Research Article A Pressure-Stabilized Lagrange-Galerkin...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 161873 13 pageshttpdxdoiorg1011552013161873

Research ArticleA Pressure-Stabilized Lagrange-Galerkin Method in a ParallelDomain Decomposition System

Qinghe Yao and Qingyong Zhu

School of Engineering Sun Yat-Sen University 510275 Guangzhou China

Correspondence should be addressed to Qingyong Zhu mcszqymailsysueducn

Received 30 April 2013 Accepted 14 June 2013

Academic Editor Santanu Saha Ray

Copyright copy 2013 Q Yao and Q Zhu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A pressure-stabilized Lagrange-Galerkin method is implemented in a parallel domain decomposition system in this work and thenew stabilization strategy is proved to be effective for large Reynolds number and Rayleigh number simulations The symmetry ofthe stiffness matrix enables the interface problems of the linear system to be solved by the preconditioned conjugate method andan incomplete balanced domain preconditioner is applied to the flow-thermal coupled problems The methodology shows goodparallel efficiency and high numerical scalability and the new solver is validated by comparing with exact solutions and availablebenchmark results It occupies less memory than classical product-type solvers furthermore it is capable of solving problems ofover 30 million degrees of freedom within one day on a PC cluster of 80 cores

1 Introduction

The Lagrange-Galerkin method raises wide concern aboutthe finite-element simulation of fluid dynamics Based on theapproximation of the material derivative along the trajectoryof fluid particle the method is natural in the simulation tophysical phenomena and it is demonstrated to be uncondi-tionally stable for a wide class of problems [1ndash5] A number ofresearches about the Lagrange-Galerkinmethodwere done inthe case of single processor element (PE) (cf [6ndash8]) the sym-metry of the matrices and good stability of the scheme werereported using a numerical integration based on a division ofeach element Rui and Tabata [9] developed a second schemefor convection-diffusion problem Massarotti et al [10] useda second-order characteristic curve method and a specialiteration was used to keep the symmetry of the stiffnessmatrixThe Lagrange-Galerkin method uses an implicit timediscretization and therefore an element searching algorithmis necessary to implement it The element searching maybecome very expensive when the geometry is complicated orthe mesh size is very small Due to its doubtable efficiencyand feasibility for complex simulations in the case of singlePE rare research has been done to implement it in parallel bywhich the enormous computation power enables us to solvemore challenging simulation problems

The present study is concentrated on improving thesolvability of the Lagrange-Galerkin method on large scaleand complex problems by domain decompositions Piecewiselinear interpolations are thus employed for velocity pressureand temperature therefore the so-called inf-sup condition[11] should be satisfied which is the first difficulty to beovercome in this work Stabilization methods for incom-pressible flow problems were reported by many researchers(cf [12ndash15]) Park and Sung proposed a stabilization forRayleigh-Benard convection by using feedback control [16]for consistently stabilized finite element methods Barth et alclassified the stabilization techniques and studied influence ofthe stabilization parameter in convergence [17] Bochev et alstated the requirements on choice of stabilization parameterif time step and mesh are allowed to vary independently[18] As far as we know it may not be enough to investigatewhat stabilization techniques are efficient for nonsteadyand nonlinear flow problems approximated by Lagrange-Galerkin methods in a domain decomposition system wherethe interface problem can be solved by preconditionedconjugate gradient (PCG) method In this paper a pressure-stabilization method which keeps the symmetry of thelinear system and is effective for high Reynolds number andRayleigh number simulations is introduced to implement

2 Abstract and Applied Analysis

the Lagrange-Galerkin method in a domain decompositionsystem

The element searching algorithm in a domain decompo-sition system using unstructured grids is the second difficultyto implement the Lagrange-Galerkin method in a domaindecomposition system (cf [5 19]) Minev et al reported anoptimized binary searching algorithm for single PE by storingthe necessary data structures in away similar to theCSR com-pact storage format however the element information datais stored distributedly in the domain decomposition systemby the skyline format and a different way needs to be foundto overcome the extra difficulty caused by the parallel com-puting algorithm This step is critical in the sense that it canbe very computationally expensive and can thus make theentire algorithm impractical

