The Material Point Method: Theory, Implementations and Applications

Preface
	Reference
Contents
1 Introduction
	1.1 Computational Sciences and Engineering
	1.2 The Role of Experiments in CSE
	1.3 One Dimensional Wave Equation
	1.4 Mesh-Based and Meshfree Methods
		1.4.1 Mesh-Based Methods
		1.4.2 Meshless Methods
	1.5 A Brief Introduction to the MPM
		1.5.1 Lagrangian Particles and Eulerian Grid
		1.5.2 The Basic MPM Algorithm
		1.5.3 Advantages and Disadvantages of the MPM
		1.5.4 Existing MPM Formulations
		1.5.5 Multiphysics MPM
		1.5.6 Contacts
		1.5.7 Fracture
		1.5.8 Fluids and Gases
		1.5.9 The MPM Versus Other Methods
		1.5.10 Coupling the MPM with Other Methods
	1.6 Applications of the MPM
		1.6.1 Large Strain Geo-Technical Engineering
		1.6.2 Fluid-Structure Interaction
		1.6.3 Image-Based Simulations
		1.6.4 Computer Graphics
		1.6.5 Other Applications
	1.7 Open Source and Commercial MPM Codes
	1.8 Layout
	1.9 Notations
	References
2 A General MPM for Solid Mechanics
	2.1 Basic Concepts of Continuum Mechanics
		2.1.1 Motion and Deformation
		2.1.2 Strain Measures
		2.1.3 Stress Measures
		2.1.4 Objective Stress Rates
		2.1.5 Conservation Equations
		2.1.6 Constitutive Models
	2.2 Strong Form
	2.3 Weak Form and Spatial Discretization
	2.4 MPM as FEM with Particles as Integration Points
	2.5 Temporal Discretization and Resulting MPM Algorithms
		2.5.1 Lumped Mass Matrix
		2.5.2 Calculation of Nodal Velocities (Momenta)
		2.5.3 Standard Formulation (USL)
		2.5.4 Modified Update Stress Last (MUSL)
		2.5.5 Update Stress First (USF)
	2.6 Total Lagrangian MPM (TLMPM)
		2.6.1 Motivation: Numerical Fracture
		2.6.2 Derivation of TLMPM
	2.7 Axi-Symmetric MPM
		2.7.1 Axi-Symmetric ULMPM
		2.7.2 Axi-Symmetric TLMPM
	2.8 Adaptive Time Step
	2.9 Particle/Element Inversion
	2.10 Adaptivity
		2.10.1 Grid Adaptive Refinement
		2.10.2 Particle Splitting and Merging
	References
3 Various MPM Formulations
	3.1 Properties of Weighting Functions
	3.2 Standard Linear Basis Functions
	3.3 Generalized Interpolation Material Point (GIMP)
		3.3.1 uGIMP
		3.3.2 cpGIMP
	3.4 B-Splines Basis Functions
		3.4.1 Recursive B-Splines
		3.4.2 Boundary Modified B-Splines
	3.5 Bernstein Functions
	3.6 Convected Particle Domain Interpolation
		3.6.1 One Dimensional Linear CPDI (CPDI-L2)
		3.6.2 Convected Particle Domain Interpolation (CPDI-R4)
		3.6.3 Quadrilateral Convected Particle Domain Interpolation (CPDI-Q4)
		3.6.4 Triangular Convected Particle Domain Interpolation (CPDI-T3)
		3.6.5 Three Dimensional Linear Tetrahedron CPDI (CPDI-Tet4)
		3.6.6 Polygonal and Polyhedral CPDI
		3.6.7 Complications in GIMP/CPDIs
	3.7 The Generalized Particle in Cell Method
		3.7.1 General Algorithms
		3.7.2 Computation of Mass and Forces on FE Meshes
		3.7.3 Finite Element Basis Functions
		3.7.4 Equivalence Between CPDI and GPIC
		3.7.5 Axi-Symmetric GPIC
	References
4 Constitutive Models
	4.1 Linear Elastic Isotropic Material
	4.2 Hyperelastic Solids
	4.3 Elasto-Plastic Materials
		4.3.1 Equation of State
		4.3.2 Johnson-Cook Flow Model
		4.3.3 Damage
		4.3.4 Algorithm
	References
5 Implementation
	5.1 Initial Particle Distribution
		5.1.1 Regular Particle Distribution
		5.1.2 Irregular Particle Distribution
		5.1.3 Particle Distribution from CAD
