Meshless Local Petrov-Galerkin (MLPG) Method for Convection-Diffusion
Problems
H. Lin, S.N. Atluri1
Abstract: Due to the very general nature of the Meshless
Local Petrov-Galerkin (MLPG) method, it is very easy and
natural to introduce the upwinding concept (even in multidimensional cases) in the MLPG method, in order to deal with
convection-dominated flows. In this paper, several upwinding
schemes are proposed, and applied to solve steady convectiondiffusion problems, in one and two dimensions. Even for very
high Peclet number flows, the MLPG method, with upwinding, gives very good results. It shows that the MLPG method is
very promising to solve the convection-dominated flow problems, and fluid mechanics problems.
keyword: MLPG, MLS, convection-dominated flow, upwinding.
1 Introduction
Although the finite element method (FEM) and the closely
related finite volume method (FVM) are well-established numerical techniques for computer modeling in engineering and
sciences, they are not without shortcomings. First of all,
their reliance on a mesh leads to complications for certain
classes of problems. The generation of good quality meshes
presents significant difficulties in the analysis of engineering
systems (especially in 3D). These difficulties can be overcome by the so-called meshless methods, which have attracted considerable interest over the past decade. A number of meshless methods have been developed by different authors, such as Smooth Particle Hydrodynamics (SPH) [Lucy
(1977)], Diffuse Element Method (DEM) [Naroles, Touzot,
and Villon (1992)], Element Free Galerkin method (EFG) [Belytcshko, Lu, and Gu (1994)], Reproducing Kernel Particle
Method (RKPM) [Liu, Jun, and Zhang (1995)], hp-clouds
method [Duarte and Oden (1996)], Finite Point Method (FPM)
[Oñate, Idelsohn, Zienkiewicz, and Taylor (1996)], Partition
of Unity Method (PUM) [Babus̃ka and Melenk (1997)], Local
Boundary Integral Equation method (LBIE) [Zhu, Zhang, and
Atluri (1998a,b)], Meshless Local Petrov-Galerkin method
(MLPG) [Atluri and Zhu (1998a,b); Atluri (1999); Atluri and
Zhu (2000)]. Most of these methods, in reality, are not really meshless method, since they use a background mesh for
the numerical integration of the weak form. To be a truly
meshless method, both interpolation and integration should be
1 Center
for Aerospace Research and Education, 7704 Boelter Hall, University of California, Los Angeles, CA 90095-1600
performed without a mesh. The finite point method (FPM)
[Oñate, Idelsohn, Zienkiewicz, and Taylor (1996)] is a truly
meshless method. A non-element interpolation scheme weighted least squares (WLS) is used and there is no integration required. However, this method is based on point collocation, and is very sensitive to the choice of collocation points.
As discussed in Atluri and Zhu (1998a,b), and in Zhu, Zhang,
and Atluri (1998a,b), MLPG and LBIE are truly meshless
methods, because, a traditional non-overlapping, continguous
mesh is not required, either for interpolation purpose or for
integration purpose. As pointed out in Atluri, Kim, and Cho
(1999b), the LBIE approach can be treated simply as a special
case of the MLPG scheme. The MLPG method is based on a
weak form computed over a local sub-domain, which can be
any simple geometry like a sphere, cube or ellipsoid in 3D. The
trial and test function spaces can be different or the same. It
offers a lot of flexibility to deal with different boundary value
problems. A wide range of problems has been solved by Atluri
and his coauthors. The objective of this paper is to extend the
MLPG method to solve steady convection-diffusion problems.
In fluid mechanics, the existence of the convection term makes
the problem non-self-ajoint. A special treatment is needed to
stabilize the numerical approximation for these kinds of problems. Schemes related to upwinding are the most general techniques to stabilize Finite Difference Method (FDM), FEM and
FVM. The same concept is needed in the meshless methods, so
as to obtain a good accuracy for convection-dominated flows.
Only very few works were reported by using the so-called
meshless methods, to solve convection-dominated flows. In
Oñate, Idelsohn, Zienkiewicz, and Taylor (1996), the FPM
method was applied, with upwinding for the first derivative
or with characteristic approximation. However, there is a significant drawback of the FPM as discussed above, and, in
multidimensional cases, it is not easy to define the critical
distance, which is important to stabilize the method and obtain good accuracy. In Liu, Jun, Sihling, Chen, and Hao
(1997), the RKPM method, combined with the Streamline
Upwind/Petrov-Galerkin (SUPG) form of variational formulation, was used. As pointed out before, the RKPM method is
not a truly meshless method, since a background mesh is used
to integrate the weak form. In fact, a truly meshless method,
such as the MLPG method, is much easier and more flexible
for introducing the upwinding concept, with a very clear physical meaning. This will be illustrated in this paper, and numeri-
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cal examples for the steady convection-diffusion problems will The local weak form provides a very clear concept for a local
be used for verification purposes.
non-element integration, which does not need any background
integration cells which are continguous over the entire domain.
Also the MLPG method leads to a natural way to construct the
2 Local Weak Form
global stiffness matrix through the integration over a local subLet Ω be a bounded region in Rnsd , where nsd is the num- domain. To solve this local weak form, some kind of meshless
ber of space dimensions, and assume that Ω has a piecewise interpolation schemes is needed. This will be discussed in the
smooth boundary Γ. Then, the governing equation for steady next section.
convection-diffusion problems in Ω can be written as:
vj
∂φ
∂x j
=
∂
∂φ
(K
)+ f
∂x j ∂x j
3 The MLS approximation scheme
(1)
In order to preserve the local character of the numerical implementation, a meshless method uses a local interpolation or
where, v j is the flow velocity, K is the diffusivity coefficient, f
approximation to represent the trial/test functions with the valis a source term, and repeated indices are to be summed. The
ues (or the fictitious values) of the unknown variable at some
boundary conditions are assumed to be:
randomly located nodes. There are a lot of local interpolation schemes, such as MLS, PUM, RKPM, hp-clouds, Shepard
Essential Boundary Conditions:
function, etc., available to achieve this aim.
