Adaptive high-order finite element solution of transient elastohydrodynamic lubrication problems

This article presents a new numerical method to solve transient line contact elastohydrodynamic lubrication (EHL) problems. A high-order discontinuous Galerkin (DG) finite element method is used for the spatial discretization, and the standard Crank-Nicolson method is employed to approximate the time derivative. An h-adaptivity method is used for grid adaptation with the time-stepping, and the penalty method is employed to handle the cavitation condition. The roughness model employed here is a simple indentation, which is located on the upper surface. Numerical results are presented comparing the DG method to standard finite difference (FD) techniques. It is shown that micro-EHL features are captured with far fewer degrees of freedom than when using low-order FD methods

Tribological investigations of the load, temperature, and time dependence of wear in sliding contact

An effort was made to study and characterize the evolution of transient tribological wear in the presence of sliding contact. Sliding contact is often characterized experimentally via the standard ASTM D4172 four-ball test, and these tests were conducted for varying times ranging from 10 seconds to 1 hour, as well as at varying temperatures and loads. A numerical model was developed to simulate the evolution of wear in the elastohydrodynamic regime. This model uses the results of a Monte Carlo study to develop novel empirical equations for wear rate as a function of asperity height and lubricant thickness; these equations closely represented the experimental data and successfully modeled the sliding contact

An adaptive finite element procedure for fully-coupled point contact elastohydrodynamic lubrication problems

This paper presents an automatic locally adaptive finite element solver for the fully-coupled EHL point contact problems. The proposed algorithm uses a posteriori error estimation in the stress in order to control adaptivity in both the elasticity and lubrication domains. The implementation is based on the fact that the solution of the linear elasticity equation exhibits large variations close to the fluid domain on which the Reynolds equation is solved. Thus the local refinement in such region not only improves the accuracy of the elastic deformation solution significantly but also yield an improved accuracy in the pressure profile due to increase in the spatial resolution of fluid domain. Thus, the improved traction boundary conditions lead to even better approximation of the elastic deformation. Hence, a simple and an effective way to develop an adaptive procedure for the fully-coupled EHL problem is to apply the local refinement to the linear elasticity mesh. The proposed algorithm also seeks to improve the quality of refined meshes to ensure the best overall accuracy. It is shown that the adaptive procedure effectively refines the elements in the region(s) showing the largest local error in their solution, and reduces the overall error with optimal computational cost for a variety of EHL cases. Specifically, the computational cost of proposed adaptive algorithm is shown to be linear with respect to problem size as the number of refinement levels grows

An optimally efficient technique for the solution of systems of nonlinear parabolic partial differential equations

This paper describes a new software tool that has been developed for the efficient solution of systems of linear and nonlinear partial differential equations (PDEs) of parabolic type. Specifically, the software is designed to provide optimal computational performance for multiscale problems, which require highly stable, implicit, time-stepping schemes combined with a parallel implementation of adaptivity in both space and time. By combining these implicit, adaptive discretizations with an optimally efficient nonlinear multigrid solver it is possible to obtain computational solutions to a very high resolution with relatively modest computational resources. The first half of the paper describes the numerical methods that lie behind the software, along with details of their implementation, whilst the second half of the paper illustrates the flexibility and robustness of the tool by applying it to two very different example problems. These represent models of a thin film flow of a spreading viscous droplet and a multi-phase-field model of tumour growth. We conclude with a discussion of the challenges of obtaining highly scalable parallel performance for a software tool that combines both local mesh adaptivity, requiring efficient dynamic load-balancing, and a multigrid solver, requiring careful implementation of coarse grid operations and inter-grid transfer operations in parallel

Algorithm xxxx: HiPPIS A High-Order Positivity-Preserving Mapping Software for Structured Meshes

Polynomial interpolation is an important component of many computational problems. In several of these computational problems, failure to preserve positivity when using polynomials to approximate or map data values between meshes can lead to negative unphysical quantities. Currently, most polynomial-based methods for enforcing positivity are based on splines and polynomial rescaling. The spline-based approaches build interpolants that are positive over the intervals in which they are defined and may require solving a minimization problem and/or system of equations. The linear polynomial rescaling methods allow for high-degree polynomials but enforce positivity only at limited locations (e.g., quadrature nodes). This work introduces open-source software (HiPPIS) for high-order data-bounded interpolation (DBI) and positivity-preserving interpolation (PPI) that addresses the limitations of both the spline and polynomial rescaling methods. HiPPIS is suitable for approximating and mapping physical quantities such as mass, density, and concentration between meshes while preserving positivity. This work provides Fortran and Matlab implementations of the DBI and PPI methods, presents an analysis of the mapping error in the context of PDEs, and uses several 1D and 2D numerical examples to demonstrate the benefits and limitations of HiPPIS

The Tribological Effects of Lubricating Oil Containing Nanometer-Scale Diamond Particles

This dissertation investigates the tribological effects of diamond nanoparticles as a lubricant mineral oil additive. A numerical code was developed that models the sliding contact observed in a standard four-ball test of sliding contact. Four-ball experimental tests were conducted both of neat mineral oil and mineral oil with the diamond nanoparticle additives, varying the trial times, temperatures, nanoparticle concentrations, and loads. The numerical results matched the experimental data remarkably by adjusting the lubricant thermal conductivity to account for the enhanced conductivity of diamond; demonstrating that thermal enhancements are the primary cause of the wear reduction properties of diamond nanoparticle additives