Pre-Shape Calculus: Foundations and Application to Mesh Quality Optimization

Abstract

Deformations of the computational mesh arising from optimization routines usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize classical shape optimization and -calculus. We define pre-shape derivatives and derive according structure and calculus theorems. In particular, tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives featuring only normal directions. Techniques from classical shape optimization and -calculus are shown to carry over to this framework. An optimization problem class for mesh quality is introduced, which is solvable by use of preshape derivatives. This class allows for simultaneous optimization of classical shape objectives and mesh quality without deteriorating the classical shape optimization solution. The new techniques are implemented and numerically tested for 2D and 3D.

Simultaneous Shape and Mesh Quality Optimization using Pre-Shape Calculus

“…In this work, we want to approach these two aspects differently. By using preshape calculus techniques introduced in [12], we propose regularization techniques for shape gradients with two goals in mind. First, the required computational burden should be limited.…”

“…An complementing literature review is found in the introduction of [12]. For this work we build on results of [12], where pre-shape calculus is rigorously established.…”

“…An complementing literature review is found in the introduction of [12]. For this work we build on results of [12], where pre-shape calculus is rigorously established. The results from [12] feature a structure theorem in the style of Hadamard's theorem for shape derivatives, which also shows the relationship of classical shape and pre-shape derivatives.…”

“…For this work we build on results of [12], where pre-shape calculus is rigorously established. The results from [12] feature a structure theorem in the style of Hadamard's theorem for shape derivatives, which also shows the relationship of classical shape and pre-shape derivatives. In particular, shape calculus and pre-shape differentiability results carry over to the pre-shape setting.…”

“…In this work, we provide an approach to regularize general shape optimization problems to increase both shape and volume mesh quality. For this, we employ pre-shape calculus as established in [12]. Existence of regularized solutions is guaranteed.…”