# A continuous perspective on modeling of shape optimal design problems

arXiv: https://arxiv.org/abs/2004.06942 · HTML: https://mlsanity.com/paper/2004.06942/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Apr 2020.

## Built on by (most cited first, 1 of 1)

- [Mesh motion in fluid-structure interaction with deep operator networks](https://mlsanity.com/paper/2402.00774/) (2024)

## Abstract (arXiv)

In this article we consider shape optimization problems as optimal control problems via the method of mappings. Instead of optimizing over a set of admissible shapes a reference domain is introduced and it is optimized over a set of admissible transformations. The focus is on the choice of the set of transformations, which we motivate from a function space perspective. In order to guarantee local injectivity of the admissible transformations we enrich the optimization problem by a nonlinear constraint. The approach requires no parameter tuning for the extension equation and can naturally be combined with geometric constraints on volume and barycenter of the shape. Numerical results for drag minimization of Stokes flow are presented.