinear energy sink with negative stiffness andsliding friction and the SMA-based damped outrigger building are discussed. To ensure the843.5. SUMMARYreliable performance of the proposed controller, the tuning parameters are determined bymaximizing structural reliability, which is evaluated using an outcrossing rate approach. Tosolve the reliability-based optimization problem, a coupled framework is used, where theouter loop solves the optimization problem and the inner loop minimizes the propagation ofuncertainties. The entire framework becomes computationally expensive. To address thisissue, a surrogate model is used to approximate the computationally expensive dynamicsolver. The major observations from this chapter are highlighted below:∗ It is found that the proposed controller is more effective at mitigating seismic vibrationas compared to cubic NES. For example, the cubic NES is able to reduce 33% of thepeak and 48% RMS responses, whereas with the proposed controller, the peak andRMS responses are reduced by 38% and 65%, respectively.∗ For SMA-based damped outrigger structure, with the optimal parameters of SMA,the proposed system can able to reduce 56% in peak and 60% in RMS responses oftop floor displacement compared with the conventional undamped outrigger system.Also, the proposed system can able to reduce the inter-storey drift, inter-storey driftangles for all floors in an efficient manner, results in reduction in base shear and anoverturning moment of the structure.85Chapter 4Reliability Analysis of Structuresusing Probability DensityEvolution MethodIn Chapter 3, the reliability of the structures is estimated using level-crossing formula-tion. The probability of failure of the structures is calculated based on the stationary as-sumption. To alleviate this assumption, the numerical integration method is suitable, whichestimates the probability density function (PDF) of the performance function for both sta-tionary and non-stationary processes. The Fokker–Planck–Kolmogorov (FPK) equation isone of the common methods for estimating the transition PDF of a nonlinear dynamicalsystem’s responses subjected to a stochastic excitation representing a Markov vector pro-cess. In general, the discretization method is used, in which the entire domain is discretizedinto several grid points and the FPK equation is solved for those grid points. Therefore,when the response is evaluated at any point other than the grid points, the interpolationtechnique has been used. This method becomes inefficient for higher-dimensional problemsdue to the curse of dimensionality. Most of the studies focus on the stationary solutionof the FPK equation. However, the transient solution is essential for calculating the firstpassage failure probability. Because the first passage failure may occur before achievingthe stationary condition. The transient solution of the FPK equation for a highly nonlin-ear system with a large number of degrees of freedom is difficult to estimate because theevolution equation is coupled with the physical system. To overcome these drawbacks ofthe FPK solution, in this chapter, the probability density evolution method (PDEM) isadopted for estimating the probability density function (PDF) of the performance function,which is applicable for both stationary and non-stationary processes. The partial differen-tial equations involved in PDEM are solved using the finite difference method with totalvariation diminishing (TVD), where a proper strategy is required to generate the repre-sentative points of the random variables present in the system. The purpose of using astrategy for generating points is that the representative points should be covered over theentire probability space, which helps PDEM estimate PDF accurately within the distribu-tion domain. The generalized F (GF) discrepancy-based method is adopted for estimatingPDF using PDEM. Though GDEE estimates PDF accuracy, it requires a large numberof representative points, which becomes inefficient for time-variant reliability analysis andreliability-based design optimization of high-fidelity models. To reduce the computationalburden, PDEM coupled with the stochastic spectral embedding method is adopted in this864.1. OBJECTIVESstudy. The stochastic spectral embedding method is an advanced version of PCE that com-bines spectral representation and adaptive domain decomposition. The entire domain isdivided into a number of subdomains. This discretization is updated adaptively until theresidual between the original and predicted responses becomes minimal. The details of theproposed framework for estimating PDF and performing numerical analysis are presentedin this chapter.4.1 Objectives∗ This chapter provides a brief description of the numerical integration method forestimating the PDF of the performance function of the structures. It also providesthe complete derivation of the PDEM.∗ The reliability analysis of the stochastic systems is carried out using the first-passagereliability method, where the probability density function of the performance functionis estimated using PDEM.∗ Develop an efficient algorithm, stochastic spectral embedding surrogate-based PDEM,for reducing the computational cost involved in the traditional PDEM.∗ The performance of the proposed method is compared with Monte-Carlo simulation.4.2 Principle of Probability ConservationThe principle of probability conservation states that the system will be a probabilisticconservative system if neither a new random factor arises nor any existing random factor isvanished in a physical process.4.2.1 Time Independent SystemLet x be the input random vector, and the output random vector is denoted by y. Usingthe principle of probability conservation, it is written as∫pY(y)dy =∫pX(x)dx (4.1)where pX and pY are the PDF of x and y, respectively. If a mapping exists from x to y,i.e.,G : x→ y⇒ y = g(x) (4.2)From Eq. 4.1, it is written aspY(y) = pX(x)dxdy= pX[g−1(y)]dxdy= |J |pX[g−1(y)](4.3)874.3. LIOUVILLE EQUATIONwhere J represents the Jacobian matrix, which is expressed asJ =dxdy=dxd[g(x)](4.4)4.2.2 Time Dependent SystemFor time dependent system, Eq. 4.1 is written as follows∫pY(y, t)dy =∫pX(x, t0)dx (4.5)where x is the input random vector introduced into the system at initial instant t = t0.Similarly, like Eq. 4.2, if a mapping exists from x to y, i.e.,y(t) = g(xs, θ(xf , t))= g(x, t) (4.6)where xs and xf are the random vector from system parameters and excitation, respectively.Therefore,pY(y, t) = |J |pX[g−1(y, t)](4.7)4.3 Liouville EquationConsider a dynamic system whose state equation is expressed as followsy˙ = A(y, t) y(t0) = y0 (4.8)where y = [y1, y2, . . . , yn]T is an n-dimensional state vector which is the solution of Eq. 4.8.The initial state vector is represented by y0 = [y1,0, y2,0, . . . , yn,0]T andA = [A1, A2, . . . , An]Tis the operator vector.Suppose, DFixed is a fixed domain with boundary ∂DFixed of the state space. Accordingto principle of probability conservation, during an interval of time [t1, t2], the incrementalchange of probability in DFixed domain is equal to the transition of probability through theboundary ∂DFixed, i.e.,∆[t1,t2]PDFixed = ∆[t1,t2]P∂DFixed (4.9)where∆[t1,t2]PDFixed =∫DFixedpY(y, t2)dy −∫DFixedpY(y, t1)dy=∫DFixed∫ t2t1∂pY(y, t)∂tdydt(4.10a)∆[t1,t2]P∂DFixed = −∫ t2t1∫∂DFixedpY(y, t)A(y, t) · nˆdSdt= −∫ t2t1∫DFixedn∑l=1∂[pY(y, t)Al(y, t)]∂yldydt(4.10b)884.4. DOSTUPOV–PUGACHEV EQUATIONwhere nˆ is the unit normal vector outward from boundary ∂DFixed. The negative sign inEq. 4.10b implies that when the probability transits outside, the probability in the domaindecreases. Substituting Eqs. 4.10a and 4.10b into Eq. 4.9 yields∂pY(y, t)∂t+n∑l=1∂[pY(y, t)Al(y, t)]∂yl= 0 (4.11)4.4 Dostupov–Pugachev EquationConsider a n degrees-of-freedom structural system whose governing equation of motionis written as followsMX¨ + f(X˙,X) = B(X, t)ξ(t), X˙(t0) = X˙0, X(t0) = X0 (4.12)where X, X˙, and X¨ represent the displacement, velocity, and acceleration vectors, respec-tively. The mass matrix is denoted by M . The internal force vector is f which includesthe damping and restoring forces. B(X, t) is the influence matrix for input force and ξ(t)is the input force. In Eq. 4.12, X0 and X˙0 denote the initial displacement and initial ve-locity, respectively. The second-order differential equation of motion for the system can beexpressed in state-space form, which is as followsY¨ = A(Y, t) +B(Y, t)ξ(t) Y (t0) = Y0 (4.13)whereA(Y, t) =[−M−1f(Y )X˙]B(Y, t) =[M−1B(X, t)0](4.14)In the above equation, Y represents the state vector which is, Y = [X˙T , XT ]T . If therandomness is included in the excitation force, i.e., ξ(t) = F (Θ, t), Eq. 4.13 is re-written asY˙ = G(Θ, Y, t) Y (t0) = Y0 (4.15)whereG(Θ, Y, t) = A(Y, t) +B(Y, t)F (Θ, t) (4.16)For a given value of θ, the random variable y(t+∆t) can be expressed asy˜ = y(t+∆t) = y(t) +G(y, θ, t)∆t+ o(∆t) (4.17)Differentiating Eq. 4.17 yields into,dy˜ =[1 +m∑l=1∂Gl(y, θ, t)∂yl]dy = |J |dy (4.18)From Eq. 4.1, it is written aspYΘ(y˜, θ, t+∆t)dy˜ = pYΘ(y, θ, t)dy (4.19)894.5. GENERALIZED DENSITY EVOLUTION METHODSubstituting Eqs. 4.17 and 4.18 into left hand side of Eq. 4.19,pYΘ(y˜, θ, t+∆t)dy˜ = pYΘ(y +G(y, θ, t)∆t+ o(∆t), θ, t+∆t)|J |dy=[pYΘ(y, θ, t+∆t) +m∑l=1∂pYΘ∂ylGl(y, θ, t)∆t][1 +m∑l=1∂Gl(y, θ, t)∂yl]dy + o(∆t)=[pYΘ(y, θ, t+∆t) +{m∑l=1∂pYΘ∂ylGl(y, θ, t)+pYΘ(y, θ, t+∆t)m∑l=1∂Gl(y, θ, t)∂yl}∆t]dy + o(∆t)(4.20)Ignoring the higher order terms of Eq. 4.20 and equating with Eq. 4.19,[pYΘ(y, θ, t+∆t) +{m∑l=1∂pYΘ∂ylGl(y, θ, t)+pYΘ(y, θ, t+∆t)m∑l=1∂Gl(y, θ, t)∂yl}∆t]dy = pYΘ(y, θ, t)dypYΘ(y, θ, t+∆t)− pYΘ(y, θ, t)∆t+m∑l=1∂pYΘ∂ylGl(y, θ, t) + pYΘ(y, θ, t+∆t)m∑l=1∂Gl(y, θ, t)∂yl= 0∂pYΘ(y, θ, t)∂t+m∑l=1∂[pYΘ(y, θ, t)Gl(y, θ, t)]∂yl= 0(4.21)4.5 Generalized Density Evolution MethodThe classical Liouville equation, as described in Eq. 4.11 and the Dostupov-Pugachevequation, as in Eq. 4.21, are high-dimensional partial differential equations due to thecoupling between the coefficient and joint PDF. It is difficult solve high-dimensional partialdifferential equation when nonlinearity is present in the system. Therefore, the applicationsof these equations to practical problems are limited. To overcome this issue, Li and Chen[LC08] proposed the generalized density evolution equation (GDEE) based on the principleof probability conservation. As defined in Eq. 4.15, the state-space form of a stochasticdynamical system is expressed asY˙ = G(Y,Θ, t) Y (t0) = Y0 (4.22)where Y0 is the initial value whose PDF is pX0(x0); the random variables are representedby Θ with a joint PDF, pΘ(θ). It is assumed that the solution of Eq. 4.22 is of the followingformY = H(Y0,Θ, t) (4.23)904.5. GENERALIZED DENSITY EVOLUTION METHODwhich should be satisfied the initial condition i.e., Y (t0) = H(Y0,Θ, t0) = Y0. Then, it canbe written asY˙ = g(Y0,Θ, t) (4.24)where g(·) = ∂H(·)/∂t. The conditional PDF of Y (t) under the condition {Y0 = y0,Θ = θ}is written as followspY |Y0,Θ(y, t|y0, θ) = δ(y −H(y0, θ, t)) (4.25)where δ(·) is the Dirac’s delta function. Differentiating above equation with respect to tyields,∂pY |Y0,Θ(y, t|y0, θ)∂t=∂∂tδ(y −H(y0, θ, t))=∂[δ(y −H(y0, θ, t))]∂(y −H(y0, θ, t)) ·∂(y −H(y0, θ, t))∂t= −∂[δ(y −H(y0, θ, t))]∂y· ∂H(y0, θ, t)∂t= −∂pY |Y0,Θ(y, t|y0, θ)∂y· g(y0, θ, t)(4.26)According to Eq. 4.25, the joint PDF of (Y, Y0,Θ) is written as followspY Y0Θ(y, y0, θ, t) = pY |Y0,Θ(y, t|y0, θ)pY0(y0)pΘ(θ) = δ(y −H(y0, θ, t))pY0(y0)pΘ(θ) (4.27)From Eqs. 4.26 and 4.27, it can be written as∂pY Y0Θ(y, y0, θ, t)∂t+ g(y0, θ, t)∂pY Y0Θ(y, y0, θ, t)∂y= 0 (4.28)Substituting Eq. 4.24 yields into,∂pY Y0Θ(y, y0, θ, t)∂t+ Y˙ (y0, θ, t)∂pY Y0Θ(y, y0, θ, t)∂y= 0 (4.29)For multi-dimensional stochastic system, Eq. 4.28 can be re-written as∂pYY0Θ(y,y0, θ, t)∂t+m∑l=1gl(y0, θ, t)∂pYY0Θ(y,y0, θ, t)∂yl= 0 (4.30)The PDF of Y (t) is estimated aspY (y, t) =∫ΩY0∫ΩΘpYY0Θ(y,y0, θ, t)dy0dθ (4.31)When the initial condition is deterministic i.e., pY0(y0) = δ(y0 − y¯0), then Eq. 4.31 iswritten aspY (y, t) =∫ΩΘpYΘ(y, θ, t)dθ (4.32)914.6. ESTIMATION OF RELIABILITYSimilarly, when the parameters are deterministic i.e., pΘ(θ) = δ(θ − θ¯), then Eq. 4.31 isexpressed aspY (y, t) =∫ΩY0pYY0(y,y0, t)dy0 (4.33)4.6 Estimation of ReliabilityThe probability of failure for a first-passage problem can be expressed asPf = Pr{|Y (t)| > YThres,∃t ∈ [0, T ]} (4.34)where Pr{·} represents the probability operator, the failure domain is denoted by ΩF , and[0, T ] represents the time duration. In Eq. 4.34, YThres denotes the threshold value of Y (t).Also, the probability of failure, Pf in Eq. 4.34 can be formulated using the principle ofequivalent extreme-value event [LCF07] which is given asPf = Pr{YEEV ≥ YThres}=∫ ∞YThrespYEEV(y)dy (4.35)where pYEEV is the PDF of equivalent extreme-value (EEV) of Y (t), expressed asYEEV = maxt∈[0,T ](|Y (t)|) (4.36)The failure probability is estimated by performing one-dimensional integration of pYEEV , asin Eq. 4.35. To estimate pYEEV , a virtual stochastic process is formulated, which is expressedasV(τ) = YEEV sin(ωτ); τ ∈ [0, 1] (4.37)where ω = 5π/2 [LCF07]. Eq. 4.37 shows that when τ = 0, V(τ) = 0 and when τ = 1,V(τ) = YEEV. Therefore, it is needed to compute the PDF of V(τ) i.e., pV(V, τ) which isobtained by solving GDEE, as in Eq. 4.29∂pVΘ(v, θ, τ)∂τ+ V˙(θ, τ)∂pVΘ(v, θ, τ)∂v= 0 ; pVΘ(v, θ, τ)|τ=0 = δ(v)pΘ(θ) (4.38a)pV(v, τ) =∫ΩΘpVΘ(v, θ, τ)dθ (4.38b)where δ(·) is the Dirac delta function. Once pV(V, τ) is obtained, the PDF of YEEV iscalculated aspYEEV(y) = pV(v, τ)|v=y,τ=1 (4.39)924.7. FINITE DIFFERENCE METHOD WITH TVD4.7 Finite Difference Method with TVDIn this section, we focus on solving the GDEE, as mentioned in Eq. 4.38a. Here, thefinite difference method with total variation diminishing (TVD) scheme is adopted [LC09],which is expressed as∂p(m+1)V,l = ∂p(m)V,l −12(λV˙(m) − ∣∣λV˙(m)∣∣)∆p(m)V,l+ 12− 12(λV˙(m) + ∣∣λV˙(m)∣∣)∆p(m)V,l− 12−12(∣∣λV˙(m)∣∣− ∣∣λV˙(m)∣∣2)(ψl+ 12∆p(m)V,l+ 12− ψl− 12∆p(m)V,l− 12) (4.40)In the above equation, p(m)V,l represents the PDF of V(τ) at V = Vl and τ = τm, i.e.,pV(Vl, τm) where Vl = l∆V (l = 0,±1,±2, . . .) and τm = m∆τ (m = 0, 1, 2, . . .). The spaceand time steps are denoted by ∆V and ∆τ , respectively. In Eq. 4.40, λ is the mesh ratio,i.e., λ = ∆τ/∆V; ∆p(m)V,l+ 12= p(m)V,l+1 − p(m)V,l and ∆p(m)V,l− 12= p(m)V,l − p(m)V,l−1. Also, factors ψl+ 12and ψl− 12are known as flux limiters, which are a function of the irregularity of pV(V, τ). Ifψl+ 12= ψl− 12= 1, Eq. 4.40 is known as the Lax-Wendroff scheme, whereas if ψl+ 12= ψl− 12= 0, Eq. 4.40 is known as the one-sided scheme. For the TVD scheme, the factors are givenbyψl+ 12(r+l+ 12, r−l+ 12)= H(− V˙(m))ψ0(r+l+ 12)+H(V˙(m))ψ0(r−l+ 12)(4.41)where H(·) is the Heaviside function. The factors r+l+ 12, r−l+ 12and ψ0 are given asr+l+ 12=p(m)V,l+2 − p(m)V,l+1p(m)V,l+1 − p(m)V,l, r−l+ 12=p(m)V,l − p(m)V,l−1p(m)V,l+1 − p(m)V,l, ψ0(r) = max(0,min(2r, 1),min(r, 2))(4.42)4.8 Selection of Representative PointsIn Eq. 4.22, the random parameters involved in a stochastic system are denoted byΘ. It is necessary to use a proper scheme to generate the representative points of randomparameters to attain the accuracy of PDEM. In this study, the generalized F-discrepancy(GF) based method [LC09] is used to select the representative points from the parameterdomain and the corresponding assigned probabilities are determined.Let Rn = {θp = (θp,1, θp,2, . . . , θp,q), p = 1, 2, . . . , n} be an q-dimensional point set withn representative points. The GF-discrepancy of Rn is expressed as followsDGF (Rn) = max1≤i≤q{sup∣∣fn,i(θ)− fi(θ)∣∣} (4.43)where fi(θ) and fn,i(θ) are the marginal and empirical cumulative distribution functions(CDF) of the i-th random variable (i.e., θi), respectively. The empirical CDF, fn,i(θ) is934.9. NUMERICAL ALGORITHMgiven byfn,i(θ) =n∑p=1Pp · I{θp,i ≤ θ}(4.44)where I(·) is the indicator function, and its value is 1 if and only if the condition is true; oth-erwise, it is 0. In Eq.4.44, Pp denotes the assigned probability of the representative sampleθp,i. The accuracy of GDEE in Eq. 4.29 can be attained by minimizing GF-discrepancy ofthe representative point set, which is expressed using extended Koksma-Hlawka inequality[CYL16], is given as∣∣∣∣ ∫Ωθg(θ)pΘ(θ)dθ −n∑p=1Pp · g(θp)∣∣∣∣ ≤ ν(q) · DGF (Rn) · TV (g) (4.45)where ν(q) is a factor which is same order of q whose value is found in [CC19]; TV (g)denotes the total variation of function g(θ), which represents its irregularity. To begin thealgorithm, an initial set of samples {θp = (θp,1, θp,2, . . . , θp,q), p = 1, 2, . . . , n} is drawn froma Sobol set {up = (up,1, up,2, . . . , up,q), p = 1, 2, . . . , n} over a unit hypercube which is offollowing formθm,i = f−1i(um,i); i = 1, . . . , q (4.46)where f−1i is the inverse CDF of θi. Then, for each dimension, a transformation is performedso that the assigned probability Pp of n points is close to each other.