Liu Chu, PhD, is an Associate Professor, School of Electronic and Information Engineering, Tongji University. Her research is primarily focused on uncertainty quantification in stochastic defects, numerical simulations for electronic packaging, and the development of advanced computational mechanics methods.

About the Author xi

Preface xiii

1 Overview 1

2 Electronic Packaging 7

2.1 Introduction 8

2.2 Geometrical Parameters 9

2.3 Material Parameters 13

2.4 Boundary Conditions 16

2.5 Stochastic Variables 18

2.6 Short Summary 20

References 20

3 Random Sampling Methods 23

3.1 Monte Carlo Sampling 24

3.2 Latin Hypercube Sampling Method 25

3.2.1 Limitations of LH Sampling in High-dimensional Spaces 27

3.2.2 Clustering or Gaps in Sample Points 27

3.3 Equal Distributed Sampling Method 27

3.4 Importance Monte Carlo Sampling 29

3.5 Directional Sampling Monte Carlo 31

3.6 Directional Importance Sampling Monte Carlo 32

3.7 Self-adaptive Monte Carlo Sampling Method 34

3.7.1 Adaptive Importance Sampling Monte Carlo Method 34

3.8 Short Summary 35

References 36

4 Random Fields and Stochastic Processes 39

4.1 Stochastic Processes 39

4.2 Random Fields 40

4.3 Discretization of Random Fields 41

4.3.1 Local Averaging of One-Dimensional Random Fields 42

4.3.2 Local Averaging of a Two-Dimensional Random Field 46

4.3.3 Three-Dimensional Locally Averaged Random Field 47

4.4 Independent Transformation of Random Variables 48

4.4.1 Cholesky Decomposition Transformation 48

4.4.2 Eigen Orthogonalization Transformation 49

4.5 Triangular Series Simulation 51

4.5.1 Simulation of Gaussian Stationary Random Process 51

4.6 Non-stationary Gaussian Random Process 53

4.7 Short Summary 54

References 54

5 Reliability Prediction 57

5.1 First-order Reliability Method 57

5.1.1 Linear Search Method 59

5.2 Second-order Reliability Method (SORM) 60

5.3 Response Surface Method 64

5.4 Moment Method 65

5.5 Maximum Entropy Method 66

5.5.1 The Concept of Information Entropy 67

5.5.2 Maximum Entropy Principle 68

5.5.2.1 Failure Probability Calculation 70

5.6 System Reliability 71

5.6.1 Reliability of Series System 72

5.6.2 Reliability of Parallel System 73

5.6.3 Reliability of Complex Systems 74

5.7 Sensitivity Analysis 76

5.8 Methods Comparison 80

5.8.1 Comparison of the Principles of Various Approximation Methods 80

5.8.2 Comparison of Computational Effort of Various Approximation Methods 80

5.8.3 Comparison of Computational Accuracy of Various Approximation Methods 81

5.9 Short Summary 82

References 82

6 Finite Element Method 85

6.1 Small Deformation 87

6.2 Large Deformation 88

6.2.1 Strain Measures 89

6.2.2 Stress Measures 91

6.3 Newmark Method for Nonlinear Equation 93

6.4 Solutions for Nonlinear Equations 94

6.5 Geometrical Nonlinear Finite Element Method 96

6.5.1 Truss Element 98

6.5.2 Beam Element 100

6.5.3 Nonlinear Geometry 105

6.6 Material Nonlinear Analysis 111

6.7 Short Summary 112

References 113

7 Nonlinear Stochastic Finite Element Method 115

7.1 Gradient Vector 116

7.2 Discretization of Random Fields 120

7.3 Spectral Decomposition Method 123

7.3.1 Karhunen-Loève Series Expansion Method 123

7.3.2 Orthogonal Series Expansion 125

7.4 Stochastic Field Nonlinear Beam Element 127

7.5 Neumann SFEM 131

7.6 Example of SFEM 134

7.6.1 Computational Framework 135

7.6.2 Deformation and Strain Results 138

7.6.3 Computation Convergence 141

7.6.4 Probability Distribution 143

7.7 Short Summary 146

References 146

8 Random Shear Stress and Thermal Temperature 151

8.1 Introduction 151

8.2 Geometrical Configuration 153

8.3 Theoretical Foundation 155

8.3.1 Constitutive Equations 155

8.3.2 Thermoelastic Matrices 156

8.3.3 Heterogeneous BGA Interconnects 157

8.4 Four Cases 158

8.5 Results and Discussion 160

8.5.1 Independent Solder Ball 160

8.5.2 Four BGA Cases 162

8.5.3 Coupled Random Shear Stress and Thermal Temperature 165

8.5.4 Parameter Discussion 167

8.5.5 Time-dependent Thermal Analysis 170

8.5.6 Thermal Creep 173

8.6 Short Summary 177

References 178

9 Material Uncertainty in Electromigration 181

9.1 Introduction 181

9.2 Model Description 183

9.2.1 Mathematical Theory 184

9.2.2 Material Properties 185

9.2.3 Monte-Carlo-Based Stochastic Finite Element Model 185

9.3 Results and Discussion 189

9.3.1 Extreme Values 189

9.3.2 Finite Element Results 192

9.3.3 Statistical Results 194

9.3.4 Correlation Analysis 197

9.4 Short Summary 199

References 200

10 Mechanical Reliability in the Replaceable Integrated Chiplet Assembly 203

10.1 Introduction 203

10.2 Fem 204

10.3 Parameter Correlation 207

10.4 Mechanical Reliability 209

10.5 Short Summary 211

References 212

11 Kriging Surrogate Model 215

11.1 Introduction 215

11.2 Method Description 217

11.2.1 Parameter Definitions 217

11.2.2 Kriging Surrogate Model 220

11.2.3 Program Implementation 221

11.3 Results and Discussion 221

11.3.1 The Deterministic Initial FEM 221

11.3.2 Material Parameter Discussion 226

11.3.3 Geometrical Parameter Discussion 226

11.3.4 Stochastic Probability Results 235

11.3.5 Computational Competence of KSM 236

11.4 Conclusion 239

References 240

12 Digital Twins Based on SFEM 243

12.1 Introduction 243

12.2 Analysis Framework 245

12.2.1 Digital Coupling 245

12.2.2 Virtual Representation 248

12.2.3 Tools in DT 249

12.2.4 Functional Output 251

12.3 Difficulties and Challenges 252

12.4 SFEM for DT Development 254

12.5 Short Summary 256

References 256

Appendix 259

Codes in Chapter 3 259

Codes in Chapter 4 261

Codes in Chapter 7 262

Codes in Chapter 9 264

Codes in Chapter 10 266

Index 269