OPTIMIZATION OF SURFACE

TEXTURE SHAPES

IN HYDRODYNAMIC CONTACTS

AGATA GUZEK

(MSc)

THIS THESIS IS PRESENTED FOR THE DEGREE OF DOCTOR OF

PHILOSOPHY OF THE UNIVERSITY OF WESTERN AUSTRALIA

SCHOOL OF MECHANICAL AND CHEMICAL ENGINEERING

2012

i

Abstract

The development of an optimization system/method for surface textures in

hydrodynamic contacts is of great importance to tribology. Such method, once

developed, would replace inefficient techniques that have been used to date in

determining the optimum surface texture shapes of mechanical components.

Various attempts at finding optimal texture shapes have been made, but most

of them have been limited to either numerical or experimental exhaustive

searches which are both costly and time consuming. Few works exist that

employ numerical optimization techniques; most of them are only applicable to

specific cases they were designed for. Thus, there is an urgent need for the

development of a new optimization approach that would be versatile and more

efficient than previously used methods.

This thesis is divided into six parts. The first part contains introduction, thesis

objectives, overview and review of, literature; in the second part the numerical

methodology is outlined, in the third part a new unified approach is designed

to optimize 1D hydrodynamic bearings governed by the Reynolds equation.

The approach is based on the theory of optimal parameter selection, control

parametrization and nonlinear programming. The method can be used to

optimize hydrodynamic bearing texture shapes to obtain maximum load

capacity or minimum friction force/coefficient. It has been validated by

comparison of obtained results to published data when possible and to the

results of performed parametric exhaustive search. The method was tested on a

number of cases, including smooth 1D journal and pad bearings, parallel 1D

ii

sliders with rectangular and elliptical dimples; bearings lubricated with non-

Newtonian fluids and lubricant viscosity changing with temperature were also

studied. Cavitation was taken into account by using the Reynolds boundary

condition. The approach was proved to yield correct results and thus formed a

foundation of our following works. This part of the thesis is described in detail

in Chapter 4.

The fourth part of the thesis contains an extension of the previously proposed

approach to enable optimization of bearings governed by the 2D Reynolds

equation. The concept of textured shape parametrization, as in the previous

chapter, was the base of the optimization procedure. The goal was to select

parameters of the surface texture so that maximum load capacity or minimum

friction force/coefficient could be obtained. For the optimization, sequential

quadratic programming (SQP) technique was used. To solve the Reynolds

equation, both commercially available software and own program codes were

applied. The codes were developed to enable solution with a mass-conserving

cavitation algorithm and viscosity varying with temperature. To validate the

method, examples of finite parallel sliders textured with rectangular and

elliptical dimples were optimized and the results compared to published data

when possible and to performed numerical exhaustive search. The results

obtained were comparable, which confirms the validity of the proposed

approach. Additionally, the influence of shifting dimples, using different

cavitation models and temperature dependent viscosity on the optimization

results was examined. This work is described in Chapter 5.

In the fifth part of this work (Chapter 6), attention was brought to the cases

when the simplifying assumptions of the Reynolds equation do not hold and

the Navier-Stokes equations have to be solved to determine the pressure

distribution in a bearing; e.g. when inertia occurs in bearings. An approach was

thus developed to optimize surface texture parameters for bearings governed

iii

by the Navier-Stokes equations. The domain on which the equations are solved

(the fluid film) was parametrized and its parameters optimized. Similarly to the

previous chapter, SQP optimization technique was used. The approach was

validated using a number of examples, including both infinitely long and finite

bearings. Inertia, non-Newtonian lubrication and temperature dependant

viscosity were taken into account. The results of optimization and numerical

exhaustive search were compared; yielding good agreement. This confirms that

the developed approach is a valuable tool for surface texture shape

optimization.

In conclusion, the uniform approaches developed in this thesis can successfully

be applied to accurately find optimal parameters of surface textures of

hydrodynamic bearings and other mechanical components. This way,

tribological characteristics of the mechanical components, such as load capacity

and friction force, can be optimized. The method proposed could be of great use

to engineering tribology and find applications in many industries, such as

automotive, microsystems and energy.

iv

Table of contents

Abstract i

Table of contents iv

Acknowledgements ix

Journal publications and conference presentations arising from this

thesis x

Statement of candidate contribution xi

Abbreviations xii

Chapter 1. Introduction ................................................................................................ 1

1. Thesis objectives.................................................................................................... 2

2. Thesis overview .................................................................................................... 4

References ................................................................................................................... 9

List of figures ............................................................................................................ 10

Chapter 2. Optimization of surface texture shapes in hydrodynamic contacts –

Literature review ......................................................................................................... 11

1. Surface texturing ................................................................................................. 11

2. Optimization of surface texture shapes ........................................................... 12

3. Conclusions ......................................................................................................... 14

References .................................................................................................................. 16

Chapter 3. Methodology ............................................................................................ 20

1. Problem formulation .......................................................................................... 20

2. Analytical solutions ............................................................................................ 21

2.1 1D Reynolds equation .................................................................................. 21

2.2 2D Reynolds equation ................................................................................. 26

3. Numerical methods ........................................................................................... 28

4. Choice of governing equations ........................................................................ 31

v

5. Conclusions ......................................................................................................... 38

References .................................................................................................................. 40

Nomenclature ............................................................................................................ 46

List of tables ............................................................................................................... 48

List of figures ............................................................................................................. 49

Chapter 4. A unified computational approach to the optimization of surface

textures: One dimensional hydrodynamic bearings ............................................ 53

1. Introduction ......................................................................................................... 55

2. Methods ............................................................................................................... 58

2.1 Texture shape optimization (TSO) ............................................................. 58

2.2 2D Combined optimal control and optimal parameter selection (COC-

OPS) ..................................................................................................................... 58

2.3 Control parametrization .............................................................................. 60

3. Results .................................................................................................................. 61

Example 1. Step bearing – load capacity optimization ............................... 61

Example 2. Partially textured parallel bearing with rectangular or

elliptical dimples – load capacity optimization .......................................... 62

Example 3. Partially textured parallel bearing with elliptical dimples

lubricated by non-Newtonian fluids – load capacity optimization ......... 64

Example 4. Partially textured parallel bearing with elliptical dimples and

lubricant viscosity changing with temperature – load capacity

optimization ..................................................................................................... 66

Example 5. Partially textured parallel bearing with elliptical dimples –

friction force optimization .............................................................................. 67

4. Discussion ............................................................................................................ 69

5. Conclusions ......................................................................................................... 72

References ................................................................................................................. 74

Nomenclature ........................................................................................................... 79

List of tables .............................................................................................................. 81

List of figures ............................................................................................................ 87

vi

Chapter 5. Optimization of textured surfaces in 2D parallel bearings

governed by the Reynolds equation including cavitation and temperature ... 92

1. Introduction ......................................................................................................... 94

2. Method description ............................................................................................ 96

2.1 Texture shape optimization (TSO) ............................................................. 96

2.2 Bearing and surface geometry ................................................................... 97

2.3 Optimal parameter selection ....................................................................... 98

2.4 Nonlinear programming ............................................................................. 98

2.5 Numerical method ....................................................................................... 99

2.6 Computational implementation ............................................................... 100

3. Results ................................................................................................................ 101

Example 1. Optimization of surface textured by rectangular (elliptical)

dimples for the maximum load capacity ...................................................... 101

Example 2. Optimization of surface textured by rectangular (elliptical)

dimples for the minimum friction coefficient .............................................. 103

Example 3. Optimization of surface textured by rectangular (elliptical)

dimples for the maximum load capacity, Reynolds equation considering

mass-conserving cavitation ............................................................................ 104

Example 4. Optimization of textured surface for the maximum load

capacity, Reynolds equation considering mass-conserving cavitation and

temperature change ......................................................................................... 105

4. Discussion .......................................................................................................... 106