θ′m,i = f−1i[n∑p=11n· I{θp,i < θm,i}+ 12· 1n](4.47)The assigned probability Pp of samples θ′p,i (p = 1, . . . , n) are estimated by Pp =∫ρ(θ′p)dθpwhere ρ(θ) = ∂sfn(θ)∂θ1...∂θsand fn(θ) =1n∑np=1 I{θp ≤ θ}. Once the probability Pp is estimated,the following transformation is used to reduce GF-discrepancyθ′′m,i = f−1i[n∑p=1Pp · I{θ′p,i < θ′m,i}+12· Pm](4.48)Finally {θ′′p = (θ′′p,1, θ′′p,2, . . . , θ′′p,q), p = 1, 2, . . . , n} is adopted as the representative pointset. Readers can find a detailed explanation of the GF-discrepancy-based selection of rep-resentative points in [CYL16]. Fig. 4.1 shows GF-discrepancy-based representative pointsof random variables following standard uniform and standard normal distributions.4.9 Numerical AlgorithmIn Algorithm 6, it is noted that a large number of representative points are required toget satisfactory accuracy of PDEM. As a result, the computational model needs to be run944.10. NUMERICAL EXAMPLES0 0.2 0.4 0.6 0.8 100.20.40.60.81(a)-3 -2 -1 0 1 2 3-3-2-10123(b)Figure 4.1: GF-discrepancy-based representative points for (a) standard uniform and (b)standard normal distributionmultiple times, which becomes computationally expensive. To reduce the cost, stochasticspectral embedding is used as a surrogate model. In this method, a small number ofrepresentative points are generated, and corresponding model evaluations are performedfor every sample. With this data, the surrogate model is trained. Once the trained modelis obtained, a large number of samples are passed through the trained model to get thecomplete response surface. The details of stochastic spectral embedding are discussed inSection 2.3.2.4.10 Numerical ExamplesThree different numerical examples are presented to illustrate the reliability analysisusing the probability density evolution method. These are: (1) a four-branch problem inwhich the probability of failure is estimated from a series system; (2) a reliability analysis ofa shear building frame; and (3) reliability-based design optimization of a moment-resistingframe coupled with a nonlinear energy sink with negative stiffness and sliding friction.To reduce the computational cost, stochastic spectral embedding (SSE) is used, whichapproximates the original response surface of an equivalent extreme event. For simulationpurposes, MATLAB software is used.954.10. NUMERICAL EXAMPLESAlgorithm 6 Estimation of failure probability using PDEM1: Generate the representative points Rn = {θp = (θp,1, θp,2, . . . , θp,q), p = 1, 2, . . . , n} ofrandom variables based on GF-discrepancy and estimate the assigned probability Pp ofsamples.2: For each value of θp, solve the governing equation of motion (i.e., Eq. 4.22) using thecomputational model. Once the response time history is obtained, estimate the EEV(i.e., YEEV) using Eq. 4.36.3: Construct the virtual stochastic process V(τ) and its derivative w.r.t τ using Eq. 4.37which is a function of YEEV.4: Solve GDEE for every value of θp as mentioned in Eq. 4.38a to obtain the PDF of V(τ)i.e., pV(V, τ). For solving GDEE, the finite difference method with TVD scheme is used,see Section 4.7.5: Estimate the PDF of YEEV i.e., pYEEV(y) by substituting τ = 1 in Eq. 4.39.6: Estimate the probability of failure by performing one-dimensional integration onpYEEV(y) mentioned in Eq. 4.35.xi(θ)tYEEV = maxt∈ 0 Tx θ, tθ 1 YEEV1θ 2 YEEV2…..…..θ n YEEVnSamples:YEEVθ1θ2𝜕p𝒱𝚯1 (𝑣, 𝛉, 𝜏)𝜕𝜏+ ሶ𝒱 YEEV1 , 𝜏𝜕p𝒱𝚯1 (𝑣, 𝛉, 𝜏)𝜕𝑣= 0ቚp𝒱1 (𝑣, 𝛉, 𝜏)τ=0= δ 𝑣 Pp1𝜕p𝒱𝚯n (𝑣, 𝛉, 𝜏)𝜕𝜏+ ሶ𝒱 YEEVn , 𝜏𝜕p𝒱𝚯n (𝑣, 𝛉, 𝜏)𝜕𝑣= 0ቚp𝒱n (𝑣, 𝛉, 𝜏)τ=0= δ 𝑣 Ppn…..…..pYEEVYEEVPfRepresentative Points Based on GF-discrepancyComputational Model Estimation of EEVStochastic Spectral EmbeddingSolve GDEEsEstimation of PDF and Reliability AnalysisM1M2MN-1MNK1K2KN-1KNx1 (θ)x2 (θ)xN-1 (θ)xN(θ)Figure 4.2: Flowchart for reliability analysis using PDEM964.10. NUMERICAL EXAMPLES4.10.1 Example 1: Four-branch ProblemAs the first example, a four-branch problem is considered whose Pf is given byPf = Pr[{− 0.1(θ1 − θ2)2 + θ1 + θ2√2≥ 3}∪{− 0.1(θ1 − θ2)2 − θ1 + θ2√2≥ 3}∪{θ1 − θ2 ≥ 7√2}∪{θ2 − θ1 ≥ 7√2}] (4.49)where θ1 and θ2 are the random variables which follow the standard Gaussian distribution.According to an equivalent extreme event, the above equation is equivalent toPf = Pr[YEEV ≥ 7/√2]=∫ ∞7/√2pYEEV(y)dy (4.50)where YEEV is given asYEEV = max−0.1(θ1 − θ2)2 + (θ1 + θ2)/√2 + 7/√2− 3−0.1(θ1 − θ2)2 − (θ1 + θ2)/√2 + 7/√2− 3(θ1 − θ2)(θ2 − θ1) (4.51)The surrogate model needs to be trained before estimating the probability density function.For this purpose, 100 representative points of random variables are generated using the GF-discrepancy algorithm. YEEV is calculated for each set of random variables using Eq. 4.51.With these output responses, the surrogate model is trained. Once training is completed,another set of 1500 representative points is generated, which is then passed through thetrained surrogate model to obtain the complete response surface. Fig. 4.3(a) shows thecontour plot of four failure domains in which training samples are shown within the domain.With these samples, SSE starts to train the model where the entire domain is divided intosubdomains sequentially. Fig. 4.3(b) shows that the SSE produces a minimum residualwhen the entire domain of random variables is divided into 8 subdomains. The residualobtained in every iteration is shown in Fig. 4.3(c). Once the surrogate model’s accuracy isdetermined, the virtual stochastic process V(τ) is calculated for each value of YEEV usingEq. 4.37. The GDEEs are solved for evaluating the PDF of V(τ) using the finite differencemethod combined with the TVD scheme. Fig. 4.4(a) shows the PDF of V(τ). The PDF ofYEEV is obtained by setting τ to 1,as illustrated in Fig. 4.4(b). Once the PDF is obtained,the probability of failure is estimated using Eq. 4.50. The failure probability is 2.5×10−3.In comparison, a Monte Carlo simulation with 106 samples yields a failure probability of2.43×10−3, which is close to the value obtained using stochastic spectral embedding.4.10.2 Example 2: 10-storey Shear BuildingHere, a 10-storey shear building frame is considered as our next problem. The stiffnessof each floor is assumed to be lognormally distributed, whose mean values are 1.962, 1.875,974.10. NUMERICAL EXAMPLES(a) (b)0 2 4 6Refinement Steps0.010.020.030.04Residual(c)Figure 4.3: (a) Contour plot of four failure domains with the training samples, (b) Subdo-mains of the training samples in SSE and (b) Degree of accuracy corresponding to increasingnumber of subdomain in SSE1.758, 1.754, 1.662, 1.662, 1.662, 1.662, 1.662, 1.662 (×108 N/m) from the bottom storeyand the coefficient of variation is assumed to be 0.1. The lumped masses of each floor are:m1 = 3.478, m2 = 3.225, m3 = 2.887, m4 = 2.667, m5 = 2.558, m6 = 2.558, m7 = 2.558,m8 = 2.558, m9 = 2.558, m10 = 2.558 (×105 kg), respectively. The height of each floor istaken as 3.6 m. The structure is subjected to a ground motion, which is given asx¨g(t) = β1x¨NS(t) + β2x¨EW (t) (4.52)984.10. NUMERICAL EXAMPLES(a)0 2 4 6 800.20.40.60.81(b)Figure 4.4: PDF of (a) virtual stochastic process,V(τ) and (b) equivalent extreme event,YEEVwhere x¨NS and x¨EW are the El Centro ground motion in N-S and E-W components, re-spectively. The coefficients β1 and β2 are normally distributed; their mean and standarddeviation are 2 and 0.2, respectively. The damping ratio is assumed to be 0.05. Hence, theproblem has 12 random variables, i.e., θ = [k1, k2, . . . , k10, β1, β2]. The probability of failureis given byPf = Pr{|Yi(θ, t)| ≥ YThres,∃t ∈ [0, T ], i = 1, 2, . . . , 10}(4.53)where Yi(θ, t) denotes interstorey drift of the i-th storey. The allowable interstorey drift isdenoted by YThres, and the duration of the ground motion is denoted by T . According tothe equivalent extreme event, Eq. 4.53 is equivalent toPf = Pr[YEEV ≥ YThres]=∫ ∞YThrespYEEV(y)dy (4.54)where YEEV is given asYEEV = max1≤i≤10{maxt∈[0,T ]∣∣Yi(θ, t)∣∣} (4.55)In this problem, the procedure for estimating failure probability is the same as in theprevious example. 100 representative samples for each random variable are generated usingGF-discrepancy. The output response in Eq. 4.55 is evaluated corresponding to each set ofrandom variables, θ. With these samples, the surrogate model is trained. Thereafter, 10,000samples are generated using GF-discrepancy as the prediction samples, which are differentfrom the training samples. The prediction samples are passed through a trained surrogatemodel to obtain the complete response surface. The virtual stochastic process, V(τ) is994.10. NUMERICAL EXAMPLESevaluated corresponding to each YEEV and GDEEs are solved to obtain the PDF of V(τ).When τ is set to 1, a PDF of YEEV is obtained, as shown in Fig. 4.5(a). The probabilityof failure is estimated using Eq. 4.54. To investigate the accuracy of the proposed method,Monte Carlo simulation (MCS) is performed with 106 samples. Fig. 4.5(b) shows the failureprobability for different threshold values for MCS and the proposed method. For instance,when the threshold value, YThres is taken as 8 mm, Pf is 0.055 using the proposed method,where as the same is 0.056 for MCS.0 5 10 15 2000.070.140.210.280.35(a)6 8 10 12 1410-1010-5100(b)Figure 4.5: (a) PDF of equivalent-extreme event, YEEV; (b) Failure probability for differentthreshold values4.10.3 Example 3: One-storey Frame with NESThe third numerical example we consider is the one-storey moment-resisting frame cou-pled with the nonlinear energy sink (NES) with negative stiffness and sliding friction. Thedetails of the structural system and the governing equations of motion are provided inSection 3.4.1.For numerical purposes, the mass, stiffness, and damping ratio of the one-storey frameare taken as 189.4 kg, 26.2 Nmm−1 and 0.35%, respectively [DTC+20]. For this problem,the source of uncertainty is considered only in ground motions. An ensemble of 100 groundmotions is simulated from the Kanai-Tajimi spectrum, defined in Eq. 3.25. The parametersrelated to the Kanai-Tajimi spectrum i.e., S0, ωg and ηg are assumed to be uniformly dis-tributed whose bounds are as follows: S0 ∼ U(0.016, 0.033) m2s−3, ωg ∼ U(7.9, 16.2) rads−1and ηg ∼ U(0.26, 0.53)%. The stiffness and compressed length of the spring, i.e., kNES andL0, are used as design variables in this problem and estimated using the PDEM-basedreliability-based design optimization method. The lower and upper bounds of the designvariables are considered as kNES = [4000, 7000] Nm−1 and L0 = [0.07, 0.15] m, respectively.1004.10. NUMERICAL EXAMPLESThe procedure used in this problem is the same as the previous. The algorithm startswith the generation of 150 sets of representative points for design variables using the GF-discrepancy method. For each set of representative points, the model evaluation is per-formed considering the ensemble of 100 ground motions. In this problem, the storey driftis chosen as the extreme event, which is expressed as YEEV = maxt∈[0,T ]|X(t)|. Therefore, foreach set of representative points, a set of 100 values of YEEV is obtained, which are usedto estimate the PDF of YEEV using PDEM as defined in Eqs. 4.38a and 4.39. Fig. 4.6(a)shows the PDF of YEEV for different sets of design variables. Once the PDF is obtained,the probability of failure is estimated corresponding to each set of design variables, whichis expressed asPf = Pr[YEEV ≥ YThres]=∫ ∞YThrespYEEV(y)dy (4.56)Here, the threshold value, i.e., YThres is taken as 0.04 m [DTC+20]. Once the failureprobability for 150 representative points is obtained, the stochastic spectral embeddingmodel is trained. Fig. 4.6(b) shows the stochastic spectral embedding model divides theentire domain into 14 sub-domains to achieve residual minimal. Additionally, a set of20,000 points of design variables, which are different from the 150 training points, aregenerated using the GF-discrepancy method and are regarded as the prediction points.The prediction points are passed through the trained surrogate model, which producesthe complete response surface, shown in Fig. 4.6(c). The optimization is performed onthe predicted response surface. The optimal values of the design variables are kNES =4808 N/m and L0 = 0.14 m, respectively where the probability of failure is minimum i.e.,Pf = 4.4× 10−8.With these optimal values, the performance of the NES is compared with the designparameters used by Chen et al. [CQC+19]. They used the values kNES = 5850 N/m andL0 = 0.14 m, respectively. Fig. 4.7(a) shows the displacement time history of the primarystructure with the parameters used by Chen et al. [CQC+19] and the optimized parametersobtained from this study. The figure reveals that the peak displacements are 14.74 mm and12.75 mm, respectively, for before and after optimization. The root mean square values ofthe same are 4.5 mm and 3.6 mm, respectively. Fig. 4.7(b) shows the amount of energy theprimary structure dissipates before and after optimization. The peak is reduced by 18%after optimization. Therefore, this example demonstrates the necessity of reliability-basedstructural optimization.4.10.4 Example 4: SMA-based Damped Outrigger Timber BuildingAs a final example, a shape memory alloy (SMA)-based damped outrigger tall tim-ber building is considered as shown in Fig. 4.8. The details of structural description arefound in [TD21]. The governing equations of motion of this structure are derived using theLagrangian formulation, whose details are provided below.1014.10. NUMERICAL EXAMPLES0 0.02 0.04 0.06020406080(a) (b)(c)Figure 4.6: (a) PDF of equivalent-extreme event, YEEV; (b) Failure probability responsescorresponding to design variablesEquations of Motion of an Outrigger SystemHere, a reduced-order model of a high-rise structure with a shape memory alloy (SMA)-based outrigger beam is considered. The purpose of the SMA spring is to reduce excessiveloads on the perimeter column. Fig. 4.8 shows the architecture of the proposed outriggersystem, where one end of the outrigger is attached to the core of the structure and theother end is connected to the SMA spring, which is attached to the perimeter column. Inthis study, the core of the structure is idealized as a cantilever beam, and each floor mass1024.10. NUMERICAL EXAMPLES0 10 20 30 40 50 60Time [s]-20-1001020Displacement [mm]Before OptimizationAfter Optimization(a)0 10 20 30 40 50 60Time [s]00.10.20.30.4Energy [J](b)Figure 4.7: (a) Displacement time history of the primary structure and (b) Amount ofenergy primary structure dissipates before and after optimizationFigure 4.8: Schematic representation of SMA-based damped outrigger structureis considered a discrete mass that acts at the junction between the floor and the core ofthe structure. The governing equations of motion of the combined system of shear core and1034.10. NUMERICAL EXAMPLESoutrigger are derived using the Lagrange formulation. There are a few assumptions thathave been considered in this study to formulate the governing equations of motion for thecombined system, i.e., the floor slabs of the system are rigid and the core and columns arefixed at the base of the structure. The outrigger beam is modeled as a torsional springhaving constant rotational stiffness, and the moment developed due to the torsional springis concentrated at the junction of the core of the structure and the outrigger, as shown inFig. 4.8. Also, in this study, the plan of the structure has been considered symmetrical,and therefore, the effect of torsion on the structure along its height has been neglected.Based on the above assumptions, the core of the structure is modeled as a continuouscantilever beam with a rectangular cross-section, and the associated equations of motion areformulated based on Hamilton’s principle. Consider the continuous beam’s flexural stiffnessEI(x) and mass per unit height m(x). Let u(x, t) be the displacement of the system, whichis assumed to be a function of position and time and is defined as followsu(x, t) = Φ(x)q(t) =∑iϕi(x)qi(t) (4.57)where ϕi represents the ith mode-shape of the system. Also, qi is the displacement of theith mode shape. Therefore, the kinetic energy of the system is given by [Mei80]T =12∫ H0m(x)[∂u(x, t)∂t]2dx︸ ︷︷ ︸Core (Continuous System)+12N∑k=1Mˆk[∂u(Hk, t)∂t]2︸ ︷︷ ︸Floor (Discrete System)(4.58)where x denotes the spatial position of any point on the structure along with the heightwhich is bounded in 0 ≤ x ≤ H. In Fig. 4.8, H represents the total height of the structure.In Eq. 4.58, the discrete floor mass is denoted by Mˆk which is acted at a height of Hk,measured from the base of the structure. N is the total number of floors in the structure.Similarly, the potential energy of the combined outrigger system can be expressed as [Mei80]V =12∫ H0EI(x)[∂2u(x, t)∂x2]2dx︸ ︷︷ ︸Core (Continuous System)+n∑j=1Krj[∂u(αjH, t)∂x]2︸ ︷︷ ︸Outrigger (Discrete System)(4.59)where Krj represents the jth equivalent rotational stiffness due to outrigger. n is the totalnumber of outriggers considered in the structure. In Eq. 4.59, αj is the normalized positionof the outrigger measured from the base of the structure and ranges 0 ≤ αj ≤ 1, as shown inFig. 