φ = φ̄ on Γφ
(2) The Moving Least Square (MLS) method is generally considered as one of the schemes to interpolate data with a reasonable
Natural Boundary Conditions:
accuracy. Therefore, the MLS scheme is chosen in this paper.
Consider the approximation of a function u(x) in a domain
(3) Ω with a number of scattered nodes fxi g, i = 1; 2; : : : ; n, the
moving least-square approximant uh (x) of u(x), 8x 2 Ω, can
be defined by
where, φ̄ and t¯ are given, n j is the outward unit normal vector
to Γ, Γφ and Γt are subsets of Γ satisfying Γφ \ Γt = 0/ (the uh (x) = pT (x)a(x) 8x 2 Ω
(7)
empty set) and Γφ [ Γt = Γ.
where, pT (x) = [ p1 (x); p2 (x); : : : ; pm(x)] is a complete monoDifferent from other meshless methods, the MLPG method is
mial basis of order m, which, for example, can be chosen as
based on a local weak form over a local sub-domain Ωs , which
linear:
is located entirely inside the global domain Ω. It is noted that
the local sub-domain can be of an arbitrary shape containing pT (x) = [1; x; y]; m = 3;
(8)
the node position x in question.
or quadratic:
To satisfy Eq. (1) in a local sub-domain Ωs with a piecewise
(9)
smooth boundary Γs , Eq. (1) is weighted by a test function pT (x) = [1; x; y; x2; xy; y2 ]; m = 6
w and integrated over the local sub-domain such that the local
for 2D problems, and a(x) is a vector containing coefficients
weighted residual equation can be written as:
a j (x), j = 1; 2; : : : ; m which are functions of the space coordiZ
nates x, and determined by minimizing a weighted discrete L2
∂φ
∂
∂φ
[v j
(K
)
f ]w dΩ = 0
(4) norm, defined as:
∂x
∂x
∂x
Ωs
j
j
j
K
∂φ
n j = t¯ on Γt
∂x j
n
for all w. By using the integration by parts, Eq. (4) is recast J(x) =
∑ wi(x)[pT (xi )a(x) ûi]2
into a local weak form as:
i=1
T
Z
Z
=
[P a(x) û] W[P a(x) û]
(10)
∂φ
∂φ ∂w
∂φ
[v j
+K
f w] dΩ
K
n j w dΓ = 0 (5)
∂x j
∂x j ∂x j
Ωs
Γs ∂x j
where wi (x) is the weight function associated with the node
i,
with wi (x) > 0 for all x in the support of wi (x), xi denotes
for all continuous trial functions φ and continuous test functhe
value of x at node i, n is the number of nodes in Ω for
tions w. In general, the boundary Γs of the local sub-domain
which
the weight functions wi (x) > 0, the matrices P and W
Ωs may intersect with the boundary of the global domain Ω.
are
defined
as
Therefore,
2 T
3
p (x1 )
Γs = ΓsI [ Γsφ [ Γst
(6)
6 pT (x2 ) 77
(11)
P=6
4 5
where, ΓsI is the part of Γs which is inside the global domain,
T
p (xn ) nm
Γsφ = Γs \ Γφ, and Γst = Γs \ Γt .
47
MLPG for Convection-Diffusion Problems
2
w1 (x) W = 4 0
3
5
and
p j (x) 2 C l (Ω); j = 1; 2; : : : ; m, then φi (x) 2 C r (Ω) with r =
(12) min(k; l ). A number of choices are available for the basis functions and the weight functions. In this paper, the linear basis
is chosen and a spline weight function as in Atluri and Zhu
(1998a) is used:
ûT
(13)
0
wn (x)
= [û1 ; û2; : : : ; ûn]
w i (x ) =
1
6( drii )2 + 8( drii )3
0
3( drii )4
0 di ri
di ri
(21)
where, ûi ; i = 1; 2; : : : ; n are the fictitious nodal values and not
the nodal values of the unknown trial function uh (x) in general. where di = jx xi j is the distance from node xi to point x, and
The stationarity of J in Eq. (10) with respect to a(x) leads to ri is the size of the support for the weight function wi . It can
be easily seen that the spline weight function is C 1 continuous
the following relation between a(x) and û:
over the entire domain.
A(x)a(x) = B(x)û
(14) From the above discussion, it shows that the MLS shape funcwhere the matrices A(x) and B(x) are defined by
n
A(x) = PT WP = ∑ wi (x)p(xi )pT (xi )
i =1
B(x) = PT W = [w1 (x)p(x1 ); w2(x)p(x2 ); : : : ;
wn (x)p(xn )]
tions don’t have the Kronecker delta property. This causes the
difficulty to impose the essential boundary conditions. Several
methods have been proposed, e.g., see Belytcshko, Krongauz,
(15) Organ, Fleming, and Krysl (1996) and Zhu and Atluri (1998).
In this paper, the penalty method used in Zhu and Atluri (1998)
and the method recently proposed by Atluri, Kim, and Cho
(1999b) are used to deal with the essential boundary condi(16) tions.