The remainder of this paper is organized as fol-lows in Section 2 the formula of the governing equationand the pressure-stabilization Lagrange-Galerkin method isdescribed Section 3 focuses on the parallel implementationof this scheme Numerical results and comparisons withclassical asymmetric product typemethods in [20] are shownin Section 4 Conclusions are drawn in Section 5

2 Formulation

21 The Governing Equations Let Ω be a three-dimensionalpolyhedral domain with the boundary 120597Ω The conservationequations of mass and momentum are governed by

120597119906

120597119905+ (119906 sdot nabla) 119906 minus 2]nabla sdot 119863 (119906) + nabla119901 = 119891

buoyancy

in Ω times (0 119905)

nabla sdot 119906 = 0 in Ω times (0 119905)

119906 = on Γ1times (0 119905)

3

sum

119895=0

120590119894119895119899119895= 0 on 120597Ω

Γ1times (0 119905)

119906 = 1199060

in Ω at 119905 = 0

(1)

where Γ1sub 120597Ω and

119891buoyancy

= 120573 (119879119903minus 119879) 119892 (2)

is the gravity force per unit mass derived on the basis ofBoussinesq approximation 119892 is the gravity [ms2] 120573 119879and 119879

119903are the thermal expansion coefficient [1119870] the

temperature [119870] and the reference temperature [119870] and119906 119905 ] and 119901 are velocity vector [ms] time [s] kinematicviscosity coefficient [m2s] and kinematic pressure [m2s2]respectively 120590

119894119895is the stress tensor [Nm2] defined by

120590119894119895(119906 119901) equiv minus119901120575

119894119895+ 2]119863

119894119895(119906)

119863119894119895(119906) equiv

1

2(120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

) 119894 119895 = 1 2 3

(3)

with the Kronecker delta 120575119894119895

The fluid is assumed to be incompressible according toBoussinesq approximation and the density is assumed to beconstant except in the gravity force term where it depends ontemperature according to the indicated linear law see (2)Theenergy equation is

120597119879

120597119905+ 119906 sdot nabla119879 minus 119886Δ119879 = 119878 in Ω times (0 119905)

119879 = on Γ2times (0 119905)

119886120597119879

120597119899= 0 on 120597Ω

Γ2times (0 119905)

119879 = 1198790

in Ω at 119905 = 0

(4)

where Γ2sub 120597Ω 119886 is the thermal diffusion coefficient [m2s]

and 119878 is the source term with the unit of [119870s]

22 The Lagrange-Galerkin Finite-Element Method Somepreliminaries are arranged for the derivation of a finiteelement scheme of (1) and (4) Let the subscript ℎ denotethe representative length of the triangulation and let I

ℎequiv

119870 denote a triangulation of Ω consisting of tetrahedralelements Given that 119892 is a vector valued function on Γ

1 the

finite element spaces are as follows

119883ℎequiv Vℎisin 1198620(Ω)3

Vℎ

1003816100381610038161003816119870isin 1198751(119870)3 forall119870 isin I

ℎ

119872ℎequiv 119902ℎisin 1198620(Ω) 119902

ℎ

1003816100381610038161003816119870isin 1198751(119870) forall119870 isin I

ℎ

119881ℎ(119892) equiv V

ℎisin 119883ℎ Vℎ(119875) = 119892 (119875) forall119875 isin Γ

1

Θℎ(119887) equiv 120579

ℎisin 119872ℎ 120579ℎ(119875) = 119887 (119875) forall119875 isin Γ

2

119881ℎequiv 119881ℎ(0) Θ

ℎequiv Θℎ(0) 119876

ℎ= 119872ℎ

(5)

Let (sdot sdot) defines the 1198712inner product the continuous

bilinear forms 119886 and 119887 are introduced by

119886 (119906 V) equiv 2] (119863 (119906) 119863 (V))

119887 (119906 V) equiv minus (nabla sdot 119906 119902)

(6)

respectivelyLet Δ119905 be the time increment and let 119873

119905equiv [119905Δ119905] be the

total step number Let the superscript 119899 denote the time stepa finite element approximation of (1) is described as followsfind (119906119899

