Meshfree and Particle Methods E-Book

Ted Belytschko

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 537

Veröffentlichungsjahr: 2023

Beschreibung

Meshfree and Particle Methods Provides thorough coverage of essential concepts and state-of-the-art developments in the field Meshfree and Particle Methods is the first book of its kind to combine comprehensive, up-to-date information on the fundamental theories and applications of meshfree methods with systematic guidance on practical coding implementation. Broad in scope and content, this unique volume provides readers with the knowledge necessary to perform research and solve challenging problems in nearly all fields of science and engineering using meshfree computational techniques. The authors provide detailed descriptions of essential issues in meshfree methods, as well as specific techniques to address them, while discussing a wide range of subjects and use cases. Topics include approximations in meshfree methods, nonlinear meshfree methods, essential boundary condition enforcement, quadrature in meshfree methods, strong form collocation methods, and more. Throughout the book, topics are integrated with descriptions of computer implementation and an open-source code (with a dedicated chapter for users) to illustrate the connection between the formulations discussed in the text and their real-world implementation and application. This authoritative resource: * Explains the fundamentals of meshfree methods, their constructions, and their unique capabilities as compared to traditional methods * Features an overview of the open-source meshfree code RKPM2D, including code and numerical examples * Describes all the variational concepts required to solve scientific and engineering problems using meshfree methods such as Nitsche's method and the Lagrange multiplier method * Includes comprehensive reviews of essential boundary condition enforcement, quadrature in meshfree methods, and nonlinear aspects of meshfree analysis * Discusses other Galerkin meshfree methods, strong form meshfree methods, and their comparisons Meshfree and Particle Methods: Fundamentals and Applications is the perfect introduction to meshfree methods for upper-level students in advanced numerical analysis courses, and is an invaluable reference for professionals in mechanical, aerospace, civil, and structural engineering, and related fields, who want to understand and apply these concepts directly, or effectively use commercial and other production meshfree and particle codes in their work.

Leseprobe

Meshfree and Particle Methods

Fundamentals and Applications

Ted Belytschko

Northwestern University

USA

J. S. Chen

University of California

USA

Michael Hillman

The Pennsylvania State University

USA

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Library of Congress Cataloging‐in‐Publication DataNames: Belytschko, Ted, 1943–2014, author. | Chen, J. S., 1958–author. | Hillman, Michael, author.Title: Meshfree and particle methods : fundamentals and applications / Ted Belytschko, J. S Chen, Michael Hillman.Description: Hoboken, NJ : Wiley, 2024. | Includes bibliographical references and index.Identifiers: LCCN 2022041519 (print) | LCCN 2022041520 (ebook) | ISBN 9780470848005 (hardback) | ISBN 9781119811152 (adobe pdf) | ISBN 9781119811138 (epub)Subjects: LCSH: Meshfree methods (Numerical analysis) | Particle methods (Numerical analysis)Classification: LCC QA297 .B379 2023 (print) | LCC QA297 (ebook) | DDC 518/.2–dc23/eng20221219LC record available at https://lccn.loc.gov/2022041519LC ebook record available at https://lccn.loc.gov/2022041520

Cover Design: WileyCover Image: Courtesy of Jonghyuk Baek, J. S. Chen, and Michael Hillman

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 Pennsylvania State University, as well as short courses offered by the second and third authors. They serve as the main introduction to these methods, focusing on linear problems. The remaining chapters do not necessarily contain more advanced materials but detail the application of these methods to unique problem domains (Chapter 7), the variety of meshfree methods formulated over the years (Chapter 8), leveraging smoothness to solve the strong form directly (Chapter 9), and the computer implementation of these methods (Chapter 10).

