Analysis of Crack Propagation in an Elastic Bar Using Meshfree Methods

ANALYSIS OF CRACK PROPAGATION IN AN ELASTIC BAR USING MESHFREE

METHODS

Introduction

Mesh free methods is used to establish a system of equations for the whole problem domain

without the use of predefined mesh. Mesh free methods use a set of nodes scattered within the

problem domain as well as the sets of scattered nodes on boundary domain to represent the

domain or the boundaries. These sets of nodes do not form a mesh, which means that there is no

relationship between nodes is required, at least for field variable interpolation.

Aim of the thesis

The aim was to implement the Element Free Galerkin method to compare the

method and its accuracy with the exact and FEM solution for crack propagation in an elastic bar.

The understanding of the basic procedure for meshfree method is developed and MATLAB code

is developed to solve the system of equations for calculating the increase in the length of crack in

the bar under the influence of point load and body force. The increase in the length due to

temperature is included in the MATLAB code. The Element Free Galerkin method was chosen

for this Master Thesis, to investigate its applicability and accuracy.

Features of mesh free methods

The some of the most important features of meshfree methods, often in comparison to

analogous properties of mesh-based methods are :-

(a) Absence of mesh

In meshfree methods the connectivity of the nodes is determined at run-

time

No mesh alignment sensitivity, this is a serious problem in mesh-based

calculations.

h-adaptivity is comparably simple with meshfree methods as only nodes

have to be added, and the connectivity is then computed at run-time

automatically. p-adaptivity is also conceptionally simpler than in mesh

based methods.

No mesh generation at the beginning of the calculations is necessary. This

is still not fully automatic may require human intervention especially in

complex domains.

No re-meshing during the calculations. Especially in problems with large

deformations of domains or moving discontinuities a frequent re-meshing

is needed in mesh based methods, which is difficult.

(b) Continuity of shape functions: The shape functions of meshfree methods may be

easily constructed to have the desired order of continuity.

Meshfree methods readily fulfill the requirement on the continuity arising

from the order of the problem under consideration.

No post-processing is required in order to determine the smooth

derivatives of the unknown functions.

(c) Convergence: For the same order of consistency numerical experiments suggest that

the convergence results of the meshfree methods are often considerably better than the results

obtained by the mesh-based shape functions.

(d) Computational effort: In practice, for a given reasonable accuracy, meshfree methods

are often considerably more time-consuming than their mesh-based counter parts.

Meshfree shape functions are of a more complex nature than the

polynomial-like shape functions of mesh-based methods. Consequently,

the number of integration points for a sufficiently accurate evaluation of

the integrals of the weak form is considerably larger in meshfree methods

than in mesh-based methods.

At each integration point the following steps are necessary to evaluate

the meshfree shape function: Neighbor search, solution of small system

of equations and small matrix-matrix and matrix –vector operations to

determine the derivatives.

The resulting global system of equations has in general a larger

bandwidth for meshfree methods than for comparable mesh-based

methods.

(e) Essential boundary conditions: Most of the meshfree methods lack the Kronecker

delta property. This is in contrast to the mesh-based methods which often posses this property.

Consequentially the imposition of the essential boundary conditions requires certain attention in

meshfree methods and may degrade the convergence of the method.

Overview to Meshfree Methods

2.1 Introduction

The Finite Element Methods used for modeling and analysis of practical problems in

many fields of engineering are well-established. With the development of the computer

technology, the Finite Element Methods have become the most popular tool due to its advantages

in computer implementation. This is a robust and thoroughly developed technique.

However, the finite element methods have some shortcomings. It is not always

advantageous in handling some problems, consider the modeling of large deformation process

(remarkable loss in accuracy happens when the elements in the mesh become extremely skewed

or compressed), the growth of cracks with arbitrary and complex paths, moving discontinuities,

fracture mechanics and phase change problems. These kinds of problems are difficult to be

solved efficiently by finite element methods because of its reliance on the pre-defined mesh.

The solution was found to re-mesh the domain of the problem at each step during the

simulation allowing the mesh-lines to remain distortion free. Mesh generation each time is a far

more time consuming and expensive task than the assembly and the solution.

Therefore, a new method was needed. Meshless methods have been developed to handle

these difficulties. These do not require a mesh to discretize the problem domain. All meshfree

methods share the common feature that the partial differential equation model can be discretized

without any structured and pre-defined meshing. Mesh-free approximations are constructed

using a set of scattered particles or nodes that have no particular pre-defined relationship or

connection among them.

