Onhp-adaptive BEM for frictional contact problems in linear elasticity

Abstract

Tresca/Coulomb frictional contact problem hp-adaptive BEM Residual and bubble a posteriori error estimates a b s t r a c t A mixed formulation for a Tresca frictional contact problem in linear elasticity is considered in the context of boundary integral equations, which is later extended to Coulomb friction. The discrete Lagrange multiplier, an approximation of the surface traction on the contact boundary part, is a linear combination of biorthogonal basis functions. In case of curved (isoparametric) elements, these are the solutions of local problems. In particular, the biorthogonality allows to rewrite the variational inequality constraints as a simple set of complementarity problems. Thus, enabling an efficient application of a semi-smooth Newton solver for the discrete mixed problem, converging locally super-linearly in the frictional case and quadratically in the frictionless case. Typically, the solution of frictional contact problems is of reduced regularity at the interface between contact to non-contact and from stick to slip. To identify the a priori unknown locations of these interfaces two a posteriori error estimations are introduced. In a first step the error is split into specific error contributions resulting from the contact and friction conditions and from the discretization error of a variational equation. For the latter a residual and a bubble error estimation are considered explicitly. The numerical experiments show the applicability of the derived error estimations and the superiority of hp-adaptivity compared to low order uniform and adaptive approaches.

Comparison of mixed hp-BEM (stabilized and non-stabilized) for frictional contact problems

“…The connection between these three formulations is as follows: 12]). The problems (4) and (5) are equivalent and are also equivalent to (1) in a distributional sense.…”

“…An important benefit of this approach with biorthogonal basis functions is the componentwise decoupling of the weak contact conditions [12]. These are equivalent to the system…”

“…In [12] we analyze the following mixed hp-boundary element formulation: [20]. That is with the canonical embedding i hp :…”

“…In [12] we consider the following variational inequality formulation on the boundary only, which is a reformulation of (1):…”

“…While mixed finite element approaches for contact problems are preferably done for low order elements [11] we focus here on recent results for the use of higher order BEM and especially on hp-adaptive procedures which we report from [12,13]. For approaches with variational inequalities (not mixed) see [8][9][10] whereas for obstacle problems we refer to [14][15][16].…”