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STRUCTURAL OPTIMIZATION USING PARAMETRIC PROGRAMMING METHOD Krupakaran Ravichandraan Clemson University, [email protected]

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STRUCTURAL OPTIMIZATION USING PARAMETRIC PROGRAMMING METHOD Krupakaran Ravichandraan Clemson University, Kravich@G.Clemson.Edu Clemson University TigerPrints All Theses Theses 12-2015 STRUCTURAL OPTIMIZATION USING PARAMETRIC PROGRAMMING METHOD Krupakaran Ravichandraan Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_theses Part of the Mechanical Engineering Commons Recommended Citation Ravichandraan, Krupakaran, "STRUCTURAL OPTIMIZATION USING PARAMETRIC PROGRAMMING METHOD" (2015). All Theses. 2264. https://tigerprints.clemson.edu/all_theses/2264 This Thesis is brought to you for free and open access by the Theses at TigerPrints. It has been accepted for inclusion in All Theses by an authorized administrator of TigerPrints. For more information, please contact [email protected]. STRUCTURAL OPTIMIZATION USING PARAMETRIC PROGRAMMING METHOD A Thesis Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Master of Science Mechanical Engineering by Krupakaran Ravichandraan December 2015 Accepted by: Dr. Georges M. Fadel, Committee Chair Dr. Margaret Wiecek Dr. Lonny L. Thompson ABSTRACT The Parametric Programming method is investigated to consider its applicability to structural optimization problems. It is used to solve optimization problems that have design variables as implicit functions of some independent input parameter(s). It provides optimal solutions as a parametric function of the input parameter(s) for the entire parameter space of interest. It does not require the detailed discrete optimizations needed at a large number of parameter values as in traditional non-parametric optimization. Parametric programming is widely used in optimal controls, model predictive control, scheduling, process synthesis and material design under uncertainty due to the above mentioned benefits. Its benefits are however, still unexplored in the field of structural optimization. Parametric programming could for instance, be used to aid designers in identifying and optimizing for uncertain loading conditions in complex systems. The first objective of this thesis is to identify a suitable multi-parametric programming algorithm among the many available ones in the literature to solve structural optimization problems. Once selected, the second goal is to implement the chosen algorithm and solve single parametric and multi-parametric sizing optimization problems, shape optimization problems, and use multi-parametric programming as a multi-objective optimization tool in structural optimization. In this regard, sizing optimization of truss structures and shape optimization of beams for load magnitude and load directions as varying parameters are solved for single and multi-parameter static and/or dynamic load cases. Parametric programming is also used to solve the multi-objective optimization of a honeycomb panel and the results are compared with those from non-parametric ii optimization conducted using commercial optimization software. Accuracy of results, and computational time are considered. From these studies, inferences are drawn about the issues and benefits of using parametric programming in structural optimization. iii DEDICATION I would like to dedicate my work to the Almighty, my parents for their continuous support and encouragement in whatever I pursue, my brother, and those who always pray for my best. iv ACKNOWLEDGMENTS I take this opportunity to sincerely thank my advisor Dr. Georges M. Fadel for all his guidance, encouragement, and moral support. He has played a great role in shaping my research acumen, in improving my professional skills and in my improvement as a person. I thank him again for always being there for me and guiding me in the right direction. I thank Dr. Margaret Wiecek for providing valuable inputs through the entire course of my research work. Her constant motivation and support has been a great driving force for my research. I also thank Dr. Lonny L. Thompson for giving me his valuable inputs and suggestions which have helped shape this research work. Finally, I thank my two seniors, Dr. Meng Xu and Dr. Jonathon Leverenz. Whenever I needed help, they were always there for me in providing their inputs and feedback. v TABLE OF CONTENTS Page TITLE PAGE .................................................................................................................... i ABSTRACT ..................................................................................................................... ii DEDICATION ................................................................................................................ iv ACKNOWLEDGMENTS ............................................................................................... v LIST OF TABLES ........................................................................................................ viii LIST OF FIGURES ......................................................................................................... x NOMENCLATURE ...................................................................................................... xv CHAPTER 1. INTRODUCTION ..................................................................................................... 1 1.1 Motivation ...................................................................................................... 1 1.2 Organization of the Thesis ............................................................................. 4 1.3 Literature Review........................................................................................... 5 2. CASE STUDY OF MULTI-PARAMETRIC PROGRAMMING ALGORITHMS ON A BENCHMARK STRUCTURAL OPTIMIZATION PROBLEM ..................................................................... 21 2.1 Multi-parametric Toolbox 3.0 and Solution ................................................ 23 2.2 Multi-parametric Quadratic/Outer Approximation Algorithm and Solution ........................................................................................... 30 2.3 Approximation Simplex Method Algorithm and Solution .......................... 35 2.3 Conclusion ................................................................................................... 47 3. APPLICATIONS OF PARAMETRIC PROGRAMMING METHOD ................... 50 3.1 Sizing Optimization Using Parametric Programming Method .................... 50 3.2 Parametric Programming as a Multi-objective Optimization Tool for Structural Optimization ........................................................... 88 3.3 Shape Optimization Using Parametric Programming Method .................. 103 vi Table of Contents (Continued) Page 4. CONCLUSION AND FUTURE WORK .............................................................. 117 4.1 Conclusion ................................................................................................. 117 4.2 Issues Pertaining To Parametric Programming and ASM Algorithm ................................................................................... 122 4.3 Future Work ............................................................................................... 123 APPENDICES ............................................................................................................. 125 A: Syntax for Problem Formulation In MPT 3.0 ............................................ 127 B: MATLAB Code for Recursive Linear Approximation .............................. 128 C: MATLAB Code for mp-Q/OA Problem Formulation ............................... 132 D: MATLAB FEA Solver for Problem 3.1.1.1 .............................................. 152 E: MATLAB FEA Code for Problem 3.1.5.1 ................................................ 155 F: MATLAB Code for ASM Algorithm for Problem 3.1.1.1 ........................ 161 G: MATLAB Code for Problem 3.1.2.1 ......................................................... 172 H: MATLAB Code for Problem 3.1.3.1 ......................................................... 182 I: MATLAB FEA Code for Problem 3.1.4.1 ................................................ 188 J: MATLAB ASM Code for Problem 3.1.5.1 ............................................... 195 K: MATLAB ASM Code for Problem 3.2.1.1 ............................................... 198 L: MATLAB Code for Problem 3.3.1 ............................................................ 202 M: MATLAB FEA Code to Extract Optimized Beam Shape ......................... 212 REFERENCES ............................................................................................................ 214 vii LIST OF TABLES Table Page 2.1 List of optimal function of objectives and decision variables for load range of 100 N to 1000 N for an error tolerance of 0.1 ......................... 26 2.2 List of optimal value functions 푥(푝) corresponding to the associated simplexes (critical regions) .................................................................... 41 2.3 Comparison of results from ASM with search tool and without search tool ........................................................................................................ 43 3.1 Table of optimal parametric function for 푥2 and 푚푎푠푠 with corresponding parameter intervals (critical regions) for error tolerance of 0.01..................................................................................... 57 3.2 Comparison of computational performance and accuracy of parametric optimization results via ASM algorithm with Non-parametric optimization ................................................................. 61 3.3 Table of optimal parametric function of 푥1 and 푥2
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