The Inclusion-Based Boundary Element Method (iBEM)

Front Cover
The Inclusion-Based Boundary Element Method (iBEM)
Copyright
Contents
List of figures
Biography
Huiming Yin
Gan Song
Liangliang Zhang
Chunlin Wu
Preface
1 Introduction
1.1 Virtual experiments
1.2 Inclusion and inhomogeneity
1.3 Equivalent inclusion method (EIM)
1.4 Boundary element method (BEM)
1.5 Inclusion-based boundary element method (iBEM)
1.6 Case study
1.7 Scope of this book
1.A Index notation of vectors and tensors
1.A.1 Index notation of vectors and tensors
1.A.2 Algebra operations of vectors and tensors
1.A.3 Calculus of vector and tensor fields
1.A.3.1 Del operator and operations
1.A.3.2 Gauss' theorem
1.A.3.3 Green's and Stokes' theorems
1.B Two generalized functions
1.B.1 Definitions of two generalized functions
1.B.2 Properties of two generalized functions
2 Fundamental solutions
2.1 Introduction to boundary value problems
2.2 Fundamental solution for elastic problems
2.3 Fundamental solution for potential flows
2.4 Fundamental solution for the Stokes flows
2.A Extension to bimaterial infinite domain
2.A.1 Elastic fundamental solution for bimaterials
2.A.2 Fundamental solution for a potential flow in bimaterials
2.A.3 Fundamental solution for a Stokes flow in bimaterials
3 Integrals of Green's functions and their derivatives
3.1 Introduction to inclusion problems
3.2 Eshelby's tensors for polynomial eigenstrains of an ellipsoidal or elliptical inclusion
3.2.1 Ellipsoidal inclusion with uniform eigenstrain
3.2.2 Ellipsoidal inclusion with a polynomial eigenstrain
3.2.3 Ellipsoidal inclusion with body force
3.2.4 Elliptical inclusion with a polynomial eigenstrain
3.3 Eshelby's tensors for polynomial eigenstrains at an polyhedral inclusion
3.3.1 Eshelby's tensor for a uniform eigenstrain on a polyhedral inclusion
3.3.2 Eshelby's tensor for higher-order eigenstrains
3.3.3 Eshelby's tensor for a uniform eigenstrain on a polygonal inclusion
3.3.4 Eshelby's tensor for higher-order eigenstrains in a polygonal inclusion
3.3.4.1 Linear Eshelby's tensor for the polygonal inclusion
3.3.4.2 Quadratic Eshelby's tensor for the polygonal inclusion
3.4 Properties of Eshelby's tensor
*3.5 Extension of the inclusion problem to a bimaterial infinite domain or a semi-infinite domain
*3.5.1 Case study for a semi-infinite domain containing spherical inclusions
*3.5.2 Domain integral of fundamental solutions for other shapes of inclusions
3.A Ellipsoidal/elliptical domain integrals of ψ and φ and their derivatives
3.A.1 Coefficients of Eshelby's tensor
3.A.2 Explicit expression of Φ, Φp, Φpq, Ψ, Ψp, and Ψpq in terms of integrals I and V
3.B Closed-form domain integral of polygonal inclusion and their derivatives
3.B.1 Partial differentiation chain rule of functions P(bf,lf-,lf+)
3.B.2 Domain integral of Φ and Ψ and their derivatives
3.B.3 Expression of the fourth derivative of Ψpq and second derivative of Φpq
3.B.4 Domain integral with Green's theorem and the components of contour part
3.C Closed-form domain integral of polyhedral inclusion and their derivatives
3.C.1 Partial differentiation chain rules in 3D transformed coordinate (TC)
3.C.2 Expressions of the fourth derivative of Ψpq and second derivative of Φpq
3.C.3 Domain integral of Φ and Ψ and their derivatives
3.C.4 Explicit forms of F(aI, bJI, le) functions
Partial derivatives of the integrand functions F1, …, F12
4 The equivalent inclusion method
4.1 Introduction to Eshelby's equivalent inclusion method
4.2 Ellipsoidal and elliptical inhomogeneities
4.2.1 General cases of inhomogeneity with a prescribed eigenstrain
4.2.2 Interface condition and the uniqueness of solution
4.2.3 Elastic solution for a pair of ellipsoidal inhomogeneities in the infinite domain
4.3 Polyhedral and polygonal Inhomogeneities with a single polynomial eigenstrain
4.4 Discretization of the polyhedral/polygonal inhomogeneities
4.4.1 Polyhedral part
4.4.2 Polygonal part
4.4.3 Implementation of particle discretization method
4.4.4 Evaluation of domain integral at interior nodes
4.5 Singularity of stress and eigenstrain in angular particles and its influence zone
*4.6 Extension to an semi-infinite domain
*4.6.1 One inhomogeneity
*4.6.2 Multiple inhomogeneities
*4.6.3 Case study – one particle
*4.6.3.1 Case study – two top-down particles
*4.6.3.2 Case study – two side-by-side particles
4.A Domain integral with quadratic shape function
4.B Domain integral with bilinear/quadratic shape function
4.C Combination of several types of elements
5 The iBEM formulation and implementation
5.1 Introduction to BIE and iBEM
5.2 Inclusion problems with both boundary and volume integrals
5.3 Equivalent inclusion method for inhomogeneity problems
5.4 The architecture of iBEM software development
5.5 Periodic boundary conditions for periodic microstructure
*5.6 Numerical verification and comparison of iBEM with FEM
*5.6.1 Boundary effects of one particle
*5.6.2 Equal-sized particle interaction
*5.6.3 Large–small particle interaction
*5.7 Virtual experiments of particulate composite with spherical particles
*5.7.1 A ``cut-off'' strategy in the simulation of a large number of particles
*5.7.2 Effective modulus under periodic boundary condition
*5.7.3 Case study of size gradation of random distribution on effective modulus
5.A Examples of particle interactions
5.A.1 Softer inhomogeneity–boundary interaction
5.A.2 One softer center inhomogeneity
