  Computational meshes arising from shape optimization routines commonly suffer
from decrease of mesh quality or even destruction of the mesh. In this work, we
provide an approach to regularize general shape optimization problems to
increase both shape and volume mesh quality. For this, we employ pre-shape
calculus (cf. arXiv:2012.09124). Existence of regularized solutions is
guaranteed. Further, consistency of modified pre-shape gradient systems is
established. We present pre-shape gradient system modifications, which permit
simultaneous shape optimization with mesh quality improvement. Optimal shapes
to the original problem are left invariant under regularization. The
computational burden of our approach is limited, since additional solution of
possibly larger (non-)linear systems for regularized shape gradients is not
necessary. We implement and compare pre-shape gradient regularization
approaches for a hard to solve 2D problem. As our approach does not depend on
the choice of metrics representing shape gradients, we employ and compare
several different metrics.