ℎ 119901119899

ℎ)119873119905

119899=1isin 119881ℎ(119892)times119876

ℎ such that for (V

ℎ 119902ℎ) isin 119881ℎtimes119876ℎ

(

119906119899

ℎminus 119906119899minus1

ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 Vℎ) + 119886 (119906

119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) = (119891

119899 Vℎ)

119887 (119906119899

ℎ 119902ℎ) = 0

(7)

where 1198831(sdot sdot) denotes a first-order approximation of a parti-

clersquos position [5] and the notation ∘ denotes the compositionof functions

Abstract and Applied Analysis 3

For the purpose of large scale computation a piecewiseequal-order interpolation for velocity and pressure is usedas can be seen from (5) Pressure stabilization is thusneeded to keep the necessary link between 119881

ℎand 119876

ℎ A

penaltyGalerkin least-squares (GLS) stabilizationmethod forpressure is proved in [12] to hold the same asymptotic errorestimates as the method of Hughes et al [21] and it is com-putationally cheap For P1P1 elements the stabilization isreduced to

sum

119870isinIℎ

120575119870ℎ2

119870(nabla119901119899

ℎ minusnabla119902ℎ)119870 (8)

which does no modification to the momentum equationbecause of vanishing of the second-order derivate termHereℎ119870denotes the maximum diameter of an element 119870 Unlike

[6 12] where a constant 120575 (gt0) is used as the stabilizationparameter an element-wise stabilization parameter

120575119870=

120572

for log10[Max 10038171003817100381710038171003817nabla119901

119899minus1

ℎ

1003817100381710038171003817100381724

119894=1] le 1

120572 times log10[Max 10038171003817100381710038171003817nabla119901

119899minus1

ℎ

1003817100381710038171003817100381724

119894=1]

otherwise

(9)

is used in this work where nabla119901119899minus1

ℎis gradient of the FEM

approximated pressure at 119905119899minus1 and 119894 is the number of the nodalpoint in a tetrahedral element Since 120572 is very important tobalance the accuracy and convergence of the scheme it isdiscussed in Section 41The localized stabilization parameteris designed to be adaptive to the pressure gradient and thusit has a better control on the pressure field

By adding (8) to (7) a pressure-stabilized FEMscheme forNavier-Stokes problems is achieved The nonsteady iterationloops for solving (1) and (4) and then reads the following

Step 1 Compute the particlersquos coordinates by

1198831(119906119899minus1

ℎ Δ119905) equiv 119909 minus 119906

119899minus1

ℎΔ119905 (10)

and search the element holding the particle at 119905119899minus1

Step 2 Find 119879119899

ℎby

(

119879119899


ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 120579ℎ) + (119886nabla119879

119899

ℎ nabla120579ℎ) = (119878

119899 120579ℎ)

(11)

Step 3 Find (119906119899

ℎ 119901119899

ℎ) by

(

119906119899


ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 Vℎ) + 1198860(119906119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) + 119887 (119906

119899

ℎ 119902ℎ)

+ sum

119870isinIℎ

120575119870ℎ2

119870(nabla119901119899

ℎ minusnabla119902ℎ)119870

= (119891119899 Vℎ) + (120573 (119879

119903minus 119879119899

ℎ) 119892 Vℎ)

(12)

Step 4 Compute the relative error by a 1198671 times 1198712times 1198671 norm

defined by

1003817100381710038171003817(119906 119901 119879)10038171003817100381710038171198671times1198712times1198671

equiv1

radicRe1199061198671(Ω)3

+1003817100381710038171003817119901

10038171003817100381710038171198712+ 1198791198671(Ω)

(13)

where Re denotes the Reynolds number and set

diff =

10038171003817100381710038171003817(119906119899 119901119899 119879119899) minus (119906

119899minus1 119901119899minus1

119879119899minus1

)100381710038171003817100381710038171198671times1198712times1198671

1003817100381710038171003817(119906119899minus1 119901119899minus1 119879119899minus1)

10038171003817100381710038171198671times1198712times1198671

le ErrNS

(14)

as the steady-state criterion if (14) is satisfied or the numberof loops reaches the maximum then stop the iterationotherwise repeat Steps 1ndash3

