Meshless Local Petrov-Galerkin (MLPG) Method for ... · [O˜nate, Idelsohn, Zienkiewicz, and Taylor...

Meshless Local Petrov-Galerkin (MLPG) Method for Convection-DiffusionProblems

H. Lin, S.N. Atluri1

Abstract: Due to the very general nature of the MeshlessLocal Petrov-Galerkin (MLPG) method, it is very easy andnatural to introduce the upwinding concept (even in multi-dimensional cases) in the MLPG method, in order to deal withconvection-dominated flows. In this paper, several upwindingschemes are proposed, and applied to solve steady convection-diffusion problems, in one and two dimensions. Even for veryhigh Peclet number flows, the MLPG method, with upwind-ing, gives very good results. It shows that the MLPG method isvery promising to solve the convection-dominated flow prob-lems, and fluid mechanics problems.

keyword: MLPG, MLS, convection-dominated flow, up-winding.

1 Introduction

Although the finite element method (FEM) and the closelyrelated finite volume method (FVM) are well-established nu-merical techniques for computer modeling in engineering andsciences, they are not without shortcomings. First of all,their reliance on a mesh leads to complications for certainclasses of problems. The generation of good quality meshespresents significant difficulties in the analysis of engineeringsystems (especially in 3D). These difficulties can be over-come by the so-called meshless methods, which have at-tracted considerable interest over the past decade. A num-ber of meshless methods have been developed by different au-thors, such as Smooth Particle Hydrodynamics (SPH) [Lucy(1977)], Diffuse Element Method (DEM) [Naroles, Touzot,and Villon (1992)], Element Free Galerkin method (EFG) [Be-lytcshko, Lu, and Gu (1994)], Reproducing Kernel ParticleMethod (RKPM) [Liu, Jun, and Zhang (1995)], hp-cloudsmethod [Duarte and Oden (1996)], Finite Point Method (FPM)[Onate, Idelsohn, Zienkiewicz, and Taylor (1996)], Partitionof Unity Method (PUM) [Babuska and Melenk (1997)], LocalBoundary Integral Equation method (LBIE) [Zhu, Zhang, andAtluri (1998a,b)], Meshless Local Petrov-Galerkin method(MLPG) [Atluri and Zhu (1998a,b); Atluri (1999); Atluri andZhu (2000)]. Most of these methods, in reality, are not re-ally meshless method, since they use a background mesh forthe numerical integration of the weak form. To be a trulymeshless method, both interpolation and integration should be

1 Center for Aerospace Research and Education, 7704 Boelter Hall, Univer-sity of California, Los Angeles, CA 90095-1600

performed without a mesh. The finite point method (FPM)[Onate, Idelsohn, Zienkiewicz, and Taylor (1996)] is a trulymeshless method. A non-element interpolation scheme -weighted least squares (WLS) is used and there is no integra-tion required. However, this method is based on point colloca-tion, 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 meshlessmethods, because, a traditional non-overlapping, continguousmesh is not required, either for interpolation purpose or forintegration purpose. As pointed out in Atluri, Kim, and Cho(1999b), the LBIE approach can be treated simply as a specialcase of the MLPG scheme. The MLPG method is based on aweak form computed over a local sub-domain, which can beany simple geometry like a sphere, cube or ellipsoid in 3D. Thetrial and test function spaces can be different or the same. Itoffers a lot of flexibility to deal with different boundary valueproblems. A wide range of problems has been solved by Atluriand his coauthors. The objective of this paper is to extend theMLPG method to solve steady convection-diffusion problems.

In fluid mechanics, the existence of the convection term makesthe problem non-self-ajoint. A special treatment is needed tostabilize the numerical approximation for these kinds of prob-lems. Schemes related to upwinding are the most general tech-niques to stabilize Finite Difference Method (FDM), FEM andFVM. The same concept is needed in the meshless methods, soas to obtain a good accuracy for convection-dominated flows.

Only very few works were reported by using the so-calledmeshless methods, to solve convection-dominated flows. InOnate, Idelsohn, Zienkiewicz, and Taylor (1996), the FPMmethod was applied, with upwinding for the first derivativeor with characteristic approximation. However, there is a sig-nificant drawback of the FPM as discussed above, and, inmultidimensional cases, it is not easy to define the criticaldistance, which is important to stabilize the method and ob-tain good accuracy. In Liu, Jun, Sihling, Chen, and Hao(1997), the RKPM method, combined with the StreamlineUpwind/Petrov-Galerkin (SUPG) form of variational formu-lation, was used. As pointed out before, the RKPM method isnot a truly meshless method, since a background mesh is usedto integrate the weak form. In fact, a truly meshless method,such as the MLPG method, is much easier and more flexiblefor introducing the upwinding concept, with a very clear phys-ical meaning. This will be illustrated in this paper, and numeri-

46 Copyright c 2000 Tech Science Press CMES, vol.1, no.2, pp.45-60, 2000

cal examples for the steady convection-diffusion problems willbe used for verification purposes.

