7

International Journal of Engineering and Management Research, Vol.-2, Issue-6, December 2012

ISSN No.: 2250-0758

Pages: 7-20

www.ijemr.net

Analysis of One Dimensional Beam Problem Using Element Free Galerkin

Method

Vijender1, Sunil Kumar Baghla

2

1M.Tech Student,

2Assistant Professor

Department of Mechanical Engineering, Yadvindra College of Engineering Punjabi University Guru Kashi

Campus Talwandi Sabo, Bathinda, Punjab, INDIA

ABSTRACT

For analysis of mechanical parts in designing it needs to

calculate mechanical properties of element or a part, for

this we start from the stress/strain calculation. Although

stress/strain problems can be solved analytically, but for

high accuracy, time to solve the problem and for solving

the typical problems (e.g. irregular shaped part, beams

with various loading conditions, moving boundary

problems etc.) process needs a method so that above

problems can be solved easily or minimized.FEM is the

primary method for simulation of parts. But there is a

problem associated with FEM that it is complicated to

solve problems of discontinuous stress and strain or

distortion or deformed body. Because re meshing

problem arises in the procedure. It consumes most of

the analysis time. So, there is a procedure to overcome

from this problem that is Mesh free Methods. This saves

the re meshing time. Because, this method is based on

nodes calculation rather than element calculation In

this work different mesh free methods are discussed and

the mainly EFGM (ELEMENT FREE GALERIKIN

METHOD) is used to solve the problem. A problem of

varying cross section is solved by this method. The

results are compared with the FEM solution as well as

with the analytical solution for the validation of results.

The main work is to validate the results after that the

effect of the weight function, domain of influence etc.

are discussed as well as compared their results are

compared.

Keywords —Element Free Galerkin Method, FEM,

Mesh free methods, one dimensional stress, varying

cross sectional beam.

I. INTRODUCTION

Mesh free methods as the name indicates there

are no mesh generation in this method as in case of

the FE method. The calculation is based on the nodal

parameters. These methods have some advantages

over FE method.

Fig 1.1 Discretization of FEM and Mesh free

methods

The Fig1.1 shows the discretization of the FEM

and Mesh free methods. In left part of figure there are

elements and these are connected with respective

nodes and create a mesh where as in the right side the

part is discretized by the nodes only no element is

present there. It clearly shows the basic difference

between FE method and mesh free method. Important

features of mesh free methods (MMs), comparing

them with the properties of mesh-based methods:

In MMs the connectivity of the nodes is determined

at run-time. No mesh alignment sensitivity. This is a

serious problem in mesh based calculations e.g. of

cracks. Conceptionally simpler than in mesh-based

methods. No mesh generation at the beginning of the

calculation is necessary. No remeshing during the

calculation. Especially in problems with large

deformations of the domain or moving discontinuities

a frequent remeshing is needed in mesh-based

8

methods, however, a conforming mesh with

sufficient quality may be impossible to maintain.

Even if it is possible, the remeshing process degrades

the accuracy considerably due to the perpetual

projection between the meshes, and the post-

processing in terms of visualization and time-

histories of selected points requires a large effort. The

shape functions of MMs may easily be constructed to

have any desired order of continuity. MMs readily

fulfill the requirement on the continuity arising from

the order of the problem under consideration. In

contrast, in mesh-based methods the construction of

even C1 continuous shape functions needed e.g. for

the solution of forth order boundary value problems

may pose a serious problem. No post-processing is

required in order to determine smooth derivatives of

the unknown functions, e.g. smooth strains. Special

cases where the continuity of the mesh free shape

functions and derivatives is not desirable, e.g. in

cases where physically justified discontinuities like

cracks, different material properties etc. exist, can be

handled with certain techniques. For the same order

of consistency numerical experiments suggest that

the convergence results of the MMs are often

considerably better than the results obtained by mesh-

based shape functions. In practice, for a given

reasonable accuracy, MMs are often considerably

more time-consuming than their mesh-based

counterparts. Mesh free shape functions are of a more

complex nature than the polynomial-like shape

functions of mesh-based methods. Number of

integration points for a sufficiently accurate

evaluation of the integrals of the weak form is

considerably larger in MMs than in mesh-based

methods. At each integration point the following

steps are often necessary to evaluate mesh free shape

functions: Neighbour search, solution of small

systems of equations and small matrix-matrix and

matrix vector operations in order to determine the

derivatives. Most MMs lack Kronecker delta

property, i.e. the mesh free shape functions Φi do not

fulfill Φi(xj) = δij . This is in contrast to mesh-based

methods which often have this property. The

imposition of essential boundary conditions requires

certain attention in MMs and may degrade the

convergence of the method.

