Computational Methods Based on Peridynamics and Nonlocal Operators. Theory and Applications

Preface
Contents
About the Authors
1 Introduction
	1.1 Overview of Meshless Method
		1.1.1 Smoothed Particle Hydrodynamics
		1.1.2 Moving Least Square (MLS) Method
		1.1.3 Reproducing Kernel Particle Method (RKPM)
		1.1.4 Essential Boundary Conditions
		1.1.5 Maximum-Entropy Meshfree Approximations
		1.1.6 Peridynamics
		1.1.7 Peridynamic Differential Operator Method
	1.2 Brief Review of Nonlocal Theories
	1.3 Energy Form and Variational Principle
	1.4 Weak Form and Weighted Residual Method
	1.5 Outline of the Book
	References
2 Dual-Horizon Peridynamics
	2.1 Conventional Peridynamics
	2.2 Ghost Force and Spurious Wave Reflection in Peridynamics
	2.3 Governing Equations Based on Horizon and Dual-Horizon
		2.3.1 Horizon and Dual-Horizon
		2.3.2 Equation of Motion for Peridynamics with Horizon Variable
		2.3.3 Proof of Basic Physical Principles
	2.4 Dual-Horizon Peridynamics
		2.4.1 Dual-Horizon Bond-Based Peridynamics
		2.4.2 Dual-Horizon Ordinary State-Based Peridynamics
		2.4.3 Dual-Horizon Non-ordinary State-Based Peridynamics
	2.5 Numerical Examples
		2.5.1 Wave Propagation in 1D Homogeneous Bar
		2.5.2 2D Wave Reflection in a Rectangular Plate
		2.5.3 Kalthoff–Winkler Experiment
		2.5.4 Adaptive Refined Peridynamics
		2.5.5 Multiple Materials
	2.6 Conclusions
	References
3 First-Order Nonlocal Operator Method
	3.1 Support, Dual-Support and Nonlocal Operators
		3.1.1 Nonlocal Operators in Support
		3.1.2 Variation of the Nonlocal Operator
	3.2 The Variational Principles Based on the Nonlocal Operator
		3.2.1 Divergence Operator
		3.2.2 Curl Operator
		3.2.3 Gradient Operator of Vector Field
		3.2.4 Gradient Operator of Scalar Field
	3.3 Operator Energy Functional
	3.4 Higher Order Operator Energy Functional
		3.4.1 Higher Order Operator Energy Functional
	3.5 Applications
		3.5.1 1D Beam and Bar Test
		3.5.2 Poisson Equation
		3.5.3 Nonlocal Theory for Linear Small Strain Elasticity
		3.5.4 Nonhomogeneous Biharmonic Equation
		3.5.5 2D Solid Beam
		3.5.6 Plate with Hole in Tension
	3.6 Conclusions
	References
4 Nonlocal Operator Method  for Computational Electromagnetic  Field and Waveguide Problem
	4.1 Brief Review of Maxwell Equations
	4.2 Basic Concepts in Nonlocal Operator Method
		4.2.1 Nonlocal Operators and Definitions Based  on the Support
		4.2.2 Variation of Nonlocal Operators
	4.3 Waveguide
	4.4 Hourglass Energy Functional
	4.5 NOM for Electromagnetic in the Time Domain
	4.6 Numerical Examples
		4.6.1 The Schrödinger Equation in 1D
		4.6.2 Electrostatic Field Problems
		4.6.3 Rectangular Waveguide Problem
	4.7 Conclusion
	References
5 Higher Order Nonlocal Operator Method
	5.1 Nonlocal Operator Method
		5.1.1 Basic Concepts
		5.1.2 Taylor Series Expansion
		5.1.3 Mathematica Code for Multi-index
		5.1.4 Higher Order Nonlocal Operator Method
	5.2 Quadratic Functional
		5.2.1 Newton–Raphson Method for Nonlinear Functional
		5.2.2 Elastic Solid Materials
	5.3 Numerical Examples by Strong Form
		5.3.1 Second-Order ODE
		5.3.2 1D Schrödinger Equation
		5.3.3 Poisson Equation
	5.4 Numerical Examples by Weak Form
		5.4.1 Poisson Equation in Higher Dimensional Space
		5.4.2 Square Plate with Simple Support
		5.4.3 Von Kármán Equations for a Thin Plate
		5.4.4 Nearly Incompressible Block
		5.4.5 Fracture Modeling by Phase Field Method
	5.5 Concluding Remarks
	References
6 Nonlocal Operator Method with Numerical Integration for Gradient Solid
	6.1 Review of Nonlocal Operator Method
	6.2 Nonlocal Operator Approximation Scheme
	6.3 Gradient Solid Theory
		6.3.1 Linear Gradient Elasticity
