Level Set Methods for Optimization Problems Involving Geometry and Constraints: I. Frequencies of a Two-Density Inhomogeneous Drum

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 devised in (Osher and Sethian, J. Comput. Phys.79, 12 (1988)), the variational level set calculus presented in (Zhao et al., J. Comput. Phys.127, 179 (1996)), and the projected gradient method, as in (Rudin et al., Physica D.60, 259 (1992)), 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. Our numerical results are quite promising. We expect to use this approach to deal with a wide class of optimal design problems in the future.