Preface ........................................................ xi
Acknowledgements ............................................. xiii
1. Introduction ................................................. 1
1.1. General Formulation of Continuum Problems ............... 1
1.2. General Formulation of Discontinuum Problems ............ 2
1.3. А Typical Problem of Computational Mechanics
of Discontinua .......................................... 4
1.4. Combined Continua-Discontinua Problems ................. 27
1.5. Transition from Continua to Discontinua ................ 28
1.6. The Combined Finite-Discrete Element Method ............ 29
1.7. Algorithmic and Computational Challenge of the
Combined Finite-Discrete Element Method ................ 32
2. Processing of Contact Interaction in the Combined Finite
Discrete Element Method ..................................... 35
2.1. Introduction ........................................... 35
2.2. The Penalty Function Method ............................ 40
2.3. Potential Contact Force in 2D .......................... 41
2.4. Discretisation of Contact Force in 2D .................. 43
2.5. Implementation Details for Discretised Contact Force
in 2D .................................................. 43
2.6. Potential Contact Force in 3D .......................... 55
2.6.1. Evaluation of contact force ..................... 57
2.6.2. Computational aspects ........................... 58
2.6.3. Physical interpretation of the penalty
parameter ....................................... 62
2.6.4. Contact damping ................................. 63
2.7. Alternative Implementation of the Potential Contact
Force .................................................. 69
3. Contact Detection ........................................... 73
3.1. Introduction ........................................... 73
3.2. Direct Checking Contact Detection Algorithm ............ 77
3.2.1. Circular bounding box ........................... 77
3.2.2. Square bounding object .......................... 78
3.2.3. Complex bounding box ............................ 79
3.3. Formulation of Contact Detection Problem for Bodies
of Similar Size in 2D .................................. 80
3.4. Binary Tree Based Contact Detection Algorithm for
Discrete Elements of Similar Size ...................... 81
3.5. Direct Mapping Algorithm for Discrete Elements of
Similar Size ........................................... 87
3.6. Screening Contact Detection Algorithm for Discrete
Elements of Similar Size ............................... 89
3.7. Sorting Contact Detection Algorithm for Discrete
Elements of a Similar Size ............................. 94
3.8. Munjiza-NBS Contact Detection Algorithm in 2D ......... 102
3.8.1. Space decomposition ............................ 103
3.8.2. Mapping of discrete elements onto cells ........ 104
3.8.3. Mapping of discrete elements onto rows and
columns of cells ............................... 104
3.8.4. Representation of mapping ...................... 104
3.9. Selection of Contact Detection Algorithm .............. 117
3.10.Generalisation of Contact Detection Algorithms to
3D Space .............................................. 118
3.10.1.Direct checking contact detection algorithm .... 118
3.10.2.Binary tree search ............................. 118
3.10.3.Screening contact detection algorithm .......... 118
3.10.4.Direct mapping contact detection algorithm ..... 120
3.11.Generalisation of Munjiza-NBS Contact Detection
Algorithm to Multidimensional Space ................... 120
3.12.Shape and Size Generalisation-Williams C-GRID
Algorithm ............................................. 128
4. Deformability of Discrete Elements ......................... 131
4.1. Deformation ........................................... 131
4.2. Deformation Gradient .................................. 132
4.2.1. Frames of reference ............................ 132
4.2.2. Transformation matrices ........................ 139
4.3. Homogeneous Deformation ............................... 141
4.4. Strain ................................................ 142
4.5. Stress ................................................ 143
4.5.1. Cauchy stress tensor ........................... 143
4.5.2. First Piola-Kirchhoff stress tensor ............ 145
4.5.3. Second Piola-Kirchhoff stress tensor ........... 149
4.6. Constitutive Law ...................................... 150
4.7. Constant Strain Triangle Finite Element ............... 156
4.8. Constant Strain Tetrahedron Finite Element ............ 166
4.9. Numerical Demonstration of Finite Rotation
Elasticity in the Combined Finite-Discrete Element
Method ................................................. 174
5. Temporal Discretisation .................................... 179
5.1. The Central Difference Time Integration Scheme ........ 179
5.1.1. Stability of the central difference time
integration scheme ............................. 182
5.2. Dynamics of Irregular Discrete Elements Subject to
Finite Rotations in 3D ................................ 185
