Parameter-free shape optimization: various shape updates for engineering applications

Abstract

In the last decade, parameter-free approaches to shape optimization problems have matured to a state where they provide a versatile tool for complex engineering applications. However, sensitivity distributions obtained from shape derivatives in this context cannot be directly used as a shape update in gradient-based optimization strategies. Instead, an auxiliary problem has to be solved to obtain a gradient from the sensitivity. While several choices for these auxiliary problems were investigated mathematically, the complexity of the concepts behind their derivation has often prevented their application in engineering. This work aims at an explanation of several approaches to compute shape updates from an engineering perspective. We introduce the corresponding auxiliary problems in a formal way and compare the choices by means of numerical examples. To this end, a test case and exemplary applications from computational fluid dynamics are considered.

Shape Transformation Approaches for Fluid Dynamic Optimization

“…As an additional step, researchers have successfully applied discrete filtering of the shape sensitivity field [12][13][14] or more involved approaches such as the Steklov-Poincaré [15], Laplace-Beltrami [8] or the recently suggested p-harmonic descent method [16,17]. A general overview of such methods for engineering applications can be found in [14,18]. These methods aim to obtain a deformation vector field with a certain regularity on the boundary, which leads to feasible shape updates.…”

“…Attention is restricted to strategies that simultaneously compute the shape and mesh updates, i.e., the Steklov-Poincaré (Hilbert space) method [15] and the p-Laplace (Banach space) method [17]. While the former is deemed more efficient, the latter is seen to preserve the quality of the mesh much better but is also harder to solve [17,18].…”