Solving EHL Problems Using Iterative, Multigrid, and Homotopy Methods

1999 ◽  
Vol 121 (1) ◽  
pp. 28-33 ◽  
Author(s):  
Elyas Nurgat ◽  
Martin Berzins ◽  
Laurence Scales
Keyword(s):  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Grid Method ◽  
Contact Problems ◽  
Test Problems ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Future Developments ◽  
Relaxation Scheme ◽  
Highly Nonlinear

The numerical solution of ElastoHydrodynamic Lubrication (EHL) point contact problems requires the solution of highly nonlinear systems of equations which pose a formidable computational challenge. Multigrid methods provide one efficient approach. EHL problems solved using a single grid and multigrid will be compared and contrasted with a homotopy method which works on the concept of deforming one problem into another by the continuous variation of a single parameter. Both the multigrid and the single grid method employ a new relaxation scheme. Numerical results on demanding test problems will be used to compare these methods and suggestions for future developments to produce robust solvers will be made.

scholarly journals A nonmonotone inexact Newton algorithm for nonlinear systems of equations

1995 ◽  
Vol 36 (4) ◽  
pp. 460-492 ◽  
Author(s):  
Yi Xiao ◽  
Eric King-Wah Chu
Keyword(s):  
Hopf Bifurcation ◽  
Nonlinear Systems ◽  
Numerical Experiments ◽  
Standard Test ◽  
Test Problems ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Special Cases ◽  
Nonmonotone Techniques ◽  
Newton's Methods

AbstractIn this paper, an inexact Newton's method for nonlinear systems of equations is proposed. The method applies nonmonotone techniques and Newton's as well as inexact Newton's methods can be viewed as special cases of this new method. The method converges globally and quadratically. Some numerical experiments are reported for both standard test problems and an application in the computation of Hopf bifurcation points.

A NEW DERIVATIVE-FREE CONJUGATE GRADIENT METHOD FOR LARGE-SCALE NONLINEAR SYSTEMS OF EQUATIONS

2017 ◽  
Vol 95 (3) ◽  
pp. 500-511 ◽  
Author(s):  
XIAOWEI FANG ◽  
QIN NI
Keyword(s):  
Nonlinear Systems ◽  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Large Scale ◽  
Test Problems ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Line Search Method ◽  
Derivative Free

We propose a new derivative-free conjugate gradient method for large-scale nonlinear systems of equations. The method combines the Rivaie–Mustafa–Ismail–Leong conjugate gradient method for unconstrained optimisation problems and a new nonmonotone line-search method. The global convergence of the proposed method is established under some mild assumptions. Numerical results using 104 test problems from the CUTEst test problem library show that the proposed method is promising.

scholarly journals Modified Conjugate Gradient Method via DFP Update for Solving Systems of Nonlinear Equations

2020 ◽  
Vol 3 (1) ◽  
pp. 43-49
Author(s):  
M K Dauda
Keyword(s):  
Large Scale ◽  
Memory Storage ◽  
Systems Of Nonlinear Equations ◽  
Test Problems ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Derivative Free ◽  
Scale Test ◽  
Derivative Free Method ◽  
Modified Conjugate Gradient Method

In this study, a fully derivative-free method for solving large scale nonlinear systems of equations via memoryless DFP update is presented. The new proposed method is an enhanced DFP (Davidon-FletcherPowell) update which is matrix and derivative free thereby require low memory storage. Under suitable conditions, the proposed method converges globally. Numerical comparisons using a set of large-scale test problems showed that the proposed method is efficient.

An Efficient, Robust, Multi-Level Computational Algorithm for Elastohydrodynamic Lubrication

1989 ◽  
Vol 111 (2) ◽  
pp. 193-199 ◽  
Author(s):  
L. Chang ◽  
T. F. Conry ◽  
C. Cusano
Keyword(s):  
Computational Algorithm ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Contact Problems ◽  
Working Environment ◽  
Ehd Lubrication ◽  
Wide Range ◽  
Direct Iteration ◽  
Multi Level ◽  
And Storage

A new computational algorithm is developed for the numerical analysis of elastohydrodynamic (EHD) lubrication problems. This algorithm combines direct-iteration, Newton-Raphson, and multigrid methods into one working environment. Accurate solutions for a wide range of steady-state, line-contact problems are obtained with a relatively small number of numerical operations. The algorithm can be used to efficiently simulate transient processes in EHD lubrication. It can also be extended to solve point-contact problems with high computational and storage efficiency.

A Generalized Thermal EHL Model for Point Contact Problems

2008 ◽  
Author(s):  
Yuchuan Liu ◽  
Q. Jang Wang ◽  
Dong Zhu ◽  
Fanghui Shi
Keyword(s):  
Surface Temperature ◽  
Thermal Model ◽  
Reynolds Equation ◽  
Energy Equation ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Contact Problems ◽  
Three Dimensions ◽  
Thermal Ehl ◽  
Generalized Reynolds Equation

A generalized thermal elastohydrodynamic lubrication (TEHL) model for point contact problems is developed based on an isothermal generalized Newtonian elastohydrodynamic (EHL) model recently developed. The thermal model couples FDM for lubricant energy equation and the DC-FFT method for surface temperature integration. A generalized Reynolds equation is derived considering the change of viscosity with respect to temperature, pressure and shear in three dimensions. Numerical cases are conducted to verify the model.