As can be seen from Steps 2 and 3 both (1) and (4)are approximated by the Lagrange-Galerkin method and thesearching algorithm only needs to be performed once in anonsteady loop It can also be seen that the solver is alsoflexible and it can solve pure Navier-Stokes problems bysetting the body force in (2) to external force and omittingStep 2

3 Implementation

31 A Parallel Domain Decomposition System To begin withthe parallel domain decomposition method the domaindecomposition is introduced briefly as follows The wholedomain is decomposed into a number of subdomainswithoutoverlapping and the solution of each subdomain is super-imposed on the equation of the inner boundary of thesubdomains By static condensation the linear system

119870119906 = 119891 (15)

is written as

[[[[[[[[[

[

119870(1)

1198681198680 sdot sdot sdot 0 119870

(1)

119868119861119877(1)

119861

0 d

d

0 sdot sdot sdot sdot sdot sdot 119870(119873)

119868119868119870(119873)

119868119861119877(119873)

119861

119877(1)119879

119861119870(1)119879

119868119861sdot sdot sdot sdot sdot sdot 119877

(119873)119879

119861119870(119873)119879

119868119861119870119861119861

]]]]]]]]]

]

[[[[[[[[[

[

119906(1)

119868

119906(119873)

119868

119906119861

]]]]]]]]]

]

=

[[[[[[[[[

[

119891(1)

119868

119891(119873)

119868

119891119861

]]]]]]]]]

]

(16)

where 119870 is the stiffness matrix 119906 denotes the unknowns (119906and 119901) and 119891 is the force vector 119877 is the restriction operatorconsists of 0-1 matrix The superscripts (119873) means the


119873th subdomain and subscript 119868 and 119861 relate to the elementof the inner boundary and interface boundary respectively

From (16) it can be observed that the interface problems

119873

sum

119894=1

119877(119894)119879

119861(119870(119894)

119861119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119870(119894)

119868119861)119877(119894)

119861119906119861

=

119873

sum

119894=1

119877(119894)119879

119861(119891(119894)

119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119891(119894)

119868)

(17)

and the inner problems

119906(119894)

119868= 119870(119894)minus1

119868119868(119891(119894)

119868minus 119870(119894)119879

119868119861119877(119894)

119861119906119861) 119894 = 1 119873 (18)

can be solved separately [22] In this work the interfaceproblems are solved first iteratively and the inner problemsare then solved by substituting 119906

119861in to (18)

The Lagrange-Galerkin method keeps the symmetry ofthe stiffness matrix and the GLS pressure-stabilization termin (8) also produces a symmetric matrix therefore119870 is sym-metric in (15) and a PCG method is employed to get the 119906

119868

in (18) and to avoid drawback of the classical domain decom-positionmethod such as Neumann-Neumann and diagonal-scaling a balanced domain decomposition preconditioner isused to prevent the growing of condition number with thenumber of subdomains An identity matrix is chosen as thecoarse matrix and the coarse problem is solved incompletelyby omitting the fill-ins in some sensitive places duringthe Cholesky factorization By using this inexact balanceddomain decomposition preconditioning the coarse matrix issparser and thus easier to be solved therefore the new solveris expected to have better solvability on large scale computa-tion models

32 The Lagrange-Galerkin Method in Parallel The elementsearching algorithm requires a global-wise element informa-tion to determine the position of one particle in the previoustime step However in the parallel domain decompositionsystem the whole domain is split into several parts oneprocessor element (PE) works only on the current part andit does not contain any element information of other partsEach part is further divided into many subdomains and thedomain decomposition is performed by the PE in charge ofthe partThis parallelity causes a computational difficulty foreach time step the particle is not limited within one parttherefore exchanging the data between different PEs isnecessary which demands the PEs to communicate in thesubdomain-wise computation

In order to know the position of a particle at 119905119899minus1 a

neighbour elements list is created at the beginning of theanalysis Based on the information of neighbour elements andthe coordinates calculated by (10) it is possible to find theelement holding this particle at 119905119899minus1 A 2-dimensional search-ing algorithm is present as follows (120582

