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A series of the work "The development of domain decomposition methods for solution of variational inequalities that model the contact between several elastic bodies"


Author:   ProkopyshynI.I.

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of NAS of Ukraine

The series consists of 12 papers and 18 conference proceedings within 7 years.

The goal of the series is the development and the mathematical justification of new parallel domain decomposition methods for solution of variational inequalities and equations that describe the problems of unilateral and ideal contact between several linear and nonlinear elastic bodies with nonlinearsurface layers, the proof of the convergence of these methods, and their numerical implementation and approbation for studying the contact interaction of the system of deformable solids.

In this series the new variational formulations of the problems of multibody contact between linear and nonlinear elastic solids through nonlinear Winkler surface layers have been obtained and the theorems on existence and uniqueness of their solution have been proved. Moreover, a new class of parallel Robin type domain decomposition algorithms for solution of such contact problems has been proposed and their mathematical justification has been given. The convergence theorems for these algorithms have been proved and the estimates of the convergence rate have been established. The numerical implementation of the proposed domain decomposition methods has been performed on the base of the finite element and the boundary element approximations. The numerical analysis of these methods has been made for a number of contact problems for several bodies and has helped to discover new mechanical phenomena.

Obtained numerical methods are effective mathematical tool for solution of many applied contact problems that arise in geophysics, construction, machinery, biomechanics, and other fields. They allow to organize the parallel computations and to use the optimal mathematical models and methods in each subdomain, provide the economy of computational resources, and can be generalized to solve more complicated contact problems.