This book is organized as follows. We first present the history of method development, the definition of a meshfree method, the key approximation properties, a demonstration of meshfree modeling, and classes of meshfree methods in Chapter 1, “Introduction to Meshfree and Particle Methods.” Next, the strong forms and weak forms (variational equations) of diffusion and elasticity problems are given in Chapter 2, “Preliminaries: Strong and Weak Forms of Diffusion, Elasticity, and Solid Continua,” with particular emphasis on the imposition of Dirichlet boundary conditions since meshfree approximations are generally kinematically inadmissible. Here, we note that the readers with intimate knowledge of strong and weak forms, constrained variational principles like the Lagrange multiplier method, and the governing equations of continua may prefer to skip this chapter, but they may still find it useful for reference later in the book. In Chapter 3, “Meshfree Approximations,” complete derivations of the popularly used moving‐least squares (MLS) approximation employed in the element‐free Galerkin (EFG) method, and the reproducing kernel (RK) approximation used in the reproducing kernel particle method (RKPM) are provided. Their associated properties and various methods to compute derivatives needed for solving PDEs are also presented. Although convergence and stability of various meshfree formulations are discussed with numerical demonstrations provided, the proofs of the mathematical properties are not described, but appropriate references are given for further information and details. Complete procedures for the discretization of PDEs using meshfree approximations with the associated enforcement of the Dirichlet boundary conditions are discussed in Chapter 4, “Solving PDEs with the Galerkin Meshfree Methods.” Various methods to construct kinematically admissible meshfree approximations are introduced in Chapter 5, “Construction of Kinematically Admissible Shape Functions,” as well as their incorporation with consistent weak formulations to achieve optimal convergence when higher‐order bases are used. In Chapter 6, “Quadrature in Meshfree Methods,” the concept of integration constraints (ICs) is first introduced as the sufficient conditions to achieve optimal convergence, and various stabilized nodal integration methods with variationally consistent corrections to meet the ICs are presented. Lagrangian and semi‐Lagrangian meshfree approximations and discretizations for solving large (finite) deformation solid mechanics problems are discussed in Chapter 7, “Nonlinear Meshfree Methods,” along with their stability conditions in transient problems and their extension to contact problems. Chapter 8, “Other Galerkin Meshfree Methods,” presents other approximations commonly used in the weak formulation, including smoothed particle hydrodynamics (SPH), the partition of unity method, h‐p clouds, and Sibson and non‐Sibson interpolation used in the natural element method. Utilizing the smoothness of the meshfree approximations to discretize PDEs directly, Chapter 9, “Strong Form Collocation Meshfree Methods,” introduces the alternative approach of using collocation rather than weak forms. The radial basis collocation method (RBCM), the reproducing kernel collocation method (RKCM), the gradient reproducing kernel collocation method (GRKCM), and the application of these approaches to solving problems with heterogeneity and discontinuities are discussed. Chapter 9 includes content republished by permission from Springer Nature: Advances in Computational Plasticity: A Book in Honour of D. Roger J. Owen, Performance Comparison of Nodally Integrated Galerkin Meshfree Methods and Nodally Collocated Strong Form Meshfree Methods, Hillman, M.; Chen, J.S., 2017. Finally, Chapter 10, “RKPM2D: A Two‐dimensional Implementation of RKPM,” details the implementation aspects of meshfree methods, with a description of an open‐source software RKPM2D with preprocessing, solver, and postprocessing integrated under the MATLAB environment. The computer code should help readers understand the programming of meshfree and particle methods and allow the implementation of advanced meshfree algorithms and extensions to other related applications. Chapter 10 includes content republished by permission from Springer Nature: Computational Particle Mechanics, An Open-Source Implementation of Nodally Integrated Reproducing Kernel Particle Method for Solving Partial Differential Equations, Tsung-Hui Huang, et al., 2019.

Suggested Use in Instruction

A one‐semester graduate course on linear meshfree methods can cover Chapters 2–6, while a one‐quarter course can cover most topics in these chapters. An instructor may also choose to cover selected topics from Chapters 7–9, such as the strong‐form collocation method, or a brief introduction to other meshfree methods, as we have done in the past. A more advanced course could also be compiled from select topics from the book. A few examples are given below:

One‐semester course: “Meshfree and Particle Methods for Linear Mechanics.”

Topics: Linear Galerkin meshfree methods (Chapters 2–6) and strong form collocation meshfree methods (Chapter 9).