119894is the barycentric coor-

dinates and 119899119890(120582119894) is the neighbour element see Figure 1)

Part interface

ei

ecurrent

R2

ne (1205821)

ne (1205822)

ne (1205823)

Figure 1 A searching algorithm

(1) initialize 1198900= 119890current

(2) iterate 119894 = 0 1 MaxloopsIf 1205821 1205822 1205823gt 0 return 119890

119894

else if 119899119890(Min1205821 1205822 1205823) = boundary

119890119894+1

= 119899119890(Min1205821 1205822 1205823)

else break(3) return 119890

119894

The request of the old solutions which is the 119906119899minus1ℎ

in (10)is relatively trivial when using single PEs or simply solving theproblem parallel using symmetric multiprocessing howeverin the domain decomposition system the particle is notlimited within one part it may pass the interface of differentparts as can be seen from Figure 1 Because one PE only hasthe elements information that belongs to the current partcommunications between PEs are necessary However thenumber of total elements in one subdomainmay be differentwhich means that some point to point communicationtechniques such as MPI SendMPI Recv or MPI Sendrecvin MPICH cannot be used in element wise computation Inthe previous research [23] a global variable to store all theold solutions is constructed This method maintains a highcomputation speed but costs a hugememory usage To reducethe memory consumption a request-response system is usedin this work In the computation the searching algorithm isperformed first and the element that contains the currentparticle in the previous time step is thus known thereforethe PE to get 119906

119899minus1

ℎfrom is also known However as the

sender does not know which PE requires message from itselfthe receiver has to send its request to the sender first afterthe request is detected the sender sends the message to thereceiver The procedure is as follows

(1) by scanning all the particles in the current subdomainan array including all the data that is needed by thecurrent PE is sent to all the other PEs

(2) All PEs check if there is any request to itself If it existsPEs will prepare an array of the needed data and sendit

(3) The current PE receives the data sent by other PEs

Data transferred byMPI communication should be pack-aged properly to avoid the overflow of MPI buffer in case


0 1000 2000 3000 4000 5000Iteration times

120575 = 0001

120575 = 0005

120575 = 001

120575 = 01

120575 = 1

120575 = 10

120575 = 100

1E+041E+031E+021E+011E+001Eminus011Eminus021Eminus031Eminus041Eminus051Eminus061Eminus07

Resid

ual

(a)

0 2000 4000 6000 8000 10000Iteration times

Resid

ual

Re = 10E6 (120575 = 0005)Re = 10E5 (120575 = 0005)Re = 10E3 (120575 = 0005)Re = 10E4 (120575 = 0005)

Re = 10E3 (120575 K)Re = 10E4 (120575 K)Re = 10E5 (120575 K)Re = 10E6 (120575 K)


(b)

Figure 2 Convergence of (a) different constant 120575 at Re = 103 (b) different Re for 120575 = 0005 and localized 120575

119870

Number of subdomains

0100200300400500600700800

90 180 270 360 450 540 630 720 810 900 990 1080

NoneCurrent

Num

ber o

f ite

ratio

ns

Figure 3 Numerical scalability

of large-scale computation Nonblocking communication isemployed and as the 3 steps are performed subsequentlythus the computation time and communication time will beoverlapped

4 Numerical Results and Discussion

The parallel efficiency of new solver is firstly evaluated inthis section and to validate the scheme exact solutions andavailable benchmark results classical computational modelsare compared The CG convergence is judged by Euclidiannorm with a tolerance of 10minus6 and for nonsteady iterationErrNS = 10

minus4 is set as the criterion using the 1198671 times 1198712times 1198671

norm defined in (13) For pure Navier-Stokes problems asimilar 119867

1times 1198712 norm which is related to velocity and

pressure is employed to judge the steady state

41 Efficiency Test The BDD serious preconditioners wereemployed in this work they are very efficient and theiriteration numbers are about 1acircĄĎ10 of the normal domaindecomposition preconditioners (cf [23]) The inexact pre-conditioner mentioned in Section 3 also shows good conver-gence and is more suitable for large scale computations [24]

It was set as the default preconditioner for all the followingcomputations of this research