II. LITERATURE REVIEW

III. Belytschko et. al [1] used the moving least-

squares interpolants to construct the trial and test

functions for the variational principle (weak form)

and weight functions. In contradistinction to DEM,

they introduced certain key differences in the

implementation to improve the accuracy. Also in

their paper, they illustrated these modifications with

the examples where no volumetric locking occurs and

the rate of convergence highly exceeded that of finite

elements. Chen et. al [2] Domain integration by

Gauss quadrature in the EFGM adds considerable

complexity to solution procedures. A strain

smoothing stabilization for nodal integration is

proposed to eliminate spatial instability in nodal

integration. For convergence, an integration

constraint (IC) is introduced as a necessary condition

for a linear exactness in EFGM. The gradient matrix

of strain smoothing is shown to satisfy IC using a

divergence theorem. No numerical control parameter

is involved in the proposed strain smoothing

stabilization. The numerical results show that the

accuracy and convergent rates in the mesh-free

method with a direct nodal integration are improved

considerably by the proposed stabilized conforming

nodal integration method. It is also demonstrated that

the Gauss integration method fails to meet IC in

mesh-free discretization. For this reason the proposed

method provides even better accuracy than Gauss

integration for EFGM as presented in several

numerical examples. In a paper a new body

integration technique is presented and applied to the

evaluation of the stiffness matrix and the body load

vector of elastostatic problems obtained by Duflot

and Dang [3] in mesh less method. It does not work

on a partition of the integration domain into small

cells, but rather on a partition of unity by a set of

moving least squares shape functions each defined on

a small patch that belongs to a set of overlapping

patches covering the domain and so leads to a truly

mesh less method and gives results that this method

is especially useful when the nodes are irregularly

scattered. Soparat et. al [4] has extended the EFGM

to include nonlinear behavior of cracks in 2D

concrete. A cohesive curved crack is modeled by

using several straight-line interface elements

connected to form the crack. The constitutive law of

the crack is considered through the use of these

interface elements. The stiffness equation of the

domain is constructed by directly including, in the

weak form of the system equation, a term related to

the energy dissipation along the interface elements.

Using the interface elements in conjunction with the

EFG method allows crack propagation to be traced

easily and without any constraint on its direction. The

proposed method is found to be an efficient method

for simulating propagation of cracks in concrete. Gu

et. al [5] discussed that EFGM is computationally

expensive for many problems. A coupled

EFG/Boundary Element (BE) method is proposed in

this paper to improve the solution efficiency. A

procedure is developed for the coupled EFG/BE

method so that the continuity and compatibility are

9

preserved on the interface of the two domains where

the EFG and BE methods are applied. The present

coupled EFG/BE method has been coded in

FORTRAN. The validity and efficiency of the

EFG/BE method are demonstrated. It is found that

the present method can take the full advantages of

both EFG and BE methods. It is very easy to

implement, and very flexible for computing

displacements and stresses of desired accuracy in

solids with or without infinite domains. Rabczuk et.

al [6] presented an in-depth presentation and survey

of mesh-free particle methods. Several particle

approximation are reviewed; the SPH method,

corrected gradient methods and the Moving least

squares (MLS) approximation. The discrete equations

are derived from a collocation scheme or as Galerkin

method. Special attention is paid to the treatment of

essential boundary conditions. A brief review over

radial basis functions is given because they play a

significant role in mesh-free methods. Finally,

different approaches for modeling discontinuities in

mesh-free methods are described. Zhang et. al [7]

used moving least-square technique to construct

shape function in the Element Free Galerkin Method

at present, but sometimes the algebra equations

system obtained from the moving least-square

approximation is ill-conditioned and the shape

function needs large quantity of inverse operation.

The weighted orthogonal functions are used as basis

ones, the application in the calculation of plate

bending shows that the improved moving least-

square approximation is effective and efficient.

