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Shape calculus concerns the calculation of directional derivatives of somequantity of interest, typically expressed as an integral. This articleintroduces a type of shape calculus based on localized dilation of boundaryfaces through perturbations of a level-set function. The calculus is tailoredfor shape optimization problems where a partial differential equation isnumerically solved using a fictitious-domain method. That is, the boundary of adomain is allowed to cut arbitrarily through a computational mesh, which isheld fixed throughout the computations. Directional derivatives of a volume orsurface integral using the new shape calculus yields purely boundary-supportedexpressions, and the involved integrands are only required to be element-wisesmooth. However, due to this low regularity, only one-sided differentiabilitycan be guaranteed in general. The dilation concept introduced here differs fromthe standard approach to shape calculus, which is based on domaintransformations. The use of domain transformations is closely linked the theuse of traditional body-fitted discretization approaches, where thecomputational mesh is deformed to conform to the changing domain shape. Thedirectional derivatives coming out of a shape calculus using deforming meshesunder domain transformations are different then the ones from theboundary-dilation approach using fixed meshes; the former are not purelyboundary supported but contain information also from the interior.

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