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The Inclusion-Based Boundary Element Method (iBEM)

1st Edition - April 14, 2022

Authors: Huiming Yin, Gan Song, Liangliang Zhang, Chunlin Wu

Paperback ISBN:

9 7 8 - 0 - 1 2 - 8 1 9 3 8 4 - 6

eBook ISBN:

9 7 8 - 0 - 1 2 - 8 1 9 3 8 5 - 3

The Inclusion-Based Boundary Element Method (iBEM) is an innovative numerical method for the study of the multi-physical and mechanical behaviour of composite materials, linear… Read more

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The Inclusion-Based Boundary Element Method (iBEM) is an innovative numerical method for the study of the multi-physical and mechanical behaviour of composite materials, linear elasticity, potential flow or Stokes fluid dynamics. It combines the basic ideas of Eshelby’s Equivalent Inclusion Method (EIM) in classic micromechanics and the Boundary Element Method (BEM) in computational mechanics.

The book starts by explaining the application and extension of the EIM from elastic problems to the Stokes fluid, and potential flow problems for a multiphase material system in the infinite domain. It also shows how switching the Green’s function for infinite domain solutions to semi-infinite domain solutions allows this method to solve semi-infinite domain problems. A thorough examination of particle-particle interaction and particle-boundary interaction exposes the limitation of the classic micromechanics based on Eshelby’s solution for one particle embedded in the infinite domain, and demonstrates the necessity to consider the particle interactions and boundary effects for a composite containing a fairly high volume fraction of the dispersed materials.

Starting by covering the fundamentals required to understand the method and going on to describe everything needed to apply it to a variety of practical contexts, this book is the ideal guide to this innovative numerical method for students, researchers, and engineers.

Cover image
Title page
Table of Contents
Copyright
Dedication
List of figures
Biography
Huiming Yin
Gan Song
Liangliang Zhang
Chunlin Wu
Preface
Chapter 1: Introduction
Abstract
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
Appendix 1.A. Index notation of vectors and tensors
Appendix 1.B. Two generalized functions
Bibliography
Chapter 2: Fundamental solutions
Abstract
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
Appendix 2.A. Extension to bimaterial infinite domain
Bibliography
Chapter 3: Integrals of Green's functions and their derivatives
Abstract
3.1. Introduction to inclusion problems
3.2. Eshelby's tensors for polynomial eigenstrains of an ellipsoidal or elliptical inclusion
3.3. Eshelby's tensors for polynomial eigenstrains at an polyhedral 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
Appendix 3.A. Ellipsoidal/elliptical domain integrals of ψ and ϕ and their derivatives
Appendix 3.B. Closed-form domain integral of polygonal inclusion and their derivatives
Appendix 3.C. Closed-form domain integral of polyhedral inclusion and their derivatives
Bibliography
Chapter 4: The equivalent inclusion method
Abstract
4.1. Introduction to Eshelby's equivalent inclusion method
4.2. Ellipsoidal and elliptical inhomogeneities
4.3. Polyhedral and polygonal Inhomogeneities with a single polynomial eigenstrain
4.4. Discretization of the polyhedral/polygonal inhomogeneities
4.5. Singularity of stress and eigenstrain in angular particles and its influence zone
*4.6. Extension to an semi-infinite domain
Appendix 4.A. Domain integral with quadratic shape function
Appendix 4.B. Domain integral with bilinear/quadratic shape function
Appendix 4.C. Combination of several types of elements
Bibliography
Chapter 5: The iBEM formulation and implementation
Abstract
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
*5.7. Virtual experiments of particulate composite with spherical particles
Appendix 5.A. Examples of particle interactions
Bibliography
Chapter 6: The iBEM implementation with particle discretization
Abstract
6.1. Introduction to iBEM for composites containing arbitrary inhomogeneities
6.2. Implementation for a polynomial eigenstrain on polygonal and polyhedral inhomogeneities
6.3. Continuity and singularity of elastic fields
6.4. Numerical verification with angular particles
*6.5. Virtual experiments for arbitrary composites
Appendix 6.A. Stress equivalent equations with boundary integral equation
Appendix 6.B. Stress contour plot of inclusion problem
Appendix 6.C. Stress contour plot of inhomogeneity problem
Appendix 6.D. Discussion on mesh strategy of polygonal inhomogeneities
Appendix 6.E. Effective modulus with multiple triangular/tetrahedral inhomogeneities
Bibliography
Chapter 7: The iBEM for potential problems
Abstract
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
Bibliography
Chapter 8: The iBEM for the Stokes flows
Abstract
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
*8.4. Virtual experiments of particle settlement in a viscous fluid
8.5. Formulation of iBEM for the Stokes flow containing elliptical particles
Appendix 8.A. Derivation and explicit expression of the tensors
Bibliography
Chapter 9: The iBEM for time-dependent loads and material behavior
Abstract
9.1. Harmonic vibration with time
9.2. Transient heat conduction problems
9.3. Time-dependent material behavior of composites
Appendix 9.A. Elastodynamic Green's functions for an isotropic infinite domain
Appendix 9.B. Green's function for transient heat conduction
Appendix 9.C. Integral of Green's function for transient heat conduction for experimental validation
Bibliography
Chapter 10: The iBEM for multiphysical problems
Abstract
10.1. Introduction to multiphysical modeling of composites
10.2. Equivalent inclusion method for magnetostatic problem
10.3. Basic theory of a steady-state magnetic problem
10.4. Particle motion in a rheological fluid
*10.5. Numerical simulation and case studies
*10.6. Validation with laboratory tests
*10.7. Virtual experiments
Bibliography
Chapter 11: Recent development toward future evolution
Abstract
11.1. Recent development of iBEM
11.2. Future research directions
Bibliography
Appendix 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.4. Installation of iBEM
A.5. Run iBEM
A.6. A case study for tutorial
A.7. Notes for redevelopment of iBEM
Bibliography
Bibliography
Bibliography
Index