The meshfree methods eliminate the difficulties experienced by finite element methods

by approximating it entirely in terms of nodes which may or may not be uniformly distributed in

the domain of interest. The connectivity between nodes is completely defined by the overlap of

the nodal domains of influence. This methodology uses only the nodes.

2.2 Literature review

Mesh free methods go back to the seventies. The major difference to finite element

methods is that the domain of interest is discretized only with nodes, often called particles. These

particles interact via meshfree shape functions in a continuum framework similar as finite

elements do although particle “connectivity” can change over the course of a simulation. The

most important advantages of mesh free methods compared to finite element methods are their

higher order continuous shape functions that can be exploited for higher smoothness; simpler

incorporation of p and h-adaptivity and certain advantages in crack problems. The most

important drawback of meshfree methods is probably their higher computational cost, regardless

of some instability that certain meshfree methods have.

One of the oldest meshfree methods is the Smooth Particle Hydrodynamics (SPH)

developed by Lucy and Gingold and Monaghan in 1977. SPH was first applied in astrophysics to

model phenomena such as supernova and was later employed in fluid dynamics. In 1993,

Petschek and Libersky extended SPH to solid mechanics. Early SPH formulations suffered from

spurious instabilities and inconsistencies that were a hot topic of investigations. Many corrected

SPH versions were developed that improved either the stability behavior of SPH or its

consistency. Consistency, often referred to as completeness in a Galerkin framework, means the

ability to reproduce exactly a polynomial of certain order.

Based on the idea of Lancaster and Salkauskas and probably motivated by the purpose

to model arbitrary crack propagation without computational expensive re-meshing, the group of

Prof. Ted Belytschko developed the Element Free Galerkin (EFG) method in 1994. The EFG

method is based on an MLS approximation and avoids inconsistencies inherent of some SPH

formulations. In 1995, the group of Prof. W.K. Liu proposed a similar method, the Reproducing

Kernel Particle Method (RKPM). Though the method is very similar to the EFG method, it

originates from wavelets rather than from curve-fitting. The first method that employed an

extrinsic basis was the hp-cloud method of Duarte and Oden. In contrast to the EFG and RKPM

method, the hp-cloud method increases the order of consistency (or completeness) by an

extrinsic basis. In other words, additional unknowns were introduced into the variational

formulation to increase the order of completeness. This idea was later adopted (and modified) in

the XFEM context through the extrinsic basis (or extrinsic enrichment) and used to describe the

crack kinematics rather than to increase the order of completeness in a p-refinement sense. The

group of Prof. Ivo Babuska discovered certain similarities between finite element methods &

meshfree methods and formulated a general framework, the Partition of Unity Finite Element

Method (PUFEM) that is similar to the generalized Finite Element Method (GFEM) of

Strouboulis and colleagues.

Another very popular meshfree method worth mentioning is the Meshless Local Petrov

Galerkin (MLPG) method developed by the group of Prof. S.N. Atluri in 1998. The main

difference of the MLPG method to all other methods mentioned above is that local weak forms

are generated over overlapping sub-domains rather than using global weak forms. The

integration of the weak form is then carried out in these local sub-domains. In this context, Atluri

introduced the notion of “truly” meshfree methods since truly meshfree methods do not need

construction of any background mesh that is needed for integration.

The issue of integration in meshfree methods was a topic of investigations since its early

times. Methods that are based on a global weak form may use different types of integration

schemes: nodal integration, stress-point integration and Gauss quadrature based on a background

mesh that does not necessarily need to be aligned with the particles. Nodal integration is from the

computational point of view, the easiest and cheapest way to build the discrete equations but

similar to finite elements, meshfree methods based on nodal integration suffer from instability

due to rank deficiency. Adding stress points to the nodes can eliminate (or at least alleviate) this

instability. The term stress-point integration comes from the fact that additional nodes were

added to the particles where only stresses are evaluated. All kinematics values are obtained from

the "original" particles. The concept of stress points was actually first introduced in one

dimension in an SPH setting by Dyka. This concept was introduced into higher order dimensions

by Randles and Libersky and the group of Prof. Belytschko. There is a subtle difference between

the stress point integration of Belytschko and Randles and Libersky. While Randles and Libersky

evaluate stresses only at the stress points, Belytschko and colleagues evaluate stresses also at the

nodes. Meanwhile, many different versions of stress point integration were developed. The most

accurate way to obtain the governing equations is Gauss quadrature. In contrast to finite

elements, integration in meshfree methods is not exact. A background mesh has to be constructed

and usually a larger number of quadrature points as in finite elements are used. For example,

while usually 4 quadrature points are used in linear quadrilateral finite elements, Belytschko

recommends use of 16 quadrature points in the EFG method.