Solving this for a(x) and substituting it into Eq. (7), we get the 4 Upwinding Schemes
MLS approximation as
4.1 Overview
n
uh (x) = ΦT (x) û = ∑ φi (x)ûi
8x 2 Ω
(17) It is well known that convection-dominated flows are some of
the most difficult problems to solve numerically. The presi=1
ence of the convection term causes serious numerical difficulwhere, the nodal shape function corresponding to nodal point
ties, appearing in the form of “wiggles” (oscillatory solutions),
xi is given by
when the convection term is dominant. This problem could be
T
T
1
Φ (x) = p (x)A (x)B(x)
(18) solved somewhat heuristicaly, by using upwinding. A number
of upwinding schemes has been developed, for FDM, FEM
It should be noted that the MLS approximation is well defined and FVM. In the one dimensional cases, optimal upwinding
only when the matrix A in Eq. (14) is non-singular. It can be schemes may be designed so as to result in exact nodal soluseen that this is the case if and only if the rank of P equals m. tions. However, in the multidimesional cases, generalizations
A necessary condition for a well-defined MLS approximation of traditional upwinding schemes were unsuccessful due to the
is that at least m weight functions are non-zero (i.e. n m) for crosswind diffusion problem (see Fig.1-Fig.2).
each sample point x 2 Ω and that the nodes in Ω will not be It was apparent that the upwind effect, arrived at by whatever
arranged in a special pattern such as on a straight line.
means, was needed only in the direction of flow. However, It
is not easy to design such methods for multidimensional cases.
The partial derivatives of φi (x) can be obtained as
Hughes and Brooks (1979) introduced the ‘Streamline Upm
wind (SU) method’, where the artificial diffusion operator is
1
1
1
φi;k = ∑ [ p j ;k(A B) ji + p j (A B;k + A;k B) ji]
(19)
constructed to act only in the flow direction, a priori eliminatj =1
ing the possibility of any crosswind diffusion. A similar idea
was described in Kelly, Nakazawa, Zienkiewicz, and Heinin which A;k 1 is given by
rich (1980) as ‘anisotropic balancing dissipation’. As shown
A;k 1 = A 1 A;k A 1
(20) in Hughes and Brooks (1979), one could obtain much better
solutions for the crosswind problem by using the streamline
and the index following a comma indicates a spatial derivative. upwind method, but several deficiencies remained. The upIt is known that the smoothness of the shape functions φi (x) winded convection term was not consistent with the centrally
is determined by that of the basis functions and of the weight weighted source and transient terms, resulting in excessively
functions. Let C k (Ω) be the space of k-th continuously dif- diffuse solutions when these terms were present (see Fig.3ferentiable functions. If wi (x) 2 C k (Ω); i = 1; 2; : : : ; n and Fig.4).
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Copyright c 2000 Tech Science Press
CMES, vol.1, no.2, pp.45-60, 2000
u (const)
φ=0
0
15
1
f(x)
0
x
−.5
(a)
Figure 3 : Pure convection with a source term: (a) Problem
Statement.
Figure 1 : Illustration of the crosswind diffusion problem with
Pe = 100 (from Leonard (1979)): (a) Exact.
EXACT
SU
SUPG
φ
2
1
0
0
5
10
15
x
(b)
Figure 4 : Pure convection with a source term: (b) Results for
SU and SUPG.
Figure 2 : Illustration of the crosswind diffusion problem with
Pe = 100 (from Leonard (1979)): (b) Optimal Upwinding.
Clearly, upwind weighting of all terms in the equation was
needed, i.e., some kind of Petrov-Galerkin method is needed
for consistency purposes. Therfore, in order to solve the
crosswind problem and be consistent with all terms, more refined versions of the upwinding concept should be based on
Petrov-Galerkin methods, which utilize upwinding only in the
streamline direction. As known, most popular among such
methods is the Streamline Upwind Petrov-Galerkin method
(SUPG)[Brooks and Hughes (1982)], which consistently introduces an additional stability term in the upwind direction.
This method has better stability and accuracy properties than
the standard Galerkin formulation for convection-dominated
flows. However, some drawbacks still remain. The choice
of the related parameters is still not so straightfoward (another
definition of the related parameters can be seen in Franca, Frey,
and Hughes (1992)). In addition, for non-regular solutions,
spurious oscillations remain in the neighborhoods containing
sharp layers. In order to prevent those ‘wiggles’, many methods have been proposed by adding some kind of discontinuitycapturing perturbation term (see e.g. Hughes, Mallet, and
Mizukami (1986), Almeida and Silva (1997) and references
therein).
For meshless methods, the same kind of consideration should
be taken to deal with convection-dominated flows. As pointed
out in the introduction, the very general nature of truly meshless methods, such as the MLPG method, makes it easier to
introduce the upwind concept more clearly and effectively. In
this paper, two upwinding schemes for the MLPG method are
proposed in the following.
4.2 MLPG Upwinding Scheme I
The MLPG method is based on the Petrov-Galerkin weightedresidual procedures. Different spaces for the test and trial
functions can be used, as shown in Fig.5.
49
MLPG for Convection-Diffusion Problems
Figure 7 : MLPG Upwinding Scheme I (US-I): Specification.
Figure 5 : The MLPG method without Upwinding.
Figure 8 : MLPG Upwinding Scheme II (US-II).
in which Pe is a local Peclet number defined as:
Pe =
uri
K
(23)
The size of the support for the trial functions also equal to ri at
xi , and the local sub-domain at xi is coincided with the support
for the test functions at xi .
Figure 6 : MLPG Upwinding Scheme I (US-I).
4.3 MLPG Upwinding Scheme II
Therefore one of the very natural ways to construct upwinding
schemes is to choose different trial and test functions. This
can be done by a lot of ways. For example, in order to apply
upwinding in the streamline direction, we can skew the test
function opposite to the streamline direction as shown in Fig.6.
For convenience, we denote this as Upwinding Scheme I (USI).