4.1 Number of dimples (m) ............................................................................. 107

4.2 Position of dimple within cell .................................................................. 108

4.3 Height ratio (Hr) .......................................................................................... 108

4.4 Area density (Ar) and textured portion of slider length (α) ................. 109

4.5 Cavitation algorithm .................................................................................. 110

4.6 Viscosity varying with temperature ........................................................ 111

4.7 Friction coefficient optimization .............................................................. 111

4.8 Limitations of the current approach and future work .......................... 112

5. Conclusion ......................................................................................................... 114

vii

References ............................................................................................................... 116

Nomenclature ......................................................................................................... 122

List of tables ............................................................................................................ 124

List of figures .......................................................................................................... 138

Chapter 6. Optimization of surface texture of parallel bearings governed by

the Navier-Stokes equations ................................................................................... 148

1. Introduction ....................................................................................................... 149

2. Problem formulation ........................................................................................ 150

2.1 Configuration of bearings ......................................................................... 150

2.2 Governing equations ................................................................................. 152

2.3 Boundary conditions .................................................................................. 153

2.4 Solving the NS equations .......................................................................... 153

2.5 Mesh independence ................................................................................... 154

2.6 Optimal parameter selection ..................................................................... 155

2.7 Optimization method ................................................................................. 156

3. Optimization examples .................................................................................... 156

Treatment of cavitation .................................................................................... 157

3.1 Examples of optimization of infinitely long bearings ........................... 158

Example 1. Optimization of a 2D bearing surface textured with

rectangular (elliptical) dimples for the maximum load capacity .......... 158

Example 2. . Optimization of a 2D bearing surface textured with

rectangular (elliptical) dimples for the minimum friction coefficient . 159

Example 3. . Optimization of a 2D bearing surface textured with

rectangular (elliptical) dimples for the maximum load capacity

including inertia effects .............................................................................. 159

Example 4. 2D bearing textured with rectangular dimples, lubricated

with non-Newtonian fluids, power-law model – load capacity

optimization ................................................................................................. 160

Example 5. Optimization of a 2D bearing textured with rectangular

dimples for the maximum load capacity; NS equations coupled with

energy equation to consider the decrease of viscosity with temperature

......................................................................................................................... 162

viii

3.2 Examples of optimization of finite-length bearings ............................. 163

Example 6. 3D bearing textured with rectangular dimples – load

capacity optimization .................................................................................. 163

Example 7. 3D bearing textured with elliptical dimples – load capacity

optimization ................................................................................................. 164

4. Discussion .......................................................................................................... 164

4.1 Influence of physical phenomena on optimal results ........................... 165

4.2 Influence of geometric texture parameters on optimal load/friction

coefficient .................................................................................................... 167

Limitations of the current approach and future work ................................ 168

5. Conclusions ....................................................................................................... 169

References ............................................................................................................... 171

Nomenclature ......................................................................................................... 178

List of tables ............................................................................................................ 180

List of figures .......................................................................................................... 196

Chapter 7. Conclusions and future work .............................................................. 199

1. Summary of findings and observations ........................................................ 199

2. General conclusions ......................................................................................... 201

3. Future work ....................................................................................................... 202

3.1 Tribology ........................................................................................................ 203

3.2 Other engineering applications .................................................................. 205

3.3 Others ............................................................................................................. 205

References ............................................................................................................... 206

ix

Acknowledgements

I would like to take this opportunity to thank everyone without whom the

completion of this PhD would not be possible.

I would like to express my deepest gratitude to my supervisors, Associate

Professor Pawel Podsiadlo and Winthrop Professor Gwidon Stachowiak for

providing me with invaluable guidance through this project, expertise,

encouragement and patience.

I would like to thank my colleagues from the UWA Tribology Laboratory:

Grazyna Stachowiak, Tomasz Woloszynski, Marcin Wolski, Mobin Salasi and

Wen-Hsi Chua for all their support, advice and friendship.

I would like to acknowledge the financial support of the Commonwealth

Government of Australia, The University of Western Australia and the

assistance of the School of Mechanical and Chemical Engineering.

A special thanks goes to my parents for their guidance, care and support they

have given me throughout this stage of my life, whom without this thesis

would not be possible.

Finally, I would like to thank my partner Nathan Jensen for all his love,

encouragement and continuous support throughout this journey.

x

Journal publications and conference presentations

arising from this thesis

Journal publications

A. Guzek, P. Podsiadlo and G.W. Stachowiak, A unified computational approach to

the optimization of surface textures: One dimensional hydrodynamic bearings,

Tribology Online, Vol. 5, No 3, 2010, pp. 150-160 (Chapter 4).

A. Guzek, P. Podsiadlo and G.W. Stachowiak, Optimization of textured surface in

2D parallel bearings governed by the Reynolds equation including cavitation and

temperature, Submitted to Tribology Online, December 2011 (Chapter 5).

Conference presentation

A. Guzek, P. Podsiadlo and G.W. Stachowiak, Optimization of Surface Texture

Shapes in Hydrodynamic Contacts: Two-Dimensional Bearings, Oral presentation at

ASIATRIB 2010 - 4th International Tribology Congress, December 2010, Perth,

Western Australia

Invited talk

A. Guzek, P. Podsiadlo and G.W. Stachowiak, Optimization of textured surface in

2D parallel bearings governed by the Reynolds equation including cavitation and

temperature effects, Oral presentation at the International Tribology Conference,

Hiroshima 2011, October-November 2011, Hiroshima, Japan

xi

Statement of candidate contribution

This thesis contains published work and work submitted for publication, which

has been co-authored. The bibliographical details of the work and where it

appears in the thesis are outlined below with a statement of per cent

contribution by the student:

1. A. Guzek (70%), P. Podsiadlo and G.W. Stachowiak, A unified

computational approach to the optimization of surface textures: One

dimensional hydrodynamic bearings, Tribology Online, Vol. 5, No 3, 2010,

pp. 150-160 (Chapter 4).

2. A. Guzek (70%), P. Podsiadlo and G.W. Stachowiak, Optimization of

textured surface in 2D parallel bearings governed by the Reynolds equation

including cavitation and temperature, Submitted to Tribology Online, 28

December 2011, Accepted 14 September 2012 (Chapter 5).

Candidate signature: ....…………………………………………….. Agata Guzek

Supervisor signature: ....…………………………………………….. A/Prof. Pawel Podsiadlo

xii

Abbreviations

1D one-dimensional

2D two-dimensional

3D three-dimensional

ALE arbitrary Lagrangian-Eulerian

BFGS Broyden–Fletcher–Goldfarb–Shanno

CFD computational fluid dynamics

COC-OPS combined optimal control and optimal parameter selection

JFO Jakobsson-Floberg-Olsson

NLMP nonlinear mathematical programming

NS Navier-Stokes

ODE ordinary differential equation

OPS optimal parameter selection

PDE partial differential equation

QP quadratic programming

RE Reynolds equation

SQP sequential quadratic programming

THD thermohydrodynamic

TSO textured surface optimization

1

Chapter 1

Introduction

The PhD project described in this thesis is presented in seven chapters. In the

Introduction (Chapter 1), a short thesis description is presented followed by

Chapter 2, where the thesis background is introduced together with

literature review. In Chapter 3 the approach/methodology is described. The

three following Chapters (4-6) form the core of this thesis; they show the

development and progression of ideas and appropriate methodologies

towards the completion of this PhD project. The thesis ends with conclusions

and suggestions for future work.

The goal of this project is to develop a unified computational approach for

the optimization of surface texture shapes in hydrodynamic lubrication

contacts. Surface shape optimization approach can be defined as a method of

finding the optimal surface shape of a hydrodynamic bearing that improves

its performance, i.e. maximizes load carrying capacity or minimizes friction

force/coefficient of friction. Although optimal surface texture shapes have

been sought in numerous studies, there is no generally accepted unified

method for surface texture shape optimization. Each of the existing methods

works only for a specific case: bearing dimensions, lubricant type and

working conditions. Therefore, an approach that could be used for any

bearing type, physical phenomena occurring and operation conditions is

required.