		5.1.4 Particle Distribution from Images
	5.2 Initial and Boundary Conditions
		5.2.1 Dirichlet Boundary Conditions
		5.2.2 Symmetric Boundary Conditions
		5.2.3 Neumann Boundary Conditions
		5.2.4 Neumann Boundary Conditions with CPDI
		5.2.5 Boundary Conditions in GPIC
		5.2.6 Rigid Bodies
	5.3 Implementation of CPDI
	5.4 MPM Using an Unstructured Grid
		5.4.1 Shape Functions
		5.4.2 Particle Registration
		5.4.3 Mixed Integration
		5.4.4 uMPM with C1 Shape Functions
	5.5 Visualization
	References
6 MPMat: A MPM Matlab Code
	6.1 Code Structure
	6.2 Background Grid
	6.3 Particle Data
	6.4 Particle Generation
		6.4.1 Particle Generation Using a Mesh
		6.4.2 Particle Generation for Simple Geometries
	6.5 Solution Algorithm
	6.6 Three Dimensions
	6.7 Implementation of (u/cp)GIMP
	6.8 B-splines MPM
		6.8.1 Recursive B-splines MPM
		6.8.2 Bézier Extraction B-splines MPM
	6.9 Implementation of CPDI-R4
		6.9.1 Data Structure for Particles
		6.9.2 Evaluation of φIp and φIp
		6.9.3 Time Advance
	6.10 Implementation of CPDI2s (CPDI-Q4, CPDI-T3)
	6.11 Implementation of CPDI-Poly
	6.12 Visualization Toolkit (VTK)
	6.13 Some Efficiency Improvements
	6.14 More Improvements Using MEX Files
	6.15 Examples
		6.15.1 One Dimensional Examples
		6.15.2 Impact of Two Elastic Disks
		6.15.3 High Velocity Impact
		6.15.4 Large Deformation Vibration of a Compliant Cantilever Beam
		6.15.5 Lateral Compression of Thin-Walled Tubes
	References
7 Karamelo: A Multi-CPU/GPU C++ Parallel MPM Code
	7.1 Karamelo in a Nutshell
	7.2 Hierarchical Class System
	7.3 Pre and Post-processing
	7.4 Input Files
	7.5 Parallelization Using MPI
	7.6 Compilation
	7.7 Extending Karamelo
	7.8 GPU Support
	7.9 Some Simulations
		7.9.1 Taylor Anvil Test
		7.9.2 Upsetting of a Cylindrical Billet
		7.9.3 Cold Spraying
		7.9.4 Scalability Tests
	7.10 Conclusions
	References
8 Contact and Fracture
	8.1 Contacts in the ULMPM
		8.1.1 Contact Without Friction
		8.1.2 Contact with Coulomb Friction
		8.1.3 Derivation
		8.1.4 Calculation of Normal Vector
		8.1.5 Algorithm
		8.1.6 Contact Between a Deformable Solid and a Rigid Wall
		8.1.7 Matlab Implementation
		8.1.8 Differences of MPM Contacts with Other Contacts
		8.1.9 Final Remarks
	8.2 Contacts in the TLMPM
		8.2.1 Enforcing Non-penetration
		8.2.2 Complete Algorithm
	8.3 Contact in GPIC
	8.4 Contact Simulations
		8.4.1 Test 1: Collision of Two Compressible Neo-Hookean Rings
		8.4.2 Test 2: High Velocity Impact of a Steel Disk Onto an Aluminum Target
		8.4.3 Test 3: Contact of a Rigid Sphere with a Half Plane
		8.4.4 Test 4: Cylinder Rolling on an Inclined Plane
		8.4.5 Test 5: Stress Wave in a Granular Material
		8.4.6 Test 6: Penetration of a Steel Sphere Into an Aluminium Cylinder
		8.4.7 Test 7: Scratch Test
	8.5 Fracture Modeling
		8.5.1 Fracture Modeling Within the MPM Framework
		8.5.2 Variational Fracture Theories
		8.5.3 Implementation of Variational Fracture Phase-Field Model
		8.5.4 Nonlocal Johnson-Cook Damage Models
	8.6 Some Fracture Simulations
		8.6.1 Tensile Test Specimen Experiencing Necking and Damage
		8.6.2 Double Circular Notched Specimen
		8.6.3 Compact Tension Specimen
		8.6.4 Machining Simulations
		8.6.5 High Velocity Impact of a Bullet Into a Steel Plate
	References
