The Inclusion-Based Boundary Element Method (iBEM)

The Inclusion-Based Boundary Element Method (iBEM)

1st Edition - April 14, 2022

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  • Authors: Huiming Yin, Gan Song, Liangliang Zhang, Chunlin Wu
  • Paperback ISBN: 9780128193846
  • eBook ISBN: 9780128193853

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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.

Key Features

  • The multidisciplinary approach used in this book, drawing on computational methods as well as micromechanics, helps to produce a computationally efficient solution to the multi-inclusion problem
  • The iBEM can serve as an efficient tool to conduct virtual experiments for composite materials with various geometry and boundary or loading conditions
  • Includes case studies with detailed examples of numerical implementation


Graduate students and researchers from civil, mechanical, aerospace and materials science and engineering disciplines

Table of Contents

  • 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

Product details

  • No. of pages: 354
  • Language: English
  • Copyright: © Academic Press 2022
  • Published: April 14, 2022
  • Imprint: Academic Press
  • Paperback ISBN: 9780128193846
  • eBook ISBN: 9780128193853

About the Authors

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

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