Adaptive finite element method for shape optimization^∗

¹ Departamento de Matemática, Facultad de Ingeniería Química and Instituto de Matemática Aplicada del Litoral, Universidad Nacional del Litoral, CONICET, Santa Fe, Argentina
pmorin@santafe-conicet.gov.ar; www.imal.santafe-conicet.gov.ar/pmorin
² Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, USA
rhn@math.umd.edu; www.math.umd.edu/˜rhn
³ Department of Mathematics and Institute for Applied Mathematics and Computational Science, Texas A&M University, College Station, 77843 TX, USA
pauletti@math.tamu.edu; www.math.tamu.edu/˜pauletti
⁴ MOX – Modelling and Scientific Computing – Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Milano, Italy
marco.verani@polimi.it; mox.polimi.it/˜verani

Received: 1 July 2011
Revised: 19 September 2011

Abstract

We examine shape optimization problems in the context of inexact sequential quadratic programming. Inexactness is a consequence of using adaptive finite element methods (AFEM) to approximate the state and adjoint equations (via the dual weighted residual method), update the boundary, and compute the geometric functional. We present a novel algorithm that equidistributes the errors due to shape optimization and discretization, thereby leading to coarse resolution in the early stages and fine resolution upon convergence, and thus optimizing the computational effort. 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 – a new paradigm in adaptivity.