Homogenization of the Signorini boundary-value problem in a thick plane junction

TL;DR

This paper analyzes the asymptotic behavior of a Poisson equation with Signorini boundary conditions in a complex junction structure as the number of thin rods increases infinitely and their thickness diminishes, transforming the boundary conditions into variational inequalities.

Contribution

It introduces a new asymptotic analysis method for Signorini problems in thick junctions, proving convergence and the transformation of boundary conditions into variational inequalities.

Findings

Proved convergence of solutions as e9  0.

Showed Signorini conditions become variational inequalities in the limit.

Established existence and uniqueness of the limit problem solution.

Abstract

We consider a mixed boundary-value problem for the Poisson equation in a plane thick junction $\Omega_{\varepsilon}$ which is the union of a domain $\Omega_0$ and a large number of $\varepsilon$-periodically situated thin rods. The nonuniform Signorini conditions are given on the vertical sides of the thin rods. The asymptotic analysis of this problem is made as $\varepsilon \to 0$, i.e., when the number of the thin rods infinitely increases and their thickness tends to zero. With the help of the integral identity method we prove the convergence theorem and show that the nonuniform Signorini conditions are transformed (as $\varepsilon \to 0$) in the limiting variational inequalities in the region that is filled up by the thin rods in the limit passage. The existence and uniqueness of the solution to this non-standard limit problem is established. The convergence of the energy integrals is proved as well.