Solution of Coercive and Semicoercive Contact Problems by FETI Domain Decomposition

A new Neumann-Neumann type domain decomposition algorithm for the solution of contact problems of elasticity and similar problems is described. The discretized variational inequality that models the equilibrium of a system of elastic bodies in contact is first turned by duality to a strictly convex quadratic programming problem with either box constraints or box and equality constraints. This step may be considered a variant of the FETI domain decomposition method where the subdomains are identified with the bodies of the system. The resulting quadratic programming problem is then solved by algorithms proposed recently by the authors. Important new features of these algorithms are efficient adaptive precision control on the solution of the auxiliary problems and effective application of projections, so that the identification of a priori unknown contact interfaces is very fast. We start our exposition by reviewing a variational inequality in displacements that describes the conditions of equilibrium of a system of elastic bodies in contact without friction. The inequality enhances the natural decomposition of the spatial domain of the problem into subdomains that correspond to the bodies of the system, and we also indicate how to refine this decomposition. After discretization, we get a possibly indefinite quadratic programming problem with a block diagonal matrix. A brief inspection of the discrete problem shows that its structure is not suitable for computations. The main drawbacks are the presence of general constraints that prevent effective application of projections, and a semidefinite or ill conditioned matrix of the quadratic form that may cause extremely expensive solutions of the auxiliary problems. A key observation is that both difficulties may be essentially reduced by the application of duality theory. The matrix of the dual quadratic form turns out to be regular, moreover its spectrum is much more favorably distributed for application of the conjugate gradient based methods than the spectrum of the matrix of the