hp-mortar Boundary Element Method and FE/BE coupling for multibody contact problems with friction.

The

hp

-methods is a very efficient and accurate tool in modern computational mechanics. For a wide range of problems

hp

-techniques provide the exponential convergence rate of the discrete solution to the exact solution, while

h

-version and

p

-version give only an algebraic convergence rate. The finite element method is used commonly for numerical simulations of contact problems [

3

]. The boundary element techniques are relatively seldom used, despite some significant advantages [

1

]. In the boundary element method only the boundaries of the bodies are discretized, which reduces the dimension of the problem by one. This simplifies e.g. mesh generation significantly. Also, the number of unknowns in the problem is reduced greatly, but in contrast to finite elements, the Galerkin matrix will be dense due to nonlocal boundary integral operators.

We introduce a new

hp

-BEM mortar technique for multibody contact problems with friction. Often, it is very convenient to use independent discretizations of the bodies, subjected to their particularities. In case of nonmatched meshes on the contact interface, independent refinement of the bodies can be applied. Following the approach of [

1

] we impose contact constraints on the discrete global set of affinely transformed Gauss-Lobatto points on the single elements. The data transfer is realized in terms of mortar projection. The problem is reformulated as a variational inequality with the Steklov-Poincaré operator of the second kind over a convex cone of admissible solutions.

We obtain an upper error bound in the energy norm. Due to nonconformity of our approach, the error is decomposed to the approximation error and the consistency error. Finally we show that the discrete solution converges to the exact solution as

$$ \mathcal{O}\left( {\left( {{h \mathord{\left/ {\vphantom {h p}} \right. \kern-\nulldelimiterspace} p}} \right)^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} } \right) $$

.

We solve the discrete problem employing the Dirichlet-to-Neumann algorithm. The original two-body formulation is rewritten as a one-body contact problem and a one-body Neumann problem [

2

]. Then the global problem is solved employing fixed point iterations. We give numerical examples for the two-body contact problem with friction which show efficiency and accuracy of our approach.