4 Chapter 2 Element Free Galerkin (EFG) method 2.1 Introduction The element free Galerkin (EFG) method is a meshless method developed by Belytschko, Lu and Gu (1994). This method only requires a set of nodes and a description of the boundaries to construct an approximation solution. The connectivity between the data points and the shape functions are constructed by the method without recourse to elements. The EFG method employs the moving least square (MLS) approximations, which are composed of three components: a weight function of compact support associated with each node, a polynomial basis and a set of coefficients that depend on position. The support of the weight function defines a node the domain of influence, which is the sub-domain over which a particular node contributes to the approximation. The overlap of the nodal influence domain defines the nodal connectivity. One useful property of MLS approximations is that their continuity is equal to the continuity of the weight function; highly continuous approximations can be generated by an appropriate choice of the weight function. Although the EFG can be considered meshless with respect to shape function construction or function approximation, a mesh will be required for solving partial differential equations by the Galerkin approximation procedure. This is because evaluation of the integrals in the weak form requires a subdivision of the domain unless nodal quadrature is used.
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4
Chapter 2
Element Free Galerkin (EFG) method
2.1 Introduction
The element free Galerkin (EFG) method is a meshless method developed by Belytschko,
Lu and Gu (1994). This method only requires a set of nodes and a description of the
boundaries to construct an approximation solution. The connectivity between the data points
and the shape functions are constructed by the method without recourse to elements.
The EFG method employs the moving least square (MLS) approximations, which are
composed of three components: a weight function of compact support associated with each
node, a polynomial basis and a set of coefficients that depend on position. The support of the
weight function defines a node the domain of influence, which is the sub-domain over which
a particular node contributes to the approximation. The overlap of the nodal influence domain
defines the nodal connectivity.
One useful property of MLS approximations is that their continuity is equal to the
continuity of the weight function; highly continuous approximations can be generated by an
appropriate choice of the weight function. Although the EFG can be considered meshless with
respect to shape function construction or function approximation, a mesh will be required for
solving partial differential equations by the Galerkin approximation procedure. This is
because evaluation of the integrals in the weak form requires a subdivision of the domain
unless nodal quadrature is used.
CHAPTER 2: ELEMENT FREE GALERKIN METHOD 5
This chapter describes the construction of MLS approximations and the resulting EFG
shape functions in two types of support domains either circle or rectangle. In my work, the
second-derivatives of the shape function are required for equilibrium model in Chapter 3. So,
there are two cases to compute the second-order derivatives of the shape functions also
presented here. The first case is referred in the PhD thesis of Duflot (2004), Duflot (2005) and
Duflot and Nguyen-Dang (2001), (2004). And the other is in Belytschko, Lu and Gu (1994),
Dolbow and Belytschko (1998) and Liu (2003). In the course of this description, the effect of
different weight functions is illustrated.
Finally, the choice of the basic functions, the support and influence domain concepts are
defined on that. The determination of the dimension of a support domain and the detail
algorithm to compute the shape function and their derivatives are also presents in this chapter
as follows.
2.2 Moving least square (MLS) approximation
In this section, I would like to present the MLS approximation, which was introduced by
Lancaster and Salkauskas (1981) for smoothing and interpolating data. Currently the MLS
method is a widely used alternative for constructing meshless shape functions for
approximation. The first, Nayroles et al (1992) were used MLS approximation to construct
shape functions for their diffuse element method (DEM), after Belytschko, Lu and Gu (1994),
who named it the EFG method, where the MLS approximation is also employed.
This method employs MLS approximants to approximate the function with as
in Liu (2003), Fries and Matthies (2004), and Belytschko, Lu and Gu (1994). We consider a
sub-domain , the neighbourhood of a point
)(xu )(xu h
xΩ x and denoted as the domain of definition of
the MLS approximation for the trial function x , which is located in the problem domain Ω .
