Asymptotic Analysis and Polarization Matrices

Polarization matrices are considered for the elasticity boundary value problems in two and three spatial dimensions. The matrices are introduced in the framework of asymptotic analysis for boundary value problems depending on small geometrical parameter, it is the size of an elastic inclusion or a defect (cavity, crack) in an elastic body. Our analysis is performed for some representative classes of boundary value problems, however the method is general and can be applied to the modelling and optimization in structural mechanics or for coupled models like piezoelectricity. The explicit properties obtained for polarization matrices are useful for mathematical analysis and for numerical solution of control, inverse and shape optimization problems with mathematical models derived by the asymptotic analysis in singularly perturbed geometrical domains. The analysis is performed by some different techniques including asymptotics in unbounded domains, singular perturbations and shape sensitivity. In particular, since the polarization matrices can be identified for some classes of shapes, we provide the formulae for numerical evaluation of such matrices for nearby shapes by means of the shape sensitivity analysis.