As an illustration, we choose a skewed weight function as the
test function. The skewed weight function is given as follows:
using the same form of weight function as in Eq. (21), we shift
the position of the maximum of wi (x) from xi to xi γri si , as
shown in Fig.7,
where, si is the unit vector of the streamline direction at xi , ri
is the size of the support for the test functions at xi , and γ is
given by
γ=
1
Pe
coth( )
2
2
1
Pe
(22)
In fact, because the MLPG method is based on a local weak
form over a local sub-domain, there is another very convenient way to design upwinding schemes. We can use the
Bubnov-Galerkin procedure to discretize the local weak form,
as done in Atluri, Kim, and Cho (1999b), and shift the local
sub-domain opposite to the streamline direction, as shown in
Fig.8. We denote this as Upwinding Scheme II (US-II).
Here, the same spaces for the trial and test functions are used,
that is, the same support and the same interpolation scheme
(MLS) for the trial functions and the test functions are employed. In this case, the local sub-domain at xi is no longer
coincident with the support for the test functions at xi , but the
size is the same. (It should be noted that in the usual MLPG
method, we usually choose the test functions such that the integration term along the boundary ΓsI equals to zero, but, in
general, this is not true for the MLPG method with US-II.
Therefore, in the local weak form, the integration term along
the boundary ΓsI should be retained.)
In particular, the shifting distance of the local sub-domain can
50
Copyright c 2000 Tech Science Press
CMES, vol.1, no.2, pp.45-60, 2000
EXACT
MLPG1
MLPG2
φ
2
1
0
Figure 9 : MLPG Upwinding Scheme II (US-II): Specification.
0
5
10
15
x
(c)
Figure 10 : Pure convection with a source term: (c) Results
be specified as γri , where, ri is the size of the support for the for MLPG1 and MLPG2.
test functions, which is equal to the size of the local domain,
at xi , and γ is given by
γ = coth(
Pe
)
2
2
Pe
(24)
K = 1; f = 0;
φ = 0 at x = 0;
φ = 1 at x = 1:
in which Pe is a local Peclet number, defined as:
Pe =
2uri
K
(25)
The direction of the shifting is opposite to the streamline direction si at xi , as shown in Fig.9.
(27)
Case 2:
K = 1; f = 100;
φ = 0 at x = 0;
φ = 0 at x = 1:
5 Numerical examples
To assess the effectiveness of the methods described herein,
a series of numerical calculations is performed. For convenience, we note the MLPG method without upwinding as
MLPG, the MLPG method with US-I as MLPG1 and the
MLPG method with US-II as MLPG2. Here, the MLPG
method without upwinding is based on the Galerkin procedure
to approximate the local weak form and the local sub-domain
coincides with the support of the test functions. Examples for
1D and 2D problems are given in the following.
Case 1:
(28)
u may have different values, leading to different Peclet numbers Pe, which is defined as:
Pe =
uL
K
(29)
For the above cases, L = 1 and K = 1, thus, Pe = u.
11 points and the value of 2:3 times nodal distance (2:3∆x) for
radius of support are used to solve these problems. Penalty
method is chosen to deal with essential boundary conditions
For 1D problems, Eq. (1) can be written as:
due to its simplicity. Results for the value of φ are shown in
Fig.11-Fig.12.
2
dφ
d φ
u
K 2 f =0
(26) These results show that both MLPG1 and MLPG2 produce
dx
dx
very good solutions when Peclet number is very high, howFirst, reconsider the
as shown in Fig.3. Here, K = ever, the MLPG method without upwinding gives oscillation8problem
< 1 14 x; (0 < x 6)
ary solutions when Peclet number becomes large. For low
1
0, u = 1 and f =
: 04;x (28; < x(6<15x ) 8) The results for Peclet number, all methods get good results. It also shows
that, in general, MLPG2 gives better solutions than MLPG1.
MLPG1 and MLPG2 are shown in Fig.10. It shows that both It can be said that the choice for the test function in MLPG1
MLPG1 and MLPG2 give very good solutions.
is not the best one and more care should be taken. So, for 2D
Then we consider another problem, with the domain 0 x 1. problems tested in the following, we will only use the MLPG2,
i.e., Upwinding Scheme II.
Two different cases are considered:
5.1 One-dimensional problems
51
MLPG for Convection-Diffusion Problems
Pe=1
Pe=1
1
12
EXACT
MLPG
MLPG1
MLPG2
0.75
0.5
φ
φ
9
6
0.25
EXACT
MLPG
MLPG1
MLPG2
3
0
0
0.25
0.5
0.75
0
1
0
0.25
0.5
EXACT
MLPG
MLPG1
MLPG2
0.75
1
Pe=10
Pe=10
1
0.75
x
x
6
4
φ
φ
0.5
0.25
EXACT
MLPG
MLPG1
MLPG2
2
0
0
0
0.25
0.5
0.75
0
1
0.25
0.5
1.5
EXACT
MLPG
MLPG1
MLPG2
0.75
0.5
1
0.75
1
Pe=100
Pe=100
1
0.75
x
x
EXACT
MLPG
MLPG1
MLPG2
1
φ
φ
0.25
0
0.5
-0.25
-0.5
0
-0.75
0
0.25
0.5
0.75
0
1
0.25
0.5
x
x
Pe=1000
1
EXACT
MLPG
MLPG1
MLPG2
0.75
0.5
Pe=1000
0.2
EXACT
MLPG
MLPG1
MLPG2
0.15
MLPG(Galerkin)
0.1
φ
φ
0.25
0
0.05
-0.25
0
-0.5
MLPG(Galerkin)
-0.75
0
0.25
0.5
0.75
1
-0.05
0
0.25
0.5
0.75
1
x
x
Figure 11 : Case 1 for 1D problems.