Another important issue regarding the stability of mesh free methods is related to the kernel

function, often called window or weighting function. The weighting function is somehow related

to the meshfree shape function. The weight function can be expressed in terms of material

coordinates or spatial coordinates. We then refer to Lagrangian or Eulerian kernels, respectively.

Early meshfree methods such as SPH use an Eulerian kernel. Many meshfree methods that are

based on Eulerian kernels have a so-called tensile instability, meaning the method gets unstable

when tensile stresses occur. In a sequence of papers by Belytschko, it was shown that the tensile

instability is caused by the use of an Eulerian kernel. Meshfree methods based on Lagrangian

kernels do not show this type of instability. Moreover, it was demonstrated that for some given

strain softening constitutive models, methods based on Eulerian kernels were not able to detect

the onset of material instability correctly while methods that use Lagrangian kernels were able to

detect the onset of material instability correctly. This is a striking drawback of Eulerian kernels

when one wishes to model fracture.

However, a general stability analysis is difficult to perform and will of course also depend on

the underlying constitutive model. Note also, that Libersky proposed a method based on Eulerian

kernels and showed stability in the tension region though he did not consider strain softening

materials. For too large deformations, methods based on Lagrangian kernels tend to get unstable

as well since the domain of influence in the current configuration can become extremely

distorted. Some recent methods to model fracture try to combine Lagrangian and Eulerian

kernels though certain aspects still have to be studied, e.g. what happens in the transition area or

how are additional unknowns treated (in case an enrichment is used).

In mesh free methods, we talk about approximation rather than interpolation since the

meshfree shape functions do not satisfy the Kronecker-delta property. This entails certain

difficulties in imposing essential boundary conditions. Probably the simplest way to impose

essential boundary conditions is by boundary collocations. Another opportunity is to use the

penalty method, Lagrange multipliers or Nitsche’s method.

Coupling to finite elements is one more alternative that was extensively pursued in the

literature-in this case; the essential boundary conditions are imposed in the finite element

domain. In the first coupling method by Belytschko, the meshfree nodes have to be located at the

finite element nodes and a blending domain is constructed such that the shape functions are zero

at the finite element boundary. In this first approach, discontinuous strains were obtained at the

meshfree-finite element interface.

Many improvements were made and methods were developed that exploit the advantage of

both meshfree methods and finite elements, e.g. the Moving Particle Finite Element Method

(MPFEM) by Su Hao et al. or the Reproducing Kernel Element Method (RKEM) developed by

the group of Prof. W.K. Liu. Meanwhile, several textbooks on mesh free methods have been

published, WK Liu and S Li, Prof T Belytschko, SN Atluri and some books by Prof. GR Liu.

Over the last years a number of different mesh free methods have been developed. Some of the

most well known methods are:

2.2.1 Smooth particle hydrodynamics (SPH) was introduced by Lucy (1977) and further

developed by Monaghan (1982). It is the simplest method, partly because it is a point collocation

method and also because the shape functions are very simple with no special cases at the

boundary. It has some problems with both stability and accuracy, but many corrections have

been made to improve the method, for example better integration scheme and correction term in

the shape functions.

2.2.2 Element free galerkin (EFG) was developed by Dr, Belytschko et al. (1994). The shape

functions are constructed with MLS approximation, and the test function is chosen as the shape

function. Boundary conditions are enforced by Lagrange multiplier and in general a lot of gauss

integration points are needed to get accurate results.

2.2.3 Reproducing kernel particle method (RKPM) was created by Liu et al. (1995). It is a

particle method, but instead of point collocation it uses a Galerkin formulation. Also the shape

function has a correction term to improve the accuracy at the boundary.

2.2.4 Truly meshfree method or Mesh less local petrov-galerkin (MLPG) was originated by

SN Atluri and Zhu (1998). Instead of a global weak form, it has a local weak form. Therefore no

integration over the domain is necessary, so no background mesh is needed as for example EFG

and RKPM. MLPG can have different shape functions and test functions and is then named as

MLPG1, MLPG2 and so on.

