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.

翻译：过去十年中，无参数方法在形状优化问题领域已发展至成熟阶段，为复杂工程应用提供了通用工具。然而，在此类梯度优化策略中，从形状导数获得的灵敏度分布无法直接用作形状更新量，而是需要求解辅助问题以从灵敏度中导出梯度。尽管这些辅助问题的多种选择已从数学角度得到深入研究，但其推导背后的概念复杂性往往阻碍了其在工程中的应用。本文旨在从工程视角阐释多种计算形状更新的方法。我们以形式化方式引入相应的辅助问题，并通过数值算例对比各方案的优劣。为此，我们以计算流体动力学中的测试案例与典型工程应用作为验证对象。