Finite Element Modeling of Elastohydrodynamic Lubrication Problems

Covers the latest developments in modeling elastohydrodynamic lubrication (EHL) problems using the finite element method (FEM)

This comprehensive guide introduces readers to a powerful technology being used today in the modeling of elastohydrodynamic lubrication (EHL) problems. It provides a general framework based on the finite element method (FEM) for dealing with multi-physical problems of complex nature (such as the EHL problem) and is accompanied by a website hosting a user-friendly FEM software for the treatment of EHL problems, based on the methodology described in the book. Finite Element Modeling of Elastohydrodynamic Lubrication Problems begins with an introduction to both the EHL and FEM fields. It then covers Standard FEM modeling of EHL problems, before going over more advanced techniques that employ model order reduction to allow significant savings in computational overhead. Finally, the book looks at applications that show how the developed modeling framework could be used to accurately predict the performance of EHL contacts in terms of lubricant film thickness, pressure build-up and friction coefficients under different configurations.

Finite Element Modeling of Elastohydrodynamic Lubrication Problems offers in-depth chapter coverage of Elastohydrodynamic Lubrication and its FEM Modeling, under Isothermal Newtonian and Generalized-Newtonian conditions with the inclusion of Thermal Effects; Standard FEM Modeling; Advanced FEM Modeling, including Model Order Reduction techniques; and Applications, including Pressure, Film Thickness and Friction Predictions, and Coated EHL.

This book:

• Comprehensively covers the latest technology in modeling EHL problems
• Focuses on the FEM modeling of EHL problems
• Incorporates advanced techniques based on model order reduction
• Covers applications of the method to complex EHL problems
• Accompanied by a website hosting a user-friendly FEM-based EHL software

Finite Element Modeling of Elastohydrodynamic Lubrication Problems is an ideal book for researchers and graduate students in the field of Tribology.

WASSIM HABCHI, PHD, is an Associate Professor of Mechanical Engineering in the Department of Industrial and Mechanical Engineering at the Lebanese American University, Lebanon. His main area of expertise is in the finite element modeling of elastohydrodynamic lubrication problems. He is a leading authority in this field, and has published in numerous tribology journals.

