Greedy algorithms for eigenvalue optimization problems in shape design of two-density inhomogeneous materials

Abstract

This paper studies the eigenvalue optimization problems in the shape design of the two-density inhomogeneous materials. Two types of greedy algorithms are proposed to solve three optimization problems in finite element discretization. In the first type, the whole domain is initialized by one density. For each problem of the eigenvalue optimizations, we define a measurement of the element, which is the criterion to determine the ‘best’ element. We change the density of the ‘best’ element to the other density. Then the algorithm repeats the procedure until the area constraint is satisfied. In the second type, the algorithm begins with the density distribution satisfying the area constraint. Also, according to the measurement of the element, the algorithm finds a pair of the ‘best’ elements and exchanges their densities between each other. Furthermore, the accelerating greedy algorithms are proposed to speed up both two types. Three numerical examples are provided to illustrate the results.

Keywords:

• greedy algorithm
• inhomogeneous drum
• the eigenvalue optimization problem
• finite element method
• measurement

2000 AMS Subject Classifications :

• 35P15
• 65K10
• 74S05

Acknowledgements

This research is supported by the National Science Foundation of China under Grant No. 10871179, National Basic Research Programme of China under Grant No. 2008CB717806 and the Research Fund for the Doctoral Program of Higher Education of China (No. 2007R10012).