Shape calculus concerns the calculation of directional derivatives of some quantity of interest, typically expressed as an integral. This article introduces a type of shape calculus based on localized dilation of boundary faces through perturbations of a level-set function. The calculus is tailored for shape optimization problems where a partial differential equation is numerically solved using a fictitious-domain method. That is, the boundary of a domain is allowed to cut arbitrarily through a computational mesh, which is held fixed throughout the computations. Directional derivatives of a volume or surface integral using the new shape calculus yields purely boundary-supported expressions, and the involved integrands are only required to be element-wise smooth. However, due to this low regularity, only one-sided differentiability can be guaranteed in general. The dilation concept introduced here differs from the standard approach to shape calculus, which is based on domain transformations. The use of domain transformations is closely linked the the use of traditional body-fitted discretization approaches, where the computational mesh is deformed to conform to the changing domain shape. The directional derivatives coming out of a shape calculus using deforming meshes under domain transformations are different then the ones from the boundary-dilation approach using fixed meshes; the former are not purely boundary supported but contain information also from the interior.

翻译：形状演算涉及对感兴趣量的方向导数进行计算，该量通常表示为积分形式。本文介绍了一种基于通过水平集函数扰动实现边界面局部膨胀的形状演算类型。该演算专门针对使用虚拟域方法数值求解偏微分方程的形状优化问题设计，即域边界可任意切割计算网格，且该网格在整个计算过程中保持固定。使用新型形状演算计算体积或表面积分的方向导数时，可得到纯边界支持的表达式，且涉及的被积函数仅需满足单元光滑性。然而，由于这种低正则性，通常只能保证单侧可微性。本文引入的膨胀概念不同于基于域变换的标准形状演算方法。域变换的使用与传统体适配离散化方法密切相关，其中计算网格发生变形以适应不断变化的域形状。在域变换下使用变形网格进行形状演算所得的方向导数，与在固定网格下使用边界膨胀方法所得的方向导数不同：前者并非纯边界支持，还包含来自内部的信息。