4.8. Using the Hamilton’s principle, the principle function of dynamics, A is expressedas the time integral of the Lagrangian, L between two time instants, t1 and t2, is given by[Mei80]A =∫ t2t1Ldt (4.60)1044.10. NUMERICAL EXAMPLESwhere Lagrangian L denotes the difference between the kinetic energy and the potentialenergy. Using Hamilton’s variational principle, the minimum value of A, as defined in theabove equation, is attained when [PP10]δA = δ∫ t2t1Ldt =∫ t2t1δLdt =∫ t2t1δ(T − V )dt = 0 (4.61)where δ(·) is the variation operator. External force causes nonconservative virtual work,which can be expressed as [Mei80]δWnc(x, t) = F (x, t)δu(x, t) (4.62)Substituting Eqs. 4.58, 4.59 and 4.62 into Eq. 4.61 yieldsδA = 12∫ t2t1∫ H0δ{m(x)[∂u(x, t)∂t]2− EI(x)[∂2u(x, t)∂x2]2}dxdt+∫ t2t1δ{12N∑k=1Mˆk[∂u(Hk, t)∂t]2−n∑j=1Krj[∂u(αjH, t)∂x]2}dt+∫ t2t1∫ H0δWncdxdt = 0(4.63)The Lagrangian density variation can be expressed as [Mei80]δLˆ = ∂Lˆ∂uδu+∂Lˆ∂u′δu′+∂Lˆ∂u′′δu′′+∂Lˆ∂u˙δu˙ (4.64)where Lˆ is the Lagrangian density which is expressed asLˆ = 12{m(x)[∂u(x, t)∂t]2 − EI(x)[∂2u(x, t)∂x2]2}(4.65)In Eq. 4.64, u′and u˙ represent the total derivative of u with respect to x and t, respectively.Also, the discrete term of the Lagrangian variation in Eq. 4.63 can be written asδLˆ0 = ∂Lˆ0∂u′δu′+∂Lˆ0∂u˙δu˙ (4.66)whereLˆ0 = 12N∑k=1Mˆk[∂u(Hk, t)∂t]2−n∑j=1Krj[∂u(αjH, t)∂x]2(4.67)Substituting Eqs. 4.62, 4.64 and 4.66 into Eq. 4.63 yields∫ t2t1[ ∫ H0(∂Lˆ∂uδu+∂Lˆ∂u′δu′+∂Lˆ∂u′′δu′′+∂Lˆ∂u˙δu˙+ Fδu)dx+ δLˆ0]dt = 0 (4.68)1054.10. NUMERICAL EXAMPLESUsing integration by parts of Eq. 4.68 with respect to both x and t, the governing equationcan be expressed in the following form [Mei80]∫ t2t1[ ∫ H0{∂Lˆ∂u− ∂∂x( ∂Lˆ∂u′)+∂2∂x2( ∂Lˆ∂u′′)− ∂∂t(∂Lˆ∂u˙)+ F}δudx+(∂Lˆ∂u+∂Lˆ0∂u)δu∣∣∣∣x=H− ∂Lˆ∂u′δu∣∣∣∣x=0]dt = 0(4.69)The governing equations of motion and the boundary conditions for the outrigger structurecan be written from Eq. 4.69, which are given by∂Lˆ∂u− ∂∂x( ∂Lˆ∂u′)+∂2∂x2( ∂Lˆ∂u′′)− ∂∂t(∂Lˆ∂u˙)+ F = 0 ; 0 < x < H (4.70a)∂Lˆ∂u′δu∣∣∣∣x=0= 0 (4.70b)(∂Lˆ∂u+∂Lˆ0∂u)δu∣∣∣∣x=H= 0 (4.70c)Substituting values of Lˆ and Lˆ0 into the above equations yields the following differentialequations in the following formm∂2u∂t2+∂2∂x2(EI∂2u∂x2)− F = 0 ; 0 < x < H (4.71a)[∂∂x(EI∂2u∂x2)]x=0= 0 (4.71b)[EI∂2u∂x2+ 2n∑j=1Krj∂u(αjH, t)∂x−N∑k=1Mˆk∂u(Hk, t)∂t]x=H= 0 (4.71c)Also, since the structure is modeled as a cantilever beam whose base is fixed, the essentialboundary conditions are as followsu(0, t) = 0∂u(0, t)∂x= 0 (4.72)The modal frequency for the outrigger building is obtained by solving Eq. 4.71 and 4.72and external force (F ) is set to zero. The closed-form solutions for the modal frequenciesand mode shapes are proposed by Malekinejad et al. [MR13].1064.10. NUMERICAL EXAMPLESRotational Stiffness due to OutriggerFor a wall-framed structure with outriggers, the outriggers are connected with a shearwall at one end, and the other end of the outriggers is connected with the column. Forsimplicity, the effect of the outrigger on the structure is modeled as a rotational spring atthe junction of the shear wall and outrigger. It is evident that the equivalent rotationalstiffness due to outrigger depends on the flexural rigidity of the core wall, the outrigger -perimeter column system [LBK08]. The equivalent rotational stiffness due to outrigger isgiven by [LBK08]Kr =1θ(4.73)where θ is the total rotation in the outrigger caused by the restraining moment, expressedasθ = θc + θo (4.74)where θc is the rotation in the outrigger caused by axial deformation of the column due torestraining forces in the perimeter column. The rotation, θc is written as [LBK08]θc =2αHd2EcAc(4.75)where d is the length of the outrigger, (αH) is the location of the outrigger from groundlevel, and EcAc is the axial stiffness of the perimeter column. In Eq. 4.74, θo is the flexuraldeformation of the outrigger system due to the action of column forces, which results inadditional drift in the floor. The rotation, θo is given by [LBK08]θo =d12EoIoe(4.76)where EoIoe is the effective flexural stiffness of the outrigger which is obtained from theactual flexural rigidity of the outrigger EoIo by converting the flexural rigidity of a wide-column beam to that of an equivalent full-span beam, which is written as [SS83]EoIoe =[1 +(bc/2)(d/2− bc/2)]3EoIo (4.77)where bc is the width of the core, as shown in Fig. 4.9. Finally, the equivalent rotationalstiffness for the undamped outrigger system is obtained by substituting Eq. 4.75 and 4.76into Eq. 4.73 yields,Kr =[2αHd2EcAc+d12EoIoe]−1(4.78)Coupled System DynamicsOnce the modal frequencies and corresponding mode shapes are estimated using Eq. 4.71,the mass and stiffness of any vibrational mode can be expressed asMi =∫ H0mϕ2i (x)dx+N∑k=1Mˆkϕi(Hk) Ki = ω2iMi (4.79)1074.10. NUMERICAL EXAMPLESFigure 4.9: Wide-column effect of core and outriggerwhere ϕ is the normalized mode shape corresponding to the ith vibrational frequency. In thisstudy, the SMA damper is installed at the junction of the outrigger and perimeter column,as shown in Fig. 4.8. The restrained moment due to the SMA-based damper outriggersystem is expressed asMo(x, t) = δ(x− αH)[2{FSMA(d2)+Kr}∂u(αH, t)∂x](4.80)where αH is the location of the outrigger from the ground level; (d/2) is the length of theoutrigger on one side. As outriggers are attached on both sides of the core wall, factor 2 isadded. Kr is the rotational stiffness due to outrigger, obtained from Eq. 4.78. The forceinduced on the SMA damper is denoted by FSMA. The details are provided in the below.In Eq. 4.80, δ(x− αH) represents the Dirac-delta function, is given byδ(x− αH) ={∞, when x = αH0, when x ̸= αH (4.81)Substituting Eq. 4.57 into Eq.4.80 yieldsMo(x, t) = δ(x− αH)[2{FSMA(d2)+Kr}∑i∂ϕi(x)∂xqi(t)](4.82)The modal force from the outrigger is expressed asFo,i(t) =∫ H0ϕi(x)f(x, t)dx =∫ H0ϕi(x)∂Mo∂xdx=∫ H0ϕi(x)∂δ(x− αH)[2{FSMA(d2)+Kr}∑i∂ϕi(x)∂x qi(t)]∂xdx= 2{FSMA(d2)+Kr}{∂ϕi(x)∂x∑i∂ϕi(x)∂x}∣∣∣∣∣x=αHqi(t)(4.83)1084.10. NUMERICAL EXAMPLESSimilarly, the modal force due to external loading is written asFg,i(t) = −∫ H0ϕi(x)f(x, t)dx = −∫ H0ϕi(x)mx¨g(t)dx = −SimHx¨g(t) (4.84)where x¨g is the ground acceleration and Si is the load coefficient of the ith vibrational mode.Finally, the governing equation of motion of the outrigger system for ith mode is given byMiq¨i(t) + Ciq˙i(t) +Kiqi(t) = Fg,i(t)− Fo,i(t) (4.85)Considering the first n modes, the above governing equation of motion can be re-written as[M ][q¨(t)] + [C][q˙(t)] + [K][q(t)] = IgmHx¨g(t)− 2{FSMA(d2)+Kr}[ϕ¯ ϕ¯T ][q(t)] (4.86)whereIg = −[S1 · · · Sn]Tϕ¯ =[∂ϕ1(x)∂x · · · ∂ϕn(x)∂x]T ∣∣∣∣x=αH(4.87a)[M ] =M1 · · · 0... . . . ...0 · · · Mn ; [K] =ω21 · · · 0.... . ....0 · · · ω2n [M ]; [C] =2ξ1ω1 · · · 0... . . . ...0 · · · 2ξnωn [M ](4.87b)It is noted that Eq. 4.71 is the partial differential equation, whereas Eq.4.86 is the ordinarydifferential equation that is much easier to solve compared to Eq. 4.71. The fourth-orderRunge-Kutta method is adopted to solve Eq.4.86 in the MATLAB environment.Constitutive Model of SMAHere, shape memory alloy (SMA) is introduced to augment the energy dissipation char-acteristics of the outrigger. Different constitutive models have been proposed in the lit-erature for the nonlinear behavior of SMA. Among them, the Graesser-Cozzarelli model[GC91, GC94] is frequently used for passive vibration control. It illustrates superelasticbehavior above austenite finish temperature following the classical Bouc-Wen hysteresis,representing the one-dimensional force vs. deformation characteristics of SMA. The de-scription of this model is given below.F˙SMA = ka[x˙s− | x˙s |∣∣∣∣FSMA − βFys∣∣∣∣n−1(FSMA − βFys)](4.88a)β = kaαs[xs − FSMAka+ fT | xs |c erf (axs)](4.88b)where ka and Fys are the initial austenite stiffness and yield force in SMA, respectively, andβ is the uni-axial back-stress. In Eq. 4.88, αs is the ratio of the transformation stiffness to1094.10. NUMERICAL EXAMPLES-0.02 -0.01 0 0.01 0.02xs-30-20-100102030F SMAFigure 4.10: Nonlinear hysteresis of SMAthe initial austenite stiffness ka of SMA, n and fT are the two constants that control thesharpness of the transition from the elastic to the inelastic phase and the size of hysteresis,respectively. The other constants, a and c control the extent of elastic recovery duringunloading and the slope of the unloading path. The error function, denoted by erf (x), isgiven byerf (x) =2√π∫ x0e−t2dt (4.89)For optimization, the combined outrigger system needs to be solved multiple times,where computational time becomes a factor. For this reason, the hysteresis model depictedin Eq. 4.88 is converted into an equivalent system, which is given byFSMA = Keffxs + Ceff x˙s (4.90)where Keff and Ceff are the equivalent effective stiffness and damping, respectively, whichare estimated from the hysteresis of SMA, shown in Fig. 4.10, are given byKeff =Fs,max − Fs,minxs,max − xs,min =1 + αs(µs − 1)µska (4.91a)Ceff = 2ξeffMωeff (4.91b)In the above equations, Fs,max and Fs,min are two extreme forces corresponding to dis-placements xs,max and xs,min, respectively, as shown in Fig. 4.10. Also, µs denotes the1104.10. NUMERICAL EXAMPLESductility ratio, which is expressed as a ratio of maximum design displacement, xs,max toyield displacement, xys of SMA. In Eq. 4.88, the yield force, Fys equals to kaxys. InEq. 4.91b, M and ωeff are the total mass of the outrigger structure and the effectivefrequency (ωeff =√Keff/M), respectively. The effective modal damping ratio, ξeff , isexpressed asξeff =WD2πKeffx2s,max(4.92)where WD, the energy loss per hysteresis cycle, equals to 2χFys(xs,max − xys). Here χ =(1- αs).Ground Motion Selection0 1 2 3 400.511.522.5Figure 4.11: Response spectra of selected ground motions and target response spectra for2% probability of exceedance in 50 yearsSeismic ground motion is the key source of uncertainty that affects the performance ofthe outrigger. To ensure the optimal performance of the controller, it is required to select anensemble of suitable ground motion records that reflect the seismic hazards of a particularregion. Site-specific spectrum-compatible ground motions are one of the key methods wherethe response spectra of the selected ground motions are matched with the target responsespectra of a particular site. Conditional mean spectrum (CMS) [Bak11] is one of the popularchoices for defining the target response spectra. CMS is a modified uniform hazard spectrumused for the seismic design of buildings [HAAR14]. In this study, the ground motions areselected using an in-house probabilistic seismic hazard analysis (PSHA) tool, proposed by1114.10. NUMERICAL EXAMPLESAtkinson and Goda [AG11] where all major components of the national seismic hazardmodel are considered, i.e., crustal, interface, and inslab events. Here, a set of 30 groundmotions (each having two horizontal components, i.e., 60 ground motions in total) at thereturn period of 2475 years is selected based on the CMS-based record selection methodfor the Vancouver site (Canada) at a time period of 2.0 s. In the set of ground motions,the numbers of crustal, interface, and inslab records are 7, 13, and 10, respectively. Themagnitude of the selected earthquake records ranges from 8.5 to 9.1; the rupture distanceis 120 km and the mean occurrence rate is 0.002 (i.e., a 1-in-500-year chance). The detailsof the selection of ground motions are found in [GA11, GT19]. Fig. 4.11 shows the responsespectra of selected ground motion records and the target response spectra for the Vancouversite.Numerical ResultsHere, a 20-storey timber building is considered. The architectural plan of the buildingcan be found in [TD21]. The beams and columns of the building are designed using glulam,and the core shear wall is cross-laminated timber (CLT). The material properties of glulamand CLT and the cross-sectional details of the beam, column, and shear wall are providedin [TD21]. Two SMA-based outriggers are considered. The objective of this study is toestimate the optimal positions of outriggers and tuning parameters of the SMA springby minimizing the probability of failure of the system. One of the design parameters isthe normalized outrigger location, α, which ranges from 0 to 1 (Fig. 4.8). Other designparameters associated with SMA hysteresis parameters include the initial austenite stiffness(ka), the maximum design displacement (xs,max), and the ductility ratio (µs) of SMA.Also, the initial austenite stiffness is estimated from the normalized transformation strengthof SMA, which is F0 = kaxys/(mH), and ranges F0 ∈ [0.1, 0.5], where xys is the yielddisplacement of SMA. The mean values of the other parameters, xs,max and µs are takenas 0.2 m and 20, respectively. The coefficient of variation of these two variables is assumedto be 20%, which follows a uniform distribution. The bounds are applied to the designparameters to avoid numerical instability during optimization. For this reason, the boundsare taken as xs,max ∈ [0.13, 0.27] and µs ∈ [13, 27], respectively. This study also takesinto account the material uncertainties, which are the modulus of elasticity of CLT inlongitudinal and transverse directions, i.e., E1 and E2, and the modulus of elasticity ofglulam beam and column (Eg). Therefore, it is seen that there are two sets of randomvariables, i.e., θ = [α1, α2, F0, µs, xs,max], which are design variables, and ϵ = [E1, E2, Eg].The variables relating to CLT and glulam material properties, namely E1, E2, and Eg, areassumed to have Gaussian distributions with mean values of 11.7, 9, and 12.4 (×109 Pa),respectively, and the c.o.v is 10%. The objective function is taken as the maximum value ofthe inter-storey drift ratio of the floors. To compute that, an ensemble of ground motionsis considered, which are selected based on the conditional mean spectrum-based recordselection method for the Vancouver site (Canada) at a natural time period of 2.0 s. The1124.11. SUMMARYprobability of failure of the system is defined as followsPf = Pr[YEEV ≥ YThres]=∫ ∞YThrespYEEV(y)dy (4.93)where YEEV is given asYEEV =1NgNg∑j=1max1≤i≤20{maxt∈[0,T ]∣∣IDRi,j(θ, ϵ, t)∣∣} (4.94)where Ng denotes the total number of ground acceleration time histories and IDR the inter-storey drift ratio. In Eq. 4.93, YThres is the allowable inter-storey drift ratio, which is takenas 2.5% according to NBC 2015 [NBC15]. The procedure to evaluate the PDF of YEEV issame as before. Once PDF is obtained corresponding to each value of θ, the probability offailure is estimated using Eq. 4.93. The obtained value of failure probability correspondingto each set of design variables, θ is used to carry out the optimization problem. Theobjective function is defined as followsθ∗ = argminθ[Pf (θ)]s.t. θll ≤ θ ≤ θul(4.95)Fig. 4.12 shows the PDF of YEEV and the response surface of the probability of failurecorresponding to design variables, θ. The optimal value is taken at the point where thefailure probability is the lowest. The optimal locations of the outriggers from the groundlevel are 21 m and 57 m, respectively. The other obtained optimal design parameters are[F0, µs, xs,max] = [0.27, 22.15, 0.18], respectively.4.11 SummaryThe numerical investigations presented in this chapter are focused on reliability analysisof stochastic systems using the PDEM. The probability of failure of the system is evaluatedbased on the PDF of an EEV. The representative points of the random variables present inthe system are generated using the generalized F-discrepancy method. To estimate the PDFof EEV accurately, a large number of representative points are required, which makes thePDEM computationally expensive for high-fidelity models. To reduce the computationalburden, stochastic spectral embedding is used so that EEV is estimated accurately using asfew representative points as possible. The efficiency of the proposed method is establishedby considering three numerical examples. The major observations from this chapter arelisted below:∗ PDEM enables to estimate the joint probability density function of the equivalent-extreme event of an MDOF system by solving GDEEs efficiently in contrast to othertraditional methods such as the Fokker-Plank equation, Liouville equation, etc.1134.11. SUMMARY0 1 2 3 4 5 6 702468(a) (b)Figure 4.12: (a) PDF of equivalent extreme event and (b) Probability of failure surfacecorresponding to design parameters∗ To compare the efficiency of the proposed stochastic spectral embedding-based PDEM,the probability of failure is compared with Monte Carlo simulation (MCS). In MCS106 samples of random variables are required, whereas with 100 to 150 samples, theprobability of failure using the proposed method matches with MCS. As a result, theproposed method reduces the model evaluation significantly without compromisingaccuracy.114Chapter 5Reliability Assessment ofDeteriorating Structures usingPhysics Informed Neural Networkbased PDEMIn Chapter 4, the reliability analysis of a stochastic system coupled with the uncertaintyrelated to the system’s parameter using probability density evolution method (PDEM)is presented. It estimates the probability density function of the structural response bysolving generalized density evolution equations (GDEEs). The advantage of PDEM is thatit is derived based on the principle of probability conservation, where GDEEs are decoupledfrom the physical system. In Chapter 4, GDEEs in PDEM are solved using a finite differencescheme in which the accuracy of the numerical solution depends on the number of temporaland spatial discretizations, making them computationally inefficient for high-fidelity models.With this in view, this chapter proposes a physics-informed neural network (PINN), anovel deep learning method-based PDEM, for solving the GDEEs. PINN utilizes physicalinformation in the form of differential equations to enhance the performance of the neuralnetworks. This method does not need any interpolation or coordinate transformation, whichis often seen in any numerical scheme, thus the computational budget is reduced. Thedetails of the proposed framework for estimating PDF and performing numerical analysisare presented in the following section.5.1 Objectives∗ Develop a physics-informed neural network-based probability density evolution frame-work for estimating the transition probability density function of the performancefunction, which is used to calculate the first-passage failure probability of the struc-ture.