1

Introduction

1.1

Defining Mesh Free Methods

Designing advanced engineering systems requires the use of computer-aided design (CAD)
tools. In such tools, computational simulation techniques are often used to model and
investigate physical phenomena in an engineering system. The simulation requires solving
the complex differential or partial differential equations that govern these phenomena.

Traditionally, such complex partial differential equations are largely solved using numer-

ical methods, such as the finite element method (FEM) and the finite difference method
(FDM). In these methods, the spatial domain where the partial differential governing
equations are defined is often

discretized

into meshes.

A

mesh

is defined as any of the open spaces or interstices between the strands of a net

that is formed by connecting nodes in a predefined manner. In FDM, the meshes used are
also often called grids; in the finite volume method (FVM), the meshes are called volumes
or cells; and in FEM, the meshes are called elements. The terminologies of grids, volumes,
cells, and elements carry certain physical meanings as they are defined for different physical
problems. However, all these grids, volumes, cells, and elements can be termed

meshes

according to the above definition of mesh. The key here is that a mesh must be predefined
to provide a certain relationship between the nodes, which is the base of the formulation
of these conventional numerical methods.

By using a properly predefined mesh and by applying a proper principle, complex

differential or partial differential governing equations can be approximated by a set of
algebraic equations for the mesh. The system of algebraic equations for the whole problem
domain can be formed by assembling sets of algebraic equations for all the meshes.

The mesh free method, abbreviated

MFree

method in this book, is used to establish a

system of algebraic equations for the whole problem domain without the use of a pre-
defined mesh. MFree methods use a set of nodes scattered within the problem domain as
well as sets of nodes scattered on the boundaries of the domain to

represent

(not discretize)

the problem domain and its boundaries. These sets of scattered nodes do not form a mesh,
which means that no information on the relationship between the nodes is required, at
least for field variable interpolation.

There are a number of MFree methods, such as the element free Galerkin (EFG) method

(Belytschko et al., 1994b), the meshless local Petrov–Galerkin (MLPG) method (Atluri and
Zhu, 1998), the point interpolation method (PIM) (Liu, G. R. and Gu, 1999), the point
assembly method (PAM) (Liu, G. R., 1999), the finite point method (Onate et al., 1996),
the finite difference method with arbitrary irregular grids (Liszka and Orkisz, 1980; Jensen,
1980), smooth particle hydrodynamics (SPH) (Lucy, 1977; Gingold and Monaghan, 1977),
reproducing kernel particle method (Liu, W. K. et al., 1993), which is an improved version
of SPH, and so forth. They all share the same feature that predefined meshes are not used,

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© 2003 by CRC Press LLC

at least for field variable interpolation. The names for the various MFree methods are still
being debated. Because the methodology is still in a rapid development stage, new names
of methods are constantly proposed. It may take some time before all the methods are
properly categorized and unified to avoid confusion in the community.

In contrast to FEM, the term

element free method

is preferred, and in contrast to FDM,

the term

finite difference method using arbitrary or irregular grids

is preferred. Some of the

MFree methods have been assembled and termed

meshless methods

(see the excellent review

paper by Belytschko et al., 1996b). The term mesh free method has also appeared in the
literature in recent years. This book uses the term mesh free method or MFree method for
the collection of all the different mesh free methods because the term carries more a
positive sense than meshless method. A reason to prefer the abbreviation MFree is because
the M can also stand for manpower and money.

M

esh free methods free

M

anpower from

the chore of meshing and free substantial

M

oney for computer-aided engineering (CAE)

projects. The question now is what qualifies as an MFree method?

The

minimum

requirement for an MFree method:

• That a predefined mesh is not necessary, at least in field variable interpolation.

The

ideal

requirement for an MFree method:

• That no mesh is necessary at all throughout the process of solving the problem

of given arbitrary geometry governed by partial differential system equations
subject to all kinds of boundary conditions.

The reality is that the MFree methods developed so far are not really ideal, and fail in one
of the following categories:

• Methods that require background cells for the integration of system matrices

derived from the weak form over the problem domain. These methods are not
truly mesh free. EFG methods belong in this category. These methods are prac-
tical in many ways, as the creation of a background mesh is generally feasible
and can always be automated using a triangular mesh for two-dimensional (2D)
domains and a tetrahedral mesh for three-dimensional (3D) domains.

• Methods that require background cells locally for the integration of system

matrices over the problem domain. MLPG methods belong in this category. These
methods are said to be essentially mesh free because creating a local mesh is
easier than creating a mesh for the entire problem domain; it is a simpler task
that can be performed automatically without any predefinition for the local
mesh.

