Numerical analysis of the problems of contact of three elastic bodies by the domain decomposition methods
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NUMERICAL ANALYSIS OF THE PROBLEMS OF CONTACT OF THREE ELASTIC BODIES BY THE DOMAIN DECOMPOSITION METHODS І. І. Prokopyshyn,1 І. І. Dyyak,2 and R. M. Martynyak1
UDC 519.6: 539.3
We study the domain decomposition methods for the numerical solution of the problems of frictionless unilateral contact of many elastic bodies of finite sizes. By using the finite-element approximations, we solve the problems of the unilateral contact of three elastic bodies compressed by rigid plates and the contact of three fixed bodies one of which is subjected to the action of an external load. The distributions of normal contact and equivalent stresses in the bodies are analyzed. Keywords: contact of elastic bodies, variational inequalities, penalty method, iterative methods, domain decomposition methods, finite-element method.
The domain decomposition methods (DDM) reduce the solution of boundary-value problems of mathematical physics in the entire domain to the solution of a sequence of problems in individual subdomains. This enables one to combine different mathematical methods and models and parallelize numerical computations. The DDM can be classified by the type of boundary conditions imposed on the common boundaries in the problems posed for individual subdomains. The conditions of order zero are called Dirichlet conditions, the conditions for the derivatives are called Neumann conditions, and the mixed conditions are called Robin or Poincaré conditions. The DDM have been extensively developed for the linear problems of mathematical physics (in particular, of the linear theory of elasticity) and, in the last decade, for the contact problems of the theory of elasticity, which are, generally speaking, nonlinear. For the solution of the problems of the unilateral contact of two elastic bodies, on can use the Signorini– Neumann-type continual domain decomposition method developed in [1] based on the consecutive solution (in each step) of the problem of unilateral contact with a rigid surface (Signorini problems) for one body and the problem of the theory of elasticity with Neumann conditions for the second body. Moreover, for the problems of contact of two bodies on the continual level, it is proposed to use the Signorini–Dirichlet-type [2] and Signorini–Signorini-type [3] domain decomposition schemes, and the DDM based on the application the augmented Lagrangian and the Uzawa block algorithm [4]. Among the discrete DDM aimed at the solution of contact problems, we can especially mention the algorithms based on the method of substructures [5] and the finiteelement tearing and interconnecting method (FETI) [6, 7]. A class of Robin–Robin-type continual DDM was proposed for the solution of the problems of unilateral contact of many elastic bodies based on the penalty method for the variational inequalities and iterative methods for nonlinear variational equations [8–12]. Their convergence was proved in [9, 10, 12]. 1
Pidstryhach Institute for Applied Problems in Mechanics and Mathematics, Ukrainian National Academy of Sciences, Lviv,
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