2 Local Weak Form

Let Ω be a bounded region in Rnsd , where nsd is the num-ber of space dimensions, and assume that Ω has a piecewisesmooth boundary Γ. Then, the governing equation for steadyconvection-diffusion problems in Ω can be written as:

v j∂φ∂x j

=∂

∂x j(K

∂φ∂x j

)+ f (1)

where, v j is the flow velocity, K is the diffusivity coefficient, fis a source term, and repeated indices are to be summed. Theboundary conditions are assumed to be:

Essential Boundary Conditions:

φ = φ on Γφ (2)

Natural Boundary Conditions:

K∂φ∂x j

n j = t on Γt (3)

where, φ and t are given, n j is the outward unit normal vectorto Γ, Γφ and Γt are subsets of Γ satisfying Γφ \Γt = /0 (theempty set) and Γφ[Γt = Γ.

Different from other meshless methods, the MLPG method isbased on a local weak form over a local sub-domain Ωs, whichis located entirely inside the global domain Ω. It is noted thatthe local sub-domain can be of an arbitrary shape containingthe node position x in question.

To satisfy Eq. (1) in a local sub-domain Ωs with a piecewisesmooth boundary Γs, Eq. (1) is weighted by a test functionw and integrated over the local sub-domain such that the localweighted residual equation can be written as:

ZΩs

[v j∂φ∂x j

∂∂x j

(K∂φ∂x j

) f ]w dΩ = 0 (4)

for all w. By using the integration by parts, Eq. (4) is recastinto a local weak form as:Z

Ωs

[v j∂φ∂x j

+K∂φ∂x j

∂w∂x j

f w]dΩZ

Γs

K∂φ∂x j

n jw dΓ = 0 (5)

for all continuous trial functions φ and continuous test func-tions w. In general, the boundary Γs of the local sub-domainΩs may intersect with the boundary of the global domain Ω.Therefore,

Γs = ΓsI [Γsφ[Γst (6)

where, ΓsI is the part of Γs which is inside the global domain,Γsφ = Γs\Γφ, and Γst = Γs\Γt .

The local weak form provides a very clear concept for a localnon-element integration, which does not need any backgroundintegration cells which are continguous over the entire domain.Also the MLPG method leads to a natural way to construct theglobal stiffness matrix through the integration over a local sub-domain. To solve this local weak form, some kind of meshlessinterpolation schemes is needed. This will be discussed in thenext section.

3 The MLS approximation scheme

In order to preserve the local character of the numerical im-plementation, a meshless method uses a local interpolation orapproximation to represent the trial/test functions with the val-ues (or the fictitious values) of the unknown variable at somerandomly located nodes. There are a lot of local interpola-tion schemes, such as MLS, PUM, RKPM, hp-clouds, Shepardfunction, etc., available to achieve this aim.

The Moving Least Square (MLS) method is generally consid-ered as one of the schemes to interpolate data with a reasonableaccuracy. Therefore, the MLS scheme is chosen in this paper.

Consider the approximation of a function u(x) in a domainΩ with a number of scattered nodes fxig, i = 1;2; : : : ;n, themoving least-square approximant uh(x) of u(x), 8x 2 Ω, canbe defined by

uh(x) = pT (x)a(x) 8x 2 Ω (7)

where, pT (x) = [p1(x); p2(x); : : : ; pm(x)] is a complete mono-mial basis of order m, which, for example, can be chosen aslinear:

pT (x) = [1;x;y]; m = 3; (8)

or quadratic:

pT (x) = [1;x;y;x2;xy;y2]; m = 6 (9)

for 2D problems, and a(x) is a vector containing coefficientsa j(x), j = 1;2; : : : ;m which are functions of the space coordi-nates x, and determined by minimizing a weighted discrete L2

norm, defined as:

J(x) =n

∑i=1

wi(x)[pT (xi)a(x) ui]2

= [P a(x) u]T W[P a(x) u] (10)

where wi(x) is the weight function associated with the nodei, with wi(x) > 0 for all x in the support of wi(x), xi denotesthe value of x at node i, n is the number of nodes in Ω forwhich the weight functions wi(x) > 0, the matrices P and Ware defined as

P =

2664

pT (x1)pT (x2)

pT (xn)

3775

nm

(11)

MLPG for Convection-Diffusion Problems 47

W =

24 w1(x) 0

0 wn(x)

35 (12)

and

uT = [u1; u2; : : : ; un] (13)

where, ui; i = 1;2; : : : ;n are the fictitious nodal values and notthe nodal values of the unknown trial function uh(x) in general.

The stationarity of J in Eq. (10) with respect to a(x) leads tothe following relation between a(x) and u:

A(x)a(x) = B(x)u (14)

where the matrices A(x) and B(x) are defined by

A(x) = PT WP =n

∑i=1

wi(x)p(xi)pT (xi) (15)

B(x) = PT W = [w1(x)p(x1);w2(x)p(x2); : : : ;

wn(x)p(xn)] (16)

Solving this for a(x) and substituting it into Eq. (7), we get theMLS approximation as

uh(x) = ΦT (x) u =n

∑i=1

φi(x)ui 8x 2 Ω (17)

where, the nodal shape function corresponding to nodal pointxi is given by

ΦT (x) = pT (x)A1(x)B(x) (18)

It should be noted that the MLS approximation is well definedonly when the matrix A in Eq. (14) is non-singular. It can beseen that this is the case if and only if the rank of P equals m.A necessary condition for a well-defined MLS approximationis that at least m weight functions are non-zero (i.e. n m) foreach sample point x 2 Ω and that the nodes in Ω will not bearranged in a special pattern such as on a straight line.

