Shape Optimization Under Uncertainty—A Stochastic Programming Perspective

Abstract

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 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.

On Shape Optimization with Stochastic Loadings

“…For a discussion of the appropriate differentiation of boundary integrals we refer to [27,62]. Here, we confine ourselves, as in [25,24], to the simpler case that homogeneous Neumann boundary conditions are assumed on the part of the boundary which is optimized. Furthermore, let us emphasize that the above classical shape derivative is only admissible in case of additional regularity for the primal solution u and the dual solution p, which holds under sufficiently strong regularity assumptions on loads and geometry of the considered shapes.…”

“…Neither Γ D nor Γ N will be subject to the shape optimization. For details on the concrete handling of the boundary conditions in the shape optimization we refer to [25]. Concerning the loading, we are in particular interested in a stochastic loads.…”

“…The optimal shape determined using the shape derivative alone may strongly depend on this choice of the initial topology (cf. [5,8,25]). This is not only due to the discretization, but also to the descent strategy itself.…”

“…Under our assumption that j(·) and k(·) are linear or quadratic functions, there is a significant algorithm shortcut at our disposal for N s 1, which we will recall here. For details we refer to [25]. Let us assume that there is a small number of (deterministic) basis volume forces f 1 , .…”

“…, N S are composed of the discrete primal and dual solutions U i,j and P i,j , respectively, based on the discrete analog of (20) and (21), respectively. For further details on the primal and dual solutions we refer to [25] and for the composite finite element approach in the context of domains described via level sets we refer to [37].…”