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Structural and Multidisciplinary Optimization

## Lecture notes : Introduction to Numerical Optimization (Prof. P. Tossings)

1. INTRODUCTION TO MATHEMATICAL PROGRAMMING THEORY: basic concepts, convexity, convergence, KKT conditions, etc.
2. ALGORITHMS FOR UNCONSTRAINED OPTIMIZATION: Gradient methods
3. LINE SEARCH TECHNIQUES: + additionnal material
4. ALGORITHMS FOR UNCONSTRAINED OPTIMIZATION: Newton, Netwon-like and quasi-Newton methods
5. QUASI UNCONSTRAINED MINIMIZATION
6. GENERAL NON LINEAR PROGRAMMING I: Dual Methods
7. GENERAL NON LINEAR PROGRAMMING II: Transofrmation methods: Barrier function, penalty method, augmented lagrangian method and linearization methods (SLP, SQP).

## Lecture notes: Structural optimization (Prof. P. Duysinx)

1. INTRODUCTION : Course objectives and organization, pedagogical commitments
2. OPTIMIZATION IN ENGINEERING: Formulation of engineering design as an optimization problem, definitions, examples of applications
3. ELEMENTARY CONCEPTS IN STRUCTURAL OPTIMIZATION
4. INTRODUCTION TO FINITE ELEMENTS METHOD: historical perspective, bar element, linear elasticity
5. FINITE ELEMENT METHOD AND OPTIMIZATION:Finite Element discretization. Principle of virtual work
6. OPTIMALITY CRITERIA: Optimality criteria, FSD, single displacement, mutiple displacement and stresses
7. FROM OC TO STRUCTURAL APPROXIMATIONS: OC are first order Taylor expansion in the reciprocal design space
8. STRUCTURAL APPROXIMATIONS: Linear approximation, reciprocal approximation, CONLIN, MMA, GCMM (Updated version)
9. GENERALIZED OPTIMALITY CRITERIA: dual maximization to solve efficiently convex seperable subproblems, Berke's expression of displacement is a first order Taylor expansion of displacement and stresses, reciprocal variables
10. CONLIN AND MMA SOLVERS: dual solvers for convex linearisation schemes, CONLIN and MMA
11. SENSITIVITY ANALYSIS: finite differences, direct and adjoin approaches in static and vibration linear problems, semi analytical sensitivity analysis
12. SHAPE OPTIMIZATION: formulation, velocity field, multidisciplinary optimization, XFEM & Level Set
13. TOPOLOGY OPTIMIZATION: FUNDAMENTALS. An introduction to topology optimization
14. TOPOLOGY OPTIMIZATION: Designing with Topology Optimization

## Recording & Podcast : Structural optimization (Prof. P. Duysinx)

• INTRODUCTION : Course objectives and organization, pedagogical commitments
• OPTIMIZATION IN ENGINEERING: Formulation of engineering design as an optimization problem, definitions, examples of applications
• ELEMENTARY CONCEPTS IN STRUCTURAL OPTIMIZATION
• INTRODUCTION TO FINITE ELEMENTS METHOD: historical perspective, bar element, linear elasticity
• FINITE ELEMENT METHOD AND OPTIMIZATION:Finite Element discretization. Principle of virtual work
• OPTIMALITY CRITERIA:
• GENERALIZED OPTIMALITY CRITERIA: dual maximization to solve efficiently convex seperable subproblems, Berke's expression of displacement is a first order Taylor expansion of displacement and stresses, reciprocal variables
• STRUCTURAL APPROXIMATIONS: Linear approximation, reciprocal approximation, CONLIN, MMA, GCMMA
• CONLIN AND MMA SOLVERS: dual solvers for convex linearisation schemes, CONLIN and MMA
• SENSITIVITY ANALYSIS: finite differences, direct and adjoin approaches in static and vibration linear problems, semi analytical sensitivity analysis
• SHAPE OPTIMIZATION: formulation, velocity field, multidisciplinary optimization, XFEM & Level Set
• TOPOLOGY OPTIMIZATION: FUNDAMENTALS. An introduction to topology optimization

1. Introduction to Math Programming (C. Fleury)
2. Unconstrained and linearly constrained minimization (C. Fleury)
3. General non linear programming (C. Fleury)
4. Reconciliation of MP and OC (C. Fleury)
5. Dual methods for structural optimization (C. Fleury)
6. Convex approximations for structural optimization (C. Fleury)

## Projects & Computer works

• Project 1: Unconstrained constrained minimization

## Practical sessions

• Optimisation des structures: Exercices (livre)