The partial derivatives of φi(x) can be obtained as

φi;k =m

∑j=1

[p j;k(A1B) ji+ p j(A1B

;k +A1;k B) ji] (19)

in which A1;k is given by

A1;k = A1A

;kA1 (20)

and the index following a comma indicates a spatial derivative.

It is known that the smoothness of the shape functions φi(x)is determined by that of the basis functions and of the weightfunctions. Let Ck(Ω) be the space of k-th continuously dif-ferentiable functions. If wi(x) 2 Ck(Ω); i = 1;2; : : : ;n and

p j(x) 2 Cl(Ω); j = 1;2; : : : ;m, then φi(x) 2 Cr(Ω) with r =min(k; l). A number of choices are available for the basis func-tions and the weight functions. In this paper, the linear basisis chosen and a spline weight function as in Atluri and Zhu(1998a) is used:

wi(x) =

16( diri)2 +8( di

ri)33( di

ri)4 0 di ri

0 di ri(21)

where di = jxxij is the distance from node xi to point x, andri is the size of the support for the weight function wi. It canbe easily seen that the spline weight function is C1 continuousover the entire domain.

From the above discussion, it shows that the MLS shape func-tions don’t have the Kronecker delta property. This causes thedifficulty to impose the essential boundary conditions. Severalmethods have been proposed, e.g., see Belytcshko, Krongauz,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-tions.

4 Upwinding Schemes

4.1 Overview

It is well known that convection-dominated flows are some ofthe most difficult problems to solve numerically. The pres-ence of the convection term causes serious numerical difficul-ties, appearing in the form of “wiggles” (oscillatory solutions),when the convection term is dominant. This problem could besolved somewhat heuristicaly, by using upwinding. A numberof upwinding schemes has been developed, for FDM, FEMand FVM. In the one dimensional cases, optimal upwindingschemes may be designed so as to result in exact nodal solu-tions. However, in the multidimesional cases, generalizationsof traditional upwinding schemes were unsuccessful due to thecrosswind diffusion problem (see Fig.1-Fig.2).

It was apparent that the upwind effect, arrived at by whatevermeans, was needed only in the direction of flow. However, Itis not easy to design such methods for multidimensional cases.Hughes and Brooks (1979) introduced the ‘Streamline Up-wind (SU) method’, where the artificial diffusion operator isconstructed to act only in the flow direction, a priori eliminat-ing the possibility of any crosswind diffusion. A similar ideawas described in Kelly, Nakazawa, Zienkiewicz, and Hein-rich (1980) as ‘anisotropic balancing dissipation’. As shownin Hughes and Brooks (1979), one could obtain much bettersolutions for the crosswind problem by using the streamlineupwind method, but several deficiencies remained. The up-winded convection term was not consistent with the centrallyweighted source and transient terms, resulting in excessivelydiffuse solutions when these terms were present (see Fig.3-Fig.4).


Figure 1 : Illustration of the crosswind diffusion problem withPe = 100 (from Leonard (1979)): (a) Exact.

Figure 2 : Illustration of the crosswind diffusion problem withPe = 100 (from Leonard (1979)): (b) Optimal Upwinding.

(a)

1

0

−.5

f(x)

x

u (const)

φ=0

0 15

Figure 3 : Pure convection with a source term: (a) ProblemStatement.

(b)

x

φ

0 5 10 150

1

2EXACTSUSUPG

Figure 4 : Pure convection with a source term: (b) Results forSU and SUPG.

Clearly, upwind weighting of all terms in the equation wasneeded, i.e., some kind of Petrov-Galerkin method is neededfor consistency purposes. Therfore, in order to solve thecrosswind problem and be consistent with all terms, more re-fined versions of the upwinding concept should be based onPetrov-Galerkin methods, which utilize upwinding only in thestreamline direction. As known, most popular among suchmethods is the Streamline Upwind Petrov-Galerkin method(SUPG)[Brooks and Hughes (1982)], which consistently in-troduces an additional stability term in the upwind direction.This method has better stability and accuracy properties thanthe standard Galerkin formulation for convection-dominatedflows. However, some drawbacks still remain. The choiceof the related parameters is still not so straightfoward (anotherdefinition 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 containingsharp layers. In order to prevent those ‘wiggles’, many meth-ods have been proposed by adding some kind of discontinuity-capturing perturbation term (see e.g. Hughes, Mallet, andMizukami (1986), Almeida and Silva (1997) and referencestherein).

For meshless methods, the same kind of consideration shouldbe taken to deal with convection-dominated flows. As pointedout in the introduction, the very general nature of truly mesh-less methods, such as the MLPG method, makes it easier tointroduce the upwind concept more clearly and effectively. Inthis paper, two upwinding schemes for the MLPG method areproposed in the following.

4.2 MLPG Upwinding Scheme I

The MLPG method is based on the Petrov-Galerkin weighted-residual procedures. Different spaces for the test and trialfunctions can be used, as shown in Fig.5.


