Efficient finite element simulation of full-system elastohydrodynamic lubrication problems

This thesis is concerned with the efficient numerical solution of problems of elastohydrodynamic lubrication (EHL). Our approach is to consider fully-coupled models in
which the governing equations for the lubricating film, the elastic deformation and the force balance are each discretized and solved as a single monolithic nonlinear system of algebraic equations. The main contributions of this work are to propose, implement and analyse a novel, optimal, preconditioner for the Newton linearization of this algebraic system, and to assess the development of efficient finite element meshes through both manual
tuning and the use of adaptive mesh refinement based upon a posteriori error estimation and control.

Throughout this work, we employ first order finite element discretizations for both the Reynolds equation (for the lubricant) and for the linear elasticity model on a finite domain. The resulting nonlinear algebraic equations are then solved using a quasi-Newton algorithm. For each linear solve a Krylov subspace method is used and a new blockwise
preconditioner is presented which is designed to exploit the specific structure that is present in this class of problem. This preconditioner combines the use of multigrid
preconditioning for the elasticity block and a separate, efficient, approximation to precondition the Reynolds block. The solver developed in this work can be distinguished into two variants based upon the use of algebraic and geometric multigrid preconditioning of
the elasticity block.

Numerical results are presented both for line and point contact problems to validate the implementations and to allow a comparison of the performance and efficiency of the
proposed solution strategies compared to the use of a state-of-the-art sparse direct solver at each Newton step. These results demonstrate that the preconditioned iterative approach is both computationally and memory superior to the sparse direct solver. Most importantly, both the computational and memory costs are seen to grow linearly with the number of unknowns.

A locally adaptive solution scheme is also developed for fully-coupled EHL point contact problems. This automates the refinement process to the regions of the domain which
exhibit large error in their solutions. Numerical results are presented which demonstrate the performance and effectiveness of the proposed procedure.