One‐quarter course: “Meshfree and Particle Methods for Linear Mechanics.”

Topics: Strong and weak formulations for linear problems (Sections 2.1–2.2), meshfree approximations (Sections 3.1–3.4), solving diffusion and elasticity (Sections 4.1–4.2), kinematically admissible meshfree methods (Sections 5.1–5.5), quadrature (Sections 6.1–6.6), and introduction to strong form collocation methods (Sections 9.1–9.3).

One‐quarter or one‐semester course: “Meshfree and Particle Methods for Linear and Nonlinear Mechanics.”

Topics: Select materials from Chapters 2–6, Chapter 7 (Sections 7.1–7.5), and select materials from Chapters 8–9.

A short course would undoubtedly need to be much more selective in the material but could refer to the book for more details.

Acknowledgements

We would like to sincerely thank the following former and current students and postdocs of the second and third authors who have contributed to proofreading chapters and generating figures for this book; their efforts are greatly appreciated:

Jonghyuk Baek, The University of California, San Diego

Sheng‐Wei Chi, The University of Illinois, Chicago

Andy Groeneveld, The Pennsylvania State University, University Park

Xiaolong He, The University of California, San Diego

Tsung‐Hui Huang, National Tsing Hua University, Taiwan

Siavash Jafarzadeh, The Pennsylvania State University, University Park

Feihong Liu, The Pennsylvania State University, University Park

Ryan Schlinkman, The University of California, San Diego

Kristen Susuki, The University of California, San Diego

Karan Taneja, The University of California, San Diego

Hui‐Ping Wang, General Motors Global Research and Development

Jiarui Wang, The Pennsylvania State University, University Park

Yanran Wang, The Pennsylvania State University, University Park

The English proofing by Jennifer Dougal is also appreciated.

J. S. Chen

University of California, San Diego

Michael Hillman

The Pennsylvania State University, University Park

1Introduction to Meshfree and Particle Methods

Meshfree methods have several origins: on one genealogical tree are the particle methods originated by Lucy [1] and Gingold and Monaghan [2], then refined and extended by Monaghan and his coworkers [3, 4]. On the other are the generalized finite difference methods, originated by Jensen [5] and refined by Perrone, Kao, Liszka, and Orkisz [6, 7]. One of the most important papers in the emergence of these methods was the work of Nayroles, Touzot, and Villon [8]. They called the method the diffuse element method. As we shall see, there are many similarities between these methods, though the initial viewpoints appear to be decidedly different.

Remarkably, a watershed in the development of these methods occurred when they were named “meshfree” (or meshless) methods, which shows that an attractive name goes a long way. The advantage of the name meshfree is that it highlights the most compelling attribute of these methods: the absence of a mesh of elements interconnected by nodes. The generation of finite element meshes for three‐dimensional problems of bodies with a variety of features is still very challenging, especially when the model must be remeshed as the solution evolves, as in solidification and dynamic fracture problems. Adaptive refinement using finite elements is also cumbersome due to compatibility requirements along element boundaries.

Both names have persisted: Larry Libersky pointed out at an early specialty meeting on these methods that the name meshfree is more marketable than meshless: we don’t call foods “fatless” or soft drinks “sugarless”; they are called “fat‐free” and “sugar‐free”! Thus, meshfree is a more attractive name for these methods.

1.1 Definition of Meshfree Method

For the purposes of this book, we define a meshfree method as any method that constructs the approximation in terms of nodal values, where the connectivity of the nodes need not be specified explicitly, and the arrangement of the nodes is arbitrary. In fact, information like connectivity is usually extracted in meshfree methods on the fly. Thus, the major distinction from finite element methods (FEMs) is that elements are not used to construct the approximation functions. For traditional finite difference methods, the arrangement of nodes is also not arbitrary. Nodes are arranged in a highly structured manner so that their indices can identify adjacent nodes required in constructing the equations. The difference between these two approaches is crystallized in Figure 1.1 by observing how a finite element discretization contrasts with a meshfree discretization: the approximation is generally associated only with nodes in the latter. The meshfree approximation function associated with node I is denoted by ΨI, and the subdomain over which it is nonzero, the support of ΨI, is denoted by wI. It can be seen that the support naturally defines connectivity rather than fixed connectivity dictated by a grid or a mesh of elements. Thus, any field variable in differential equations can be approximated using a point cloud without any particular structure.