Preface xiii

Nomenclature xvii

About the CompanionWebsite xxv

Part I Introduction 1

1 Elastohydrodynamic Lubrication (EHL) 3

1.1 EHL Regime 3

1.2 Governing Equations in Dimensional Form 7

1.2.1 Generalized Reynolds Equation 9

1.2.2 FilmThickness Equation 15

1.2.3 Linear Elasticity Equations 18

1.2.4 Load Balance Equation 24

1.2.5 Energy Equations 24

1.2.6 Shear Stress Equations 28

1.3 Governing Equations in Dimensionless Form 28

1.3.1 Dimensionless Parameters 29

1.3.2 Generalized Reynolds Equation 31

1.3.3 FilmThickness Equation 32

1.3.4 Linear Elasticity Equations 33

1.3.5 Load Balance Equation 34

1.3.6 Energy Equations 34

1.3.7 Shear Stress Equations 36

1.4 Lubricant Constitutive Behavior 36

1.4.1 Pressure and Temperature Dependence 37

1.4.1.1 Density 37

1.4.1.2 Viscosity 39

1.4.1.3 Thermal Conductivity and Heat Capacity 41

1.4.2 Shear Dependence of Viscosity 41

1.4.3 Limiting Shear Stress 43

1.5 Dimensionless Groups 44

1.6 Review of EHL Numerical Modeling Techniques 46

1.7 Conclusion 52

References 52

2 Finite ElementMethod (FEM) 59

2.1 FEM:The Basic Idea 59

2.2 Model PDE 61

2.3 Steady-State Linear FEM Analysis 63

2.3.1 Elementary Integral Formulations 64

2.3.1.1 Weighted-Residual Form 64

2.3.1.2 Weak Form 65

2.3.2 Solution Approximation 66

2.3.2.1 Meshing and Discretization 67

2.3.2.2 Lagrange Linear Elements 69

2.3.2.3 Lagrange Quadratic Elements 73

2.3.3 Galerkin Formulation 75

2.3.4 Integral Evaluations: Mapping between Reference and Actual Elements 78

2.3.5 Connectivity of Elements 85

2.3.6 Assembly Process and Treatment of B.C.’s 86

2.3.7 Resolution Process 90

2.3.8 Post-Processing of the Solution 91

2.3.9 One-Dimensional Example 92

2.4 Steady-State Nonlinear FEM Analysis 99

2.4.1 Newton Methods for Nonlinear Systems of Equations 99

2.4.1.1 Newton Method 100

2.4.1.2 Damped-NewtonMethod 102

2.4.2 Nonlinear FEM Formulation 105

2.5 Transient FEM Analysis 109

2.5.1 Space-Time Discretization 110

2.5.2 Time-Dependent FEM Formulation 111

2.6 Multi-Physical FEM Analysis 112

2.6.1 Multi-Physical FEM Formulation 113

2.6.2 Assembly Process 115

2.6.3 Coupling Strategies 116

2.6.3.1 Weak Coupling 117

2.6.3.2 Full/Strong Coupling 117

2.7 Stabilized FEM Formulations 118

2.7.1 Isotropic Diffusion 120

2.7.2 Streamline Upwind Petrov–Galerkin 121

2.7.3 Galerkin Least Squares 121

2.8 Conclusion 123

References 123

Part II Finite ElementModeling Techniques 125

3 Steady-State Isothermal Newtonian Line Contacts 127

3.1 Contact Configuration 127

3.2 Geometry, Computational Domains, and Meshing 128

3.2.1 Geometry 128

3.2.2 Computational Domains 128

3.2.3 Meshing and Discretization 130

3.3 Governing Equations and Boundary Conditions 132

3.3.1 Reynolds Equation 133

3.3.2 Linear Elasticity Equations 136

3.3.3 Load Balance Equation 138

3.4 FEM Model 138

3.4.1 Connectivity of Elements 139

3.4.2 Weak Form Formulation 139

3.4.3 Elementary Matrix Formulations 141

3.4.3.1 Elastic Part 142

3.4.3.2 Hydrodynamic Part 144

3.4.3.3 Load Balance Part 145

3.4.4 Stabilized Formulations 146

3.5 Overall Solution Procedure 150

3.6 Model Calibration and Preliminary Results 153

3.6.1 Mesh Sensitivity Analysis 153

3.6.2 Penalty Term Tuning 153

3.6.3 Solid Domain Size Calibration 156

3.6.4 Preliminary Results 157

3.7 Conclusion 161

References 161

4 Steady-State Isothermal Newtonian Point Contacts 165

4.1 Contact Configuration 165

4.2 Geometry, Computational Domains, and Meshing 166

4.2.1 Geometry 166

4.2.2 Computational Domains 166

4.2.3 Meshing and Discretization 169

4.3 Governing Equations and Boundary Conditions 170

4.3.1 Reynolds Equation 171

4.3.2 Linear Elasticity Equations 173

4.3.3 Load Balance Equation 174

4.4 FEM Model 175

4.4.1 Connectivity of Elements 175

4.4.2 Weak Form Formulation 176

4.4.3 Elementary Matrix Formulations 177

4.4.3.1 Elastic Part 178

4.4.3.2 Hydrodynamic Part 180

4.4.3.3 Load Balance Part 182

4.4.4 Stabilized Formulations 183

4.5 Overall Solution Procedure 187

4.6 Model Calibration and Preliminary Results 190

4.6.1 Mesh Sensitivity Analysis 190

4.6.2 Penalty Term Tuning 191

4.6.3 Preliminary Results 192

4.7 Conclusion 196

References 196

5 Steady-State Thermal Non-Newtonian Line Contacts 199

5.1 Contact Configuration 199

5.2 Geometry, Computational Domains, and Meshing 200

5.2.1 Geometry 200

5.2.2 Computational Domains 200

5.2.3 Meshing and Discretization 201

5.3 Governing Equations and Boundary Conditions 203

5.3.1 Generalized Reynolds Equation 204

5.3.2 Linear Elasticity Equations 205

5.3.3 Load Balance Equation 205

5.3.4 Energy Equations 205

5.3.5 Shear Stress Equation 207

5.4 FEM Model 208

5.4.1 Connectivity of Elements 208

5.4.2 Weak Form Formulation 210

