IMAL   13325
INSTITUTO DE MATEMATICA APLICADA DEL LITORAL "DRA. ELEONOR HARBOURE"
Unidad Ejecutora - UE
artículos
Título:
Discrete Gradient Flows for Shape Optimization and Applications
Autor/es:
G. DOGAN; PEDRO MORIN; R.H. NOCHETTO; M. VERANI.
Revista:
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Editorial:
ELSEVIER
Referencias:
Lugar: Amsterdam, The Netherlands; Año: 2007 vol. 196 p. 3898 - 3914
ISSN:
0045-7825
Resumen:
Abstract: 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, 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. Keywords: Shape optimization, scalar product, gradient flow, semi-implicit discretization, finite elements, surface diffusion, image segmentation. AMS Subject Classifications: 49Q10, 65M60, 65K10