Figure 1.1 (a) Patching of finite element shape functions from local element domains and (b) meshfree approximation functions constructed directly at the nodes in the global coordinate, employing circular supports. The boundary of the domain is denoted as Γ.

Source: Adapted from Chen and Belytschko [9], figure 1, p. 887. Reproduced with permission of Springer Nature.

1.2 Key Approximation Characteristics

Meshfree methods possess several key characteristics that make them a highly unique class of numerical methods for solving differential equations. First, by the definition we have laid out, the approximation is formed without the need for a mesh or grid, and the connectivity is instead defined naturally. Meshfree approximations can also be constructed with arbitrary smoothness, and they can be much smoother or rougher than the finite element shape functions. In fact, the order of continuity and completeness can be made independent from one another in most meshfree methods, as opposed to finite element or isogeometric analysis, where increasing continuity requires increasing the polynomial order (completeness).

With smoother approximations, quantities involving derivatives such as strains in continuum mechanics and fluxes in heat transfer are also much smoother. This attribute increases the accuracy of the solution when the approximated function is smooth. Since solutions of diffusion equations and elasticity are usually very smooth when the material coefficients are continuous, this advantage applies to many useful classes of problems. Several of these properties, and other special properties are detailed in Chapter 3 “Meshfree Approximations.”

Utilizing the smoothness of the approximation, it is also possible to solve the strong form of a problem directly without resorting to the weak formulation. Second‐order derivatives are often required; these are simple to compute based on the smoothness of the meshfree shape functions and can also be approximated very efficiently. Chapter 9 “Strong Form Collocation Meshfree Methods” describes these approaches. This advantage of smoothness also applies to the weak formulation of thin beams, plates, and shells, where the global continuity required is easily achieved without the substantial effort required in FEM.

The ease of adaptive refinement is another attractive feature of meshfree methods. In finite elements, approximation functions are constructed in a local parent domain, and thus, compatibility is required along the element boundaries in local adaptive refinement. On the other hand, meshfree methods only rely on nodal locations in the global coordinates to construct the approximation, and enforcing compatibility is avoided in adaptive refinement.

A major distinction between meshfree and finite element approximations is that meshfree shape functions are usually not interpolants. Finite elements are usually interpolatory, and in fact, they are Lagrange interpolants for most of the commonly used elements. On the other hand, meshfree approximation functions are usually not, and we will instead call them simply approximation functions or shape functions. They are sometimes called interpolation functions, but this terminology ignores the fundamental definition of interpolants. This property of meshfree approximations makes the direct application of boundary conditions on the primary variable, often called essential boundary conditions or Dirichlet boundary conditions, more difficult. Chapter 2 “Preliminaries: Strong and Weak Forms of Diffusion, Elasticity and Solid Continua” lays out constrained variational principles to weakly enforce Dirichlet conditions such as the Lagrange multiplier method, the penalty method, and Nitsche’s method. Special techniques can also be introduced to construct meshfree shape functions so that the traditional strong approach can be employed, which are described in Chapter 5 “Construction of Kinematically Admissible Shape Functions.”

Another fundamental difference between meshfree and finite element methods is the relationship between the support of the shape functions and the subdomains over which quadrature is carried out. The element and quadrature domains are the same in finite elements, which unifies the approximation of field variables and numerical integration. This is clearly not the case for meshfree methods as the supports are defined simply by the nodes themselves, as seen in Figure 1.1b. At the same time, this offers some unique possibilities for meshfree quadrature, yet implementing a practical approach is nontrivial. Numerical integration for meshfree methods is discussed in Chapter 6, “Quadrature in Meshfree Methods.”