2.2.5 Natural element method (NEM) was developed by Prof N Sukumar et al. (1998). NEM

solves a Galerkin formulation of the problem; here the shape functions are constructed in a

different fashion. The domain is divided in Voroni cells so the NEM fulfills the Kronecker-delta

property and therefore it is straightforward to implement essential boundary conditions.

There are uncountable methods that exist in the literature. During the last two decades,

several meshfree methods for seeking approximate solutions of partial differential equations

have been proposed; we can have a look on the Table 2.1. All of these methods, except for the

MLPG, the SPH and the collocation method are not truly meshfree since the use of shadow

elements are required for evaluating integrals appearing in the governing weak formulations.

TABLE 2.1

Some Meshfree methods developed and their features

S.

no.

Method Reference Approximation Function

1 Diffuse element method Nayroles et al., 1992 MLS approximation,

Galerkin method

2 Element free Galerkin

Method (EFG)

Belytschko et al.

(1994)

MLS approximation,

Galerkin method

3 Meshless local Petrov-

Galerkin Method

(MLPG)

Atlury and Zhu, 1998 MLS approximation,

Petrov-Galerkin method

4 Finite Point Method Onate et al., 1996;

Liszka and

Orkisz,1980;Jensen,19

80

Finite Differential

Representation (taylor

series), MLS

approximation

5 Smooth particle

hydrodynamics

Lucy,1977; Gingold

and Monagan,1977

Integral representation

6 Reproducing kernel

particle method

Liu WK et al.1993 Integral representation

(RKPM)

7 hp-clouds Odan and Abani, 1994;

Armando and

Oden,1995

Partition of unity, MLS

8 Partition of unity FEM Babuska and Melenk, Partition of unity, MLS

1995

9 Point interpolation

method

Liu GR and Gu, 1999,

2000, 2001

Point interpolation

10 Boundary node method Mukherjee and

Mukherjee 1997

MLS

11 Boundary point

interpolation method

Liu GR and Gu, 2000;

Gu and Liu GR, 2001

Point interpolation

2.3 Classification of the meshfree methods

The meshfree methods may be classified in three classes shown by flowchart 2.1. These

are classified by the construction procedure for partition of unity, based upon this the

approximation function is selected. The choice of the test function forms the last step in the

characterization of the meshfree methods:

(a) Construction of the partition of unity

(b) Choice of an approximation function

(c) Choice of the test unction

FEMMesh-based methods

MLS RKPM Meshfree methods

PU

Construction ofPartition of Unity

Intrinsic basis only Additional extrinsic basis

Choice of approximation

Choice of test function

1. SPH2. EFG3. LSMM4. MLPG 15. MLPG 26. MLPG 3 etc.

1. PUM2. XFEM3. hp-clouds

1. FDM2. FVM3. Least square FEM

Flowchart 2.1 Classification of meshfree methods

2.4 Closure

In this chapter, the historical background and the classification regarding the meshfree

methods has been presented in brief. The most of referred papers considered the elastic bar with

uniform cross-section subjected to body force and Timoshenko beam as an example to verify and

compare the results of meshfree methodology.

Fundamentals of Meshfree Methods

3.1 Introduction

One of the most important things engineers and designers do is to model physical

phenomena. Virtually every phenomena in nature, whether mechanical, chemical or biological

can be described with the aid of laws of physics in terms of algebraic, differential or integral

equations relating the various quantities of interest.

This analytical description of physical phenomena and process is called mathematical

model. The use of a numerical method and a computer to evaluate the mathematical model of a

process and estimate its characteristics is called numerical simulation. There are various

numerical simulation methods, such as finite difference method (FDM), finite element method

(FEM) and variational methods etc. for analysis of any physical phenomenon occurring in

nature. The development of various methods of analysis is presented by the flowchart 3.1 in a

chronological order. In this chapter, we will discuss finite element method and compare the

meshfree methodology with help of the flowchart 3.2. The various terms in meshfree methods

are also defined.

3.2 Meshfree method in comparison to FEM

A brief introduction to the differences between finite element method and meshfree

methods is as discussed below.

