Held Shape Optimization under Uncertainty from a Stochastic Programming Point of View

2009
ISBN: 978-3-8348-9396-3
Verlag: Vieweg & Teubner
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 148 Seiten, eBook

Reihe: Stochastic Programming

Optimization problems are relevant in many areas of technical, industrial, and economic applications. At the same time, they pose challenging mathematical research problems in numerical analysis and optimization.

Harald Held considers an elastic body subjected to uncertain internal and external forces. Since simply averaging the possible loadings will result in a structure that might not be robust for the individual loadings, he uses techniques from level set based shape optimization and two-stage stochastic programming. Taking advantage of the PDE's linearity, he is able to compute solutions for an arbitrary number of scenarios without significantly increasing the computational effort. The author applies a gradient method using the shape derivative and the topological gradient to minimize, e.g., the compliance and shows that the obtained solutions strongly depend on the initial guess, in particular its topology. The stochastic programming perspective also allows incorporating risk measures into the model which might be a more appropriate objective in many practical applications.

Dr. Harald Held completed his doctoral thesis at the Department of Mathematics at the University of Duisburg-Essen. He is now a Research Scientist at Siemens AG, Corporate Technology.

Zielgruppe

Research

Autoren/Hrsg.

Held, Harald

Inhaltsverzeichnis

1;Foreword;6
2;Acknowledgments;7
3;Abstract;8
4;Contents;9
5;Symbol Index;10
6;1 Introduction;11
6.1;1.1 The Elasticity PDE;14
6.1.1;1.1.1 Variational Formulation;16
6.2;1.2 Shape Optimization Problems;23
6.3;1.3 Two-Stage Stochastic Programming;27
6.3.1;1.3.1 Expected Value;30
6.3.2;1.3.2 Risk Measures;32
7;2 Solution of the Elasticity PDE;35
7.1;2.1 Composite Finite Elements;38
7.1.1;2.1.1 Construction for the Neumann Boundary;38
7.1.1.1;2.1.1.1 Implementational Remarks;43
7.1.2;2.1.2 Construction for the Dirichlet Boundary;47
7.1.2.1;2.1.2.1 Implementational Remarks;50
7.1.2.2;2.1.2.2 Simple 1D Example;53
7.1.3;2.1.3 Mixed Boundary Conditions;54
7.1.4;2.1.4 Computation of the System Matrix and the Right-Hand Side Vector;57
8;3 Stochastic Programming Perspective;59
8.1;3.1 Stochastic Shape Optimization Problem;60
8.1.1;3.1.1 Two-Stage Stochastic Shape Optimization Problem;61
8.1.2;3.1.2 Dual Problem and Saddle Point Formulation;64
8.2;3.2 Reformulation and Solution Plan for the Expectation-Based Model;71
8.3;3.3 Expected Excess;80
8.3.1;3.3.1 Barrier Method;81
8.3.2;3.3.2 Smooth Approximation;82
8.4;3.4 Excess Probability;83
9;4 Solving Shape Optimization Problems;87
9.1;4.1 Level Set Formulation;88
9.1.1;4.1.1 Computation of the Mean Curvature;90
9.2;4.2 Shape Derivative;91
9.3;4.3 Topological Derivative;99
9.4;4.4 Steepest Descent Algorithm;104
9.4.1;4.4.1 Regularized Descent Direction;108
10;5 Numerical Results;111
10.1;5.1 Deterministic and Expectation-Based Results;112
10.1.1;5.1.1 VSS and EVPI;122
10.2;5.2 Risk Aversion;124
11;A Appendix;131
11.1;A.1 Notation;131
11.2;A.2 Important Facts and Theorems;134
12;References;137

Solution of the Elasticity PDE.- Stochastic Programming Perspective.- Solving Shape Optimization Problems.- Numerical Results.

Leseproben

4 Solving Shape Optimization Problems (S. 77-78)

This chapter is dedicated to the actual numerical solution techniques we implemented to solve the (random) shape optimization problems described in Chapter 3. As noted in the beginning, we employed a steepest descent algorithm (see Section 4.4) together with a level set method (see Section 4.1).

The necessary function evaluations are done according to Algorithm 3.16, whereas the computation of the descent direction is described here in this chapter, making use of the shape derivative (see Section 4.2) and also the topological derivative (see Section 4.3). There are various methods that aim to solve shape optimization problems, and before we start describing our particular level set approach, we brie?y mention some of these methods. For example, there is the homogenization method (cf. Allaire [All02]) whose physical idea in principle consists of averaging heterogeneous media in order to derive effective properties. In [All02, Chapter 4], the method is applied to optimal design problems with linear elasticity in form of two-phase optimization problems.

The task is then to ?nd an optimal distribution of two elastic materials, i.e. there are no void areas. This results in an ill-posed optimization problem, which, however, homogenization theory provides a relaxation to by introducing generalized designs. Numerical examples can also be found in [HN97]. Another approach, namely topology optimization by the material distribution method, is described in the book by Bendsøe and Sigmund [BS03]. Each point in the design can have material or not1. In a discrete setting, there is a grid where each grid cell, or “pixel”, is either ?lled with material, or there is none.

This leads to nonlinear optimization problems with binary variables which indicate the presence or absence of material in the grid cells, respectively. In [SS03] for example, they show that certain nonlinear 0-1 topology optimization problems can be equivalently formulated as linear mixed 0-1 programs, which can be solved as such—at least on quite coarse grids. The idea described in [BS03], however, is to replace the integer variables with continuous ones, resulting in a density function with values between 0 and 1, and then to penalize intermediate values. This yields the so-called SIMP-model2. Various solution methods are mentioned in [BS03].

Claudia Stangl implemented this model in her diploma thesis [Sta08], also incorporating stochastic forces for the expectation-based problem, and solved it using IPOPT (cf. [WB06]). Maar and Schulz [MS00] describe the application of an interior point multigrid method for this type of problem. Newton’s method, involving second order shape derivatives (cf. [NR]), has been applied to some shape optimization problem for example in [NP02]. Level set methods provide another approach to tackling shape optimization problems. This is the method we applied to our problems, so we will describe it in more detail in the following section.

Über Autor(innen)

Dr. Harald Held completed his doctoral thesis at the Department of Mathematics at the University of Duisburg-Essen. He is now a Research Scientist at Siemens AG, Corporate Technology.