A Monotonic Algorithm for Eigenvalue Optimization in Shape Design Problems of Multi-Density Inhomogeneous Materials (Communications in Computational Physics, 2010)
Many problems in engineering shape design involve eigenvalue optimizations.The relevant difficulty is that the eigenvalues are not continuously differentiablewith respect to the density. In this paper, we are interested in the case of multi-densityinhomogeneous materials which minimizes the least eigenvalue. With the finite elementdiscretization, we propose a monotonically decreasing algorithm to solve theminimization problem. Some numerical examples are provided to illustrate the efficiency of the present algorithm as well as to demonstrate its availability for the caseof more than two densities. As the computations are sensitive to the choice of thediscretization mesh sizes, we adopt the refined mesh strategy, whose mesh grids are25-times of the amount used in [S. Osher and F. Santosa, J. Comput. Phys., 171 (2001),pp. 272-288]. We also show the significant reduction in computational cost with thefast convergence of this algorithm.