∗ The performance of the proposed framework is compared with the FPK equation andMonte-Carlo simulation.∗ The three different applications are demonstrated to establish the proposed frameworkfor reliability assessment. A Van-der-Pol oscillator subjected to Gaussian white noise,1155.2. NUMERICAL ALGORITHM FOR PINN-BASED PDEM FRAMEWORKis considered as first example to show the accuracy of the proposed framework. Theother two applications are the reliability analysis of a one-storey moment resistingframe coupled with a nonlinear energy sink with negative stiffness and sliding frictionand a high-rise timber building coupled with shape memory alloy-based outriggers.In this chapter, the material properties of steel frame and outrigger-timber structureare considered to be deteriorating with time.5.2 Numerical Algorithm for PINN-based PDEMFrameworkIn this section, the numerical algorithm for PINN-based PDEM framework is discussed.The reliability estimation using PDEM is discussed in Section 4.6 in which the GDEE issolved using finite difference method and fitted with stochastic spectral embedding surrogatemodel to obtain the response surface. In this chapter, we focus on the solution of GDEE inEq. 4.38a using PINN. The details of PINN can be found in Section 2.3.7. It is seen fromEq. 4.38a that the coefficient V˙(θ, τ) is known for a given θ and τ because it is a functionof YEEV which is estimated independently from the model for a given θ. Here, the randomvariables associated with the system are represented by θ. Therefore, the loss functionsassociated with GDEE in Eq. 4.38a can be expressed in the following form:Lp(λ) = 1NpNp∑i=1∣∣∣∣∣∂pVΘ(vip,θ, τ ip;λ)∂τ + V˙(θ, τ ip)∂pVΘ(vip,θ, τ ip;λ)∂v∣∣∣∣∣2(5.1a)L0(λ) = 1N0N0∑i=1∣∣∣pVΘ(vi0,θ, 0;λ)| − δ(v)pΘ(θ)∣∣∣2 (5.1b)Therefore, the weighted loss function is expressed as L(λ) = Lp(λ)+β1L0(λ). The rest pro-cedure is elaborated in Section 2.3.7. In this chapter, we use the generalized F-discrepancy(GF)-based method to generate the collocation points. The details of the GF-discrepancymethod is discussed in Section 4.8. The numerical algorithm used in this study is depictedin Algorithm 7. The flowchart for the reliability analysis using PINN-based PDEM is shownin Fig. 5.1.5.3 Numerical ExamplesIn this section, three different numerical examples are investigated to demonstrate thereliability analysis of the structural system, in which the PDF of the performance functionis evaluated using the proposed PINN-based PDEM framework. These are: (1) a Van-der-Pol oscillator subjected to a Gaussian white noise; (2) a one-storey moment resisting frameintegrating a nonlinear energy sink with negative stiffness and sliding friction; and (3) ahigh-rise timber building coupled with a shape memory alloy-based outrigger. The first1165.3. NUMERICAL EXAMPLESAlgorithm 7 Estimation of failure probability using PDEM1: Generate the collocation samples of random variables based on the GF-discrepancy andestimate the assigned probability of the samples.2: For each collocation sample, solve the governing equation of motion using the computa-tional model. Once the response time history is obtained, the equivalent extreme event(i.e., YEEV) is evaluated.3: Construct the virtual stochastic process V(τ) and its derivative w.r.t τ using Eq. 4.37.4: Initialize the weight and bias of the neural network for solving GDEE using PINN.5: for j = 1, 2, . . ., epochs do6: Construct the neural network using training samples and estimate the output ∂pVΘ.7: Estimate the derivative with respect to τ and v using automatic differentiation.8: Construct the loss function :L(λ) = 1NpNp∑i=1∣∣∣∣∣∂pVΘ(vip,θ, τ ip;λ)∂τ + V˙(θ, τ ip)∂pVΘ(vip,θ, τ ip;λ)∂v∣∣∣∣∣2+ β1{1N0N0∑i=1∣∣∣pVΘ(vi0,θ, 0;λ)| − δ(v)pΘ(θ)∣∣∣2(λ)}9: Update the neural network by minimizing L(λ) using Adam optimization10: Update the weight and bias of neural network : λj+1 ← λj − η∇λ11: end for12: Estimate the PDF of YEEV i.e., pYEEV(y) by substituting τ = 1.13: Calculate the probability of failure by performing one-dimensional integration onpYEEV(y).1175.3. NUMERICAL EXAMPLESxi(θ)tYEEV = extt∈ 0 Tx θ, tpYEEVYEEVPfRepresentative Points Based on GF-discrepancyComputational Model Estimation of EEVEstimation of PDF Physics-Informed Neural NetworkTime-Dependent Failure ProbabilityFigure 5.1: Flowchart for reliability analysis using PINN-based PDEMexample is investigated to establish the accuracy of the proposed framework for estimatingthe transition PDF. For this reason, the results are compared with the FPK equation andMonte-Carlo simulation. Once the proposed framework is established, the remaining twoexamples are considered for reliability analysis under ground motion uncertainty, in whichthe material properties of the structural system deteriorate with time. For simulationpurposes, python using TensorFlow [ABC+16] and MATLAB software is used.5.3.1 Example 1: Van-der-Pol OscillatorAs the first example, a Van-der-Pol oscillator is considered, whose PDF using the pro-posed framework is compared with that estimated by the FPK equation. The governingequation of motion of the Van-der-Pol oscillator subjected to a Gaussian white noise iswritten as follows:x¨(t) + 2ξ[x(t)2 − 1]x˙(t) + x(t) = √2ση(t) (5.2)where x(t), x˙(t) and x¨(t) are the displacement, velocity, and acceleration of the oscillator,respectively; the damping ratio of the oscillator is denoted by ξ. In Eq. 5.2, η(t) is theGaussian white noise with spectral density σ whose first two moments can be expressed asfollowsE[η(t)]= 0; E[η(t)η(t+ τ)]= δ(τ) (5.3)1185.3. NUMERICAL EXAMPLESThe governing equation of motion of the oscillator, written in Eq. 5.2, can be expressed inthe state-space form, which is given by[x˙1x˙2]=[x2−2ξx21x2 + 2ξx2 − x1]+[0√2σ]η(t) (5.4)where x1 = x and x2 = x˙, respectively. The PDF of a stochastic process, x(t) can bedescribed by the FPK equation, which is as follows∂p(x, t)∂t= Lp(x, t) := ∇ · [p(x, t)∇V (x, t)]+∇ · [∇ · (p(x, t)D(x, t))], ∀(x, t) ∈ Rn ×R+(5.5a)∫Rnp(x, t)dx = 1; p(x, t) ≥ 0; p(x, 0) = p0(x) (5.5b)In the above equation, p(x, t) is the PDF of x(t) which is unknown, and p0(x) is the PDFof x(t) at t = 0. Also, in Eq. 5.5a, the partial differential operator is denoted by L. Thepotential function and diffusion matrix are represented by V (x, t) and D(x, t), respectively,which are expressed as∇V (x, t) = −µ(x, t); D(x, t) = 12G(x, t)G(x, t)T (5.6)where µ(x, t) and G(x, t) are the two terms of the following stochastic differential equationdx = µ(x, t)︸ ︷︷ ︸Driftdt+ G(x, t)︸ ︷︷ ︸Diffusiondη (5.7)Comparing Eq. 5.4 and Eq. 5.7, the FPK equation written in Eq. 5.5a, can be expressedfor the Van-der-Pol oscillator as∂p∂t= −2∑i=1∂[µip]∂xi+122∑i=12∑j=1∂2[(GσGT )ijp]∂xi∂xj= −∂[x2p]∂x1− ∂[(−2ξx21x2 + 2ξx2 − x1)p]∂x2+ σ∂2p∂x22= −x2 ∂p∂x1+ 2ξ(x21 − 1)p+(2ξx21x2 − 2ξx2 + x1) ∂p∂x2+ σ∂2p∂x22(5.8)The stationary solution of Eq. 5.8 is estimated when ∂p∂t becomes zero, i.e.,2ξ(x21 − 1)p− x2 ∂p∂x1+(2ξx21x2 − 2ξx2 + x1) ∂p∂x2+ σ∂2p∂x22= 0 (5.9)The exact solution of the above equation is given by [CS16]pXX˙(x, x˙) = C exp{2ξσ[x2 + x˙22− (x2 + x˙2)216]}(5.10)1195.3. NUMERICAL EXAMPLESwhere C is a normalization constant. In this study, Eq. 5.8 is solved using the alternatingdirection implicit method. In this method, the finite difference steps in each direction arecomputed separately. The finite difference in x1 direction is expressed aspm+1/2i,j = pmi,j +∆t2[− x2(pm+1/2i+1,j − pm+1/2i−1,j2∆x1)+ 2ξ(x21 − 1)pmi,j+(2ξx21x2 − 2ξx2 + x1)(pmi,j+1 − pmi,j−12∆x2)+ σ(pmi,j+1 − 2pmi,j + pmi,j−1∆x22)](5.11)Once the above equation yields pm+1/2i,j , then finite difference is applied in the x2 direction,as followspm+1i,j = pm+1/2i,j +∆t2[− x2(pm+1/2i+1,j − pm+1/2i−1,j2∆x1)+ 2ξ(x21 − 1)pm+1/2i,j+(2ξx21x2 − 2ξx2 + x1)(pm+1i,j+1 − pm+1i,j−12∆x2)+ σ(pm+1i,j+1 − 2pm+1i,j + pm+1i,j−1∆x22)](5.12)For numerical demonstration purposes, the parameters associated with Eq. 5.2 are con-sidered to be ξ = 0.05 and σ = 0.5. We aim to estimate PDF of x(t) by solving FPKequation as given in Eq. 5.8. To solve that, an alternate directions implicit (ADI) method,which is a finite difference scheme, is adopted. The state-space is discretized on the domain[-10, 10] × [-10, 10] which is divided into 100 elements per state. The initial conditions ofthe states are assumed to follow a normal distribution, whose mean and standard deviationare taken as 4 and 0.5, respectively. The step length has been set to ∆t = 0.01. Fig. 5.2shows the evolution of the solution of Eq. 5.8 where the joint PDF, pXX˙(x, x˙, t) for differenttime instant is shown. To compare the accuracy of the solution from the FPK equation,the stationary solution of the oscillator from the FPK equation is compared with the exactstationary solution, defined in Eq. 5.10. To compare the results, the marginal PDF of theprocess is easier than the joint PDF. Fig. 5.3(a) shows the marginal PDF of the displace-ment response of the oscillator, whereas the marginal PDF of the velocity response is shownin Fig. 5.3(b). It is seen from Fig. 5.3 that the agreement between the analytical stationarysolution from FPK equation and exact stationary solution is well matched.Once the accuracy of the stationary solution is established, we focus on the solution oftransition PDF of the oscillator using PINN-based PDEM framework. The architecture ofthe network in PINN is set to four hidden layers with 20 neurons in each hidden layer. Toestimate the weight and bias of the network by minimizing the loss function, the Adamoptimization algorithm is used in this study. The learning rate is set to be 0.001. Fig. 5.4shows the marginal PDF of the displacement response of the oscillator estimated usingthe proposed PINN-based PDEM. It also compares the marginal PDF of the displacement1205.3. NUMERICAL EXAMPLESFigure 5.2: Propagation of joint PDF between displacement and velocity of oscillator esti-mated from FPK Equation;(a) t = 0 s, (b) t = 20 s, (c) t = 40 s and (d) t = 60 sresponse obtained from the FPK equation and Monte Carlo simulation (MCS). It is seenfrom Fig. 5.4 that the solution of the transition PDF using the proposed framework isconsistent with the solution using the FPK equation and MCS.1215.3. NUMERICAL EXAMPLES-10 -5 0 5 1000.050.10.150.2-10 -5 0 5 1000.050.10.150.2Figure 5.3: Stationary solution of marginal PDF of (a) displacement and (b) velocity esti-mated from FPK Equation and exact solution5.3.2 Example 2: One-storey Frame with NESHere, a one-storey moment-resisting frame coupled with a nonlinear energy sink (NES)with negative stiffness and sliding friction is considered as our next problem. The structureis subjected to earthquake loading, which is assumed to be a stationary process. Fig. 3.2shows the arrangement of the NES in which two elastic springs are arranged in such a waythat when they are aligned in a vertical position, they are compressed [DTC+20]. Thenegative stiffness arises in the NES due to the arrangement of the springs, as shown inFig. 3.2(b). To prevent instability such as distortions of hinged springs in the negativestiffness region, the sliding friction [DTC+20] is used here. Also, in this numerical study,the nonlinearity in the primary structure is not considered. The details of the structuralsystem and the governing equations of motion are provided in Section 3.4.1.In this study, a power model is used to describe the deterioration of the primary structurewith time. The deterioration may occur due to corrosion. As in this study, the columnsof the primary structure is made of steel, the sizes of the column are decreased due toatmospheric corrosion. For long-term corrosion of steel, the power model is widely used[FDS+11], which is given byρ(t) = β1tβ2 (5.13)where the corrosion loss after t years is denoted by ρ(t); β1 and β2 are the corrosion lossafter one year and a constant, respectively.The structural properties, i.e., mass, damping, and stiffness of the primary structure,are taken as 189.4 kg, 26.2 Nmm−1 and 0.35%, respectively [DTC+20]. In this prob-lem, the reliability analysis of the structure coupled with the NES is performed under1225.3. NUMERICAL EXAMPLES-10 -5 0 5 1000.20.40.63 4 50.40.50.6-10 -5 0 5 1000.050.10.150.20.250 1 2 30.150.20.25-10 -5 0 5 1000.050.10.150.20.25-10 -5 0 5 1000.050.10.150.20.25Figure 5.4: Comparison between the solution using PINN-based PDEM, FPK equation andMCS for the marginal PDF of the displacement response of the oscillator at time instants:(a) t = 0 s, (b) t = 20 s, (c) t = 40 s and (d) t = 60 sground motion uncertainty. For this reason, an ensemble of 100 ground motions, whichare zero-mean stationary Gaussian processes, is simulated using the power spectral den-sity function, defined in Eq. 3.25. The parameters related to the Kanai-Tajimi spectrum,i.e., S0, ωg and ηg are assumed to be uniformly distributed, whose bounds are as follows:S0 ∼ U(0.016, 0.033) m2s−3, ωg ∼ U(7.9, 16.2) rads−1 and ηg ∼ U(0.26, 0.53)%. The stiff-ness and compressed length of the spring, i.e., kNES and L0 are taken as 5656.4 Nm−11235.3. NUMERICAL EXAMPLESand 0.13 m [DTC+20]. With these properties, the reliability analysis of the structure isperformed, where the properties of the column of the structure deteriorate with time. Inthis study, the probability of failure is estimated at 5, 10, 15, and 20 years after the design.It is obvious that the size of the column reduces due to corrosion of the steel, which resultsin a reduced moment of inertia and stiffness of the column. The power model is used fordeterioration of the stiffness of the column over the time, defined in Eq. 5.13. The twoconstants, β1 and β2 in Eq. 5.13 are assumed to be 29.1 µm and 0.54, respectively [Ha17].With this in view, the probability of failure of the structure is estimated using a PINN-based PDEM for 0 to 20 years, individually. Here 0 refers to the time at the design phase. Asan ensemble of 100 ground motions is considered in this study, the numerical algorithm startswith generating 100 collocation points of the random variables using the GF-discrepancymethod. For each ground motion, the system’s response is evaluated. The storey driftis chosen as the extreme event, which is expressed as YEEV = maxt∈[0,T ]|X(t)|. Once theextreme event is computed for all collocation points, the PDF of YEEV is estimated usingPDEM, as defined in Eq. 4.38 and Eq. 4.39. The partial differential equation in Eq. 4.38ais solved using PINN. The architecture of the neural network is set to 8 hidden layers with30 neurons