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Development of a numerical optimization methodology for the aluminum alloy wheel casting process Duan, Jianglan 2016

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    DEVELOPMENT OF A NUMERICAL OPTIMIZATION METHODOLOGY FOR THE ALUMINUM ALLOY WHEEL CASTING PROCESS by Jianglan Duan MASc., The University of British Columbia, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate and Postdoctoral Studies (Materials Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2016 © Jianglan Duan 2016 ii  Abstract Aluminum alloy wheel manufacturers face on-going challenges to produce high quality wheels and increase production rates. Improvements are generally realized by modifying the wheel and die designs and continually improving the manufacturing processes. Conventionally, these improvements have been realized by trial-and-error, building on past practice or experience. This approach typically results in long design lead times, high scrap rates and less than optimal production rates. The work presented in this study seeks to reduce the reliance on trial-and-error techniques by developing a new methodology to optimize the wheel casting process through the combination of a casting process model and open-source numerical optimization algorithms.  The casting process model utilized in this method was developed in the commercial finite element package Abaqus™ and was validated through plant trials. An open source optimization module Python Scipy.optimize has been employed to perform the optimization. The work focuses on optimizing the cooling conditions in a low-pressure die-casting (LPDC) process used to produce automotive wheels. Specifically cooling channel timing was selected because of the critical role heat extraction plays on casting quality, both in terms of dendrite cell size and the formation and growth of porosities.  The methodology was first developed with a series of test problems ending with an L-shaped geometry that employed the major features of the wheel casting process. The most suitable approach, based on the test problems, was then applied to the optimization of a 2-D axisymmetric prototype wheel die structure. The outcome revealed that numerical optimization coupled with a state-of-the-art process model has the potential to  iii  dramatically improve the method of determining cooling channel timings while also improving the product quality and process performance. The utility of the optimization methodology was found to depend on the accuracy of the casting process model. Significant challenges remain before widespread implementation of this methodology can occur in industry. Possible directions for further developments have been identified. In summary, this study represents one of the initial applications of a numerical optimization methodology to wheel casting, and that with further development; it will become an effective tool for process and die design optimization.  iv  Preface This thesis describes original work that I performed to develop a tool to optimize the wheel casting process. In performing this work, I was fortunate to draw on a number important pieces of work developed by others. The work of others that I used and my contributions include:  I developed the models of the various test problems including the L-shape geometry described in detail in Chapter 4 – Development of an Optimization Methodology.  The casting process model of the prototype wheel, introduced in Chapter 5 – Optimization of a 2D Axisymmetric Prototype Die, was developed by Dr. Carl Reilly (Research Associate in the Department of Materials Engineering at UBC) as part of a larger research program funded by Automotive Partnerships Canada.   I developed the code that allowed the prototype die model to be run cyclically, including the automatic extraction of the appropriate results at steady state, the automatic input of a refined set of cooling parameters and the automatic restarting of the code subject to whether or not a desired set of conditions (optimization) had been achieved. I also developed the analysis modules that processed the extracted data into a form suitable for input to the optimization code.  The basic optimization algorithm is part of the Python open source software package. As part of my work, I determined which algorithm within the software package was best suited for the wheel casting problem and I also determined the most suitable objective function and constraints.  v  A number of papers have been written from this work. I am the primary contributing author of these papers and was responsible for writing the first draft and performing the analysis presented in the papers. My co-authors provided editorial comments and suggestions on analysis interpretation.  Within this thesis, the discussion of the optimization settings in Chapter 4 Section 4.7 is similar to that presented in the publications [1,2,3]. The X-ray image, shown in Chapter 5 Figure 5.12, was also used in a conference paper [2]. The details of the publications are: 1. J. Duan, C. Reilly, D.M. Maijer, S.L.Cockcroft, A.B.Phillion, Development of an optimization methodology for the aluminum alloy wheel casting process, Metall. Mater. Trans. B (accepted). 2. J. Duan, C. Reilly, D.M. Maijer, S.L.Cockcroft, A.B.Phillion, Application of numerical optimization to aluminum alloy wheel casting, Model. Cast. , Weld. Adv. Solidif. Processes XII, Proc. Int. Conf., XIV ,2015, Japan. 3. J. Duan, C. Reilly, D.M. Maijer, S.L.Cockcroft, A.B.Phillion, Optimization of die cooling in the aluminum alloy wheel casting process, CIM An. Conf. (COM), 2014.     vi  Table of Contents Abstract .................................................................................................................................... ii Preface ..................................................................................................................................... iv Table of Contents ................................................................................................................... vi List of Tables ........................................................................................................................... x List of Figures ........................................................................................................................ xii List of Symbols .................................................................................................................... xvii Acknowledgements ............................................................................................................ xviii 1 Introduction .......................................................................................................................... 1 1.1 Low Pressure Die Casting Process ................................................................................. 3 1.2 The Cooling System in a Typical LPDC Water-cooled Die and the Challenges in Design ................................................................................................................................... 4 1.3 Optimal Cooling Conditions ........................................................................................... 7 2 Literature Review .............................................................................................................. 11 2.1 Casting Process Modeling ............................................................................................. 11 2.1.1 Material Properties ................................................................................................ 12 2.1.2 Boundary Conditions ............................................................................................. 13 2.2 Casting Process Optimization ....................................................................................... 16 2.2.1 Trial-and-error Method ......................................................................................... 16 2.2.2 Evolutionary Algorithm ......................................................................................... 17  vii  2.2.3 Inverse Technique .................................................................................................. 19 2.2.4 Numerical Optimization Technique ....................................................................... 21 3 Scope and Objectives ......................................................................................................... 25 3.1 Development Stage ....................................................................................................... 26 3.2 Application Stage .......................................................................................................... 27 4 Development of an Optimization Methodology .............................................................. 28 4.1 Initial Development with the Inverse Technique .......................................................... 28 4.1.1 Description of the Inverse Technique Based Methodology ................................... 29 4.1.2 Preliminary Results Using the Inverse Technique Based Methodology ................ 30 4.1.2.1 Bar Geometry ..............................................................................................................30 4.1.2.2 Wheel Geometry ..........................................................................................................33 4.1.3 Summary ................................................................................................................ 37 4.2 Structure of the Numerical Optimization Methodology ............................................... 38 4.3 Casting Process Model – Simplified Test Problem ...................................................... 42 4.3.1 Geometry of the Test Problem ............................................................................... 43 4.3.2 Mesh ....................................................................................................................... 44 4.3.3 Material Properties ................................................................................................ 44 4.3.4 Initial Conditions ................................................................................................... 45 4.3.5 Boundary Conditions ............................................................................................. 45 4.4 The Optimizer ............................................................................................................... 47 4.4.1 The Basic Algorithm .............................................................................................. 48 4.4.1.1 Update of Search Direction 𝑺......................................................................................49  viii  4.4.1.2 Update of the Step Length 𝜶 ........................................................................................50 4.4.1.3 Convergence Criterion ................................................................................................51 4.5 Analysis Modules .......................................................................................................... 52 4.6 Control Module ............................................................................................................. 53 4.7 Implementation of the Optimization Methodology for LPDC Wheel Casting ............. 55 4.7.1 Determination of Convergence Criterion .............................................................. 60 4.8 Preliminary Results and Exploration of Optimization Configuration .......................... 62 4.8.1 Objective Functions ............................................................................................... 63 4.8.2 Constraint Functions ............................................................................................. 73 4.9 Computational Intensity ................................................................................................ 79 5 Optimization of a 2D Axisymmetric Prototype Die ........................................................ 81 5.1 Casting Process Model of a Prototype Wheel ............................................................... 81 5.1.1 Model Geometry ..................................................................................................... 82 5.1.2 Mesh ....................................................................................................................... 83 5.1.3 Material Properties ................................................................................................ 83 5.1.4 Initial Conditions ................................................................................................... 85 5.1.5 Boundary Conditions ............................................................................................. 85 5.2 Implementation of the Optimization Methodology ...................................................... 86 5.3 Results and Discussion ................................................................................................. 87 5.3.1 Simple Interface Description ................................................................................. 88 5.3.1.1 Case 1 - Optimization starting from non-optimum cooling conditions .......................88 5.3.1.2 Case 2 - Optimization starting from near-optimum cooling conditions ......................96  ix  5.3.2 Enhanced Wheel/Die Interface and Water Channel Boundary Conditions ......... 102 5.3.3 Discussion ............................................................................................................ 111 6 Conclusions and Future Work ........................................................................................ 113 6.1 Conclusions ................................................................................................................. 114 6.2 Future Work ................................................................................................................ 117 Bibliography ........................................................................................................................ 119  x  List of Tables  Table 4.1 – Thermal physical properties of wheel and die materials used in the model [13,27,71]. ...................................................................................................... 44 Table 4.2 – Summary of cooling times for a simplified representation of the wheel rim section, using the default value (10 − 6) for convergence criteria. Times presented as of On – Off times relative to the cycle time. ........................................................ 61 Table 4.3 – Summary of cooling times for a simplified representation of the wheel rim section, using tightened value (10 − 8) for convergence criteria. Times presented as of On – Off times relative to the cycle time. ............................................................. 62 Table 4.4– Comparison of the initial and the calculated cooling rates at locations (P1 – P5) with linear objective function. ............................................................................ 69 Table 4.5– Comparison of the target and the calculated cooling rates at locations (P1 – P5) with quadratic objective function. ...................................................................... 69 Table 4.6– Comparison of the cooling rates at locations (P1 – P5) among cases with 16, 29, and 54 evaluation points. .................................................................................... 78 Table 5.1 – Thermal physical properties of additional die materials used in the model [13,71,72] ....................................................................................................... 84 Table 5.2 – Heat transfer coefficient at the wheel/die interfaces ...................................... 85 Table 5.3 – Heat transfer coefficient at cooling channel walls when water is on ............. 86 Table 5.4– Summary of cooling rates achieved at locations P1 – P6 (simple interface description starting from non-optimum cooling conditions). ................................... 95  xi  Table 5.5 – Summary of cooling rates achieved at locations P1 – P6 (simple interface description starting from trial-and-error based initial conditions). ......................... 100 Table 5.6 – Heat transfer coefficient at the wheel/die interfaces (updated). .................. 104 Table 5.7 – Summary of cooling rates at locations P1 – P6 (enhanced description, baseline optimized initial). ...................................................................................... 110  xii  List of Figures Figure 1.1. Schematic of the LPDC casting machine ......................................................... 4 Figure 1.2. Typical LPDC water-cooled die configuration with approximate locations of cooling channels shaded in blue and the heaters shaded in red [10] .......................... 5 Figure 1.3. Illustration of desired solidification direction in wheel .................................... 8 Figure 1.4. Photograph of sectioned wheel showing the presence of macro-porosity in the spoke-rim junction [13] ............................................................................................... 9 Figure 1.5. Micro-CT images (a) and pore size distributions (b) for a 0.5 min degassed sample at different distances from a copper chill [15] .............................................. 10 Figure 4.1 – Simple bar geometry used to construct test the inverse technique based methodology. ............................................................................................................ 31 Figure 4.2 – Temperature and heat transfer coefficient curves of the initial testing of the inverse technique based methodology on simple bar geometry, showing cyclic behavior and the effect of updating heat transfer coefficient. .................................. 32 Figure 4.3 – Axisymmetric wheel and die assembly for further development of the inverse technique based methodology. ..................................................................... 34 Figure 4.4 – Solidification sequence for the simplified 2-D wheel and die geometry ..... 36 Figure 4.5 – Predicted heat transfer coefficients over 89 cycles which contains 25 updates of the coefficients. h1 ~ h5 represents the heat transfer coefficient for Channel 1 ~ Channel 5 respectively. ............................................................................................. 37 Figure 4.6 – The structure of the optimization methodology ........................................... 39 Figure 4.7 – The geometry of the test problem. ................................................................ 43 Figure 4.8 – The flow chart of the optimization methodology. ........................................ 54  xiii  Figure 4.9 – The locations used for evaluation of the objective (P1-5) and the constraint (white dots) functions in the test problem. ................................................................ 57 Figure 4.10 – A simple geometry used to assess the convergence criterion ..................... 60 Figure 4.11 – Cooling times for the test problem before and after optimization with a linear objective function. .......................................................................................... 65 Figure 4.12 – Cooling times for the test problem before and after optimization with a quadratic objective function. ..................................................................................... 66 Figure 4.13 – Comparison of the casting and die temperature for the cooling conditions corresponding to the initial (1st row), optimized with the linear objective function (2nd row), and optimized with the quadratic objective function (3rd row). ............... 68 Figure 4.14 – Evolutions of the cooling rates corresponding to the condition optimized with linear objective function ................................................................................... 71 Figure 4.15 – Evolutions of the cooling rates corresponding to the condition optimized with quadratic objective function .............................................................................. 72 Figure 4.16 – Evolution of the objective function for optimization with linear objective function. .................................................................................................................... 72 Figure 4.17 – Evolution of the objective function for optimization with quadratic objective function. ..................................................................................................... 73 Figure 4.18 – Illustration of the locations of the constraint evaluation points, with the configurations of 29 points (Base case), 16 points (Reduced), and 54 points (Increased). ................................................................................................................ 74 Figure 4.19 – Cooling times for the test problem following optimization with 16 constraint evaluation points. ..................................................................................... 75  xiv  Figure 4.20 – Cooling times for the test problem following optimization 54 constraint evaluation points. ...................................................................................................... 75 Figure 4.21 – Comparison of the optimized results from the cases with 16 evaluation points (2nd row), 29 evaluation points (3rd row), and 54 evaluation points (4th row).77 Figure 4.22 - Evolution of the objective function for constraint function configurations with 16 (left) and 54 (right) evaluation points. ......................................................... 78 Figure 5.1 – 2-D axisymmetric wheel geometry approximating a prototype wheel geometry and approximate locations of the seven cooling channels embedded in the die. ............................................................................................................................. 83 Figure 5.2 – Cross-section of the wheel geometry with approximate locations of the seven cooling channels and showing the locations used for evaluation of the objective (P1-6) and the constraint (white dots) functions. ............................................................. 87 Figure 5.3 – Cooling times for 2-D axisymmetric prototype wheel (simple interface description starting from non-optimum cooling conditions). ................................... 90 Figure 5.4 – Cooling times evolutions over optimization iterations for Channel TD_CC_1, located above the hum-spoke junction (simple interface description starting from trial-and-error based initial conditions). .............................................. 91 Figure 5.5 – Cooling time evolution for Channel TD_CC_2, located in the lower rim (simple interface description starting from trial-and-error based initial conditions). ................................................................................................................ 92 Figure 5.6 – The initial (1st row) and optimized (2nd row) solidification sequence for the water-cooled, prototype wheel casting process (simple interface description starting from non-optimum cooling conditions). ................................................................... 94  xv  Figure 5.7 - Evolution of the objective function during optimization of the prototype wheel casting process (simple interface description starting from non-optimum cooling conditions). ................................................................................................... 