Commun. Comput. Phys., 8 (2010), pp. 565-584.


A Monotonic Algorithm for Eigenvalue Optimization in Shape Design Problems of Multi-Density Inhomogeneous Materials

Zheng-fang Zhang 1, Ke-wei Liang 1*, Xiao-liang Cheng 1

1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China.

Received 19 March 2009; Accepted (in revised version) 20 October 2009
Available online 15 April 2010
doi:10.4208/cicp.190309.201009a

Abstract

Many problems in engineering shape design involve eigenvalue optimizations. The relevant difficulty is that the eigenvalues are not continuously differentiable with respect to the density. In this paper, we are interested in the case of multi-density inhomogeneous materials which minimizes the least eigenvalue. With the finite element discretization, we propose a monotonically decreasing algorithm to solve the minimization problem. Some numerical examples are provided to illustrate the efficiency of the present algorithm as well as to demonstrate its availability for the case of more than two densities. As the computations are sensitive to the choice of the discretization mesh sizes, we adopt the refined mesh strategy, whose mesh grids are 25-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 the fast convergence of this algorithm.

AMS subject classifications: 35P15, 65K10, 74S05

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Key words: Multi-density, inhomogeneous materials, the least eigenvalue, optimization problem, finite element method, monotonic algorithm.

*Corresponding author.
Email: sophiazzf@126.com (Z. Zhang), matlkw@zju.edu.cn (K. Liang), xiaoliangcheng@zju.edu.cn (X. Cheng)
 

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