  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.