IV. METHODOLOGY In the methodology two case studies are being

discussed with their analytical, FEM and EFGM

solution methodology. The case studies are stated

following:

3.1 Case Study 1

In this problem a regular cross section beam is

selected. The beam is loaded with axial concentrated

load. The problem is given as:

A steel rod of length 1m is subjected to an axial

load of 5KN and area of cross section is 250mm. take

E=2*10^5 N/mm2. [9] Now we will find the solution

by analytical, FEM and EFGM.

L=1M

P=5kN

Fig.4.1 steel bar

Where P is the concentrated load and L is the length

of the steel bar.

Calculate:

1. Displacement

2. Stress at different points of the bar.

Analytical Solution

The bar is discritized in the four equal parts,

to calculate the required result at different nodal

points. Because there are 5 nodal points so there will

be five results of displacements as well as for the

stress.

The calculation for the analytical results starts from

the Hooke’s law

(3.1)

Where is stress,

(3.2)

E is the modulas of elasticity of the steel bar;

is strain in the material.

(3.3)

u is Displacement,

(3.4)

Now from the above given equations

analytical results can be found easily at different

nodes. Equation 3.2 gives the stress at five nodes of

the steel bar and equation3.4 gives the displacement

at the consecutive nodal points.

The calculated results of the analytical

solution are in the next chapter results and discussion.

These results are the landmark for further work,

because these results are compared with the FEM

results later on with the EFGM results.

FEM results

The bar is divided in four equal elements

(250mm) and the area of each element is regular

throughout the bar. After that a separate stiffness

matrix is generated for each element using the

formula

(3.5)

So K1; K2; K3; K4 are generated using the

given values of area (A), modulas of elasticity (E)

and length of element (L). These values are getting

together in a stiffness matrix K.

250mm

10

And the stiffness equation is generated

KU=P (3.6)

Where U is displacement vector given as;

U= (3.7)

And P is force vector given as;

P= (3.8)

U=K-1

P (3.9)

(3.9)

Using the equation3.5 the stiffness matrix for

each element is generated. Now generated stiffness

matrix is assembled into one stiffness matrix K. The

equation3.8 is load vector in this the given

concentrated load is being put here P5 is 5kN because

load is at extreme point and other nodes are not

loaded. Now the result is calculated using the

equation3.9. This will give the displacement of each

element.

(3.10)

(3.11)

Equation3.10 gives stress at first points.

Like this equation3.11 gives the stress at second point

and so on the stress is calculated up to . The

results calculated by above equations are compared

with analytical results. Now next step is to calculate

the results with the help of EFGM.

3.2 Case Study 2

The second problem is selected a varying

cross sectional beam whose area is decreasing from

fixed end to free end. The load is again concentrated

at the free end.

The equation for stress analysis is shown in

equation 1 [10]. It is simple stress problem and given

as. Consider an linear elastic bar [10] of length L=1m

with a varying cross section A(x), where Ar(x),E(x)

Fig.3.2 Elastic bar of varying cross section

; [19] (3.12)

A0= 0.01 m2; (3.13)

Young's modulus ;

The bar is rigidly supported at the left end and at the

right a concentrated force P is applied;

P= 10000N

Calculate:

1. Displacement

2. Stress at different points of the bar.

Analytical results

They can be calculated using the

equation3.1, 3.2 and 3.3. For the varying cross

section the equations for the displacement and stress

becomes as under

Displacement,

[10] (3.14)

And stress as,

[10] (3.15)

The beam is discritized in three equal elements. The

displacement and stress at consecutive nodes is

calculated analytically using the equations 3.14 and

3.15

FEM results

The bar is divided in three equal elements of

.33333 m and the area of each element is calculated

at the center of the element so that three parts can be

made of equal cross section.

After that a separate stiffness matrix is

generated for each element using the formula

(3.16)

So K1; K2; K3 are generated using the given

values of area (A(x)), modulas of elasticity (E) and

length of element (L). These values are getting

together in a stiffness matrix K.

And the stiffness equation is generated

KU=P (3.17)

Where U is displacement vector given as;

10000

N

x

X

=

0

X

=

L

11

U= (3.18)

And P is force vector given as;

P= (3.19)

(3.19)

U=K-1

P (3.20) (3.20)

The equation 3.20 gives the displacement of

the each element. Now stress is calculated as below

B= [1 -1] (3.21)

(3.22)

The equation 3.22 gives the stress at three

nodes putting the corresponding values. The results

are compared with the analytical solution.