• Methods that do not require a mesh at all, but that are less stable and less

accurate. Collocation methods and finite difference methods using irregular grids
belong to this category. Selection of nodes based on the type of a physical problem
is still important for obtaining stable and accurate results (Song et al., 1999; Xu
and Liu, 1999; Cheng and Liu, 1999). Automation of nodal selection and improv-
ing the stability of the solution are some of the challenges in these kinds of
methods. This type of method has a very significant advantage: it is very easy
to implement, because no integration is required.

• Particle methods that require a predefinition of particles for their volumes or

masses. The algorithm will then carry out the analyses even if the problem
domain undergoes extremely large deformation and separation. SPH methods

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© 2003 by CRC Press LLC

belong to this category. This type of method suffers from problems in the impo-
sition of boundary conditions. In addition, predefining the particles still techni-
cally requires some kind of mesh.

1.2

Need for MFree Methods

FEM is robust and has been thoroughly developed for static and dynamic, linear or
nonlinear stress analysis of solids, structures, as well as fluid flows. Most practical engi-
neering problems related to solids and structures are currently solved using a large number
of well-developed FEM packages that are commercially available. However, the following
limitations of FEM are becoming increasingly evident:

1. Creation of a mesh for the problem domain is a prerequisite in using FEM

packages. Usually the analyst spends the majority of his or her time in creating
the mesh, and it becomes a major component of the cost of a simulation project
because the cost of CPU (central processing unit) time is drastically decreasing.
The concern is more the manpower time, and less the computer time. Therefore,
ideally the meshing process would be fully performed by the computer without
human intervention.

2. In stress calculations, the stresses obtained using FEM packages are discontinuous

and less accurate.

3. When handling large deformation, considerable accuracy is lost because of the

element distortions.

4. It is very difficult to simulate both crack growth with arbitrary and complex

paths and phase transformations due to discontinuities that do not coincide with
the original nodal lines.

5. It is very difficult to simulate the breakage of material into a large number of

fragments as FEM is essentially based on continuum mechanics, in which the
elements formulated cannot be broken. The elements can either be totally
“eroded” or stay as a whole piece. This usually leads to a misrepresentation of
the breakage path. Serious error can occur because the nature of the problem is
nonlinear, and therefore the results are highly path dependent.

6. Re-mesh approaches have been proposed for handling these types of problems

in FEM. In the re-mesh approach, the problem domain is re-meshed at steps
during the simulation process to prevent the severe distortion of meshes and to
allow the nodal lines to remain coincident with the discontinuity boundaries.
For this purpose, complex, robust, and adaptive mesh generation processors have
to be developed. However, these processors are only workable for 2D problems.
There are no reliable processors available for creating hexahedral meshes for 3D
problems due to the technical difficulty.

7. Adaptive processors require “mappings“ of field variables between meshes in

successive stages in solving the problem. This mapping process often leads to addi-
tional computation as well as a degradation of accuracy. In addition, for large
3D problems, the computational cost of re-meshing at each step becomes very
high, even if an adaptive scheme is available.

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© 2003 by CRC Press LLC

8. FDM works very well for a large number of problems, especially for solving

fluid dynamics problems. It suffers from a major disadvantage in that it relies
on regularly distributed nodes. Therefore, studies have been conducted for a
long time to develop methods using irregular grids. Efforts in this direction are
still ongoing.

1.3 The Idea of MFree Methods

A close examination of these difficulties associated with FEM reveals the root of the
problem: the need to use elements, which are the building block of FEM. A mesh with
predefined “connectivity” is required to form the elements. As long as elements must be
used, the problems mentioned above will not be easy to solve. Therefore, the idea of
eliminating the elements and hence the mesh has evolved naturally. The concept of element
free or mesh free methods has been proposed, in which the domain of the problem is
represented by a set of arbitrarily distributed nodes.

The MFree method has great potential for solving the difficult problems mentioned

above. Adaptive schemes can be easily developed, as there is no mesh, and hence no
connectivity concept involved. Thus, there is no need to provide

a priori

any information

about the relationship of the nodes. This provides flexibility in adding or deleting points/
nodes whenever and wherever needed. For stress analysis of a solid domain, for example,
there are often areas of stress concentration, even singularity. One can relatively freely
add points in the stress concentration area without worrying about their relationship with
the other existing nodes. In crack growth problems, nodes can be easily added around
the crack tip to capture the stress concentration with desired accuracy. This nodal refine-
ment can be moved with a propagation crack through a background arrangement of nodes
associated with the global geometry. Adaptive meshing for a large variety of problems,
2D or 3D, including linear and nonlinear, static and dynamic stress analysis, can be very
effectively treated in MFree methods in a relatively simple manner.

