Scalar boundary value problems on junctions of thin rods and plates

Abstract

We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative thickness, appears in the differential equation, too, while the asymptotic structures crucially depend on the contrastness ratio. Asymptotic error estimates are derived in anisotropic weighted Sobolev norms.

Scalar problems in junctions of rods and a plate

“…As it was observed in [4], an asymptotic expansion of a solution of the Poisson scalar mixed boundary-value problem in a junction of thin rods and a thin plate in a certain range of physical parameters gains the rational dependence on the big parameter | ln h| where h > 0 is a small parameter characterizing the diameters of the rods and the thickness of the plate. Similar asymptotic forms had been discovered for other elliptic problems stated in domains with singular perturbations of the boundaries, see the books [16,11,9].…”

“…Similar asymptotic forms had been discovered for other elliptic problems stated in domains with singular perturbations of the boundaries, see the books [16,11,9]. The aim of this paper is to study the scalar spectral problem associated to the Poisson problem studied in [4]. Based on the asymptotic procedure in [16, §2.2.4, §5.5.2, §9.…”

“…Based on the asymptotic procedure in [16, §2.2.4, §5.5.2, §9. 1.3] and [17,4], it is quite predictable that the asymptotic expansions of the eigenpairs "eigenvalue/eigenfunction" become much more complicated than the one obtained for the solution of the Poisson problem and purchase the holomorphic dependence on the parameter | ln h| −1 . Such sophistication of the asymptotic expansions and the lack of algorithms allowing to clarify the appearing holomorphic functions make them almost useless in applications, especially for combined numerical and asymptotic methods.…”

“…The asymptotic analysis which has been presented at length, for example, in [4,5,6] and, in particular, includes the dimension reduction procedure, converts the differential equations (1.6) and (1.7) with the Neumann boundary conditions (1.8) and (1.9), respectively, into the limit equations…”

“…As was shown in [4], this observation requires to consider solutions of the problem (1.20), (1.22) with logarithmic singularities at the points in the set P. We also will take in the sequel such singular solutions into account, however further considerations in this paper diverge from the asymptotic analysis used in [4]. Indeed, we will provide two abstract but applicable formulations of a spectral problem in the hybrid domain in fig.…”