3.3 EFGM solution

Consider a bar whose cross sectional area is

Ar(x) and its length is L. Now assume that any force

is applied in the direction of extension of body or in

the longitudinal axis (x-axis). That force is point load

P [N] and distributed load b(x) [N/m]. The body is in

only tensile loading no shear force or bending loads

will be considered. Only stretching in its direction

resulting increase in its length. E(x) is the Young’s

modulus of elasticity of the material. Now there will

be displacement in the bar due to applied loads that is

u(x) that will be in one direction so we can say it is

the One Dimensional elastic problem. The problem

will involve displacement u(x), stress σ(x) and strain

ε(x). This can be calculated by applying boundary

conditions. Now refer the fig3.2 and Consider a small

elementary strip of length ∆x

Fig.3.4 small element of bar

By equilibrium law force can be written as

P(x+∆x)-P(x)+b(x)=0 (3.23)

By Taylor’sformula,

(3.24)

We know that,

P(x) =σ(x)Ar(x) (3.25)

And by hook’s law,

σ(x)=E(x)ε(x) (3.26)

so equation (1) can be written as,

(3.27)

And further more equation becomes,

(3.28)

Strain in the strip,

(3.29)

So,The strong form is given as,

(3.30)

The equation 3.30 gives the strong form for

the elastic problems of cantilever when it is loaded

with point load axially at free end. So this equation is

for both case studies.

3.3.1 Moving Least Squares

It is the starting point of the Element Free

Galerkin Method. For the approximation of u(x)

which is a domain Ω by

u(x)= =pT(x)a(x) , (3.31)

j=1,2,…,m

where p1(x)=1 and pj(x)aremonomial basis

in the space coordinates xT=[x, y] and is known and

complete basis, A linear and quadratic basis in one

dimension can be given by

pT(x)=[1,x], m=2 (3.32)

pT(x)=[1,x,x

2], m=3 (3.33)

and linear and quadratic for two dimensions can be

given as

pT(x)=[1,x,y], m=3 (3.34)

pT(x)=[1,x,y,x

2,xy,y

2], m=6 (3.35)

The aj(x) is unknown coefficients, which is

solved by moving least squares procedure using

nodal points, a(x) is obtained at x by minimizing a

weighted discrete Least-Squares norm J as

(3.36)

P(x+∆x) b(x)∆x

P(x)

12

here n is the number of points in the neighborhood of

x for which the weight function w(x-xi)≠0 and uiis the

nodal value of u at x= xi. and this neighborhood of x

is called domain of influence of x (or domain of

influence of node i). The relation between a(x) and u

can be written in the linear equation which is:

A(x)a(x)=B(x)u (3.37)

Or equation can be molded as

a(x)=A-1

(x)B(x)u (3.38)

here m is the number of EFG nodes and

these nodes include the domain of influence x. now

A(x) and B(x) are in matrix form and defined as

A(x)= ∑wti(x)pT(xi)p(xi) (3.39)

Where i=1 to n andwti=w(x- xi),

B(x)=[ w(x- x1) p(x1), ….., w(x- xn) p(xn)]

or

B(x)=[ wt1 p(x1), wt2 p(x2),……….., wtn p(xn)]

(3.40)

So the equation (1) becomes

u(x)= ∑ Φi(x)ui (3.41)

(where i=1 to n)

Now MLS shape function Φi(x) can be defined as

Φi(x)= ∑ pj(x) (A-1

(x)B(x))ji (3.42)

The continuity of the shape function is Φi(x) is

defined by the continuity of basis function pj;

depends on the smoothness of the matrices A-1

(x) and

B(x) and choice of the weight function. The partial

derivativeofΦi(x) can be calculated as

(3.43)

3.3.2 Weight Functions

The weight function (wt(x)=w(x- xi )) plays

an important role in the EFGM’s performance. It

should be selected so that a unique solution a(x) can

be generated. Its value should decrease in magnitude

as the distance increases from x to xi. Now dmxis the

support size for the weight function, dis=sign(x-xi),

and d=dis/dmx. The most commonly used weight

functions are given as:

The cubic and quadratic spline weight

function is more favorable because they provide

continuity and less computationally less demanding.