Because there is no need to create a mesh, and the nodes can be created by a computer

in a fully automated manner, the time an engineer would spend on conventional mesh
generation can be saved. This can translate to substantial cost and time savings in modeling
and simulation projects.

There have been a number of MFree methods developed thus far. The major features

of these methods are listed in

Table 1.1.

A software package MFree2D

©

with its own pre-

and postprocessor has also been developed by G. R. Liu and co-workers (Liu, G. R. and
Tu, 2001).

1.4 Outline of the Book

This book provides an introduction to MFree methods, and their application to various
mechanics problems. This book covers the following types of problems:

• Mechanics for solids (2D solids)
• Mechanics for structures (beams, plates, and shells)
• Fluid mechanics (fluid flow, convection flow, and hydrodynamics)

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© 2003 by CRC Press LLC

The bulk of the material in the book is the result of the intensive research work by G. R.

Liu and his research team in the past 6 years. Works of other researchers are also intro-
duced. The significance of this book is as follows:

1. This is the first book published that comprehensively covers MFree methods.
2. The book covers, in a systematic manner, basic theories, principles, techniques,

and procedures in solving mechanics problems using MFree methods. It will be
very useful for researchers entering this new area of research on MFree methods,
and for professionals and engineers developing computer codes for the next
generation of computational mechanics.

3. Readers will benefit from the research outcome of G. R. Liu’s research team and

their work on an award-winning project on MFree methods founded by the
Singapore government. Many materials in this book are the results of ongoing
projects, and have not been previously published.

TABLE 1.1

Some MFree Methods Developed Thus Far and Their Features

Method

Ref.

System Equation

to be Solved

Method of Function

Approximation

Chapter

Covering the

Method

Diffuse element

method

Nayroles et al., 1992

Weak form

MLS

approximation,
Galerkin method

5, 6

Element free

Galerkin (EFG)
method

Belytschko et al., 1994b

Weak form

MLS

approximation,
Galerkin method

5, 6, 11, 12,

14–16

Meshless local

Petrov–Galerkin
(MLPG) method

Atluri and Zhu, 1998

Local weak form

MLS

approximation,
Petrov–Galerkin
method

5, 7, 9–11, 14

Finite point

method

Onate et al., 1996;

Liszka and Orkisz,
1980; Jensen, 1980

Strong form

Finite differential

representation
(Taylor series),
MLS
approximation

5 (brief only)

Smooth particle

hydrodynamics

Lucy, 1977; Gingold

and Monaghan, 1977

Strong form

Integral

representation

5, 9

Reproducing

kernel particle
method

Liu, W. K. et al., 1993

Strong form or

weak form

Integral

representation
(RKPM)

5 (brief only)

hp

-clouds

Oden and Abani, 1994;

Armando and Oden,
1995

Weak form

Partition of unity,

MLS

5 (introduction

only)

Partition of unity

FEM

Babu ka and Melenk,

1995

Weak form

Partition of unity,

MLS

5 (introduction

only)

Point

interpolation
method

Liu, G. R. and Gu,

1999, 2000b,
2001a,b,c,d

Weak form and

local weak form

Point interpolation

5, 8–14

Boundary node

methods

Mukherjee and

Mukherjee, 1997a,b

Weak form and

local weak form

MLS

13 (brief only)

Boundary point

interpolation
methods

Liu, G. R. and Gu,

2000d; Gu and Liu, G.
R., 2001a,e

Weak form and

local weak form

Point interpolation

13, 14

s

∨

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© 2003 by CRC Press LLC

4. A software package named MFree2D

©

with its own pre- and postprocessors,

which has been developed by G. R. Liu’s team, is also introduced.

5. A large number of examples with illustrations are provided for validating, bench-

marking, and demonstrating MFree methods. These examples can be useful
reference materials for other researchers.

The book is written for senior university students, graduate students, researchers, and

professionals in engineering and science. Mechanical engineers and practitioners and
structural engineers and practitioners will also find the book useful. Knowledge of FEM
is not required but would help a great deal in understanding many concepts and proce-
dures of MFree methods. Basic knowledge of mechanics is also helpful in reading this
book smoothly. A very brief introduction on mechanics is provided to prepare readers
who are not familiar with the basics of mechanics.

The MFree method is a very new area of research. There exist many problems, problems

that offer ample opportunities for research to develop the next generation of numerical
methods. The method is also in a rapidly developing and growing stage. Different tech-
niques are developed every day. This book addresses some of the current important issues,
both positive and negative, related to MFree methods, which should prove beneficial to
researchers, engineers, and students who are interested in venturing into this area of
research.

