structural & multidisciplinary optimization · structural & multidisciplinary optimization...
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STRUCTURAL & MULTIDISCIPLINARY OPTIMIZATION
Pierre DUYSINX
Patricia TOSSINGS
Department of Aerospace and Mechanical Engineering
Academic year 2017-2018
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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 papers2
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
– …
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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
– Convex functions, convex sets
PRIMAL AND DUAL PROBLEM STATEMENT
– Lagrange function
– Lagrangian problem
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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
LINEARLY CONSTRAINED MINIMIZATION
– Generalized steepest descent concept
– Gradient projection methods
– Active set strategy
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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
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Outline
FROM OPTIMALITY CRITERIA TO SEQUENTIAL CONVEX PROGRAMMING
– Optimality Criteria
Fully stresses design
Optimality criteria with one displacement constraint
Optimality criteria with multiple displacement constraints
– Generalized optimality criteria
– Linearized mathematical programming methods
– Unified approach to structural optimization
– Examples
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Outline
SENSITIVITY ANALYSIS– Linear static problems– Natural vibrations– Linear stability
STRUCTURAL APPROXIMATIONS
– 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…
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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
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Outline
SHAPE OPTIMIZATION
Parametric approach
Sensitivity analysis wrt to boundary variables
Velocity field calculation
Extension to XFEM and Level set description
Examples
TOPOLOGY optimization
Optimal material distribution problem formulation
Microstructures and homogenization
Sensitivity analysis
Compliance design
Strength design
Examples
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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
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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, 2017 (12:00 AM)
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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, 120–127
– Available at www.topopt.dtu.dk/
Evaluation:
– Report & Power Point presentation (max 20 pages)
– Program MATLAB code
– Deadline: November 6, 2017 (12:00 AM)
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Computer work 3: Constrained minimization
Solve a generally constrained minimization problem using sequential transformation methods experimenting with the optimization methods presented during the lectures
Build up your code to MATLAB solve:
– Solve a sequence of linearly constrained problems
– General non linear minimization using a transformation method (e.g. augmented Lagrangian)
Evaluation:
– Report (max 25 pages)
– MATLAB code
– Deadline: January 7, 2018 (12:00 AM)
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Computer work 4: Solve 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, 2018 (12:00 AM)
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Exams and assessment
Oral exam: theory
– Two questions of theory
– Two questions about computer projects
– In January
Computer works & projects
– Reports & Program
– Oral presentation
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Lecture notes
Copy of the slides
– Lecture slides by Pierre Duysinx & Patricia Tossings
Web site: www.ingveh.ulg.be > cours > MECA0027
References (recommended but not mandatory!)
– Programmation mathématique : théorie et algorithmes (Tome 1). M.Minoux. Dunod, Paris, 1983.
– 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, 2003.
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Agenda
Date Hour Lecture
21/09 9h Introduction (P. Duysinx)
10h00 Engineering design using structural optimization (P. Duysinx)
28/09 9h Fundamentals of Math Programming (P. Tossings)
10h30 Basics of Topology Optimization (P. Duysinx)
05/10 9h Unconstrained Optimization I - Gradient Methods (P. Tossings)
11h Computer Work 1: Unconstrained minimization
12/10 9h Unconstrained Optimization II - Gradient Methods & Line Search (P. Tossings)
11h Computer Work 2: Topology optimization
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Agenda
Date Hour Lecture
19/10 9h Unconstrained Optimization III – Newton & Quasi Newton (P. Tossings)
11h Computer Work 3: Unconstrained minimization
26/10 9h Linearly Constrained Minimization (P. Tossings)
11h Computer Work 4: Topology Optimization
02/11 9h HOLLIDAY
06/11 12h Deadline Project 1 Unconstrained minimization & Project 2 Basics of Topology optimization
09/11 9h From Optimality Criteria to Sequential Convex Programming (P. Duysinx)
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Agenda
Date Hour Lecture
16/11 9h General nonlinear programming : Duality (P. Tossings)
11h Computer Work 5: Constrained Optimization
23/11 9h Computer Work 6: Topology Optimization (NX)
Computer Work 7: Topology Optimization (NX)
30/11 9h Structural approximations and dual solvers: CONLIN, MMA (P. Duysinx)
11h Computer Work 8: Constrained Optimization
07/12 9h Solving Sequential Convex Problems: Transformation methods (Barrier, Penalty, Augmented Lagrangian) (P. Tossings)
11h Solving Sequential Convex Problems: Transformation methods, SLP, SQP, Dual solvers (P. Duysinx)
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Agenda
Date Hour Lecture
14/12 9h Sensitivity analysis (P. Duysinx)
11h Computer Work 9: Topology Optimization
21/12 9h Shape and Topology Optimization (P. Duysinx)
11h Computer Work 10: Constrained Optimization
07/01 12h Deadline project 2 Topology optimization
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Contacts
Pierre Duysinx
– LTAS-Automotive Engineering
– Institute de Mécanique B52 0/514
– Tel 04 366 9194
– Email: [email protected]
Patricia TOSSINGS
– Mathématiques Générales B37 0/57
– Institut de Mathématique
– Tél: 04 366 9373
– Email. [email protected]
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Contacts
Assistants :
– Maxime COLLET
LTAS-Automotive Engineering
Institute de Mécanique B52 0/512
Tel 04 366 9273
Email:[email protected]
– Eduardo FERNANDEZ SANCHEZ
LTAS-Automotive Engineering
Institute de Mécanique B52 0/512
Tel 04 366 9273
Email: [email protected]
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