in each hidden layer. The hyperbolic tangent function is used as an activationfunction. The Adam optimization algorithm is used to compute the weight and bias of thenetwork, in which the learning rate is assumed to be 0.001. Fig. 5.5(a) shows the PDF ofthe virtual stochastic process, V(τ) obtained from PINN, whereas the loss function in PINNis estimated in every iteration, as shown in Fig. 5.5(b). Once the PDF of V(τ) is obtained,the PDF of YEEV is estimated by substituting τ equals to 1 in PDF of V(τ), as in Eq. 4.39.Fig. 5.5(c) shows the PDF of YEEV at design phase, at 10 and 20 years from design phase.Once the PDF is obtained, the probability of failure is estimated, which is expressed asPf = Pr[YEEV ≥ YThres]=∫ ∞YThrespYEEV(y)dy (5.14)Here, the threshold value, i.e., YThres is taken as 0.04 m [DTC+20]. The time-dependentprobability of failure of the structure is shown in Fig. 5.5(d). It is seen from Fig. 5.5(d) thatat the design phase, the probability of failure is 2.8E-5, whereas after 20 years, the proba-bility of failure of the structure increases to 4.5E-2 due to deterioration of the structure.5.3.3 Example 3: SMA-based Damped Outrigger Timber BuildingAs a last example, the reliability analysis of a tall timber building coupled with a shapememory alloy (SMA)-based damped outrigger is demonstrated. The outrigger is generallyplaced between the column and shear wall with the aim of reducing the deformation ofthe tall structure when the structure is subjected to dynamic external excitation. Theincreased load demand is frequently observed on the column connected to the outrigger,which increases member size and raises construction costs. A SMA spring is used in betweenthe outrigger and column, which dissipates the energy due to an external load and, hence,reduces the load demand on the column connected to the outrigger. Fig. 4.8 shows the1245.3. NUMERICAL EXAMPLES0 0.025 0.05 0.075 0.1102030400 5 10 15 2000.010.020.030.040.05(c) (d)Figure 5.5: (a) PDF of the virtual stochastic process, V(τ); (b) Convergence history ofPINN; (c) PDF of equivalent extreme event, YEEV when age of the structure is 0, 10 and20 years and (d) Time dependent probability of failure of the structurearrangement of the SMA-based outrigger for timber structure. The governing equations ofmotion of this structure are derived using the Lagrangian formulation, whose details areprovided in Section 4.10.4.Here, a 20-storey timber building with two SMA-based outriggers is taken into consid-eration. The beams and columns of the structure are made of glulam, and the shear wallis made of cross-laminated timber (CLT). The material properties of glulam and CLT, the1255.3. NUMERICAL EXAMPLESsizes of the beams, columns, and shear wall, and the architectural plan of the structureare taken from [DT22]. The outriggers are located at 35% and 95% of the total heightof the structure [DT23]. The material properties of SMA springs are the initial austenitestiffness (ka), the maximum design displacement (xs,max), and the ductility ratio (µs) ofSMA. Also, the initial austenite stiffness is estimated from the normalized transformationstrength of SMA, which is F0 = kaxys/(mH) where xys is the yield displacement of SMA.These values are set to F0 = 0.27, µs = 22.15 and xs,max = 0.18 [DT23]. In this numericalexample, two types of uncertainties are considered, which are ground motion uncertaintyand material property uncertainty. It is assumed that the modulus of elasticity of glulambeam (Eg), modulus of elasticity of CLT wall in longitudinal (E1) and transverse (E2) layersare random variables that follow the Gaussian distribution with mean values of 12.4, 11.7,and 9 GPa, respectively, and a coefficient of variation of 10%. Also, it is assumed that themodulus of elasticity of glulam and CLT deteriorate with the time following the power lawi.e., E(t) = (1 − k1T k2)E0, where E0 is the initial modulus of elasticity. The constants k1and k2 are assumed to be 0.007 and 0.85, respectively [FAL17]. The maximum value ofthe inter-storey drift ratio of the floors is taken as the objective function. An ensemble ofground motions is taken, which are selected based on the conditional mean spectrum-basedrecord selection method for the Vancouver site (Canada) at a natural time period of 2.0 s.The probability of failure of the system is defined as followsPf = Pr[YEEV ≥ YThres]=∫ ∞YThrespYEEV(y)dy (5.15)where YEEV is given asYEEV =1NgNg∑j=1max1≤i≤20{maxt∈[0,T ]∣∣IDRi,j(θ, t)∣∣} (5.16)where Ng denotes the total number of ground acceleration time histories and IDR theinter-storey drift ratio. The numerical procedure for this example is the same as the pre-vious. Firstly, a set of 150 collocation points of random variables is generated using GF-discrepancy. For each collocation sample, the inter-storey drift is calculated for Ng groundmotion using the model evaluation. The extreme event, defined in Eq. 5.16, is estimatedfor every collocation sample. Using these data, the GDEE in PDEM is solved using PINNfor estimating the PDF of YEEV. The architecture of the neural network is set to 10 hiddenlayers with 20 neurons in each hidden layer. The hyperbolic tangent function is used as anactivation function. Fig. 5.6(a) shows the PDF of YEEV for different time instant. Also, ”0”yr in Fig. 5.6(a) refers the time at the design phase. The probability of failure is estimatedusing Eq. 5.15 which is shown in Fig. 5.6(b). It is seen from Fig. 5.6(b) that the probabilityof failure of the structure is 1.38E-4, whereas after 20 years, the failure probability becomes4.70E-3 due to the deterioration of the material properties of the timber structure.1265.4. SUMMARY1 1.2 1.4 1.6024680 5 10 15 20012345 10-3Figure 5.6: (a) PDF of equivalent extreme event, YEEV and (b) Time dependent probabilityof failure of the outrigger structure5.4 SummaryThis chapter proposes a framework for reliability analysis of stochastic systems usinga PINN-based PDEM. The failure probability of the system is evaluated based on theprobability density function of an equivalent-extreme event, which is characterized by theGDEEs. The GDEEs are solved using PINN, a novel deep learning method. The collocationsamples of the random variables are generated using the generalized F-discrepancy method,which is used to train the neural network. Three numerical examples are illustrated toestablish the proposed PINN-based PDEM framework for reliability analysis of the system,which are a Van-der-Pol oscillator subjected to Gaussian white noise, a one-storey momentresisting frame coupled with a nonlinear energy sink with negative stiffness and slidingfriction, and a high-rise timber building coupled with shape memory alloy-based outriggers.The major observations from this study are listed below:∗ The accuracy of the proposed framework is validated with the FPK equation andMonte Carlo simulation. It is difficult to estimate the transition PDF using theFPK equation for higher degrees-of-freedom systems with system’s nonlinearity. Forthis reason, a Van-der-Pol oscillator is examined whose exact stationary solution andnumerical solution of the transition state using the FPK equation are established. It isseen from the numerical investigation that the solution using the proposed frameworkis consistent with the FPK equation and Monte Carlo simulation.∗ Once the proposed framework is established, the reliability analysis of the structuresis performed from the civil engineering application’s perspective. The material prop-1275.4. SUMMARYerties of the structural system deteriorate with time. Therefore, the problem becomesone of time-dependent reliability analysis. It is seen that 150 collocation samples ofthe random variables are required to get satisfactory performance of PINN, whichleads to a reduced computational cost compared to any numerical method such as thefinite difference method.128Chapter 6Conclusion6.1 Summary of the FindingsThis thesis investigates the reliability analysis and design of two structures: a one-storeymoment resisting frame coupled with a nonlinear energy sink with negative stiffness andsliding friction and a shape memory alloy-based damped outrigger for a high-rise timberbuilding.Chapter 2 elaborates on the importance of the selection of the design of experimentstrategy in predicting the output using surrogate models. The surrogate model predicts theoutput based on the training data by minimizing the residuals between the output corre-sponding to the training data and predicted responses. An extensive survey of the differentDoE strategies is presented in Chapter 2. A review of classical and modern experimentalstrategy design is also provided. This chapter is only focused on the performance of thephysics-informed neural network. In this context, a few salient observations are highlightedbelow:∗ In this chapter, five different partial differential equations, which are Viscous Burger’sequation, Shro¨dinger equation, one-dimensional heat equation, Allen-Cahn equationand Korteweg-de Vries equation are considered for investigation purposes. To solvethese equations, a physics-based deep neural network method is used, which is trainedusing the training data generated by different DoE schemes. It is seen that quasi-random sampling techniques, especially the Hammersley sampling scheme, yield betterresults compared to classical DoE schemes. This argument is established based oncomparing the absolute error between the actual and predicted solutions. This reflectsthe fact that the Hammersley sampling covers the whole design space better than otherDoE strategies.∗ The size of the DoE set affects the accuracy of the surrogate model. For high-fidelitymodels where the computational budget is expensive, the smallest size of the trainingset is needed to reduce the computational budget as well as the accuracy of theprediction.Chapter 3 investigates the effectiveness of the nonlinear energy sink (NES) with negativestiffness and sliding friction and a shape memory alloy (SMA)-based damped outriggerstructure for seismic vibration mitigation. The tuning parameters of the NES for one-storey frame and SMA-based outrigger system are determined by maximizing structural1296.1. SUMMARY OF THE FINDINGSreliability, which is evaluated using an outcrossing rate approach. To solve the reliability-based optimization problem, a coupled framework is used, where the outer loop solves theoptimization problem and the inner loop minimizes the propagation of uncertainties. Theentire framework becomes computationally expensive. To address this issue, a surrogatemodel is used to approximate the computationally expensive dynamic solver. The majorcontributions of the investigation presented in Chapter 3 are listed below:❏ From “one-storey frame with NES” investigations:∗ This study highlights the optimization of tuning parameters for NES in thelight of uncertainty by minimizing the probability of failure of the structure. Inthat case, Polynomial-chaos-based Kriging is used to handle the computationalexpense, and the optimization is solved by the genetic algorithm in an adaptivemanner.∗ With the optimized design variables, a number of case studies have been per-formed to investigate the performance of the proposed controller over cubic NES.It is found that the proposed controller is more effective at mitigating seismic vi-bration as compared to cubic NES. For example, the cubic NES is able to reduce33% of the peak and 48% RMS responses, whereas with the proposed controller,the peak and RMS responses are reduced by 38% and 65%, respectively. The neg-ative stiffness interval between the two stable equilibrium positions can rapidlytrigger long strokes with friction sliding of the proposed NES mass, which can ableto the dissipation of earthquake-induced vibration. Also, a sensitivity analysis isperformed with varying stiffness, which reflects the transient internal resonancebetween the proposed NES and primary structure over a wide frequency range.❏ From “SMA-based damped outrigger system” investigations:∗ To show the effectiveness of SMA-based damped outrigger, an optimization tech-nique is adopted to estimate the optimal design parameters by minimizing thefirst-passage probability of failure of the structure. As this method is compu-tationally expensive, a Kriging based meta-model is adopted to approximatethe original objective function. The accuracy of surrogate-based optimizationdepends on the Kriging surrogate. For this purpose, efficient global optimiza-tion method with expected improvement as the adaptive sampling for Krigingsurrogate is employed.∗ By performing efficient global optimization, it is seen that the optimal location ofoutrigger is found at the 29th floor, i.e., approximately 72% of the total buildingheight. With the optimized design parameters, a time history analysis is car-ried out to compare the top floor displacement time history for a bare structure(i.e., without outrigger), a conventional undamped, and an SMA-based outrig-ger structure. The proposed SMA based outrigger can able to reduce 56% inpeak and 60% in RMS responses of top floor displacement compared with the1306.1. SUMMARY OF THE FINDINGSconventional undamped outrigger system. Also, the proposed system can able toreduce the inter-storey drift, inter-storey drift angles for all floors in an efficientmanner, results in reduction in base shear and an overturning moment of thestructure.Chapter 4 focused on the reliability analysis of stochastic systems using the probabilitydensity evolution method (PDEM). The probability of failure of the system is evaluatedbased on the probability density function (PDF) of an equivalent-extreme event (EEV). Therepresentative points of the random variables present in the system are generated using thegeneralized F-discrepancy method. To estimate the PDF of EEV accurately, a large numberof representative points are required, which makes the PDEM computationally expensive forhigh-fidelity models. To reduce the computational burden, stochastic spectral embedding isused so that EEV is estimated accurately using as few representative points as possible. Theefficiency of the proposed method is established by considering three numerical examples.The major observations from this study are listed below:∗ PDEM enables to estimate the joint probability density function of the equivalent-extreme event of an MDOF system by solving GDEEs efficiently in contrast to othertraditional methods such as the FPK equation, Liouville equation, etc.∗ To compare the efficiency of the proposed stochastic spectral embedding-based PDEM,the probability of failure is compared with Monte Carlo simulation (MCS). In MCS,106 samples of random variables are required, whereas with 100 to 150 samples, theprobability of failure using the proposed method matches with MCS. As a result, theproposed method reduces the model evaluation significantly without compromisingaccuracy.Chapter 5 proposed a framework for reliability analysis of stochastic systems usinga physics-informed neural network (PINN)-based PDEM. The failure probability of thesystem is evaluated based on the probability density function of an equivalent-extremeevent, which is characterized by the generalized density evolution equations (GDEEs). TheGDEEs are solved using PINN, a novel deep learning method. The collocation samplesof the random variables are generated using the generalized F-discrepancy method, whichis used to train the neural network. Three numerical examples are illustrated to establishthe proposed PINN-based PDEM framework for reliability analysis of the system, i.e., aVan-der-Pol oscillator subjected to Gaussian white noise, a one-storey moment resistingframe coupled with a nonlinear energy sink with negative stiffness and sliding friction, anda high-rise timber building coupled with shape memory alloy-based outriggers. The majorobservations from this study are listed below:∗ The accuracy of the proposed framework is validated with the FPK equation andMonte Carlo simulation. It is difficult to estimate the transition PDF using theFPK equation for higher degrees-of-freedom systems with system’s nonlinearity. Forthis reason, a Van-der-Pol oscillator is examined whose exact stationary solution and1316.2. SCOPE OF FUTURE WORKnumerical solution of the transition state using the FPK equation are established. It isseen from the numerical investigation that the solution using the proposed frameworkis consistent with the FPK equation and Monte Carlo simulation.