The singular weight function allows the direct

imposition of essential boundary conditions. So in the

case of singular weight function there is no need of

Lagrange multipliers.

Singular:

(3.44)

Cubic spline

(3.45)

Quadratic spline:

(3.46)

3.3.3 EFGM weak formulation

And the weak formulation is

(3.47)

Integrating by parts to above equation

Where the natural boundary is condition

and is the domain boundary

(3.48)

13

(3.49)

Rearranging equation,we get

(3.50)

The above equation can be written as in the following

form,

[K]U=f (3.51)

Where the matrices [K] and f given by

(3.52)

(3.53)

(3.54)

(3.55)

(3.56)

IV. RESULTS AND DISCUSSION 4.1 Case Study 1

Using the methodology discussed the results

are calculated for the case study 1. The calculated

results for the case study 1 are;

Analytical results

Using the formula for displacement and stress given

in the methodology chapter the analytical results are

obtained. Which is shown in Table4.1 the Table4.1

shows the results of case study 1. The column 2nd

shows the displacement in beam at different nodes

displacement is in mm and similarly the 3rd

column

shows the stress in the beam at consecutive nodes the

stress is in N/mm2. The result seems continuous.

Table 4.1 Analytical results

Node U Σ

1 0 0

2 0.025 20

3 0.05 20

4 0.075 20

5 0.1 20

FEM Results

For the FEM results the equations given in

the methodology section is used. The displacement at

different nodes and stress at corresponding nodes are

calculated by those equations. Table 4.2 shows the

results of FEM

Table 4.2 FEM results

Node U σ (N/mm2)

1 0 0

2 0.025 20

3 0.05 20

4 0.075 20

5 0.1 20

EFGM Results

For this a MATLAB programme is

generated using the EFGM weak formulation results.

Like the FEM the stiffness matrix is generated but in

this weight function, shape function is added in the

stiffness matrix. After that the Lagrange multiplier is

used in the boundary conditions. The equation

becomes in the form of KU=P. The results are of

nodes. In the results the displacement is calculated

and the stress is calculated using the displacement

results. The table 4.3 shoes the EFGM displacement

and stress at different nodes.

14

Table 4.3 EFGM results

Node u σ

1 0 0

2 0.0251 20.0742

3 0.05 19.9258

4 0.0749 19.9258

5 0.1 20.0742

Validation for case study 1

Now analytical, FEM & EFGM results have

been calculated according to the methodology. These

results are compared for the validation of the EFGM

method whether the results are liked to analytical and

FEM results or not.

Fig 4.1 Comparison b/w Analytical, FEM and EFGM

displacement

The Fig4.2 shows the comparison between

analytical, FEM, EFGM results. These results are of

only displacement results of three methods at

different nodes of the beam. By the comparison a

person can see that the EFGM results obtained are

continuous and regular as in case of analytical and

FEM.

After comparing the displacement results now

the stress results of EFGM are compared with the

analytical, FEM results the following figure shows

the comparison between three methods.

Fig 4.2 Comparison b/w Analytical, FEM and EFGM

stress

These results shows that the result obtained

in the EFGM method are continuous in case of

displacement as well as in case of stress also. The

results are accurate and equal to other methods. So

this method satisfies the one dimensional elastic

problem.

4.2 Case Study 2

Using the methodology discussed the results are

calculated for the case study 1. The calculated results

for the case study 1 are;

Analytical results

Using the formula for displacement and

stress given in the methodology chapter the analytical

results are obtained. Which is shown in Table4.1 the

Table4.1 shows the results of case study 1. The

column 2nd

shows the displacement in beam at

different nodes displacement is in mm and similarly

the 3rd

column shows the stress in the beam at

consecutive nodes the stress is in N/mm2. The result

seems continuous.

Table 4.4 Analytical results

NODE Area u*(1*10^-5) σ*(10^7)

1 .0095 0 0

2 .0084 0.2 0.119

15

3 .0056 0.5 0.1785

4 .0034 1 0.2777

FEM Results

For the FEM results the equations given in

the methodology section is used. The displacement at

different nodes and stress at corresponding nodes are

calculated by those equations.

