Adaptive high-order discontinuous galerkin solution of elastohydrodynamic lubrication point contact problems

ManuscriptThis paper describes an adaptive implementation of a high order Discontinuous Galerkin (DG) method for the solution of elastohydrodynamic lubrication (EHL) point contact problems. These problems arise when modelling the thin lubricating film between contacts which are under sufficiently high pressure that the elastic deformation of the contacting elements cannot be neglected. The governing equations are highly nonlinear and include a second order partial differential equation that is derived via the thin-film approximation. Furthermore, the problem features a free boundary, which models where cavitation occurs, and this is automatically captured as part of the solution process. The need for spatial adaptivity stems from the highly variable length scales that are present in typical solutions. Results are presented which demonstrate both the effectiveness and the limitations of the proposed adaptive algorithm

High-order discontinuous Galerkin method for elastohydrodynamic lubrication line contact problems

In this paper a high-order discontinuous Galerkin method is used to solve steady-state isothermal line contact elastohydrodynamic lubrication problems. This method is found to be stable across a wide range of loads and is shown to permit accurate solutions using just a small number of degrees of freedom provided suitable grids are used. A comparison is made between results obtained using this proposed method and those from a very large finite difference calculation in order to demonstrate excellent accuracy for a typical highly loaded test problem

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

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

A Study of Preconditioned Jacobian-Free Newton-Krylov Discontinuous Galerkin Method for Compressible Flows on 3D Hexahedral Grids

Storage requirement and computational efficiency have always been challenges for the efficient implementation of discontinuous Galerkin (DG)methods for real life applications. In this paper, a fully implicit Jacobian-Free Newton-Krylov (JFNK) method is developed in the context of DG discretizations for the three-dimensional compressible Euler and Navier-Stokes equations. Compared with the Jacobian-based methods, the Jacobian-Free approach saves the storage for the Jacobian matrix which can be of great importance for DG methods. Three types of preconditioners are investigated in which the block diagonal preconditioner requires the least storage, while the block LU-SGS and ILU0 preconditioners require more storage but are more computationally efficient. An implicit time-stepping strategy is adopted for the stability of the current solver,which is based upon a hexahedral spatialmesh and the nonlinear solver package Kinsol is used to improve the computational efficiency and robustness. Numerical results demonstrate that the preconditioned JFNK-DG solver can substantially reduce the storage requirement compared with the Jacobian based method without significantly compromising accuracy or efficiency. Furthermore, as a good compromise between efficiency and storage requirement, the ILU0 preconditioner shows the best choice of the preconditioners presented

Numerical simulations for ball bearings considering fluid-structure interaction problems and non-Newtonian fluids with a free boundary

In this thesis a novel approach for the simulation of elastohydrodynamic problems is established. It is novel in modeling the fluid as non-Newtonian with a free boundary and considering fluid-structure interaction at the same time. We present a robust numerical algorithm for the simulation of problems of such type. Furthermore, we analyze the difficulties and propose some improvements for treating the free boundary as well as the non-Newtonian flow numerically. We adress modeling aspects and model uncertainties with respect to material models, parameters and experimental investigations. For validation, we compare the numerical approximations with a widely accepted analytic approximation from engineering literature and achieve a reasonable accuracy. We conclude by bringing the thesis in relation with interesting further points of investigation on the basis of the established methods which rely on the novel modeling in this thesis

Effects of Non-Newtonian Lubricants on Surface Roughness in Point Contacts

Tato dizertační práce je zaměřena na studium deformace nerovnosti uvnitř elastohydrodynamicky mazaného (EHD) kruhového kontaktu. Práce se zabývá studiem přechodu příčné nerovnosti přes kontaktní oblast, která je modelována pomocí numerických metod. Model dále uvažuje nenewtonské chování maziva. Použitý matematický model se skládá z parciální diferenciální rovnice druhého řádu pro řešení tlaku a integro-diferenciální rovnice pro řešení elastických deformací. Pro řešení tohoto modelu je použitá takzvaná multigrid (vícesíťová) metoda. Práce obsahuje popis matematického modelu EHD kontaktu a aplikované numeriké metody. Výsledky simulací jsou porovnány s experimentálně stanovenýma hodnotama tloušťky mazacího filmu. Deformace nerovnosti uvnitř kontaktní oblasti je studována pro různé provozní podmínky (střední rychlost, poměr proklzu) a různá vlastnosti maziva.This PhD thesis focuses on the study of roughness deformation inside an elastohydrodynamically lubricated point contact. The passage of the roughness feature through the contact zone is modeled using numerical techniques. A single transverse ridge is assumed in the transient EHL model which presents a complex problem with a second order partial differential equation an integro-differential equation. Non-Newtonian fluid behavior is assumed in the model which further increases its complexity. In order to solve the system of equations the multigrid techniques are applied. The thesis contains the mathematical model describing the problem and a detailed description of the multigrid method. The results obtained by the simulations are compared to experimentally evaluated film thickness values. The roughness deformation is observed for a wide range of operating conditions as well as for different lubricant parameters. The effect of these lubricant parameters on the deformation is studied as well.

Tribological optimisation of the internal combustion engine piston to bore conjunction through surface modification

Internal combustion (IC) engines used in road transport applications employ pistons to convert gas pressure into mechanical work. Frictional losses abound within IC engines, where only 38- 51% of available fuel energy results in useful mechanical work. Piston-bore and ring-bore conjunctions are fairly equally responsible for circa 30% of all engine friction - equivalent to 1.6% of the input fuel each. Therefore, reduction in piston assembly friction would have a direct impact on specific performance and / or fuel consumption. In motorsport, power outputs and duty cycles greatly exceed road applications. Consequently, these engines have a shorter useful life and a high premium is placed on measures which would increase the output power without further reducing engine life. Reduction of friction offers such an opportunity, which may be achieved by improved tribological design in terms of reduced contact area or enhanced lubrication or both. However, the developments in the motorsport sector are typically reactive due to a lack of relative performance or an ad-hoc reliance, based upon a limited number of actual engine tests in order to determine if any improvement can be achieved as the result of some predetermined action. A representative scientific model generally does not exist and as such, investigated parameters are often driven by the supply chain with the promise of improvement. In cylinder investigations are usually limited to bore surface finish, bore and piston geometrical form, piston skirt coatings and the lubricant employed. Of these investigated areas newly emerging surface coatings are arguably seen as predominate. This thesis highlights a scientific approach which has been developed to optimise piston-bore performance. Pre-existing methods of screening and benchmarking alterations have been retained such as engine testing. However, this has been placed in the context of validation of scientifically driven development. A multi-physics numerical model is developed, which combines piston inertial dynamics, as well as thermo-structural strains within a thermoelastohydrodynamic tribological framework. Experimental tests were performed to validate the findings of numerical models. These tests include film thickness measurement and incylinder friction measurement, as well as the numerically-indicated beneficial surface modifications. Experimental testing was performed on an in-house motored engine at Capricorn Automotive, a dynamometer mounted single-cylinder ‘fired’ engine at Loughborough University, as well as on other engines belonging to third party clients of Capricorn. The diversity of tests was to ascertain the generic nature of any findings. The multi-physics multi-scale combined numerical-experimental investigation is the main contribution of this thesis to knowledge. One major finding of the thesis is the significant role that bulk thermo-structural deformation makes on the contact conformity of piston skirt to cylinder liner contact, thus advising piston skirt design. Another key finding is the beneficial role of textured surfaces in the retention of reservoirs of lubricant, thus reducing friction

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