A level-set method for shape optimization. (English. Abridged French version) Zbl 1115.49306

Summary: We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh.


49Q10 Optimization of shapes other than minimal surfaces
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
74P10 Optimization of other properties in solid mechanics
Full Text: DOI


[1] Allaire, G., Shape Optimization by the Homogenization Method (2001), Springer-Verlag: Springer-Verlag New York
[2] Allaire, G.; Bonnetier, E.; Francfort, G.; Jouve, F., Shape optimization by the homogenization method, Numer. Math., 76, 27-68 (1997) · Zbl 0889.73051
[3] Allaire, G.; Kohn, R. V., Optimal design for minimum weight and compliance in plane stress using extremal microstructures, European J. Mech. A Solids, 12, 6, 839-878 (1993) · Zbl 0794.73044
[4] Bendsoe, M., Methods for Optimization of Structural Topology, Shape and Material (1995), Springer-Verlag: Springer-Verlag New York · Zbl 0822.73001
[5] Bendsoe, M.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71, 197-224 (1988) · Zbl 0671.73065
[6] A. Chambolle, A density result in two-dimensional linearized elasticity and applications, Preprint CEREMADE 121, Université Paris-Dauphine, 2001; A. Chambolle, A density result in two-dimensional linearized elasticity and applications, Preprint CEREMADE 121, Université Paris-Dauphine, 2001 · Zbl 1030.74007
[7] Chenais, D., On the existence of a solution in a domain identification problem, J. Math. Anal. Appl., 52, 189-289 (1975) · Zbl 0317.49005
[8] Cherkaev, A., Variational Methods for Structural Optimization (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0956.74001
[9] Murat, F.; Simon, S., Études de problèmes d’optimal design, (Lecture Notes in Comput. Sci., 41 (1976), Springer-Verlag: Springer-Verlag Berlin), 54-62 · Zbl 0334.49013
[10] Osher, S.; Santosa, F., Level set methods for optimization problems involving geometry and constraints: frequencies of a two-density inhomogeneous drum, J. Comput. Phys., 171, 272-288 (2001) · Zbl 1056.74061
[11] Osher, S.; Sethian, J. A., Front propagating with curvature dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 78, 12-49 (1988) · Zbl 0659.65132
[12] Pironneau, O., Optimal Shape Design for Elliptic Systems (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0496.93029
[13] Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Science (1999) · Zbl 0973.76003
[14] Sethian, J.; Wiegmann, A., Structural boundary design via level set and immersed interface methods, J. Comput. Phys., 163, 489-528 (2000) · Zbl 0994.74082
[15] Simon, J., Differentiation with respect to the domain in boundary value problems, Numer. Funct. Anal. Optimz., 2, 649-687 (1980) · Zbl 0471.35077
[16] Sokolowski, J.; Zolesio, J. P., Introduction to Shape Optimization: Shape Sensitity Analysis. Introduction to Shape Optimization: Shape Sensitity Analysis, Springer Ser. Comput. Math., 10 (1992), Springer: Springer Berlin · Zbl 0761.73003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.