Total FETI for contact problems with additional nonlinearities

Abstract

The paper is concerned with application of a new variant of the Finite Element Tearing and Interconnecting (FETI) method, referred to as the Total FETI (TFETI), to the solution to contact problems with additional nonlinearities. While the standard FETI methods assume that the prescribed Dirichlet conditions are inherited by subdomains, TFETI enforces both the compatibility between subdomains and the prescribed displacements by the Lagrange multipliers. If applied to the contact problems, this approach not only transforms the general nonpenetration constraints to the bound constraints, but it also generates an enriched natural coarse grid defined by the a priori known kernels of the stiffness matrices of the subdomains exhibiting rigid body modes. We combine our in a sense optimal algorithms for the solution to bound and equality constrained problems with geometric and material nonlinearities. The section on numerical experiments presents results of solution to bolt and nut contact problem with additional geometric and material nonlinear effects. Outline of theoryThe FETI class of nonoverlapping spatial domain decompositions was introduced by Farhat and Roux in their landmark paper [1]. The basic idea of the FETI methods is that the compatibility between subdomains, into which the original domain is partitioned, is enforced by the Lagrangian multipliers with physical meaning of forces in this mechanical context. Within the framework of the FETI concept, these forces are called the dual variables unlike the displacements which are referred to as the primal variables.After eliminating the primal variables, the original problem is reduced to a small, relatively well conditioned, typically equality constrained quadratic programming problem that is solved iteratively, see e.g. Dostál et al. [2]. The newly formulated problem exhibits both numerical and parallel scalabilities [4], which is essential for effective application to high performance computers.If concept of the FETI method is applied to solution to the contact problems, basically the same methodology can be used to prescribe conditions of nonpenetration between bodies in contact [2]. Total FETIConsider a steady-state contact problem of solid deformable bodies, which constitutes a 2nd order elliptic boundary value problem. The mathematical formulation of the problem is given by the governing equations expressing equilibrium conditions of the system, along with the boundary conditions, see e.g. [3, Chapter 2].The process of partitioning original domain into subdomains in general generates some subdomains, which are well constrained, others which are only partly constrained, and the remaining, which are totally unconstrained. All the subdomains which are not well constrained can move like rigid bodies, or they exhibit rigid body modes, so that their stiffness matrices are singular with dimensions of their kernels, or their defects, ranging from 1 to 6 for 3D problems.The original FETI method assumes that the Dirichlet boundary conditions are inherited from the ...