Efficient PDE Constrained Shape Optimization Based on Steklov--Poincaré-Type Metrics

Abstract

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincaré type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor, so far involved in the derivation of shape derivatives.

Parallel 3d shape optimization for cellular composites on large distributed-memory clusters

“…An expressions for the shape derivative dJ(Ω) [w] in direction w of the objective J in (1) is presented in [44,45] for both the elastic energy and the geometric quantities. We closely follow the approach in [43] to represent the Algorithm 1 Gradient-penalized optimization algorithm from [44] 1: Choose initial domain Ω 0 ; choose ν elast , ν vol , ν peri ; choose ν penalty , b, t 2: for k = 0, 1, 2, . .…”

Shape optimization: what to do first, optimize or discretize?

“…[4]). The representative of the shape derivative with respect to the Riemannian metric (also known as the shape gradient) is given as the solution of the so-called deformation equation.…”

“…[3]). The connection of PDE-constrained shape optimization and the differential geometric structure of this shape space is enabled for example in [4]. By choosing the Steklov-Poincaré metric associated with the linear elasticity operator, it is possible to control the mesh quality to some degree through the choice of the Lamé parameters.…”

Gradient based biobjective shape optimization to improve reliability and cost of ceramic components

“…The existence of optimal shapes has been studied in [10,26,32] -for the specific objective function f of compliance see [1]. On the algorithmic side, the adjoint approach to shape calculus has led to efficient strategies to calculate shape gradients, see e.g., [12,16,22,23,34,46,47,49]. While theory and numerical algorithms of shape calculus are highly developed mathematically, most publications in the field neither deal with multiobjective optimization problems, nor directly consider mechanical integrity as one of the objective functions, see [32,2,20,39] for some remarkable exceptions.…”