analysis of one dimensional beam problem using element ...€¦ · analysis of one dimensional beam...
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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
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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.
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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
[1]. T. Belytschko, Y.Y. Lu, L. Gu. (1994),
“Element-free Galerkin methods”.
International Journal of Numerical Methods
in Engineering, vol. 37, pp. 229-256
[2]. J. S. Chen, C. T. Wu, S. Yoon, Y. You
(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
Element method for stress analysis of two-
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”,
Encyclopedia of Computational Mechanics,
[7]. Y. Zhang, M. Xia, Y. Zhai (2009), “
Analyzing Plane-plate Bending with
20
EFGM”, Journal of Mathematics Research,
vol.-1, no-1
[8]. T.R. Chandrupatla and A.D. Belegundu
(2002), “Introduction to Finite Element in
Engineering” Ed. 3rd
[9].
http://www.scribd.com/doc/17702968/UNIT
4COMPUTER-AIDED-DESIGN (TIME
17:34, 17/08/2012)
[10]. B. Torstenfelt (2007), “An Introduction to
Elasticity and Heat Transfer”, Ed 2007