BIBLIOS

Document type
Book chapters


Title
Optimization of eigenvalues and eigenmodes by using the adjoint method

Participants in the publication
Anca-Maria Toader (Author)
Dep. Matemática
CMAFcIO
Cristian Barbarosie (Author)
Dep. Matemática
CMAFcIO

Summary
Application of the adjoint method has proven successful in shape optimization and topology optimization. In the present chapter the adjoint method is applied to the optimization of eigenvalues and eigenmodes (eigenvectors). The general case of an arbitrary cost function depending on the first n eigenvalues and eigenmodes is detailed. The direct problem does not involve a bilinear form and a linear form as usual in other applications. However, it is possible to follow the spirit of the method and deduce n adjoint problems and obtain n adjoint states, where n is the number of eigenmodes taken into account for optimization. An optimization algorithm based on the derivative of the cost function is developed. This derivative depends on the derivatives of the eigenmodes and the adjoint method allows one to express it in terms of the the adjoint states and of the solutions of the direct eigenvalue problem. The formulas hold for the case when the eigenvalues are simple. A section is dedicated to discussions on the case when there are multiple eigenvalues. The same procedures are applied to optimization of microstructures, modeled by Bloch waves. The results obtained hold for general functionals depending on the eigenvalues and on the eigenmodes of the microstructure. However, the wave vector k? is a more delicate case of optimization parameter. The derivative of a general functional with respect to k? is obtained,which has interesting implications in band-gap maximization problems.

Editor(s)
Maïtine Bergounioux, Édouard Oudet, Martin Rumpf, Guillaume Carlier, Thierry Champion, Filippo Santambrogio


Institution
Johann Radon Institute for Computational and Applied Mathematics (RICAM)

Where published
Topological Optimization and Optimal Transport

Publication Identifiers
ISSN - 3110439263 / 9783110439267
ISBN - 9783110430417

Organizers
Édouard Oudet and Martin Rumpf

Publisher
Walter De Gruyter Inc

Collection
Radon Series on Computational and Applied Mathematics



Document Identifiers
URL - http://dx.doi.org/10.1515/9783110430417-006
DOI - https://doi.org/10.1515/9783110430417-006
ISBN - 9783110430417t

Rankings
SCOPUS (2017) -
CITESCORE (until 2019) (2017) -