5.2.1. Frames of reference ............................ 185
5.2.2. Kinematics of the discrete element in
general motion ................................. 186
5.2.3. Spatial orientation of the discrete element .... 186
5.2.4. Transformation matrices ........................ 187
5.2.5. The inertia of the discrete element ............ 188
5.2.6. Governing equation of motion ................... 188
5.2.7. Change in spatial orientation during
a single time step ............................. 190
5.6.1. Change in angular momentum due to external
loads .......................................... 191
5.6.2. Change in angular velocity during a single
time step ...................................... 193
5.6.3. Munjiza direct time integration scheme ......... 194
5.3. Alternative Explicit Time Integration Schemes ......... 203
5.3.1. The Central Difference time integration
scheme (CD) .................................... 203
5.3.2. Gear's predictor-corrector time integration
schemes (PC-3, PC-4, and PC-5) ................. 204
5.3.3. CHIN integration scheme ........................ 205
5.3.4. OMF30 time integration scheme .................. 206
5.3.5. OMF32 time integration scheme .................. 206
5.3.6. Forest & Ruth time integration scheme .......... 207
5.4. The Combined Finite-Discrete Element Simulation of
the State of Rest ..................................... 211
6. Sensitivity to Initial Conditions in Combined Finite-
Discrete Element Simulations ............................... 219
6.1. Introduction .......................................... 219
6.2. Combined Finite-Discrete Element Systems .............. 220
7. Transition from Continua to Discontinua .................... 231
7.1. Introduction .......................................... 231
7.2. Strain Softening Based Smeared Fracture Model ......... 232
7.3. Discrete Crack Model .................................. 239
7.4. A Need for More Robust Fracture Solutions ............. 254
8. Fluid Coupling in the Combined Finite-Discrete Element
Method ..................................................... 255
8.1. Introduction .......................................... 255
8.1.1. CFD with solid coupling ........................ 255
8.1.2. Combined finite-discrete element method with
CFD coupling ................................... 257
8.2. Expansion of the Detonation Gas ....................... 259
8.2.1. Equation of state .............................. 259
8.2.2. Rigid chamber .................................. 259
8.2.3. Isentropic adiabatic expansion of detonation
gas ............................................ 262
8.2.4. Detonation gas expansion in a partially
filled non-rigid chamber ....................... 264
8.3. Gas Flow Through Fracturing Solid ..................... 266
8.3.1. Constant area duct ............................. 267
8.4. Coupled Combined Finite-Discrete Element Simulation
of Explosive Induced Fracture and Fragmentation ....... 270
8.4.1. Scaling of coupled combined finite-discrete
element problems ............................... 274
8.5. Other Applications .................................... 276
9. Computational Aspects of Combined Finite-Discrete
Element Simulations ........................................ 277
9.1. Large Scale Combined Finite-Discrete Element
Simulations ........................................... 277
9.1.1. Minimising RAM requirements .................... 278
9.1.2. Minimising CPU requirements .................... 279
9.1.3. Minimising storage requirements ................ 279
9.1.4. Minimising risk ................................ 279
9.1.5. Maximising transparency ........................ 280
9.2. Very Large Scale Combined Finite-Discrete Element
Simulations ........................................... 280
9.3. Grand Challenge Combined Finite-Discrete Element
Simulations ........................................... 281
9.4. Why the С Programming Language? ....................... 283
9.5. Alternative Hardware Architectures .................... 283
9.5.1. Parallel computing ............................. 283
9.5.2. Distributed computing .......................... 285
9.5.3. Grid computing ................................. 288
10.Implementation of some of the Core Combined Algorithms ..... 291
10.1.Portability, Speed, Transparency and Reusability ...... 291
10.1.1.Use of new data types .......................... 291
10.1.2.Use of MACROS .................................. 291
10.2.Dynamic Memory Allocation ............................. 292
10.3.Data Compression ...................................... 294
10.4.Potential Contact Force in 3D ......................... 294
10.4.1 Interaction between two tetrahedrons ........... 294
10.5.Sorting Contact Detection Algorithm ................... 303
10.6.NBS Contact Detection Algorithm in 3D ................. 304
10.7.Deformability with Finite Rotations in 3D ............. 313
Bibliography .................................................. 319
Index ......................................................... 331
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