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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 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.

Keywords

Shape optimizationadaptivitymesh refinement/coarseningsmoothing
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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References

G. Allaire, Conception optimale de structures. Springer-Verlag, Berlin (2007).
Alotto, P., Girdinio, P., Molfino, P. and Nervi, M., Mesh adaption and optimization techniques in magnet design. IEEE Trans. Magn. 32 (1996) 2954–2957. Google Scholar
W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations, Birkhäuser (2003)
Banichuk, N.V., Falk, A. and Stein, E., Mesh refinement for shape optimization, Struct. Optim. 9 (1995) 46–51. Google Scholar
Bänsch, E., Morin, P. and Nochetto, R.H., Surface diffusion of graphs : variational formulation, error analysis and simulation. SIAM J. Numer. Anal. 42 (2004) 773–799. Google Scholar
Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1–102. Google Scholar
Bello, J.A., Fernandez-Cara, E., Lemoine, J. and Simon, J., The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow. SIAM J. Control Optim. 35 (1997) 626–640. Google Scholar
Bonito, A. and Nochetto, R.H. and Pauletti, M.S., Geometrically consistent mesh modification. SIAM J. Numer. Anal. 48 (2010) 1877–1899. Google Scholar
Burger, M., A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5 (2003) 301–329. Google Scholar
Céa, J., Conception optimale ou identification de formes : calcul rapide de la dérivée directionnelle de la fonction coût. RAIRO Modél. Math. Anal. Numér. 20 (1986) 371–402. Google Scholar
de Gournay, F., Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45 (2006) 343–367. Google Scholar
M.C. Delfour and J.-P. Zolésio, Shapes and Geometries. SIAM Advances in Design and Control 22 (2011).
Demlow, A., Higher-order finite element methods and pointwise error estimates for elliptic problems on surfaces. SIAM J. Numer. Anal. 47 (2009) 805–827. Google Scholar
Demlow, A. and Dziuk, G., An adptive finite element method for the Laplace-Beltrami operator on implicitly defined surfaces. SIAM J. Numer. Anal. 45 (2007) 421–442. Google Scholar
Dogan, G., Morin, P., Nochetto, R.H. and Verani, M.. Discrete gradient flows for shape optimization and applications. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3898–3914. Google Scholar
Giles, M. and Süli, E., Adjoint methods for PDEs : a posteriori error analysis and postprocessing by duality. Acta Numer. 11 (2002) 145–236. Google Scholar
M. Giles, M. Larson, J.M. Levenstam and E. Süli, Adaptive error control for finite element approximation of the lift and drag coefficients in viscous flow. Technical Report 1317 (1997) http://eprints.maths.ox.ac.uk/1317/.
V. Girault and P.A. Raviart, Finite Element Methods for Navier-Stokes Equations : Theory and Algorithms, Springer Series in Computational Mathematics 5. Springer-Verlag, Berlin (1986)
A. Henderson, ParaView Guide, A Parallel Visualization Application. Kitware Inc. (2007).
Lei, M., Archie, J.P. and Kleinstreuer, C., Computational design of a bypass graft that minimizes wall shear stress gradients in the region of the distal anastomosis. J. Vasc. Surg. 25 (1997) 637–646. Google ScholarPubMed
Mekchay, K., Morin, P., and Nochetto, R.H., AFEM for Laplace Beltrami operator on graphs : design and conditional contraction property. Math. Comp. 80 (2011) 625–648. Google Scholar
B. Mohammadi, O. Pironneau, Applied shape optimization for fluids. Oxford University Press, Oxford (2001).
M.S. Pauletti, Parametric AFEM for geometric evolution equations and coupled fluid-membrane interaction. Ph.D. thesis, University of Maryland, College Park, ProQuest LLC, Ann Arbor, MI (2008)
M.S. Pauletti, Second order method for surface restoration. Submitted.
Pironneau, O., On optimum profiles in Stokes flow. J. Fluid Mech. 59 (1973) 117–128. Google Scholar
Pironneau, O., On optimum design in fluid mechanics. J. Fluid Mech. 64 (1974) 97–110. Google Scholar
Quarteroni, A. and Rozza, G., Optimal control and shape optimization of aorto-coronaric bypass anastomoses. Math. Models Methods Appl. Sci. 13 (2003) 1801–1823. Google Scholar
G. Rozza, Shape design by optimal flow control and reduced basis techniques : applications to bypass configurations in haemodynamics. Ph.D. thesis, École Polytechnique Fédèrale de Lausanne (2005).
J.R. Roche, Adaptive method for shape optimization, 6th World Congresses of Structural and Multidisciplinary Optimization. Rio de Janeiro (2005).
Schleupen, A., Maute, K. and Ramm, E., Adaptive FE-procedures in shape optimization. Struct. Multidisc. Optim. 19 (2000) 282–302. Google Scholar
A. Schmidt and K.G. Siebert, Design of Adaptive Finite Element Software, The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering 42. Springer, Berlin (2005).
J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Springer-Verlag, Berlin (1992).
Taylor, R.S., Loh, A., McFarland, R.J., Cox, M. and Chester, J.F., Improved technique for polytetrafluoroethylene bypass grafting : long-term results using anastomotic vein patches. Br. J. Surg. 79 (1992) 348–354. Google ScholarPubMed