STRUCTURAL & MULTIDISCIPLINARY OPTIMIZATION
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1 STRUCTURAL & MULTIDISCIPLINARY OPTIMIZATION Pierre DUYSINX Patricia TOSSINGS Department of Aerospace and Mechanical Engineering Academic year
2 Course objectives To become familiar with the introduction of optimization concepts in design and engineering processes. To be able to formulate your design problem as an optimization problem To present a systematic and critical overview of various numerical methods available to solve optimization problems. The basic concepts are illustrated throughout the course by solving simple optimization problems. (Generally structural problems but also multidisciplinary problems) To be able to read, implement and exploit scientific papers 2
3 Course objectives Several examples of application to real-life design problems are offered to demonstrate the high level of efficiency attained in modern numerical optimization methods. Remark: Although most examples are taken in the field of structural optimization, using finite element modeling and analysis, the same principles and methods can be easily applied to other design problems and simulation methods arising in various engineering disciplines. Electromagnetics Fluid flows Mechanical Engineering Chemical Engineering 3
4 Outline INTRODUCTION Optimization in Engineering Design Assumptions and definitions Problem statement Iterative optimization Process INTRODUCTION TO MATHEMATICAL PROGRAMMING Optimality conditions of single variable and multiple variable functions (KKT conditions) Convex functions, convex sets PRIMAL AND DUAL PROBLEM STATEMENT Lagrange function Lagrangian problem 4
5 Outline UNCONSTRAINED OPTIMIZATION Descent methods for minimization The method of steepest descent Quasi unconstrained problem: optimality conditions Line search techniques Newton type methods Conjugate direction methods Quasi Newton methods 5
6 Outline GENERAL NON LINEAR PROGRAMMING METHODS Pure dual methods Lagrange function KKT conditions Introduction to duality Weak duality Strong duality Properties of dual function in Strong duality Application to quadratic problems st linear constraints Treatment of side constraints Dual solutions of MMA subproblems 6
7 Outline SENSITIVITY ANALYSIS Linear static problems Natural vibrations Linear stability STRUCTURAL APPROXIMATIONS OC criteria as a first order expansion of displacements Linear approximation Reciprocal approximations Convex Linearization (CONLIN) Method of Moving asymptotes (MMA) Generalized Method of Moving Asymptotes The MMA family schemes Second order approximations Diagonal SQP, GMMA 7
8 Outline SOLVING SEQUENTIAL CONVEX PROGRAMMING Transformation methods: Barrier functions, Penalty function, Augmented Lagrangian methods Direct (primal) methods Gradient projection methods Method of feasible directions Linearization methods Sequential Linear Programming SLP Recursive Quadratic programming Sequential Quadratic Programming SQP Dual solvers With convex and separable approximation CONLIN and MMA Subproblem formulation Dual solvers for MMA and CONLIN 8
9 Outline TOPOLOGY optimization: Introduction Optimal material distribution problem formulation Microstructures and homogenization Sensitivity analysis Compliance design Strength design Examples TOPOLOGY OPTIMIZATION: advanced topics Geometrical restrictions: min/ max size, perimeter, etc. Stress constraints in topology optimization Multiple material topology optimization Topology optimization and additive manufacturing 9
10 Outline SHAPE OPTIMIZATION Representation of geometries Parametric curves and surfaces Constructive geometry Level set description Parametric approach of CAD systems Sensitivity analysis wrt to boundary variables Material derivatives Velocity field concept Velocity field calculation Extension to XFEM and Level set description Finite element error control Industrial software tools Examples 10
11 Outline COMPOSITE STRUCTURE OPTIMIZATION Parameterization of composites Problem formulation Sensitivity analysis Solution aspects META HEURISTIC ALGORITHMS Introduction to meta heuristic optimization algorithms Genetic Algorithms Simulated Annealing Particle Swarm Optimization 11