		6.3.2 Numerical Implementation
	6.4 Numerical Examples
		6.4.1 Static Rod in Tension
		6.4.2 Infinite Plate with Hole
		6.4.3 2D Plate with Holes
		6.4.4 Bending of 3D Block
	6.5 Concluding Remarks
	References
7 Dual-Support Smoothed Particle Hydrodynamics in Solid: Variational Principle and Implicit Formulation
	7.1 Introduction
	7.2 Variational Derivation of Dual-Support SPH
	7.3 Functional of Hourglass Energy
	7.4 Numerical Implementation
	7.5 Material Constitutions
	7.6 Numerical Examples
		7.6.1 3D Cantilever Loaded at the End
		7.6.2 Plate Under Compression
		7.6.3 3D Cantilever Tension Test
		7.6.4 Influence of Smoothing Length
		7.6.5 Rubber Pull Test
		7.6.6 Large Deformation Problem
	7.7 Conclusions
	References
8 Nonlocal Strong Forms of Thin Plate, Gradient Elasticity, Magneto–Electro-Elasticity and Phase Field Fracture by Nonlocal Operator Method
	8.1 Second-Order Nonlocal Operator Method
		8.1.1 Support and Dual-Support
		8.1.2 Dual Property of Dual-Support
		8.1.3 A Simple Example to Illustrate Dual-Support
		8.1.4 Nonlocal Gradient and Hessian Operator
		8.1.5 Stability of the Second-Order Nonlocal Operators
	8.2 Nonlocal Governing Equations Based on NOM
		8.2.1 Nonlocal Form for Hyperelasticity
		8.2.2 Nonlocal Thin Plate Theory
		8.2.3 Nonlocal Gradient Elasticity
		8.2.4 Nonlocal Form of Magneto–Electro-Elasticity
		8.2.5 Nonlocal Form of Phase Field Fracture Method
	8.3 Instability Criterion for Fracture Modeling
	8.4 Numerical Implementation
	8.5 Numerical Examples
		8.5.1 Accuracy of Nonlocal Hessian Operator
		8.5.2 Square Thin Plate Subject to Pressure
		8.5.3 Single-Edge Notched Tension Test
		8.5.4 Out-of-Plane Shear Fracture in 3D
	8.6 Conclusion
	References
9 Nonlocal Operator Method for Dynamic Brittle Fracture Based on an Explicit Phase Field Model
	9.1 Nonlocal Operator Method
		9.1.1 Basic Principle
		9.1.2 Nonlocal Form of Linear Elasticity
		9.1.3 Operator Energy Functional for Vector Field  and Scalar Field
	9.2 Outline of Phase Field Fracture Model
		9.2.1 Phase Field Model
		9.2.2 Evolution Equations in Gradient Damage Mechanics
		9.2.3 Phase Field Evolution with and without Threshold
		9.2.4 Explicit Phase Field Model with Sub-Step
	9.3 Nonlocal Form of the Phase Field Model
	9.4 Numerical Implementation
	9.5 Numerical Examples
		9.5.1 Convergence of Sub-Step Scheme
		9.5.2 Single-Edge Notched Tension Test
		9.5.3 Dynamic Crack Branching
		9.5.4 Kalthoff–Winkler Experiment in 2D
		9.5.5 Cylinder Under Impact
	9.6 Conclusion
	References
10 A Nonlocal Operator Method for Finite Deformation Higher-Order Gradient Elasticity
	10.1 Higher Order Gradient Solid with Finite Deformation
	10.2 Governing Equations of Second-Gradient Solid
		10.2.1 Integration by Parts on Close Surface
		10.2.2 Variational Derivation of Second-Gradient Solid
	10.3 Numerical Implementation
		10.3.1 Review of Nonlocal Operator Method
		10.3.2 Newton-Raphson Method
	10.4 Numerical Examples
		10.4.1 Convergence of Strain Energy in E3 Elasticity
		10.4.2 2D Plate with Uniform Deformation
		10.4.3 2D Plate Subjected to Point Force
		10.4.4 Plate with a Hole: Influence of Length Scales
		10.4.5 Large Deformation of 2D Plate with a Hole
		10.4.6 Large Deformation of 3D Plate Subjected to Line Load
	10.5 Conclusions
	References
Appendix A Preliminary of Mathematica
A.1 Preliminary of Mathematica
A.1.1 Function Compile
A.1.2 Velocity Verlet Algorithm
A.1.3 Fast Taylor Series Expansion
A.1.4 NOM Functions
Appendix B Higher Order Tensors and Their Symmetry
B.1 Symmetry of Higher Order Tensors
B.2 Matrix Form of Strain Gradient Energy by Voigt Notations
	References