3.2.1 Finite element method

The finite element method is a powerful numerical method for solving engineering

problems and can analyze complex, structural, mechanical and electrical system. The finite

element method is used to analyze both linear and nonlinear system and to solve problems in

static and dynamic systems. Because of its various applications such as heat transfer,

Methods of Analysis

Experimentation on actual model

Experimentation on Prototype

Mathematical model

Numerical Simulationor

Computer simulation

Classical Variational methods

1. Rayleigh-Ritz method2. Galerkin method

3. Collocation method, etc

Approximation methods

1. Finite difference method (FDM)2. Finite volume method (FVM)3. Finite element method (FEM)

4. Meshfree methods

electrostatic potential, fluid mechanics, vibration analysis and so on, the finite element method

has become very popular.

Flowchart 3.1 Development of methods of analysis

GEOMETRY GENERATION

Element Mesh Generation Nodal Mesh Generation

FEM MeshfreeFEM

Shape Function CreationBased on Element Predefined

Shape Function Creation BasedOn Nodes in a Local Domain

System Equation for Elements System Equation for Nodes

Global Matrix Assembly

Essential Boundary Condition

Support Specification

Solution for displacements

Computation of Strains andStresses from Displacements

Results Assessment

Flowchart 3.2 Flowchart for FEM and meshfree method procedures

In many physical problems, it is very difficult to find an analytical solution for complex

geometry with complex boundary conditions. In the finite element method, a complex region

defining a continuum is discretized into simple geometric shapes called finite elements that are

connected at specified node points as shown in Figure 3-1. The shapes of the elements are

intentionally made as simple as possible, such as triangles and quadrilaterals in two-dimensional

domains, and tetrahedral, pentahedral and hexahedra in three dimensions. The entire mosaic-like

pattern of elements is called a mesh.

Figure 3.1 Boundary representations in FEM and meshfree methods

The material properties and the governing relationships are considered over these

elements and expressed in terms of unknown field values at element corners. These equations

describe the physical problem that is to be analyzed and generally can be expressed using the

weak form based on variational principle.

3.2.2 Meshfree method

Creation of a mesh for the problem domain is a prerequisite in FEM packages and the

analysts spend the majority of their time in creating the mesh. The stresses obtained using FEM

packages are discontinuous and less accurate if the mesh density and quality is poor. When

handling large deformation, considerable accuracy is lost because of the element distortion. The

root of these difficulties was the element mesh and the idea of eliminating the mesh has evolved

to counter this problem.

The meshfree method is used to establish a system of algebraic equations for the whole

problem domain without using a predefined mesh. Meshfree methods use a set of nodes scattered

within the problem domain as well as node scattered on the boundaries of the domain to

represent 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. In Mesh Free method (Liu 2003) [2], adaptive schemes can be

easily developed, as there is no mesh, no connectivity is involved. This provides flexibility in

adding or deleting points/ nodes whenever and wherever needed. The analyst can save the time

spend on conventional mesh generation because there is no mesh, and the nodes can be created

by a computer in a fully automated manner. This can translate into major cost and time savings

in modeling and simulation projects.

The fundamental difference between FEM and Meshfree method is the construction of

the shape functions. In FEM, the shape functions are constructed using elements. These shape

functions are therefore predetermined for different types of elements before the finite element

analysis starts. In Meshfree method, the shape functions constructed are usually only for a

particular point of interest, and the shape function changes as the location of the point of interest

changes. The construction of the element free shape function is performed during the analysis.

Modeling the geometry is also a difference between these two methods. In FEM, curved parts of

the geometry and its boundary can be modeled using curves and curved surfaces using high-

order elements. If linear elements are used, these curves and surfaces are straight lines or flat

surfaces. Figure 3-1 shows an example of smooth boundary approximated in the finite element

model by the straight line edges of triangular elements. The accuracy of representation of the

curved parts is controlled by the number of elements and the order of the approximation.

In Mesh Free methods, the boundary is represented by nodes as shown in Figure 3-1. At

any point between two nodes on the boundary, one can interpolate using Meshfree shape

functions. Because the Meshfree shape functions are created using nodes in a moving local

domain, the curved boundary can be approximated very accurately even if linear polynomial

basis functions are used.

3.2.2.1 Meshfree terminology

The common terms used in the meshfree methodology are concisely defined and

presented for easy assimilation.

3.2.2.2 Support domain

The support domain is defined as the domain/area or field that is affected or influenced

by any point of interest xQ, in the problem domain Ω, this point of interest may or may not be a

node. The influence domain is defined as the domain that is affected or influenced by any node

of interest xi in the domain. The influence domain is defined for each node in the problem

domain, and it can be different from node to node to represent the area of influence of any node.

