Level set methods for optimization problems involving geometry and constraints. I: Frequencies of a two-density inhomogeneous drum. (English) Zbl 1056.74061

From the summary: Many problems in engineering design involve optimizing the geometry to maximize a certain design objective. Geometrical constraints are often imposed. In this paper, we use the level set method, the variational level set calculus and the projected gradient method to construct a simple numerical approach for problems of this type. We apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry.


74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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[1] Adalsteinsson, D.; Sethian, J., A fast level set method for propagating interfaces, J. Comput. Phys., 118, 269 (1995) · Zbl 0823.65137
[2] G. Allaire, The homogenization method for topology and shape optimization, in, Topology Optimization in Structural Mechanics, edited by, Rozvany, CISM, 1997.; G. Allaire, The homogenization method for topology and shape optimization, in, Topology Optimization in Structural Mechanics, edited by, Rozvany, CISM, 1997. · Zbl 0885.73049
[3] Bendsoe, M.; Mota Soares, C., Topology Design of Structures (1993) · Zbl 0789.73004
[4] Bendsoe, M., Optimization of Structural Topology, Shape and Material (1997)
[5] Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. Comput. Phys., 135, 8 (1997) · Zbl 0889.65133
[6] Cox, S., The two phase drum with the deepest bass note, Japan J. Indust. Appl. Math., 8, 345 (1991) · Zbl 0755.35029
[7] Cox, S., Generalized gradient at a multiple eigenvalue, J. Func. Anal., 130, 30 (1995) · Zbl 0884.47008
[8] Cox, S.; Dobson, D., Band structure optimization of two-dimensional photonic crystals in H-polarization, J. Comput. Phys., 158, 214 (2000) · Zbl 0949.65115
[9] Cox, S.; McLaughlin, J., Extremal eigenvalue problems for composite membranes, I and II, Appl. Math. Optimizat., 22, 153 and 169 (1990) · Zbl 0709.73045
[10] Garabedian, P., Partial Differential Equations (1964) · Zbl 0124.30501
[11] Gilbarg, D.; Trudinger, N., Elliptic Partial Differential Equations of Second Order (1997) · Zbl 0691.35001
[12] Jiang, G.; Peng, D., Weighted ENO schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput., 21, 2126 (2000) · Zbl 0957.35014
[13] Osher, S.; Sethian, J., Front propagation with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79, 12 (1988) · Zbl 0659.65132
[14] Osher, S.; Shu, C., High order essentially nonoscillatory schemes for Hamilton-Jacobi equations, SIAM J. Numer. Anal., 28, 907 (1991) · Zbl 0736.65066
[15] Peng, D.; Merriman, B.; Osher, S.; Zhao, H.; Kang, M., A PDE-based fast local level set method, J. Comput. Phys., 155, 410 (1999) · Zbl 0964.76069
[16] Pironeau, O., Optimal Shape Design for Elliptic Systems (1984) · Zbl 0534.49001
[17] Rozvany, I., Topology Optimization in Structural Mechanics (1997) · Zbl 0873.00017
[18] Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259 (1992) · Zbl 0780.49028
[19] Santosa, F., A level-set approach for inverse problems involving obstacles, Control, Optimizat. Calculus Variat., 1, 17 (1996) · Zbl 0870.49016
[20] Sethian, J.; Wiegmann, A., Structural boundary design via level set and immersed interface methods, J. Comput. Phys., 163, 489 (2000) · Zbl 0994.74082
[21] Sokolowski, J.; Zolesio, J.-P., Introduction to Shape Optimization. Shape Sensitivity Analysis (1992) · Zbl 0761.73003
[22] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. Comput. Phys., 114, 146 (1994) · Zbl 0808.76077
[23] Zhao, H.; Chan, T.; Merriman, B.; Osher, S., A variational level set approach to multiphase motion, J. Comput. Phys., 127, 179 (1996) · Zbl 0860.65050
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