Figure 5 : The MLPG method without Upwinding.

Figure 6 : MLPG Upwinding Scheme I (US-I).

Therefore one of the very natural ways to construct upwindingschemes is to choose different trial and test functions. Thiscan be done by a lot of ways. For example, in order to applyupwinding in the streamline direction, we can skew the testfunction opposite to the streamline direction as shown in Fig.6.For convenience, we denote this as Upwinding Scheme I (US-I).

As an illustration, we choose a skewed weight function as thetest function. The skewed weight function is given as follows:using the same form of weight function as in Eq. (21), we shiftthe position of the maximum of wi(x) from xi to xi γrisi, asshown 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 isgiven by

γ =12

coth(Pe2) 1

Pe(22)

Figure 7 : MLPG Upwinding Scheme I (US-I): Specification.

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 atxi, and the local sub-domain at xi is coincided with the supportfor the test functions at xi.

4.3 MLPG Upwinding Scheme II

In fact, because the MLPG method is based on a local weakform over a local sub-domain, there is another very conve-nient way to design upwinding schemes. We can use theBubnov-Galerkin procedure to discretize the local weak form,as done in Atluri, Kim, and Cho (1999b), and shift the localsub-domain opposite to the streamline direction, as shown inFig.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 em-ployed. In this case, the local sub-domain at xi is no longercoincident with the support for the test functions at xi, but thesize is the same. (It should be noted that in the usual MLPGmethod, we usually choose the test functions such that the in-tegration term along the boundary ΓsI equals to zero, but, ingeneral, this is not true for the MLPG method with US-II.Therefore, in the local weak form, the integration term alongthe boundary ΓsI should be retained.)

In particular, the shifting distance of the local sub-domain can


Figure 9 : MLPG Upwinding Scheme II (US-II): Specifica-tion.

be specified as γri, where, ri is the size of the support for thetest functions, which is equal to the size of the local domain,at xi, and γ is given by

γ = coth(Pe2) 2

Pe(24)

in which Pe is a local Peclet number, defined as:

Pe =2uri

K(25)

The direction of the shifting is opposite to the streamline di-rection si at xi, as shown in Fig.9.

5 Numerical examples

To assess the effectiveness of the methods described herein,a series of numerical calculations is performed. For con-venience, we note the MLPG method without upwinding asMLPG, the MLPG method with US-I as MLPG1 and theMLPG method with US-II as MLPG2. Here, the MLPGmethod without upwinding is based on the Galerkin procedureto approximate the local weak form and the local sub-domaincoincides with the support of the test functions. Examples for1D and 2D problems are given in the following.

5.1 One-dimensional problems

For 1D problems, Eq. (1) can be written as:

udφdx

Kd2φdx2 f = 0 (26)

First, reconsider the problem as shown in Fig.3. Here, K =

0, u = 1 and f =

8<:

1 14 x; (0 < x 6)

14 x2; (6 < x 8)0; (8 < x 15)

The results for

MLPG1 and MLPG2 are shown in Fig.10. It shows that bothMLPG1 and MLPG2 give very good solutions.

Then we consider another problem, with the domain 0 x 1.Two different cases are considered:

(c)

x

φ

0 5 10 150

1

2EXACTMLPG1MLPG2

Figure 10 : Pure convection with a source term: (c) Resultsfor MLPG1 and MLPG2.

Case 1:

K = 1; f = 0;φ = 0 at x = 0;φ = 1 at x = 1:

(27)

Case 2:

K = 1; f = 100;φ = 0 at x = 0;φ = 0 at x = 1:

(28)

u may have different values, leading to different Peclet num-bers Pe, which is defined as:

Pe =uLK

(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) forradius of support are used to solve these problems. Penaltymethod is chosen to deal with essential boundary conditionsdue to its simplicity. Results for the value of φ are shown inFig.11-Fig.12.

These results show that both MLPG1 and MLPG2 producevery good solutions when Peclet number is very high, how-ever, the MLPG method without upwinding gives oscillation-ary solutions when Peclet number becomes large. For lowPeclet number, all methods get good results. It also showsthat, in general, MLPG2 gives better solutions than MLPG1.It can be said that the choice for the test function in MLPG1is not the best one and more care should be taken. So, for 2Dproblems tested in the following, we will only use the MLPG2,i.e., Upwinding Scheme II.


x

φ

0 0.25 0.5 0.75 10

0.25

0.5

0.75

1

EXACTMLPGMLPG1MLPG2

Pe=1

x

φ

0 0.25 0.5 0.75 1

0

0.25

0.5

0.75

1

EXACTMLPGMLPG1MLPG2

Pe=10

x

φ

0 0.25 0.5 0.75 1-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

EXACTMLPGMLPG1MLPG2

Pe=100

x

φ

0 0.25 0.5 0.75 1-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

EXACTMLPGMLPG1MLPG2

Pe=1000

MLPG(Galerkin)

Figure 11 : Case 1 for 1D problems.

x

φ

0 0.25 0.5 0.75 10

3

6

9

12

EXACTMLPGMLPG1MLPG2

Pe=1

x

φ

0 0.25 0.5 0.75 10

2

4

6

EXACTMLPGMLPG1MLPG2

Pe=10

x

φ

0 0.25 0.5 0.75 1

0

0.5

1

1.5 EXACTMLPGMLPG1MLPG2

Pe=100

x

φ

0 0.25 0.5 0.75 1-0.05

0

0.05

0.1

0.15

0.2

EXACTMLPGMLPG1MLPG2

Pe=1000

MLPG(Galerkin)

Figure 12 : Case 2 for 1D problems.