Figure 12 : Case 2 for 1D problems.
52
Copyright c 2000 Tech Science Press
CMES, vol.1, no.2, pp.45-60, 2000
GFEM
Y
PHI
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
X
SUPG
PHI
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
Y
Figure 13 : Convection skew to the mesh-I: problem statement.
5.2 Two-dimensional problems
For 2D problems, Eq. (1) can be written as:
u
∂φ
∂φ
+v
∂x
∂y
K
∂2 φ
∂x2
K
∂2 φ
∂y2
f
=0
(30)
X
MLPG (No Upwinding)
Y
PHI
0.937496
0.874992
0.812489
0.749985
0.687481
0.624977
0.562473
0.499969
0.437466
0.374962
0.312458
0.249954
0.18745
0.124947
0.0624428
X
MLPG2
PHI
0.937484
0.874968
0.812452
0.749936
0.68742
0.624904
0.562388
0.499872
0.437356
0.37484
0.312324
0.249808
0.187292
0.124777
0.0622606
Y
In this section, several cases are considered in a unit square domain given by 0 x; y 1. These problems have been widly
studied in literature, e.g., see Hughes, Mallet, and Mizukami
(1986) and Almeida and Silva (1997), as tests for the accuracy of various numerical schemes. The first two cases are
used to assess the performance of the MLPG and MLPG2
methods for different Peclet numbers. The last three cases involve very high Peclet numbers, and are used to assess solutions which are essentially of pure convection flows. For the
purpose of comparison, the standard Galerkin finite element
method (GFEM) and the streamline-upwind Petrov-Galerkin
method (SUPG) have been included in the results. Linear finite elements are used for GFEM and SUPG. The parameters
for SUPG are chosen as those in Brooks and Hughes (1982).
Except for the last case, a regular mesh (SUPG) or node distribution (MLPG) with 11 11 nodes is used. Unless specified,
the value of 2:1 times nodal distance h (2:1h) for radius of
support is used.
5.2.1
Convection skew to the mesh-I
In this case, the source term is assumed to be zero. The flow is
unidirectional and constant with velocity components:
u = cos(π=4);
v = sin(π=4):
See Fig.13 for the problem statement.
(31)
X
Figure 14 : Convection skew to the mesh-I: Pe = 1.
53
MLPG for Convection-Diffusion Problems
GFEM
GFEM
PHI
2951.09
2738.19
2525.28
2312.38
2099.47
1886.56
1673.66
1460.75
1247.84
1034.94
822.031
609.125
396.219
183.313
-29.5938
Y
Y
PHI
1.82906
1.70712
1.58519
1.46325
1.34131
1.21937
1.09744
0.9755
0.853562
0.731625
0.609687
0.48775
0.365812
0.243875
0.121937
X
X
SUPG
Y
PHI
0.9375
0.875
0.8125
0.75
0.6875
0.625
0.5625
0.5
0.4375
0.375
0.3125
0.25
0.1875
0.125
0.0625
X
X
MLPG (No Upwinding)
MLPG (No Upwinding)
PHI
2.73206
2.54313
2.35419
2.16525
1.97631
1.78738
1.59844
1.4095
1.22056
1.03163
0.842688
0.65375
0.464813
0.275875
0.0869375
Y
Y
PHI
1.05464
0.984276
0.913915
0.843553
0.773191
0.702829
0.632468
0.562106
0.491744
0.421382
0.351021
0.280659
0.210297
0.139935
0.0695736
X
X
MLPG2
PHI
0.935189
0.870777
0.806366
0.741955
0.677544
0.613132
0.548721
0.48431
0.419899
0.355487
0.291076
0.226665
0.162254
0.0978425
0.0334312
X
Figure 15 : Convection skew to the mesh-I: Pe = 100.
PHI
1.09204
1.01608
0.940124
0.864165
0.788206
0.712247
0.636289
0.56033
0.484371
0.408412
0.332454
0.256495
0.180536
0.104577
0.0286188
Y
MLPG2
Y
PHI
1.00125
0.9345
0.86775
0.801
0.73425
0.6675
0.60075
0.534
0.46725
0.4005
0.33375
0.267
0.20025
0.1335
0.06675
Y
SUPG
X
Figure 16 : Convection skew to the mesh-I: Pe = 106 .
Copyright c 2000 Tech Science Press
CMES, vol.1, no.2, pp.45-60, 2000
GFEM
Y
PHI
0.0690094
0.0644088
0.0598081
0.0552075
0.0506069
0.0460063
0.0414056
0.036805
0.0322044
0.0276038
0.0230031
0.0184025
0.0138019
0.00920125
0.00460063
PHI
1.49625
1.3965
1.29675
1.197
1.09725
0.9975
0.89775
0.798
0.69825
0.5985
0.49875
0.399
0.29925
0.1995
0.09975
Y
GFEM
X
X
SUPG
Y
PHI
0.0689812
0.0643825
0.0597837
0.055185
0.0505862
0.0459875
0.0413887
0.03679
0.0321912
0.0275925
0.0229937
0.018395
0.0137962
0.0091975
0.00459875
X
X
MLPG (No upwinding)
PHI
0.067478
0.062976
0.058474
0.0539719
0.0494699
0.0449679
0.0404659
0.0359639
0.0314619
0.0269598
0.0224578
0.0179558
0.0134538
0.00895179
0.00444977
PHI
0.882421
0.823542
0.764663
0.705784
0.646905
0.588026
0.529146
0.470267
0.411388
0.352509
0.29363
0.234751
0.175872
0.116993
0.0581139
Y
MLPG (No Upwinding)
Y
PHI
0.843844
0.787587
0.731331
0.675075
0.618819
0.562562
0.506306
0.45005
0.393794
0.337537
0.281281
0.225025
0.168769
0.112512
0.0562562
Y
SUPG
X
X
MLPG2
PHI
0.0687647
0.0641694
0.0595741
0.0549788
0.0503835
0.0457882
0.0411929
0.0365976
0.0320023
0.027407
0.0228117
0.0182164
0.0136211
0.0090258
0.0044305
X
Figure 17 : Convection with a source term-I: Pe = 1.