Figure 3.2 Circular and rectangular domains of Influence

The concept of influence domain is clarified by the figure 3.2; node 1 has an influence

domain of radius r1, and node 2 has an influence of radius r2. These domains of influence may

have different shapes and dimensions. Most commonly used shapes are circular or rectangular

ones. The figure 3.3 depicts the circular domain of influence of a node represented by Ω I,

whereas the problem domain is shown by Ω.

Figure 3.3 Circular domains of influence

3.2.2.3 Node connectivity

In case of finite element methods the nodes are connected by the elements whereas in

case of meshfree methods the node connectivity is established by the overlapping domains of

influence. Figure 3.4 shows

the overlapping domains of influence and local node numbering at point of interest xQ.

Figure 3.4 Overlapping domains of influence and node numbering at point xQ

3.2.2.4 Dimension of support domain

The dimensions of domain of influence affects the accuracy of the interpolation at the

point of interest, therefore the selection of suitable dimension of support domain is very

important.[2,8] To define the support domain for a point xQ, the dimension of the support domain

ds is determined by

d s=α s dc 3.1

Where αs is the dimensionless size of the support domain. And dc is the characteristic

length that relates to the nodal spacing near the point xQ. If the nodes are uniformly distributed dc

is simply the distance between the two neighboring nodes.

3.2.2.5 Weight function

The weight function plays an important role in the performance of approximate solution.

These are monotonically decreasing functions with respect to distance from x to xi which implies

that the magnitude decreases as the distance from x to xi increases. It is obvious that weight

function should be continuous and positive in its domain of support. Different types of weight

functions are used in EFG method. In this work quartic spline weight function has been used. [2,

8, 9] The function is expressed as:

W ( x−x i )=1−6 r 2+8 r3−3 r4 ,0

r≤1r>1 3.2

Where r is given by r=

||x−x i||d i , di, is the size of the domain of support of node i.

The plot of quartic weight function and its derivative is shown in figure 3.5. The value of

radius of support domain of influence is taken to be equal to 2.0, the domain is given by Ω= (0,

5). Only one curve has been plotted each for weight function and the derivative of weight

function for the central node.

Figure 3.5 Quartic spline weight function and its derivative

3.2.2.6 Effect of dimension of support domain

The smoothening of the weighting function, because of the selection of the support

domain of influence for the node i (central node in our plots) can be visualized by the plot in the

figure 3.6. The figure plots the Quartic weight function and its derivatives for different values of

support domain i.e. di= 1.5, 2.0 and 2.4. With the increase of the dimension of the support

domain the curves are smoother which produces the smooth shape functions.

Figure 3.6 Effect of support domain on weight function and its derivative

3.3 Closure

The primary difference between finite element method and meshfree method were

discussed in brief in this chapter. The terminology concerned to meshfree methods is defined.

The effect of the support domain on the weighting function is presented by the plot with different

values.

Element Free Galerkin Method

4.1 Introduction

The Element Free Galerkin method is one of the most used meshfree methods

based on the diffuse elements method proposed by Nayroles et al. (1992) and

developed further by Belytscho et al (1994)[7]. The major features of it are as follows:

Moving least square approximation (MLS) is used for the construction of the

shape function.

Galerkin procedure is employed to derive the discrete equations from the

weak form.

Integration is performed with background cells or mesh for calculation of

system matrices.

4.2 Basic procedure of meshfree analysis

The problem of elastic bar with uniform cross-sectional area will be used to

illustrate the basic steps involved in the meshfree analysis of any solid mechanics

problem. The elastic bar with loading conditions is represented by figure 4.1. The

displacement of any point ‘x’ on the bar is given by u(x). The governing equilibrium

equation in terms of displacement u(x) as given by Reddy [4] is:

ddx (EA

dudx )+b( x )=0

, 0<x<1 4.1

Subject to boundary conditions

u(0) =0

and

Representation,Number of nodes

Weight function

Approximation &Shape Function

Weak Form

Material property,Loading

Nodal Discrete Equations

Global Matrix Assembly

Solve for Nodal Displacement Parameters

Basis function

[EAdudx ]

x=1=P

Flowchart 4.1 Meshfree solution procedure

L

x

P

dx

Body force b(x)

