The Inclusion-Based Boundary Element Method (iBEM)

By Huiming Yin, Gan Song, Liangliang Zhang and Chunlin Wu

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.

• 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

• Language: English
• Publisher: Academic Press
• Release date: Apr 14, 2022
• ISBN: 9780128193853

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.

Book preview

Preface

Huiming Yin     

This book stems from the course on Micromechanics of Composite Materials that I started in 2012 at Columbia University while I found the significant needs from students and researchers on composite material designs and modeling. At that time the classic micromechanics-based methods used the concept of representative volume element (RVE) to link the effective material behavior with the local solution. Therefore Eshelby's theory on a single particle in an infinite domain can play a role to predict the effective material behavior of a composite, which links mechanical behavior between the microscale of particles and the marcoscale of composites. However, with the advancement of experimental approach, we can control and manipulate the microstructure very well and fabricate very small specimens for advanced material characterization and optimization. Actually, the scale differentiation is not clear in experiments with small testing samples. Linking the fundamental material properties of each material phase with the sample testing results and then the effective composite material behavior becomes a challenging problem, which creates an urgent need for cross-scale modeling and virtual experiments that are crucial for new material development and discovery.

Then we came up with an idea to generalize Eshelby's method to consider the boundary effect on material behavior of composites in some experiments by changing the Green's function and indeed obtained some interesting results. However, it is not practical to find Green's function for every experiment. Eventually, I turned to the boundary integral method for general virtual experiments. At that time the second author Dr. Gan Song was still my PhD student. He implemented the algorithm with the boundary element method code, and we named the software package iBEM in 2015 as an abbreviation of the inclusion-based boundary element method. He applied iBEM to solve some challenging problems and completed an excellent PhD dissertation. He won the highest award in the Department, the Mindlin Award.

After Dr. Song graduated from Columbia, Dr. Liangliang Zhang took the main responsibility to maintain and develop the iBEM program, and Dr. Chunlin Wu joined the team in research and development of the algorithm. Because the early version of iBEM was developed for ellipsoidal particles only, the application is relatively narrow. Dr. Wu extended the program to two-dimensional (2D) case and generalized it to arbitrary particle shape for both 2D and 3D problems. Therefore iBEM can be a truly practical software for material engineers on material design and analysis, which brought significant interests from industry as well. The favorable response from both researchers and engineers encouraged us to work together and document the method into a book as a systemic record and study resource. Dr. Wu also won the Mindlin Award when he graduated in 2021.

Surely, this book does not intend to cover everything in this dynamic research area, but serves a start for a long journey of further development and applications of the iBEM. Although readers with micromechanics and boundary element method will feel easier to read the book, no particular background is required of readers because some necessary mathematics, mechanics, and physics are explained in the text and Appendices. Although we tried to be fair in citing the literature, due to the very rich research in this area, we have to apologize if some papers do not receive proper credit or are missing in citation. The chapters are arranged in a sequence of the learning process, and we recommend to read them one by one for beginners. However, the sections or subsections with an asterisk can be skipped as they could be in a branch area, which is relatively independent from other part, or an advanced topic for more focused or advanced readers.

During the preparation of this book, we have received help and support from our colleagues, students, and collaborators. First, we thank Yingjie Liu, Xin He, Fangliang Chen, Tengxiang Wang, Zifeng Yuan, and Zhenyu Shou for their contribution in this research topic at Columbia University. Junhe Cui read through the book and provided her feedback, which was very helpful for improving this book. I am grateful for my advisors and mentors, Lizhi Sun, Minzhong Wang, Glaucio Paulino, Bill Buttlar, George Weng, Frank DiMaggio, Rene Testa, and Jiun-Shyan (JS) Chen, who have provided guidance and encouragement in the course of my research and career progress toward the achievement of this book. I would like to thank my colleagues and collaborators, including Yuhong Wang, Yang Gao, Yingtao Zhao, Gautam Dasgupta, Raimondo Betti, Haim Waisman, Qiang Du, and Marco Giometto, for the insightful discussion. Some parts of this book are based on the publications and collaborative work with them.

In addition, this book has mainly been written in our spare time at night or weekend. We had weekly zoom meeting on Saturday morning in the past two years during the COVID pandemic period. We highly appreciate the support and consideration from our families.

Finally, we acknowledge the financial support from National Science Foundation (NSF) – Divisions of Civil, Mechanical and Manufacturing Innovation (CMMI) and Industrial Innovation and Partnerships (IIP), National Natural Science Foundation of China, Department of Defense (DOD) – Air Force Office of Scientific Research (AFOSR) and Office of Naval Research (ONR), and US Department of Agriculture – National Institute of Food and Agriculture. Without the research support, the work is impossible to be done.

August 23, 2021, at Alpine, New Jersey

Chapter 1: Introduction

Virtual experiments with iBEM

Abstract

The inclusion-based boundary element method (iBEM) is a powerful tool, which is developed to efficiently solve the boundary value problems (BVPs) for a finite domain with multiple particles for modeling of composite materials. It is particularly suitable for virtual experiments with computer to simulate physical experiments and predict the test results. The equivalent inclusion method (EIM) and boundary element method (BEM) provide the foundation of the iBEM. Unlike traditional BEM, which requires to mesh all the surfaces of particles, each particle is treated as a source with an eigenfield, such as eigenstrain for elasticity, eigenstrain rate for Stokes' flow, or others for different BVPs, and the response can be obtained by analytical volume integrals for rapid computation and exactness of the integral. Different from discrete element method (DEM), iBEM is based on continuum mechanism and provides detailed local fields for analysis and design of advanced materials. Since iBEM adopts analytical integrals over the inclusion, it does not need to mesh the matrix domain and is able to converge to the exact solution more rapidly than the finite element method (FEM). Moreover, because iBEM solves the field variables on the boundary, which can directly describe the effective material response, iBEM can be an ideal tool for virtual experiments of composite materials and