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Discrete gradient flows for shape optimization and applications. (English) Zbl 1173.49307

Summary: We present a variational framework for shape optimization problems that establishes clear and explicit connections among the continuous formulation, its full discretization and the resulting linear algebraic systems. Our approach hinges on the following essential features: shape differential calculus, a semi-implicit time discretization and a finite element method for space discretization. We use shape differential calculus to express variations of bulk and surface energies with respect to domain changes. The semi-implicit time discretization allows us to track the domain boundary without an explicit parametrization, and has the flexibility to choose different descent directions by varying the scalar product used for the computation of normal velocity. We propose a Schur complement approach to solve the resulting linear systems efficiently. We discuss applications of this framework to image segmentation, optimal shape design for PDE, and surface diffusion, along with the choice of suitable scalar products in each case. We illustrate the method with several numerical experiments, some developing pinch-off and topological changes in finite time.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65K10 Numerical optimization and variational techniques
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[1] Almgren, F.; Taylor, J. E.; Wang, L., Curvature-driven flows: a variational approach, SIAM J. Control Optim., 31, 2, 387-438 (1993) · Zbl 0783.35002
[2] Ambrosio, L., Minimizing movements, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl., 19, 5, 191-246 (1995) · Zbl 0957.49029
[3] Ambrosio, L.; Gigli, N.; Savaré, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures. Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich (2005), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1090.35002
[4] Asaro, R. J.; Tiller, W. A., Surface morphology development during stress corrosion cracking: Part i: via surface diffusion, Metall. Trans., 3, 1789-1796 (1972)
[5] Aubert, G.; Kornprobst, P., Mathematical Problems in Image Processing. Mathematical Problems in Image Processing, Applied Mathematical Sciences, vol. 147 (2002), Springer-Verlag: Springer-Verlag New York, Partial differential equations and the calculus of variations, with a foreword by Olivier Faugeras · Zbl 1019.94002
[6] Bänsch, E.; Morin, P.; Nochetto, R. H., A nite element method for surface diffusion: the parametric case, J. Comput. Phys., 203, 1, 321-343 (2005) · Zbl 1070.65093
[7] Burger, M., A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces Free Bound., 5, 3, 301-329 (2003) · Zbl 1081.35134
[8] Cahn, J.; Taylor, J., Surface motion by surface diffusion, Acta Metall. Mater., 42, 1045-1063 (1994)
[9] Caselles, V.; Kimmel, R.; Sapiro, G., Geodesic active contours, IJCV, 22, 1, 61-79 (1997) · Zbl 0894.68131
[10] Céa, J., Numerical methods for optimal shape design, (Haug, E. J.; Céa, J., Proceedings of the NATO Advanced Study Institute on Optimization of Distributed Parameter Structural Systems (1981), Martinus Nijho Publishers) · Zbl 0173.44401
[11] Delfour, M. C.; Zolésio, J.-P., Shapes and Geometries. Shapes and Geometries, Advances in Design and Control (2001), Society for Industrial and Applied Mathematics (SIAM): Society for Industrial and Applied Mathematics (SIAM) Philadelphia, PA · Zbl 1002.49029
[12] G. Dogˇan, Topological changes and adaptivity for curve evolution, in preparation.; G. Dogˇan, Topological changes and adaptivity for curve evolution, in preparation.
[13] Dziuk, G., An algorithm for evolutionary surfaces, Numer. Math., 58, 6, 603-611 (1991) · Zbl 0714.65092
[14] Geometry Center at the University of Minnesota, Geomview, Interactive 3D Viewing Program. www.geomview.org; Geometry Center at the University of Minnesota, Geomview, Interactive 3D Viewing Program. www.geomview.org
[15] Henrot, A.; Pierre, M., Variation et Optimisation de Formes, Mathematiques et Applications, vol. 48 (2005), Springer · Zbl 1098.49001
[16] Henrot, A.; Villemin, G., An optimum design problem in magnetostatics, M2AN Math. Model. Numer. Anal., 36, 2, 223-239 (2002) · Zbl 1054.49030
[17] Hintermüller, M.; Ring, W., A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., 64, 2, 442-467 (2003/04) · Zbl 1073.68095
[18] Luckhaus, S., Solutions for the two-phase Stefan problem with the Gibbs-Thomson law for the melting temperature, Eur. J. Appl. Math., 1, 2, 101-111 (1990) · Zbl 0734.35159
[19] Mohammadi, B.; Pironneau, O., Applied Shape Optimization for Fluids. Applied Shape Optimization for Fluids, Numerical Mathematics and Scientific Computation (2001), The Clarendon Press, Oxford University Press, Oxford Science Publications: The Clarendon Press, Oxford University Press, Oxford Science Publications New York · Zbl 0970.76003
[20] Novruzi, A.; Roche, J. R., Newton’s method in shape optimisation: a three-dimensional case, BIT, 40, 1, 102-120 (2000) · Zbl 0956.65054
[21] Pironneau, O., Optimal Shape Design for Elliptic Systems. Optimal Shape Design for Elliptic Systems, Springer Series in Computational Physics (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0534.49001
[22] Schmidt, A.; Siebert, K., Design of Adaptive Finite Element Software. Design of Adaptive Finite Element Software, Lecture Notes in Computational Science and Engineering, vol. 42 (2005), Springer-Verlag: Springer-Verlag Berlin, The finite element toolbox ALBERTA, With 1 CD-ROM (Unix/Linux) · Zbl 1068.65138
[23] Shewchuk, J. R., Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator, (Lin, M.; Manocha, D., Applied Computational Geometry: Towards Geometric Engineering. Applied Computational Geometry: Towards Geometric Engineering, Lecture Notes in Computer Science, vol. 1148 (1996), Springer-Verlag)
[24] Sokołowski, J.; Zolésio, J.-P., Introduction to Shape Optimization. Introduction to Shape Optimization, Springer Series in Computational Mathematics, vol. 16 (1992), Springer-Verlag: Springer-Verlag Berlin, Shape sensitivity analysis · Zbl 0765.76070
[25] Spencer, B. J.; Davis, S. H.; Voorhees, P. W., Morphological instability in epitaxially-strained dislocation-free solid films: nonlinear evolution, Phys. Rev. B, 47, 2, 9760-9777 (1993)
[26] Zolésio, J.-P., The material derivative (or speed method for shape optimization), (Haug, E. J.; Céa, J., Proceedings of the NATO Advanced Study Institute on Optimization of Distributed Parameter Structural Systems (1981), Martinus Nijhoff Publishers) · Zbl 0517.73097
[27] Zunino, P., Multidimensional pharmacokinetic models applied to the design of drug eluting stents, J. Cardiovasc. Engrg., 4, 2 (2004)
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