Preface

We dedicate this book "Meshfree and Particle Methods: Fundamentals and Applications" to the late Professor Ted Belytschko for his vision, leadership, and remarkable contributions in this field. The book project was initiated quite a few years before Ted's illness. In the beginning and before Ted's passing in 2014, the efforts by the first two authors were on the fundamental formulation of meshfree methods, and the progress was initially slow, while the topics in the book simultaneously evolved due to the active research in meshfree methods and other related fields. Through the addition of the third author, this book project was finally brought to completion. A two-dimensional MATLAB implementation of the reproducing kernel particle method for solving linear elasticity (RKPM2D) was also attached to this book to illustrate the programming of meshfree methods.

History seems to repeat itself indeed. Like finite difference and finite element methods, meshfree and particle methods originated as fundamental research topics in academia, and eventually found their way into industrial applications. The first workshops, titled "Workshop on Meshfree Methods," sponsored by the National Science Foundation and organized by the University of Iowa, were held in 2000 and 2001. Afterward, workshops called "Workshop on Meshless Methods, Generalized Finite Element Methods, and Related Approaches" were held at the University of Maryland from 2005 to 2009. Around the same time, the biennial "International Workshop on Meshfree Methods for Partial Differential Equations" was initiated by the University of Bonn, Germany, in 2001 and then held every odd year. The biennial US version, "USACM Thematic Conference on Meshfree and Particle Methods" (slightly different names were given to each), started in 2014 and has been held every even year since.

This book is an attempt to present both the fundamentals of meshfree and particle methods, as well as the state-of-the-art of several topics that we feel are very practical for engineering applications. It does not reflect the breadth and amount of research activities across the field but instead focuses on subjects where these methods are most advantageous. It would, of course, be impossible to cover the entirety of the state of research in one book.

Our intent then is to provide a comprehensive discussion on the fundamental concepts, basic formulations, numerical algorithms, computer implementation, and the application of meshfree and particle methods to challenging engineering and scientific problems. We expect it to be a useful introduction and reference for engineers and scientists across academia, industry, and government, and suitable for the instruction of graduate courses at universities. We present the basic formulation of meshfree methods through the moving least squares and reproducing kernel approximations, to demonstrate the unique approximation and discretization properties for solving both diffusion and linear and nonlinear mechanics problems.

The Utility of Meshfree Methods

For the reader who is not intimately familiar with these approaches, we first describe why they are useful. We begin by defining meshfree methods: a class of numerical techniques that do not rely on any mesh, grid, or structured discretization, aside from a set of points. That is, the connectivity between the points (called nodes) does not have to be dictated a priori in an adjoining fashion, and only needs to satisfy some minimum requirements. The Galerkin class of meshfree methods was designed to inherit the main advantages of the finite element method, such as the compact support of shape functions, good approximation properties, and mathematical foundations in variational and related principles. At the same time, they overcome the main disadvantages of the finite element method, such as the strong tie between mesh quality and approximation quality, difficulties in constructing discontinuous or highly continuous approximations, tedious adaptive refinement, solution sensitivity to mesh distortion, and solution divergence due to mesh entanglement in large deformation problems.

The nodes, which form patches of supports, only need to cover the domain of interest. This feature obviates the conforming requirement in the finite element method. Therefore, constructing a model for engineering analysis is much less burdensome than the traditional mesh-based approach: one does not need to be concerned with "high quality" elements. When the Galerkin class of these meshfree methods spawned an explosion of research in the 1990s, this feature was highly celebrated. The tedious and time-consuming task of generating a mesh suitable for analysis can be circumvented entirely.

Several other features of these methods are quite remarkable and perhaps even more appealing. First, the order of completeness in the approximation is not only arbitrary but uncoupled from the order of continuity. This is in contrast to most formulations based on conforming polynomials, where to increase continuity, one must also increase the order. Thus low-order methods with high-order smoothness are possible, and vice versa. This feature is very practical for solving problems in mechanics, where the governing equations can involve high-order derivatives such as thin shells. It is also no longer necessary to employ the weak formulation to reduce the order of differentiation to accommodate the low order of global continuity of traditional finite elements. The most significant advantage of meshfree methods is this flexibility in customizing the approximation functions for desired regularity and ability to capture essential physics and features of particular problems of interest by embedding special functions. Adaptivity and multiple-scale solution strategies also can be implemented with relative ease.

The last unique aspect that we will highlight here is that during the simulation, significant distortions (even fluid-like material flow), fracture, and surface closure, are easily accommodated since no mesh is employed. Historically, much of the method development has been driven by large-deformation plasticity problems (metal forming and earthmoving were among the first industrial applications of the Galerkin version), high-rate defense simulations, and elastomeric devices. To this day, these remain the primary domains where these methods are applied.

Over the years, it has become clear that meshfree methods provide considerable advantages over the conventional finite element methods in solving problems involving moving discontinuities, evolving interfaces, multiple-scale phenomena, large material distortion and structural deformation, and fracture and damage processes. The overall extreme versatility has opened up seemingly limitless possibilities in method development, and there appears to be an ever-present interest in these methods despite nearly three decades of development.

What is Unique About this Book

A handful of books have been published on meshfree and particle methods, so we would like to highlight some unique aspects of this book:

• Detailed descriptions of essential issues and how to address them, not covered in detail in other books, organized over several dedicated chapters: essential boundary condition enforcement, numerical integration, and nonlinear meshfree methods.
• Up-to-date and complete information about the state-of-the-art in Galerkin and collocation meshfree and particle methods, covering the fundamental theories and applications.
• The inclusion of many meshfree methods, such as the Galerkin type, collocation type, partition of unity methods, and kernel estimate of conservation equations (smoothed particle hydrodynamics).
• The topics are integrated with an open-source code, with a chapter describing the code in detail that cross-references the methods described in the book.

Another key feature is that it can serve as both an introduction and a valuable reference to students, engineers, and scientists who either want to learn about meshfree methods or are working in this area already.

Level and Background

This book is designed for readers without prior experience with meshfree and particle methods, but it still requires some basic knowledge of numerical analysis and mechanics. In particular, readers will greatly benefit from an understanding of the linear and nonlinear finite element method, and have a deeper understanding and appreciation for the materials presented. An introductory course in mechanics or elasticity covering indicial and tensor notation is a prerequisite.

The primary audience includes practitioners and researchers in the mechanical, aerospace, civil, and structural engineering industries. A secondary audience is graduate students in these fields, and students of applied mathematics. This book provides fundamental theories, mathematical formulations, numerical algorithms, and code implementation steps to learn the fundamentals and help develop meshfree codes for performing research and analysis.

Content and Structure of this Book

The first six chapters of this book have been compiled with the help of lecture notes (in particular the example problems) from SE 279 "Meshfree Methods for Linear and Nonlinear Mechanics" at The University of California, San Diego, and CE 597 "Meshfree Methods and Advanced Computational Solid Mechanics" at The...