96 Figure 5.8 – Cooling times for 2-D axisymmetric prototype wheel (simple interface description starting from trial-and-error based initial conditions). ........................... 97 Figure 5.9 – Cooling times evolutions over optimization iterations for Channel TD_CC_2, located in the lower rim (simple interface description starting from trial-and-error based initial conditions). ........................................................................... 98 Figure 5.10 – The initial (1st row) and optimized (2nd row) solidification sequence for the water-cooled, prototype wheel casting process (simple interface description starting from trial-and-error based initial cooling conditions). ............................................ 100 Figure 5.11 - Evolution of the objective function during optimization of the prototype wheel casting process (simple interface description starting from trial-and-error based initial conditions). ......................................................................................... 101 Figure 5.12 – X-ray image showing distributed shrinkage porosity in the top rim at the top of the wheel rim ................................................................................................ 103 Figure 5.13 – Thermocouple data measured at the hub (TC1), mid rim (TC2), and top rim (TC3), showing that the hub area is the last to fill. ................................................. 106 Figure 5.14 – Cooling times for 2-D axisymmetric prototype wheel (enhanced description, baseline optimized initial). .................................................................. 108 Figure 5.15 – The solidification sequence predicted for initial (1st row) trial-and-error based starting conditions and after the 17th optimization iteration (2nd row) for the  xvi  water-cooled, prototype wheel casting process (enhanced description, baseline optimized initial). .................................................................................................... 110 Figure 5.16 - The optimization history / the evolution of the objective function (enhanced description, trial-and-error optimized initial). ........................................................ 111  xvii  List of Symbols Latin Symbols Description Units 𝑻 temperature °C 𝑻𝒍𝒊𝒒𝒖𝒊𝒅𝒖𝒔 liquidus temperature °C 𝑻𝒔𝒐𝒍𝒊𝒅𝒖𝒔 solidus temperature °C ?̇?𝒊𝒕𝒂𝒓𝒈𝒆𝒕 target cooling rate °C/s ?̇?𝒊𝒄𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒆𝒅 calculated cooling rate °C/s ?̇?𝒊𝒂𝒗𝒆𝒓𝒂𝒈𝒆 average cooling rate °C/s 𝒕𝒇𝒌 freezing time of Node k s 𝒕𝒋𝟓𝟕𝟓℃ time reaching 575°C for Node j s 𝒕𝑫𝒊𝒆𝑶𝒑𝒆𝒏 die open time s 𝒕𝑺𝒐𝒍𝒊𝒅𝒊𝒇𝒊𝒄𝒂𝒕𝒊𝒐𝒏 solidification time s 𝒕𝒊𝒍𝒊𝒒𝒖𝒊𝒅𝒖𝒔 time reaching liquidus temperature for Node i  s 𝒕𝒊𝒔𝒐𝒍𝒊𝒅𝒖𝒔 time reaching solidus temperature for Node i s 𝒉 heat transfer coefficient W/m2/K 𝒉𝒎𝒂𝒙 maximum heat transfer coefficient W/m2/K k thermal conductivity W/m/K Cp specific heat J/kg/K L latent heat kJ/kg 𝑺 search direction     Greek Symbols Description Units 𝛒 density kg/m3 𝜶 step length  𝝁 weights used in the penalty function  𝒂𝒊 weights used in the objective function   xviii  Acknowledgements Over the past few years, I have received invaluable help from my supervisors, colleagues, friends and families, which I am deeply grateful for. The first person I would like to thank is my supervisor Prof. Daan Maijer. Without his help and guidance, this thesis work, as well as my master thesis work, could not have been completed. He has provided me with the right amount of challenges and has always motivated me to pursue better results. He has great patience for his students and is willing to spend a great amount of time to help them progress. I am grateful for all the instructions and suggestions he has given me regarding my research work, as well as his careful revision of all my papers, theses and presentations (detailed down to the use of punctuations), and his comments and suggestions on my performance after every academic presentation. I am blessed to have not only one but two intelligent and resourceful professors as supervisors. My co-supervisor, Prof. Steven Cockcroft, has also been greatly involved in supervising my work since my first day in graduate school. He has provided many critical advices regarding my research work and has spent a lot of time teaching me how to think critically, how to write concisely, and how to present effectively. There are many interesting and profound conversations I, as well as many other students, have with him on the critical qualities of a good researcher, on career development, and on culture differences and blend, which I have benefited from and have always valued.  I would like to thank Dr. Carl Reilly for his involvement and his advice during my studies. He sets a good example of what a proactive and efficient researcher should be.  xix  Thanks to Prof. Andre Phillion and Dr. Matt Roy, for sharing their experience. And thanks to my colleagues and friends, Lu Yao, Jun Ou, Xiaodan Wei, Sara Moayedinia, and many others for sharing the good times and the bad times, and for being always supportive and joyful. Thanks to my dear friends, Yajun Wang, Jingfei Zhang, Phoebe Li and more for the wonderful journey we have had together in Vancouver.  I would also like to acknowledge NSERC and Canadian Autoparts Toyota Inc. (CAPTIN) for the financial support. Special thanks to Venerable Bhikku Kaiyin, a great Buddhism monk, for his teaching in Loving-Kindness and meditation practices, which I found beneficial in both work and life. Finally my deepest gratitude goes to my family, for their unconditional love. Jianglan Duan August 2015  1  1 Introduction The aluminum alloy automotive wheel industry is very competitive and manufacturers are constantly under pressure to improve quality and reduce cost. These factors drive manufacturers to increase production rates while attempting to also maintain, or preferably reduce, scrap rates.  Production rates may be increased by reducing the time required to produce each part. Aluminum alloy wheels are generally produced using two variants of the low pressure die casting (LPDC) technique:  the conventional air-cooled version, which uses compressed air as the cooling media, and the water-cooled version, which uses water as the cooling media. The water-cooled LPDC process offers enhanced cooling and solidification rates, thus reducing the time required to produce a wheel. Higher cooling rates also provide the added benefit of finer microstructure [1], which in turn leads to superior fatigue performance [2,3]. However, being a relatively new technique, the water-cooled LPDC technique still faces challenges from higher operational costs and higher scrap rates, as will be discussed in Section 1.2. The defects that can lead to scrap that are commonly found in aluminum alloy wheels include porosity, oxide films, and exogenous inclusions [4].  Oxide films and exogenous inclusions can be effectively eliminated by inducing quiescent die filling and using filters [5,6]. Porosity is however very difficult to control and completely remove. Porosity impacts both the mechanical performance and cosmetic appearance (through paint-based defects [7-10]), and is a prominent defect of concern. The formation and growth of  2  porosity can be traced back to the casting process, and the severity can be significantly reduced via the proper design and operation of die cooling systems. Historically, the design of the die, including the cooling systems and the associated operational parameters, were arrived at through a combination of experience and trial-and-error optimization. This design methodology typically involves long lead times, and multiple stages of prototyping / preproduction trials before the die is qualified for full production. Depending on the success of this approach, initial production runs can result in high scrap rates and less than optimum production rates. Mathematical modeling offers the ability to enhance design decisions with quantitative information and to reduce development times by replacing plant trials with simulations run in a computer. In recent years, there has been an increased usage of numerical optimization techniques for engineering design problems. Numerical optimization provides a systematized and versatile procedure for arriving at a design solution. It can reduce the time required to generate production-ready designs, removes experiential biases from the design process, and virtually always yields some design improvement [11]. Though the benefits are obvious, relatively little effort has been devoted to applying numerical optimization technique to casting processes in general, not to mention the wheel casting process. Currently, the wheel manufacturing industry is still relying on time-consuming and expensive trial-and-error, in–plant trials. Against this background, this study aims to develop an optimization methodology that improves the quality and productivity of the LPDC wheel casting process.   3  1.1 Low Pressure Die Casting Process The LPDC technique is generally used in the production of rotationally symmetric parts [12]. It can produce large volumes of high-quality, near-net-shape aluminum components. As shown in Figure 1.1, a typical LPDC casting machine is comprised of a die assembly sitting above an electrically heated furnace, which contains a reservoir of liquid metal (shown in blue). For wheel production, the die assembly typically contains one top die, two side dies and one bottom die. The LPDC process is cyclic where each cycle starts with the pressurization of the holding furnace. The melt is pushed up into the die cavity via the transfer tube and sprue. The melt is then cooled and solidified by heat transfer from the melt to the die and then from the die to the cooling systems and/or to the surrounding air by convection and radiation. After solidification is complete, the side dies open and the top die is raised vertically. The wheel moves up with the top die before being ejected onto a tray, and the die closes for the next cycle.  At the start of an LPDC casting campaign, the die is preheated either off-line in an oven or in the casting machine via torches. Once casting starts, the die temperature distribution evolves considerably from cycle-to-cycle, as the die moves toward cyclic steady state (cyclic steady state is reached when the die temperature distribution at the end of the cycle is equal to the die temperature at the beginning of the cycle). The wheels cast from these first cycles generally contain significant defects and are thus discarded. Depending on the mass of the die and the cooling conditions, the die will take 5 – 10 casting cycles to approach cyclic steady. Optimal operation / productivity of an LPDC process occurs  4  when the process reaches cyclic steady-state quickly and maintains stable operation throughout the casting campaign.   Figure 1.1. Schematic of the LPDC casting machine 1.2 The Cooling System in a Typical LPDC Water-cooled Die and the Challenges in Design As mentioned in the previous section, there are two routes to remove the heat that enters the die from the liquid metal. The first is by heat transfer to cooling media (air or water) flowing in channels within the die and the other is by convective and radiative heat transfer to the environment surrounding the die. Between the two, the cooling system (cooling media in the channels) is responsible for removing the majority of the heat. An  5  example of an LPDC wheel die, configured for water cooling, is shown in Figure 1.2 [10]. The die structure is complex, containing multiple cooling channels (shown in blue) and sometimes heaters (shown in red). Water is passed through the cooling channels for predefined periods during each cycle, and at the end of each period, water may be purged from the channels by blowing compressed air through the system. The predefined periods are comprised of a start time and end time within each cycle and are generally different for each channel. In addition, the water flow rate is also typically different for each channel. In combination with differences in channel cross section, the flow rate determines an effective heat transfer rate for each channel when switched on. The combined parameters used for each cooling channel represent a cooling program, which is the key factor in determining the overall effect of cooling on the die and casting.  Figure 1.2. Typical LPDC water-cooled die configuration with approximate locations of cooling channels shaded in blue and the heaters shaded in red [10]   6  The recent interest in adopting water-cooling in place of air-cooling presents a significant challenge to foundry engineers as there is a limited experiential base to draw on. Moreover, as the heat extraction rates are typically much larger for water than for air, there is less room for error in establishing the optimum cooling parameters. Basing water-cooled die designs on air-cooled ones has a number of drawbacks that will be discussed based on the example die shown in Figure 1.2. Firstly, the increased cooling intensity in the top die, resulting from water-cooling, causes solidification to start at a location in the middle of the rim in the wheel. The middle of the rim is the thinnest cross-section in the wheel and thus the fastest to solidify. This requires that extra metal be added to the casting as a small riser on the top of the inboard rim flange to compensate for the volume change associated with solidification (4% for aluminum alloy A356) [4]. This adds to in-plant scrap production as additional machining is required. Finally, heaters are required to prevent the die from being too cold. If the die becomes too cold, liquid metal can solidify before it has a chance to fill the die cavity and leads to what is known as a misrun. The use of heaters adds both maintenance and operational costs.  To improve this water-cooling die, the riser should be removed to reduce in-plant scrap recycling provided good directional solidification can be maintained to avoid shrinkage-based porosity. In addition, the heaters should be removed to reduce maintenance and operational costs. Finally, high cooling rates are preferable to refine the as-cast structure and improve wheel fatigue performance. A redesign of the cooling system/die and the cooling parameters to achieve these improvements is not straightforward and is proving to be a significant challenge using only experience to draw on. For example, one challenge relates to how to balance the desire for rapid or aggressive cooling against the  7  opposite need to maintain a hot die to facilitate filling of the wheel cavity to avoid misruns. Additionally, the changing cross-sectional area in the wheel and the large mismatch in the thermal properties between the wheel and the die add another level of complexity. Finally, the thermal state of an operational LPDC wheel casting die can take multiple casting cycles to fully respond to an operational change – i.e. for the change to fully propagate throughout the die. All these challenges complicate the design process, especially when there is a lack of experience in the area. An accurate simulation tool is needed to enable designers to have access to quantitative data in order to make good design decisions, and to reduce the need for extensive in-plant trial-and-error testing. There is, however, an additional piece missing from this approach. In the case of the cyclic LPDC process, the system is sufficiently complex that virtual-based design inevitably devolves to a trial-and-error based approach, requiring many, many, time consuming simulations to be run. What is ultimately needed is a numerically based optimization methodology.    1.3 Optimal Cooling Conditions The optimal cooling conditions may be defined as those that lead to directional solidification in the wheel while achieving short cycle time (high cooling rates). As illustrated in Figure 1.3, directional solidification in a wheel refers to solidification starting at the inboard-rim flange, progressing down through the rim, across the spokes, and ending just below the hub in the top of the sprue. If this directional solidification pattern is not achieved, shrinkage porosity will form.  8   Figure 1.3. Illustration of desired solidification direction in a wheel Shrinkage porosity forms when liquid metal fails to feed regions of a casting to offset the volumetric contraction that occurs during the liquid to solid transformation. This typically occurs when a region of liquid metal is isolated or encapsulated by solid metal. In wheel castings, shrinkage porosity is commonly found at the hub-spoke junction and the spoke-rim junction [4,13,14] where there is large thermal mass and a transition in the solidification front geometry. The combination of these two phenomena can result in these areas becoming hot spots where local solidification is delayed. The delay in solidification results in a loss of connection to the liquid supply leading to void formation. Figure 1.4 shows an example of shrinkage porosity in the spoke-rim junction.   9   Figure 1.4. Photograph of sectioned wheel showing the presence of macro-porosity in the spoke-rim junction [13] Cooling rates are also important in determining the quality of the wheel. Not only does cooling rate affect the dendrite cell size but it also affects the size and volume fraction of gas-based pores [15-17].  Figure 1.5 illustrates the effect of cooling rate variation (distance from a water-cooled copper chill) on the pore size distribution in an A356 aluminum alloy casting. The maximum pore size increases significantly with distance from the chill (decreasing cooling rate) [15]. This result reveals a close relation between micro-porosity size and cooling rate.    10    (a). 2-D projection of 3-D porosity determined by micro-CT (b). Porosity size distribution (radius is equivalent spherical pore of the same volume) Figure 1.5. Micro-CT images (a) and pore size distributions (b) for a 0.5 min degassed sample at different distances from a copper chill [15] In the next chapter, the state of mathematical models developed to simulate casting processes will be introduced and a comprehensive discussion of the major optimization techniques will follow, as these two areas of study form the technical foundation for this thesis.   11  2 Literature Review Casting process modeling has become a powerful tool to study filling and solidification, and may be employed to predict the location of internal defects such as shrinkage porosity and cold shuts. It can be used for trouble shooting existing castings, and for developing new castings without the need for extensive preproduction trials. Recent advances in numerical optimization theory have also been leveraged to develop a variety of design/optimization strategies. These strategies allow large numbers of variables to be considered relative to traditional methods.  This chapter will first review the state-of-the-art in casting process modeling, focusing on die casting, as the accuracy of the casting simulation model will influence its ability to be applied in optimization activities. Following that, the available techniques for casting process optimization will be discussed. 2.1 Casting Process Modeling Casting process models are now commonly used to study the solidification behavior of a wide variety of shaped castings. In the context of the die casting processes, the development of process models capable of predicting die filling has been studied by a number of researchers [18-23]. Initially, these models focused on the free-surface phenomena during filling excluding heat transfer. More recent studies have shown that it is now possible to couple die filling simulations with heat transfer predictions to more accurately describe the liquid metal temperature distribution at the end of filling. However, commercial software that excels at modeling one aspect may not give  12  satisfactory results on the other. Depending on the problem at hand, researchers often choose to focus on either fluid flow or heat transfer. For example, under conditions where only limited heat transfer is expected to occur, fluid flow-only models can be used to study the free surface behaviour of the die/mould filling process and to qualitatively predict the formation of gas entrainment defects [19-22,24,25]. In another example, when accurate characterization of shrinkage related phenomena is of the upmost interest, researchers may choose to focus only on the heat transfer modeling [13,26]. Providing accurate material property data is available, the single most critical aspect limiting the predictive capabilities of a heat transfer focused casting process models is the boundary conditions. These boundary conditions are generally temperature-, spatial-dependent and/or time-dependent, which makes for a highly nonlinear problem.  2.1.1 Material Properties The realization of the importance of accurate and reliable data for the thermophysical properties involved in the heat and fluid flow modeling emerged over a decade ago. Extensive experimental efforts have since been devoted to obtaining that data. Well-established experimental techniques have been used such as dilatometry for density measurements and differential scanning calorimetry for heat capacity measurements [27]. Nowadays comprehensive material data for most commercial alloys is available in standard literature such as [27,28].  In general, material properties vary with temperature. Occasionally these properties may be altered so as to approximate other phenomena of importance that are not directly addressed within the formulation of the model. For example, in the development of a  13  thermal only model for the solidification process that considers only the diffusive transport of heat, the local fluid flow induced by natural convection in the liquid is not modeled. However, measures can be taken to approximate this effect by artificially increasing the thermal conductivity values of the metal above its liquidus temperature [13,29].  2.1.2 Boundary Conditions Characterization of boundary conditions has been an on-going challenge for process modellers as they can be very difficult to quantify and they are typically case-dependent. The boundary conditions commonly used in casting problems can be categorized into three groups: 1) interfacial heat transfer, which describes the exchange of heat between  two surfaces in proximity to one another; 2) environmental heat transfer, which describes the exchange of heat between a surface and the environment; and 3) adiabatic boundaries, associated with the adoption of any symmetry planes, or where heat transfer can be ignored - e.g. in the presence of insulating material [13]. Specifically in the wheel casting process model:  1) interfacial boundary conditions are used to describe heat transport between the wheel and die, as well as the various interfaces between different die components; 2) environmental heat transfer is also used to describe transport between the cooling channel walls (die) and the cooling medium (water or air); 3) environmental heat transfer is used to describe transport between the die and the surrounding environment, on the outside surfaces of the die, and between the  14  wheel and the surrounding environment when the wheel is lifted away from the side and bottom dies (note this includes both natural convection to air and radiation). The boundary conditions in the wheel casting process are further complicated by the different stages occurring during the process, namely 1) die filling, 2) wheel solidification and cooling, 3) die opening, 4) wheel ejection, and 5) die cooling while the die is open prior to the start of the next cycle. The boundary conditions applied in the model need to be varied to reflect the changing heat transfer conditions occurring during each stage. For example, during die filling, as the liquid metal gradually fills the die, the wheel/die interface boundary condition needs to be varied with time to reflect the height of the liquid metal in the die at a given time (this is an approximation to account for die filling). During wheel solidification, the wheel/die interface condition shifts to one dependent on the surface temperature of the wheel to account for variation in heat transfer associated with a transition from liquid metal contact to gradual formation of a gap and/or pressure. After solidification is complete, the side dies open and the top die rises up. The wheel is attached to the top die due to thermal contraction. At this time, the wheel/side die and wheel/bottom die interfaces no longer are applicable and the boundary conditions on these surfaces switches to transport to the environment.   