Recent extensions of mortar-based contact formulations: Lubrication modeling and parallel implementations

Mortar element methods have recently been successfully applied to large deformation dry-frictional contact problems. In this contribution, we summarize two recent extensions of the mortar based contact approach: solution of lubricated contact problems between deformable solid bodies, and implementation of mortar methods in parallel equation solving frameworks. Lubricated contact problems, in which thin fluid films exist between solid contact surfaces, are widely found in engineering and sciences. Based on Reynolds equations, different numerical methods have been developed for solving lubricated contact problems between rigid or linearly elastic solid bodies (see, for example, Christopherson [2] and Hamrock et al. [6]). In our approach, the fluid film thickness is computed from a least squares projection based on dual basis functions [12]. The free boundary problem for the fluid phase is regularized with a penalty method. The solid and the fluid phase unknowns are solved in a fully coupled system of equations, based on the linearization of the weak forms of the governing equations for the solid and the fluid phases. The performance and potential of the proposed method is demonstrated here in a two dimensional example of elastohydrodynamic lubrication. Another important issue for mortar based contact formulations lies in determining their effectiveness within parallel computing architectures. In general, contact problems result in potentially ill-conditioned and highly unsymmetric stiffness matrices, so that direct methods have been preferred (particularly in highly nonlinear large deformation applications). However, direct methods require memory and computation time up to O (n(1.5)) and O (n(2)), respectively, so that their applicability to very large scale contact simulations appears to be limited. Iterative methods require memory and computation time of O (n) and O (n(1.5)), making their use in large-scale numerical simulations tempting. We consider here a conceptually simple implementation which uses iterative methods to solve the linear systems generated by consistent linearization of the mortar contact formulation, on distributed memory machines, and examine performance in a representative numerical example.