Figure 13 : Convection skew to the mesh-I: problem state-ment.

5.2 Two-dimensional problems

For 2D problems, Eq. (1) can be written as:

u∂φ∂x

+ v∂φ∂yK

∂2φ∂x2 K

∂2φ∂y2 f = 0 (30)

In this section, several cases are considered in a unit square do-main given by 0 x;y 1. These problems have been widlystudied in literature, e.g., see Hughes, Mallet, and Mizukami(1986) and Almeida and Silva (1997), as tests for the accu-racy of various numerical schemes. The first two cases areused to assess the performance of the MLPG and MLPG2methods for different Peclet numbers. The last three cases in-volve very high Peclet numbers, and are used to assess solu-tions which are essentially of pure convection flows. For thepurpose of comparison, the standard Galerkin finite elementmethod (GFEM) and the streamline-upwind Petrov-Galerkinmethod (SUPG) have been included in the results. Linear fi-nite elements are used for GFEM and SUPG. The parametersfor SUPG are chosen as those in Brooks and Hughes (1982).Except for the last case, a regular mesh (SUPG) or node distri-bution (MLPG) with 1111 nodes is used. Unless specified,the value of 2:1 times nodal distance h (2:1h) for radius ofsupport is used.

5.2.1 Convection skew to the mesh-I

In this case, the source term is assumed to be zero. The flow isunidirectional and constant with velocity components:

u = cos(π=4); v = sin(π=4): (31)

See Fig.13 for the problem statement.

X

Y

PHI0.93750.8750.81250.750.68750.6250.56250.50.43750.3750.31250.250.18750.1250.0625

GFEM

X

Y

PHI0.93750.8750.81250.750.68750.6250.56250.50.43750.3750.31250.250.18750.1250.0625

SUPG

X

Y

PHI0.9374960.8749920.8124890.7499850.6874810.6249770.5624730.4999690.4374660.3749620.3124580.2499540.187450.1249470.0624428

MLPG (No Upwinding)

X

Y

PHI0.9374840.8749680.8124520.7499360.687420.6249040.5623880.4998720.4373560.374840.3123240.2498080.1872920.1247770.0622606

MLPG2

Figure 14 : Convection skew to the mesh-I: Pe = 1.


X

Y

PHI1.829061.707121.585191.463251.341311.219371.097440.97550.8535620.7316250.6096870.487750.3658120.2438750.121937

GFEM

X

Y

PHI0.93750.8750.81250.750.68750.6250.56250.50.43750.3750.31250.250.18750.1250.0625

SUPG

X

Y

PHI1.054640.9842760.9139150.8435530.7731910.7028290.6324680.5621060.4917440.4213820.3510210.2806590.2102970.1399350.0695736

MLPG (No Upwinding)

X

Y

PHI0.9351890.8707770.8063660.7419550.6775440.6131320.5487210.484310.4198990.3554870.2910760.2266650.1622540.09784250.0334312

MLPG2


X

Y

PHI2951.092738.192525.282312.382099.471886.561673.661460.751247.841034.94822.031609.125396.219183.313

-29.5938

GFEM

X

Y

PHI1.001250.93450.867750.8010.734250.66750.600750.5340.467250.40050.333750.2670.200250.13350.06675

SUPG

X

Y

PHI2.732062.543132.354192.165251.976311.787381.598441.40951.220561.031630.8426880.653750.4648130.2758750.0869375

MLPG (No Upwinding)

X

Y

PHI1.092041.016080.9401240.8641650.7882060.7122470.6362890.560330.4843710.4084120.3324540.2564950.1805360.1045770.0286188

MLPG2



X

Y

PHI0.06900940.06440880.05980810.05520750.05060690.04600630.04140560.0368050.03220440.02760380.02300310.01840250.01380190.009201250.00460063

GFEM

X

Y

PHI0.06898120.06438250.05978370.0551850.05058620.04598750.04138870.036790.03219120.02759250.02299370.0183950.01379620.00919750.00459875

SUPG

X

Y

PHI0.0674780.0629760.0584740.05397190.04946990.04496790.04046590.03596390.03146190.02695980.02245780.01795580.01345380.008951790.00444977

MLPG (No Upwinding)

X

Y

PHI0.06876470.06416940.05957410.05497880.05038350.04578820.04119290.03659760.03200230.0274070.02281170.01821640.01362110.00902580.0044305

MLPG2

Figure 17 : Convection with a source term-I: Pe = 1.