One of the challenges with the various interface boundaries in the LPDC process has been to obtain the quantitative relationships used to describe the interfacial heat transfer occurring between the casting and die and between various die components, as these have been found to play a major role in the temperature evolution in the casting during solidification [30-33]. As previously described, in the case of the wheel/die interface the  15  state of the interface can vary from being liquid/solid, solid/solid with varying pressure, to solid/solid with varying gap size and hence factors such as the roughness of the die surface, the conductivity of the gas in the gap, and the thermo-physical properties of both the casting and the die can impact on heat transport [30]. For the sake of simplicity, early versions of computer-based thermal models disregarded this complexity and used constant heat transfer coefficients at the casting/die interfaces [34,35]. More recently, sophisticated temporally- and spatially- dependent boundary conditions have been implemented, which greatly enhances the accuracy of casting process models [18,30,36].  The state-of-the-art of wheel casting process modeling has been reviewed in the literature [13,26]. In an earlier version, Zhang et al. [13] developed a comprehensive 3-D thermal model of the LPDC process, employing both temporally and spatially dependent boundary conditions, which was capable of simulating the cyclic nature of the casting process. This model has been validated extensively using temperature measurements collected during normal casting operations. In addition to accurately reproducing the measured temperatures, the model also predicted liquid encapsulation in the rim/spoke junction area of the wheel being analyzed, consistent with where macro-porosity is observed in an industrially produced wheel. Since the original publication, this model has been enhanced to include the effect of heat loss during die filling, the effect of wheel/die deformation on interfacial heat transfer and an improved description of the cooling conditions in the cooling channel to account for the effect of boiling. Some of the details can be found in [26] and will also be discussed in this thesis (please refer to the methodology development and results sections for more details).   16  2.2 Casting Process Optimization In general, optimization of casting processes can either focus on the topological aspects – e.g mould system/die geometry, or the operational aspects, such as liquid metal temperature, mould filling rates and or cooling parameters, if applicable. Both aspects require a great deal of experience and fundamental understanding of the process to arrive at an optimized solution that is practical. The objectives of optimizing each of these aspects of the process are however quite similar: increasing production rates, reducing material consumption, and minimizing or eliminating casting defects. Until recently, most casting process improvements were arrived at using a trial-and-error methodology and there are only a few examples where numerical optimization has been applied. The numerical methodologies that have been applied include the evolutionary algorithm [37-41], the inverse technique [42-44], and the classical gradient based optimization method [45-49].  2.2.1 Trial-and-error Method The first and most straightforward optimization technique is the trial-and-error method, which relies heavily on experience [50-54]. This approach generally involves four steps: 1) develop an initial design, 2) perform simulations and/or experiments with this design, 3) evaluate the design based on defined objectives, and 4) alter the design.  Steps 2 – 4 are then repeated until the objectives are reached. Traditionally, the trial-and-error optimization process is associated with experimental trials; however, with the continued  17  development of computational models, the trial-and-error method is now commonly used with computational models. Otsuka et al. [53] used process models to select the initial temperature of a die and the metal being cast, injection speed and cast shape that yields the best die filling sequences and solidification patterns. Other similar work has been published that includes the application of a thermal – fluid flow model to improve the pressurisation sequence in wheel casting [55], and the application of a thermal model to study the influence of process factors (such as the material in side cores) and metallurgical factors (such as a near-eutectic alloy vs. A356) on the production of cast aluminum alloy wheels [56]. These studies show the potential benefits of using process models to investigate and improve the casting process for aluminum alloy wheels.  The trial-and-error approach is easy to understand and implement. However, as the process becomes more complex and multiple competing factors coexist, such a strategy typically requires long lead times, becomes prohibitively expensive [57] and is unlikely to provide an optimum solution. 2.2.2 Evolutionary Algorithm Evolutionary algorithms are inspired by biological evolution associated with reproduction, such as mutation, recombination, and selection and constitute a family of algorithms. With this family there are three main streams, namely the evolution strategy, the genetic algorithm and evolutionary programming. The specific implementation depends on the algorithm but, generally, evolutionary algorithms start with a number of designs, mimicking a population of individuals in the evolution process. The number is typically  18  on the order of hundreds or thousands. Each design is represented by a vector of design variables, and is randomly chosen to allow the entire range of possible solutions. The performance of each design is evaluated and the best ones are selected to produce the next generation. Both recombination and mutation operators are applied to generate new designs, with the aim of maintaining good features in the ‘parental’ generations and at the same time achieving diversity and/or exploring other possibilities for new generations. The new generation of candidate solutions is then used in the next iteration of the algorithm. The process iterates until a termination criteria is fulfilled or a fixed number of generations is reached [40]. Evolutionary algorithms have been applied to analyze various casting scenarios [37-39,41].  For example, they have been used to improve the grain structure in a simplified turbine blade geometry cast via the Bridgeman process [38]. The Bridgeman process is a casting technique used to produce single crystal or directionally solidified castings, in which the shell mould containing the molten metal is pulled from a hot zone into a cold zone to produce a single-crystal solidification structure. The withdrawal velocity is a critical variable in controlling the grain structure as well as the process time. At the same time, it can be easily controlled, and thus it was selected as the design variable to modify. The evolutionary algorithm was used to optimize the velocity profile to achieve superior casting quality with no stray grains and a refined grain size across the aerofoil. Another interesting feature of this work was a comparison of convergence speed achieved using an evolution strategy versus the gradient method. The results showed that the evolution strategy converged faster than the gradient method (steepest–descent gradient). One key aspect the author of this thesis noted was that the steepest-descent gradient method has  19  been claimed as the worst available gradient method to use in numerical optimization, and it is noted to have convergence problems [11]. There are other more advanced gradient methods available such as the Newton or quasi-Newton methods that have better convergence behaviour.  Evolutionary algorithms focus directly on the parameters of individual designs. This feature has three advantages: i) function continuity is not required, ii) parallel computation can be easily implemented [37], and iii) these algorithms are suitable for the global exploration of solutions. The evolutionary approach to optimization requires a relatively high number of computational experiments in order to approach the optimum and is thus computationally intensive and time-consuming.  2.2.3 Inverse Technique The inverse technique refers to the method for converting observed measurements into information about a physical object or system, normally the boundary conditions [58-60]. For example, a typical inverse problem in heat conduction is the estimation of the surface heat flux history - i.e. the boundary conditions - given one or more measured temperature histories inside a conducting body [58]. The technique works to minimize the difference between the calculated temperatures and the measured temperatures via adjusting the magnitude of the heat flux at each time interval defined. If the measurements are substituted with the ideal or target data - i.e. the inverse technique is used to calculate the boundary conditions needed to achieve desired conditions - a design/optimization problem is formulated.  20  Zabaras et al. [42-44] proposed a continuum formulation to optimize mould cooling/heating conditions in order to achieve directional solidification with a stable interface of a binary aqueous solution in a rectangular mold. They used the absence of constitutional undercooling in the liquid melt as the necessary condition for interface stability and tried to minimize the discrepancy of the calculated temperature from the concentration-dependent liquidus temperature at the interface. The temperature discrepancy was computed at the continuum level - i.e. integrated along the solid-liquid interface rather than at discrete points. Drezet and Rappaz [54] applied the inverse technique to optimize the shape of a mould for producing sheet ingots via the direct chill casting process for aluminum alloys. Their approach sought to minimize the calculated height difference for nodes on the outer surface of a cross-section. Starting with an initial guess for each of the shape parameters, the resulting ingot cross section was computed using a 3D thermo-mechanical model. Then each of the shape parameters was perturbed by a small increment to enable the calculation of a matrix of sensitivity coefficients, which in turn allows the calculation of the corrections needed to the initial parameters. Proceeding in this manner, they were able to determine a mould shape that reduced the camber of ingot surface to within allowable limits. The inverse technique is conceptually straightforward to understand and the literature reviewed has shown that it can be applied to diverse optimization situations. However, one of the distinct disadvantages of this method is that the simple mathematical formulation and the lack of constraints causes this problem to be ill-posed, meaning that there are often multiple numerically viable solutions to a given problem and determining a physically realistic solution may be difficult. Sometimes the solutions oscillate and  21  become unstable. Regularization methods have been developed to address these instabilities [58]. Another disadvantage is that they are CPU-intensive, especially when multiple locations or variables are being compared. Finally, in optimization problems with additional constraints, it is not clear how these constraints can be imposed, as this technique only builds a relationship or sensitivity between the design variables and the measured or the targeted data. 2.2.4 Numerical Optimization Technique Numerical optimization refers to a group of gradient-based methods that formed the basis of early optimization theory. It is essentially a procedure for finding the minimum or the maximum of an objective function, which mathematically describes some attribute of the product or the process being optimized. Often, constraints are added to account for additional limitations imposed on the design. The general mathematical expressions of an objective function and constraint functions are presented in Equation 2.1 and Equation 2.2 [61]:  max (min)       𝑓(𝑥)     𝑥 = (𝑥1, 𝑥2, … , 𝑥𝑛)         Equation 2.1 subject to   𝑔𝑖(𝑥) = 0    𝑓𝑜𝑟 𝑖 = 1,2, … , 𝑛     ℎ𝑗(𝑥) ≤ 0     𝑓𝑜𝑟 𝑗 = 1,2, … , 𝑚         Equation 2.2  where 𝑥 is a vector containing all of the design variables that may be modified in order to achieve the optimization goal, 𝑓(𝑥) is an objective function, and 𝑔𝑖(𝑥)  and ℎ𝑗(𝑥)  are equality and inequality constraints, respectively.   22  To use this technique, one needs to carefully choose the design variables and, based on the optimization goal and practical limitations, carefully design the objective and constraint functions. Fortunately, there are some example applications for casting optimization. For example, McDavid and Dantzig [45] used the numerical optimization technique to modify the shape of a casting runner system. Their goal was to reduce the occurrence of air entrainment in the main runner caused by fluid backflow. Eight parameters characterizing the shape of the runner, or more specifically, the thickness of the runner at eight different locations, were selected as the design variables. The objective function was set to be the sum of the amount of fluid in contact with the runner wall in the region of concern during filling. Constraints were imposed to limit the range of the design variables. By maximizing the objective function, the amount of liquid in contact with the runner wall was maximized, thus restricting the formation of an air pocket. This approach improved the filling pattern and reduced the severity of backflow appreciably. Similar work can be found in literature [46,47,62].   In other applications, Dantzig and his colleagues [48] used numerical optimization to minimize riser volume while maintaining directional solidification. To reduce material consumption, the volume of the riser was defined as the objective function and minimized. To avoid the formation of shrinkage and obtain sound castings, constraints were set to ensure that the nodes nearer to the riser solidified later than those that are farther away ensuring that directional solidification was achieved. The constraints, based on the freezing times - i.e. the time to reach the solidus temperature - were applied at select nodes. The mathematical expression used for these constraints is given as Equation 2.3 [48].  23  𝐹𝑘 = 𝑡𝑓𝑘 − 𝑡𝑓𝑘+1 Equation 2.3 where 𝑡𝑓𝑘 and 𝑡𝑓𝑘+1 are the freezing times for element k and k+1. Element k+1 is closer to the riser than element k, thus if the constraint value is positive it implies that the location in the riser solidified prior to the location in the casting. Positive values of the constraints may lead to isolated liquid pockets and shrinkage porosity in this example.  Singh et al. [14,18,49] employed a similar approach to minimize casting cycle time while applying directional solidification constraints for LPDC of aluminum alloy wheels. The deviation of the predicted temperatures from a desired cooling curve at two locations in the casting was used to formulate an objective function. As the deviation was minimized the calculated cooling curve approached the desired cooling. A series of constraint functions based on the freezing times were implemented to achieve directional solidification. In this study, the design variables were the thermo-physical properties of die sections, i.e. thermal conductivity and heat capacity. The sections of the die in direct contact with the wheel were divided into 32 different regions. The map of optimum thermal-physical properties was then used to determine locations for cooling and insulation. The results were implemented in an industrial low-pressure die-casting process for aluminum alloy wheel production and lead to an 80% increase in production capacity (from 10 to 18 wheels per hour) and a 15% reduction in the design lead time.  The work of both Dantzig et al. and Singh et al. are good examples of numerical optimization of castings. Both Dantzig et al. and Singh et al. chose to set freezing time-based constraints to enforce directional solidification from the casting into the riser. However, freezing time may not be a good representation of whether directional  24  solidification is achieved or not, as the microstructure and properties evolve continuously during solidification and the ability to feed regions of the casting will be affected by the local microstructure.  A recent paper by Chiesa et al. [63] studied the critical solid fraction used in a computational model to predict the formation of shrinkage porosity. Their work suggested that a solid fraction fs equal to 0.35 was best for A356 based on comparison with measurements. Note that their study was on a tilt poured permanent mold casting process. In the case of the low-pressure die-cast process, the critical solid fraction should be higher as the feeding pressure is higher. From the literature, it can be seen that there have been successful applications of numerical optimization to casting processes, some of which are similar to the system that is studied in this thesis. The numerical optimization technique is a reliable and relatively efficient method compared to other optimization techniques and it has been used in this work.  25  3 Scope and Objectives The objective of this study is to develop a methodology to optimize the cooling parameters in the LPDC wheel casting process. The variables considered for optimization are the operational parameters for the cooling channels, namely the on and off timing. The optimal solution will be assessed based on improved solidification conditions and increased solidification rates.  The following tasks have been undertaken in order to achieve the overall objective: 1. Evaluate candidate optimization techniques in order to select one that is suitable for optimizing the operational cooling parameters in the wheel casting process;  2. Develop optimization modules based on the chosen technique. The development work includes, but is not limited to, the development of analysis modules that are needed to connect an available computational model and the optimization algorithm; 3. Assess the capabilities of the optimization methodology using test problems and a set of suitable optimization configurations; 4. Apply the developed optimization methodology to a 2-D axisymmetric geometry representative of a prototype wheel casting process, and assess the optimization outcomes to determine the effectiveness of the methodology. Overall, this work contains a development stage and an application stage, which are described as follows.  26  3.1 Development Stage Tasks 1 to 3 are focused on developing a suitable optimization package for wheel casting processes. Candidate strategies are compared based on their ability to achieve a desired solution, the computational effort they require, and ease of implementation associated with geometry or system changes. It is also valuable if the optimization strategy requires minimal user interference and/or trial-and-error steps in setting up the optimization process.  Two optimization techniques have been tested in this study, namely the inverse and numerical optimization techniques.  These two techniques have had several casting related applications that were discussed in Chapter 2 - Literature Review. In terms of searching for new control parameters (the design variables), these two techniques have the advantage of relying on mathematical algorithms to guide the search, as compared to the trial-and-error method and the evolutionary algorithm, which in theory will lessen the computational load.  After the initial round of developments and comparison, the numerical optimization technique is selected based on its ability to find a solution.  In applying the numerical optimization technique, the optimization methodology has been developed to link an optimization program with a computational model of the casting process. A script, or wrapper, has been used as a bridge between both the computational model and the optimization program. To achieve this, analysis modules have been developed to extract the appropriate data from the process model and facilitate its input, to the optimization algorithm - i.e. to process the data coming from one module in a format that can be understood by the next module.  27  To test the optimization methodologies examined during development, various test problems have been formulated culminating in a model of a simple casting. Although geometrically simple, this model includes more than one component - i.e. both a casting and die - to include heat transfer between the two components and to assess the impact of their associated thermal masses on the cyclic response of the system. The optimizer has been developed to optimize operational cooling parameters such as cooling intensity and / or cooling duration.  3.2 Application Stage Once developed using simplified geometry, the optimization methodology will be applied to wheel casting. To avoid overcomplicating the problem and to focus on applying the methodology, a 2-D axisymmetric geometry will be used. The cooling parameters from an existing prototype die have been selected as the starting point. The 2-D axisymmetric geometry has been based on a 2-D profile extracted from an existing prototype die. The computational model has been validated to the extent possible against data collected from a production die. The optimization methodology has then been used to determine the optimal operational cooling parameters for this geometry.    28  4 Development of an Optimization Methodology In developing a numerical methodology to optimize a casting process, the first issue that must be addressed is to select a suitable optimization algorithm. Depending on the algorithm selected, there may be a commercial optimization program and/or an open-source package available to use. The second issue is to identify and connect the representation of the casting process that the optimization algorithm will work on. Normally, numerical optimization is paired with a computational model of some type. The computational model acts as a virtual experiment or process to test the effects of design variable changes. Additional analysis modules may be needed to extract the relevant data from the process model output and convert it for use by the optimization algorithm.  