Homogenization of the Signorini boundary-value problem in a thick junction and boundary integral equations for the homogenized problem

Abstract

We consider a mixed boundary-value problem for the Poisson equation in a thick junction X e which is the union of a domain X 0 and a large number of e-periodically situated thin cylinders. The non-uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as e → 0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non-uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as e → 0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non-standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in X 0 and an appropriate postprocessing. The equations in X 0 finally are also treated with boundary integral equations.

Homogenization of some degenerate pseudoparabolic variational inequalities

“…Taking in (25) the limit as N, m → ∞ and using convergence results in (26), together with the continuity of nonlinear functions, we obtain…”

“…In [42] the multiscale analysis of a parabolic variation inequality corresponding to the Stefan problem was performed using the H-convergence method [33]. Homogenization of variational inequalities in domains with thick junctions, for which standard extension results do not hold, was studied in [26,27,28] using the method of monotone operators and construction of appropriate auxiliary functions.…”

Boundary augmented Lagrangian method for the Signorini problem

“…The research has been supported by China Scholarship Council, the Natural Science Foundation Project of CQ CSTC of China (Grant Nos. convergent, accurate and efficient methods for the numerical simulation of Signorini problems is still a very active field of research, and we mention selected contributions [5], [11], [14], [16], [18], [23], [25], [26].…”