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Adaptive finite element method for shape optimization∗

Published online by Cambridge University Press:  16 January 2012

Pedro Morin ,
Ricardo H. Nochetto ,
Miguel S. Pauletti  and
Marco Verani
Pedro Morin
Affiliation:
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
Ricardo H. Nochetto
Affiliation:
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
Miguel S. Pauletti
Affiliation:
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
Marco Verani
Affiliation:
MOX – Modelling and Scientific Computing – Dipartimento di Matematica “F. Brioschi”, Politecnico di Milano, Milano, Italy; marco.verani@polimi.it; mox.polimi.it/˜verani
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Abstract

We examine shape optimization problems in the context of inexact sequential quadraticprogramming. Inexactness is a consequence of using adaptive finite element methods (AFEM)to approximate the state and adjoint equations (via the dual weightedresidual method), update the boundary, and compute the geometric functional. We present anovel algorithm that equidistributes the errors due to shape optimization anddiscretization, thereby leading to coarse resolution in the early stages and fineresolution upon convergence, and thus optimizing the computational effort. We discuss theability of the algorithm to detect whether or not geometric singularities such as cornersare genuine to the problem or simply due to lack of resolution – a new paradigm inadaptivity.

Keywords

Shape optimizationadaptivitymesh refinement/coarseningsmoothing

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Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 1122 - 1149
Copyright
© EDP Sciences, SMAI, 2012

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