On Shape Optimization with Stochastic Loadings

Abstract

Abstract. This article is concerned with different approaches to elastic shape optimization under stochastic loading. The underlying stochastic optimization strategy builds upon the methodology of two-stage stochastic programming. In fact, in the case of linear elasticity and quadratic objective functional our strategy leads to a computational cost which scales linearly in the number of linearly independent applied forces, even for a large set of realization of the random loading. We consider, besides minimization of the expectation value of suitable objective functionals, also two different risk-averse approaches, namely the expected excess and the excess probability. Numerical computations are performed using either a level-set approach representing implicit shapes of general topology in combination with composite finite elements to resolve elasticity in two and three dimensions, or a collocation boundary element approach, where polygonal shapes represent geometric details attached to a lattice and describing a perforated elastic domain. Topology optimization is performed using the concept of topological derivatives. We generalize this concept, and derive an analytical expression which takes into account the interaction between neighboring holes. This is expected to allow efficient and reliable optimization strategies of elastic objects with a large number of geometries details on a fine scale. Mathematics Subject Classification (2000). 90C15, 74B05, 65N30, 65N38, 34E08, 49K45.Keywords. shape optimization in elasticity, two-stage stochastic programming, risk averse optimization, level set method, boundary element method, topological derivative.

A posteriorimodeling error estimates in the optimization of two-scale elastic composite materials

“…As we typically consider mixed boundary value problems equation (5) has to be rearranged according to known and unknown values. Further details can be found in [5]. The coefficients of the effective material tensor can then be evaluated as in (3) using a boundary integral formulation.…”

“…The boundary integral operators U and V are only applied to piecewise affine functions on the boundary, therefore their application to the basis functions has been computed analytically. As we typically consider mixed boundary value problems equation (5) has to be rearranged according to known and unknown values. Further details can be found in [5].…”

“…As we typically consider mixed boundary value problems equation (5) has to be rearranged according to known and unknown values. Further details can be found in [5].…”

Stochastic approximation for optimization in shape spaces

“…Increasingly, stochastic models are being used in shape optimization with the goal of obtaining more robust solutions. A number of works has focused on structural optimization with either random Lamé parameters or forcing [3,10,11,13,36]. Stochastic models have also handled uncertainty in the geometry of the domain [7,25,33].…”