Chapter 1

2

To the best of the author’s knowledge, no mathematically sound, universal

approach for finding optimal textured parameters in hydrodynamic bearings

has been shown so far. Several attempts were reported in the literature

aiming to use optimization methods for determining the shape of textured

surfaces, however only limited work has been conducted to date. Existing

approaches lack mathematical foundation and do not account for physical

phenomena occurring in bearings such as inertia, temperature change and

non-Newtonian flow. These issues will be addressed in this thesis.

This PhD project is directed towards the development and validation of a

new uniform optimization approach for surface textures in hydrodynamic

bearings. The approach proposed is based on nonlinear programming

methods and accounts for phenomena such as cavitation, non-Newtonian

flow, temperature change and inertia. In order to satisfy these objectives,

optimization approaches for bearings governed by the Reynolds and Navier-

Stokes equations are presented.

1. Thesis objectives

The objectives of the thesis are following:

I. Development of an approach for optimizing surfaces in hydrodynamic

contacts governed by the one-dimensional (1D) Reynolds equation:

• Use of nonlinear programming optimization and surface texture

parametrization,

• Optimization for the maximum load capacity and minimum friction force,

• Reynolds cavitation boundary condition used,

• Extensions of the approach: non-Newtonian fluids, viscosity varying with

temperature,

• Validation of the approach by comparison against the results of conducted

exhaustive search and published data when available.

Chapter 1

3

II. Extension of the approach for optimizing surfaces in hydrodynamic

contacts governed by the two-dimensional (2D) Reynolds equation:

• Optimization of bearings governed the 2D Reynolds equation; using both

commercial FEM software and own computer program codes,

• Development of optimization approach with a mass-conserving

cavitation model,

• Coupling the Reynolds equation with the energy equation for viscosity

varying with temperature,

• Comparison of results obtained with different cavitation models,

• Validation of results against published data and exhaustive search.

III. Development of an approach for optimizing surfaces in hydrodynamic

contacts governed by the Navier-Stokes equations:

• Identification of cases for which the Reynolds equation is not valid, e.g.

when inertia effects occur,

• Development of the optimization approach for the two- and three-

dimensional (2D and 3D) Navier-Stokes equations; optimizing the domain

shape versus given parameters,

• Extension of the method to account for other phenomena occurring in

bearings, i.e. non-Newtonian fluids lubrication, temperature-dependent

viscosity,

• Validation of results against published data and exhaustive search.

Chapter 1

4

2. Thesis overview

A schematic overview of the thesis is shown in Figure 1. Additionally, a

short summary of each chapter is provided below.

2.1. Chapter 2: Optimization of surface texture shapes in hydrodynamic

contacts - Literature review

This chapter explains why surface texturing in tribology is important and

provides an overview of methods used so far to determine optimal surface

texture shapes in hydrodynamic contacts. These methods include: numerical

and experimental exhaustive searches and heuristic techniques, such as

genetic algorithms. A literature review of these methods is presented.

2.2. Chapter 3: Methodology

This chapter outlines the developments achieved to date in analytical and

numerical bearing shape optimization and confirms the necessity of

developing a new approach. Furthermore, the importance of choosing

proper lubrication equations is underlined.

In conclusion, recommendations for the development of a new unified

approach to bearing surface texture optimization are given. The significance

of this development stems from the fact that previously applied techniques

do not offer a uniform, systematic approach for bearing optimization and are

specific to particular bearing type, dimensions and operating conditions.

2.3. Chapter 4: A unified computational approach to the optimization of

surface textures: One dimensional hydrodynamic bearings

In this chapter, a new method of optimizing textured surfaces in

hydrodynamic contacts governed by the 1D Reynolds equation is proposed

and the theory that forms the foundation of this approach is presented.

The texture shape optimization (TSO) is treated as a mathematical

programming problem since the texture shape of a bearing is a mathematical

Chapter 1

5

function of spatial coordinates. To solve the optimization problem, an

approach based on the concept of combined optimal control and optimal

parameter selection (COC-OPS) is developed.

The approach allows for optimization the shapes of 1D contacts in a wide

range of bearings, such as journal bearings, step bearings and surface

textured bearings. Examples of the use of the approach are provided for

different cases, e.g. load capacity and friction force optimization in a variety

of operating conditions, such as viscosity varying with temperature and for

non-Newtonian fluids for which the Rabinowicz model [1] can be applied.

The data acquired through the optimization procedure show that the method

is valid and gives results comparable to those that can be found in literature

and obtained from numerical exhaustive search. The approach has potential

to be a valuable tool in the shape optimization of mechanical components.

An extension of the optimization into 2D cases is addressed in our following

works.

2.4. Chapter 5: Optimization of textured surfaces in 2D parallel bearings

governed by the Reynolds equation including cavitation and temperature

In this work, the optimization approach presented in the preceding chapter

is extended to account for hydrodynamic bearings governed by the 2D

Reynolds equation. With an objective function defined as the load capacity of

the bearing or the friction force/friction coefficient, optimization is

performed using sequential quadratic programming. Optimal texture

parameters sought are height ratio, area density and textured portion of

slider length. To calculate the objective function, it is required to solve the

2D Reynolds equation which is a partial differential equation. The methods

applied to solve it include the use of a commercial finite element package [2]

and development of own finite difference and finite volume program codes.

Chapter 1

6

Results of optimization are presented for a number of examples of sliders

textured with rectangular or ellipsoidal dimples. Optimization is performed

either taking the load capacity or the friction coefficient as the objective

function. The approach is extended to account for different cavitation models

and with viscosity decreasing with temperature.

The approach is validated by comparison with previously published studies

and exhaustive search conducted. The results obtained are in good

agreement with other data for similar bearings. Influence of the texture

parameters on bearing performance is in agreement with literature.

Other investigated issues include the influence of different dimple positions

on bearing load capacity, viscosity decreasing with temperature and

different cavitation boundary conditions. The data obtained show that the

application of a mass-conserving cavitation model, which can be

computationally costly, gives results similar to the simple half-Sommerfeld

model. Therefore for the tested case it is sufficient to apply the latter model.

Calculated examples confirm that the developed approach can be

successfully applied to optimize hydrodynamic bearings governed by the 2D

Reynolds equation. However, to relax the simplifying assumptions of this

equation, an extension of the approach is needed to enable optimization of

bearings governed by the Navier-Stokes equations. This will be presented in

the following chapter.

2.5. Chapter 6: Optimization of surface texture of parallel bearings

governed by the Navier-Stokes equations

This chapter extends previous work that was based on the Reynolds

equation so that optimization of a bearing governed by the Navier-Stokes

equations can be performed. The two- and three-dimensional Navier-Stokes

equations for incompressible flow are used. The domain on which the

Chapter 1

7

equations are solved is the fluid film. Thus, in order to optimize the surface

texture parameters, the domain shape has to be optimized. This is

accomplished by parametrizing the fluid film shape with respect to surface

texture parameters and then optimizing its parameters to obtain maximum

load/minimum friction coefficient. Computations are performed using

sequential quadratic programming techniques for optimization and finite

element methods for the Navier-Stokes equations solution.

Using the approach developed, optimization is conducted for a number of

cases, examples of which include infinitely long and finite bearings; inertia,

temperature and non-Newtonian fluid flow are taken into account. Bearings

are textured with rectangular or elliptical dimples. Optimization parameters

are height and length ratios for infinitely long bearings and height ratio, area

density and textured portion of slider length for finite length bearings. The

textured surfaces were optimized to either maximize the load capacity or

minimize the friction coefficient.

The optimization results obtained show the importance of including inertia

in the calculations when it occurs, as inertia can have negative impact on the

load capacity of the bearing.

Comparison of the results obtained in the study with exhaustive search and

other findings confirms the validity of the approach. Also, the data obtained

for the cases researched in previous chapters show that there is a good

agreement of the Reynolds and the Navier-Stokes equations optimization

results provided that the simplifying assumptions of the Reynolds equation

hold. The Navier-Stokes equation should be used whenever this is not the

case, e.g. when inertia effects are of significance.

Chapter 1

8

In the future, the approach could be further verified by comparing the

obtained solutions with experimental data collected from testing real

bearings.

