Finite Element Methods for Shape Optimization and Applications


Advanced Numerical Methods for Scientific Computing
MOX 72
Finite Element Methods for Shape Optimization and Applications
Monday 12th December 2005
Dogan, G.; Morin, P.; Nochetto, R.H., Verani, Marco
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We present a variational framework for shape optimization problems that establishes and clarifies 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 exibility 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 e_ciently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.