The absence of a mesh circumvents element distortion and entanglement issues, making meshfree methods quite suitable for arbitrarily large deformations and distortions in Lagrangian continuum mechanics. Chapter 7 “Nonlinear Meshfree Methods” summarizes their implementation in this class of problems, leveraging several of the special properties of meshfree approximations.

1.3 Meshfree Computational Model

A meshfree computational model consists of a set of nodes and a description of the model’s surfaces. We will call the set of nodes and the boundary description the grid, so as to avoid the word “mesh” in a “meshfree method.” It can be seen in Figure 1.1b that the arrangement of the nodes can be quite arbitrary, although uniform nodal distributions are often used when domain geometry is regular. The boundaries can be described by a variety of methods, for example, level set functions and CAD methods. It is noteworthy that in contrast to FEM, the geometric description of the body must usually be given, rather than the geometry defined directly by the mesh topology.

The generation of a meshfree grid for computational analysis is shown in Figure 1.2. Starting with a geometric definition of the model, points are generated inside the volumes while the surface definitions are retained. Nodes can be generated to fill the volume in a variety of ways. A simple way is to simply use a triangulation of the associated domain, retaining only the vertices. In fact, any finite element meshing technology can be employed, with only the nodes retained from the mesh generation. On the other hand, relying on meshing technology is not necessary or even desirable, as meshfree methods obviate meshing which is a key advantage. A wide variety of sphere and ellipsoidal packing algorithms exist [10], which can be used to avoid meshing altogether. In addition, triangulation algorithms or Voronoi diagrams can be used to obtain nodal volumes, and thus the only requirement for discretization is the generation of nodes.

Figure 1.2 Constructing a meshfree computational grid for a bullet penetration analysis: (a) CAD geometry and (b) the meshfree grid, which fills the CAD volumes with nodes and retains the surfaces for applying boundary conditions.

1.4 A Demonstration of Meshfree Analysis

Figure 1.3 demonstrates several key characteristics of a meshfree solution. A penetration analysis associated with a meshfree grid (see Figure 1.2) is performed for a concrete target. The computation proceeds without the difficulties of element distortion and entanglement. The material break‐up and ensuing multi‐body contact with free surface formation can be handled by node‐based algorithms (see Chapter 7). As the analysis proceeds, a meshfree error detector [11] embedded in the meshfree approximation indicates where additional accuracy is needed, and points are added in these areas on‐the‐fly with multiple levels of adaptive refinement. The refinement is performed simply by inserting additional points without issues such as “hanging nodes” as in FEM. Thus, the framework of meshfree analysis provides a flexible tool for challenging scientific and engineering problems.

1.5 Classes of Meshfree Methods

Meshfree methods have been developed under two general branches of formulations which are covered in this book:

The Galerkin meshfree methods based on the weak form of partial differential equations (PDEs).

While no mesh is needed in the construction of the approximation, domain integration is required, and special techniques to enforce essential boundary conditions are needed. Domain integration and enforcement of boundary conditions are discussed in

Chapters 4

–

6

; and

Figure 1.3 Meshfree simulation of a bullet penetration event with adaptive model refinement. Top to bottom: back view, side view, and a front perspective; left to right: as the simulation proceeds.

The collocation meshfree methods based on the strong form of PDEs.

Because of the ease in constructing smooth meshfree approximations, PDEs can be solved using the strong form directly at collocation points without special domain integration and essential boundary condition procedures, as will be presented in

Chapter 9

.

Table 1.1 shows the wide variety of the meshfree methods that have been proposed; Table 1.2 lists the common acronyms for these methods. Essentially, one may classify these methods based on the approximation and how it is employed to solve the governing equations at hand. As can be seen, meshfree methods have generally been developed using the classical weak formulation.

In this book, we do not go into too much detail on the meshfree implementation of Peridynamics [27] or optimal transportation meshfree methods [48], which are based on governing equations other than the classical differential ones. We also do not include the Lagrangian–Eulerian type methods of particle in cell [56, 57] and material point methods [12, 13], which employ Lagrangian points with an Eulerian computational mesh. Instead, we refer the interested reader to the review by Li and Liu [58] for a broad overview that includes meshfree methods that do not strictly fall under the categories of Galerkin or collocation.