Huiming Yin

Huiming Yin is an associate professor in the Department of Civil Engineering and Engineering Mechanics at Columbia University, and the director of the NSF Center for Energy Harvesting Materials and Systems at Columbia Site. His research specializes in the multiscale/physics characterization of civil engineering materials and structures with experimental, analytical, and numerical methods. His research interests are interdisciplinary and range from structures and materials to innovative construction technologies and test methods. He has taught courses in energy harvesting, solid mechanics, and composite materials at Columbia University.

Affiliations and expertise

Associate Professor, Department of Civil Engineering and Engineering Mechanics, Columbia University, NY, USA

Gan Song

Dr. Gan Song obtained his Ph.D. in the Department of Civil Engineering and Engineering Mechanics at Columbia University. His research interest focuses on numerical simulation of the mechanical behaviour of civil engineering materials. He develops this innovative numerical method – iBEM under the advice of Professor Yin, which is a powerful tool to characterize mechanical property of composite material containing various sizes, shapes and types of particles within affordable computational cost. The method is able to be extended to analyse fluid mechanics, potential flow, and other multi-physical problems as well.

Affiliations and expertise

Department of Civil Engineering and Engineering Mechanics, Columbia University, USA

Liangliang Zhang

Liangliang Zhang is an Associate Research Scientist in the Department of Civil Engineering and Engineering Mechanics at Columbia University. He earned his Ph. D. in Engineering Mechanics at China Agricultural University. Before joining Columbia University in 2017, he worked as an engineer in company for two years and obtained multidisciplinary engineering experience covering innovative structural design and materials. His research interests are focus on the advanced smart materials and composite structures.

Affiliations and expertise

Associate Research Scientist, Department of Civil Engineering and Engineering Mechanics, Columbia University, USA

Chunlin Wu

Chunlin Wu, PhD from Civil Engineering and Engineering Mechanics, specializes in the iBEM software developing. He received a BS in civil engineering from Tongji University, China in 2017, MS in engineering mechanics from Columbia University, 2018, and PhD in engineering mechanics from Columbia University in October 2021. He received the Mindlin's scholar award in Civil engineering and Engineering mechanics for his PhD studies.

Affiliations and expertise

Columbia University, NY, USA