Table 4.5 FEM resul

EFGM Results

For this a MATLAB programme is

generated using the EFGM weak formulation results.

Like the FEM the stiffness matrix is generated but in

this weight function, shape function is added in the

stiffness matrix. After that the Lagrange multiplier is

used in the boundary conditions. The equation

becomes in the form of KU=P. The results are of

nodes. These are the discretized equations of the

EFGM

Table 4.6 EFGM results

Node Area u*(1*10^-

5) σ

1 0.0095 0 0

2 0.0084 0.1589 0.06658

3 0.0056 0.5293 0.19529

4 0.0034 0.9528 0.32399

Validation for case study 2

For the validation the results are compared

firstly with analytical results. This shows results are

favorable; continuous; consistent and like. A

combined table for the displacement is made for the

comparison. The displacement of the EFGM and

Analytical solution is same. We can see the results in

Table 4.7 at each node the displacement values are

approximately same with negligible differences.

Table 4.7 Comparison b/w Analytical and EFGM

displacement

Fig. 4.3 Comparison b/w Analytical and EFGM

displacement

The Fig. 4.3 comparing the results of analytical

and EFGM lines are overlaping with minor

diffrences. Displacement variation at each node

is plotted for both of the methods. In Table 4.8 and

Fig. 4.4 the comparisn between FEM and EFGM

Node Area u*(1*10^-

5) σ*(10^7)

1 0.0095 0 0

2 0.0084 0.1983 0.119

3 0.0056 0.4946 0.1778

4 0.0034 0.9844 0.2939

Node Area

EFGM

u*(1*10^-

5)

ANALYTICAL

u*(1*10^-5)

1 0.0095 0 0

2 0.0084 0.1589 0.2

3 0.0056 0.5293 0.5

4 0.0034 0.9528 1

16

shown these results are also favourable and like

results.

Table 4.8 Comparison b/w EFGM and FEM

displacement

Node Area EFGM

u*(1*10^-5)

FEM

u*(1*10^-

5)

1 0.0095 0 0

2 0.0084 0.1589 0.1983

3 0.0056 0.5293 0.4946

4 0.0034 0.9528 0.9844

Fig. 4.4 Comparison b/w EFGM and FEM

displacement

Now the composite results for three methods

are generated to check their difference. So a

displacement and stress table is generated below;

Table 4.9 Comparison between Analytical, FEM and

EFGM displacement

Fig 4.5 Comparison b/w Analytical, FEM and EFGM

displacement

Composite results are also favorable all

values are approaching to the required results and

overlapped the FEM and EFGM line in fig 4.4

Table 4.10 Comparison b/w Analytical, FEM and

EFGM STRESS

Node Area ANALY-

TICAL FEM EFGM

1 0.0095 0 0 0

2 0.0084 0.2 0.1983 0.1589

3 0.0056 0.5 0.4946 0.5293

4 0.0034 1 0.9844 0.9528

17

Fig. 4.6 Comparison b/w FEM and EFGM STRESS

In the case of stress at node 2 it is different

but all other are showing good results for EFGM.

Fig. 4.7 Comparison b/w Analytical, FEM and

EFGM STRESS

In Table 4.11 comparison for the stress have

been done successfully. The results are positive for

the EFGM. Here I can say the results are positive so

the method i choose is valid for the selected problem

.further the results may be verified. Results like

domain of influence if changed effect in the results or

change in weight function gives same results or

results get changed due to the weight function. So for

that following attempt have been done

4.3 Effect of dmx on the result

Table 4.12 Effect of dmx on displacement

Node dmx=2 dmx=2.4 dmx=3

1 0 -0.0002 -

0.0026

2 0.1064 0.1125 0.0261

3 0.2115 0.2071 -0.002

4 0.3177 0.3209 0.0647

5 0.4235 0.4214 0.0057

6 0.5293 0.5323 0.0962

7 0.6351 0.6309 0.0207

8 0.7413 0.749 0.1154

Node Area

ANALY-

TICAL

σ*(10^7)

FEM

σ*(10^7)

EFGM

σ*(10^7)

1 0.0095 0 0 0

2 0.0084 0.119 0.119 0.06658

3 0.0056 0.1785 0.1778 0.19529

4 0.0034 0.2777 0.2939 0.32399

18

9 0.8464 0.8369 0.0589

10 0.9528 0.9537 0.1029

Fig. 4.8 Effect of dmx on displacement

Fig. 4.9 Effect of dmx on stress

Now by the observation of fig 4.7 it is clear

that the results are same if the dmx is changed 2 to

2.4 but if it increased further the results get distorted.

