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STOCHASTIC DOMINANCE CONSTRAINTS IN ELASTIC shape optimization

SIAM JOURNAL ON CONTROL AND optimization 2018年第4期56卷 3021-3034页

作者： Conti, Sergio Rumpf, Martin Schultz, Ruediger Toelkes, Sascha Univ Bonn Inst Appl Math D-53115 Bonn Germany Univ Bonn Inst Numer Simulat D-53115 Bonn Germany Univ Duisburg Essen Fac Math D-45127 Essen Germany

This paper deals with shape optimization for elastic materials under stochastic loads. It transfers the paradigm of stochastic dominance, which allows for flexible risk aversion via comparison with benchmark random variables, from finite-dimensional stochastic programming to shape optimization. Rather than handling risk aversion in the objective, this enables risk aversion by including dominance constraints that single out subsets of nonanticipative shapes which compare favorably to a chosen stochastic benchmark. This new class of stochastic shape optimization problems arises by optimizing over such feasible sets. The analytical description is built on risk-averse cost measures. The actual optimized cost functional measures the volume and perimeter of the structure. In the implementation, shapes are represented by a phase field which permits an easy estimate of a regularized perimeter. The analytical description and the numerical implementation of dominance constraints are built on risk-averse measures for the cost functional. A suitable numerical discretization is obtained using finite elements both for the displacement and the phase field function. Different numerical experiments demonstrate the potential of the proposed stochastic shape optimization model and in particular the impact of high variability of forces or probabilities in the different realizations.

关键词： shape optimization in elasticity stochastic optimization risk aversion stochastic dominance

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RISK AVERSE shape optimization

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SIAM JOURNAL ON CONTROL AND optimization 2011年第3期49卷 927-947页

作者： Conti, Sergio Held, Harald Pach, Martin Rumpf, Martin Schultz, Ruediger Univ Bonn Inst Angew Math D-53115 Bonn Germany Univ Duisburg Essen Fak Math D-47057 Duisburg Germany Univ Bonn Inst Numer Simulat D-53115 Bonn Germany

Risk averse optimization has attracted much attention in finite dimensional stochastic programming. In this paper, we propose a risk averse approach in the infinite dimensional context of shape optimization. We consider elastic materials under stochastic loading. As measures of risk awareness we investigate the expected excess and the excess probability. The developed numerical algorithm is based on a regularized gradient flow acting on an implicit description of the shapes based on level sets. We incorporate topological derivatives to allow for topological changes in the shape optimization procedure. Numerical results in two dimensions demonstrate the impact of the risk averse modeling on the optimal shapes and on the cost distribution over the set of scenarios.

关键词： shape optimization in elasticity two-stage stochastic programming risk aversion level set method topological derivative

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shape optimization UNDER UNCERTAINTY-A STOCHASTIC PROGRAMMING PERSPECTIVE

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SIAM JOURNAL ON optimization 2009年第4期19卷 1610-1632页

作者： Conti, Sergio Held, Harald Pach, Martin Rumpf, Martin Schultz, Ruediger Univ Duisburg Gesamthsch Dept Math D-47048 Duisburg Germany Univ Bonn Inst Numer Simulat D-53115 Bonn Germany

We present an algorithm for shape optimization under stochastic loading and representative numerical results. Our strategy builds upon a combination of techniques from two-stage stochastic programming and level-set-based shape optimization. In particular, usage of linear elasticity and quadratic objective functions permits us to obtain a computational cost which scales linearly in the number of linearly independent applied forces, which often is much smaller than the number of different realizations of the stochastic forces. Numerical computations are performed using a level set method with composite finite elements both in two and in three spatial dimensions.

关键词： two-stage stochastic programming shape optimization in elasticity level set method

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