2.6. Chapter 7: Concluding remarks and recommendations for future

work

This chapter presents a summary of the entire research project. The approach

developed allows for successful optimization of surface texture shapes in

various types of hydrodynamic contacts, governed either by the Reynolds or

the Navier-Stokes equations, with different contact geometries and physical

phenomena occurring such as cavitation, inertia, temperature-dependent

viscosity and non-Newtonian lubrication. The approach was validated by

comparison of obtained results to experimental and numerical data

published by other researchers and performed parametric exhaustive search.

The approach was primarily developed for use in tribology and lubrication

technology. However, it could also be tailored to different technical and non-

technical (e.g. biomedical) applications in which optimization of surface

texture shapes is important. Some suggestions for future research are given

in this chapter.

Chapter 1

9

References

1. Lin, J.-R., Non-Newtonian Effects on the Dynamic Characteristics of One-

Dimensional Slider Bearings: Rabinowitsch Fluid Model. Tribology Letters,

2001, 10 (4), 237-243.

2. COMSOL Multiphysics 3.5a, User Manual, COMSOL Inc.: 2009.

Chapter 1

10

List of figures

Fig. 1. Thesis overview diagram

11

Chapter 2

Optimization of surface texture shapes in hydrodynamic

contacts - Literature review

The application of surface texturing in technology is not a new development

as surface textures have been used for instance to improve the aerodynamic

characteristics of golf balls [1], or on ski surfaces to improve sliding on snow

[2]. The benefits of texturing in mechanical components were discovered in

1960s [3] which has lead to research of its potential applications in tribology.

1. Surface texturing

Due to wear and side leakage that can cause malfunction of the analytically

determined optimal step bearing [4], efforts have been made to find

alternative surface shapes that would give similar benefits to the optimal

step shape, but without these negative effects. It has been proved that

surface textures, in form of dimples, grooves, holes, chevrons and other

shapes can improve bearing performance [5-9]. Subsequently, research

efforts were undertaken to examine the influence of textured surfaces on the

operation of hydrodynamic contacts. Various forms of such textures have

been investigated for their possible reduction of friction and wear, increase

of load-carrying capacity and avoidance of seizure and damage of contact

surfaces.

Chapter 2

12

2. Optimization of surface texture shapes

To find the optimal parameters of hydrodynamic bearing surface texture

shapes, both numerical and experimental methods were applied.

To perform experimental search, it is necessary to manufacture bearings with

various textured surfaces and conduct experiments, measuring tribological

characteristics given by different surface texture parameters. Experimental

data are available in numerous published studies. Examples include: a study

by Wakuda et al. [10] where the influence of surface texture dimensions on

the friction coefficient under lubricated sliding contact was tested; research

performed by Sinanoglu et al. [11] on the influence of different shaft surface

texture shapes on the pressure buildup in a journal bearing; a study by

Marian et al. [12] who tested a thrust bearing textured with square dimples

using photolitographic wet etching, taking into account thermal effects;

investigation of the influence of shape parameters of elliptical dimples

engraved on the surface of a parallel slider on the friction coefficient by Ma

and Zhu [13]. Extensive experimental research on the influence of textures in

hydrodynamic lubrication was also performed by Etsion and his co-workers.

In their studies, the laser surface texturing (LST) technology was used to

produce microasperities on component surfaces and the effects of different

texture geometries on the load and friction of mechanical seals [14-16], piston

rings [17-19] and thrust bearings [20] were experimentally investigated. Each

of the above studies offered some new developments in finding optimal

surface textures. However, the experimental approach is time-consuming

and the cost of performing many tests is often prohibitively high.

In addition to the experimental work, with the increasing computing power

available, there has been growing interest in numerical studies. Research

efforts were mainly focused on parametric search, i.e. solving the governing

equations numerically multiple times, for every possible set of texture

Chapter 2

13

parameters, and selecting the configuration that yields the best results.

Examples of such research on bearings governed by the Reynolds equation

include an investigation of the influence of different asperity shapes on the

pressure and friction coefficient in a thrust bearing by Siripuram and Stevens

[21], a study of the influence of spherical dimples on the load capacity of a

parallel slider by Brizmer et al. [22], a mass-conserving study of a parallel

slider textured with trapezoidal dimples by Dobrica et al. [23], a study of a

slider with an elliptical texture by Ma and Zhu [13], a study of a textured

journal bearing by Qiu and Khonsari [24], and a parametric study of the

influence of surface texturing on the performance of converging bearings by

Fowell et al. [25]. Due to the complexity of solving the Navier-Stokes

equations, there have been fewer studies of textured bearings governed by

these equations. Such studies were conducted e.g. by Sahlin et al. [26] and

Cupillard et al. [27-29].

Performing many numerical evaluations can be time-consuming and

expensive as a huge amount of computing power is needed. Therefore, the

idea of using numerical optimization methods in the search for optimal

surface texture parameters arose. However, so far only heuristic methods

such as genetic algorithms have been used. Genetic algorithms were applied

by Buscaglia et al. [30] in optimization of size and depth of microdimples in

a finite slider governed by the Reynolds equation for the minimum friction

force. The slider was square and covered with square dimples arranged in a

square pattern. Optimization was conducted by assigning a random

combination of the parameters to every population member and then

evaluating values of their respective objective functions. Subsequently, a

selection procedure was carried out to choose the surviving individuals and

then crossover between each pair was performed. Finally, a random

mutation of individuals was performed. By following this procedure, dimple

heights and sizes for each group of 4 dimples were obtained for a predefined

Chapter 2

14

number of 140 dimples and a given location of the dimpled zone. However, a

large number of function evaluations was required in this study. Also,

although a reduction of up to 5.2% in the friction force was obtained (for

18000 function evaluations), it cannot be proved whether the solution was

optimum.

Also, Papadopoluos et al. [31] optimized texture parameters in infinitely

long micro-thrust bearings governed by the Navier-Stokes equations with

the use of genetic algorithms aided by local search to improve convergence

rates. The bearing was optimized for the maximum load carrying capacity

and the trapezoidal texture parameters sought were: dimple area density,

dimple height, left and right length of the trapezoidal dimple and untextured

length at bearing outlet. This study was then extended to finite micro-thrust

bearings [32] with rectangular grooves. The concept of Pareto dominance

was applied to enable optimization with two objectives: maximum load and

minimum bearing clearance. With the use genetic algorithms, bearing

parameters that gave significant improvement were obtained.

However, the use of heuristic techniques such as genetic algorithms is

widely criticized; they provide a shape that gives an improvement from the

initial value but they do not guarantee to find the optimum and do not have

mathematical foundation [33].

3. Conclusions

From the literature review conducted, the conclusion can be drawn that

textured surfaces are mostly studied by trial and error. This procedure can

be costly and time consuming, both when conducting experimental studies

and when using numerical parametric searches which require adequate

computing power and have long running times.

Chapter 2

15

There have been some attempts to apply numerical optimization techniques,

however, only heuristic algorithms have been used to date. These methods

do not guarantee finding the optimum and lack mathematical foundation;

there has been no mathematically proven optimization method used for

texture shapes in hydrodynamic contacts.

Also, each of the presented optimization approaches works only for a

specific case: specific bearing geometry, operating conditions and texture

shape. No uniform approach to find the optimum texture is given that could

provide solutions for any bearing shape.

Thus, the methodology proposed in this thesis aims to design a unified

optimization approach that could be applicable to a wide range of

hydrodynamic contacts and would be based on mathematically proven

optimization theories.

Chapter 2

16

References

1. Davies, J. M., The Aerodynamics of Golf Balls. Journal of Applied

Physics, 1949, 20 (9), 821-828.

2. Eaton, E. V. Ski Bottom Finishing Method. US3652102, 1972.

3. Hamilton, D. B.; Walowit, J. A.; Allen, C. M., A Theory of Lubrication by

Micro-Asperities. ASME Journal of Basic Engineering, 1966, 88 (1), 177-

185.

4. Stachowiak, G. W.; Batchelor, A., Engineering Tribology. 2005.

5. Moore, D. F., A History of Research on Surface Texture Effects. Wear,

1969, 13, 381-412.