9 Stability, Accuracy and Recent Improvements
	9.1 Energy and Momenta Conservation
		9.1.1 Linear Momentum Conservation
		9.1.2 Angular Momentum Conservation
		9.1.3 Total Energy Conservation
	9.2 The Method of Manufactured Solutions (MMS)
		9.2.1 An One Dimensional Manufactured Solution
		9.2.2 A Two Dimensional MMS
		9.2.3 Generalized Vortex Problem
		9.2.4 Norms
		9.2.5 Convergence Rate
		9.2.6 Convergence Rate of the MPM
	9.3 Moving Least Square MPM
		9.3.1 Least Square Approximations
		9.3.2 Velocity Projection
		9.3.3 One Point Quadrature
		9.3.4 Implementation
		9.3.5 Improved Implementation
	9.4 The Affine Particle in Cell (APIC)
		9.4.1 The Gradient Enhancement Technique
		9.4.2 Derivation
		9.4.3 Implementation
		9.4.4 Momenta Conservation
		9.4.5 Energy Conservation
	9.5 Convergence Tests
		9.5.1 One Dimensional Convergence Test
		9.5.2 Generalized Vortex Problem
	9.6 Volumetric Locking
		9.6.1 Overview of the F-bar Method
		9.6.2 F-bar Method in MPM: Cell Averaging
		9.6.3 F-bar Method in MPM: Nodal Averaging
	References
10 Other Topics: Modeling of Fluids, Membranes and Temperature Effects
	10.1 Fluids and Gases
		10.1.1 Fluids
		10.1.2 Gases
		10.1.3 Some Examples
	10.2 Modeling Membranes
		10.2.1 York's MPM Algorithm for Membranes
		10.2.2 A Coupled FEM-MPM for Modeling Membranes
	10.3 Thermo-Mechanical Problems
		10.3.1 Thermal Problem
		10.3.2 Coupled Thermo-Mechanical MPM
		10.3.3 Verification Tests
	10.4 Fluid-Structure Interaction
	References
Appendix A Strong Form, Weak Form and Completeness
A.1 Weak Formulation
A.1.1 Strong Form to Weak Form
A.1.2 Weak Form to Strong Form
A.2  Completeness
Appendix B Derivation of CPDI Basis Functions
B.1 CPDI-L2 Basis
B.2 CPDI-L3 Basis
B.3 CPDI-Q4 Basis
B.4 Derivation of CPDI-T3 Basis
B.5 Derivation of CPDI-Tet4 Basis
Appendix C Utilities
C.1 Scripts to Plot Basis Functions
C.2  Symbolic Calculus
C.3 Derivation of B-Spline Basis Functions
C.3.1 Cubic B-Splines
C.3.2 Quadratic B-Splines
C.4 Running Simulations Using a Remote Machine
C.5 Units
Appendix D Explicit Lagrangian Finite Elements
D.1  Updated Lagrangian Finite Elements
D.1.1 General Flowchart
D.1.2 Computation of Internal Force
D.2  Total Lagrangian Finite Elements
D.3  Implementation
D.4  Examples
D.4.1 One Dimensional Convergence Test
D.4.2 Two Dimensional Convergence Test
D.4.3 Large Deformation Vibration of a Cantilever Beam
Appendix E Implicit Lagrangian Finite Elements
E.1  Implicit Dynamics FEM
E.1.1 General Case
E.1.2 Linear Case
E.2 Implementation
E.3 Examples
Appendix F Implementing the Material Point Method Using Julia
F.1 A Short Introduction to Julia
F.1.1 Julia: Installation and Code Editor
F.1.2 Using `for' Loops Versus Vectorization
F.1.3 Composite Types Versus Arrays
F.1.4 Arrays
F.1.5 Sets and Dictionaries
F.1.6 Memory Allocation
F.1.7 Types and Multiple Dispatch
F.1.8 Type Stability
F.1.9 Modules
F.2 A Simple MPM Code in Julia
F.2.1 Code Organization
F.2.2 Data Structures
F.2.3 Solution Phase
F.2.4 Examples
F.3 A More Efficient Julia MPM Code
F.3.1 Grid and Particle Data Structure
F.3.2 Basis Functions
F.3.3 Material
F.3.4 Problem
F.3.5 Graphics
F.3.6 A Complete Example
F.3.7 Implementation of GPIC
F.3.8 Performance Tests
F.4 Tweaks for Speed
	References
Index