Eigenvalue Problem in a Solid with Many Inclusions: Asymptotic Analysis

Abstract

We construct the asymptotic approximation to the first eigenvalue and corresponding eigensolution of Laplace's operator inside a domain containing a cloud of small rigid inclusions. The separation of the small inclusions is characterised by a small parameter which is much larger compared with the nominal size of inclusions. Remainder estimates for the approximations to the first eigenvalue and associated eigenfield are presented. Numerical illustrations are given to demonstrate the efficiency of the asymptotic approach compared to conventional numerical techniques, such as the finite element method, for three-dimensional solids containing clusters of small inclusions.

On meso-scale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions

“…Meso-scale approximations for eigenvalue problems in domains with clusters of many inclusions were analysed in [6]. Fundamental ideas of the method of compound asymptotic expansions [7] in domains with singularly perturbed boundaries were used.…”

Asymptotic behavior of u-capacities and singular perturbations for the Dirichlet-Laplacian

“…By elliptic regularity theory (see for instance Theorem 1.2, page 205, in [22]), u N is analytic in a neighborhood of 0. Next we note that by (35) we have…”

“…We note that in the last years the investigation of this type of problems has been carried out in many different directions. Maz'ya, Movchan, and Nieves have [35] have constructed the asymptotic approximation to the first eigenvalue and corresponding eigenfunctions of Laplace operator inside a domain containing a cloud of small rigid inclusions. Lanza de Cristoforis [32] has considered a Neumann eigenvalue problem and shown representation formulas in terms of analytic maps and log ε (depending on the dimension n).…”

“…where H (0,0) and N are as in formula (12). Finally, we note that the if we look at the first summand in the right hand side of equality (35), then the information on the function u is in the numerator, whereas the geometry of Ω and ω is taken into account in the denominator.…”

Asymptotic Analysis of Solutions to Transmission Problems in Solids with Many Inclusions

“…Low-frequency vibration problems for solids with arrays of small inclusions have also been addressed using a modification of the methods of compound and meso-scale asymptotic approximations. Asymptotics of the first eigenvalue and corresponding eigenfunction for domains with a cloud of rigid inclusions have appeared in [27]. Applications of the method of meso-scale approximations have also appeared in [5,6], where the scattering problems for many small obstacles in the infinite space were considered.…”