For this study, the initial development of an optimization methodology focused on using an inverse technique to adjust the cooling intensity, which is related to the flow rate of the cooling media in the cooling channel. The methodology and the significant challenges encountered are discussed in Section 4.1. Following that, Sections 4.2 - 4.8 will provide the details of the numerical optimization technique that was eventually selected for the optimization methodology, the process model, and their integration to enable the optimization of casting processes. 4.1 Initial Development with the Inverse Technique As mentioned in the Literature Review, the inverse technique generally refers to calculating the boundary conditions needed to achieve a solution that matches the  29  experimental measurements. It can also be formed to calculate the boundary conditions required to achieve a desired condition. The latter can be regarded as a design/optimization problem.  4.1.1 Description of the Inverse Technique Based Methodology  In this strategy, the objective function is based on the differences between target cooling rates and calculated cooling rates at selected points in the wheel. The cooling intensity (heat transfer coefficient on the surface of a cooling channel) was used as the design variable, which was updated based on the sensitivity of the cooling rates to the cooling intensity and the differences between the target and calculated cooling rates. To facilitate directional solidification, a novel temperature-based constraint on cooling start and end times was implemented. In this methodology, a cooling channel turns on only after the temperature at a selected adjacent point in the wheel falls below a predefined temperature. This point in the wheel is termed the “switch-on control point” and the predefined temperature is termed the “switch-on temperature in this study”. Meanwhile, to prevent the die from becoming too cold, causing premature solidification in the next cycle, cooling channels were switched off when the temperature at a selected adjacent point in the die falls below another predefined temperature. This point was defined as the “switch-off control point” and the predefined temperature in this context is termed the “switch-off temperature”. In this method, the cooling duration (amount of time cooling is active in a cycle) is dependent on the switch-on and switch-off temperatures. If multiple cooling channels are present, each channel will be controlled separately to achieve directional solidification.   30  4.1.2 Preliminary Results Using the Inverse Technique Based Methodology Modules were built around this technique and the methodology was tested with a series of models with increasing complexity, and the results will be discussed in Section 4.1.2.1 and 4.1.2.2.   4.1.2.1 Bar Geometry The optimization package was first applied to a simple bar geometry as shown in Figure 4.1. The purpose of using this geometry was to test the package’s ability to reach a set target.  The model contains 2-D, axisymmetric components, a die and a casting, each 10 mm long. The bar was placed 200 mm away from the origin of the coordinate system to mimic a slice of the rim section. For the sake of simplicity at the initial stage of this study, constant values were used for the material properties and the die / casting interfacial heat transfer coefficient (1300 W/m2 K). The model was run for multiple cycles with a fixed cycle time of 100 s. The initial condition used at the beginning of each cycle for the casting was a uniform temperature of 700°C. For the die, the temperature for the first cycle was uniformly 500°C. In subsequent cycles, the initial temperature in the die was taken to be the predicted temperature distribution at the end of the previous cycle. At the die/cooling channel interface, a convective heat transfer boundary condition was used. At the die/casting interface, an interface-type boundary condition was used to enable the implementation of an effective heat transfer coefficient, which combines both conductive and radiative heat transfer. Adiabatic conditions were assumed to apply to the top, bottom and right hand side surfaces.    31  The algorithm was formulated to achieve a target cooling rate, set to 15°C/s, by either increasing or decreasing the heat transfer coefficient at the die/cooling channel interface from an initial guess of 200 W/(m2K). The magnitude of the heat transfer coefficient was chosen to lie in an intermediate range between air and water as the cooling, the two fluid variants commonly used in industry. The update to the heat transfer coefficient was performed after cyclic steady-state results were achieved to allow the effect from the last update to propagate throughout the die. The cyclic steady-state condition was achieved when the die temperature at the start of the cycle differs less than 1°C from the die temperature at the end of the cycle. Being close to a 1-D heat transfer problem, this criterion was only checked at the leftmost and the rightmost points of the die. The switch-on control point and the switch-off control point for the temperature-based constraint were placed as shown in Figure 4.1. Initially, cooling was set to start at the beginning of the cycle, which is equal to a switch-on temperature of 700°C. The switch-off temperature was set to 400°C. The whole optimization package including the FE model was developed using Fortran.  Figure 4.1 – Simple bar geometry used to construct test the inverse technique based methodology.  The temperature evolution over 28 cycles at locations in the casting and die are plotted in Figure 4.2. The evolution of the predicted heat transfer coefficient is also plotted in Figure 4.2. Temperatures on the casting-side of the casting / die interface are higher than  32  the die-side which is in turn higher than temperature at the die/cooling interface. This is expected as heat is being extracted from the casting to the die before it leaves the system via convection at the die/cooling interface.  Figure 4.2 – Temperature and heat transfer coefficient curves of the initial testing of the inverse technique based methodology on simple bar geometry, showing cyclic behavior and the effect of updating heat transfer coefficient. To bring the predicted cooling rate in line with the target cooling rate, the heat transfer coefficient was updated after each cyclic steady-state condition was achieved. With each new coefficient, the model was allowed to run to steady state again before the next update. Five heat transfer coefficient updates were required for the predicted cooling rate to match the target cooling rate. Note: that the number of cycles needed to reach steady state after each update of heat transfer coefficient decreases, and that the correction to the  33  coefficient is getting smaller. These results indicate that the in-house optimization strategy is able to achieve stable solutions on the simple bar geometry.  4.1.2.2 Wheel Geometry After testing the in-house optimization strategy on the simple bar geometry, the algorithm was first applied to a simplified, hypothetical, 2-D axisymmetric wheel and die assembly shown in Figure 4.3. The FE model in the case was developed in the commercial FE package Abaqus, instead of the in-house Fortran code, as Abaqus is superior in handling complex boundary conditions especially the interfacial heat transfer BC. The switch to use Abaqus necessitated the development and implementation of external codes to handle the cyclic runs and the optimization algorithm. These external codes were written in Python (a scripting language).  The 2-D axisymmetric geometry was used to reduce computational load. There are four circular cooling channels located in the top die. The initial cooling parameters used in the implementation of the temperature-based constraint on this model are indicated in Figure 4.3. The switch-on temperatures for most of the cooling channels are the same (550°C) to initiate sequential cooling. The switch-on temperature was set to be higher in the top cooling channel (700°C) to drive solidification down from the inboard rim flange area (top of wheel in Figure 4.3). The switch-off temperatures were chosen according to the temperature data extracted from an experimentally validated thermal model developed by colleagues. As before, the design variables are the heat transfer intensity in the cooling channels and the objective functions are set to achieve target cooling rates at five selected locations,  34  indicated in yellow in Figure 4.3. The update calculation was performed after cyclic steady-state results were achieved. However, as this problem was more complex than the previous one, several modifications were required to allow the algorithm to run. First, bounds were established to constrain the calculated heat transfer coefficients. The lower bound was 5.0 W/m2K for stagnant air, and the upper bound was 7000 W/m2K (calculated forced water convection correlations, based on the maximum water flow rates expected in production dies). Second, the temperature-based constraints to turn cooling on or off were not used during sensitivity evaluation. Instead, the cooling start and finish times were based on the cooling times from the last steady-state solution. Third, the maximum allowable change of the calculated heat transfer coefficient was restricted to 10% of the last heat transfer coefficient. The last two modifications were found to be helpful in preventing the solution going fully saturating (either 7000 W/m2/K or 5 W/m2/K).  Figure 4.3 – Axisymmetric wheel and die assembly for further development of the inverse  35  technique based methodology.  The algorithm was applied to optimize the cooling intensity in 5 top die cooling channels.  As shown in Figure 4.4, the predicted solidification images are presented as temperature contours at 4 times during a casting cycle. The 575 °C isotherm, which represents approximately 40% solid fraction, has been highlighted in these plots.   If the isotherm encloses on itself, this indicates that liquid encapsulation will occur leading to shrinkage defects. The solidification images have shown liquid encapsulation at two well-known problematic areas: 1) the inboard rim flange as the wheel starts solidifying halfway along the rim causing solidification fronts to proceed upward and downward; and 2) the rim-spoke junction. In an effort to eliminate liquid encapsulation, several tests were performed with different switch-on and off temperatures and the locations of the switch-on and off points, with no improvement in the results. Liquid encapsulations at these two locations are difficult to avoid because: 1) the small thermal mass of the rim section (much thinner compared to the inboard rim flange) makes it easier to solidify prior to other parts of the wheel and thus cuts off the liquid supply to the solidifying metal in the inboard rim flange; and 2) the large thermal mass in the rim-spoke junction makes it difficult to solidify before the solidification in the spoke starts.   36   Figure 4.4 – Solidification sequence for the simplified 2-D wheel and die geometry The evolution in the calculated heat transfer coefficients for all five channels is shown in Figure 4.5.  The magnitude of the calculated heat transfer coefficients appear to be reasonable, but the oscillations in the solution indicates a lack of stability. The oscillation in solution is likely due to the fact that cooling intensity (the magnitude of the heat transfer coefficient) and cooling duration are interrelated. For example, when the cooling intensity is increased, the die temperature reaches the switch-off temperature faster thus reducing the cooling duration. Therefore, the cooling times and the cooling intensity are constantly changing making determination of an optimum heat transfer coefficient problematic. Furthermore, a second look at the literature revealed two studies that showed that the effect of water flow rate on heat transport in a bench-scale die simulator was limited [64,65]. Therefore, it would appear to be more appropriate to switch to other cooling parameters such as the on and off timing, which is also more consistent with what is practically often controlled in the industrial LPDC processes.  37   Figure 4.5 – Predicted heat transfer coefficients over 89 cycles, which contains 25 updates of the coefficients. h1 – h5 represents the heat transfer coefficient for Channel 1 –  Channel 5 respectively. 4.1.3 Summary In summary, the findings from the investigation of the inverse optimization method and further research on cooling water heat transfer in the LPDC process are as follows:  i) The directional solidification constraints were not directly imposed, rather directional solidification was encouraged by careful selection of the switch-on and –off temperatures. The consequence of this was that directional solidification was not enforced and was often difficult to achieve. Additionally, there was no quantitative measure to assess whether directional solidification was achieved or not. As it is unclear that a viable solution exists with this configuration, this was not deemed to be a reason to abandon this approach; and  38  ii) The solution was sometimes unstable due to the ill-posed nature of the inverse problem. This more troubling aspect of this approach relates to what is presented in Figure 4.5 – e.g. after in excess of 80 updates to the heat transfer coefficients, the results do not appear to be converging and in fact appear in some instances to be diverging. The outcomes of the initial round of development revealed some issues that led to a decision to abandon the inverse optimization method using heat transfer coefficients as the design variable. The focus of this study then shifted to using the numerical optimization technique with the cooling on and off times selected as the design variables. In the following sections, an overview of the structure and the flow of the numerical optimization methodology developed for this study will be introduced, followed by a detailed discussion on the formulation of each software component. An example of the implementation of the methodology using a simple test problem will be presented and used to assess the effects of different optimization configurations.    4.2 Structure of the Numerical Optimization Methodology The structure of the optimization methodology developed for this study is shown in Figure 4.6. The casting process model simulates the thermal history in the wheel and dies during each stage of the casting process - i.e. die close, wheel solidification and cooling,  39  wheel ejection, and die open. The casting process model was developed within the framework of the general-purpose, commercial FEA software, AbaqusTM.   Figure 4.6 – The structure of the optimization methodology The three main functions of the optimizer are: i) check if the design criteria, based on user-defined objective and constraint function values, have been met; ii) calculate the changes to the design variables if the design criteria has not been realized; iii) generate requests for additional data from the FEA model that are needed to calculate the next iteration of the design variables. Several algorithms are available to build an optimizer. The Sequential Quadratic Programming (SQP) method is an advanced method [11] and it has arguably become the most successful method for solving nonlinearly constrained optimization problems [66]. Schittkowski [67], for example, has implemented and tested this method and found it outperforms every other tested method in terms of efficiency, accuracy, and percentage of successful solutions, over a large number of test problems. Based on these reviews, the SQP method was selected.                                                  TM Abaqus is a trademark of Dassault Systèms    40  An overview of the SQP method can be found in references [68,69]. The SQP method is an iterative method. It uses the Lagrangian function, which combines the objective and the constraint functions, to compute the changes in the design variable. At each major iteration, an approximation is made of the Hessian of the Lagrangian function using a quasi-Newton updating method (the Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field, and describes the local curvature of the function). This is then used to generate a quadratic programming (QP) sub-problem whose solution is used to form a search direction for a line search procedure. The general method can be stated as follows. Given the problem description in Equation 2.1 and Equation 2.2, the Lagrangian for this problem is given in Equation 4.1, such that if 𝑓(𝑥0) is a maximum or a minimum of the original constrained problem, then there exists 𝜆0  and 𝜎0  such that ( 𝑥0, 𝜆0, 𝜎0 ) is a stationary point for the Lagrange function. The method of using the Lagrange function yields a necessary condition for optimality in constrained problems.           ℒ(𝑥, 𝜆, 𝜎) = 𝑓(𝑥) − ∑ 𝜆𝑖𝑔𝑖(𝑥) −  ∑ 𝜎𝑗ℎ𝑗(𝑥) 𝑚𝑗=1𝑛𝑖=1          Equation 4.1 where ℒ(𝑥, 𝜆, 𝜎) is the Lagrangian function, 𝜆 and 𝜎 are the Lagrange multipliers, 𝑛 is the number of equality constraints and 𝑚 is the number of the inequality constraints. The Lagrangian function is used to generate a QP sub-problem of the form min 12𝑆𝑇𝐵𝑘𝑆 + ∇𝑓(𝑥𝑘)𝑇𝑆 Equation 4.2 subject to ∇𝑔(𝑥𝑘)𝑇𝑆 + 𝑔(𝑥𝑘) = 0 ∇ℎ(𝑥𝑘)𝑇𝑆 + ℎ(𝑥𝑘) ≤ 0  41  where  𝐵𝑘  is the approximation of the Hessian of the Lagrangian function, 𝑆  is the solution to the quadratic programming subproblem – the search direction. There are sophisticated open-source packages that implement the SQP algorithm, and with some modifications these packages can be tailored to specific applications. In this study, the open-source Python package, Scipy.optimize, was selected [70]. Because the casting process model, developed in the commercial FE software, Abaqus, can be manipulated through a convenient Python interface it allowed for straightforward communication between the casting process model and the optimizer.  The analysis modules are needed to generate the data required by the optimizer. They were developed with Python and also contain commands specific to the FE package AbaqusTM to allow data extraction from the FEA model.  The major operations are:  i) extracting data from the casting process model, ii) calculating the objective and constraint functions and periodically (upon the request of the optimizer) calculating the sensitivity of the objective and constraint functions to design variable changes, and iii) arranging the data obtained from ii), based on the specific form the optimizer requires. The control module controls the execution of the various modules, including the execution of the casting process model in cyclic mode. It also manages the                                                  TM Abaqus is a trademark of Dassault Systèms    42  communication between other modules and the optimizer. Details of each of the modules will be presented in the following sections.  4.3 Casting Process Model – Simplified Test Problem The methodology employed to develop the casting process model is similar to that discussed in [13,26]. In order to develop the optimization methodology, a casting process model of a simplified test problem was developed first.  Each of the casting process models presented in this thesis are thermal-only models – i.e. only diffusional heat transport is considered. The fluid flow occurring during die filling (macro-scale) is not directly considered in these models. Previous experience in fluid flow modeling have shown that accurate modeling of the die filling process is very time consuming [71] and therefore impractical for implementation in the optimization methodology at this stage. However, a number of techniques have been implemented to approximate the effects of flow related phenomena, which are important in improving the overall accuracy of the LPDC model. The effect of the progressive contact development between the liquid metal (wheel) and die during filling has been approximated (details given in Chapter 5 Section 5.14 and 5.15). Additionally, once the die is filled the effect of buoyancy is approximated by increasing the thermal conductivity of the liquid metal (refer to Section 4.3.3). The models employing these techniques have been shown to agree well with the plant trial data [13,26].   43  4.3.1 Geometry of the Test Problem The geometry of the test problem is shown in Figure 4.7. The wheel is represented by an L-shaped geometry with the rim and spoke-like elements labeled. The scale of the geometry is similar to an automotive wheel casting, and the number and location of the cooling channels were based on current production die configurations. Similar to the production dies, the simplified die geometry has three sections: a top die, a side die and a bottom die. The shape, location, and naming convention for the cooling channels are also shown. Each cooling channel is controlled separately except for the side die channels. Due to an actual casting machine limitation, the two cooling channels in the side die (SD_CC_1 and SD_CC_2) were connected in series and thus share the same cooling on and off times.    Figure 4.7 – The geometry of the test problem.  44  4.3.2 Mesh The test problem geometry has been meshed with Abaqus CAE, using quad elements based around an average element length of 3 mm. There are in total 3653 nodes and 3173 elements.  4.3.3 Material Properties The majority of cast automotive wheels are made from aluminum alloy A356. The major of the die sections (top, sides, and bottom) are made from H13 tool steel. The material properties for these materials are temperature-dependent and the values used in the model were based on those reported in the literature [13,27,72]. Table 4.1 summarizes the material properties used in the simplified test model and in the full process model.  For temperatures above the liquidus temperature (614°C), before solidification starts, the thermal conductivity for A356, the wheel material, has been artificially increased to 400 W/m/K to enhance heat transfer for the reasons previously described [13,29]. In the  model formulation employed, the domain is fixed and hence the volume of the casting does not change. Therefore, the density has been held constant to avoid the corresponding increase in mass that would occur in the domain if the density changed with temperature. The latent heat released during the liquid to solid phase transformation in A356 has been included as a source term and is released throughout a series of discrete steps over temperatures ranging from 613.2°C to 533.0°C, in proportion to the evolution of fraction solid [13,72]. The total latent heat released is 397.5 kJ/kg.  Table 4.1 – Thermal physical properties of wheel and die materials used in the model [13,27,72].  