Penalty Robin-Robin Domain Decomposition Schemes for Contact Problems of Nonlinear Elasticity

Abstract

In this paper we propose on continuous level several domain decomposition methods to solve unilateral and ideal multibody contact problems of nonlinear elasticity. We also present theorems about convergence of these methods.

Domain Decomposition Methods for Problems of Unilateral Contact Between Elastic Bodies with Nonlinear Winkler Covers

“…where Φ : V × V → R is a functional, which is linear in v, but nonlinear in u, and Y : V → R is linear continuous form. For numerical solution of (27) consider the next nonstationary iterative method [5,11]:…”

“…Suppose that conditions (4), (6) and (7) hold. Then the contact problem (1)-(3), (5), (8)- (11) is equivalent to problem (1)-(3), (5), (8) with the following nonlinear boundary value conditions on the possible contact areas:…”

“…Suppose that conditions ( 4), ( 6), (7) hold. Then problem ( 1)-( 3), ( 5), ( 8)- (11) has an alternative weak formulation as the following minimization problem:…”

“…These DDMs are based on abstract nonstationary iterative methods for variational equations in Banach spaces. They are the generalization of domain decomposition methods, proposed by us earlier in [4,5,10] for unilateral contact problems without covers. Some particular cases of proposed DDMs can be viewed as a modification of semismooth Newton method [7].…”

Convergence of penalty Robin-Robin domain decomposition methods for unilateral multibody contact problems of elasticity

“…In works [28,29,30,31] we proposed on the continuous level a class of penalty parallel Robin-Robin type domain decomposition methods for the solution of unilateral multibody contact problems of elasticity. These methods are based on the penalty method for variational inequalities and some stationary and nonstationary iterative methods for nonlinear variational equations.…”