PHI
0.798251
0.744003
0.689754
0.635505
0.581256
0.527008
0.472759
0.41851
0.364261
0.310013
0.255764
0.201515
0.147266
0.0930175
0.0387688
Y
MLPG2
Y
54
X
Figure 18 : Convection with a source term-I: Pe = 100.
55
MLPG for Convection-Diffusion Problems
GFEM
Y
PHI
4692.19
4379.37
4066.56
3753.75
3440.94
3128.12
2815.31
2502.5
2189.69
1876.87
1564.06
1251.25
938.437
625.624
312.812
PHI
0.472849
0.438498
0.404146
0.369795
0.335444
0.301093
0.266741
0.23239
0.198039
0.163688
0.129336
0.094985
0.0606338
0.0262825
-0.00806875
Y
GFEM
X
X
SUPG
Y
PHI
1.06969
0.998375
0.927063
0.85575
0.784438
0.713125
0.641813
0.5705
0.499188
0.427875
0.356563
0.28525
0.213938
0.142625
0.0713125
X
X
MLPG (No Upwinding)
Y
PHI
2.61546
2.39191
2.16837
1.94482
1.72128
1.49774
1.27419
1.05065
0.827106
0.603563
0.380019
0.156475
-0.0670687
-0.290612
-0.514156
PHI
0.540519
0.504437
0.468356
0.432274
0.396193
0.360112
0.32403
0.287949
0.251867
0.215786
0.179704
0.143623
0.107542
0.0714602
0.0353788
Y
MLPG (No Upwinding)
X
X
MLPG2
PHI
0.815539
0.759377
0.703216
0.647055
0.590894
0.534732
0.478571
0.42241
0.366249
0.310087
0.253926
0.197765
0.141604
0.0854425
0.0292812
X
Figure 19 : Convection with a source term-I: Pe = 106 .
PHI
0.466077
0.434955
0.403832
0.37271
0.341587
0.310464
0.279342
0.248219
0.217096
0.185974
0.154851
0.123729
0.0926059
0.0614833
0.0303606
Y
MLPG2
Y
PHI
0.594281
0.554662
0.515044
0.475425
0.435806
0.396187
0.356569
0.31695
0.277331
0.237712
0.198094
0.158475
0.118856
0.0792375
0.0396187
Y
SUPG
X
Figure 20 : Convection with a source term-II: Pe = 106 .
56
Copyright c 2000 Tech Science Press
CMES, vol.1, no.2, pp.45-60, 2000
0.8
Figure 22 : Convection skew to the mesh-II: problem statement.
MLPG2
0.7
0.6
0.5
5.2.2 Convection with a source term-I
φ
0.4
The problem is given by
0.3
u = 1; v = 0; f = 1;
φ = 0 along the boundary.
0.2
0.1
0
-0.1
0
0.2
0.4
0.6
0.8
1
y
(b)
Figure 21 : Convection with a source term-II (Pe = 108 ): (a)
results from Almeida and Silva (1997); (b) results for MLPG2.
(33)
K is also varied so as to get different Peclet number Pe, which
is equal to 1=K. Again, the penalty method is used to deal
with the boundary conditions. Contours for φ are shown in
Fig.17-Fig.19.
As before, all methods give good results for low Peclet number. But, for high Peclet numbers, MLPG2 and SUPG are
much superior to MLPG and GFEM. Furthermore, it shows
that MLPG2 gives better solutions than the SUPG, when
Peclet number is very high.
5.2.3 Convection with a source term-II
K is varied in order to get different Peclet number Pe. Here,
Pe is given by
Pe =
The problem is given by
1; (0 < x 12 )
1; ( 12 < x < 1)
φ = 0 along the boundary.
u = 1;
jjujjL
(32)
K
p
where, jjujj = u2 + v2 . In this case, jjujj = 1 and L = 1, so
Pe = 1=K. It has smooth boundary conditions, so the penalty
method is used for simplicity. Contours for φ are shown in
Fig.14-Fig.16.
As in 1D problems, all methods give good results for low
Peclet number. But, for high Peclet numbers, MLPG2 and
SUPG are much superior to MLPG and GFEM. It also shows
that MLPG2 is somewhat better than the currently most popular method, the SUPG.
v = 0;
f
=
;
(34)
The diffusivity K is 10 6 or Peclet number Pe is 106 . Again,
the penalty method is used to deal with the boundary conditions. Contours for φ are shown in Fig.20.
It shows that MLPG2 and SUPG are better than GFEM and
MLPG, and MLPG2 is superior to SUPG. Actually, the solutions of MLPG2 are comparable (even better) to the results
produced by Almeida and Silva (1997) (see Fig.21).
However, Almeida and Silva (1997) introduced a very complicated modification for SUPG of Hughes and his co-workers,
to achieve better results.
57
MLPG for Convection-Diffusion Problems
Z
SUPG
Z
MLPG2 (r=1.2h)
X
Z
MLPG2 (r=2.1h)
X
X
Y
Y
Y
Figure 23 : Convection skew to the mesh-II: Pe = 106 and θ = π8 .
Z
SUPG
Z
MLPG2 (r=1.2h)
X
Z
MLPG2 (r=2.1h)
X
X
Y
Y
Y
Figure 24 : Convection skew to the mesh-II: Pe = 106 and θ = π4 .