6. Costa, H. L.; Hutchings, I. M., Hydrodynamic Lubrication of Textured

Steel Surfaces under Reciprocating Sliding Conditions. Tribology

International, 2007, 40, 1227–1238.

7. Etsion, I., State of the Art in Laser Surface Texturing. ASME Journal of

Tribology, 2005, 127 (1), 248-253.

8. Li, Y.; Menon, A. K., The Development and Implementation of Discrete

Texture for the Improvement of Tribological Performance. ASME Journal of

Tribology, 1995, 117 (2), 279-284.

9. Etsion, I., Improving Tribological Performance of Mechanical Components

by Laser Surface Texturing. Tribology Letters, 2004, 17 (4), 733-737.

10. Wakuda, M.; Yamauchi, Y.; Kanzaki, S.; Yasuda, Y., Effect of Surface

Texturing on Friction Reduction between Ceramic and Steel Materials under

Lubricated Sliding Contact. Wear, 2003, 254, 356–363.

11. Sinanoglu, C.; Nair, F.; Baki Karamis, M., Effects of Shaft Surface Texture

on Journal Bearing Pressure Distribution. Journal of Materials Processing

Technology, 2005, 168 (2), 344-353.

12. Marian, V. G.; Kilian, M.; Scholz, W., Theoretical and Experimental

Analysis of a Partially Textured Thrust Bearing with Square Dimples.

Chapter 2

17

Proceedings of the Institution of Mechanical Engineers. Part J, Journal

of engineering tribology, 2007, 221 (7), 771-778.

13. Ma, C.; Zhu, H., An Optimum Design Model for Textured Surface with

Elliptical-Shape Dimples under Hydrodynamic Lubrication. Tribology

International, 2011, 44 (9), 987–995.

14. Etsion, I.; Burstein, L., A Model for Mechanical Seals with Regular

Microsurface Structure. Tribology Transactions, 1996, 39 (3), 677 - 683.

15. Etsion, I.; Kligerman, Y.; Halperin, G., Analytical and Experimental

Investigation of Laser-Textured Mechanical Seal Faces. Tribology

Transactions, 1999, 42 (3), 511-516.

16. Etsion, I.; Halperin, G., A Laser Surface Textured Hydrostatic Mechanical

Seal. ASLE Transactions, 2002, 45 (3), 430-434.

17. Ryk, G.; Kligerman, Y.; Etsion, I., Experimental Investigation of Laser

Surface Texturing for Reciprocating Automotive Components. Tribology

Transactions, 2002, 45 (4), 444-449.

18. Ryk, G.; Kligerman, Y.; Etsion, I.; Shinkarenko, A., Experimental

Investigation of Partial Laser Surface Texturing for Piston-Ring Friction

Reduction. Tribology Transactions, 2005, 48 (4), 583-588.

19. Ryk, G.; Etsion, I., Testing Piston Rings with Partial Laser Surface

Texturing for Friction Reduction. Wear, 2006, 261 (7-8), 792-796.

20. Etsion, I., Experimental Investigation of Laser Surface Textured Parallel

Thrust Bearings. Tribology Letters, 2004, 17 (2), 295-300.

21. Siripuram, R. B.; Stephens, L., Effect of Deterministic Asperity Geometry

on Hydrodynamic Lubrication. ASME Journal of Tribology, 2004, 126 (3),

527-534.

22. Brizmer, V.; Kligerman, Y.; Etsion, I., A Laser Surface Textured Parallel

Thrust Bearing. Tribology Transactions, 2003, 46 (3), 397-403.

23. Dobrica, M. B.; Fillon, M.; Pascovici, M. D.; Cicone, T., Optimizing

Surface Texture for Hydrodynamic Lubricated Contacts Using a Mass-

Chapter 2

18

Conserving Numerical Approach. Proc. IMechE Part J: Journal of

Engineering Tribology, 2010, 224 (Special Issue), 737-750.

24. Qiu, Y.; Khonsari, M. M., On the Prediction of Cavitation in Dimples

Using a Mass-Conservative Algorithm. Trans. ASME: Journal of

Tribology, 2009, 131 (3), 041702.

25. Fowell, M. T.; Medina, S.; Olver, A. V.; Spikes, H. A.; Pegg, I. G.,

Parametric Study of Texturing in Converging Bearings. Tribology

International, 2012, 52, 7-16.

26. Sahlin, F.; Glavatskih, S. B.; Almqvist, T.; Larsson, L., Two-Dimensional

CFD-Analysis of Micro-Patterned Surfaces in Hydrodynamic Lubrication.

ASME Journal of Tribology, 2005, 127 (1), 96-102.

27. Cupillard, S.; Glavatskih, S.; Cervantes, M. J., Computational Fluid

Dynamics Analysis of a Journal Bearing with Surface Texturing.

Proceedings of the Institution of Mechanical Engineers Part J: Journal

of Engineering Tribology, 2008, 222 (2), 97-107.

28. Cupillard, S.; Glavatskih, S.; Cervantes, M. J., Inertia Effects in Textured

Hydrodynamic Contacts. Proc. IMechE Part J: Journal of Engineering

Tribology, 2010, 224, 751-756.

29. Cupillard, S.; Glavatskih, S.; Cervantes, M. J., 3D Thermohydrodynamic

Analysis of a Textured Slider. Tribology International, 2009, 42, 1487–

1495.

30. Buscaglia, G. C.; Ciuperca, I.; Jai, M., On the Optimization of Surface

Textures for Lubricated Contacts. Journal of Mathematical Analysis and

Applications, 2007, 335 (2), 1309-1327.

31. Papadopoulos, C. I.; Nikolakopoulos, P. G.; Kaiktsis, L., Evolutionary

Optimization of Micro-Thrust Bearings With Periodic Partial Trapezoidal

Surface Texturing. Journal of Engineering for Gas Turbines and Power,

2011, 133 (1), 012301.

Chapter 2

19

32. Papadopoulos, C. I.; Efstathiou, E. E.; Nikolakopoulos, P. G.; Kaiktsis,

L., Geometry Optimization of Textured Three-Dimensional Micro-Thrust

Bearings. ASME Journal of Tribology, 2011, 133 (4), 041702.

33. Reeves, C. R.; Rowe, J. E., Genetic Algorithms: Principles and Perspectives:

a Guide to GA Theory Kluwer Academic Publishers: Norwell, MA, USA,

2002.

20

Chapter 3

Methodology

The search of an optimal shape that could improve performance of

hydrodynamic components such as bearings has been an ongoing subject of

research for decades. In this chapter the surface texture optimization

problem is formulated, historical approaches to solving this problem are

described and a new approach that would not exhibit the limitations of the

previously used methods is proposed.

1. Problem formulation

The optimization problem can be stated as:

Find the optimum shape h(x,y) that maximizes (or minimizes) an objective

functional

)),(( yxhg

subject to the following constraints:

• System dynamics: a differential equation governing the pressure in a

hydrodynamic bearing,

• Boundary conditions that the pressure vanishes on the bearing edges,

where g is an integral that represents a friction force or a load or other

optimization objectives.

Chapter 3

21

2. Analytical solutions

2.1. One-dimensional (1D) Reynolds equation

The earliest studies in this field were conducted analytically. In 1918, Lord

Rayleigh obtained the one-dimensional optimal step bearing shape that has

the maximum load capacity using the calculus of variations [1].

For the optimization problem, the pressure in the slider bearing was

governed by the dimensionless 1D Reynolds equation:

063 =

+ hdx

dph

dx

d

subject to the boundary conditions 0)1()0( == pp ,

where h≥ h0, h0 is a given minimum film thickness.

The objective is to find h which maximizes the load capacity ∫=1

0

pdxW .

To achieve this, the first variation of W is obtained:

{ }xdxhhdxh

xdxhx

h

hW∫

∫∫ −

−−=−

−

23

12 3

3

4

δδ,

where δh is a variation of h, h is the integration constant given by

∫∫=1

03

1

02 h

dx

h

dxh .

