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

## TL;DR

This paper reviews various parameter-free shape optimization methods used in engineering, explaining auxiliary problems for shape updates, and compares their effectiveness through numerical examples in fluid dynamics.

## Contribution

It provides a formal explanation of different auxiliary problems for shape updates and compares their performance in engineering applications.

## Key findings

- Different auxiliary problems influence shape update effectiveness
- Numerical examples demonstrate practical differences in CFD applications
- Parameter-free methods are versatile for complex engineering problems

## 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.