Non-Newtonian Thermal Analyses of Point EHL Contacts Using the Eyring Model

2005 ◽  
Vol 127 (1) ◽  
pp. 70-81 ◽  
Author(s):  
Xiaoling Liu ◽  
Ming Jiang ◽  
Peiran Yang ◽  
Motohiro Kaneta
Keyword(s):  
Effective Viscosity ◽  
Numerical Solutions ◽  
Thermal Analyses ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Contact Problems ◽  
Point Contacts ◽  
Eyring Model ◽  
Thermal Ehl

A non-Newtonian numerical solution system for the thermal elastohydrodynamic lubrication (EHL) problems in point contacts has been developed. The Eyring rheology model has been used to describe the non-Newtonian flow of the lubricant. An effective viscosity has been defined for the Eyring fluid. The Newtonian solver can be applied easily to the non-Newtonian problems when the viscosity of the Newtonian fluid is replaced by the effective viscosity. A novel technique for the determination of the effective viscosity is proposed. Numerical solutions for the conventional point contact and normally crossing cylinders contact problems are presented and the effects of the entraining velocity, the load, the slide-roll ratio, and the characteristic shear stress of the Eyring fluid on the lubricating performance are discussed. The results indicate that the non-Newtonian thermal EHL theory predicts more realistic film temperatures and traction coefficients.

scholarly journals Convergence properties of the Broyden-like method for mixed linear–nonlinear systems of equations

2021 ◽  
Author(s):  
Florian Mannel
Keyword(s):  
Linear Independence ◽  
Affine Subspace ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
Convergence Properties ◽  
Initial Matrix ◽  
First Time ◽  
Special Case ◽  
Dimensional Mapping ◽  
Finite Number Of Iterations

AbstractWe consider the Broyden-like method for a nonlinear mapping $F:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}$ F : ℝ n → ℝ n that has some affine component functions, using an initial matrix B0 that agrees with the Jacobian of F in the rows that correspond to affine components of F. We show that in this setting, the iterates belong to an affine subspace and can be viewed as outcome of the Broyden-like method applied to a lower-dimensional mapping $G:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d}$ G : ℝ d → ℝ d , where d is the dimension of the affine subspace. We use this subspace property to make some small contributions to the decades-old question of whether the Broyden-like matrices converge: First, we observe that the only available result concerning this question cannot be applied if the iterates belong to a subspace because the required uniform linear independence does not hold. By generalizing the notion of uniform linear independence to subspaces, we can extend the available result to this setting. Second, we infer from the extended result that if at most one component of F is nonlinear while the others are affine and the associated n − 1 rows of the Jacobian of F agree with those of B0, then the Broyden-like matrices converge if the iterates converge; this holds whether the Jacobian at the root is invertible or not. In particular, this is the first time that convergence of the Broyden-like matrices is proven for n > 1, albeit for a special case only. Third, under the additional assumption that the Broyden-like method turns into Broyden’s method after a finite number of iterations, we prove that the convergence order of iterates and matrix updates is bounded from below by $\frac {\sqrt {5}+1}{2}$ 5 + 1 2 if the Jacobian at the root is invertible. If the nonlinear component of F is actually affine, we show finite convergence. We provide high-precision numerical experiments to confirm the results.

scholarly journals A thermal resistances-based approach for thermal-elastohydrodynamic calculations in point contacts

2017 ◽  
Vol 232 (11) ◽  
pp. 2088-2102 ◽  
Author(s):  
Eduardo de la Guerra Ochoa ◽  
Javier Echávarri Otero ◽  
Enrique Chacón Tanarro ◽  
Benito del Río López
Keyword(s):  
Friction Coefficient ◽  
Film Thickness ◽  
Thermal Effects ◽  
Good Accuracy ◽  
Computational Cost ◽  
Point Contact ◽  
Elastohydrodynamic Lubrication ◽  
Point Contacts ◽  
New Approach ◽  
Ball On Disc

This article presents a thermal resistances-based approach for solving the thermal-elastohydrodynamic lubrication problem in point contact, taking the lubricant rheology into account. The friction coefficient in the contact is estimated, along with the distribution of both film thickness and temperature. A commercial tribometer is used in order to measure the friction coefficient at a ball-on-disc point contact lubricated with a polyalphaolefin base. These data and other experimental results available in the bibliography are compared to those obtained by using the proposed methodology, and thermal effects are analysed. The new approach shows good accuracy for predicting the friction coefficient and requires less computational cost than full thermal-elastohydrodynamic simulations.

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