Z
SUPG
Z
MLPG2 (r=1.2h)
X
X
Y
Z
MLPG2 (r=2.1h)
X
Y
Figure 25 : Convection skew to the mesh-II: Pe = 106 and θ =
Y
3π
8 .
58
Copyright c 2000 Tech Science Press
Z
CMES, vol.1, no.2, pp.45-60, 2000
MLPG(r=1.2h)
(No Upwinding)
GFEM
Y
Z
MLPG(r=2.1h)
(No Upwinding)
Y
X
X
Figure 26 : Convection in a rotating flow field: K = 10
Z
MLPG2(r=1.2h)
SUPG
Y
X
6
(Continued).
Z
MLPG2(r=2.1h)
Y
X
Z
Y
Z
Y
X
X
Figure 27 : Convection in a rotating flow field: K = 10 6 .
5.2.4
Convection skew to the mesh-II
MLPG2. This can be explained as following: as pointed out
in Atluri, Cho, and Kim (1999a), larger size of support gives
Now we consider another case, which involves discontinuous
better accuracy for smooth solutions because it includes more
boundary conditions and causes not only the sharp boundary
points and works like higher order schemes, but when the
layer but also an internal sharp layer. In this case, the source
solution is discontinuous, it is well known that higher order
term is assumed to zero. The flow is unidirectional and conschemes produce more oscillatory results. Again,it seems that
stant with velocity components:
the MLPG2 with small size of support gives somewhat better
u = cos θ; v = sin θ:
(35) solutions than SUPG at the internal discontinuity, but at the
boundary, MLPG2 gives somewhat worse results than SUPG.
where, θ is the angle between the flow direction and the posi- This may be caused by the method of imposing boundary conditions, which is a very strong constraint.
tive x direction. See Fig.22 for the problem statement.
Due to the discontinuous boundary conditions, it is difficult
5.2.5 Convection in a rotating flow field
to impose the essential boundary conditions by using penalty
method. (Actually it results in oscillation at boundary.) So we In this case, the flow velocity components are given by
use the method developed by Atluri, Kim, and Cho (1999b).
(36)
Just as before, for low Peclet numbers, all methods give good u = y + 0:5; v = x 0:5
results. Therefore, only the plots of φ with K = 10 6 or Pe =
106 and different flow directions are shown in Fig.23-Fig.25. and there is no source term f = 0. Along the external boundary
Here, the results for GFEM and MLPG are not shown due to φ = 0 and along the internal boundary (AB),
their very large oscillations. Two different sizes of radius of
1
)) along x = 0:5; 0 y 0:5
(37)
support (1:2h and 2:1h) have been used for MLPG2. Gen- φ = cos(2π(y
4
erally, MLPG2 and SUPG gives much better solutions than
GFEM and MLPG. Due to the internal discontinuity, it ap- See Fig.28 for the problem statement. The diffusivity is chosen
pears that a smaller size of support gives better solutions for as K = 10 6 and a uniform mesh or node distribution with 21 MLPG for Convection-Diffusion Problems
Figure 28 : Convection in a rotating flow field: problem statement.
59
to get nonoscillatory solutions. This property makes MLPG2
very flexible to deal with different problems, such as problems
with discontinuity. It is very easy to change the size of support
so that it is possible that, for smooth part of solutions, high
accuracy can be obtained, but for the discontinuous part, nonoscillatary solutions can be produced. Further research about
this, and about boundary conditions, is needed.
The MLPG method with upwinding introduced in this paper is
very general. It is very promising for more general fluid mechanics problems, such as Navier-Stokes equation, due to its
simplicity and meshlessness. Further results will be published
soon by the authors.
Acknowledgement: The support of this research by ONR
and NASA-Langley is sincerely appreciated. Useful discussion with Drs. Y.D.S. Rajapakse and I.S. Raju are also thank21 nodes is employed. Due to the internal boundary condition,
fully acknowledged.
the method proposed by Atluri, Kim, and Cho (1999b) is used
to cope with the boundary conditions. The solution for φ is
References
plotted in Fig.26-Fig.27 for different methods.
It shows that both GFEM and MLPG give oscillationary re- Almeida, R. C.; Silva, R. S. (1997): A stable petrov-galerkin
sults but SUPG and MLPG2 produce nonoscillatory solutions. method for convection-dominated problems. Comput. MethAgain, the effect of the size of support should be noted. For ods Appl. Mech. Engrg., vol. 140, pp. 291–304.
MLPG, when the size of support increases, the result becomes
Atluri, S. N. (1999): Truly meshless methods in compuworse due to the oscillation; however, for MLPG2, larger size
tational mechanics. In Aravas, N.; Katsikadelis, J. T.(Eds):
of support gives more accurate solutions and smaller size of
Proc. 3rd Greek National Congress on Computational Mesupport produces more spurious crosswind diffusion effect.
chanics, pp. 3–25, 24-26 June, 1999, Vol. 1.
Atluri, S. N.; Cho, J. Y.; Kim, H. (1999a): Analysis of
thin beams, using the meshless local petrov-galerkin method,
The MLPG method has been extended to solve the convection- with generalized moving least squares interpolations. Comdiffusion problems. Just as GFEM, MLPG works well for put. Mech., vol. 24, pp. 334–347.
low Peclet number flows, but is not good for high Peclet numAtluri, S. N.; Kim, H.; Cho, J. Y. (1999b): A critical asber flow. To deal with convection-dominated flow, two kinds
sessment of the truly meshless local petrov-galerkin (mlpg),
of natural upwinding schemes have been proposed. The Upand local boundary integral equation (lbie) methods. Comput.
winding Scheme I (US-I) is very flexible. 1D numerical tests
Mech., vol. 24, pp. 348–372.
show it is possible to design a good scheme by using this
idea. Both 1D and 2D examples show, however, that the Up- Atluri, S. N.; Zhu, T. (1998a): A new meshless local petrovwinding Scheme II (US-II) is very successful in dealing with galerkin (mlpg) approach in computational mechanics. Comconvection-dominated flows.
put. Mech., vol. 22, pp. 117–127.
