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Multiplicity of the maximum eigenvalue in structural optimization problems

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Abstract

Many problems in structural optimization can be formulated as a minimization of the maximum eigenvalue of a symmetric matrix. In practise it is often observed that the maximum eigenvalue has multiplicity greater than one close to or at optimal solutions. In this note we give a sufficient condition for this to happen at extreme points in the optimal solution set. If, as in topology optimization, each design variable determines the amount of material in a finite element in the design domain then this condition essentially amounts to saying that the number of elements containing material at a solution must be greater than the order of the matrix.

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Notes

  1. To see that all of these problems can be formulated as a minimization of the maximum eigenvalue, note that λ 1(A) = −λ n (−A), where λ 1 and λ n is the largest and, respectively, smallest eigenvalue, of the symmetric matrix A, and that, for a positive definite A, if λ 1 is the maximum eigenvalue for the generalized eigenvalue problem (B−λ A)u = 0, then it plays the same role in the eigenvalue problem (A −1/2 B A −1/2−λ I)u = 0.

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Acknowledgments

Thanks to Prof. Anders Klarbring for valuable discussions and the anonymous reviewers for their comments. This research was supported in part by the Swedish Foundation for Strategic Research Grant No. AM13-0029.

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  1. Division of Solid Mechanics, Department of Management and Engineering, Linköping University, SE 581 83, Linköping, Sweden

    Carl-Johan Thore

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  1. Carl-Johan Thore
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Correspondence to Carl-Johan Thore.

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Thore, CJ. Multiplicity of the maximum eigenvalue in structural optimization problems. Struct Multidisc Optim 53, 961–965 (2016). https://doi.org/10.1007/s00158-015-1380-3

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