QUALITATIVE AND NUMERICAL ANALYSES OF ELASTOHYDRODYNAMIC LUBRICATION PROBLEMS WITH PENALTY METHOD (ERROR ESTIMATE, CONTACT PROBLEM, FREE BOUNDARY)

The strongly nonlinear Reynolds-Hertz equations and the classical free-boundary problems of elastohydrodynamic lubrication are formulated in a variational inequality framework and then regularized with the penalty method. The operator in the governing equations is shown to be bounded, coercive and pseudomonotone. Then the existence of solutions is proved. A penalty method is introduced to control the free boundary of cavitation, both in theoretical analysis and in finite element algorithms. Finite element codes with iteration schemes are generated for both line contact and point contact problems. The numerical results show that the penalty method works excellently in locating the free boundary. Further study shows the convergence of the penalty method as well as the finite-dimensional approximations. The regularity of solutions of one-dimensional case is proved and confirmed by the numerical experiments. An a-priori error estimate for finite element solutions is proposed. An adaptive refinement scheme is developed for generating optimal meshes. The scheme is shown to work satisfactorily for a class of problems with light contact pressure, however it is unstable for heavy contact loads.