The complexity of thermal elastohydrodynamic lubrication (TEHL) problems has led to a variety of specialised numerical approaches ranging from finite difference based direct and inverse iterative methods such as Multilevel MultiIntegration solvers, via differential deflection methods, to finite element based fullsystem approaches. Hence, not only knowledge of the physical and technical relationships but also knowledge of the numerical procedures and solvers is necessary to perform TEHL simulations. Considering the state of the art of multiphysics software, the authors note the absence of a commercial software package for solving TEHL problems embedded in larger multiphysics software. By providing guidelines on how to implement a TEHL simulation model in commercial multiphysics software, the authors want to stimulate the research in computational tribology, so that, hopefully, the research focus can be shifted even more on physical modelling instead of numerical modelling. Validations, as well as result examples of the suggested TEHL model by means of simulated coefficients of friction, coated surfaces, and nonsmooth surfaces, highlight the flexibility and simplicity of the presented approach.
Measurements, numerical simulations, and derived analytical solutions have, for decades, created detailed insights into the behaviour of thermal elastohydrodynamically lubricated (TEHL) contacts. Due to the large number of numerical investigations, it is astonishing that, so far, no commercial software package embedded in larger multiphysics software has been available to solve TEHL problems.
The simulation of TEHL contacts is, mainly due to elastic deformations and the large pressureinduced increase of lubricant viscosity, highly nonlinear and has led to various specialised numerical simulation approaches. Great computational efforts, and the tendency to instabilities at high loads when using computational fluid dynamics (CFD) [
Considering the large number of selfdeveloped approaches and solvers, the modelling of TEHL contacts is hardly transparent. Hence, the promotion of using commercial software packages for solving TEHL problems is necessary to make simulation techniques available to a broader audience of the tribology society and to focus research more on physical relationships rather than on numerical procedures. Accordingly, the authors provide guidelines for the implementation and solving of TEHL problems in the commercial software, COMSOL Multiphysics, to stimulate and accelerate research in computational tribology. The possibility of developing a complex TEHL model with moderate effort is demonstrated in this paper.
In this section, the theory and governing equations for simulation of TEHL contacts are addressed. For simplification, a twodimensional line contact is considered.
Contact conditions of, for example, gears are characterised by varying load, motion, and geometry along the path of contact. This transient contact can be interpreted as a transient model contact of two rolling elements with varying load, motion, and geometry for each meshing point. In TEHL simulations, the transient contact is usually further simplified to an equivalent contact between a single (inelastic) roller and an elastic flat body. This is adopted in this study and exemplarily shown for a line contact of a spur gear in Figure
Line contact in a spur gear and its simplification to a model (disc) contact and to an equivalent contact.
Characteristic quantities of the equivalent contact are described by the Hertzian contact parameters, for example,
Derived from the twodimensional compressible NavierStokes equations by considering reasonable assumptions for TEHL contacts [
The calculation of the elastic deformation of the equivalent body is based on the finite element method as introduced by Habchi [
The film thickness equation describes the height of the separating lubricant film and consists of the constant parameter
Similarly to the generalised Reynolds equation, the transient energy equations are simplified with reasonable assumptions typically used in TEHL calculations [
Pressure, temperature, and shear rate distributions in TEHL contacts have a significant influence on lubricant properties. The accompanying changes in viscosity have the largest influence on the TEHL contact itself. The models for pressure and temperature dependency of the viscosity as suggested in [
A guideline for the implementation of a TEHL simulation model in COMSOL Multiphysics (abbr. by COMSOL) and MathWorks MATLAB [
For a convenient numerical solution procedure of the nonlinear system of equations with good conditioning, the variables are transferred to a dimensionless form. Following Habchi [
The numerical solution scheme shown in Figure
Numerical solution scheme.
FEMmodel (
After reading all the required input parameters, an initial solution based on a simple steadystate isothermal Newtonian approach is calculated in the FEMmodel (
Within the time loop, a global loop is launched for each timestep. Firstly the pressure and film thickness distribution under consideration of nonNewtonian fluid behaviour is calculated for a given temperature distribution
After the convergence of
Finally, the converged pressure distribution and temperature distribution in the lubricant of two consecutive global loops are compared. Again, convergence is assumed when the maximum absolute difference between two consecutive solutions of pressure and temperature distributions is smaller than
In the following, the characteristics of the FEMmodel (
Figure
The elastic deformation of the equivalent body in
The pressure and film thickness distributions
The mesh in the FEMmodel (
Figure
The boundary conditions in Section
The communication between the MATLAB sequence controller and the FEMmodel (
For solving the nonlinear system of equations, two separate COMSOL solvers for the FEMmodel (
In order to show the functionality and the plausibility of the TEHL simulation model described in Section
Hartinger et al. [
Comparison of simulated pressure and film thickness distributions for a thermal (TEHL) and an isothermal (EHL) contact between the TEHL simulation model of this study (a) and the CFD model of Hartinger et al. [
Venner [
Comparison of simulated pressure and film thickness distribution for a transient isothermal Newtonian EHL contact between the TEHL simulation model of this study (a) and the solution of Venner [
The validation of the TEHL simulation model shows its functionality and plausibility.
In this section, examples of possible application opportunities, such as simulating coated surfaces, coefficients of friction, and nonsmooth surfaces, of the described TEHL simulation model are demonstrated.
Firstly, the possibility of investigating the effect of thin surface coatings on the TEHL contact behaviour is shown. Results based on the TEHL model described in this study with slightly different lubricant properties have already been published by Lohner et al. [
Figure
Pressure and film thickness (a) and temperature distribution (b) of a TEHL contact with coated (SiO_{2}) upper solid acc. to Lohner et al. [
For comparing measurement and simulation results, the coefficient of friction is a suitable quantity with integral character. Coefficient of friction measurements in fluid film lubrication regime can be performed at the FZG twin disk test rig with, for example, cylindrical discs (
Lubricant and bulk material properties (
Bulk material properties  