The chapter-by-chapter description of this book is as follows:

Chapter 1:

Provides a brief introduction to MFree methods including background

and motivations that have led to the development of MFree methods, as well as
this book.

Chapter 2:

Describes MFree procedures by comparing them with conventional FEM.

The basic steps involved in using MFree methods for solving engineering prob-
lems are listed. Some important common terms frequently used in MFree meth-
ods are defined.

Chapter 3:

Presents a brief introduction to the basics of mechanics for solids, beams,

and plates. Readers who are familiar with mechanics may skip this chapter.

Chapter 4:

Briefly introduces some principles and weak forms that will be used for

creating discretized system equations. Constrained weak forms that are useful
for MFree formulation are also presented. Readers who are familiar with varia-
tional, energy, and weighted residual methods may skip this chapter.

Chapter 5:

Provides a detailed description of the methods for constructing MFree

shape functions. This is one of the core chapters of this book discussing one of
the most important issues of MFree methods. A new classification of theories of
function approximation methods is presented. Methods of approximation of func-
tions are then systematically introduced. Issues related to constancy, reproduction,
representation, compatibility, and convergence of an approximation are dis-
cussed. Traditional methods, SPH and MLS, and new methods of point interpo-
lation are formulated in detail for the construction of MFree shape functions.

Chapter 6:

Introduces the element free Galerkin method (EFG), one of the MFree

methods that has created a great difference. Very detailed formulation based on
the constrained Galerkin weak form is presented, and a number of techniques
that are used for handling essential boundary conditions are discussed in detail.
Issues related to the background integration are also examined.

Chapter 7:

Introduces the meshless local Petrov–Galerkin (MLPG) method, which

requires only local cells of a background mesh for integration and which is a

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© 2003 by CRC Press LLC

significant advance. The constrained local Petrov–Galerkin method is used to
formulate the MLPG method for both static and dynamic problems. Issues on
types of local domains, effects of the dimension of these domains, handling local
integration, and procedures in dealing with essential boundary conditions are
discussed.

Chapter 8:

Introduces the point interpolation methods (PIM), which are formulated

based on both the Galerkin formulation and the local Petrov–Galerkin formula-
tion. Different methods of PIM, both conforming and nonconforming, are for-
mulated, examined, and benchmarked, issues related to the selection of
parameters are discussed, and a large number of examples are provided.

Chapter 9:

Introduces MFree methods for computational fluid dynamics problems.

SPH, MLPG, and PIM are formulated and applied to solve a number of bench-
marking and application problems. Issues related to iterative solution procedures
for MFree methods in solving fluid flow problems are discussed.

Chapter 10:

Introduces PIM methods developed for analysis of beams. Both thin

beams governed by the Euler–Bernoulli beam theory and thick beams governed
by the Timoshenko beam theory are used in the formulations. Shear-locking
issues for beams in using MFree methods based on shear deformable beams are
also discussed.

Chapter 11:

Introduces MFree methods developed for analysis of plates. Both thin

plates governed by the Mindlin plate theory and thick plates governed by the
third-order shear deformation theory are used to develop the MFree methods.
Three methods—EFG, PIM, and MLPG—are formulated. Shear-locking issues
for plates are also discussed in great detail.

Chapter 12:

Introduces MFree methods developed for analysis of shells. Both thin

shells governed by the Kirchhoff–Love theory and thick shells are used to formu-
late the MFree methods. Two methods—EFG and PIM—are formulated for static
and dynamic problems. Advantages of using MFree methods for shells are dis-
cussed. Shear- and membrane-locking issues for shells are also briefly discussed.

Chapter 13:

Formulates two boundary-type MFree methods—BPIM and radial

BPIM. Procedures of using the MFree concept for solving boundary integral
equations are provided. A number of examples are presented to demonstrate the
advantages of these methods in solving problems of infinite domain.

Chapter 14:

Introduces methods that are formulated by coupling domain and bound-

ary types of MFree methods. MFree methods that couple with the traditional
FEM and boundary element method (BEM) are also formulated and benchmarked.

Chapter 15:

Discusses a number of implementation issues in coding with MFree

methods, including node generation, background cell generation, determination of
nodal influence domain, node selection, error estimations, and adaptive procedures.

Chapter 16:

Presents MFree2D

©

, which has been developed by G. R. Liu and co-

workers with pre- and postprocessors for adaptive analysis of 2D solids. Its
functions and usage are introduced.

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