Point Load

Figure 4.1 One-dimensional bar subjected to point-load and body force

The equation 4.1 can be written as, using the equilibrium of forces acting

on a small element dx of the bar, Reddy [4]:

dσdx

+b( x )=0 4.2

where

Stress = σ = εE 4.3

Strain = ε=du

dx ¿¿ 4.4

Considering the same problem for elastic bar of length L=1, with uniform cross-

sectional Area A, subjected to body force b due to density ρ, Young’s modulus E and a

point load P with consistent units. The exact analytical solution of the governing

equation 4.1 subject to boundary conditions is given by:

u( x )= 1EA (Px+b( x− x2

2 )) 4.5

4.2.1 Domain representation

In meshfree method we model the problem and represent it by the set of nodes

scattered in the problem domain and its boundary. Specify the boundary and loading

conditions. The density and distribution of the nodes depends upon the accuracy

requirement and the resource availability. These nodes are often called field nodes as

these will carry the value of the field variables in the meshfree formulation. Figure 4.2

represents the One-dimensional meshfree representation along with the boundary

conditions of elastic bar using 11 nodes.

.

Figure 4.2 Meshfree re-presentation of one-dimensional elastic bar

4.2.2 Displacement interpolation

The field variable, say a component of displacement u at any point at x within the

problem domain is interpolated using the displacements at its nodes within the support

domain of the point at xi, and is given by:

4.6

Where n is the number of nodes included in the support domain of the point x

and is the shape function of the ith node determined using the nodes that are

included in the small support domain of x.

4.2.3 Formation of system equations

The discrete equations of a meshfree method are formulated using the shape

function and weak form of governing differential equations. These equations are written

in nodal matrix form and are assembled into the global system matrices for the entire

problem domain. The derivation of meshfree discrete equations or the algebraic

equations of meshfree approximation involves following:

a) Approximate solution over the domain

b) Weak form

c) Derivation of discrete equations

4.2.3.1 Approximate solution over the domain

The approximate solutions are sought over the problem domain

Ω = (0, L) at once in meshfree methods the approximate solution is given by the form of

approximation u(x) [2, 7]

4.7

Where p(x) is the complete polynomial of order m and a(x) is given by:

4.8

4.9

The order of polynomial is defined as the order of basis function, in one

dimension the complete linear basis function is given as

pT ( x )=[1 x ] 4.10

On putting equation 4.9 and 4.10 in equation 4.7 for approximate solution we get

u( x )=a0 ( x )+a1 ( x )x 4.11

The unknown coefficient ai of a(x) varies with x. These approximations are

known as Moving Least Square (MLS) approximations in curve and surface fitting and

were first described by Lancaster and Salkauskas [2, 9]. The MLS method is now

widely used as an alternative for constructing meshfree shape functions for

approximation. It was used in Element Free Galerkin method by Belytschko et al.

(1994). MLS approximation has two major features that make it popular: 1) the

approximated field function is continuous and smooth in the entire problem domain and

2) it is capable of producing an approximation with the desired order of consistency.

Figure 4.3 Data fitting using least square method

The unknown coefficients ai of a(x) at any point are determined by minimizing the

difference between the local approximation at that point and nodal parameters ui.

4.12

Where

is the quartic spline weight function, and solving equation 4.12 to minimize the

functional J

and 4.13

We get 4.14

This is further rearranged and can be written compactly as:

4.15

4.16

We get the reduced form of Equation-4.14 as

A( x )a ( x )=B( x )u 4.17

On manipulation for evaluating a(x)

a (x )=A−1 ( x )B( x )u 4.18

Recalling the Equation-4.7 of approximate solution and substituting the values

we get this equation for approximation:

4.19

Hence, the approximate solution can be written concisely as

. 4.20

Where the moving least square shape function or is defined by

4.21

Thus, we get the equation of shape function to be used for substituting in the

weak form to establish the discrete equations. The obtained plots of shape function for a

set of 13 nodes in the problem domain Ω = (0, 1) are shown in the figure 4.4.