∗ Once the proposed framework is established, the reliability analysis of the structuresis performed from the civil engineering application’s perspective. The material prop-erties of the structural system deteriorate with time. Therefore, the problem becomesone of time-dependent reliability analysis. For one simulation, finite difference methodtakes 270 s cpu time, whereas only 95 s cpu time is required for PINN. It is seen that150 collocation samples of the random variables are required to get satisfactory per-formance of PINN, which leads to a reduced computational cost compared to anynumerical method such as the finite difference method.6.2 Scope of Future WorkIn terms of future work, there are many potential research paths to follow:∗ In this study, a one-storey frame coupled with the NES system is considered, and theproposed method can be easily adopted for large structures. However, it is requiredto study the interaction between NES and the complex model of multi-degree freedomstructures.∗ In this study, the high-rise timber building is designed in ETABS software, and thecross-section of the beam, column, and shear wall is determined based on the estimatedload in ETABS. With these properties, a reduced-order model of the SMA-baseddamped outrigger timber building is derived based on the Lagrangian formulation.However, the connection details are not investigated properly. To capture the behaviorof connections, detailed finite element modeling is required.∗ In Chapter 4, the reliability analysis is carried out using stochastic spectral embedding-based PDEM, in which the number of samples used to train the surrogate model isevaluated by the trial-and-error method, which can be modified using an adaptivesampling technique to obtain the exact sample size. The efficacy of the proposedmethod has not been investigated for high-dimensional reliability problems. In thefuture, the proposed method can be extended to high-dimensional reliability problems.∗ Chapter 5 demonstrates the effectiveness of the proposed physics-informed neuralnetwork-based PDEM framework for the time-dependent reliability analysis of thestructure under uncertainty. Though this method shows accurate results, the accu-racy is highly dependent on the architecture of the neural network, i.e., the numberof hidden layers and the number of neurons in each hidden layer, which are obtainedby the trial-and-error method. This can be improved by using a graph neural net-work, in which the architecture of the neural network is trained by the physics ofthe problem. Also, the effectiveness of the proposed framework is not investigated1326.2. SCOPE OF FUTURE WORKfor high-dimensional reliability analysis, where the convergence of physics-informedneural networks is challenging. In this perspective, the proposed framework can beextended using PINN with a Fourier neural operator or PINN with a wavelet neuraloperator.∗ In Chapter 5, the proposed PINN-based PDEM framework does not consider anysequential method where one sample is added to the DoE set used in the previousiteration based on some performance measures. There are a few limitations of PINNthat have been found that can guide future research. There is no single strategyfor the selection of the weight parameters in a loss function, and it is not clear howthe weight parameters influence the gradients. For this reason, PINN with existingoptimization algorithms may suffer from non-convergence issues and lower efficiency.These limitations of PINN can be directions for future research.∗ In this study, deterioration of the material properties of the structure is assumed tofollow the power law. But, in reality, sometimes it does not reflect. For this reason, thestructural performance at any time is available through experimental investigation,and if one wants to include the data with the model, the PDEM framework used inthis study will not be used directly. This PDEM framework should be extended withBayesian statistics, where the model parameters need to be updated using prior andlikelihood. Also, real structures may not produce the same performance as the com-puter simulation. Like, the behavior of soil may not capture in computer simulationwhich may differ the performance from real structure. In this perspective, the digitaltwin concept is useful, in that the data from the physical twin will be used to updatethe digital twin so that the computer simulation can produce the same performanceas the real structures.133Bibliography[AB01] S.K. Au and J.L. Beck. Estimation of small failure probabilities in high dimen-sions by subset simulation. Probabilistic Engineering Mechanics, 16(4):263–277, 2001. → pages 2[ABC+16] M. Abadi, P. Barham, J. Chen, Z. Chen, A. Davis, J. Dean, M. Devin, S. Ghe-mawat, G. Irving, and M. Isard. TensorFlow: A System for Large-Scale Ma-chine Learning. In 12th {USENIX} Symposium on Operating Systems Designand Implementation ({OSDI} 16), pages 265–283, 2016. → pages 38, 118[ADLF21] A.H.S. Ang, D. De Leon, and W. Fan. Optimal reliability-based design ofcomplex structural systems. Structural Safety, 90:102048, 2021. → pages 3[AG11] G.M. Atkinson and K. Goda. Effects of Seismicity Models and New Ground-Motion Prediction Equations on Seismic Hazard Assessment for Four CanadianCities. Bulletin of the Seismological Society of America, 101(1):176–189, 2011.→ pages 112[AM74] V.L. Anderson and R.A. McLean. Design of Experiments: A Realistic Ap-proach, 1974. → pages 10[Ant14] J. Antony. Design of Experiments for Engineers and Scientists, 2014. → pages10[ANWK93] I.D. Aiken, D.K. Nims, A.S. Whittaker, and J.M. Kelly. Testing of PassiveEnergy Dissipation Systems. Earthquake Spectra, 9(3), 1993. → pages 72[AS14] M.A. AL-Shudeifat. Highly efficient nonlinear energy sink. Nonlinear Dynam-ics, 76(4):1905–1920, 2014. → pages 4[ASM+16] Y. Araki, K.C. Shrestha, N. Maekawa, Y. Koetaka, T. Omori, and R. Kainuma.Shaking table tests of steel frame with superelastic Cu–Al–Mn SMA tensionbraces. Earthquake Engineering & Structural Dynamics, 45(2):297–314, 2016.→ pages 6[ASWQ+13] M.A. Al-Shudeifat, N. Wierschem, D.D. Quinn, A.F. Vakakis, L.A. Bergman,and B.F. Spencer Jr. Numerical and experimental investigation of a highlyeffective single-sided vibro-impact non-linear energy sink for shock mitigation.International Journal of Non-Linear Mechanics, 52:96–109, 2013. → pages 4134Bibliography[AW17] T. Asai and Y. Watanabe. Outrigger tuned inertial mass electromagnetictransducers for high-rise buildings subject to long period earthquakes. Engi-neering Structures, 153:404–410, 2017. → pages 5[Bak11] J.W. Baker. Conditional Mean Spectrum: Tool for Ground-Motion Selection.Journal of Structural Engineering, 137(3):322–331, 2011. → pages 111[BCM04] M. Benesˇ, V. Chalupecky`, and K. Mikula. Geometrical image segmentation bythe Allen–Cahn equation. Applied Numerical Mathematics, 51(2-3):187–205,2004. → pages 45[BDL11] J.M. Bourinet, F. Deheeger, and M. Lemaire. Assessing small failure probabil-ities by combined subset simulation and Support Vector Machines. StructuralSafety, 33(6):343–353, 2011. → pages 2[BF88] P. Bratley and B.L. Fox. Algorithm 659: Implementing Sobol’s quasirandomsequence generator. ACM Transactions on Mathematical Software (TOMS),14(1):88–100, 1988. → pages 18[BG04] H.J. Bungartz and M. Griebel. Sparse grids. Acta Numerica, 13(1):147–269,2004. → pages 26[BNR00] V. Barthelmann, E. Novak, and K. Ritter. High dimensional polynomial inter-polation on sparse grids. Advances in Computational Mathematics, 12(4):273–288, 2000. → pages 26[Bon83] J.V. Bondar. Universal optimality of experimental designs: Definitions and acriterion. Canadian Journal of Statistics, 11(4):325–331, 1983. → pages 17[Bre91] L. Breiman. Discussion: Multivariate Adaptive Regression Splines. The An-nals of Statistics, 19(1):82–91, 1991. → pages 17[Bri93] L.C. Brinson. One-Dimensional Constitutive Behavior of Shape Memory Al-loys: Thermomechanical Derivation with Non-Constant Material Functionsand Redefined Martensite Internal Variable. Journal of Intelligent MaterialSystems and Structures, 4(2):229–242, 1993. → pages 72[BS10] G. Blatman and B. Sudret. An adaptive algorithm to build up sparse poly-nomial chaos expansions for stochastic finite element analysis. ProbabilisticEngineering Mechanics, 25(2):183–197, 2010. → pages 29[BS11] G. Blatman and B. Sudret. Adaptive sparse polynomial chaos expansion basedon least angle regression. Journal of Computational Physics, 230(6):2345–2367,2011. → pages 29[BSO+08] M.A. Bezerra, R.E. Santelli, E.P. Oliveira, L.S. Villar, and L.A. Escaleira.Response surface methodology (RSM) as a tool for optimization in analyticalchemistry. Talanta, 76(5):965–977, 2008. → pages 16135Bibliography[BW51] G.E.P. Box and K.B. Wilson. On the Experimental Attainment of OptimumConditions. Journal of the Royal Statistical Society, Series B (Methodological),13(1):1–45, 1951. → pages 12, 14[CC19] J. Chen and J. Chan. Error estimate of point selection in uncertainty quan-tification of nonlinear structures involving multiple nonuniformly distributedparameters. International Journal for Numerical Methods in Engineering,118(9):536–560, 2019. → pages 94[CG95] S. Chib and E. Greenberg. Understanding the Metropolis-Hastings Algorithm.The American Statistician, 49(4):327–335, 1995. → pages 12[CHHW13] R.B. Chen, D.N. Hsieh, Y. Hung, and W. Wang. Optimizing Latin hypercubedesigns by particle swarm. Statistics and Computing, 23(5):663–676, 2013. →pages 18[CL08] J.B. Chen and J. Li. Strategy for selecting representative points via tangentspheres in the probability density evolution method. International Journal forNumerical Methods in Engineering, 74(13):1988–2014, 2008. → pages 3[CLE+19] A.G. Carlon, R.H. Lopez, L.F.D.R. Espath, L.F.F. Miguel, and A.T. Beck.A stochastic gradient approach for the reliability maximization of passivelycontrolled structures. Engineering Structures, 186:1–12, 2019. → pages 54, 55[CLMT19] F. Carraro, R.H. Lopez, L.F.F. Miguel, and A.J. Torii. Monte Carlo integra-tion with adaptive variance selection for improved stochastic efficient globaloptimization. Structural and Multidisciplinary Optimization, 60(1):245–268,2019. → pages 54[CMW05] H. Chi, M. Mascagni, and T. Warnock. On the optimal Halton sequence.Mathematics and Computers in Simulation, 70(1):9–21, 2005. → pages 18[CMW+10] Y. Chen, D.M. McFarland, Z. Wang, B.F. Spencer Jr., and L.A. Bergman.Analysis of tall buildings with damped outriggers. Journal of Structural En-gineering, 136(11):1435–1443, 2010. → pages 7[Cow66] G.R. Cowper. The Shear Coefficient in Timoshenko’s Beam Theory. Journalof Applied Mechanics, 33(2):335–340, 1966. → pages 71[CQC+19] Y. Chen, Z. Qian, K. Chen, P. Tan, and S. Tesfamariam. Seismic performanceof a nonlinear energy sink with negative stiffness and sliding friction. StructuralControl and Health Monitoring, 2019. → pages 4, 59, 101[CR11] S. Chakraborty and B.K. Roy. Reliability based optimum design of TunedMass Damper in seismic vibration control of structures with bounded uncer-tain parameters. Probabilistic Engineering Mechanics, 26(2):215–221, 2011.→ pages 63136Bibliography[Cra02] S.B. Crary. Design of Computer Experiments for Metamodel Generation.Analog Integrated Circuits and Signal Processing, 32(1):7–16, 2002. → pages10[CS16] L. Chen and J.Q. Sun. The Closed-Form Steady-State Probability DensityFunction of van der Pol Oscillator under Random Excitations. Journal ofApplied Nonlinear Dynamics, 5(4):495–502, 2016. → pages 119[CTBA02] V.C.P. Chen, K.L. Tsui, R.R. Barton, and J.K. Allen. A review of designand modeling in computer experiments. Handbook in Statistics: Statistics inIndustry, 2002. → pages 10[CTN+15] A.A. Chojaczyk, A.P. Teixeira, L.C. Neves, J.B. Cardoso, and C.G. Soares.Review and application of Artificial Neural Networks models in reliability anal-ysis of steel structures. Structural Safety, 52:78–89, 2015. → pages 2[CW09] Z.Q. Chen and Z.H. Wang. A novel passive energy dissipation system forframe-core tube structure. In The Seventh Asia-Pacific Conference on WindEngineering, Taipei: Taiwan, 2009. → pages 7[CW19] J. Chen and Z. Wan. A compatible probabilistic framework for quantifica-tion of simultaneous aleatory and epistemic uncertainty of basic parametersof structures by synthesizing the change of measure and change of randomvariables. Structural Safety, 78:76–87, 2019. → pages 3[CYL16] J. Chen, J. Yang, and J. Li. A GF-discrepancy for point selection in stochasticseismic response analysis of structures with uncertain parameters. StructuralSafety, 59:20–31, 2016. → pages 3, 94[Daf05] C.M. Dafermos. Hyperbolic Conservation Laws in Continuum Physics, 2005.→ pages 38[DC01a] M. Dolce and D. Cardone. Mechanical behaviour of shape memory alloysfor seismic applications 1. Martensite and austenite NiTi bars subjected totorsion. International Journal of Mechanical Sciences, 43(11):2631–2656, 2001.→ pages 72[DC01b] M. Dolce and D. Cardone. Mechanical behaviour of shape memory alloys forseismic applications 2. Austenite NiTi wires subjected to tension. InternationalJournal of Mechanical Sciences, 43(11):2657–2677, 2001. → pages 72[DCM00] M. Dolce, D. Cardone, and R. Marnetto. Implementation and testing of passivecontrol devices based on shape memory alloys. Earthquake Engineering &Structural Dynamics, 29(7):945–968, 2000. → pages 7137Bibliography[DFG99] Q. Du, V. Faber, and M. Gunzburger. Centroidal Voronoi Tessellations: Ap-plications and Algorithms. SIAM Review, 41(4):637–676, 1999. → pages 12,18, 19[DGQ20] D.D. Domenico, E. Gandelli, and V. Quaglini. Adaptive isolation system com-bining low-friction sliding pendulum bearings and SMA-based gap dampers.Engineering Structures, 212:110536, 2020. → pages 6[DHCL21] H. Dong, M. He, C. Christopoulos, and Z. Li. Quasi-static tests and parametricsimulations of hybrid steel frame and light wood shear walls with frictionaldampers. Engineering Structures, 228:111485, 2021. → pages 6[DMSW90] T.W. Duerig, K.N. Melton, D. Stockel, and C.M. Wayman. Engineering As-pects of Shape Memory Alloys, 1990. → pages 6[Doe70] D.H. Doehlert. Uniform Shell Designs. Journal of the Royal Statistical Society:Series C (Applied Statistics), 19(3):231–239, 1970. → pages 15[DPLX14] K. Deng, P. Pan, A. Lam, and Y. Xue. A simplified model for analysis ofhigh-rise buildings equipped with hysteresis damped outriggers. The StructuralDesign of Tall and Special Buildings, 23(15):1158–1170, 2014. → pages 4[DS19] K.L. Du and M.N.S. Swamy. Multilayer Perceptrons: Architecture and ErrorBackpropagation. In Neural Networks and Statistical Learning, pages 97–141.Springer, 2019. → pages 35[DT22] S. Das and S. Tesfamariam. Multiobjective design optimization of multi-outrigger tall-timber building: Using SMA-based damper and Lagrangianmodel. Journal of Building Engineering, 51:104358, 2022. → pages 126[DT23] S. Das and S. Tesfamariam. Reliability based design optimization of damped-outrigger timber structure using stochastic spectral embedding based proba-bility density evolution method. In Seismic Isolation, Energy Dissipation andActive Vibration Control of Structures: 17th World Conference on SeismicIsolation (17WCSI), pages 273–284. Springer, 2023. → pages 126[DTC+20] S. Das, S. Tesfamariam, Y. Chen, Z. Qian, P. Tan, and F. Zhou. Reliability-based optimization of nonlinear energy sink with negative stiffness and slidingfriction. Journal of Sound and Vibration, 485:115560, 2020. → pages 100, 101,122, 124[DW09] H. Dai and W. Wang. Application of low-discrepancy sampling method instructural reliability analysis. Structural Safety, 31(1):55–64, 2009. → pages19138Bibliography[DWW18] J. Ding, S. Wang, and H. Wu. Seismic performance analysis of viscous damp-ing outrigger in super high-rise buildings. The Structural Design of Tall andSpecial Buildings, 27(13):e1486, 2018. → pages 4[Elf52] G. Elfving. Optimum Allocation in Linear Regression Theory. The Annals ofMathematical Statistics, pages 255–262, 1952. → pages 17[ER93] S. Engelund and R. Rackwitz. A benchmark study on importance samplingtechniques in structural reliability. Structural Safety, 12(4):255–276, 1993. →pages 2[FAL17] W.L. Fan, A.H.S. Ang, and Z.L. Li. Reliability assessment of deteriorat-ing structures using Bayesian updated probability density evolution method(PDEM). Structural Safety, 65:60–73, 2017. → pages 126[Fan12] C.J. Fang. Seismic performance and dynamic reliability of a novel dampedoutrigger system of super high-rise building. Guangzhou, China: GuangzhouUniversity, 2012. → pages 7[Fau82] H. Faure. Discre´pance de suites associe´es a` un syste`me de nume´ration (endimension s). Acta Arithmetica, 41(4):337–351, 1982. → pages 12, 19, 24[FBF+07] S.L.C. Ferreira, R.E. Bruns, H.S. Ferreira, G.D. Matos, J.M. David, G.C.Branda˜o, E.G.P. da Silva, L.A. Portugal, P.S. Dos Reis, A.S. Souza, andW.N.L. dos Santos. Box-Behnken design: An alternative for the optimiza-tion of