Pre-shape calculus and its application to mesh quality optimization

Deformations of the computational mesh, arising from optimization routines, usually lead to decrease of mesh quality or even destruction of the mesh. We propose a theoretical framework using pre-shapes to generalize the classical shape optimization and calculus. We define pre-shape derivatives and derive corresponding structure and calculus theorems. In particular, tangential directions are featured in pre-shape derivatives, in contrast to classical shape derivatives, featuring only normal directions. Techniques from classical shape optimization and calculus are shown to carry over to this framework. An optimization problem class for mesh quality is introduced, which is solvable with the use of pre-shape derivatives. This class allows for simultaneous optimization of the classical shape objectives and mesh quality without deteriorating the classical shape optimization solution. The new techniques are implemented and numerically tested for 2D and 3D.