45  Materials Density Thermal conductivity Specific heat Latent heat  ρ (kg/m3) T (°C) k (W/m/K) T (°C) Cp (J/kg/K) T (°C) L (kJ/kg) A356 2380 25 163 25 880 563.6>T≥533.0 36.2   100 165 100 921 567.2>T≥563.6 170.4   200 162 200 967 588.2>T≥567.2 48.7   300 155 300 1011 610.7>T≥588.2 91.2   380 153 380 1046 613.2>T≥610.7 51.0   400 153 400 1055     500 145 500 1098     567 134 567 1127     614 400* (65.8) 614 1190     700 400* (67.9) 700 1190           H13 7800 20 24.6 23 458.8  N/A   200 26.3 200 518.5     500 27.3 400 587.8     600 27.8 600 726.2     800 28.1 700 905.4   Note: The thermal conductivities of A356 at temperatures above 614°C (marked with an asterisk) have been augmented intentionally to account for the effect of convection in the liquid. The nominal values have been provided in parenthesis for comparison.  4.3.4 Initial Conditions To capture the cyclic behavior of the LPDC casting process, the temperature distribution in the die at the start of each cycle is taken as the temperature distribution from the end of the previous cycle (apart from the first cycle when the die is set uniformly to 500°C). In the full casting process model developed by Dr. Carl Reilly, the initial temperature of the wheel varies from the rim to the sprue to account for the heat loss from the wheel to the die as the liquid metal is filling the die cavity. However, in the test problem the initial temperature of wheel is set to 700°C uniformly for the sake of simplicity. 4.3.5 Boundary Conditions Two major boundary conditions of significant importance in the wheel casting process are the interfacial boundary conditions between the wheel and the die components, and  46  the heat transfer conditions in the cooling channels. The interfacial heat transfer between the wheel and die components is defined as temperature-dependent. This has been done to approximate the effects of changing contact resistance at the interface during solidification. The functions defining this behavior can be different for the various interfaces  - i.e. wheel/side die, wheel/top die, or side die/bottom die - to account for differences in the evolution of the gap at the interface. The temperature dependent interfacial heat transfer coefficients between the wheel and the different die sections are defined as follows in terms of the various temperature regimes [13]: 𝑅𝑒𝑔𝑖𝑚𝑒 𝐼:    T ≥  614 °C  ℎ = ℎ𝑚𝑎𝑥 Equation 4.3 𝑅𝑒𝑔𝑖𝑚𝑒 𝐼𝐼: 614 ≥  𝑇 ≥  567 °C  ℎ= ℎ𝑚𝑎𝑥 × [𝑇 − 567614 − 567× (1 − 𝑓𝑙𝑖𝑚𝑖𝑡) + 𝑓𝑙𝑖𝑚𝑖𝑡] Equation 4.4 𝑅𝑒𝑔𝑖𝑚𝑒 𝐼𝐼𝐼: T ≤  567 °C  ℎ = ℎ𝑚𝑎𝑥 × 𝑓𝑙𝑖𝑚𝑖𝑡 Equation 4.5 where ℎ𝑚𝑎𝑥 is the maximum heat transfer coefficient, which is applicable in Regime I, which occurs while the local wheel interface temperature is above the liquidus temperature (614 °C). Below the liquidus temperature, but above the eutectic temperature of 567 °C, in Regime II, the heat transfer coefficient is ramped linearly with temperature from hmax to 𝑓𝑙𝑖𝑚𝑖𝑡 × ℎ𝑚𝑎𝑥, which is the lower limit to the interfacial heat transfer. In Regime III the heat transfer coefficient is held constant at 𝑓𝑙𝑖𝑚𝑖𝑡 × ℎ𝑚𝑎𝑥,   For the test problem the value of ℎ𝑚𝑎𝑥 is set to 3000 W/m2/K and 𝑓𝑙𝑖𝑚𝑖𝑡 sets to 0.3 for all of the casting/die interfaces. As there are no die-die interfaces in the simplified test problem geometry, no boundary condition was defined.  47  The heat transfer coefficient in the water cooling channels has been assumed to be constant when on and has been described using a heat transfer coefficient. Based on industrial operational practice, when the water is switched off, there is a period of time where compressed air is used to purge water from the cooling channels. After the purge, the channel is left with stagnant air until the end of the cycle. During the purging procedure, the heat transfer coefficient in the cooling channel decrease linearly from the value when water is flowing to a minimum value representing stagnant air. In the test problem, the former is set to 6000 W/m2/K and the latter is set to 5 W/m2/K for all channels.  The simplified casting model is run repeatedly (cyclically) for each set of input conditions until cyclic steady state is reached, to reflect what is occurring in the industrial process. Numerically, cyclic steady state is defined to have been reached when the largest temperature difference at a location within the die between the start and the end of a cycle is less than a user specified maximum. For this work, cyclic steady state was assumed to occur when the largest temperature difference in the die was less than 5°C.  4.4 The Optimizer The open-source Python optimization package Scipy.optimize has been used in this study. The main code in this package, the SQP algorithm, which calculates the updates to the design variables, is written in Fortran. Python is used as a wrapper to interface with external analysis modules. The interconnection between the Fortran code and the Python wrapper is realized through a Python extension built from the Fortran code. The extension needs to be rebuilt if any change is made to the Fortran code. The Python  48  wrapper constitutes part of the control module and its function will be introduced in Section 4.6. In this section, the main part of the optimizer - i.e. the Fortran code, will be discussed.   4.4.1 The Basic Algorithm In the optimizer, the line search method is used to update the design variables. The line search method first determines a descent direction along which the objective function is reduced and then computes a step length that determines how far along the defined direction the update should move.  In the line search method, the design variable vector is updated as follows [11]: 𝑥𝑘+1 = 𝑥𝑘 + α𝑆 Equation 4.6 where 𝒙𝑘 is a vector containing the design variables at the kth iteration and 𝒙𝑘+1 is the new design variable vector at the (k+1)th iteration,  𝑆 is the search direction, and α is the step length. The basic optimization algorithm is: i) Obtain the search direction 𝑆. ii) Choose the step length α and set 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑆. iii) Repeat ii) until the user-defined maximum trial number has been reached, or until a certain condition is reached (to be discussed in Section 4.4.1.2). 𝑓(𝑥𝑘 + 𝛼𝑆)  ≤  𝑓(𝑥𝑘) + 𝑐𝛼∇𝑓𝑘𝑆 Equation 4.7 iv) Check convergence; stop and output the optimum solution if converged.  49  v) Set 𝑘 = 𝑘 + 1; go to i). 4.4.1.1 Update of Search Direction 𝑺 The search direction is determined by solving the quadratic sub-problem with the quasi-Newton method, as discussed in Section 4.2. The quasi-Newton method has been reported to have better convergence behavior than the Steepest Descent method, which uses only the negative of the gradient of the objective function as the search direction, and it is more efficient than Newton’s method as it only requires information on first-order derivatives. Detailed discussion on these three methods can be found in [11].  The quasi-Newton method requires the information related to how sensitive the objective and the constraint functions are to a change, or perturbation, in the design variables. Based on this information, the optimization algorithm modifies the design variables to achieve a minimum in the objective function. There are three techniques described in the literature for evaluating the sensitivity coefficients: finite difference, direct differentiation [46-48] and adjoint methods [42,44]. The first technique assesses the sensitivities numerically, while the latter two techniques determine the sensitivities analytically. The finite difference method has been used in this study, as no analytical relationship exists to describe the variation in temperature with time in the wheel casting process. The finite difference procedure is straightforward to couple with the casting process model and the optimizer and the formulation is shown below in Equation 4.8:  ∇𝑖𝐹(𝑥) =𝐹(𝑥 + ∆𝑥) − 𝐹(𝑥)∆𝑥 Equation 4.8  50  where ∇𝑖𝐹 is the ith component of the gradient ∇𝐹, which is the sensitivity to the ith design variable, 𝐹(𝑥) in this case is the objective or constraint function,  x is the design variable vector, and ∆𝑥 is the design perturbation. After the sensitivity information is computed, it is passed back to the QP sub-problem as shown in Equation 4.2 to update the search direction. 4.4.1.2 Update of the Step Length 𝜶 The search direction 𝑆  is one of the vectors needed to generate the next trial solution 𝒙𝑘+1, the other is a step length 𝛼. Separate from the procedure to update the search direction, the update of the step length only requires the values of the objective and the constraint functions. The step length can be determined either exactly, meaning finding the minima of the objective function in a given direction 𝑆, or inexactly, meaning finding an α that leads to an acceptable descent amount. The latter approach has been implemented in the optimization algorithm used in this study. In the course of determining 𝛼, a penalty function is constructed as a way of measuring progress towards a solution, in which a reduction of this function implies an acceptable step α𝑆 has been taken [66]. In the optimizer, the 𝑙1 penalty function is used and it is based on the sum of the constraint violations as a penalty to the objective function. The penalty function is presented in Equation 4.9.  𝛷 =  𝑓(𝑥)   + 𝜇 ∑ 𝑚𝑎𝑥(−ℎ𝑖(𝑥), 0) Equation 4.9  51  where 𝛷 is the penalty function, 𝑓(𝑥) is the objective function, ℎ𝑖(𝑥) are the inequality constraint functions, where negative values indicate constraint violations, 𝜇  is a parameter that weights the drive to decrease 𝑓 with the need to satisfy the constraints. As discussed before, the search for the step length 𝛼 is iterative. Every time a new 𝛼 is computed, the casting process model will be run and the objective and the constraint functions will be assessed. The information will be used in the 𝑙1 penalty function to check whether it satisfies the Armijo condition [3].  The Armijo condition, which ensures the calculated α give sufficient decrease in the penalty function, is shown in Equation 4.7.  If Armijo condition is not met, the process will stop when the user-defined maximum trial number has been reached. The default maximum number of trials to determine 𝛼  in the optimizer is 11. 4.4.1.3 Convergence Criterion After a proper value for the step length has been selected or the maximum trial number has been exceeded, convergence will be checked to assess whether an optimum solution is reached. Convergence is deemed achieved based on the following conditions: i) The change in the objective function between two consecutive iterations is less than a tolerance, or the Euclidean norm of all the changes in the design variables is less than a tolerance, and ii) The sum of all constraint violations is less than a tolerance. The tolerance for the convergence criterion is 10−6  for all three terms within Scipy.optimize. If this tolerance is not met, the optimization process will move to the next  52  iteration and compute a new search direction. The process iterates until the iteration number reaches the predefined maximum, which is set to 10 in default. However, later in Section 4.7.1 it will be shown that the criterion for assessing convergence can be a challenge and must be determined in the context of the problem at hand.  4.5 Analysis Modules After the casting process model reaches steady-state, the control module calls a series of analysis modules to extract and post-process the modeling results. These modules include: i) A module that extracts the temperature histories at predefined discrete points in the casting; ii) A module that calculates the average cooling rate during the solidification interval from the extracted temperature data, and then calculates the objective function based on this data; and iii) A module that calculates the time taken to reach the set constraint temperatures using interpolation and then calculates the constraint functions. And when function sensitivities are needed: iv) A module that computes the sensitivities of the objective and constraint functions to the cooling parameters.    53  4.6 Control Module The control module consists of an in-house Python code and a Python wrapper that comes with the Python package. The in-house Python code controls the execution of the casting process model and the analysis modules as follows: i) It sets up the form of the objective and the constraint functions; ii) It starts the optimization run, calculating the initial values of objective and constraint functions; iii) It controls the cyclic running of the casting process model, which includes setting the initial conditions (temperature field) at the beginning of a cycle to be the temperature distribution at the end of the previous cycle and determining whether the steady-state criterion has been met achieved; iv) After cyclic steady-state has been achieved, it calls the analysis modules to extract the model results; v) It calls the analysis modules to calculate the values of the objective and constraint functions; vi) When sensitivity information for the objective and constraint functions is requested, it sets up and executes parallel computational runs for the perturbation processes that are needed to obtain the sensitivity matrix; and vii)  Updates the casting process model when new design variables have been calculated by the optimizer. The Python wrapper communicates with the optimizer - i.e. receives requests from the optimizer and passes the required data back into the optimizer.  54  A flow chart of the overall optimization methodology is shown in Figure 4.8. To start the optimization process, the initial values of the design variables, and the objective and the constraint functions need to be provided. With the initial parameters, the casting process model is run to steady-state, and then the objective function, constraints, and the sensitivities of these functions to design variables are evaluated. Subsequent to the first pass through the optimizer, whether the sensitivities of these functions will be assessed or not depends on whether a search direction update is required. The optimization process iterates until an optimal condition has been reached, or until a user-specified maximum number of iterations has been met.  Figure 4.8 – The flow chart of the optimization methodology.  55  4.7 Implementation of the Optimization Methodology for LPDC Wheel Casting The focus of this study is to develop a methodology to optimize the cooling conditions in the LPDC process for automotive wheels. The considerations that guide this optimization include maximizing the cooling rates or matching predefined cooling rate targets, while at the same time achieving directional solidification. The former is desired to both increase productivity (reduce the overall cycle time) and produce a finer microstructure with better overall fatigue performance. The latter is needed to eliminate/reduce shrinkage porosity, which, if present, would negate the positive benefits of a reduced cycle time on fatigue performance. The start time and duration (or on and off timing) during each cycle for each cooling channel are selected as the design variables in this optimization because they directly affect the cooling conditions in the die and are easy to control/change. In total there are twelve design variables – i.e. 1 start time and 1 end time for each of the top die, bottom die and side die (pair) cooling channels, as the two side die channels are connected in series. To achieve the optimum conditions, the objective and constraint functions must be carefully designed. The objective and constraint functions are based on the solidification conditions in the wheel. It is neither necessary nor practical to examine the solidification conditions at all points in the wheel. A small number of carefully selected points in the wheel can provide sufficient information to characterize the solidification path. As a baseline, a series of points, called evaluation points, shown in Figure 4.9 as white dots,  56  were selected to assess solidification conditions. The points are located along the centerline of the wheel and their position generally follows the wheel profile.  Two objective functions have been formulated for evaluation: 1) the quadratic form and; 2) the linear form. In the quadratic form, the objective function is based on the sum of the square of differences between the predicted cooling rate and a target cooling at selected points. The selected points are labeled P1 through P5 in Figure 4.9. The formulation of the quadratic objective function is presented in Equation 4.10. In the linear form, shown in Equation 4.11, the objective function has been formulated based on the cooling rates at the selected evaluation points, P1 through P5. With the quadratic form, the optimization algorithm is run to minimize the absolute value of the differences between the calculated and the target cooling rates. In this manner, the optimum solution is one in which the cooling timing yields calculated cooling rates that approach the target ones to within a tolerance. This approach is suitable for cases where the target cooling rate at specific locations is known or when a target dendrite arm spacing, which is linked to cooling rate, is desired. If the linear form is used, the optimization algorithm seeks to achieve the maximum cooling rate. In both cases, the constraints related to directional solidification are being applied.   57   Figure 4.9 – The locations used for evaluation of the objective (P1-5) and the constraint (white dots) functions in the test problem. 𝑓(𝑥) = ∑(𝑎𝑖 ×  (?̇?𝑖𝑡𝑎𝑟𝑔𝑒𝑡 − ?̇?𝑖𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑))2n𝑖=1 Equation 4.10 𝑓(𝑥) = ∑ 𝑎𝑖 × (− ?̇?𝑖𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑) 𝑛𝑖=1 Equation 4.11 where f(x) is the objective function, x is a vector containing the design variables (the on and off timings of the cooling channels), ?̇?𝑖𝑡𝑎𝑟𝑔𝑒𝑡is the target cooling rate at the ith point, ?̇?𝑖𝑐𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒𝑑 is the calculated average cooling rate within the solidification range at the ith point, 𝑛 is the total number of evaluation points used to calculate the objective function, and ai are constants used to adjust the relative weights of the objective function. In the quadratic form, ai is set to 10.0 to correct for the effect of squaring (the magnitude of the difference between the target cooling rates and the calculated cooling rates is ~10-2 and after squaring this will decrease to 10-4). In the linear form a value of 1.0 is used for ai.   58  The constraint functions should be formulated in such a way that requires regions further away from the supply of liquid metal (the top of the sprue) to solidify, or reach a critical temperature, before regions that are closer to the liquid metal supply. This imposes a directional solidification constraint that, if satisfied, will eliminate shrinkage-based porosity. The critical temperature for evaluating directional solidification should be based on the solid fraction above which mass feeding becomes difficult. In his book, Campbell [6] asserts that mass feeding can be blocked when solid fractions reaches anywhere between 20 and 50 percent, depending on the pressure differential driving the flow, and depending on state or morphology of the dendrite structure. Based on a series of tilt pouring experiments, Chiesa et al. [63] (as discussed in Chapter 2 – Literature Review) suggested a solid fraction of 0.35 to assess the end of mass feeding and ultimately to predict the occurrence of encapsulation. Considering the increased pressures of the low-pressure die-cast process compared to the tilt pour casting process, a solid fraction of 40% has been used in this study. Correspondingly, the time to reach the critical temperature of 575°C was selected to assess the constraints functions during the optimization process. The constraint functions, shown in Equation 4.12, were formulated based on the difference in time to reach 575°C between two adjacent points, j and j-1, and they are evaluated at the points shown as white dots in Figure 4.9. These constraints were required to be greater than or equal to zero. The location corresponding to j=1 is at the top of the inboard rim flange and j=29 for example, is the point close to the top of the sprue (center of the wheel). These points are placed in the center of a given section of the the 2-D axisymmetric wheel and follow the profile of the wheel. Additional points have been carefully added in areas known to be problematic, in order to accurately reflect the  59  solidification sequence at these locations. Determining the number of points required to evaluate the constraints was not straightforward, and will be discussed further in a following section.   A second type of constraint has been developed to ensure that the wheel completely solidifies during the casting cycle. This constraint, Equation 4.13, is evaluated at the last evaluation point (i.e. j=29) before the die opens. The difference between the time when the die opens and the solidification time was normalized with the die open time because the difference without normalizing is several orders of magnitude larger than the other constraint functions. Numerical difficulties arise when one function is of a substantially different magnitude or changes more rapidly than the other and therefore dominates the optimization process [11]. ℎ𝑗1(𝑥) = 𝑡𝑗575℃ − 𝑡𝑗−1575℃  ≥ 0 ,   𝑗 = 2,3, … 29 Equation 4.12 ℎ2(𝑥) = (𝑡𝐷𝑖𝑒𝑂𝑝𝑒𝑛 −  𝑡𝑆𝑜𝑙𝑖𝑑𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛)/𝑡𝐷𝑖𝑒𝑂𝑝𝑒𝑛  ≥ 0 Equation 4.13 where  ℎ𝑗1(𝑥) is the first set of constraints to ensure directional solidification, and ℎ2(𝑥) is the second type of constraint to ensure complete solidification in the wheel, 𝑡𝐷𝑖𝑒𝑂𝑝𝑒𝑛 is the cycle time when the die opens, and 𝑡𝑠𝑜𝑙𝑖𝑑𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 is the time taken for solidification to reach the middle of the hub. Within the framework developed for this study, the objective and constraint functions are evaluated based on the temperature outputs from the casting process model once the cyclic steady-state condition has been achieved. The cyclic steady-state condition has been used to ensure that the changes in die temperature associated with cooling parameter  60  changes have stabilized. The propagation of cooling parameter changes can take several casting cycles to occur owing to the relatively large thermal mass of the die.  4.7.1 Determination of Convergence Criterion As discussed in Section 4.4.1.3, the criterion for assessing convergence must be determined in the context of the problem at hand.  To conduct a preliminary assessment of whether the default values are suitable for the die casting problem, the optimizer was applied to a simplified version of test problem to facilitate running a large number of simulations. The geometry for the simplified test problem is shown in Figure 4.10 and contains only two active channels, TD_CC and SD_CC. The evaluation points are placed as shown. The linear form of objective function is employed and only the cooling rate of the lowest point is used to evaluate the objective function.  