Table of Contents
Part I Finite Element Formulation
1 Introduction 3
1.1 Basic Ideas of FEM 3
1.2 Formulation of Finite Element Equations 4
1.2.1 Galerkin Method 5
1.2.2 Variational Formulation 8
Example of Shape-function Determination 9
Problems 10
2 Finite Element Equations for Heat Transfer 13
2.1 Problem Statement 13
2.2 Finite Element Discretization of Heat Transfer Equations 14
2.3 Different Type Problems 16
2.4 Triangular Element 17
Problems 19
3 FEM for Solid Mechanics Problems 21
3.1 Problem Statement 21
3.2 Finite Element Equations 23
3.3 Stiffness Matrix of a Triangular Element 26
3.4 Assembly of the Global Equation System 27
3.5 Example of the Global Matrix Assembly 29
Problems 30
4 Finite Element Program 33
4.1 Object-oriented Approach to Finite Element Programming 33
4.2 Requirements for the Finite Element Application 34
4.2.1 Overall Description 34
4.2.2 User Description 35
4.2.3 User Interface 35
4.2.4 Functions 35
4.2.5 Other Requirements 36
4.3 General Structure of the Finite Element Code 36
Problems 38
Part II Finite Element Solution
5 Finite Element Processor 43
5.1 Class Structure 43
5.2 Problem Data 49
5.2.1 Data Statements 49
5.2.2 Model Data 51
5.2.3 Load Specification 52
5.2.4 Data Example 54
5.3 Data Scanner 57
Problems 61
6 Finite Element Model 63
6.1 Data for the Finite Element Model 63
6.2 Class for the Finite Element Model 66
6.3 Adding New Data Item 72
Problems 72
7 Elastic Material 75
7.1 Hooke's Law 75
7.2 Class for a Material 76
7.3 Class for Elastic Material 79
Problems 81
8 Elements 83
8.1 Element Methods 83
8.2 Abstract Class Element 84
8.2.1 Element Data 84
8.2.2 Element Constructor 85
8.2.3 Methods of Particular Elements 87
8.2.4 Methods Common to All Elements 88
8.2.5 Container for Stresses 90
8.3 Adding New Element Type 91
Problems 92
9 Numerical Integration 93
9.1 Gauss Integration Rule 93
9.2 Implementation of Numerical Integration 95
Problems 99
10 Two-dimensional Isoparametric Elements 101
10.1 Shape Functions 101
10.2 Strain-Displacement Matrix 104
10.3 Element Properties 107
10.4 Nodal Equivalent of the Surface Load 108
10.5 Example: Computing Nodal Equivalents of a Distributed Load 109
10.6 Calculation of Strains and Stresses 110
Problems 111
11 Implementation of Two-dimensional Quadratic Element 113
11.1 Class for Shape Functions and Their Derivatives 113
11.1.1 Element Degeneration 114
11.1.2 Shape Functions 115
11.1.3 Derivatives of Shape Functions 116
11.1.4 One-dimensional Shape Functions and Their Derivatives 118
11.2 Class for Eight-node Element 118
11.2.1 Stiffness Matrix 119
11.2.2 Displacement Differentiation Matrix 121
11.2.3 Thermal Vector 122
11.2.4 Nodal Equivalent of a Distributed Load 123
11.2.5 Equivalent Stress Vector 125
11.2.6 Extrapolation from Integration Points to Nodes 126
11.2.7 Other Methods 127
Problems 128
12 Three-dimensional Isoparametric Elements 129
12.1 Shape Functions 129
12.2 Strain-Displacement Matrix 131
12.3 Element Properties 133
12.4 Efficient Evaluation of Element Matrices and Vectors 134
12.5 Calculation of Nodal Equivalents for External Loads 134
12.6 Example: Nodal Equivalents of a Distributed Load 136
12.7 Calculation of Strains and Stresses 138
12.8 Extrapolation of Strains and Stresses 138
Problems 139
13 Implementation of Three-dimensional Quadratic Element 141
13.1 Class for Shape Functions and Their Derivatives 141
13.1.1 Element Degeneration 141
13.1.2 Shape Functions 143
13.1.3 Derivatives of Shape Functions 144
13.1.4 Shape Functions and Their Derivatives for an Element Face 147
13.2 Class for Twenty-node Element 149
13.2.1 Stiffness Matrix 150
13.2.2 Thermal Vector 152
13.2.3 Nodal Equivalent of a Distributed Load 153
13.2.4 Equivalent Stress Vector 154
13.2.5 Extrapolation from Integration Points to Nodes 155
13.2.6 Other Methods 156
Problems 158
14 Assembly and Solution 161
14.1 Disassembly and Assembly 161
14.1.1 Disassembly of Vectors 161
14.1.2 Assembly of Vectors 163
14.1.3 Assembly Algorithm for Matrices 164
14.2 Displacement Boundary Conditions 166
14.2.1 Explicit Specification of Displacement Boundary Conditions 166
14.2.2 Method of Large Number 167
14.3 Solution of Finite Element Equations 167
14.4 Abstract Solver Class 168
14.5 Adding New Equation Solver 170
Problems 171
15 Direct Equation Solver 173
15.1 LDU Solution Method 173
15.2 Assembly of Matrix in Symmetric Profile Format 174
15.3 LDU Solution Algorithm 178
15.4 Tuning of the LDU Factorization 182
Problems 186
16 Iterative Equation Solver 187
16.1 Preconditioned Conjugate Gradient Method 187
16.2 Assembly of Matrix in Sparse-row Format 188
16.3 PCG Solution 193
Problems 196
17 Load Data and Load Vector Assembly 199