5.4.3 Elementary Matrix Formulations 213

5.4.3.1 Elastic Part 215

5.4.3.2 Hydrodynamic Part 215

5.4.3.3 Load Balance Part 218

5.4.3.4 Thermal Part 219

5.4.3.5 Shear Stress Part 224

5.4.4 Stabilized Formulations 225

5.5 Overall Solution Procedure 227

5.6 Model Calibration and Preliminary Results 228

5.6.1 Mesh Sensitivity Analysis 230

5.6.2 Full versusWeak Coupling 230

5.6.3 Preliminary Results 239

5.7 Conclusion 240

References 241

6 Steady-State Thermal Non-Newtonian Point Contacts 243

6.1 Contact Configuration 243

6.2 Geometry, Computational Domains, and Meshing 244

6.2.1 Geometry 244

6.2.2 Computational Domains 244

6.2.3 Meshing and Discretization 245

6.3 Governing Equations and Boundary Conditions 247

6.3.1 Generalized Reynolds Equation 248

6.3.2 Linear Elasticity Equations 249

6.3.3 Load Balance Equation 249

6.3.4 Energy Equations 249

6.3.5 Shear Stress Equations 252

6.4 FEM Model 252

6.4.1 Connectivity of Elements 253

6.4.2 Weak Form Formulation 255

6.4.3 Elementary Matrix Formulations 258

6.4.3.1 Elastic Part 260

6.4.3.2 Hydrodynamic Part 261

6.4.3.3 Load Balance Part 264

6.4.3.4 Thermal Part 264

6.4.3.5 Shear Stress Part 270

6.4.4 Stabilized Formulations 273

6.5 Overall Solution Procedure 274

6.6 Model Calibration and Preliminary Results 275

6.6.1 Mesh Sensitivity Analysis 276

6.6.2 Preliminary Results 276

6.7 Conclusion 280

References 280

7 Transient Effects 281

7.1 Contact Configuration 281

7.2 Geometry, Computational Domains, and Meshing 281

7.3 Governing Equations, Boundary, and Initial Conditions 282

7.3.1 Reynolds Equation 282

7.3.2 Linear Elasticity Equations 284

7.3.3 Load Balance Equation 284

7.4 FEM Model 284

7.4.1 Connectivity of Elements 285

7.4.2 Weak Form Formulation 285

7.4.3 Elementary Matrix Formulations 286

7.4.3.1 Elastic Part 288

7.4.3.2 Hydrodynamic Part 288

7.4.3.3 Load Balance Part 289

7.5 Overall Solution Procedure 289

7.6 Preliminary Results 291

7.7 Conclusion 295

References 295

8 Model Order Reduction (MOR) Techniques 297

8.1 Introduction 297

8.2 Reduced Solution Space Techniques 299

8.2.1 Modal Reduction 302

8.2.2 Ritz-Vector-Like Method 303

8.2.3 EHL-Basis Technique 304

8.2.3.1 Typical Test Case Results 306

8.2.3.2 Performance Analysis: Reduced versus Full Model 310

8.3 Static Condensation with Splitting (SCS) 313

8.3.1 Static Condensation 315

8.3.2 Splitting 316

8.3.3 Overall Numerical Procedure 316

8.3.4 Results and Discussion 320

8.3.4.1 Typical Test Cases 320

8.3.4.2 Splitting Algorithm Tuning 321

8.3.4.3 Preservation of Solution Scheme Generality 327

8.3.4.4 Performance Analysis 329

8.4 Conclusion 335

References 337

Part III Applications 339

9 Pressure and Film Thickness Predictions 341

9.1 Introduction 341

9.2 Qualitative Parametric Analysis 341

9.2.1 Isothermal Newtonian Conditions 342

9.2.2 Thermal Non-Newtonian Conditions 345

9.3 Quantitative Predictions 348

9.4 Analytical FilmThickness Predictions 351

9.4.1 Numerical Experiments 352

9.4.2 Correction Factors and FilmThickness Formulas 353

9.4.3 Experimental Validation 355

9.5 Conclusion 357

References 359

10 Friction Predictions 361

10.1 Introduction 361

10.2 Quantitative Predictions 363

10.3 Friction Regimes 369

10.3.1 Relevant Dimensionless Numbers 370

10.3.1.1 Weissenberg Number 370

10.3.1.2 Nahme–Griffith Number 370

10.3.1.3 LSS Number 370

10.3.1.4 Roller Compliance Number 370

10.3.2 Delineation of Friction Regimes 371

10.3.2.1 Linear Regime 375

10.3.2.2 Nonlinear Viscous Regime 376

10.3.2.3 Plateau Regime 377

10.3.2.4 Thermoviscous Regime 378

10.3.3 Friction Regimes Chart 378

10.4 Conclusion 380

References 381

11 Coated EHL Contacts 383

11.1 Introduction 383

11.2 Modeling Subtleties 385

11.3 Influence of Coating Properties on EHL Contact Performance 388

11.3.1 Pressure and FilmThickness 389

11.3.2 Friction 391

11.3.3 Discussion 394

11.3.3.1 Influence of Coating Mechanical Properties 394

11.3.3.2 Influence of Coating Thermal Properties 396

11.4 Conclusion 402

References 403

Appendices 405

A Numerical Integration 407

A.1 Line Elements 412

A.2 Triangular Elements 412

A.3 Rectangular Elements 413

A.4 Tetrahedral Elements 414

A.5 Prism Elements 415

B Sparse Matrix Storage 417

B.1 Triplet Storage (TS) 418

B.2 Compressed Row Storage (CRS) 419

B.3 Compressed Column Storage (CCS) 419

C Shell T9 Lubricant Properties 423

C.1 Pressure and Temperature Dependence of Density 423

C.2 Pressure and Temperature Dependence of Viscosity 424

C.3 Shear Dependence of Viscosity 425

C.4 Pressure Dependence of Limiting Shear Stress 426

C.5 Pressure and Temperature Dependence ofThermal Properties 427

References 429

Index 431

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