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Topological derivatives for semilinear elliptic equations

Iguernane M., Nazarov S. A., Roche J. R., Sokolowski J., Szulc K.

International Journal of Applied Mathematics and Computer Science

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2009

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Vol. 19, no 2

191-205

EN

The form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in the L [...] norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

2

Shape sensitivity analysis of eigenvalues revisited

Nazarov S. A., Sokolowski J.

Control and Cybernetics

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2008

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Vol. 37, no 4

999-1012

EN

The paper can be considered as a complement to previous papers of the authors. An insight into applied asymptotic analysis of boundary value problems in singularly perturbed domains is presented. As a result, the asymptotic expansions of eigenvalues are obtained and discussed in terms of integral attributes of the geometrical perturbations including the virtual mass tensor, polarization tensor etc. The results are presented in such a way that can be easily employed in numerical methods for shape optimization and inverse problems.

3

Self-adjoint extensions of differential operators and exterior topological derivatives in shape optimization

Nazarov S. A., Sokołowski J.

Control and Cybernetics

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2005

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Vol. 34, no 3

903-925

EN

Self-adjoint extensions are constructed for a family of boundary value problems in domains with a thin ligament and an asymptotic analysis of a Lq-continuous functional is performed. The results can be used in numerical methods of shape and topology optimization of integral functionals for elliptic equations. At some stage of optimization process the singular perturbation of geometrical domain by an addition of thin ligament can be replaced by its approximation denned for the appropriate self-adjoint extension of the elliptic operator. In this way the topology variation of current geometrical domain can be determined and used e.g., in the level-set type methods of shape optimization.

4

Selfadjoint extensions for the elasticity system in shape optimization

Nazarov S. A., Sokołowski J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2004

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Vol. 52, nr 3

237-248

EN

Two approaches are proposed to modelling of topological variations in elastic solids. The first approach is based on the theory of selfadjoint extensions of differential operators. In the second approach function spaces with separated asymptotics and point asymptotic conditions are introduced, and a variational formulation is established. For both approaches, accuracy estimates are derived.