Figure 4.4 MLS shape functions at the gauss points

4.2.3.2 Weak form

The partial differential equation that governs the solid mechanics problem can be

represented as equilibrium equation, Liu GR [2]

LT σ+b=0 4.22

where b is the vector of external body forces in the x, y, and z directions

and σ is the stress tensor.

b=bx

b y

bz

4.23

and L is a differential operator matrix:

L=[d /dx

00

d /dy00

00

0d /dz

d /dzd / dy

d /dzd /dy

0d /dx

d /dx0

] 4.24

The constrained Galerkin weak form with the implementation of Lagrange

multipliers λ is used, which is developed & given by Liu GR [2] and can be written as:

∫0

L

δ( Lu )T (cLu )dx−∫0

L

δuT b dx−∫Γt

δuT P dΓ−∫Γu

δλT (u−u )dΓ−∫Γu

δuT λ dΓ=0

……….4.25

where

Stress tensor

L= differential operator

u= displacement

b= body force vector

P= point load on the natural boundary

∫0

L

δ( Lu )T (cL u )d x−∫0

L

δu T b d x−∫Γt

δu T P dΓ −∫Γu

δ λT (u−u )dΓ −∫Γu

δu T λ dΓ =0 = prescribed displacement on essential boundary

The imposition of Lagrange multiplier λ is necessary for satisfying the boundary

conditions and implementation of solution for meshfree methods. The Lagrange

multiplier λ can be interpreted as the reaction forces needed to fulfill the displacement

conditions at the essential boundary.

4.2.3.3 Derivation of discrete equations

The discrete nodal equations are obtained by using the weak form of equilibrium

equation and imposing the boundary conditions. Weak form, of problem under

consideration is given by equation-4.25, is recalled and approximate solution equation

4.20 is substituted for u, which yields the following system of linear algebraic equations

in matrix form:

[ K GGT 0 ][uλ ]=[ fq ]

4.26

Where

K ij=∫0

L

EAdφ

iT

dx

dφ j

dxdx

Gik=−∫Γu

L

φk dΓui

f i=∫Γ t

φi P dΓ t +∫0

L

φ ibdx

q i=−∫Γu

u dΓ u

These discrete nodal equations are assembled into global matrix, to solve for the

nodal displacement parameter values.

4.3 Imposition of boundary conditions

The Element Free Galerkin (EFG) shape functions obtained using moving least

square approximation do not satisfy the Kronecker delta property:

4.27

Therefore, one can conclude that they are not truly interpolants

that is why these are called approximation shape functions. This means that the found

values of are not the nodal values. The approximation at th node is dependent

upon the nodal parameters as well as the nodal parameters through ,

corresponding to all other nodes within the domain of influence of node . This makes

the imposition of essential boundary conditions difficult. In this thesis work, Lagrange

multiplier technique, Belytscho et al. [2, 7] is used to enforce the essential boundary

conditions.

4.4 Solutions of discrete equations for nodal parameters

The major difficulty in the solution of meshfree discrete equations is the

numerical integration of the weak form. This is due to the non-polynomial form of

meshfree shape functions including Moving least square approximation (MLS).

Therefore, exact integration is the most difficult to perform for meshfree methods.

Many techniques have been developed, in this work Gauss-Legendre

Quadrature technique has been used to perform the integration over the

Domain Ω = (0, 1), GR Liu [2], Reddy [4]. The Gauss Legendre quadrature formula is

given by:

∫a

b

F ( x )dx=∫ F (ξ )dξ≈∑I=1

r

F (ξ I )w I 4.28

Where wI are the weight factors and ξI are the base points.

Gauss-Legendre Quadrature technique utilizes a background mesh; the method

is ideally applicable to small and moderate deformations. The considered problem has

been solved by developing the MATLAB language code, using the uniformly distributed

set of 13 nodes. MATLAB is the trade mark of math-works, software developers. The

obtained nodal displacement parameters have been compared to exact analytical

solution. The plot obtained considering the all the variable material properties to be unity

is shown in figure 4.5. The effect of the dimensionless size of the support domain is

studied by plotting the curves with different values, figure 4.6 & 4.7. A portion of figure

4.6 has been enlarged to bring forward the clarity between the curves. It is decided to

have further programming with this value equal to 2.0, which gives better approximation.

The material properties have been considered here only as the representative values

for the development of the MATLAB program code.

Figure 4.5 Displacement vs length for bar of uniform area of cross-section

Figure 4.6 Effect of dimensionless size of support domain αS

Figure 4.7 Effect of dimensionless size of support domain αS (enlarged)

4.5 Closure

This chapter described the basic formulation of Element Free Galerkin method

and approximation of field variable (displacement). Application of Lagrange multipliers is

briefly described for imposing the essential boundary conditions. For evaluating the

global matrices Gauss-Legendre Quadrature technique with background cell integration

is used. The results for exact and meshfree method are plotted. In the next chapter the

application of Element Free Galerkin Method to elastic tapering bar is presented.