For an optimum to exist, Wδ has to vanish for all variations δh. This is

satisfied when

∫∫

−

−

=dxh

xdxhx

3

3

or hh2

3= .

It was postulated that if hδ vanishes for h>h0, xϵ[L 1,1] and is negative when

h=h0, xϵ[0,L1] , any positive δh diminishes W. This is achieved when the two

conditions are satisfied:

• ,132 2 >− )k(k

Chapter 3

22

• ,)k(kLLhLhL

hLhLhh 323

2 2123

22311

222

211

2 −=⇒++==

where k=h1/h0, h0 and h1 are film heights of two parts of the bearing step, L1 is

the location of the step as shown in Fig. 1 and L2 =1- L1.

Substituting these conditions into the equation for W yields:

)(1

321

3212)23(

120

2320

2

32212

120

kfhkk

k

hk

LLLkL

hW =

−+−=

+

+−= .

The maximum of )(kf is 0.2026 and it occurs when k=h1/h0=1.87and 12 LL =

2.588. These dimensions fully describe the optimal Rayleigh step bearing as

shown in Fig. 1. This result was later confirmed by Maday [2] using the

bounded variable method.

The slider bearing was also optimized analytically by Rohde [3] for the

minimum coefficient of friction,][

][][

hW

hFhff == , where the load capacity ][hW

and the friction force ][hF are:

dxxhxh

hxhW ∫

−−=

1

023 )(

1

)(

16][

dxxh

hxh

hF ∫

−=

1

02 )(

13

)(

4][ ,

∫∫=1

03

1

02 )()( xh

dx

xh

dxh ,

Using the calculus of variations, the first variation of f was obtained:

hdxhkxW

Fxhhxh

hWf δδ

−+−

−−= ∫ 23

33)(2

3)(

14 *1

04 ,

where δh is any variation such that 1)( ≥+ hxh δ and ∫∫=1

03

1

03

*

)()( xh

dx

xh

xdxk .

Chapter 3

23

In order for h to minimize f, it is necessary that 0≥fδ for all non-vanishing

variations δh and 0=fδ for all vanishing variations δh. This is fulfilled if one

of the two conditions is satisfied, i.e.:

• hxh2

3)( = or

• hkW

Fx

W

F)x(h *

2

333 +−= .

Basing on the above conditions, the following optimal shape was derived:

0,

1)(3

)(3

0.1

)(

434

433

3

>

≤≤−

≤≤−

≤≤

=W

F

LLLW

F

LLLxW

F

L

xh

x

x

x0

where *

3

34

32

330.1

2

30.1)(3

kW

FhL

W

F

hLLW

F

−=−

=+−

where: L3=0.1822, L4=0.2660, W

F=3.994, h1=2.00 (Fig. 2). It was confirmed that

this shape minimizes f by checking that the second variation of f is positive.

Using the above method, Rohde [4] optimized an infinitely long journal

bearing governed by the Reynolds equation in cylindrical coordinates (x

becomes θ):

,063 =

+ hd

dp)(h

d

d

θθ

θ .020 == )(p)(p π

The objective functional to be maximized is the dimensionless load capacity:

.

22

0

22

0

2

+

= ∫∫

ππθθθθθθ dcos)(pdsin)(p]h[W

The first variation of W with respect to h in the

−ψπ2

direction is:

)(h

h)h)(h())cos(]h[b(W

θδθψθδ

π

4

2

02

312 −×+−−= ∫ ,

where ,∫∫+=

ππ

φφ

φφψφ

2

03

2

03 )(h

d

)(h

d)cos(]h[b ,

= ∫∫

ππ

φφ

φφ

2

03

2

02 )(h

d

)(h

dh

πψ 20 ≤≤ and δh

is any piecewise continuous function for πθ 20 ≤≤ such that 1≥+ hh δ .

Chapter 3

24

hopt maximizing W exists if:

• 0h , pressure gradient will be negative and H will be maximized for

negative 2λ . Taking into account these derivations and boundary conditions,

Chapter 3

25

the optimal solution found was a step journal bearing with the following

parameters: ,208.1=h θ0=31.00o θk=211.00o, h0=1, h1=1.812. The same solution

was obtained by Rohde [4] (Fig. 3).

In the aforementioned studies, analytical optimization was undertaken for

bearings lubricated with fluids of constant viscosity. However, this

assumption is not always valid as viscosity may vary e.g. with pressure or

temperature in bearing. Some analytical solutions were obtained for the

cases of pressure-dependent viscosity.

Maday [2] optimized a 1D slider bearing with varying viscosity. The bearing

was governed by the 1D Reynolds equation:

063 =

− hdx

dph

dx

d η

∫

∫= 1

03

1

02

h

dxh

dx

h η

η

The objective functional to be maximized was:

∑∫=

+=2

1

1

0 iii c)x(pdxW λ

where iλ are undetermined Lagrange multipliers and ic are constraints.

The Euler-Lagrange conditions and the Weierstrass condition for this

problem were used:

0

0

032

6

161

1

2

2431

3211

≥−+=

=+

−

∂∂

−−=

)'p)'pp((

h

h

h

ph

h

h'

δλγλ

ληλ

ηλλ

where γ is a real function of x.

Chapter 3

26

If the conditions are satisfied and 0=∂∂

p

η, the solution is the classical Rayleigh

step shape. For 0≠∂∂

p

η, the solution is a step shape where location of the step

can be found by solving the derived equations for a given relationship

between η and p.

An analysis of the optimal load carrying capacity of 1D sliders with

pressure-dependent viscosity was also performed by Charnes et al. [6]. The

pressure-viscosity relation was defined as peαηη 0= , where α was a constant

and η0 viscosity at reference pressure. Variational calculus methods were

used to obtain an optimum step shape of Rayleigh, and then to derive a

formula denoting ratio between load capacities in the pressure dependent

and independent case:

G

)Gln(G

G

W

W p

β

ββ

βα

212

211 −−+

=

where20

0

h

Uαηβ = , 20603329

4.)(G =−= . Values of the ratio were then plotted

for different fluid viscosity coefficients β .

2.2. Two-dimensional (2D) Reynolds equation

Optimization shown in the previous section has only been performed for

bearings governed by the 1D Reynolds equation, which is an ordinary

differential equation. To perform the optimization in 2D, the methods of the

calculus of variations have to be supported by numerical techniques as the

governing partial differential equations cannot be solved analytically.

A finite slider profile governed by the 2D Reynolds equation was optimized

by Rohde and McAllister [7]. The optimal film shape in a square slider

bearing for the maximum load capacity was found using a method designed

by McAllister and Rohde [8]. In the method, starting from a given initial film

Chapter 3

27

shape, film height values increasing the load were found analytically, using

the calculus of variations, in an iterative manner until the difference between

subsequent iterations was smaller than a given threshold value. Gradients

and partial differential equation solutions were obtained numerically, using

finite differences. The optimum load was obtained for a pocket shape.

Recently, film height optimization in finite hydrodynamic slider bearings

governed by the 2D Reynolds equation was undertaken by Ostayen et al.

[9, 10]. Variational calculus approach was used together with COMSOL finite

element software [11]. Constrained optimization with the adjoint approach

was applied to obtain optimal film shape function for the maximum load.

This was conducted by calculating the Lagrange functional:

∫Ω

⋅Λ+−= dx)h,p()h,p(WL RE

where Λ is the adjoint variable, RE(p,h) is the Reynolds equation, and the

first order optimality conditions are:

equation DecisionDE

equation Adjoint AD

RE

02

1

12

3

0112

0

2

3

=⋅∇+∇⋅∇=Λ=∂∂

=⋅∇=Λ=∂∂

==Λ∂

∂

UΛpΛh

-)h,,p(h

L

)-h

(- )h,(p

L

)h,p(L

The optimization procedure began with an initial guess for p, Λ, h. The given

equations were solved one after another for new values of p, h, Λ and then

the cycle was repeated until convergence was achieved. Optimal film shape h

was obtained for three different configurations: a square parallel slider

bearing (results agreed with the data reported by Rohde [12]), a circular

parallel slider bearing, and a square slider with sliding speed oriented at 45o

relative to its edges.