Compared with SUPG, the computational cost of MLPG2 is Atluri, S. N.; Zhu, T. (1998b): A new meshless local petrovstill much higher, because cost-effective integration schemes galerkin (mlpg) approach to nonlinear problems in computer
are not yet found; but the concept of MLPG2 is very clear modeling and simulation. Computer Modeling and Simulation
and it is very easy to implement it for multi-dimensional flow in Engrg., vol. 3, pp. 187–196.
problems. Further, it may be argued that in a large-scale problem, the additional computational cost (due to the meshless Atluri, S. N.; Zhu, T. (2000): New concepts in meshless
integration) may be offset by the savings in human-resource methods. Int. J. Num. Meth. Eng., vol. 47, pp. 537–556.
cost (involved in gnerating the mesh) in the truly meshless
Babus̃ka, I.; Melenk, J. (1997):
The partition of unity
MLPG method. Numerical tests also show that, in general,
method. Int. J. Num. Meth. Eng., vol. 40, pp. 727–758.
MLPG2 gives better solutions than SUPG. Furthermore, numerical examples show that the size of support plays a signifi- Belytcshko, T.; Krongauz, Y.; Organ, D.; Fleming, M.;
cant role in MLPG2. Larger sizes of support result in better ac- Krysl, P. (1996): Meshless methods: an overview and recuracy for smooth solutions and smaller sizes of support pro- cent developments. Comput. Methods Appl. Mech. Engrg.,
duce larger spurious viscous effect, which sometimes is useful vol. 139, pp. 3–47.
6 Concluding Remarks
60
Copyright c 2000 Tech Science Press
CMES, vol.1, no.2, pp.45-60, 2000
Belytcshko, T.; Lu, Y. Y.; Gu, L. (1994):
Element-free Oñate, E.; Idelsohn, S.; Zienkiewicz, O. C.; Taylor, R. L.
galerkin methods. Int. J. Num. Meth. Eng., vol. 37, pp. 229– (1996): A finite point method in computational mechanics.
256.
application to convective transport and fluid flow. Int. J. Num.
Meth. Eng., vol. 39, pp. 3839–3866.
Brooks, A. N.; Hughes, T. J. R. (1982): Streamline upwind
petrov-galerkin formulations for convection-dominated flows Zhu, T.; Atluri, S. N. (1998): A modified collocation & a
with particular emphasis on the incompressible navier-stokes penalty formulation for enforcing the essential boundary conequations. Comput. Methods Appl. Mech. Engrg., vol. 32, pp. ditions in the element free galerkin method. Comput. Mech.,
vol. 21, pp. 211–222.
199–259.
Duarte, C. A.; Oden, J. T. (1996): An h-p adaptive method Zhu, T.; Zhang, J. D.; Atluri, S. N. (1998a): A local boundusing clouds. Comput. Methods Appl. Mech. Engrg., vol. 139, ary integral equation (lbie) method in computational mechanics, and a meshless discretization approach. Comput. Mech.,
pp. 237–262.
vol. 21, pp. 223–235.
Franca, L. P.; Frey, S. L.; Hughes, T. J. R. (1992): StaZhu, T.; Zhang, J. D.; Atluri, S. N. (1998b):
A meshbilized finite element methods: I. application to the advectiveless local boundary integral equation (lbie) method for solving
diffusive model. Comput. Methods Appl. Mech. Engrg., vol.
nonlinear problems. Comput. Mech., vol. 22, pp. 174–186.
92, pp. 253–276.
Hughes, T. J. R.; Brooks, A. (1979): A muti-dimensional
upwind scheme with no crosswind diffusion. In Hughes, T.
J. R.(Ed): Finite element methods for convection dominated
flows, AMD Vol. 34, ASME, New York.
Hughes, T. J. R.; Mallet, M.; Mizukami, A. (1986): A new
finite element formulation for computational fluid dynamics:
Ii. beyond supgs. Comput. Methods Appl. Mech. Engrg., vol.
54, pp. 341–355.
Kelly, D. W.; Nakazawa, S.; Zienkiewicz, O. C.; Heinrich,
J. C. (1980): A note of upwinding and anisotropic balancing
dissipation in finite element approximations to convective diffusion problems. Int. J. Num. Meth. Eng., vol. 15, pp. 1705–
1711.
Leonard, B. (1979): A survey of finite difference of opinion on numerical muddling of the incomprehensible defective
confusion confusion equation. In Hughes, T. J. R.(Ed): Finite
element methods for convection dominated flows, AMD Vol.
34, ASME, New York.
Liu, W. K.; Jun, S.; Sihling, D. T.; Chen, Y.; Hao, W.
(1997): Multiresolution reproducing kernel particle method
for computational fluid mechanics. Int. J. Num. Meth. Fluids,
vol. 24, pp. 1391–1415.
Liu, W. K.; Jun, S.; Zhang, Y. (1995): Reproducing kernel
particle methods. Int. J. Num. Meth. Fluids, vol. 20, pp. 1081–
1106.
Lucy (1977):
A numerical approach to the testing of the
fission hypothesis. The Astro. J., vol. 8, pp. 1013–1024.
Naroles, B.; Touzot, G.; Villon, P. (1992): Generalizing
the finite element method: diffuse approximation and diffuse
elements. Comput. Mech., vol. 10, pp. 307–318.