17.1 Data Describing the Load 199
17.2 Load Data Input 201
17.3 Load Vector Assembly 207
17.4 Element Face Load 209
Problems 211
18 Stress Increment, Residual Vector and Results 213
18.1 Computing Stress Increment 213
18.2 Residual Vector 215
18.3 Results 217
18.4 Solution of a Simple Test Problem 219
Problems 220
19 Elastic-Plastic Problems 223
19.1 Constitutive Relations for Elastic-Plastic Material 223
19.2 Computing Finite Stress Increments 225
19.2.1 Determining Elastic Fraction of Stress Increment 226
19.2.2 Subincrementation for Computing Stress Increment 226
19.3 Material Deformation Curve 227
19.4 Implementation of Elastic-Plastic Material Relations 228
19.5 Midpoint Integration of Constitutive Relations 234
19.6 Nonlinear Solution Procedure 239
19.6.1 Newton-Raphson Method 240
19.6.2 Initial Stress Method 241
19.6.3 Convergence Criteria 242
19.7 Example: Solution of an Elastic-Plastic Problem 243
Problems 245
Part III Mesh Generation
20 Mesh Generator 249
20.1 Block Decomposition Method 249
20.2 Class Structure 250
20.3 Mesh-generation Modules 252
20.4 Adding New Module 253
Problems 254
21 Two-dimensional Mesh Generators 257
21.1 Rectangular Block 257
21.2 Mesh Inside Eight-node Macroelement 261
21.2.1 Algorithm of Double-quadratic Transformation 261
21.2.2 Implementation of Mesh Generation 264
21.3 Example of Mesh Generation 269
Problems 270
22 Generation of Three-dimensional Meshes by Sweeping 271
22.1 Sweeping Technique 271
22.2 Implementation 272
22.2.1 Input Data 272
22.2.2 Node Numbering 275
22.2.3 Element Connectivities and Nodal Coordinates 276
22.3 Example of Mesh Generation 279
Problems 281
23 Pasting Mesh Blocks 283
23.1 Pasting Technique 283
23.2 Implementation 284
23.2.1 Data Input 284
23.2.2 Finding Coincident Nodes 286
23.2.3 Pasting 287
Problems 288
24 Mesh Transformations 289
24.1 Transformation Relations 289
24.2 Implementation 291
24.2.1 Input Data 291
24.2.2 Performing Transformations 293
24.3 Example of Using Transformations 295
Problems 296
25 Copying, Writing and Reading Mesh Blocks 297
25.1 Copying 297
25.2 Writing Mesh to File 299
25.3 Reading Mesh from File 300
Problems 302
Part IV Visualization of Meshes and Results
26 Introduction to Java 3D™ 305
26.1 Rendering Three-dimensional Objects 305
26.2 Scene Graph 306
26.3 Scene Graph Nodes 307
26.3.1 Group Nodes 307
26.3.2 Leaf Nodes 308
26.4 Node Components 309
26.4.1 Geometry 309
26.4.2 Appearance and Attributes 311
Problems 311
27 Visualizer 313
27.1 Visualization Algorithm 313
27.2 Surface of the Finite Element Model 314
27.3 Subdivision of Quadratic Surfaces 315
27.4 Class Structure of the Visualizer 315
27.5 Visualizer Class 316
27.6 Input Data 318
27.6.1 Input Data File 318
27.6.2 Class for Data Input 319
Problems 322
28 Visualization Scene Graph 325
28.1 Schematic of the Scene Graph 325
28.2 Implementation of the Scene Graph 326
28.3 Shape Objects 328
Problems 331
29 Surface Geometry 333
29.1 Creating Geometry of the Model Surface 333
29.2 Surface Faces 335
29.3 Surface Edges and Nodes 338
29.4 Modification of Nodal Coordinates 340
Problems 342
30 Edge and Face Subdivision 343
30.1 Subdivision for Quality Visualization 343
30.2 Edge Subdivision 344
30.3 Face Subdivision 347
Problems 352
31 Surface Subdivision 353
31.1 Subdivision of the Model Surface 353
31.2 Subdivision of Faces into Triangles 356
31.3 Arrays for Java 3D 359
Problems 362
32 Results Field, Color Scale, Interaction and Lights 363
32.1 Results Field 363
32.2 Color Scale 368
32.3 Mouse Interaction 370
32.4 Lights and Background 372
32.5 Visualization Example 373
Problems 375
A Data for Finite Element Solver 377
A.l Data Statements 377
A.1 1 Data Statement 377
A.1.2 Comment Statement 377
A.1.3 Including File 377
A.1.4 End Statement 378
A.2 Model Data 378
A.2.1 Parameters 378
A.2.2 Material Properties 378
A.2.3 Finite Element Mesh 379
A.2.4 Displacement Boundary Conditions 379
A.3 Load-Specification 380
A.3.1 Load Step Name 380
A.3.2 Parameters 380
A.3.3 Nodal Forces 381
A.3.4 Surface Forces 381
A.3.5 Surface Forces Inside a Box 381
A.3.6 Nodal Temperatures 382
B Data for Mesh Generation 383
B.1 Mesh-generation Modules 383
B.2 Rectangular Mesh Block 384
B 3 Mesh Inside Eight-node Macroelement 384
B 4 Three-dimensional Mesh by Sweeping 384
B.5 Reading Mesh from File 385
B.6 Writing Mesh to File 385
B.7 Copying Mesh 385
B.8 Mesh Transformations 385
B.9 Connecting Two Mesh Blocks 386
C Data for Visualizer 387
C.1 Visualization Data 387
C.2 Input Data 387
D Example of Problem Solution 389
D.1 Problem Statement 389
D.2 Mesh Generation 390
D.3 Problem Solution 391
D.4 Visualization 393
References 397
Index 399