5.A.3 Side-by-side particle interaction
5.A.4 7-particle interaction
5.A.5 Examples of virtual experiments with large number of spherical particles
5.A.5.1 Effect of number of inhomogeneities on effective modulus
5.A.5.2 Effect of random distribution on effective modulus
6 The iBEM implementation with particle discretization
6.1 Introduction to iBEM for composites containing arbitrary inhomogeneities
6.2 Implementation for a polynomial eigenstrain on polygonal and polyhedral inhomogeneities
6.2.1 Equivalent inclusion method for a bounded domain
6.2.2 Implementation of the iBEM algorithm
6.3 Continuity and singularity of elastic fields
6.4 Numerical verification with angular particles
6.4.1 Inclusion problem for a subdomain with prescribed eigenstrain
6.4.2 Triangular/tetrahedral inhomogeneity under a uniaxial load
*6.5 Virtual experiments for arbitrary composites
*6.5.1 Effective modulus for one triangular/tetrahedral inhomogeneity
*6.5.2 Effective modulus under periodic boundary condition
*6.5.3 Virtual experiments of size gradation and random distribution of particles on the effective elasticity
6.A Stress equivalent equations with boundary integral equation
6.B Stress contour plot of inclusion problem
6.C Stress contour plot of inhomogeneity problem
6.D Discussion on mesh strategy of polygonal inhomogeneities
6.E Effective modulus with multiple triangular/tetrahedral inhomogeneities
7 The iBEM for potential problems
7.1 Generalization of the EIM to boundary value problems with inhomogeneities
7.2 The iBEM for potential flow – heat conduction
7.3 Boundary effect on the heat flow
7.4 Particle interactions in steady-state heat conduction
*7.5 Homogenization of particle reinforcement composites by iBEM toward virtual experiments
*7.6 Heat flow of an infinite bimaterial domain containing inhomogeneities
8 The iBEM for the Stokes flows
8.1 Equivalent inclusion method for the Stokes flows
8.2 Particle motion in a Stokes flow
8.3 Boundary effect on the Stokes flow
8.3.1 One particle moving in the semi-infinite fluid
8.3.2 Multiple particles moving in a semi-infinite fluid
*8.4 Virtual experiments of particle settlement in a viscous fluid
*8.4.1 Sedimentation of one particle
*8.4.2 Sedimentation of multiple particles toward FGM manufacturing
8.5 Formulation of iBEM for the Stokes flow containing elliptical particles
8.5.1 Verification of iBEM
8.5.2 Simulation of the colloidal system with rods and disks like particles under shear
8.A Derivation and explicit expression of the tensors
9 The iBEM for time-dependent loads and material behavior
9.1 Harmonic vibration with time
9.1.1 Fundamental solution for harmonic excitation
9.1.2 Inclusion problem for the infinite domain
9.1.3 Inclusion problem for a bounded domain (time harmonic)
9.1.4 Equivalent inclusion method for the bounded domain (time harmonic)
9.1.5 Extension to heterogeneous materials under harmonic temperature variation
9.2 Transient heat conduction problems
9.2.1 Boundary element method for transient heat conduction
9.2.2 Inclusion problem for infinite and finite domain
9.2.3 Phase change process
9.2.4 Inclusion-based boundary element method for transient heat conduction with phase change process
9.2.5 Implementation of EIM
9.2.6 Equivalent heat flux condition with the heat capacity method
9.2.7 Assembly of the global matrix of iBEM
9.3 Time-dependent material behavior of composites
9.3.1 Laplace transform-based correspondence to elastic problems
9.3.2 Fourier transform-based correspondence to elastic problems
9.A Elastodynamic Green's functions for an isotropic infinite domain
9.A.1 Derivation of the elastodynamic Green's function for a harmonic point force
9.A.2 Derivation of the elastic Green's function for an impulse point force
9.A.3 Summary of elastodynamic Green's functions
9.B Green's function for transient heat conduction
9.C Integral of Green's function for transient heat conduction for experimental validation
9.C.1 Explicit time-integral form of G,T and discussion of singularity
9.C.2 Comparison of the mesh for the experiment verification
10 The iBEM for multiphysical problems
10.1 Introduction to multiphysical modeling of composites
10.2 Equivalent inclusion method for magnetostatic problem
10.2.1 One ellipsoidal inhomogeneity
10.2.2 Multiple ellipsoidal inhomogeneities
10.3 Basic theory of a steady-state magnetic problem
10.3.1 Local magnetic field
10.3.2 Magnetic force
10.4 Particle motion in a rheological fluid
*10.5 Numerical simulation and case studies
*10.5.1 Case study for solving the magnetostatic problem by EIM
*10.5.2 Case study for calculating magnetic force between two particles
*10.5.3 Particle chain rotating the magnetic field
*10.6 Validation with laboratory tests
*10.7 Virtual experiments
11 Recent development toward future evolution
11.1 Recent development of iBEM
11.2 Future research directions
A Introduction and documentation of the iBEM software package
A.1 Overview of iBEM
A.2 Structure of iBEM software
A.3 Key classes
A.3.1 Main class Config – Preprocess service
A.3.2 Class BEMBuilder – iBEMSystem Service
A.3.3 Class integrator – iBEMSystem Service
A.3.4 Class postprocessor – SystemSolver and PostProcessor Service
A.4 Installation of iBEM
A.5 Run iBEM
A.6 A case study for tutorial
A.7 Notes for redevelopment of iBEM
Bibliography
Index
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