Figure 4.10 – A simple geometry used to assess the convergence criterion  61  Two starting points were input to see whether a unique solution was attainable. The initial cooling on and off times were set to 0 s and 200 s  for TD_CC and SD_CC, respectively,  for Case 1 and 40 s and 200 s for Case 2. The results, following application of the optimizer, are shown in Table 4.2. As can been seen, the two cases resulted in different outcomes. An “optimized solution” was obtained for Case 1 after 1 iteration, and for Case 2, the maximum iteration number was exceeded prior to achieving convergence criterion. There were no violated constraints for both cases. Clearly, not a satisfactory result. Table 4.2 – Summary of cooling times for a simplified representation of the wheel rim section, using the default value (𝟏𝟎−𝟔) for convergence criteria. Times presented as of On – Off times relative to the cycle time. Case Cooling Channel Initial (s) Optimized (s) Optimization Status Case 1 TD_CC 0 – 200 0 – 200 Optimization completed after 1 iteration SD_CC 0 – 200 0 – 200 Case 2 TD_CC 40 – 200 0 – 234 Maximum iteration number exceeded SD_CC 40 – 200 0 – 222  In a second application of the optimizer, the convergence criteria were tightened to 10−8, and correspondingly the maximum iteration number has been increased from 10 to 20. The two cases were rerun and the outcome is shown in Figure 4.8.        62   As can be seen, tightening the convergence criteria allowed the optimization algorithm to achieve the same solution for both initial starting conditions. However, also as seen, the maximum number of iterations was exceeded in both Cases. Measures have been taken to deal with this issue are discussed in Section 4.8. 4.8 Preliminary Results and Exploration of Optimization Configuration The more complex test problem (but still simplified), has been used to assess whether the methodology that has been developed can successfully optimize the cooling parameters and to determine the influence of different objective and constraint functions on performance. The optimization performance has been evaluated in terms of the predicted optimum cooling times, the resulting solidification conditions, the cooling rates at selected points, and the evolution of the objective function. The preliminary results will be presented along with discussion of optimization settings. As discussed Section 4.7.1, the default convergence settings needed to be refined in order to get a more accurate result. One issue with using the refined convergence settings is that it becomes difficult to find a solution in which all of the criterion are strictly met within a Table 4.3 – Summary of cooling times for a simplified representation of the wheel rim section, using tightened value (𝟏𝟎−𝟖) for convergence criteria. Times presented as of On – Off times relative to the cycle time. Case Cooling Channel Initial (s) Optimized (s) Optimization Status Case 3 TD_CC 0 – 200 0 – 240 Maximum iteration number exceeded SD_CC 0 – 200 0 – 240 Case 4 TD_CC 40 – 200 0 – 240 Maximum iteration number exceeded SD_CC 40 – 200 0 – 240  63  reasonable number of iterations. Therefore, it has been necessary to introduce the concept of “viable” and “nonviable” solutions. A viable solutions is defined as being a optimization result where all of the directional solidification constraints and the solidification requirement are satisfied. A nonviable solution, in contrast, is a result where one or more of the constraints or solidification requirement are/is violated. Further discussion on this is presented later in Section 4.8.1 in the context of the more complex test problem. 4.8.1 Objective Functions  As discussed in Section 4.7, two possible objective functions were originally proposed e.g. linear and quadratic. The linear form encourages the cooling rate to be maximized at a given location or locations, while the quadratic form can be used to achieve specified or target cooling rate at a location or locations. To explore both approaches the initial cooling parameters employed with the test process model were selected so as to induce liquid encapsulation and to showcase the ability of optimization methodology to improve the solidification conditions. The initial timings for each cooling channel were to start cooling at 100 s into the casting cycle and finish at 110 s.  The objective functions was evaluated based on the selected points P1 through P5, previously defined – see Figure 4.9.  The optimized cooling timings in comparison to the initial are shown in Figure 4.11, for the linear objective function implementation, and in Figure 4.12, for the quadratic objective function implementation. The data is presented Figure 4.11 and Figure 4.12 using bar graphs showing the on and off times for each cooling channel before and after optimization. The x-axis identifies the cooling channels and the y-axis is the timing. The  64  lower limit of each bar represents the cooling start-time and the higher limit  the cooling end-time. The bars with the blue diagonal shading are the initial cooling timings input to optimization routine and the bars with the grey horizontal shading are the cooling times obtained following optimization. The position on the x-axis in which the channels values are presented is based on distance from the centerline of the wheel, with distance increasing from left-to-right. For example, Bottom Die Cooling Channel 1 is closest to the centerline of the wheel (referring to Figure 4.9), thus it is first, followed by Top Die Cooling Channel 1. Top Die Cooling Channel 3 is located at the top right of the top die and is the farthest from the centerline of the wheel, thus it is presented the far right. The intent of choosing this order is that it is in accord with the desired solidification direction from the top of the rim (near channel TD_CC_3) all the way to end up in the hub (near channel TD_CC_1 & BD_CC_1).  Comparing the optimized to the initial cooling times shows that for most of the channels, the cooling duration has been increased. The increased cooling durations lead to faster solidification times. The exceptions to this observation are the inner most top and bottom die cooling channels (TD_CC_1 and BD_CC_1), which experience reduced cooling durations following optimization. These two channels are located around the center of the pseudo wheel (near the end of the pseudo spoke). Meanwhile the cooling in the channels around the rim-spoke junction (start of the pseudo spoke), namely TD_CC_2, SD_CC_1 and BD_CC_2, started earlier than the other channels and stay on for longer durations. Consequently, the rim-spoke junction experiences enhanced cooling conditions while the center of the wheel experiences reduced cooling. The cooling times obtained through the linear and the quadratic objective settings are close to each other. The largest difference is  65  seen in the side die channels SD_CC_1 & SD_CC_2, in which the cooling of the quadratic case starts 8 seconds earlier and ends 13 seconds earlier. Comparing to the initial cooling times however, both cases start cooling earlier and cool for a longer period of time.   Figure 4.11 – Cooling times for the test problem before and after optimization with a linear objective function.  66   Figure 4.12 – Cooling times for the test problem before and after optimization with a quadratic objective function. The corresponding solidification images are shown in Figure 4.13. The results are presented as temperature contour plots at different times during the casting cycle. The 575 °C isotherm, which represents approximately 40% solid fraction, has been highlighted in these plots to aid the visualization of liquid encapsulation. Consistent with this visualization, 575 °C has been used in the directional solidification constraints.  Moving from left to right in each series, the results have been plotted for cycle times corresponding to when solidification occurs at the top of the rim, the rim-spoke junction, and the middle and end of the pseudo spoke. The first row shows that results for the initial cooling times input to the optimizer, the second row for the linear objective function and the third row for the quadratic objective.  Employing the initial cooling conditions, the solidification sequence is fine in the pseudo rim area, but a large volume of liquid encapsulation is observed in the middle of the  67  spoke, indicated by the enclosure of the 575 °C isotherm at 56 s. In addition, the casting has not fully solidified by the time the die opens at 170 s (not shown). After optimization, the cooling intensity is increased at the end of the spoke and the rim-spoke junction, as discussed above, which leads to directional solidification in the spoke eliminating encapsulation. Optimization also leads to acceleration in the solidification rate. For example, with the linear objective function setting, the isotherm reaches the rim-spoke junction at 25 s compared to 45 s with the initial timing, and it reaches the mid-spoke at 36 s compared to 53 s. Generally, the average temperature of the dies is lower in the optimized condition than in the initial condition. Comparing the total solidification time for the two objective function formulations reveals that the linear objective function resulted in a faster solidification by approximately 13%. The cooling rates achieved with the linear objective function are listed in Figure 4.14, along with the cooling rates achieved with the initial cooling timings. The comparison reveals that the cooling rates have been increased following optimization at each location. The increased cooling rates range from 0.11 °C/s near the center of the wheel (P5) to 0.61°C/s in the rim (P2).  68   Figure 4.13 – Comparison of the casting and die temperature for the cooling conditions corresponding to the initial (1st row), optimized with the linear objective function (2nd row), and optimized with the quadratic objective function (3rd row).    69  Table 4.4– Comparison of the initial and the calculated cooling rates at locations (P1 – P5) with linear objective function. Location Initial (°C/s) Optimized           (°C/s) Difference (°C/s) P1 1.73 2.33 +0.6 P2 0.72 1.33 +0.61 P3 0.61 0.97 +0.36 P4 0.44 0.70 +0.26 P5 0.47 0.58 +0.11  The cooling rates achieved with the quadratic formulation are listed Table 4.5. The target cooling rates are given for comparison. In the example, the cooling rate targets were defined for Points P1 – P5 (locations shown in Figure 4.9). The target cooling rates were selected based on values obtained from execution of the process model in order that there they would have a high probability of being physically achievable. Nonetheless, the initial target  cooling rates of 2.6 °C/s at P1 and 0.6 °C/s at P5 were found to be too aggressive for the test problem - i.e. as the optimization drove the solidification rates up, the directional solidification constraints could not be achieved. After additional trial-and-error testing,  more moderate cooling rates ranging from 2.2 °C/s at P1 to 0.5 °C/s were selected as the targets. Compared to the cooling rates achieved with the linear objective function, shown in Figure 4.4, the quadratic objective setting has lower cooling rates. Table 4.5– Comparison of the target and the calculated cooling rates at locations (P1 – P5) with quadratic objective function. Location Target (°C/s) Optimized           (°C/s) |Difference| (°C/s) P1 2.20 2.25 0.05 P2 1.20 1.15 0.05 P3 0.90 0.85 0.05 P4 0.60 0.62 0.02 P5 0.50 0.51 0.01   70  To understand how the cooling conditions in the die change during the optimization, the evolution of the cooling rates at Points P1 – P5 (locations shown in Figure 4.9) and the objective function have been plotted for each iteration (linear: Figure 4.14 and Figure 4.16, respectively; quadratic: Figure 4.15 and Figure 4.17, respectively). Overall for both cases the cooling rates increase, as shown in Figure 4.14 and Figure 4.15, with P1 (top of the rim) experiencing the largest increase and P5 (spoke near the center of the pseudo wheel) experiencing the least increase.  In the plots of objective function evolution (Figure 4.16 and Figure 4.17), the solutions have been divided into viable solution, shown in solid yellow line, and nonviable solution shown in dashed red line, and a viable solution have been achieved after 4 iterations (as a reminder, a viable solutions is defined as being a optimization result where all of the directional solidification constraints and the solidification requirement are satisfied. A nonviable solution, is a result where one or more of the constraints or solidification requirement are/is violated. Notice that in Figure 4.16 for the linear function setting, the value of objective function keeps decreasing despite the fact that one or more constraints are violated. Recall from Section 4.2, that the Lagrangian function, which is comprised of the objective function and the constraints, is used to calculate the search direction. A close examination to the violated constraint function shows that the magnitude of the violation is very small (varies from 10-2 to 10-5), while the decrease in the objective function from iteration to iteration is on the order of 10-1. Thus, with the small magnitude of the constraint violation, the optimization algorithm continues to reduce the objective function.  71  For the quadratic objective function setting, the cooling rates stabilize near the target cooling rates after the 4th iteration. Similar to the increase and stabilization of the cooling rates, the objective function, shown in Figure 4.17, decreases and stabilizes near zero by the 4th iteration. The objective function is based on the square of the difference between the calculated cooling rates and the target cooling rates, and it is a measure of how far the calculated cooling rates deviate from the target ones. The stabilization of the objective function near zero indicates that the targeted cooling rates have been achieved.  Another interesting observation is that the evolution of the objective function appears to be a reflection of the cooling rate at P1. The cooling rate at this location is the largest term incorporated into the objective function. This suggests that the location with the highest cooling rate predominately influences how the objective function behaves.  Figure 4.14 – Evolutions of the cooling rates corresponding to the condition optimized with linear objective function  72   Figure 4.15 – Evolutions of the cooling rates corresponding to the condition optimized with quadratic objective function  Figure 4.16 – Evolution of the objective function for optimization with linear objective function.  73   Figure 4.17 – Evolution of the objective function for optimization with quadratic objective function. The quadratic objective function gives a more stable performance for the example shown. However, it should be noted that it requires significant effort to determine a set of reasonable target cooling rates, which may be impossible to achieve with more complex systems such the wheel casting process.  4.8.2 Constraint Functions The effect of using different numbers of constraint functions on the optimization result has been explored with the linear form of the objective function employed. Figure 4.18 shows the 3 cases used to assess the effect of constraint points. The 1st image in Figure 4.18 shows the base case configuration with 29 evaluation points, which has about 4 mesh elements (~ 10 mm) between the neighboring evaluation points. This configuration was used to study the effect of the objective function formulation and the results are presented in the previous section (please refer to Figure 4.11, Figure 4.13, Figure 4.14, Figure 4.16, Table 4.4 and Table 4.5). The 2nd and 3rd configurations have 16 (8 elements,  74  ~ 23 mm apart) and 54 (2 elements, 6 mm apart) constraint evaluation points, respectively. Uniform on/off times are again used as the initial cooling times.  Figure 4.18 – Illustration of the locations of the constraint evaluation points, with the configurations of 29 points (Base case), 16 points (Reduced), and 54 points (Increased). The cooling times achieved with 16 and 54 evaluation points are compared to the cooling times achieved with the 29 evaluation points, i.e. the base case in Figure 4.19 and Figure 4.20 respectively. No correlation has been found between cooling times and the number of constraint evaluation points. For example, the cases with 16 and 54 evaluation points have longer cooling durations in the TD_CC_2 channel compared to the base case with 29 points; however, for the two side die channels (SD_CC_1 & SD_CC_2), it is the case with 16 points that has the longest cooling duration, then the case with 29 points and finally the case with 54 points.    75   Figure 4.19 – Cooling times for the test problem following optimization with 16 constraint evaluation points.  Figure 4.20 – Cooling times for the test problem following optimization 54 constraint evaluation points.  76  The solidification images, shown in Figure 4.21, reveal some differences between the configurations. A directional solidification sequence occurs in the base case with 29 evaluation points (the third row). No liquid encapsulation was observed anywhere in the wheel. Despite satisfying the constraint conditions, the case with 16 evaluation points (1st row) exhibits liquid encapsulation at the top of the rim. A close examination of the location of the encapsulation shows that the isotherm narrows between the 2nd and the 3rd evaluation points. Since this occurs between evaluation points, the constraints fail to capture the encapsulation event. Adding extra point/points between the two should avoid this condition. However, the case with 54 evaluation points (3rd row), with much smaller distances between evaluation points, also seems prone to encapsulation; in this case, in the spoke near the rim-spoke junction. These results suggest that using 16 evaluations points is insufficient to accurately track the solidification conditions and it is not possible to achieve satisfactory solidification conditions using 54 evaluations points.  The cooling rates achieved with the three constraint configurations are listed in Table 4.6. After optimization, the cooling rates have increased appreciably, regardless of the constraint settings. The case with 16 evaluation points has higher cooling rates than the base case with 29 points, but as discussed, this configuration is under constrained. The case with 54 evaluation points results in cooling rates that are similar to the base case. A similar difference is also shown in the objective function evolution in Figure 4.22 where the objective function reaches a lower value in the case with 16 points compared to the base case with 29 points and the 54 point case.  77   Figure 4.21 – Comparison of the optimized results from the cases with 16 evaluation points (2nd row), 29 evaluation points (3rd row), and 54 evaluation points (4th row).    78  Table 4.6– Comparison of the cooling rates at locations (P1 – P5) among cases with 16, 29, and 54 evaluation points. Location Initial (°C/s) 16 points (°C/s) 29 points         (°C/s) 54 points (°C/s) P1 1.73 2.46 2.33 2.30 P2 0.72 1.63 1.33 1.28 P3 0.61 1.19 0.97 0.90 P4 0.44 0.78 0.70 0.73 P5 0.47 0.63 0.58 0.62    Figure 4.22 - Evolution of the objective function for constraint function configurations with 16 (left) and 54 (right) evaluation points. The two rounds of analysis with a simplified test problem suggest that the optimization technique developed performs adequately with either the linear or the quadratic objective functions and 29 constraint evaluation points. However, the quadratic objective function suffers from the fact that a set of target cooling rates must be preselected. Selecting a poor set of targets can lead to either constraints being impossible to satisfy or the maximum cooling rates not being achievable. For wheel casting, specifying target cooling rates, meaning achieving cooling rates that are not any more or less than a target, seems impractical. Therefore, the linear objective function was chosen for use in subsequent studies.  79  4.9 Computational Intensity The computational intensity of the numerical optimization approach employed in this work results from three numerical operations: 1. Identifying the search direction: Each optimization was run for 20 iterations, which means that the search direction, 𝑆, was updated 20 times. During each search direction update, the solution gradient or the sensitivities of the objective and the constraint functions with respect to each design variable must be assessed. There are 12 design variables (on and off times for 6 cooling channels). To determine the sensitivity, each parameter is individually perturbed and the changes in solidification conditions are evaluated. 2. Updating the step length: Given a search direction, there are one or more updates to the step length, α, depending on whether the Armijo condition [3] is reached. Recalling Equation 4.6, each α update alters the design variables and the casting process model is rerun with the new set of parameters to assess the result. The maximum number of update evaluations was set to 11 based on the default suggested in the documentation of the Python optimization package.  3. Achieving cyclic steady-state: Every time the cooling parameter changes, the casting process model must be re-run to steady-state. Depending on the extent of the cooling parameter changes, it normally takes from 1 to 8 cycles to reach cyclic steady-state. There are three sources that alter the cooling parameters, namely the change of the search direction, the change of the step length, and during the perturbation process in sensitivity computation (please refer to Section 4.4.1.1).  80  For the casting optimization examples shown in this work, the number of times the casting process model is run with a new cooling conditions is on the order of 260 ~ 460 for the 20 optimization iterations. Considering that multiple cycles are needed to reach cyclic steady-state, there are over a thousand casting cycles simulated over the course of an optimization run, which is computationally intensive. The shared-memory multi-processor computational features of Abaqus have been enabled in order to accelerate individual casting cycle runs and multiple casting cycle runs are started during the search direction update to reduce the overall computational time required. With the current formulation, it takes approximately 3 days to finish 20 iterations, running on 4 Intel® Xeon® 2.40 GHz processors. This first application of the optimization methodology using a test casting problem has shown that numerical optimization can be used to improve solidification conditions resulting in increased cooling rates and the desired solidification sequence. Follow-on studies have been focused on applying the developed methodology to wheel casting scenarios.    81  5 Optimization of a 2D Axisymmetric Prototype Die After assessing the performance of the optimization methodology and tuning the optimization settings with a test problem, the optimization methodology has been applied to a prototype wheel. As mentioned in the Preface, the development of the wheel casting process model used in this work and the associated experimental validation were conducted by Dr. Carl Reilly, a Research Associate at the University of British Columbia. The modeling techniques have been applied to predict the temperature evolution in a variety of production dies and these predictions have been validated through extensive plant trials. These techniques have been briefly reviewed in Chapter 4 with the test problem, and only the critical features of the wheel model will be presented in this chapter. 