Chapter 3

28

3. Numerical methods

Analytical solutions shown were obtained for relatively simple cases of 1D

and 2D bearings. However, for realistic and complex bearing configurations

numerical methods were used. The methods can be grouped into heuristic

techniques and nonlinear programming.

Heuristic methods are used to give an improvement from an initial

candidate solution with no guarantee of optimality. They are usually applied

when no mathematically sound, efficient method to approach the

optimization problem is known. These methods are often based on common

sense rules or biological processes such as evolution and genetics [13].

An example of the use of such methods in bearing optimization is a study by

Rohde [12], who sought optimal square slider shape parameters for several

classes of shapes, as shown in Fig. 4. Film shape parameters optimized were

the step height and the coordinates of the vertices of the polygons. A

heuristic numerical scheme based on the Rosenbrock’s optimization method

was used. This method prescribes an algorithm of searching for an

improvement to the objective function, where the way to determine the

length of step is determined by success/failure of the previous step and the

step direction is found by trial and error [14]. Using this method, optimal

values of shape parameters were determined. For solving the 2D Reynolds

equation, a finite element method was applied.

The other numerical method used is nonlinear programming, i.e.:

Minimize (or maximize) an objective functional

( )xg x

min

subject to constraints:

,...,mj,)x(t

,...,ni,)x(s

j

i

10

10

=≥==

where the objective g and/or constraints s, t are nonlinear functions.

Chapter 3

29

Nonlinear programming was applied for optimization in tribology by Wang

et al. [15]. Optimal film shapes for elliptical and slider bearings governed by

the 2D Reynolds equation x

hU

y

ph

yx

ph

x ∂∂=

∂∂

∂∂+

∂∂

∂∂ η633 were found using

golden search, simplex method and lattice search. Optimization methods

used did not require any derivative evaluations. In this study, optimization

of an elliptical bearing was conducted for three different objectives:

• Maximize the minimum film thickness, where optimized parameters

were bearing ellipticity ratio and orientation angle.

• Maximize the sum of peak square pressures with the same

optimization parameters.

• Minimize the frictional torque. Optimized parameters were ellipticity

ratio, orientation angle and radial clearance.

The first two optimizations were performed twice, using simplex and lattice

search methods and in the third one, lattice search was used.

Both simplex and lattice methods rely on a geometric with vertices

representing different parameter sets. The search is conducted in a direction

determined from the value of the objective function at vertices (and in the

center in the case of lattice search). Although the lattice method requires

more function evaluations per geometric, the search path obtained with the

use of this method in this study was more straightforward and thus

computational time was lower than for the simplex method.

A finite slider was also optimized in this study with the maximum load

capacity taken as objective. Several types of sliders were investigated with

their geometric variables taken as optimization parameters:

• Step bearing with the following parameters: height and location of the

step. Optimization was performed using lattice search.

Chapter 3

30

• Pocket slider with the following parameters: three points describing

the contour of the pocket and the film height. Lattice search was

applied.

• Taper bearing, where the parameter was the slope of the inclined

surface of the slider. In this one-parameter case, optimization was

conducted using golden section method, where in each calculation the

search region was narrowed down to 61.8% of the previous solution.

The methods applied in the study provide an improvement in computational

efficiency as compared to exhaustive search. However, constraints on

parameter values cannot be introduced and therefore an unrealistic solution

may be found. Thus, these methods are not always reliable in bearing

optimization as there are usually some constraints on bearing geometric

variables.

This limitation can be overcome by applying nonlinear optimization

methods that can handle nonlinear constraints, such as sequential quadratic

programming. The procedure of solving a nonlinear problem is simplified to

solving a sequence of quadratic subproblems subject to linearized

constraints.

This method was first applied in bearing optimization by Hashimoto [16],

who optimized dimensions of a short journal bearing governed by the

Reynolds equation. The optimization objective was the weighted sum of

fluid film temperature rise and supply lubricant quantity. The optimized

parameters were the radial clearance, bearing length to width ratio and

average lubricant viscosity. The results obtained were promising for bearing

design; however the study was conducted for a specific case and under the

short bearing assumption, therefore its applicability is limited.

Chapter 3

31

Recently, sequential quadratic programming method produced optimum

film heights of finite width sectorial thrust bearings [17]. The optimization

objective was the maximum load capacity and three cases were studied:

• Film thickness in the radial direction was constant. Film thickness in

the circumferential direction was optimized.

• Film thickness in a quadrilateral shaped domain was constant and

optimum location of each corner of the domain and the film height

were sought.

• Optimum film thickness on the entire bearing surface was searched.

This method can give optimal solutions in terms of arbitrary film thickness,

however its application is limited as a very fine grid would be needed to

accurately capture the surface of a bearing. With every element of the grid

corresponding to an optimization parameter, the computational cost of such

an optimization could be prohibitive.

4. Choice of governing equations

When optimizing hydrodynamic contacts, the choice of equations governing

the pressure distribution in a hydrodynamically lubricated component is of

importance.

In general, the pressure in a lubricating fluid film is governed by the Navier-

Stokes equations. However, the pressure distribution in a hydrodynamic

bearing can be approximated by the equation derived by Reynolds [18]

provided that the simplifying assumptions listed in Table 1 hold.

The derivation of the Reynolds equation from the Navier-Stokes equations

was presented e.g. by Dowson in 1962 [19]. This generalized Reynolds

equation neglects phenomena such as the pressure variation through film

thickness and inertia [20]. Also, in the derivation of this equation, the so-

Chapter 3

32

called “thin film assumption” was made: the scales along the fluid film are

three orders of magnitude larger than the scale across the fluid film [21].

The influence of inertia and the limits of the “thin film” hypothesis were

determined in a study of a 1D Rayleigh step bearing comparing the Reynolds

and the Navier-Stokes based models, performed by Dobrica et al. [20]. Film

thickness in the simulation was gradually increased and different bearing

operating speeds were tested. It was found that the operating speed

influences the applicability of the Reynolds equation. For the runner velocity

of 30 m/s, relative difference between load capacities obtained from the

Navier-Stokes and Reynolds solutions was less than 5% for film thicknesses

of around 100µm but increased to 20% for film thicknesses equal to 500µm.

For a smaller velocity (10 m/s) there was also a difference between the two

models increasing with the film thickness, but it was not that pronounced. It

was concluded that the Reynolds equation is not satisfactorily accurate for

large film thicknesses (above 200µm), because of a pressure drop right after

the discontinuity line due to inertia effects, and that neglecting inertia

produces errors of more than 5%. For film thicknesses of 500 µm and above,

the errors are so pronounced that the Navier-Stokes equations should be

applied.

As the operational speed of the bearing influences the modified Reynolds

number (defined as B

UhRe

ηρ 20= ), in several studies the range of Re numbers

for which inertia effects are significant and cannot be neglected was

determined. Tichy and Chen [22] experimentally and analytically analysed

the influence of the number on fluid inertia in an infinite slider bearing. It

was concluded that fluid inertia effects are important for Re numbers greater

than 10. For Re below one, fluid inertia is negligible and the Reynolds

equation can be applied. In a study of finite width journal bearings governed

by the Navier-Stokes equations, Nassab [23] stated that neglecting inertia

Chapter 3

33

forces is only justified for small values of the Reynolds number (of the order

of 1). The pressure field was influenced more by inertia than by temperature

effects. Even with low Reynolds numbers (of the order of 10), inertia can

have significant effects on thermohydrodynamic characteristics if a journal

bearing is run under low load, high speed and large clearance conditions.