Let be the function of the field variable defined in the domain )(xu xΩ . The approximation of
at point )(xu x is denoted . The MLS approximation first writes the field function in
the form
)(xu h
(2.1) ∑ Τ==m
iii
h apu )()()()()( xaxpxxx
where is the number of terms of monomials, is a vector of basis functions that
consists most often of monomials of the lowest orders to ensure minimum completeness.
m )(xp
CHAPTER 2: ELEMENT FREE GALERKIN METHOD 6
The coefficient are also functions of )(xia x , in the equation (2.1) is obtained at
any point
)(xia
x by performing a weighted least square fit for the local approximation, which is
obtained by minimizing the difference between the local approximation and the function. The
discrete norm as follows: 2L
∑=
−−=n
II
hI xuuxxwJ
1
2))()()(( x
(2.2) 2
1])()([)( II
n
II uxxxw −−= Τ
=∑ xap
where is a weight function with compact support and n is the number of points in
the neighbourhood of
)( Ixxw −
x , for which the weight function xΩ 0)( ≠− Ixxw , and is the
nodal value of at .
Iu
u Ixx =
iu
ix
)( ih xu
)(xu hu
x 0
• ••
••
•
Figure 2.1: The approximation function and the nodal parameters in the MLS approximation )(xu hiu
Equation (2.2) can be rewritten in the form
(2.3) ))(()( uPaxWuPa −−= ΤJ
there
(2.4) ),,,( 21 nuuu K=Τu
(2.5)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
)()()(
)()()()()()(
21
22221
11211
nmnn
m
m
xpxpxp
xpxpxpxpxpxp
L
MOMM
L
L
P
CHAPTER 2: ELEMENT FREE GALERKIN METHOD 7
and
(2.6)
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−
=
)(00
0)(000)(
)( 2
1
nxxw
xxwxxw
L
MOM
L
L
xW
The functional can be minimized by setting the derivative of with respect to equal
to zero i.e.,
J J a
0=∂∂
aJ . The following system of equation results: m
0])()()[(2)(0
0])()()[(2)(0
0])()()[(2)(0
1
12
2
11
1
∑
∑
∑
=
Τ
=
Τ
=
Τ
=−−⇔=∂∂
=−−⇔=∂∂
=−−⇔=∂∂
n
IIIImI
m
n
IIIII
n
IIIII
uxxxpxxwaJ
uxxxpxxwaJ
uxxxpxxwaJ
ap
ap
ap
MMM
(2.7)
This is in vector notation
∑ (2.8) =
Τ =−−n
IIIII uxxxxxw
10])()()[(2)( app
(2.9) 0)()()()()()(21
=−−−∑=
ΤIIII
n
III uxxxwxxxxxw papp
Eliminating the constant factor and separating the right hand side given
(2.10) ∑∑==
Τ −=−n
IIIII
n
III uxxxwxxxxxw
11)()()()()()( papp
or
(2.11) PuxWxaPPxW )()()( =Τ
So, we have
(2.12) uxBxAxa )()()( 1−=
with (2.13a) ∑=
ΤΤ −==n
IIII xxxxw
1)()()()()( ppPPxWxA
is often called moment matrix, and
])()()()()()([)()( 2211 nn xxxwxxxwxxxw pppPxWxB −−−== L (2.13b)
CHAPTER 2: ELEMENT FREE GALERKIN METHOD 8
Finally, we substitute the equations (2.12), (2.13a) and (2.13b) into (2.1) and obtaining an
approximation of the form
(2.14) uxBxApx )()()()( 1−Τ= xu h
or more detail
(2.15) I
n
III
n
IIII
h uxxxwxxxxwxu ∑∑=
−
=
ΤΤ −⎥⎦
⎤⎢⎣
⎡−=
1
1
1)()()()()()()( ppppx
This can be written shortly as
(2.16) uxxx )()()(1
Τ
=
Φ== ∑n
III
h uu φ
where the shape function is defined by )(xΦ
(2.17) )()]()[()( 1 xBxAxpx −ΤΤ =Φ
and thus for one certain shape function Iφ at a point x
(2.18) )()()]()[()( 1III xxxw pxAxpx −= −Τφ
or (2.19) )()()()( xpxcx III wxΤ=φ
with (2.20) )()()( 1 xpxAxc −=
To compute the shape functions from (2.17) it is necessary to invert the A matrix. In one
dimension, this operation is not computationally expensive, but here we need to compute in
two or three dimensions it becomes burdensome. In this section, we present two cases to
compute the shape functions and their derivatives as following.
Case one: According to the PhD thesis of Duflot (2004), Duflot (2005), and Duflot
and Nguyen-Dang (2001), (2004). We have the first-order derivatives of the MLS shape