X

Y

PHI1.496251.39651.296751.1971.097250.99750.897750.7980.698250.59850.498750.3990.299250.19950.09975

GFEM

X

Y

PHI0.8438440.7875870.7313310.6750750.6188190.5625620.5063060.450050.3937940.3375370.2812810.2250250.1687690.1125120.0562562

SUPG

X

Y

PHI0.8824210.8235420.7646630.7057840.6469050.5880260.5291460.4702670.4113880.3525090.293630.2347510.1758720.1169930.0581139

MLPG (No upwinding)

X

Y

PHI0.7982510.7440030.6897540.6355050.5812560.5270080.4727590.418510.3642610.3100130.2557640.2015150.1472660.09301750.0387688

MLPG2



X

Y

PHI4692.194379.374066.563753.753440.943128.122815.312502.52189.691876.871564.061251.25938.437625.624312.812

GFEM

X

Y

PHI1.069690.9983750.9270630.855750.7844380.7131250.6418130.57050.4991880.4278750.3565630.285250.2139380.1426250.0713125

SUPG

X

Y

PHI2.615462.391912.168371.944821.721281.497741.274191.050650.8271060.6035630.3800190.156475

-0.0670687-0.290612-0.514156

MLPG (No Upwinding)

X

Y

PHI0.8155390.7593770.7032160.6470550.5908940.5347320.4785710.422410.3662490.3100870.2539260.1977650.1416040.08544250.0292812

MLPG2


X

Y

PHI0.4728490.4384980.4041460.3697950.3354440.3010930.2667410.232390.1980390.1636880.1293360.0949850.06063380.0262825

-0.00806875

GFEM

X

Y

PHI0.5942810.5546620.5150440.4754250.4358060.3961870.3565690.316950.2773310.2377120.1980940.1584750.1188560.07923750.0396187

SUPG

X

Y

PHI0.5405190.5044370.4683560.4322740.3961930.3601120.324030.2879490.2518670.2157860.1797040.1436230.1075420.07146020.0353788

MLPG (No Upwinding)

X

Y

PHI0.4660770.4349550.4038320.372710.3415870.3104640.2793420.2482190.2170960.1859740.1548510.1237290.09260590.06148330.0303606

MLPG2

Figure 20 : Convection with a source term-II: Pe = 106.


(b)

y

φ

0 0.2 0.4 0.6 0.8 1-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

MLPG2

Figure 21 : Convection with a source term-II (Pe = 108): (a)results from Almeida and Silva (1997); (b) results for MLPG2.

K is varied in order to get different Peclet number Pe. Here,Pe is given by

Pe =jjujjL

K(32)

where, jjujj=p

u2 + v2. In this case, jjujj= 1 and L = 1, soPe = 1=K. It has smooth boundary conditions, so the penaltymethod is used for simplicity. Contours for φ are shown inFig.14-Fig.16.

As in 1D problems, all methods give good results for lowPeclet number. But, for high Peclet numbers, MLPG2 andSUPG are much superior to MLPG and GFEM. It also showsthat MLPG2 is somewhat better than the currently most popu-lar method, the SUPG.

Figure 22 : Convection skew to the mesh-II: problem state-ment.

5.2.2 Convection with a source term-I

The problem is given by

u = 1; v = 0; f = 1;φ = 0 along the boundary.

(33)

K is also varied so as to get different Peclet number Pe, whichis equal to 1=K. Again, the penalty method is used to dealwith the boundary conditions. Contours for φ are shown inFig.17-Fig.19.

As before, all methods give good results for low Peclet num-ber. But, for high Peclet numbers, MLPG2 and SUPG aremuch superior to MLPG and GFEM. Furthermore, it showsthat MLPG2 gives better solutions than the SUPG, whenPeclet number is very high.

5.2.3 Convection with a source term-II

The problem is given by

u = 1; v = 0; f =

1; (0 < x 1

2 )1; ( 1

2 < x < 1);

φ = 0 along the boundary.(34)

The diffusivity K is 106 or Peclet number Pe is 106. Again,the penalty method is used to deal with the boundary condi-tions. Contours for φ are shown in Fig.20.

It shows that MLPG2 and SUPG are better than GFEM andMLPG, and MLPG2 is superior to SUPG. Actually, the so-lutions of MLPG2 are comparable (even better) to the resultsproduced by Almeida and Silva (1997) (see Fig.21).

However, Almeida and Silva (1997) introduced a very compli-cated modification for SUPG of Hughes and his co-workers,to achieve better results.


X

Y

ZSUPG

X

Y

ZMLPG2 (r=1.2h)

X

Y

ZMLPG2 (r=2.1h)

Figure 23 : Convection skew to the mesh-II: Pe = 106 and θ = π8 .

X

Y

ZSUPG

X

Y

ZMLPG2 (r=1.2h)

X

Y

ZMLPG2 (r=2.1h)

Figure 24 : Convection skew to the mesh-II: Pe = 106 and θ = π4 .

X

Y

ZSUPG

X

Y

ZMLPG2 (r=1.2h)

X

Y

ZMLPG2 (r=2.1h)

Figure 25 : Convection skew to the mesh-II: Pe = 106 and θ = 3π8 .


Y

X

ZGFEM

Y

X

ZMLPG(r=1.2h)(No Upwinding) Y

X

ZMLPG(r=2.1h)(No Upwinding)

Figure 26 : Convection in a rotating flow field: K = 106 (Continued).

