Interior-exterior penalty approach for solving elasto-hydrodynamic lubrication problem: Part I

TL;DR

This paper introduces a novel interior-exterior penalty method within a discontinuous Galerkin finite volume framework for solving complex elasto-hydrodynamic lubrication problems, achieving optimal error estimates and handling realistic conditions.

Contribution

It develops a new penalty approach for quasi-variational inequalities in EHL problems without simplifying assumptions, with proven solution existence and comprehensive algorithms.

Findings

Achieved optimal error estimates in H^1 and L^2 norms.

Successfully solved moderate load conditions in EHL problems.

Provided a foundation for extending the method to highly loaded conditions.

Abstract

A new interior-exterior penalty method for solving quasi-variational inequality and pseudo-monotone operators arising in two-dimensional point contact problem has been analyzed and developed in discontinuous Galerkin finite volume method environment. In this article, we proved the existence of solution for the more realistic model problem without taking any constant assumption in viscosity or density of the lubricant. We have shown that optimal error estimate of $H^{1}$ and $L^{2}$ norm can be achieved under a light load non-dimensional parameter condition. In addition, we provided a complete algorithm to tackle all numerical complexities appear in the solution procedure. We obtained results for moderate loaded conditions which have been discussed at the end of the section. Furthermore, results give a hope for the further development of the scheme for highly loaded condition appeared in a more realistic operating situation which will be discussed in part II. This method is well suited for solving elasto-hydrodynamic lubrication line as well as point contact problems and can probably be treated as commercial software.