We will focus primarily on meshfree methods of the Galerkin type that employ shape functions with compact support, which are covered in Chapters 3–8. Some collocation methods and approximation functions with both global and compact supports will be discussed in Chapter 9. A compact support is essential if the governing equations are to be sparse; large systems of equations that are not sparse are extremely expensive. This observation also applies to solutions by explicit time integration. When the supports are not compact, the acceleration at any node depends on the nodal values of all nodes of the model. This makes the computation of the acceleration very expensive. The situation is akin to molecular dynamics. Potentials that depend on all atoms in a system are very slow, whereas potentials that only involve the nearest neighbors are very fast.

Table 1.1 Various Galerkin and collocation‐based meshfree methods.

Solution scheme (Discretization)

Weak form

Strong form

Approximation

Local Polynomial

Eulerian mesh with Lagrangian points

MPM [

12

,

13

], PFEM‐2 [

14

,

15

]

Reconstructed

PFEM [

16

,

17

]

Discontinuous/meshless

FPM

[18]

Moving least squares and reproducing kernel

Direct derivatives

EFG

[19]

, RKPM [

20

,

21

]

FP

[22]

, RKCM [

23

,

24

]

Diffuse/implicit derivatives

DEM

[8]

GRKCM [

25

,

26

], GFD [

5

,

7

]

Non‐local derivatives

PD

[27]

, ULPH

[28]

, RKPD

[29]

Smoothed derivatives

SCNI

[30]

, RKGS

[31]

GSCM

[32]

Enriched

XEFG [

33

,

34

]

Petrov–Galerkin

MLPG

[35]

, VCI

[36]

Eulerian mesh with Lagrangian points

Improved MPM

[37]

Reconstructed meshfree

SLRKPM [

38

,

39

]

Partition of unity

Polynomial enrichment

hp

C [

40

,

41

], MFS

[42]

General enrichment

PUM [

43

,

44

], PPU

[45]

MaxEnt

Lagrangian

MaxEnt [

46

,

47

]

Reconstructed

OTM

[48]

Natural neighbor

Lagrangian

NEM [

49

,

50

]

Reconstructed

MFEM

[51]

Radial basis functions

RPIM

[52]

RBCM [

53

,

54

]

Kernel approximation

SPH [

1

,

2

]

MPS

[55]

Table 1.2 Common acronyms of meshfree methods.

DEM

Diffuse element method

EFG

Element-free Galerkin

FPM

Fragile points method

FP

Finite point

GFD

Generalized finite difference

GRKCM

Gradient reproducing kernel collocation method

GSCM

Gradient smoothing collocation method

hp

C

h

‐

p

clouds

MaxEnt

Maximum entropy

MFEM

Meshless finite element method

MFS

Method of finite spheres

MLPG

Meshless local Petrov–Galerkin

MPM

Material point method

MPS

Moving particle semi‐implicit

NEM

Natural element method

OTM

Optimal transportation meshfree

PD

Peridynamics

PFEM

Particle finite element method

PFEM‐2

Particle finite element method, second generation

PPU

Particle partition of unity

PU

Partition of unity

RBCM

Radial basis collocation method

RKCM

Reproducing kernel collocation method

RKGS

Reproducing kernel gradient smoothing

RKPD

Reproducing kernel peridynamics

RKPM

Reproducing kernel particle method

RPIM

Radial point interpolation method

SCNI

Stabilized conforming nodal integration

SLRKPM

Semi‐Lagrangian reproducing kernel particle method

ULPH

Updated Lagrangian particle hydrodynamics

VCI

Variationally consistent integration

XEFG

Extended element‐free Galerkin

1.6 Applications of Meshfree Methods

Meshfree methods are particularly well‐suited for large deformation applications where FEM fails due to mesh entanglement and other mesh‐related issues. They have been applied to many different solid mechanics problems such as large deformation of hyperelastic materials [59, 60], metal forming [61, 62], geotechnical analysis [63, 64