Y

X

ZSUPG

Y

X

ZMLPG2(r=1.2h)

Y

X

ZMLPG2(r=2.1h)

Figure 27 : Convection in a rotating flow field: K = 106.

5.2.4 Convection skew to the mesh-II

Now we consider another case, which involves discontinuousboundary conditions and causes not only the sharp boundarylayer but also an internal sharp layer. In this case, the sourceterm is assumed to zero. The flow is unidirectional and con-stant with velocity components:

u = cosθ; v = sinθ: (35)

where, θ is the angle between the flow direction and the posi-tive x direction. See Fig.22 for the problem statement.

Due to the discontinuous boundary conditions, it is difficultto impose the essential boundary conditions by using penaltymethod. (Actually it results in oscillation at boundary.) So weuse the method developed by Atluri, Kim, and Cho (1999b).Just as before, for low Peclet numbers, all methods give goodresults. Therefore, only the plots of φ with K = 106 or Pe =106 and different flow directions are shown in Fig.23-Fig.25.

Here, the results for GFEM and MLPG are not shown due totheir very large oscillations. Two different sizes of radius ofsupport (1:2h and 2:1h) have been used for MLPG2. Gen-erally, MLPG2 and SUPG gives much better solutions thanGFEM and MLPG. Due to the internal discontinuity, it ap-pears that a smaller size of support gives better solutions for

MLPG2. This can be explained as following: as pointed outin Atluri, Cho, and Kim (1999a), larger size of support givesbetter accuracy for smooth solutions because it includes morepoints and works like higher order schemes, but when thesolution is discontinuous, it is well known that higher orderschemes produce more oscillatory results. Again,it seems thatthe MLPG2 with small size of support gives somewhat bettersolutions than SUPG at the internal discontinuity, but at theboundary, MLPG2 gives somewhat worse results than SUPG.This may be caused by the method of imposing boundary con-ditions, which is a very strong constraint.

5.2.5 Convection in a rotating flow field

In this case, the flow velocity components are given by

u =y+0:5; v = x0:5 (36)

and there is no source term f = 0. Along the external boundaryφ = 0 and along the internal boundary (AB),

φ = cos(2π(y 14)) along x = 0:5;0 y 0:5 (37)

See Fig.28 for the problem statement. The diffusivity is chosenas K = 106 and a uniform mesh or node distribution with 21


Figure 28 : Convection in a rotating flow field: problem state-ment.

21 nodes is employed. Due to the internal boundary condition,the method proposed by Atluri, Kim, and Cho (1999b) is usedto cope with the boundary conditions. The solution for φ isplotted in Fig.26-Fig.27 for different methods.

It shows that both GFEM and MLPG give oscillationary re-sults but SUPG and MLPG2 produce nonoscillatory solutions.Again, the effect of the size of support should be noted. ForMLPG, when the size of support increases, the result becomesworse due to the oscillation; however, for MLPG2, larger sizeof support gives more accurate solutions and smaller size ofsupport produces more spurious crosswind diffusion effect.

6 Concluding Remarks

The MLPG method has been extended to solve the convection-diffusion problems. Just as GFEM, MLPG works well forlow Peclet number flows, but is not good for high Peclet num-ber flow. To deal with convection-dominated flow, two kindsof natural upwinding schemes have been proposed. The Up-winding Scheme I (US-I) is very flexible. 1D numerical testsshow it is possible to design a good scheme by using thisidea. Both 1D and 2D examples show, however, that the Up-winding Scheme II (US-II) is very successful in dealing withconvection-dominated flows.

Compared with SUPG, the computational cost of MLPG2 isstill much higher, because cost-effective integration schemesare not yet found; but the concept of MLPG2 is very clearand it is very easy to implement it for multi-dimensional flowproblems. Further, it may be argued that in a large-scale prob-lem, the additional computational cost (due to the meshlessintegration) may be offset by the savings in human-resourcecost (involved in gnerating the mesh) in the truly meshlessMLPG method. Numerical tests also show that, in general,MLPG2 gives better solutions than SUPG. Furthermore, nu-merical examples show that the size of support plays a signifi-cant role in MLPG2. Larger sizes of support result in better ac-curacy for smooth solutions and smaller sizes of support pro-duce larger spurious viscous effect, which sometimes is useful

to get nonoscillatory solutions. This property makes MLPG2very flexible to deal with different problems, such as problemswith discontinuity. It is very easy to change the size of supportso that it is possible that, for smooth part of solutions, highaccuracy can be obtained, but for the discontinuous part, non-oscillatary solutions can be produced. Further research aboutthis, and about boundary conditions, is needed.

The MLPG method with upwinding introduced in this paper isvery general. It is very promising for more general fluid me-chanics problems, such as Navier-Stokes equation, due to itssimplicity and meshlessness. Further results will be publishedsoon by the authors.

Acknowledgement: The support of this research by ONRand NASA-Langley is sincerely appreciated. Useful discus-sion with Drs. Y.D.S. Rajapakse and I.S. Raju are also thank-fully acknowledged.

References

Almeida, R. C.; Silva, R. S. (1997): A stable petrov-galerkinmethod for convection-dominated problems. Comput. Meth-ods Appl. Mech. Engrg., vol. 140, pp. 291–304.