analytical methods. Analytica Chimica Acta, 597(2):179–186, 2007. →pages 12, 15[FCAL12] M. Follador, M. Cianchetti, A. Arienti, and C. Laschi. A general methodfor the design and fabrication of shape memory alloy active spring actuators.Smart Materials and Structures, 21(11):115029, 2012. → pages 7[FDS+11] D. Fuente, I. Dı´az, J. Simancas, B. Chico, and M. Morcillo. Long-term at-mospheric corrosion of mild steel. Corrosion Science, 53(2):604–617, 2011. →pages 122[Fis26] R.A. Fisher. The Arrangement of Field Experiments. Ministry of Agriculture,33:503–513, 1926. → pages 10[FLS05] K.T. Fang, R. Li, and A. Sudjianto. Design and Modeling for ComputerExperiments, 2005. → pages 17[FLZ+20] C. Fang, D. Liang, Y. Zheng, M.C.H. Yam, and R. Sun. Rocking bridgepiers equipped with shape memory alloy (SMA) washer springs. EngineeringStructures, 214:110651, 2020. → pages 6139Bibliography[Fra05] T.D. Frank. Nonlinear Fokker-Planck Equations: Fundamentals and Applica-tions. Springer Science & Business Media, 2005. → pages 2[FS13] G. Falsone and D. Settineri. Explicit solutions for the response probabilitydensity function of linear systems subjected to random static loads. Proba-bilistic Engineering Mechanics, 33:86–94, 2013. → pages 2[FS14] G. Falsone and D. Settineri. On the application of the probability trans-formation method for the analysis of discretized structures with uncertainproprieties. Probabilistic Engineering Mechanics, 35:44–51, 2014. → pages 2[FSJX+19] C. Fang, B.F. Spencer Jr., J. Xu, P. Tan, and F. Zhou. Optimization of dampedoutrigger systems subject to stochastic excitation. Engineering Structures,191:280–291, 2019. → pages 70[FTCZ15] C.J. Fang, P. Tan, C.M. Chang, and F.L. Zhou. A general solution for per-formance evaluation of a tall building with multiple damped and undampedoutriggers. The Structural Design of Tall and Special Buildings, 24(12):797–820, 2015. → pages 7[FWR+18] C. Fang, W. Wang, J. Ricles, X. Yang, Q. Zhong, R. Sause, and Y. Chen.Application of an innovative SMA ring spring system for self-centering steelframes subject to seismic conditions. Journal of Structural Engineering,144(8):04018114, 2018. → pages 6[FXXQ20] D.C. Feng, S.C. Xie, J. Xu, and K. Qian. Robustness quantification of rein-forced concrete structures subjected to progressive collapse via the probabilitydensity evolution method. Engineering Structures, 202:109877, 2020. → pages3[FYLX14] C. Fang, M.C.H. Yam, A.C.C. Lam, and L. Xie. Cyclic performance of ex-tended end-plate connections equipped with shape memory alloy bolts. Journalof Constructional Steel Research, 94:122–136, 2014. → pages 72[FYLZ15] C. Fang, M.C.H. Yam, A.C.C. Lam, and Y. Zhang. Feasibility study of shapememory alloy ring spring systems for self-centring seismic resisting devices.Smart Materials and Structures, 24(7):075024, 2015. → pages 6[FZO+16] C. Fang, X. Zhou, A.I. Osofero, Z. Shu, and M. Corradi. Superelastic SMABelleville washers for seismic resisting applications: Experimental study andmodelling strategy. Smart Materials and Structures, 25(10):105013, 2016. →pages 6[GA11] K. Goda and G.M. Atkinson. Seismic performance of wood-frame houses insouth-western British Columbia. Earthquake Engineering & Structural Dy-namics, 40(8):903–924, 2011. → pages 112140Bibliography[Gam08] R. Gamaliel. Frequency-based response of damped outrigger systems for tallbuildings. PhD thesis, Massachusetts Institute of Technology, 2008. → pages69, 77, 81[GC91] E.J. Graesser and F.A. Cozzarelli. Shape-Memory Alloys as New Materialsfor Aseismic Isolation. Journal of Engineering Mechanics, 117(11):2590–2608,1991. → pages 72, 109[GC94] E.J. Graesser and F.A. Cozzarelli. A Proposed Three-Dimensional Constitu-tive Model for Shape Memory Alloys. Journal of Intelligent Material Systemsand Structures, 5(1):78–89, 1994. → pages 72, 109[Gel00] A.E. Gelfand. Gibbs Sampling. Journal of the American Statistical Associa-tion, 95(452):1300–1304, 2000. → pages 12[Gen01] O.V. Gendelman. Transition of Energy to a Nonlinear Localized Mode in aHighly Asymmetric System of Two Oscillators. Nonlinear Dynamics, 25(1-3):237–253, 2001. → pages 4[Gen03] J.E. Gentle. Random Number Generation and Monte Carlo Methods, 2003.→ pages 17[GLP07] E. Gourdon, C.H. Lamarque, and S. Pernot. Contribution to efficiency ofirreversible passive energy pumping with a strong nonlinear attachment. Non-linear Dynamics, 50(4):793–808, 2007. → pages 4[GT19] K. Goda and S. Tesfamariam. Financial risk evaluation of non-ductile rein-forced concrete buildings in eastern and western Canada. International Journalof Disaster Risk Reduction, 33:94–107, 2019. → pages 112[GTS14] B. Gaspar, A.P. Teixeira, and C.G. Soares. Assessment of the efficiency ofKriging surrogate models for structural reliability analysis. Probabilistic En-gineering Mechanics, 37:24–34, 2014. → pages 2[HA03] P.F. Hokayem and C.T. Abdallah. Quasi-Monte Carlo Methods in RobustControl Design. In 42nd IEEE International Conference on Decision andControl (IEEE Cat. No. 03CH37475), volume 3, pages 2435–2440. IEEE, 2003.→ pages 19[Ha17] C.L. Ha. Time-Dependent Reliability Analysis for Deteriorating StructuresUsing Imprecise Probability Theory. PhD thesis, 2017. → pages 124[HAAR14] S. Halchuk, T.I. Allen, J. Adams, and G.C. Rogers. Fifth generation seismichazard model input files as proposed to produce values for the 2015 NationalBuilding Code of Canada. Geological Survey of Canada Open File, 7576:15,2014. → pages 111141Bibliography[Hal60] J.H. Halton. On the efficiency of certain quasi-random sequences of points inevaluating multi-dimensional integrals. Numerische Mathematik, 2(1):84–90,1960. → pages 12, 23[HD03] J.C. Helton and F.J. Davis. Latin hypercube sampling and the propagation ofuncertainty in analyses of complex systems. Reliability Engineering & SystemSafety, 81(1):23–69, 2003. → pages 12[Hib12] D.B. Hibbert. Experimental design in chromatography: A tutorial review.Journal of Chromatography B, 910:2–13, 2012. → pages 15[HJ21] E. Haghighat and R. Juanes. SciANN: A Keras/TensorFlow wrapper for sci-entific computations and physics-informed deep learning using artificial neu-ral networks. Computer Methods in Applied Mechanics and Engineering,373:113552, 2021. → pages 38[HKM19] Y.M. Hashemi, M. Kadkhodaei, and M.R. Mohammadzadeh. Fatigue analysisof shape memory alloy helical springs. International Journal of MechanicalSciences, 161:105059, 2019. → pages 6[HM00] A. Haldar and S. Mahadevan. Probability, Reliability, and Statistical Methodsin Engineering Design. John Wiley & Sons Incorporated, 2000. → pages 2[HMW+19] P. Hao, R. Ma, Y. Wang, S. Feng, B. Wang, G. Li, H. Xing, and F. Yang.An augmented step size adjustment method for the performance measure ap-proach: Toward general structural reliability-based design optimization. Struc-tural Safety, 80:32–45, 2019. → pages 3[Hoe08] J.C.D. Hoenderkamp. Second outrigger at optimum location on high-rise shearwall. The Structural Design of Tall and Special Buildings, 17(3):619–634, 2008.→ pages 7[Hol10] M. Holtz. Sparse Grid Quadrature in High Dimensions with Applications inFinance and Insurance, 2010. → pages 26[HSS99] A.S. Hedayat, N.J.A. Sloane, and J. Stufken. Orthogonal Arrays: Theory andApplications, 1999. → pages 12, 17[HX17] Y. Huang and M. Xiong. Probability density evolution method for seismicliquefaction performance analysis of earth dam. Earthquake Engineering &Structural Dynamics, 46(6):925–943, 2017. → pages 3[Ibr09] R.A. Ibrahim. Vibro-impact Dynamics: Modeling, Mapping and Applications,volume 43. Springer Science & Business Media, 2009. → pages 4[IK94] J. Inaudi and J. Kelly. Experiments on tuned mass dampers using viscoelastic,frictional and shape-memory alloy materials. In First World Conference onStructural Control, pages 127–136, 1994. → pages 72142Bibliography[IT03] P.L.N. Inverardi and A. Tagliani. Maximum entropy density estimationfrom fractional moments. Communications in Statistics-Theory and Methods,32(2):327–345, 2003. → pages 2[JCS05] R. Jin, W. Chen, and A. Sudjianto. An efficient algorithm for constructingoptimal design of computer experiments. Journal of Statistical Planning andInference, 134(1):268–287, 2005. → pages 17[JDG02] L. Ju, Q. Du, and M. Gunzburger. Probabilistic methods for centroidalVoronoi tessellations and their parallel implementations. Parallel Computing,28(10):1477–1500, 2002. → pages 19, 20[JLZ17] H. Jiang, S. Li, and Y. Zhu. Seismic performance of high-rise buildingswith energy-dissipation outriggers. Journal of Constructional Steel Research,134:80–91, 2017. → pages 4[JMY90] M.E. Johnson, L.M. Moore, and D. Ylvisaker. Minimax and maximin distancedesigns. Journal of Statistical Planning and Inference, 26(2):131–148, 1990.→ pages 12, 19[JSW98] D.R. Jones, M. Schonlau, and W.J. Welch. Efficient global optimization ofexpensive black-box functions. Journal of Global Optimization, 13(4):455–492,1998. → pages 76[JWWH18] C. Jiang, X.P. Wei, B. Wu, and Z.L. Huang. An improved TRPD method fortime-variant reliability analysis. Structural and Multidisciplinary Optimiza-tion, 58(5):1935–1946, 2018. → pages 3[Kam21] C. Kamath. Intelligent Sampling for Surrogate Modeling, HyperparameterOptimization, and Data Analysis. Technical report, Lawrence Livermore Na-tional Lab.(LLNL), Livermore, CA (United States), 2021. → pages 21[Kan57] K. Kanai. Semi-empirical formula for the seismic characteristics of the ground.Bulletin of the Earthquake Research Institute, 35:309–325, 1957. → pages 57[Kay05] I. Kaymaz. Application of Kriging method to structural reliability problems.Structural Safety, 27(2):133–151, 2005. → pages 31[KHM17] B. Keshtegar, P. Hao, and Z. Meng. A self-adaptive modified chaos controlmethod for reliability-based design optimization. Structural and Multidisci-plinary Optimization, 55(1):63–75, 2017. → pages 3[Kie61] J. Kiefer. Optimum experimental designs V, with applications to systematicand rotatable designs. In Proceedings of the Fourth Berkeley Symposium onMathematical Statistics and Probability, Volume 1: Contributions to the The-ory of Statistics, pages 381–405. University of California Press, 1961. → pages17143Bibliography[KLF91] R.N. Kacker, E.S. Lagergren, and J.J. Filliben. Taguchi’s Orthogonal ArraysAre Classical Designs of Experiments. Journal of Research of the NationalInstitute of Standards and Technology, 96(5):577, 1991. → pages 17[KO96] J.R. Koehler and A.B. Owen. Computer experiments. Handbook of Statistics,13:261–308, 1996. → pages 10[Koh18] J.S. Koh. Design of Shape Memory Alloy Coil Spring Actuator for ImprovingPerformance in Cyclic Actuation. Materials, 11(11):2324, 2018. → pages 7[KST18] K.R. Khusnutdinova, Y.A. Stepanyants, and M.R. Tranter. Soliton solutionsto the fifth-order Korteweg–de Vries equation and their applications to surfaceand internal water waves. Physics of Fluids, 30(2):022104, 2018. → pages 47[KTB13] D.P. Kroese, T. Taimre, and Z.I. Botev. Handbook of Monte Carlo Methods,2013. → pages 12, 17[KW59] J. Kiefer and J. Wolfowitz. Optimum Designs in Regression Problems. TheAnnals of Mathematical Statistics, pages 271–294, 1959. → pages 12, 16[KW05] A. Klimke and B. Wohlmuth. Computing expensive multivariate functions offuzzy numbers using sparse grids. Fuzzy Sets and Systems, 154(3):432–453,2005. → pages 26[LA93] T. Lorenzen and V. Anderson. Design of Experiments: A No-Name Approach,1993. → pages 10[LBK08] J. Lee, M. Bang, and J.Y. Kim. An analytical model for high-rise wall-framestructures with outriggers. The Structural Design of Tall and Special Buildings,17(4):839–851, 2008. → pages 107[LC04] J. Li and J.B. Chen. Probability density evolution method for dynamic re-sponse analysis of structures with uncertain parameters. Computational Me-chanics, 34(5):400–409, 2004. → pages 2[LC07] J. Li and J.B. Chen. The number theoretical method in response analysisof nonlinear stochastic structures. Computational Mechanics, 39(6):693–708,2007. → pages 3[LC08] J. Li and J. Chen. The principle of preservation of probability and the gen-eralized density evolution equation. Structural Safety, 30(1):65–77, 2008. →pages 2, 90[LC09] J. Li and J.B. Chen. Stochastic Dynamics of Structures. John Wiley & Sons,2009. → pages 3, 93144Bibliography[LCF07] J. Li, J.B. Chen, and W.L. Fan. The equivalent extreme-value event andevaluation of the structural system reliability. Structural Safety, 29(2):112–131, 2007. → pages 92[LDWH17] Z. Li, H. Dong, X. Wang, and M. He. Experimental and numerical investi-gations into seismic performance of timber-steel hybrid structure with supple-mental dampers. Engineering Structures, 151:33–43, 2017. → pages 6[LHX+16] J. Liu, Y. Hu, C. Xu, C. Jiang, and X. Han. Probability assessments ofidentified parameters for stochastic structures using point estimation method.Reliability Engineering & System Safety, 156:51–58, 2016. → pages 2[Li02] R. Li. Model selection for analysis of uniform design and computer experi-ment. International Journal of Reliability, Quality and Safety Engineering,9(04):367–382, 2002. → pages 12[Li16] J. Li. Probability density evolution method: Background, significance andrecent developments. Probabilistic Engineering Mechanics, 44:111–117, 2016.→ pages 2[Liu01] J.S. Liu. Monte Carlo Strategies in Scientific Computing, 2001. → pages 17[LLC+19] X. Lu, W. Liao, Y. Cui, Q. Jiang, and Y. Zhu. Development of a novelsacrificial-energy dissipation outrigger system for tall buildings. EarthquakeEngineering & Structural Dynamics, 2019. → pages 5[LMO08] H. Li, C.X. Mao, and J.P. Ou. Experimental and theoretical study on twotypes of shape memory alloy devices. Earthquake Engineering & StructuralDynamics, 37(3):407–426, 2008. → pages 7[LNS02] S.N. Lophaven, H.B. Nielsen, and J. Søndergaard. DACE – A MATLABKRIGING TOOLBOX, volume 2. Citeseer, 2002. → pages 76[LR97] C. Liang and C.A. Rogers. Design of Shape Memory Alloy Springs with Ap-plications in Vibration Control. Journal of Intelligent Material Systems andStructures, 8(4):314–322, 1997. → pages 7[LS04] L.D. Lutes and S. Sarkani. Random Vibrations: Analysis of Structural andMechanical Systems. Butterworth-Heinemann, 2004. → pages 73[LTM18a] P.C. Lin, T. Takeuchi, and R. Matsui. Seismic performance evaluation of sin-gle damped-outrigger system incorporating buckling-restrained braces. Earth-quake Engineering & Structural Dynamics, 47(12):2343–2365, 2018. → pages5[LTM+18b] L. Liu, P. Tan, H. Ma, W. Yan, and F. Zhou. A novel energy dissipationoutrigger system with rotational inertia damper. Advances in Structural En-gineering, 21(12):1865–1878, 2018. → pages 5145Bibliography[LWA21] S. Li, J.Q. Wang, and M.S. Alam. Seismic performance assessment of a multi-span continuous isolated highway bridge with superelastic shape memory alloyreinforced piers and restraining devices. Earthquake Engineering & StructuralDynamics, 50(2):673–691, 2021. → pages 7[LWH+14] J. Luo, N.E. Wierschem, S.A. Hubbard, L.A. Fahnestock, D.D. Quinn, D.M.McFarland, B.F. Spencer Jr., A.F. Vakakis, and L.A. Bergman. Large-scaleexperimental evaluation and numerical simulation of a system of nonlinearenergy sinks for seismic mitigation. Engineering Structures, 77:34–48, 2014.