The Combined Finite-Discrete Element Method

Acknowledgements.

1 Introduction.

1.1 General Formulation of Continuum Problems.

1.2 General Formulation of Discontinuum Problems.

1.3 A Typical Problem of Computational Mechanics of Discontinua.

1.4 Combined Continua-Discontinua Problems.

1.5 Transition from Continua to Discontinua.

1.6 The Combined Finite-Discrete Element Method.

1.7 Algorithmic and Computational Challenge of the Combined Finite-Discrete Element Method.

2 Processing of Contact Interaction in the Combined Finite Discrete Element Method.

2.1 Introduction.

2.2 The Penalty Function Method.

2.3 Potential Contact Force in 2D.

2.4 Discretisation of Contact Force in 2D.

2.5 Implementation Details for Discretised Contact Force in 2D.

2.6 Potential Contact Force in 3D.

2.6.1 Evaluation of contact force.

2.6.2 Computational aspects.

2.6.3 Physical interpretation of the penalty parameter.

2.6.4 Contact damping.

2.7 Alternative Implementation of the Potential Contact Force.

3 Contact Detection.

3.1 Introduction.

3.2 Direct Checking Contact Detection Algorithm.

3.2.1 Circular bounding box.

3.2.2 Square bounding object.

3.2.3 Complex bounding box.

3.3 Formulation of Contact Detection Problem for Bodies of Similar Size in 2D.

3.4 Binary Tree Based Contact Detection Algorithm for Discrete Elements of Similar Size.

3.5 Direct Mapping Algorithm for Discrete Elements of Similar Size.

3.6 Screening Contact Detection Algorithm for Discrete Elements of Similar Size.

3.7 Sorting Contact Detection Algorithm for Discrete Elements of a Similar Size.

3.8 Munjiza-NBS Contact Detection Algorithm in 2D.

3.8.1 Space decomposition.

3.8.2 Mapping of discrete elements onto cells.

3.8.3 Mapping of discrete elements onto rows and columns of cells.

3.8.4 Representation of mapping.

3.9 Selection of Contact Detection Algorithm.

3.10 Generalisation of Contact Detection Algorithms to 3D Space.

3.10.1 Direct checking contact detection algorithm.

3.10.2 Binary tree search.

3.10.3 Screening contact detection algorithm.

3.10.4 Direct mapping contact detection algorithm.

3.11 Generalisation of Munjiza-NBS Contact Detection Algorithm to Multidimensional Space.

3.12 Shape and Size Generalisation–Williams C-GRID Algorithm.

4 Deformability of Discrete Elements.

4.1 Deformation.

4.2 Deformation Gradient.

4.2.1 Frames of reference.

4.2.2 Transformation matrices.

4.3 Homogeneous Deformation.

4.4 Strain.

4.5 Stress.

4.5.1 Cauchy stress tensor.

4.5.2 First Piola-Kirchhoff stress tensor.

4.5.3 Second Piola-Kirchhoff stress tensor.

4.6 Constitutive Law.

4.7 Constant Strain Triangle Finite Element.

4.8 Constant Strain Tetrahedron Finite Element.

4.9 Numerical Demonstration of Finite Rotation Elasticity in the Combined Finite-Discrete Element Method.

5 Temporal Discretisation.

5.1 The Central Difference Time Integration Scheme.

5.1.1 Stability of the central difference time integration scheme.

5.2 Dynamics of Irregular Discrete Elements Subject to Finite Rotations in 3D.

5.2.1 Frames of reference.

5.2.2 Kinematics of the discrete element in general motion.

5.2.3 Spatial orientation of the discrete element.

5.2.4 Transformation matrices.

5.2.5 The inertia of the discrete element.

5.2.6 Governing equation of motion.

5.2.7 Change in spatial orientation during a single time step.

5.6.8 Change in angular momentum due to external loads.

5.6.9 Change in angular velocity during a single time step.

5.6.10 Munjiza direct time integration scheme.

5.3 Alternative Explicit Time Integration Schemes.

5.3.1 The Central Difference time integration scheme (CD).

5.3.2 Gear’s predictor-corrector time integration schemes (PC-3, PC-4, and PC-5).

5.3.3 CHIN integration scheme.

5.3.4 OMF30 time integration scheme.

5.3.5 OMF32 time integration scheme.

5.3.6 Forest & Ruth time integration scheme.

5.4 The Combined Finite-Discrete Element Simulation of the State of Rest.

6 Sensitivity to Initial Conditions in Combined Finite-Discrete Element Simulations.

6.1 Introduction.

6.2 Combined Finite-Discrete Element Systems.

7 Transition from Continua to Discontinua.

7.1 Introduction.

7.2 Strain Softening Based Smeared Fracture Model.

7.3 Discrete Crack Model.

7.4 A Need for More Robust Fracture Solutions.

8 Fluid Coupling in the Combined Finite-Discrete Element Method.

8.1 Introduction.

8.1.1 CFD with solid coupling.

8.1.2 Combined finite-discrete element method with CFD coupling.

8.2 Expansion of the Detonation Gas.

8.2.1 Equation of state.

8.2.2 Rigid chamber.

8.2.3 Isentropic adiabatic expansion of detonation gas.

8.2.4 Detonation gas expansion in a partially filled non-rigid chamber.

8.3 Gas Flow Through Fracturing Solid.

8.3.1 Constant area duct.

8.4 Coupled Combined Finite-Discrete Element Simulation of Explosive Induced Fracture and Fragmentation.

8.4.1 Scaling of coupled combined finite-discrete element problems.

8.5 Other Applications.

9 Computational Aspects of Combined Finite-Discrete Element Simulations.

9.1 Large Scale Combined Finite-Discrete Element Simulations.

9.1.1 Minimising RAM requirements.

9.1.2 Minimising CPU requirements.

9.1.3 Minimising storage requirements.

9.1.4 Minimising risk.

9.1.5 Maximising transparency.

9.2 Very Large Scale Combined Finite-Discrete Element Simulations.

9.3 Grand Challenge Combined Finite-Discrete Element Simulations.

9.4 Why the C Programming Language?

9.5 Alternative Hardware Architectures.

9.5.1 Parallel computing.

9.5.2 Distributed computing.

9.5.3 Grid computing.

10 Implementation of some of the Core Combined Finite-Discrete Element Algorithms.

10.1 Portability, Speed, Transparency and Reusability.

10.1.1 Use of new data types.

10.1.2 Use of MACROS.

10.2 Dynamic Memory Allocation.

10.3 Data Compression.

10.4 Potential Contact Force in 3D.

10.4.1 Interaction between two tetrahedrons.

10.5 Sorting Contact Detection Algorithm.

10.6 NBS Contact Detection Algorithm in 3D.

10.7 Deformability with Finite Rotations in 3D.

Bibliography.

Index.