16MnCr5  

206000 

0.30 

7760 

44 

431 


Lubricant properties  
MIN100  



95 

10.5 

885 

0.137 

1921 

6 
A comparison of measured and simulated coefficients of the friction of lubricant MIN100 with very good correlation is shown in Figure
Simulated and measured coefficients of friction
Surface features have a significant influence on TEHL contact behaviour [
Schematic representation of the deterministically structured surface.
The considered lubricant and material parameters are shown in Table
Temperature (a) and pressure and film thickness distribution (b) of a transient TEHL contact with a deterministically structured surface shown in Figure
von Mises stress in the lower solid body (a) and in the upper solid body (b) correlating to Figure
In this paper, guidelines for the implementation of a TEHL model in commercial multiphysics software are presented. The introduced approach can be implemented with moderate effort, and the resulting TEHL model is very easy to extend for various applications. The TEHL model divided in a sequence controller in MathWorks MATLAB and in two selfdeveloped decoupled COMSOL Multiphysics FEMmodels is also capable of calculating the transient TEHL contact along the path of contact of gears. Even more challenging tasks, too, such as the extension of the described model to mixed lubrication regimes, are possible. By providing implementation guidelines for a TEHL simulation model in commercial multiphysics software, the authors are confident that the research in computational tribology has been stimulated and accelerated.
Coefficients of lubricant heat capacity model
Coefficients of lubricant Vogel temperature model
Hertzian contact halfwidth in m,
Compliance matrix in Pa
Lubricant specific heat capacity in J/(kg
Specific heat capacity of solids 1 and 2 in J/(kg
Height of solids in the FEMmodel (
Pressure coefficients of lubricant thermal conductivity model in 1/Pa
Radius of solids 1 and 2 in m
Coefficients of lubricant Bode density model
Young’s Modulus of solids 1 and 2 in Pa
Coefficients of lubricant pressureviscosity coefficient model in 1/Pa and K
Equivalent Young’s Modulus in Pa,
Reduced Young’s Modulus in Pa,
Normal force in N
Friction force in N
Film thickness in m
Constant parameter of film thickness in m
Effective contact length in width direction in m
Pressure in Pa
Hertzian pressure in Pa
Coefficient Roelands equation (
Reduced radius in m,
Deviation from the smooth profile in m
Time in s
Temperature in K
Bulk temperature in K
Displacement vector in m
Sum velocity in m/s
Velocity of solids 1 and 2 in m/s
Sliding velocity in m/s
Lubricant velocity distribution in m/s
Space coordinate in gap length direction in m
Left and right boundary of
Space coordinate in gap height direction in m
Roelands pressureviscosity parameter.
Pressureviscosity coefficient in 1/Pa
Coefficient of lubricant Bode density model in 1/K
Shear rate in 1/s
Deformation of the equivalent body in m
Strain tensor
Lubricant viscosity in Pas
Bulk temperature in °C
Oil inlet temperature in °C
Lubricant thermal conductivity in W/(m
Thermal conductivity of solids 1 and 2 in W/(m
Coefficient of friction
Poisson’s ratios of solids 1 and 2
Equivalent Poisson’s ratio
Lubricant density in kg/m^{3}
Density of solids 1 and 2 in kg/m^{3}
Coefficient of the lubricant Bode density model in kg/m^{3}
Stress tensor of equivalent body in Pa
Shear stress in Pa,
Eyring shear stress in Pa
Kinematic viscosity in mm^{2}/s
Lubricant domain in the FEMmodel (
Solid domains 1 and 2 in the FEMmodel (
Domain of Reynolds equation in the FEMmodel (
Equivalent body domain in the FEMmodel (
Upon request, the authors are pleased to offer to provide all the input parameters for recalculation of the result examples in Section
The authors declare that there is no conflict of interests regarding the publication of this paper.