Atluri, S. N. (1999): Truly meshless methods in compu-tational mechanics. In Aravas, N.; Katsikadelis, J. T.(Eds):Proc. 3rd Greek National Congress on Computational Me-chanics, pp. 3–25, 24-26 June, 1999, Vol. 1.

Atluri, S. N.; Cho, J. Y.; Kim, H. (1999a): Analysis ofthin beams, using the meshless local petrov-galerkin method,with generalized moving least squares interpolations. Com-put. Mech., vol. 24, pp. 334–347.

Atluri, S. N.; Kim, H.; Cho, J. Y. (1999b): A critical as-sessment of the truly meshless local petrov-galerkin (mlpg),and local boundary integral equation (lbie) methods. Comput.Mech., vol. 24, pp. 348–372.

Atluri, S. N.; Zhu, T. (1998a): A new meshless local petrov-galerkin (mlpg) approach in computational mechanics. Com-put. Mech., vol. 22, pp. 117–127.

Atluri, S. N.; Zhu, T. (1998b): A new meshless local petrov-galerkin (mlpg) approach to nonlinear problems in computermodeling and simulation. Computer Modeling and Simulationin Engrg., vol. 3, pp. 187–196.

Atluri, S. N.; Zhu, T. (2000): New concepts in meshlessmethods. Int. J. Num. Meth. Eng., vol. 47, pp. 537–556.

Babuska, I.; Melenk, J. (1997): The partition of unitymethod. Int. J. Num. Meth. Eng., vol. 40, pp. 727–758.

Belytcshko, T.; Krongauz, Y.; Organ, D.; Fleming, M.;Krysl, P. (1996): Meshless methods: an overview and re-cent developments. Comput. Methods Appl. Mech. Engrg.,vol. 139, pp. 3–47.


Belytcshko, T.; Lu, Y. Y.; Gu, L. (1994): Element-freegalerkin methods. Int. J. Num. Meth. Eng., vol. 37, pp. 229–256.

Brooks, A. N.; Hughes, T. J. R. (1982): Streamline upwindpetrov-galerkin formulations for convection-dominated flowswith particular emphasis on the incompressible navier-stokesequations. Comput. Methods Appl. Mech. Engrg., vol. 32, pp.199–259.

Duarte, C. A.; Oden, J. T. (1996): An h-p adaptive methodusing clouds. Comput. Methods Appl. Mech. Engrg., vol. 139,pp. 237–262.

Franca, L. P.; Frey, S. L.; Hughes, T. J. R. (1992): Sta-bilized finite element methods: I. application to the advective-diffusive model. Comput. Methods Appl. Mech. Engrg., vol.92, pp. 253–276.

Hughes, T. J. R.; Brooks, A. (1979): A muti-dimensionalupwind scheme with no crosswind diffusion. In Hughes, T.J. R.(Ed): Finite element methods for convection dominatedflows, AMD Vol. 34, ASME, New York.

Hughes, T. J. R.; Mallet, M.; Mizukami, A. (1986): A newfinite 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 balancingdissipation in finite element approximations to convective dif-fusion problems. Int. J. Num. Meth. Eng., vol. 15, pp. 1705–1711.

Leonard, B. (1979): A survey of finite difference of opin-ion on numerical muddling of the incomprehensible defectiveconfusion confusion equation. In Hughes, T. J. R.(Ed): Finiteelement 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 methodfor computational fluid mechanics. Int. J. Num. Meth. Fluids,vol. 24, pp. 1391–1415.

Liu, W. K.; Jun, S.; Zhang, Y. (1995): Reproducing kernelparticle methods. Int. J. Num. Meth. Fluids, vol. 20, pp. 1081–1106.

Lucy (1977): A numerical approach to the testing of thefission hypothesis. The Astro. J., vol. 8, pp. 1013–1024.

Naroles, B.; Touzot, G.; Villon, P. (1992): Generalizingthe finite element method: diffuse approximation and diffuseelements. Comput. Mech., vol. 10, pp. 307–318.

Onate, E.; Idelsohn, S.; Zienkiewicz, O. C.; Taylor, R. L.(1996): A finite point method in computational mechanics.application to convective transport and fluid flow. Int. J. Num.Meth. Eng., vol. 39, pp. 3839–3866.

Zhu, T.; Atluri, S. N. (1998): A modified collocation & apenalty formulation for enforcing the essential boundary con-ditions in the element free galerkin method. Comput. Mech.,vol. 21, pp. 211–222.

Zhu, T.; Zhang, J. D.; Atluri, S. N. (1998a): A local bound-ary integral equation (lbie) method in computational mechan-ics, and a meshless discretization approach. Comput. Mech.,vol. 21, pp. 223–235.

Zhu, T.; Zhang, J. D.; Atluri, S. N. (1998b): A mesh-less local boundary integral equation (lbie) method for solvingnonlinear problems. Comput. Mech., vol. 22, pp. 174–186.

Meshless Local Petrov-Galerkin (MLPG) Method for ... · [O˜nate, Idelsohn, Zienkiewicz, and Taylor...

Documents

Transcript of Meshless Local Petrov-Galerkin (MLPG) Method for ... · [O˜nate, Idelsohn, Zienkiewicz, and Taylor...