Incorporating topological derivatives into level set methods

Journal of Computational Physics

Volumn 194, Issue 1, 2004, Pages 344-362

  Burger, Martin ^a   Hackl, Benjamin ^b   Ring, Wolfgang ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b Numerical Symbolic Sci Computing   (Austria)

^c UNIVERSITY OF GRAZ   (Austria)

Author keywords

Hamilton Jacobi equations; Level set method; Shape derivative; Shape reconstruction; Topological derivative; Topology optimization

Indexed keywords

DROP BREAKUP; HYDROGELS; ITERATIVE METHODS; LEVEL MEASUREMENT; NUMERICAL METHODS; TOPOLOGY;

HAMILTON - JACOBI EQUATIONS; LEVEL SET METHOD; MODEL PROBLEMS; NEW APPROACHES; RECONSTRUCTION PROBLEMS; SHAPE DERIVATIVES; SHAPE RECONSTRUCTION; SHAPE-OPTIMIZATION; TOPOLOGICAL DERIVATIVES; TOPOLOGY OPTIMISATION;

SHAPE OPTIMIZATION;

EID: 0842312288     PISSN: 00219991     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jcp.2003.09.033     Document Type: Article

References (31)

1. R.A. Adams, Sobolev Spaces, Academic Press, New-York, 1975.
2. G. Allaire, Shape optimization by the homogenization method, Springer-Verlag, New York, 2002.
3. G. Allaire, F. Jouve, A.M. Toader, A level-set method for shape optimization, C.R. Acad. Sci. Paris Ser. I 334 (2002) 1125-1130.
4. G. Allaire, F. Jouve, A.M. Toader, Structural optimization using sensitivity analysis and a level-set method, Preprint (CMAP École Polytechnique, Paris, 2003).
5. H. Benameur, M. Burger, B. Hackl, Level set methods for geometric inverse problems in linear elasticity, Preprint, 2003.
6. M.P. Bendsøe, O. Sigmund, Topology Optimization, Springer, Berlin, 2002.
7. B. Bourdin, A. Chambolle, Design-dependent loads in topology optimization, ESAIM Control Optim. Calc. Var. 9 (2003) 19-48.
8. M. Burger, A level set method for inverse problems, Inverse Problems 17 (2001) 1327-1356.
9. M. Burger, A framework for the construction of level set methods for shape optimization and reconstruction, Interfaces and Free Boundaries 5 (2003) 301-329.
10. M.C. Delfour, J.P. Zolésio, Shapes and Geometries. Analysis, Differential Calculus, and Optimization, SIAM, Philadelphia, 2001.
11. O. Dorn, E.M. Miller, C.M. Rappaport, A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets, Inverse Problems 16 (2000) 1119-1156.
12. O. Dorn, Shape reconstruction in 2D from limited-view multifrequency electromagnetic data, Preprint, 2000.
13. S. Garreau, P. Guillaume, M. Masmoudi, The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim. 39 (2001) 1756-1778.
14. M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser, Basel, 1993.
15. P. Guillaume, K. Sid Idris, The topological asymptotic expansion for the dirichlet problem, SIAM J. Control Optim. 41 (2002) 1042-1072.
16. F. Hettlich, W. Rundell, Iterative methods for the reconstruction of an inverse potential problem, Inverse Problems 12 (1996) 251-266.
17. M. Hintermüller, W. Ring, A second order shape optimization approach for image segmentation, SIAM J. Appl. Math., in press.
18. M. Hintermüller, W. Ring, An inexact Newton-CG-type active contour approach for the minimization of the Mumford-Shah functional, J. Math. Imag. Vision 20 (2003) in press.
19. K. Ito, K. Kunisch, Z. Li, Level-set function approach to an inverse interface problem, Inverse Problems 17 (2001) 1225-1242.
20. G.S. Jiang, D. Peng, Weighted ENO-schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 21 (2000) 2126-2143.
21. A. Litman, D. Lesselier, F. Santosa, Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set, Inverse Problems 14 (1998) 685-706.
22. S. Osher, F. Santosa, Level set methods for optimization problems involving geometry and constraints. I. Frequencies of a two-density inhomogeneous drum, J. Comput. Phys. 171 (2001) 272-288.
23. S. Osher, J.A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys. 79 (1988) 12-49.
24. S. Osher, R.P. Fedkiw, The Level Set Method and Dynamic Implicit Surfaces, Springer, New York, 2002.
25. F. Santosa, A level-set approach for inverse problems involving obstacles, ESAIM: Control, Optimisation Calculus Variations 1 (1996) 17-33.
26. A. Schumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpositionierungskriterien, PhD Thesis, Universität-Gesamthochschule-Siegen, 1995.
27. J. Sokolowski, A. Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim. 37 (1999) 1251-1272.
28. J. Sokolowski, A. Zochowski, Topological derivatives for elliptic problems, Inverse Problems 15 (1999) 123-134.
29. J. Sokolowski, J.P. Zolesio, Introduction to Shape Optimization, Springer, Berlin, 1992.
30. M. Stolarska, D.L. Chopp, N. Moes, T. Belytschko, Modelling crack growth by level sets in the extended finite element method, Int. J. Numer. Meth. Eng. 51 (2001) 943-960.
31. J.P. Zolesio, The material derivative (or speed) method for shape optimization, in: Optimization of Distributed Parameter Structures, vol. II, NATO Adv. Study Inst. Ser. E, Appl. Sci. 50 (1981) 1089-1151.