An adaptive finite element method for shape optimization problems

MORIN, PEDRO; NOCHETTO, RICARDO H.; PAULETTI, M. SEBASTIÁN; VERANI, MARCO

Santiago de Chile

Congreso; III Congreso Latinoamericano de Matemáticos; 2009

Unión Matemática de América Latina y el Caribe

    We examine shape optimization problems in the context of inexact se-quential quadratic programming. Inexactness is a consequence of usingadaptive finite element methods (AFEM) to approximate the state equa-tion, compute the geometric functional, and using polygons/polyhedra toapproximate the smooth underlying shapes. We present a novel algorithm that equidistributes the errors due to approximation of the shape functional, and discretization of the state equation and geomerty, thereby leading to coarse resolution in the early stages and fine resolution upon convergence. We discuss the ability of the algorithm to detect whether or not geometric singularities such as corners are genuine to the problem or simply due to lack of resolution. Numerical experiments that ilustrate the performance the adaptive algorithm are presented.