5.1 Casting Process Model of a Prototype Wheel The casting process model used in this thesis is a cross-section extracted from a prototype wheel and die geometry that is currently undergoing testing in the plant. This approach was utilized to reduce the computational load and thus development time. This simplification (3-D to 2-D axisymmetric) means that features such as the windows and bolt-holes cannot be represented in the model; thus the spoke geometry becomes a solid disk. However, for the sake of simplicity, it will still be called the spoke throughout this thesis.  82  5.1.1 Model Geometry The 2-D axisymmetric geometry is shown in Figure 5.1. The die structure being used is proprietary and hence some details cannot be disclosed. Approximate locations of the water-cooling channels and insulation are shown but the geometry and size of the channels shown does not represent the reality. There are in total three die sections included in the model: one top die including a top die drum core, one side die and one bottom die. Eight cooling channels have been utilized in the prototype die design: three in the top die (TD_CC_1 - 3), two in the side die (SD_CC_1 - 2), two in the bottom die (BC_CC_1 - 2) and one in the central pin. The central pin cooling channel is connected to TD_CC_1 in series. Preliminary plant trials with the cooling channel configuration described above resulted in the over-cooling of the side dies and consequently led to mis-runs (i.e. the wheel solidified prior to the completion of die filling). Therefore, during the application phase of this study, a configuration where the cooling in the upper side die cooling channel is eliminated has been assessed. Insulation is typically added at locations along the rim in the top and side dies and near the thinnest section of the spoke in the bottom die, to adjust the cooling intensity and encourage directional solidification in the wheel adjacent to these locations.  83   Figure 5.1 – 2-D axisymmetric wheel geometry approximating a prototype wheel geometry and approximate locations of the seven cooling channels embedded in the die. 5.1.2 Mesh The geometry was meshed with quad elements using ICEM, an ANSYS mesh generation software. It was then converted to an Abaqus readable format. A finer mesh was used in the wheel compared to the die sections with average element lengths equal to ~2 mm and ~4 mm, respectively. The mesh includes a total of 13,170 nodes and 11,893 elements.  5.1.3 Material Properties The properties of aluminum alloy A356 and H13 tool steel has been tabulated in Table 4.1. As the structure of the prototype die is much more complex than that in the test problem, additional die components and materials exist and their properties are updated  84  in Table 5.1. These additional components include one sprue and multiple insulation blocks. The sprue, which directs liquid metal from the holding furnace into the die cavity, is made from a combination of cast iron and a small amount of tungsten carbide. The insulation material placed at select locations is a silica-based material. The thermophysical properties of these materials are also temperature-dependent. Table 5.1 – Thermal physical properties of additional die materials used in the model [13,72,73] Materials Density Thermal conductivity Specific heat Latent heat  ρ (kg/m3) T (°C) k (W/m/K) T (°C) Cp (J/kg/K) T (°C) L (kJ/kg) Cast iron 7200 25 49 25 490  N/A   100 48 100 510     200 46 200 555     300 43 300 600     400 42 400 640     500 41 500 700     600 38 600 785     700 35 700 1000           Tungsten carbide 15600 20 84.2 25 166  N/A    100 183       200 196       300 205       400 211       500 217       600 221       700 224           Insulation 100 0 821 24 0.038  N/A   100 900 150 0.059     200 969 260 0.091     300 1027 800 0.300     400 1075       500 1111       600 1137              85  5.1.4 Initial Conditions The initial condition for the die components at the beginning of a cast cycle simulation is the temperature distribution carried over from the end of the previous cycle to capture the cyclic behavior of the LPDC casting process, as mentioned in Chapter 4 Section 4.3.4. The initial temperature in the wheel is assumed to vary from 620 °C at the top of the rim to 700 °C at the inlet from the sprue, to account for the heat loss from the A356 to the die during die filling. The values were estimated from thermocouple data collected during plant trials and were performed by Dr. Carl Reilly. 5.1.5 Boundary Conditions Unlike the test problem discussed in Chapter 4 Section 4.3.5, the interfacial boundary condition between the wheel and die is activated sequentially at the start of each casting cycle based on the expected filling sequence to approximate the effects of filling on heat transfer to the die. The values of ℎ𝑚𝑎𝑥 and 𝑓𝑙𝑖𝑚𝑖𝑡 also vary. The interfacial heat transfer conditions are summarized in Table 5.2. Table 5.2 – Heat transfer coefficient at the wheel/die interfaces  Interface ℎ𝑚𝑎𝑥 𝑓𝑙𝑖𝑚𝑖𝑡 (W/m2/K)  Top die/wheel 1000 0.3 Top die drum core/wheel 2000 0.3 Side die/wheel 1000 0.3 Bottom die/wheel 2500 0.3 Sprue/cast sprue 1000 0.3  The water cooling heat transfer coefficients have been updated according to Table 5.3. These heat transfer coefficient values were calculated using an Excel model, developed as  86  part of a companion M.A.Sc. project performed by Ms. Sara Moayedinia, that considers the effects of the channel dimensions, water flow rates and fluid properties [74]. The temperature of the cooling water is assumed to be constant during the casting process and is set to 25°C. Table 5.3 – Heat transfer coefficient at cooling channel walls when water is on Cooling channel Heat transfer coefficient (W/m2/K) TD_CC_1 & Central pin cooling 4720 TD_CC_2 5379 TD_CC_3 2071 SD_CC_1 2869 SD_CC_2 OFF BD_CC_1 6993 BD_CC_2 2426  5.2 Implementation of the Optimization Methodology The optimization approach, demonstrated in Chapter 4 with a test problem, has been applied to a prototype wheel casting process. Figure 5.2 shows the geometry and the locations for the evaluation points (white dots). Details of the optimization settings can be found in Chapter 4 Section 4.7 and are briefly reiterated here. The linear objective function is used to maximize the cooling rates. The objective function is based on the sum of the negatives of the cooling rates assessed at Points P1 – P6 as shown in Figure 5.2. There are a total of 36 evaluations points, placed ~ 10 mm apart from each other, that have been used to evaluate the constraint functions.      87   Figure 5.2 – Cross-section of the wheel geometry with approximate locations of the seven cooling channels and showing the locations used for evaluation of the objective (P1-6) and the constraint (white dots) functions. A solidification requirement, shown in Equation 4.13, is also evaluated at the middle of the hub (i.e. j=31) before the die opens. An experimentally validated 3-D version of the casting process model developed by a research associate at the University of British Columbia, Dr. Carl Reilly, has shown that when this requirement is met, the wheel structure can be maintained when die opens. 5.3 Results and Discussion This section discusses the optimization outcomes of the prototype wheel casting process. The first example presented has been performed with the model discussed above and will be referred to as the model with simple interface description, as compared to an updated casting process model, which has enhanced boundary conditions that will be described later.    88  5.3.1 Simple Interface Description To assess the performance of the optimization starting from different initial cooling conditions, and to get a sense of the effect of the initial condition on the optimization outcome, two starting conditions have been tested:  1) non-optimum cooling condition 2) near-optimum cooling condition The term ‘optimum’ here does not necessarily indicate the best result. In problems of practical interest, it is seldom possible to ensure that the absolute optimum solution will be found subject to multiple constraints [11]. More often than not, multiple solutions exist to the optimization problem, all leading to improved results. Therefore, in this study, an optimum solution refers to a viable solution that satisfies all constraints, which include the directional constraints and the solidification requirement. The non-optimum cooling conditions are educated guesses when information about the process is limited. The near-optimum cooling conditions are based on the cooling timing result obtained from trial-and-error runs conducted with the model, where desired solidification conditions have been approached.  5.3.1.1 Case 1 - Optimization starting from non-optimum cooling conditions  When developing the cooling timing for a new wheel model, rough estimates may be used as a starting point to begin refinement. In this case, the initial cooling timings for each cooling channel, shown in Figure 5.3, were sequenced to start with those channels furthest away from the metal source (TD_CC_3), followed by the channels near the rim-spoke junction (TD_CC_2, SD_CC_1, BD_CC_2), and finally the channels near the hub- 89  spoke junction (TD_CC_1, BD_CC_1). This sequence is intended to encourage directional solidification as explained previously. However, there is little expectation that directional solidification will occur as the specific numbers for cooling on and off times have been established based on a overly simplified rationale. The range within which the cooling timings may be changed is from 7.0 seconds after the start of the casting to its end. The lower bound of 7.0 s is based on a limitation of the casting equipment considered in this study – the PLC in the equipment requires 7 s to perform interlock checks at the start of the casting cycle and no control moves can be performed during this time.  The cooling times following optimization have also been summarized in Figure 5.3. The upper side die channel (SD_CC_2) is not active in cooling the die as discussed in Section 5.1.1, thus it is not shown in the figure. The initial cooling times are shown shades in blue and the optimized cooling times are shown in grey. For the initial cooling times, most of the channels have similar cooling durations, except for channel TD_CC_2, which is located in the lower rim. The cooling start times are sequenced from the channel located on the top of the rim (TD_CC_3) all the way to channel near the hub (TD_CC_1 & BD_CC_1).  After optimization, the cooling durations of most channels have been reduced. The cooling in the top die channel near the middle of the rim (TD_CC_2) and the one above the hub area (TD_CC_1) has been turned off. The duration of the cooling in the bottom channel of the side die (SD_CC_1) is only on for 9s, compared to 50s initially.  The net effect of these changes should be an increase in the local temperature of the top and side dies, thus reducing the heat transfer rate between these die sections and the wheel, and  90  consequently reducing the solidification rate. The durations of cooling in the top die channel above the rim (TD_CC_3) and the outside channel in the bottom die (BD_CC_2) have been increased, which should drive the solidification front down the rim and across the spoke.   Figure 5.3 – Cooling times for 2-D axisymmetric prototype wheel (simple interface description starting from non-optimum cooling conditions).  Focusing on the two channels that have been switched off following optimization, TD_CC_1 and TD_CC_2, the evolution in the cooling times for these two channels have been plotted out in Figure 5.4 and Figure 5.5. An examination of the record of violated constraints provides insight to the evolution in the cooling channel timing driven by the optimizer. The initial cooling duration for channel TD_CC_1, shown in Figure 5.4, was 50 s. However, this condition resulted in multiple constraints in the spoke being violated. The optimizer then reduces the duration from iteration 1 to 2 to remove the constraint violations in the spoke. At iteration 2, the spoke is free of constraint violations, however, due to the reduced cooling duration, the wheel is not fully solidified within the specified  91  time, resulting in violation of the solidification requirement constraint. To deal with this, the optimization algorithm then seeks to increase the cooling duration until another violated constraint appears at the rim-spoke junction at the 4th iteration. The algorithm then takes a large step to reduce the cooling duration at 5th iteration (91% reduction), where it remains switched off for the balance of operation of the optimizer.   Figure 5.4 – Cooling time evolutions over optimization iterations for Channel TD_CC_1, located above the hum-spoke junction (simple interface description starting from trial-and-error based initial conditions). Channel TD_CC_2 shown in Figure 5.5 switches off in the 2nd iteration, however both the cooling start and end times are moved earlier in the cycle in the next iteration, despite essentially having a zero difference between the two. A constraint has been placed in the code to require the cooling end time to be larger than or equal to the cooling start time,  92  thus as the algorithm moved the cooling end time to earlier in the cycle the start time followed.  Figure 5.5 – Cooling time evolution for Channel TD_CC_2, located in the lower rim (simple interface description starting from trial-and-error based initial conditions). Figure 5.6 shows a series of temperature contour plots at different times in the cycle obtained for the initial and optimized cooling parameters timings. The results are presented overlaid with the 575 °C (0.4 fs) isotherm to illustrate the solidification path. For the initial cooling parameters, large pockets of liquid (indicated by the bounding isothermal lines) are encapsulated at the top of the rim, the rim-spoke junction, and the hub-spoke junction. These locations are known to be prone to shrinkage porosity. The top rim is prone to shrinkage because the middle of the rim solidifies rapidly, as it is the thinnest section of the wheel, thus, cutting off the supply of liquid metal to the top of the rim. The rim-spoke and the hub-spoke junctions are prone to shrinkage porosity because  93  of their large thermal mass - i.e. these locations can easily become hot spots during solidification [13].  From the plot of the optimized results, the algorithm has successfully eliminated the liquid encapsulation and the potential for solidification shrinkage in all three of these areas. Note that due to the reduced cooling durations for the channels around the rim and the hub-spoke junction, the solidification time has been increased from 205 s to 282 s, representing a 38% increase. Although the increased cycle time represents a decrease in productivity, the optimized cooling conditions have led to the directional solidification conditions necessary to produce high quality wheels. The wheels produced with the initial cooling times would very likely be rejected due to shrinkage porosity.  The average cooling rate during solidification at different locations in the wheel has also been evaluated. The average cooling rates were calculated using Equation 4.11 and the values are summarized in Table 5.4. The cooling rates have been evaluated at the points used to assess the objective function within the wheel.  94   Figure 5.6 – The initial (1st row) and optimized (2nd row) solidification sequence for the water-cooled, prototype wheel casting process (simple interface description starting from non-optimum cooling conditions).  ?̇?𝑖𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =𝑇𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠 −  𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠𝑡𝑖𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠 − 𝑡𝑖𝑠𝑜𝑙𝑖𝑑𝑢𝑠,     𝑖 = 1,2, … 6 Equation 5.1 where ?̇?𝑖𝑎𝑣𝑒𝑟𝑎𝑔𝑒is the average cooling rate at location i, 𝑇𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠 is the liquidus temperature, 𝑇𝑠𝑜𝑙𝑖𝑑𝑢𝑠 is the solidus temperature, 𝑡𝑖𝑙𝑖𝑞𝑢𝑖𝑑𝑢𝑠 is the time for location i to reach the liquidus temperature, and 𝑡𝑖𝑠𝑜𝑙𝑖𝑑𝑢𝑠  is the time for location i to reach the solidus temperature. The cooling rates predicted with the optimized cooling parameters have decreased relative to the initial cooling parameter results, which corresponds with decreased  95  solidification rates. Locations P1 and P4 exhibit the lowest cooling rate reduction, as a result of the fact that they are close to the two channels that have increased cooling durations – i.e. TD_CC_3 and BD_CC_2, respectively.  It may be possible to modify the size and shape of the cooling channels in the die and the location and thickness of insulation near the spoke to enhance cooling rates while achieving directional solidification; however, investigating topological or die geometry improvements is beyond the scope of this study. Table 5.4– Summary of cooling rates achieved at locations P1 – P6 (simple interface description starting from non-optimum cooling conditions). Location Initial (°C/s) Optimized (°C/s) Change           (°C/s) P1 0.93 0.90 - 0.03  P2 0.76 0.65 - 0.1 P3 0.59 0.47 - 0.12 P4 0.45 0.36 -0.09 P5 0.46 0.34 - 0.12 P6 0.42 0.29 - 0.13  The evolution of the objective function as a function of optimization iteration is shown in Figure 5.7. The dotted portion of the line indicates iterations where the constraints were violated. The solid portion of the line represents a viable solution in which all of the constraints were satisfied. During the first portion of the optimization, the constraints related to directional solidification conditions near the top of the rim and the rim/spoke and spoke/hub junctions were violated (refer to Figure 5.6). The optimization algorithm attempted to sequence solidification in the wheel, and thereby eliminate the occurrence of liquid encapsulation, which led to an increase in the objective function. A solution was reached at the 10th iteration, where all constraints were satisfied.  The objective function is relatively stable after this point, with the maximum change less than 2% compared to  96  the value at the 10th iteration. Continuing to reduce the objective function, resulted in one or more constraints being violated. The optimized solidification images shown in Figure 5.6 are for the viable solution iteration with the lowest objective function value.   Figure 5.7 - Evolution of the objective function during optimization of the prototype wheel casting process (simple interface description starting from non-optimum cooling conditions). 5.3.1.2 Case 2 - Optimization starting from near-optimum cooling conditions The initial cooling conditions used for this case, shown in Figure 5.8, were based on operational experience with a similar wheel casting process and several rounds of trial-and-error based optimization with the process model. The cooling start times were sequenced to achieve directional solidification, as in the previous example. Cooling in the top die channels beside the rim (TD_CC_2) and above the hub area (TD_CC_1) is not activated, or is activated for very short duration. Under these conditions, solidification in the hub area of the wheel is encouraged by cooling from the nearby bottom die channel (BD_CC_1). Midway through the cycle, solidification is driven along the spoke by cooling that occurs in the outer bottom die channel (BD_CC_2).   97  The cooling times following optimization have been summarized in Figure 5.8. The red bars represent the trial-and-error based cooling times that are used as the starting point for the optimization. The yellow bars represent the cooling times after optimization. Despite starting from a trial and error based solution, significant changes have occurred during optimization. In the bottom die, the cooling duration in channel BD_CC_2 was increased and the cooling start time was delayed for channel BD_CC_1. The cooling duration in the lower side die channel (SD_CC_1) decreased from 43s to 9s and the cooling start time was postponed form 7s to 45s. The cooling parameters did not change for channel TD_CC_2, which remains switched-off. Channel TD_CC_1 however, changed from completely switched-off to a cooling duration of 27 s.   Figure 5.8 – Cooling times for 2-D axisymmetric prototype wheel (simple interface description starting from trial-and-error based initial conditions). The switching-off of channel TD_CC_2 in both this case and the previous one suggest that this channel should be excluded from the die design. As a note: although the  98  optimization is focused on optimization of the cooling parameters and not die topology, it would appear nonetheless that there is insight gained into the latter. The evolution of the cooling timing for channel TD_CC_2 have been plotted in Figure 5.9.  Cooling has been switched off since the 1st iteration. Again however, as in the previous case, the cooling start time has been moved earlier. After successive successful attempts to eliminate constraint violations and increase cooling rates, an attempt has been made to switch on the cooling again to further increase cooling rates. However, further increase of cooling duration of channel TD_CC_2 has led to violated constraints in the top rim region, thus in iteration 18 and 19, this channel has been switched off again.  Figure 5.9 – Cooling times evolutions over optimization iterations for Channel TD_CC_2, located in the lower rim (simple interface description starting from trial-and-error based initial conditions).  99  The solidification conditions corresponding to the trial-and-error and optimized cooling conditions are presented in Figure 5.10. The 1st row shows the results using the initial cooling times, which are based on trial-and-error analysis using the model. At 49 s, there is a region of liquid / semi-solid metal, indicated by the 40% fraction solid contour, which is encapsulated near the top of the rim, which could result in shrinkage porosity. This is a well-known problem area in the water-cooled version of the LPDC process (note: this is occurring in an area that will be accurately described using the 2-D axisymmetric model). However, there is no liquid encapsulation predicted in the rim-spoke junction (73s), which is also an area prone to solidification shrinkage [4,13].  The results obtained following numerical optimization are presented in the 2nd row of images in Figure 5.10. As can be seen, numerical optimization has eliminated the liquid encapsulation issue in the upper rim, while in other regions the solidification condition remains similar. The final solidification times are similar, with the optimized case being slightly smaller. The optimization reached a set of cooling conditions that satisfied all constraints in the 5th iteration. Cooling rates extracted from temperature histories predicted for both the initial trial-and-error based parameters and the numerically optimized cooling parameters are summarized in Table 5.5. There are small increases in all locations except location P3, which experiences a slight decrease in the cooling rate. P3 is located at the rim-spoke junction and the decrease in the cooling rate may be due to the reduced cooling duration in the nearby channel SD_CC_1.  