Abstract

Dedicated to Ivo Babuˇska on the occasion of his 80th birthday We present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE,

Citations