So for this problem the dmx 2 or 2.4 is best selected

domain of influence. Although the displacement gets

changed a lot but in fig 4.8 one can see the results of

stresses do not change drastically. It changes little bit

to the previous results.

4.4 Effect of Weight Functions on results

Fig. 4.10 Effect of weight function on displacement

Now the change in the weight function is

attempted to see the effect of weight function which

one is better for this problem. But after the

comparison in fig 4.9 it is observed that the result are

as continuous and same as they were so there is not

any drastic change in the displacement results due to

the weight function.

There is not a big change due to the quadric

spline weight function from the results of cubic

spline. In the fig 4.10 stress results are as they were

even at some points they are with same results. So we

can say for this type of problem both the weight

functions can hold good command over the results.

So results are checked and verified by the

FEM results. And the effect of change in parameters

has been studied in this section.

19

Fig. 4.11 Effect of weight function on stress

V. CONCLUSION

EFG method is an achievement in the

improvement of mesh free methods. In this thysis a

MATLAB program has been developed to analyze

plane stress problem by EFG method. The obtained

results are compared with analytical and FEM

results.

In this study, the varying cross section

problem has been solved. In this the cantilever beam

is subjected to the simply point load at the free end.

The applied load is tensile in nature which results in

extension or displacement of the bar.

The results obtain in the form of

displacement are equal to the FEM as well as to the

analytical method. Same in case of stress. So we can

say EFGM as a better alternate for the problems.

Change in weight function in programme

even does not alter the results the results are same

with same continuity. Second change in domain of

influence effect the result when it changed to value 3.

But between 2-2.7 it holds good results for the

problem .The problem is solved by varying the

number of nodes which gives the continuity of the

results. The results show that EFG method gives the

satisfactory results. Which can be seen in any case

applied to it. It has been observed that EFG gives

accurate results when compared to FEM. Although it

is not perfectly mesh less because we need back mesh

for integration but re meshing problem is obsolete

here as in case of FEM. The time consumed in the

EFGM is more as in case of FEM. But the total time

for solution is less because not only cost the most of

the time is consumed in the mesh generation. So, the

EFG method can be better alternate to analyze

structure problems. If the better no of nodes and

quadrature points are selected the best results can be

achieved.

VI. FUTURE SCOPE

The extension of EFG to bending problems

such as beams of varying cross section loaded with

self-weight or uniformly distributed load can be done

by this generated code. For more work different

material models such as laminated composites, and

nonlinear problems, temperature stresses problem on

these types of problems can be analyzed. More over

this dynamic problem on this can be done .the

problem here analyzed can be extended to 2D and 3D

work to check the results whether it holds same

relation or not. One more topic we can add to future

work is temperature stresses on this type of problems

and temperature distribution in with conduction,

convection or radiation. So for this topic there is a

vast area to work one can say it’s the infancy period

of this field so it need more work to do.

REFERENCES

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“Element-free Galerkin methods”.

International Journal of Numerical Methods

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(2001), “A Stabilized conforming nodal

integration for galerkin mesh free methods”,

International Journal of Numerical Methods

in Engineering, vol. 50, pp. 435-466

[3]. M. Duflot, N. Dang (2002), “A truly

meshless Galerkin method based on a

moving least squares quadrature”, vol:18,1-9

[4]. P. Soparat, P. Nanakorn,” Analysis of crack

growth in concrete by the element free

galerkin method”, pp. 42-46

[5]. L. Gu, Y. Tong, G. R. Liu. (2001), “A

coupled Element Free Galerkin /Boundary

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dimensional solids”, Computer Methods in

Applied Mechanics and Engineering 190,

pp. 4405-4419.

[6]. A. Huerta, T. Belytschko, S. F. Mendez, T.

Rabczuk (2004), “Meshfree Methods”,

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Analyzing Plane-plate Bending with

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EFGM”, Journal of Mathematics Research,

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