100   Figure 5.10 – The initial (1st row) and optimized (2nd row) solidification sequence for the water-cooled, prototype wheel casting process (simple interface description starting from trial-and-error based initial cooling conditions). Table 5.5 – Summary of cooling rates achieved at locations P1 – P6 (simple interface description starting from trial-and-error based initial conditions). Location Initial (°C/s) Optimized (°C/s) Change           (°C/s) P1 0.81 0.87 + 0.06 P2 0.64 0.66 + 0.02 P3 0.54 0.51 - 0.03 P4 0.39 0.39 0.0 P5 0.36 0.37 + 0.01 P6 0.30 0.32 + 0.02  The evolution of the objective function during the numerical optimization is shown in Figure 5.11. Compared to the previous case, that started from non-optimum conditions,  101  there are smaller variations in the objective function in the current case. Another obvious advantage of running the optimization starting from values closer to an optimum is that it needs less time to reach a viable solution (i.e. 5th iteration compared to the 10th iteration in the previous case). The number of iterations with viable solutions is also larger. Finally, the time taken to complete solidification is also smaller in the case starting with near-optimum cooling times (262 s in Figure 5.10 compared to 282 s in Figure 5.6). These facts suggest that starting from near-optimum condition saves optimization time and is more likely to achieve a viable solution. However, when determining the total benefit, the time taken to obtain the near-optimum solutions should also be considered.   Figure 5.11 - Evolution of the objective function during optimization of the prototype wheel casting process (simple interface description starting from trial-and-error based initial conditions). It is important to note that although each set of cooling parameters result in different isotherm profiles, both sets of optimized cooling parameters achieve directional solidification with good solidification rates. As each set of cooling parameters have led to an overall improvement in solidification conditions, the parameters sets are accepted as  102  local optima. Additional investigation of this issue has shown that unique global optimums are achievable in simpler problems - e.g. simple rectilinear geometry subject to one cooling channel. However, with multiple channels and complicated wheel and die geometry, attempts to obtain a unique optimized solution were unsuccessful. This is a recognized issue in gradient-based optimization [11]. 5.3.2 Enhanced Wheel/Die Interface and Water Channel Boundary Conditions Based on the optimized cooling times, a plant trial was performed on a production low pressure die-casting machine to verify the performance of the casting process calculated with the optimization package (note: the water-cooling timing actually used in the industrial casting trial was further refined using a 3-D model of the LPDC casting process to address limitations in the 2-D axi-symmetric model). The prototype die assembly that was the basis for the 2-D axi-symmetric prototype wheel casting process model was used in the trials. The prototype die was heavily instrumented with thermocouples at key locations within the die, and where possible within the wheel, to measure the evolution of temperature during a series of casting cycles including, under steady-state cyclic conditions. Wheels cast under steady-state conditions were selected for post-casting X-ray imaging to check of porosity. The results of the X-ray imaging revealed the presence of distributed shrinkage porosity in the upper rim, see Figure 5.12. Given the fact that encapsulation was not predicted in the casting process model with the optimized cooling times, two possibilities exist: 1) the optimization algorithm does work; or 2) the process thermal model is not accurate enough  103  to allow for a practical solution to be obtained. Note: the assumption of axisymmetry should not negatively impact on the predictions of the model in the upper rim.  Figure 5.12 – X-ray image showing distributed shrinkage porosity in the top rim at the top of the wheel rim  Wheel/die interface behaviour In conjunction with this work, research efforts at UBC have revealed the importance and complexity of some of the process boundary conditions, including the heat transfer at the wheel/die interface. The wheel die interface exhibits complex spatial and temporal behaviour during a casting cycle with air gaps forming at some locations, contact pressure increasing in others and both gap formation followed by gap closure and pressure development in still others [75]. Following the approach reported by Wei [75], each of the wheel/die interfaces in the wheel casting process model has been divided into multiple sections, and the interfacial heat transfer descriptions have been based on whether the section is in good contact indicating limited or no gap formation, or in poor contact indicating appreciable gap formation. Based on a comparison with plant trial thermocouple data, the magnitudes of the maximum heat transfer coefficients have been  104  increased significantly, with the values between the wheel and the major die components changing from 2500 W/m2/K to 4000 - 7000 W/m2/K. Wei also determined that that gap formation occurs at temperatures lower than the liquidus temperature (614 °C) [75], therefore the temperature interval over which the interfacial heat transfer varies has been changed. The details of the revised wheel/die interfacial heat transfer coefficient formulations are summarized in Table 5.6.  Table 5.6 – Heat transfer coefficient at the wheel/die interfaces (updated). Interface Temperature Heat transfer coefficient  (°C) (W/m2/K)   Good contact Bad contact Top die/wheel T≥574 4000 4000  574≥T≥524 4000 56T – 28144   T<524 4000 1200     Top die drum core/wheel T≥574 7000 574≥T≥524 42T - 17108  T<524 4900     Side die/wheel T≥574 5000 5000  574≥T≥524 5000 80T – 40920   T<524 5000 1000     Bottom die/wheel T≥574 7000 7000  574≥T≥524 42T - 17108 70T – 33180   T<524 4900 3500     Sprue/cast sprue T≥574 1000  574≥T≥524 10T – 4740   T<524 500 Note: The gap formation along the top die drum core/wheel and the sprue/cast sprue were expected to be nearly uniform, thus these two interfaces were not divided into good or poor contact versions.  In addition, in an effort to include the effects of delayed heat transfer to the die resulting from filling, the wheel/die interfacial heat transfer coefficients were activated based on the estimated maximum height of the liquid metal during die filling. It is assumed that filling takes place sequentially, moving from the sprue, to the spoke and finally to the rim. As filling progresses, the air that originally takes up the die cavity exits through multiple  105  vents and the space is taken by liquid metal. However, plant trial thermocouple data clearly shows that the hub is the last place to fill in the die cavity. The thermocouple data collected from the hub (TC1), the mid rim (TC2) and the top rim (TC3) are plotted in Figure 5.13. The temperatures at these locations initially decrease gradually because of heat loss from the die to the environment, and then rise rapidly as filling progresses and the hot metal contacts the surfaces. Consequently, the time when the curve starts to ramp up (indicated by triangles in Figure 5.13) shows when the hot metal reaches the thermocouple location. Clearly the hub area where thermocouple TC1 is located is the last place to contact the hot metal.  The delay is likely due to inadequate venting in the hub area which gives rise to the development of a significant amount of backpressure that slows down the filling in this region. This phenomenon has been reported and analyzed previously using a 3-D die filling model developed by the casting research group in the University of British Columbia [71]. To reflect this behaviour in this study, the heat transfer between the wheel/die has been delayed in the hub area to account for this phenomenon.  106   Figure 5.13 – Thermocouple data measured at the hub (TC1), mid rim (TC2), and top rim (TC3), showing that the hub area is the last to fill. Cooling channels The heat transfer behavior in the cooling channels is also complex, exhibiting transient periods of boiling heat transfer when the water first enters the channel and again when the flow is terminated to the channel [74]. The heat transfer coefficients between the water and the cooling channels were constant in baseline version of the model. In the updated model, the heat transfer coefficients applied in the cooling channels are based on the instantaneous calculation that takes account of the cooling channel geometry, the local surface temperature as well as the process parameters - i.e. water flow rate. The calculated heat transfer coefficients vary from over 10,000 W/m2/K to ~200 W/m2/K over the course of a casting cycle. Details of the equations and implementation can be found in the M.A.Sc. thesis of S. Moayedinia [74].    107  The optimization was re-run with the updated boundary conditions starting from baseline optimized cooling times (note: baseline is defined as the optimization timing obtained with the optimization algorithm prior to revision and updating of the various boundary conditions described above). The yellow bars represent the baseline optimized cooling times that are used as the starting point for this optimization. The green bars represent the cooling times after optimization with the updated casting simulation model boundary conditions. With the enhanced boundary conditions, the baseline result reveals the encapsulation at the top rim, which is now consistent with the results of the industrial casting trial. Unfortunately, the optimization does not achieve a viable solution that satisfies all the constraints. The solutions obtained after the 17th iteration are slightly better than the previous iterations, with only one constraint in the upper rim violated and the violation is in the order of 10-3. The solution from the 17th iteration has been used in the discussion that follows to evaluate the results of the optimization.  Examining the cooling parameters of this near optimal solution (refer to Figure 5.14), the optimization algorithm increased the cooling durations of channel TD_CC_3 and channel BD_CC_2. This behavior is similar to the results presented from the two previous cases indicating that for the process models used, the upper channel (TD_CC_3) in the top die and the outer channel (BD_CC_2) in the bottom die are important for driving the solidification down the rim and across the spoke, respectively. Secondly, the cooling durations in the channel at the lower side die SD_CC_1 have been reduced, and channel TD_CC_2 remains off. Again it suggests that channel TD_CC_2 should be removed. These trends were also observed previously, which suggests that to achieve directional solidification conditions in the rim, the cooling durations of these two channels must be  108  reduced. The only differences from previous cases are that the cooling duration of the top die channels around the hub (TD_CC_1 & the central pin cooling) have been increased, and that cooling in the inner bottom die cooling channel (BD_CC_1) has been delayed appreciably. This change may be related to the changes in the magnitude of the interfacial heat transfer coefficients applied at the wheel/top die drum core interface and the wheel/bottom die interface.  In previous cases, the wheel/bottom die interface had higher heat transfer coefficients than the wheel/top die drum core interface, whereas in the current case, the opposite now occurs. The relative magnitude of interfacial heat transfer coefficients at other interfaces remains similar.   Figure 5.14 – Cooling times for 2-D axisymmetric prototype wheel (enhanced description, baseline optimized initial). The solidification images of the model with enhanced boundary conditions are presented in Figure 5.15. Focusing first on the initial results, the model predicts shrinkage porosity in the upper rim. This encapsulation will lead to shrinkage porosity in the final casting. The solidification images of the process conditions after 17 iterations of optimization  109  show that the severity of the encapsulation in the upper rim has been significantly reduced. There was no encapsulation identified in the mid-spoke region based on the constraints, however a closer examination of the solidification sequence in this area revealed that a small region of partially solidified material (indicated by the 571 °C isotherm) that has been encapsulated from the surface of the wheel in contact with the bottom die in the mid-spoke (not shown). The optimization algorithm did not identify this issue because it did not violate the constraint based on the 575 °C isotherm as the encapsulation occurs below the points used to assess the constraint function values. The cooling rates obtained with the cooling parameters identified by the 17th optimization iteration are summarized in Table 5.7. The cooling rate calculated at location P5 is larger than that of P4 and correlates with the occurrence of the mid-spoke encapsulation. The evolution of the objective function as a function of iteration number is shown in Figure 5.16. In general, the objective function is decreasing.  110   Figure 5.15 – The solidification sequence predicted for initial (1st row) trial-and-error based starting conditions and after the 17th optimization iteration (2nd row) for the water-cooled, prototype wheel casting process (enhanced description, baseline optimized initial). Table 5.7 – Summary of cooling rates at locations P1 – P6 (enhanced description, baseline optimized initial). Location Initial (°C/s) 17th iteration (°C/s) Change           (°C/s) P1 3.71 3.81 + 0.1 P2 1.83 1.88 + 0.05 P3 1.16 1.20 + 0.04 P4 0.65 0.68 + 0.03 P5 0.81 0.92 + 0.11 P6 0.59 0.58 - 0.01   111   Figure 5.16 - The optimization history / the evolution of the objective function (enhanced description, trial-and-error optimized initial).  5.3.3 Discussion The initial application of the optimization methodology to a 2-D axisymmetric prototype wheel and die structure showed successful results from the standpoint of eliminating shrinkage based porosity associated with encapsulation of liquid and achieving reasonable cooling rates. However, for the methodology to applied industrially it must first be validated.  The results from the initial industrial casting trial (with the cooling times obtained from the optimization algorithm) revealed a number of shortcomings in  the initial formulation of several key process boundary conditions. A second attempt at optimizing the cooling timing with enhanced boundary conditions incorporated in the casting process model resulted in a reduction in liquid encapsulation in the upper rim but not its elimination. From this result, it may inferred that the current die topology is not “optimizable” through manipulation of the cooling timing alone and needs to be redesigned with respect to the  112  placement and size of some of the cooling channels. In other words, the topology of the die needs to be optimized.  In an effort to quickly assess this, additional optimization test cases have been run in which the distribution of cooling in two of the top die channels (TD_CC_2 & 3) were changed by eliminating the cooling on portions of the channel surface – effectively changing their geometry within a limited scope. The optimization quickly (i.e. the 6th iteration) reached a solution where no encapsulation was observed in the upper rim. These results suggest that modifying the cooling channel geometry / location may offer significant benefit. Though promising, further work in this area is beyond the scope of this thesis.  113  6 Conclusions and Future Work A methodology has been developed to optimize the cooling channel timing in the low pressure die casting process used to produce automotive wheels. The goals were to achieve directional solidification as well as high solidification rates. The methodology was first formulated using a carefully designed test problem, and then was applied to a   2-D axisymmetric prototype die structure. The development was limited to determining the optimal timing sequence for water-cooling in the die, as cooling is critical for controlling the temperature history in the die and consequently wheel quality. This work constitutes the first step toward a more general optimizer that would include topology optimization.  The optimization methodology required the development of a control framework that coupled a commercial FEA package with an open source optimizer. Key challenges included: 1. the ability to execute cyclic model runs with the FEA package in batch mode including the modification of model setup file; 2. extraction of model results from the FEA package in the format required by the optimizer; and  3. communication with the open-source optimizer including understanding the signals associated with data requests from the optimizer. This study is one of the first to utilize a free open-source optimizer (Python Scipy.optimize) to construct the optimization methodology. Using an open-source  114  package has provided direct access to the optimization algorithm ensuring an understanding of the optimizer operation and enabling changes to be made to customize the implementation. Additionally, use of open-source code was economical since a commercial optimization package did not have to be purchased.   The development of this framework was undertaken in the scripting language, Python. The specific commands required will vary between software packages; however, with a few modifications, the developed framework can be readily coupled to other FEA packages. The work on the optimization methodology included the development and testing of various formulations for the objective and constraints functions. The resulting objective function and constraints are specific to low-pressure die casting technology for the production of automotive wheels and embody optimization of the cooling timing in order to minimize shrinkage defect formation and minimize casting cycle time. Different starting points – i.e. initial values for the cooling parameters – have also been tested to explore the robustness of the optimization methodology.  The major findings and suggestions for future work are summarized below. 6.1 Conclusions During the course of this study it has been found that:  It is important to carefully design the objective and constraint functions to achieve an optimization goal. It was demonstrated that for casting processes, where solidification is of primary importance, both the objective and constraint functions  115  can be based on solidification conditions. The objective function used in this study was based on the cooling rates at selected points distributed along the wheel, and the constraints were formulated as the time to reach a critical temperature in the wheel.   Theoretically, the objective function can be formulated in either linear form or quadratic form depending on the specific optimization goal – the former maximizes the cooling rates and the latter achieves target cooling rates. In this study, the linear format was shown to be superior as it does not require significant trail-and-error effort to determine a set of reasonable target cooling rates, which may often turn out to be an underestimation or an overestimation of the achievable maximum cooling rates.   The critical temperature used to identify the occurrence of liquid encapsulation was set to 575°C, which corresponds to 40% fraction solid. This temperature was verified through comparison of modeling results and plant trial data, both of which show encapsulation, or evidence of an encapsulation event, at the same location in the top rim.  Using a 2-D axisymmetric prototype die structure, two sets of starting points were investigated to assess their effect on the optimum cooling conditions – one being a non-optimum cooling condition and the other being a near-optimum cooling condition. This approach was taken because in problems of practical interest, it is seldom possible to ensure that the absolute optimum design will be found subject to multiple constraints [11]. This revealed at least three important findings:  116  o The optimization methodology was able to achieve viable solutions for both starting points; o Different starting points led to different optimized cooling parameters, which is consistent with results reported by other researchers that a global optimum is seldom available for practical problems; and  o Starting from a near-optimum cooling condition requires fewer optimization iterations to reach viable solutions, which saves computational time  After improving the boundary condition description in the model, the model predicted the formation of the industrially observed defect.  However, follow-up optimization failed to find a set of cooling conditions that eliminated the defect. Thus, this analysis has identified a limitation in the underlying geometric design of the die section with respect to the placement and size of some of the cooling channels.  Overall, the results show that the developed optimization methodology is able to achieve directional solidification and / or improve the solidification pattern, while also achieving increased or targeted solidification rates. This methodology requires minimal user intervention and/or trial-and-error testing to start it off. The approach demonstrated is flexible in that it can be applied to different geometries or systems, and could be easily adapted to other optimization applications. A final note: as the casting industry moves to adopt more complex models for the simulation of casting processes including those with optimization capabilities, there will inevitably be a tendency to assume that they will be more accurate and that therefore less  117  work and effort is required to validate them. This is a false assumption and in fact the opposite is true. The efficacy of the optimization methodology developed and demonstrated in this work was shown to depend critically on accuracy of the process model and the underlying formulation of key process boundary conditions.  6.2 Future Work Looking forward, the results of this study point the way for follow-on work in several areas: 1. The evaluation point locations for constraint functions should be studied. One of the drawbacks of placing the evaluation points in the middle of the wheel as done in this study is liquid encapsulation or the loss of directional solidification can be missed if the encapsulated volume is small or if it is away from the evaluation points. Potential options to improve the performance are to place extra rows of points in problematic regions or to shift the original points lower or higher in these regions. 2. Automotive wheels are not axisymmetric due to the presence and design of spokes (as well as the windows located between them) and other features (i.e. lightener pockets and bolthole blanks). Thus, going forward the optimization methodology should be applied to a 3-D model of a production die in order to assess its capabilities on these types of problems. Using a 3-D geometry will provide a more accurate casting process model, which in turn will make the optimization results more reliable.   118  3. The optimization of dies for industrial applications will ultimately require the development techniques to consider the die structure (topology, in conjunction with process timing optimization. It is foreseen that this development poses formidable computational challenges, as the location, shape, size and timing of the cooling in each channel are interrelated aspects of die design and should considered concurrently. This greatly increases the number of design variables. 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