12 Computer work 1: Unconstrained Minimization Solve optimization unconstrained minimization problems using computer and MATLAB to experiment the optimization methods presented during the lectures Group of two students Build up your MATLAB code to solve: Unconstrained minimization of quadratic and non quadratic functions Evaluation: Report and Power Point presentation (max 20 pages) MATLAB code Deadline: November 6, 2018 (12:00 AM) 12
13 Computer work 2: Basics of Topology Optimization Building up your own topology optimization solver to experiment with the method Group of two students Goal: solve efficiently your topology optimization problem Write your MATLAB code Ref: A 99 line topology optimization code written in Matlab. O. Sigmund. Struct Multidisc Optim 21, Available at Evaluation: Report & Power Point presentation (max 20 pages) Program MATLAB code Deadline: December 22, 2018 (12:00 AM) 13
14 Computer work 3: Solve an industrial problem using topology optimization Introduction to an industrial topology optimization tool (NX TOPOL) Group of two students Getting starting with TOPOL Goal: getting starting to and being able to solve industrial applications Getting started with TOPOL Understand the TOPOL parameter settings Solve an industrial benchmark Evaluation: Report and Power Point Presentation (max 20 pages) Deadline January 7, 2019 (12:00 AM) 14
15 Exams and assessment Oral exam: theory Two questions of theory Two questions about computer projects In January Computer works & projects Reports & Program Oral presentation 15
16 Lecture notes Copy of the slides Lecture slides by Pierre Duysinx & Patricia Tossings Web site: > cours > MECA0027 References (recommended but not mandatory!) Programmation mathématique : théorie et algorithmes (Tome 1). M.Minoux. Dunod, Paris, Foundations of Structural Optimization: A Unified Approach. A.J. Morris. John Wiley & Sons Ltd, 1982 Haftka, R.T. and Gürdal, Z., Elements of Structural Optimization, 3rd edition, Springer, 1992 Topology Optimization, Theory, Methods, and Applications. M.P. Bendsoe and O. Sigmund, Springer Verlag, Berlin,
17 Agenda Date Hour Lecture 18/09 14h Fundamentals of Math Programming including KKT conditions (P. Tossings) 25/09 14h Introduction to Engineering Design using Structural Optimization (P. Duysinx) Optimality Criteria approach of structural optimization (P. Duysinx) 02/10 14h Unconstrained Optimization I - Gradient Methods (P. Tossings) Computer Work 1: Unconstrained minimization 09/10 14h Unconstrained Optimization II - Gradient Methods & Line Search (P. Tossings) Computer Work 2: Unconstrained Minimization 17
18 Agenda Date Hour Lecture 16/10 14h Unconstrained Optimization III Newton & Quasi Newton (P. Tossings) Computer Work 3: Unconstrained minimization 23/10 14h Topology optimization. Introduction (P. Duysinx) 30/10 HOLLIDAY Computer Work 4: Topology Optimization using MATLAB 07/11 12h Deadline Project 1 Unconstrained minimization 07/11 14h Sensitivity analysis of linear structural systems (P. Duysinx) Computer Work 5: Topology Optimization using MATLAB 18
19 Agenda Date Hour Lecture 13/11 14h General nonlinear programming : Duality (P. Tossings) Computer Work 6: Topology Optimization (MATLAB) 20/11 14h Structural approximations and dual solvers: CONLIN, MMA (P. Duysinx) Computer Work 7: Topology Optimization (MATLAB) 27/11 14h Solving Sequential Convex Problems: Transformation methods (Barrier, Penalty, Augmented Lagrangian) (P. Tossings) Computer Work 8: Topology optimization using NX TOPOL 04/12 14h Solving Sequential Convex Problems: Transformation methods, SLP, SQP, Dual solvers (P. Duysinx) Computer Work 9: Topology optimization (NX TOPOL) 19
20 Agenda Date Hour Lecture 11/12 14h Shape optimization (P. Duysinx) Computer Work 10: Topology Optimization (NX TOPOL) 18/12 Topology optimization and additive manufacturing (P. Duysinx, E. Fernandez) Computer Work 11: Topology Optimization (NX TOPOL) 22/12 12h Deadline project 2 Topology optimization using MATLAB 07/01 12h Deadline project 3 Topology optimization using NX TOPOL 20
21 Contacts Pierre Duysinx LTAS-Automotive Engineering Institute de Mécanique B52 0/514 Tel Patricia TOSSINGS Mathématiques Générales B37 0/57 Institut de Mathématique Tél:
22 Contacts Assistants : Pablo ALARCON LTAS-Automotive Engineering Institute de Mécanique B52 0/512 Tel palarcon@uliege.be Eduardo FERNANDEZ SANCHEZ LTAS-Automotive Engineering Institute de Mécanique B52 0/512 Tel efsanchez@uliege.be 22
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