→ pages 4[LWH+19] Z. Li, X. Wang, M. He, W. Dong, and H. Dong. Seismic performance oftimber–steel hybrid structures. I: Subassembly testing and numerical mod-eling. Journal of Structural Engineering, 145(10):04019113, 2019. → pages6[MA10] T.J. Mackman and C.B. Allen. Investigation of an adaptive sampling methodfor data interpolation using radial basis functions. International Journal forNumerical Methods in Engineering, 83(7):915–938, 2010. → pages 2[MBN20] L. McClenny and U. Braga-Neto. Self-Adaptive Physics-Informed Neural Net-works using a Soft Attention Mechanism. arXiv preprint arXiv:2009.04544,2020. → pages 37[MBTYP18] M. Morales-Beltran, G. Turan, U. Yildirim, and J. Paul. Distribution of strongearthquake input energy in tall buildings equipped with damped outriggers.The Structural Design of Tall and Special Buildings, 27(8):e1463, 2018. →pages 4[Mei80] L. Meirovitch. Computational Methods in Structural Dynamics, volume 5.Springer Science & Business Media, 1980. → pages 104, 105, 106[MG15] R.K. Mohammadi and A. Ghasemof. Performance-based design optimizationusing uniform deformation theory: A comparison study. Latin American Jour-nal of Solids and Structures, 12:18–36, 2015. → pages 62[MGC13] S.K. Mishra, S. Gur, and S. Chakraborty. An improved tuned mass damper(SMA-TMD) assisted by a shape memory alloy spring. Smart Materials andStructures, 22(9):095016, 2013. → pages 72, 81[MGH03] R.L. Mason, R.F. Gunst, and J.L. Hess. Statistical Design and Analysis ofExperiments With Applications to Engineering and Science, 2003. → pages10[MGM+03] L.I. Manevitch, O. Gendelman, A.I. Musienko, A.F. Vakakis, and L. Bergman.Dynamic interaction of a semi-infinite linear chain of coupled oscillators with a146Bibliographystrongly nonlinear end attachment. Physica D: Nonlinear Phenomena, 178(1-2):1–18, 2003. → pages 4[MHZ05] L. Mathelin, M.Y. Hussaini, and T.A. Zang. Stochastic Approaches to Un-certainty Quantification in CFD Simulations. Numerical Algorithms, 38(1-3):209–236, 2005. → pages 24[MKK+05] D.M. McFarland, G. Kerschen, J.J. Kowtko, Y.S. Lee, L.A. Bergman, and A.F.Vakakis. Experimental investigation of targeted energy transfers in stronglyand nonlinearly coupled oscillators. The Journal of the Acoustical Society ofAmerica, 118(2):791–799, 2005. → pages 4[Mon17] D.C. Montgomery. Design and Analysis of Experiments, 2017. → pages 10,12, 13[MR13] M. Malekinejad and R. Rahgozar. An analytical approach to free vibrationanalysis of multi-outrigger–belt truss-reinforced tall buildings. The StructuralDesign of Tall and Special Buildings, 22(4):382–398, 2013. → pages 106[MRC13] S.K. Mishra, B.K. Roy, and S. Chakraborty. Reliability-based-design-optimization of base isolated buildings considering stochastic system param-eters subjected to random earthquakes. International Journal of MechanicalSciences, 75:123–133, 2013. → pages 81[MS18] S. Marelli and B. Sudret. An active-learning algorithm that combines sparsepolynomial chaos expansions and bootstrap for structural reliability analysis.Structural Safety, 75:67–74, 2018. → pages 2[MS21] N. Mulay and A. Shmerling. Analytical approach for the design and opti-mal allocation of shape memory alloy dampers in three-dimensional nonlinearstructures. Computers & Structures, 249:106518, 2021. → pages 6[MWLS21] S. Marelli, P.R. Wagner, C. Lataniotis, and B. Sudret. Stochastic SpectralEmbedding. International Journal for Uncertainty Quantification, 11(2):25–47, 2021. → pages 29, 31[NBC15] NBCC. National Building Code of Canada, 2015. → pages 113[NLC+18] F.F.S. Nascentes, R.H. Lopez, J.E.S. Cursi, R. Sampaio, and L.F.F. Miguel.An Efficient Global Optimization Approach for Reliability Maximization ofFriction-Tuned Mass Damper-Controlled Structures. Shock and Vibration,2018, 2018. → pages 62[Nor38] W. Norbert. The Homogeneous Chaos. American Journal of Mathematics,60(4):897–936, 1938. → pages 28147Bibliography[NPS16] S.R.K. Nielsen, Y.B. Peng, and M.T. Sichani. Response and reliability analysisof nonlinear uncertain dynamical structures by the probability density evolu-tion method. International Journal of Dynamics and Control, 4(2):221–232,2016. → pages 3[Owe92] A.B. Owen. Orthogonal arrays for computer experiments, integration andvisualization. Statistica Sinica, pages 439–452, 1992. → pages 17[Owe98] A.B. Owen. Scrambling Sobol’ and Niederreiter–Xing Points. Journal of Com-plexity, 14(4):466–489, 1998. → pages 18[PP10] A.G. Piersol and T.L. Paez. Harris’ Shock and Vibration Handbook, 2010. →pages 105[PSKC07] H.J. Pradlwarter, G.I. Schueller, P.S. Koutsourelakis, and D.C. Charmpis. Ap-plication of line sampling simulation method to reliability benchmark prob-lems. Structural Safety, 29(3):208–221, 2007. → pages 2[PSP93] A. Parkinson, C. Sorensen, and N. Pourhassan. A General Approach for Ro-bust Optimal Design. Journal of Mechanical Design, 115(1):74–80, 1993. →pages 17[Puk06] F. Pukelsheim. Optimal Design of Experiments. SIAM, 2006. → pages 17[PZL21] Y. Peng, T. Zhou, and J. Li. Surrogate modeling immersed probability den-sity evolution method for structural reliability analysis in high dimensions.Mechanical Systems and Signal Processing, 152:107366, 2021. → pages 3[QZ17] C. Qiu and S. Zhu. Shake table test and numerical study of self-centeringsteel frame with SMA braces. Earthquake Engineering & Structural Dynamics,46(1):117–137, 2017. → pages 6[Rai18] M. Raissi. Deep Hidden Physics Models: Deep Learning of Nonlinear PartialDifferential Equations. The Journal of Machine Learning Research, 19(1):932–955, 2018. → pages 38[RBGP06] V.J. Romero, J.V. Burkardt, M.D. Gunzburger, and J.S. Peterson. Compar-ison of pure and “Latinized” centroidal Voronoi tessellation against variousother statistical sampling methods. Reliability Engineering & System Safety,91(10-11):1266–1280, 2006. → pages 18[RC04] C.P. Robert and G. Casella. Monte Carlo Statistical Methods, 2004. → pages17[RC17] A.K. Rathi and A. Chakraborty. Reliability-based performance optimizationof TMD for vibration control of structures with uncertainty in parameters andexcitation. Structural Control and Health Monitoring, 24(1):e1857, 2017. →pages 4148Bibliography[RPK19] M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural net-works: A deep learning framework for solving forward and inverse problemsinvolving nonlinear partial differential equations. Journal of ComputationalPhysics, 378:686–707, 2019. → pages 41[RX04] S. Rahman and H. Xu. A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probabilistic Engineering Me-chanics, 19(4):393–408, 2004. → pages 2[SAKK11] I.M. Sobol’, D. Asotsky, A. Kreinin, and S. Kucherenko. Construction andComparison of High-Dimensional Sobol’ Generators. Wilmott, 2011(56):64–79, 2011. → pages 18[SF14] D. Settineri and G. Falsone. A method for the evaluation of the responseprobability density function of some linear dynamic systems subjected to non-gaussian random load. Probabilistic Engineering Mechanics, 38:165–172, 2014.→ pages 2[SHC10] L. Sun, W.M. Huang, and J.Y. Cheah. The temperature memory effect andthe influence of thermo-mechanical cycling in shape memory alloys. SmartMaterials and Structures, 19(5):055005, 2010. → pages 72[Shi72] M. Shinozuka. Monte Carlo solution of structural dynamics. Computers &Structures, 2(5-6):855–874, 1972. → pages 2[SM95] M.A. Savi and E.N. Mamiya. Passive vibration control using pseudoelastic ma-terials DINAME. In Proc. 4th Symposium on Dynamic Problems of Mechanics(Caxambu, 1995), pages 230–3, 1995. → pages 72[Sob67] I.M. Sobol’. On the distribution of points in a cube and the approximateevaluation of integrals. USSR Computational Mathematics and MathematicalPhysics, 7(4):86–112, 1967. → pages 12, 19[SQVB12] T.P. Sapsis, D.D. Quinn, A.F. Vakakis, and L.A. Bergman. Effective stiffeningand damping enhancement of structures with strongly nonlinear local attach-ments. Journal of Vibration and Acoustics, 134(1):011016, 2012. → pages4[SS81] B.S. Smith and I. Salim. Parameter study of outrigger-braced tall buildingstructures. Journal of the Structural Division, 107(10):2001–2014, 1981. →pages 4[SS83] B.S. Smith and I. Salim. Formulae for optimum drift resistance of outriggerbraced tall building structures. Computers & Structures, 17(1):45–50, 1983.→ pages 107149Bibliography[SSB16] A. Shah, M. Sabir, and P. Bastian. An efficient time-stepping scheme fornumerical simulation of dendritic crystal growth. European Journal of Com-putational Mechanics, 25(6):475–488, 2016. → pages 45[SSM16] R. Scho¨bi, B. Sudret, and S. Marelli. Rare Event Estimation Using Polynomial-Chaos Kriging. ASCE-ASME Journal of Risk and Uncertainty in EngineeringSystems, Part A: Civil Engineering, 3(2):D4016002, 2016. → pages 32[SSQB18] A. Shah, M. Sabir, M. Qasim, and P. Bastian. Efficient numerical scheme forsolving the Allen-Cahn equation. Numerical Methods for Partial DifferentialEquations, 34(5):1820–1833, 2018. → pages 45[SSW89] J. Sacks, S.B. Schiller, and W.J. Welch. Designs for Computer Experiments.Technometrics, 31(1):41–47, 1989. → pages 10, 31[SSW15] R. Schobi, B. Sudret, and J. Wiart. Polynomial-chaos-based Kriging. Inter-national Journal for Uncertainty Quantification, 5(2), 2015. → pages 32[Ste87] M. Stein. Large Sample Properties of Simulations Using Latin HypercubeSampling. Technometrics, 29(2):143–151, 1987. → pages 18[SW07] R.J. Smith and M.R. Willford. The damped outrigger concept for tall build-ings. The Structural Design of Tall and Special Buildings, 16(4):501–517, 2007.→ pages 4, 69[SWNW03] T.J. Santner, B.J. Williams, W.I. Notz, and B.J. Williams. The Design andAnalysis of Computer Experiments, 2003. → pages 10[SY11] A. Shah and L. Yuan. Numerical solution of a phase field model for incom-pressible two-phase flows based on artificial compressibility. Computers &Fluids, 42(1):54–61, 2011. → pages 45[Tag59] G. Taguchi. Linear graphs for orthogonal arrays and their applications toexperimental designs, with the aid of various techniques. Report of StatisticalApplications Research, Japanese Union of Scientists and Engineers, 6:1–43,1959. → pages 17[Taj60] H. Tajimi. A statistical method of determing the maximum response of abuilding structure during an earthquake. In Proc. of the 2nd World ConferenceEarthquake Engineering, Tokyo, Japan, pages 781–797, 1960. → pages 57[Tat96] M.A. Tatang. Direct incorporation of uncertainty in chemical and environmen-tal engineering systems. PhD thesis, Massachusetts Institute of Technology,1996. → pages 24[TD21] Solomon Tesfamariam and Sourav Das. Resilient tall timber building design:Damped-outrigger system. 2021. → pages 101, 112150Bibliography[TNT95] K. Tanaka, F. Nishimura, and H. Tobushi. Transformation start lines in TiNiand Fe-based shape memory alloys after incomplete transformations inducedby mechanical and/or thermal loads. Mechanics of Materials, 19(4):271–280,1995. → pages 6[To00] C.W.S. To. Nonlinear Random Vibration: Analytical Techniques and Appli-cations. CRC Press, 2000. → pages 2[TT91] H. Tobushi and K. Tanaka. Deformation of a shape memory alloy helicalspring: Analysis based on stress-strain-temperature relation. JSME Interna-tional Journal. Ser. 1, Solid Mechanics, Strength of Materials, 34(1):83–89,1991. → pages 7[TWW73] D.L. Thomas, J.M. Wilson, and R.R. Wilson. Timoshenko beam finite ele-ments. Journal of Sound and Vibration, 31(3):315–330, 1973. → pages 70[Vak01] A.F. Vakakis. Inducing passive nonlinear energy sinks in vibrating systems.Journal of Vibration and Acoustics, 123(3):324–332, 2001. → pages 4[Vap98] V.N. Vapnik. Statistical Learning Theory. Wiley New York, 1998. → pages32[VGB+08] A.F. Vakakis, O.V. Gendelman, L.A. Bergman, D.M. McFarland, G. Kerschen,and Y.S. Lee. Nonlinear Targeted Energy Transfer in Mechanical and Struc-tural Systems, volume 156. Springer Science & Business Media, 2008. → pages4[Via16] F.A.C. Viana. A Tutorial on Latin Hypercube Design of Experiments. Qualityand Reliability Engineering International, 32(5):1975–1985, 2016. → pages 18,19[WC16] Z. Wang and W. Chen. Time-variant reliability assessment through equivalentstochastic process transformation. Reliability Engineering & System Safety,152:166–175, 2016. → pages 3[WCC16] G. Weerasinghe, H. Chi, and Y. Cao. Particle Swarm Optimization Simulationvia Optimal Halton Sequences. Procedia Computer Science, 80:772–781, 2016.→ pages 18[WCLA20] Z. Wan, J.B. Chen, J. Li, and A.H.S. Ang. An efficient new PDEM-COM basedapproach for time-variant reliability assessment of structures with monoton-ically deteriorating materials. Structural Safety, 82:101878, 2020. → pages3[WDG+14] C. Wang, Q. Duan, W. Gong, A. Ye, Z. Di, and C. Miao. An evaluation ofadaptive surrogate modeling based optimization with two benchmark prob-lems. Environmental Modelling & Software, 60:167–179, 2014. → pages 62151Bibliography[Wen80] Y.K. Wen. Equivalent linearization for hysteretic systems under random ex-citation. Journal of Applied Mechanics, 47(1):150–154, 1980. → pages 72[WG17] J. Wu and X. Geng. Inverse scattering transform and soliton classification ofthe coupled modified Korteweg-de Vries equation. Communications in Non-linear Science and Numerical Simulation, 53:83–93, 2017. → pages 47[WLH97] T.T. Wong, W.S. Luk, and P.A. Heng. Sampling with Hammersley and Haltonpoints. Journal of Graphics Tools, 2(2):9–24, 1997. → pages 12[WSL21] D. Wang, W. Sun, and J. Li. An RKPM-based formulation of the general-ized probability density evolution equation for stochastic dynamic systems.Probabilistic Engineering Mechanics, 66:103152, 2021. → pages 3[WWSJL14] J. Wang, N.E. Wierschem, B.F. Spencer Jr., and X. Lu. Track nonlinearenergy sink for rapid response reduction in building structures. Journal ofEngineering Mechanics, 141(1):04014104, 2014. → pages 4[WWSJL15] J. Wang, N.E. Wierschem, B.F. Spencer Jr., and X. Lu. Experimental studyof track nonlinear energy sinks for dynamic response reduction. EngineeringStructures, 94:9–15, 2015. → pages 4[WYP22] S. Wang, X. Yu, and P. Perdikaris. When and why PINNs fail to train: A neu-ral tangent kernel perspective. Journal of Computational Physics, 449:110768,2022. → pages 37[WZGF98] K. Wilde, Y. Zheng, P. Gardoni, and Y. Fujino. Experimental and analyticalstudy on a shape memory alloy damper. In Smart Structures and Materials1998: Smart Systems for Bridges, Structures, and Highways, volume 3325,pages 182–191. International Society for Optics and Photonics, 1998. → pages72[WZJ+21] J. Wu, D. Zhang, C. Jiang, X. Han, and Q. Li. On reliability analysis methodthrough rotational sparse grid nodes. Mechanical Systems and Signal Process-ing, 147:107106, 2021. → pages 27[XCL12] J. Xu, J.B. Chen, and J. Li. Probability density evolution analysis of engineer-ing structures via cubature points. Computational Mechanics, 50(1):135–156,2012. → pages 3[XF19] J. Xu and D.C. Feng. Stochastic dynamic response analysis and reliabilityassessment of non-linear structures under fully non-stationary ground motions.Structural Safety, 79:94–106, 2019. → pages 3[XH05] D. Xiu and J.S. Hesthaven. High-Order Collocation Methods for Differen-tial Equations with Random Inputs. SIAM Journal on Scientific Computing,27(3):1118–1139, 2005. → pages 24152Bibliography[XK02] D. Xiu and G.E. Karniadakis. The Wiener–Askey polynomial chaos forstochastic differential equations. SIAM Journal on Scientific Computing,24(2):619–644, 2002. → pages 28[XL17] J. Xu and Z.H. Lu. Evaluation of moments of performance functionsbased on efficient cubature formulation. Journal of Engineering Mechanics,143(8):06017007, 2017. → pages 2[XSG21] J. Xian, C. Su, and H. Guo. Seismic reliability analysis of energy-dissipationstructures by combining probability density evolution method and explicittime-domain method. Structural Safety, 88:102010, 2021. → pages 3[XXY20] J. Xu, S. Xu, and Z. Yuan. Probabilistic seismic analysis of single-layer retic-ulated shell structures controlled by viscoelastic dampers with an effectiveplacement. Engineering Structures, 222:111052, 2020. → pages 3[YN00] X. Yan and J. Nie. Response of SMA superelastic systems under randomexcitation. Journal of Sound Vibration, 238:893–901, 2000. → pages 72, 73[Zhu06] W. Q. Zhu. Nonlinear Stochastic Dynamics and Control in Hamiltonian For-mulation. Applied Mechanics Reviews, 59(4):230–248, 07 2006. → pages 2[ZL07] Y.G. Zhao and Z.H. Lu. Fourth-moment standardization for structural reli-ability assessment. Journal of Structural Engineering, 133(7):916–924, 2007.→ pages 2[ZMM04] N.A. Zeidabadi, K. Mirtalae, and B. Mobasher. Optimized use of the outriggersystem to stiffen the coupled shear walls in tall buildings. The StructuralDesign of Tall and Special Buildings, 13(1):9–27, 2004. → pages 7[ZP13] X. Zhang and M.D. Pandey. Structural reliability analysis based on the con-cepts of entropy, fractional moment and dimensional reduction method. Struc-tural Safety, 43:28–40, 2013. → pages 2[ZP21] T. Zhou and Y. Peng. Active learning and active subspace enhancementfor PDEM-based high-dimensional reliability analysis. Structural Safety,88:102026, 2021. → pages 3[ZP22a] T. Zhou and Y. Peng. A new active-learning function for adaptive Polynomial-Chaos Kriging probability density evolution method. Applied MathematicalModelling, 106:86–99, 2022. → pages 3[ZP22b] T. Zhou and Y. Peng. Efficient reliability analysis based on deep learning-enhanced surrogate modelling and probability density evolution method. Me-chanical Systems and Signal Processing, 162:108064, 2022. → pages 3153Bibliography[ZP22c] T. Zhou and Y. Peng. Reliability analysis using adaptive Polynomial-ChaosKriging and probability density evolution method. Reliability Engineering &System Safety, 220:108283, 2022. → pages 3[ZP22d] T. Zhou and Y. Peng. A two-stage point selection strategy for probabilitydensity evolution method-based reliability analysis. Structural and Multidis-ciplinary Optimization, 65(5):1–18, 2022. → pages 3[ZZL16] Y. Zhou, C. Zhang, and X. Lu. An inter-story drift-based parameter analysisof the optimal location of outriggers in tall buildings. The Structural Designof Tall and Special Buildings, 25(5):215–231, 2016. → pages 7154
Thesis/Dissertation
eng
Civil Engineering
Vancouver : University of British Columbia Library
Graduate
Tesfamariam, Solomon
Surrogate assisted reliability analysis and probabilistic design of structures under uncertainty
Text