Shape calculus for fitted and unfitted discretizations: domain transformations vs. boundary-face dilations

TL;DR

This paper introduces a boundary-face dilation-based shape calculus tailored for unfitted discretizations in shape optimization, providing boundary-supported derivatives suitable for fixed meshes and low-regularity functions.

Contribution

It presents a novel shape calculus based on localized boundary-face dilations, contrasting with traditional domain transformation methods, and applicable to unfitted mesh discretizations.

Findings

Boundary-supported derivatives derived from boundary-face dilations.

Applicable to low-regularity functions with element-wise smoothness.

Differentiation results differ from traditional domain transformation approaches.

Abstract

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.