Binary level set methods for topology and shape optimization of a two-density inhomogeneous drum

We optimize eigenvalues in optimal shape design using binary level set methods. The interfaces of subregions are represented implicitly by the discontinuities of binary level set functions taking two values 1 or -1 at convergence. A binary constraint is added to the original model problems. We propose two variational algorithms to solve the constrained optimization problems. One is a hybrid type by coupling the Lagrange multiplier approach for the geometry constraint with the augmented Lagrangian method for the binary constraint. The other is devised using the Lagrange multiplier method for both constraints. The two iterative algorithms are both largely independent of the initial guess and can satisfy the geometry constraint very accurately during the iterations. Intensive numerical results are presented and compared with those obtained by level set methods, which demonstrate the effectiveness and robustness of our algorithms.