Research Article

Adaptive finite element method for shape optimization ^∗

Pedro Morin^a1, Ricardo H. Nochetto^a2, Miguel S. Pauletti^a3 and Marco Verani^a4

^a1 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

^a2 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

^a3 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

^a4 MOX – Modelling and Scientific Computing – Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Milano, Italy; marco.verani@polimi.it; mox.polimi.it/˜verani

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.

(Received July 1 2011)

(Revised September 19 2011)

(Online publication January 16 2012)

Key Words:

• Shape optimization;
• adaptivity;
• mesh refinement/coarsening;
• smoothing

Mathematics Subject Classification:

• 49M25;
• 65M60

Footnotes

^∗  Partially supported by UNL through GRANT CAI+D 062-312, by CONICET through Grant PIP 112-200801-02182, by MinCyT of Argentina through Grant PICT 2008-0622 and by Argentina-Italy bilateral project “Innovative numerical methods for industrial problems with complex and mobile geometries”. Partially supported by NSF grants DMS-0505454 and DMS-0807811. Partially supported by NSF grants DMS-0505454 and DMS-0807811, and by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). Partially supported by Italian MIUR PRIN 2008 “Analisi e sviluppo di metodi numerici avanzati per EDP” and by Argentina-Italy bilateral project “Innovative numerical methods for industrial problems with complex and mobile geometries”.