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A LEVEL SET BASED SHAPE OPTIMIZATION METHOD FOR AN ELLIPTIC OBSTACLE PROBLEM

Martin Burger

Institute for Computational and Applied Mathematics, University of Münster, Einsteinstr. 62, D-48149 Münster, Germany

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Norayr Matevosyan

DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

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, and

Marie-Therese Wolfram

DAMTP, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

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https://doi.org/10.1142/S0218202511005155Cited by:9 (Source: Crossref)

Abstract

In this paper, we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape.

Using the shape sensitivities, we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem.

Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments.

Keywords:

AMSC: 35R35, 49Q10, 65K10, 49Q12

References

R. A. Adams , Sobolev Spaces ( Academic Press , 1975 ) . Google Scholar
G. Allaire, F. Jouve and A. M. Toader, C. R. Acad. Sci. Paris, Ser. I 334, 1125 (2002). Crossref, Web of Science, Google Scholar
G. Barles, H. M. Soner and P. E. Souganidis, SIAM J. Control Optim. 31, 439 (1993). Crossref, Web of Science, Google Scholar
E. N. Barron and R. Jensen, Comm. PDE 15, 1713 (1990). Google Scholar
H. Ben Ameur, M. Burger and B. Hackl, Inverse Problems 20, 673 (2004). Crossref, Web of Science, Google Scholar
M. Burger, Inverse Problems 17, 1327 (2001). Crossref, Web of Science, Google Scholar
M. Burger, Interfaces Free Bound. 5, 301 (2003). Crossref, Web of Science, Google Scholar
M. Burger, Inverse Problems 20, 259 (2004). Crossref, Web of Science, Google Scholar
M. Burger and S. Osher, Eur. J. Appl. Math. 16, 263 (2005). Crossref, Web of Science, Google Scholar
L. A. Caffarelli, J. Fourier Anal. Appl. 4, 383 (1998). Crossref, Web of Science, Google Scholar
P. Cardaliaguet and O. Ley, Arch. Rational Mech. Anal. 183, 21 (2007). Crossref, Web of Science, Google Scholar
P. Cardaliaguet and O. Ley, Interfaces Free Bound. 10, 223 (2008). Crossref, Web of Science, Google Scholar
G. Q. Chen and B. Su, Chin. Ann. Math. 21B, 165 (2000). Link, Web of Science, Google Scholar
M. Dambrine and M. Pierre, M2AN Math. Model. Numer. Anal. 34, 811 (2000). Crossref, Web of Science, Google Scholar
M. C. Delfour and J. P. Zolésio , Shapes and Geometries. Analysis, Differential Calculus, and Optimization ( SIAM , 2001 ) . Google Scholar
H. Egger and J. Schöberl, A hybrid mixed discontinuous Galerkin method for convection-diffusion problems, IMA J. Numer. Anal., to appear . Google Scholar
C. M. Elliott and H. Garcke, Adv. Math. Sci. Appl. 7, 465 (1997). Web of Science, Google Scholar
L. C. Evans , Partial Differential Equations , Graduate Studies in Mathematics 19 ( Amer. Math. Soc. , 1998 ) . Google Scholar
L. C. Evans and J. Spruck, J. Differ. Geom. 33, 635 (1991). Crossref, Web of Science, Google Scholar
F. Hettlich and W. Rundell, Inverse Problems 12, 251 (1996). Crossref, Web of Science, Google Scholar
M. Hintermüller and W. Ring, SIAM J. Appl. Math. 64, 442 (2003). Crossref, Web of Science, Google Scholar
K. Ito and K. Kunisch , Lagrange Multiplier Approach to Variational Problems and Applications ( SIAM , 2008 ) . Crossref, Google Scholar
G. Barles, P. Cardaliaguet, O. Ley and A. Monteillet, Uniqueness results for nonlocal Hamilton–Jacobi equations, J. Funct. Anal., to appear . Google Scholar
G. Barles, P. Cardaliaguet, O. Ley and A. Monteillet, Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations, Nonlinear Anal. TMA, to appear . Google Scholar
O. Ley, Evolution de fronts avec vitesse non-locale et equations de Hamilton–Jacobi, Habilitation Diriger des Recherches, Universite de Tours, 2008 . Google Scholar
P. L. Lions , Generalized Solutions of Hamilton–Jacobi Equations ( Pitman , 1982 ) . Google Scholar
N. Matevosyan, Comm. Partial Differential Equations 30, 1205 (2005). Crossref, Web of Science, Google Scholar
A. C. G. Mennucci , Level-Set and PDE-Based Reconstruction Methods , eds. M. Burger and S. Osher ( Springer , 2009 ) . Google Scholar
C. B. Morrey Jr. , Multiple Integrals in the Calculus of Variations ( Springer , 1966 ) . Crossref, Google Scholar
S. J. Osher and R. P. Fedkiw , The Level Set Method and Dynamic Implicit Surfaces ( Springer , 2002 ) . Google Scholar
S. Osher and F. Santosa, J. Comput. Phys. 171, 272 (2001). Crossref, Web of Science, Google Scholar
S. J. Osher and J. A. Sethian, J. Comput. Phys. 79, 12 (1988). Crossref, Web of Science, Google Scholar
F. Santosa, ESAIM: Control, Optim. Calculus of Variations 1, 17 (1996). Crossref, Google Scholar
J. Simon, Control and Estimation of Distributed Parameter Systems (Vorau, 1988), Internat. Ser. Numer. Math. 91 (Birkhäuser, 1989) pp. 361–378. Google Scholar
J. Sokolowski and J. P. Zolesio , Introduction to Shape Optimization ( Springer , 1992 ) . Crossref, Google Scholar
M. Y. Wang, X. M. Wang and D. M. Guo, Comp. Meth. Appl. Mech. Eng. 192, 227 (2003). Crossref, Web of Science, Google Scholar
J. P. Zolesio, Weak set evolution and variational applications, Shape Optimization and Optimal Design. Proc. of the IFIP Conference, eds. J. Cagnolet al. (Marcel Dekker, 2001) pp. 415–439. Google Scholar