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

International Journal of Computer Mathematics (Impact Factor: 0.82). 01/2011; 88(1):183-195. DOI: 10.1080/00207160903365891
Source: DBLP


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.

10 Reads
  • [Show abstract] [Hide abstract]
    ABSTRACT: This paper investigates optimization of the least eigenvalue of −Δ with the constraint of one-dimension Hausdorff measure of Dirichlet boundary. We propose the boundary piecewise constant level set (BPCLS) method based on the regularity technique to combine two types of boundary conditions into a single Robin boundary condition. We derive the first variation of the least eigenvalue w.r.t. the BPCLS function and propose a penalty BPCLS algorithm and an augmented Lagrangian BPCLS algorithm. Numerical results are reported for experiments on ellipse and L-shape domains.
    Journal of Computational Physics 01/2011; 230(2):458-473. DOI:10.1016/ · 2.43 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: Designing an acoustic drum can be categorized into a class of eigenvalue optimization problems in the structural engineering area. In this paper, we propose an algorithm that is based on the Gâteaux derivative of the objective function with respect to the density functions and analyze our algorithm in detail. In the algorithm, we deal with the geometry constraint by exchanging the densities of two domains occupied by two kinds of different materials. Finally we apply this algorithm to some practical examples frequently used by the researchers and present some numerical results to show its feasibility, stability and efficiency.
    Computers & Mathematics with Applications 08/2013; 66(1):62–80. DOI:10.1016/j.camwa.2013.04.012 · 1.70 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We propose two algorithms for elliptic boundary value problems in shape optimization. With the finite element method, the optimization problem is replaced by a discrete variational problem. We give rules and use them to decide which elements are to be reserved. Those rules are determined by the optimization; as a result, we get the optimal design in shape. Numerical examples are provided to show the effectiveness of our algorithms.
    Mathematical Problems in Engineering 01/2015; 2015(29). DOI:10.1155/2015/241837 · 0.76 Impact Factor