Issue |
ESAIM: COCV
Volume 18, Number 4, October-December 2012
|
|
---|---|---|
Page(s) | 1122 - 1149 | |
DOI | https://doi.org/10.1051/cocv/2011192 | |
Published online | 16 January 2012 |
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
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.
Mathematics Subject Classification: 49M25 / 65M60
Key words: Shape optimization / adaptivity / mesh refinement/coarsening / smoothing
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”.
© EDP Sciences, SMAI, 2012
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