In another study on the importance of inertia effects, Sahlin et al. [24]

compared the Navier-Stokes and the Stokes solutions (i.e. solutions with

advective/inertia terms truncated) for different Reynolds numbers in a 2D

parallel bearing with a cylindrical groove. For Re=10, the difference between

the pressure profiles obtained with the two equations was low, but for

Re=160 it was significant. The Navier-Stokes solution gave much higher

pressure jump at the groove and much larger maximal pressure. Also, Arghir

et al. [25] conducted an analysis of incompressible, laminar and isothermal

flow where inertia effects are of importance, i.e. when Stokes model cannot

be used e.g. for some modern applications with the characteristic Reynolds

number is of the order of 104-105 (high speed applications with low viscosity

fluids). It was stated that the pressure jump at fluid film discontinuity is an

effect of the inertia forces, which deems the use of the Stokes equations

insufficient. Cupillard et al. [26] determined that inertia in infinitely long

textured parallel sliders is important for Re>60 and can decrease the load

capacity of the bearings, especially for deep dimples.

In addition to the importance of inertia effects depending on film thickness

and Re number, further conclusions with respect to bearing dimensions can

be made. Dobrica and Fillon [27] investigated the influence of the dimple

aspect ratio (ratio between dimple length and dimple depth) and the

Reynolds number on the presence of inertia effects. For small dimple aspect

ratios and large Re numbers, inertia had large influence on bearing

performance and it was deemed necessary to use the Navier-Stokes

equations. For small aspect ratios and small Re numbers, inertia effects were

Chapter 3

34

negligible, so it was stated that in this case the Stokes approximation can be

used.

In conclusion, the Navier-Stokes equations should be used whenever inertia

effects can be of importance: when the thin film assumption does not hold,

when dimple aspect ratio is low and when Re number is large, i.e. for high

runner velocities and/or low viscosity fluids. Therefore, it is important to

cover these cases in bearing surface texture optimization by designing an

optimization approach for the Navier-Stokes equations.

Another phenomenon that can have significant influence on the pressure in a

hydrodynamic bearing is cavitation. Because the governing equations

describe lubricated areas in bearings, other methods have to be incorporated

to account for the pressure loss in cavitated areas. It has been proved that

cavitation occurs in bearings with textured surfaces [28, 29], therefore it

cannot be neglected in current work. One of the most straightforward ways

to deal with cavitation is the half-Sommerfeld boundary condition which

stipulates that all negative pressures predicted in the bearing are neglected.

This condition, although simple, does not account for the continuity of flow

and is therefore unphysical [30].

Another simple and thus commonly used cavitation model is the Reynolds

boundary condition. This condition stipulates that at the boundary between

zero and non-zero pressure, 0==xd

dpp applies [30]. In effect, negative

pressures are avoided by replacing them with zero. This boundary condition

has been used in many studies on textured surfaces [28, 29, 31]. However,

the Reynolds condition is also unphysical, as it correctly predicts fluid film

rupture, but neglects film reformation [32]. It was shown to be inaccurate in

textured bearings, due to its underestimation of the cavitated area [33].

Therefore, employment of an approach that would preserve the principle of

the conservation of mass is preferred. The mass-conserving cavitation theory

Chapter 3

35

was established by Jakobsson, Floberg and Olsson [34]. A numerical

algorithm following this development was first presented by Elrod [35, 36].

A universal Reynolds equation proposed was valid throughout the fluid

film, with an additional variable changing its value in cavitated and non-

cavitated areas. This algorithm was then used in numerous studies,

including those on textured surfaces. Ausas et al. applied it for. 1- and 2D

slider and journal bearings under steady and dynamic loading [33, 37]; a

comparison of the use of this method to the Reynolds boundary condition

proved that the Elrod’s algorithm is preferred in the case of microtextured

bearings as it yielded more accurate results. A mass-conserving algorithm

was also used in a study of parallel dimpled sliders by Dobrica and Fillon

[38] where parametric search was conducted to obtain optimal trapezoidal

structure parameters. Another mass-conserving study was performed by Qiu

and Khonsari [39] for a textured mechanical seal/thrust bearing, basing on

Vijayaraghavan’s modification of Elrod’s algorithm [40]. Estimates of

cavitation in dimples on surfaces of components showed that applying the

JFO theory gave more realistic results than other cavitation boundary

conditions. In a recent parametric study of 1D textured bearings, Fowell et al.

[41] stated that cavitation is of great importance in bearing analysis because

it may significantly influence the optimum load carrying capacity and cause

variations in bearing performance, giving positive results when the inlet

suction mechanism occurs or negative results when an extended region with

minimum pressure is present along the cavitated length.

The studies described above show that cavitation should be taken into

account when developing optimization methods for surface textured

bearings and that a mass-conserving algorithm which gives most accurate

results should be applied.

In most studies of bearings with textured surfaces, analysed lubricants are

assumed to be Newtonian. However, lubricants with additives such as

Chapter 3

36

polymers are frequently used in bearings to reduce viscosity change due to

temperature [30]. Studies have shown that non-Newtonian lubrication may

have effect on the load carrying capacity and thus is important to include in

present work as it may influence optimization results. There are many

models describing the nonlinear relationship between the shear stress and

the shear rate in non-Newtonian fluids. For example, non-Newtonian

lubrication can be solved with the Reynolds equation by using the

Rabinowicz model. This was performed in a study by Lin [42], who

examined the influence of pseudoplastic and dilatant fluids on 1D slider

bearings. It was found that the lubricant behaviour affects the pressure in

bearings and thus changes the load carrying capacity, which is higher for

dilatant and lower for pseudoplastic fluids. A commonly used method of

accounting for non-Newtonian behaviour in tribological calculations is the

power-law model which was applied in a study of infinitely wide slider

bearings lubricated with non-Newtonian fluids with the Navier-Stokes

equations conducted by Das [43]. It was found that higher power-law indices

give higher load capacities. The influence of power-law lubricants was also

studied by Buckholz [44] for a finite width plane bearing. Other non-

Newtonian models used in tribology include the Eyring model, applied e.g.

by Tayal et al. [45]; multi-grade oil model proposed by Gecim et al. [46] and

modified by Paranjpe [47]; couple stress model [48] and viscoelastic models

[49] such as the Maxwell model [50, 51].

Because in real-life applications, temperature increase in working mechanical

components occurs, it is necessary to account for the change in viscosity due

to temperature rise in an operating bearing. The loss of viscosity caused by

heating may lead to significant reduction in bearing load capacity [30]. To

take thermal effects in a bearing into consideration, an equation describing

temperature changes – the energy equation – needs to be solved

simultaneously with an equation governing the hydrodynamics of the

Chapter 3

37

bearing. Thermohydrodynamic (THD) analysis of bearings was conducted

already in the 1970s, using the generalized Reynolds equation accounting for

variations in viscosity derived by Dowson [19]. Early numerical models were

proposed by Ezzat and Rohde, using a finite difference method [52] and

Tichy, using a finite element method [53]. Those methods were later

improved to account for reverse flow at inlet and elastic effects by

Boncompain et al. [54], where optimal size of a pocket in a slider bearing

governed by the 2D Reynolds equation was obtained through a parametric

analysis. A THD study of a textured slider was also performed by Marian et

al. [55] who analysed a partially textured thrust bearing by numerically

solving the Reynolds and energy equations and validated the results

experimentally. A finite-volume method for solving THD 1D step bearings

was proposed in [20]. A THD analysis of a 3D bearing governed by the NS

equations was conducted by Cupillard et al. [56]. A textured bearing with

three dimples and a fore-region was examined, the influence of temperature

change on the pressure distribution was observed. The above studies show

that including the temperature effect can have influence on optimization

results and thus should not be neglected.

When solving the governing hydrodynamic lubrication equations

numerically, the question arises whether the used grid size is fine enough to

give accurate solution. In order to test grid sizes and determine their

suitability, methods of checking the grid independence of the solution have

been developed. A common technique used in fluid dynamics calculations is

the Richardson extrapolation [49]. It is applied by solving the equations on

subsequently refined grids and using the obtained solutions to estimate the

grid convergence error and/or grid convergence index [57] from the

Richardson extrapolation. However, the use of this method has not been

reported in many studies in the field of tribology. It was applied by

Cupillard et al. [56] who used it to estimate numerical error and choose

Chapter 3

38

appropriate grid size in a study of 3D textured sliders governed by the

Navier-Stokes equati