The penalty methods are not consistent since the sub-stitution of an exact solution into the discrete equations(12) leaves a residual that is proportional to the penaltyparameter (cf [17]) therefore 120575

119870should be determined

carefully Numerical experiments of a lid-driven cavity flowwere tested and the mesh size was 62 times 62 times 62 The totaldegrees of freedom (DOF) are 1000188 and the resultsare given by Figure 2 For the purpose of higher accuracy120575119870is expected to be small however the convergence turns

worse when 120575119870goes small as can be seen from Figure 2(a)

In Figure 2(b) a constant 120575 = 0005 is used for differentReynolds numbers and no convergence is achieved within10000 PCG loops for Re = 10

6 and the comparison showsthat the 120575

119870performs much better than a constant 120575 = 0005

when 120572 is set to 0005The parallel efficiency is assessed firstly by freezing the

mesh size of test problem and refining the domain decom-position by decreasing the subdomain size and thereforeincreasing the number of subdomains the comparison of thenumerical scalability of the current scheme with and withoutthe preconditioner is assessed by Figure 3 It can be seen thatwith the preconditioning technique the iterative procedure


0

400

800

1200

1600

2000

2400

2800

5000 35000 65000 95000 125000 155000 185000

Elap

sed

time (

s)

DOF

Current

BiCGSTAB (ADV sFlow 05)BiCGSTAB2 (ADV sFlow 05)GPBiCG (ADV sFlow 05)

(a)

5000 35000 65000 95000 125000 155000 185000DOF

Current


0

100

200

300

400

500

Tota

l mem

ory

usag

e (G

B)(b)

Figure 4 Time and memory usages

12345678

8 16 24 32 40 48 56 64

Model1 (DOF 148955)Model2 (DOF 1134905)Model3 (DOF 12857805)Ideal

Number of PEs

Spee

d-up


of current scheme converges rapidly and the convergence isindependent of the number of subdomains

Based on the paralyzed Lagrange-Galerkin method thenew solver makes a symmetric stiffness matrix thereforeonly the lowerupper triangular matrix needs to be savedMoreover nonblocking MPI communication is used insteadof constructing global arrays to keep the old solutionsand the current solver is expected to reduce the memoryconsumption without sacrificing the computation speedTheusage of time and memory of solving the thermal drivencavity problem by different solvers is compared and theresults are given by Figure 4

The test problem was solved by the new solver andthe ADV sFlow 05 [25] which contains some nonsym-metric product-type solvers like GPBiCG BiCGSTAB andBiCGSTAB2 [20] The ADV sFlow 05 employed a domaindecomposition system similar to the work however noprecondition technique is used because of the non-symmetryof the stiffness matrix in (15) The comparisons of elapsedtime and memory occupation of the new solver and that ofproduct-type solvers in ADV sFlow 05 are show in Figures4(a) and 4(b) As can be seen the current scheme reducesthe demand of computation time and memory consumptionremarkably and it is more suitable for large scale problemsthan product-type solvers

The parallel scalability of the searching algorithm isalso a concern for us as it characterizes the ability of analgorithm to deliver larger speed-up using a larger numberof PEs To know this the number of subdomains in one partis fixed and computations on the test problem of variousmesh sizes are performed by the new scheme The speed-up is shown in Figure 5 Three models were tested by thesearching algorithm With an increase in the mesh sizeof the computation model the parallel scalability of thesearching algorithm tends to be better An explanation to thisis that when the DOF increase the number of elements inone subdomain is also increasing therefore the searchingalgorithm is accelerated more efficiently However too manyelements in one subdomain will occupy more memory and atrade-off strategy is necessary for parameterization

42 Validation Tests In this section a variety of test problemshave been presented in order to prove the capability of theparallel Lagrange-Galerkin algorithm Benchmarks test of


x1

x2

x3

0

1

1

4

b

Figure 6 A plane Couette flow model

0010203040506070809

1

0 02 04 06 08 1 12 14

Exac Val(minus3) Exac Val(minus2)Exac Val(minus1) Exac Val(0)Exac Val(1) Exac Val(2)Exac Val(3) Num Res 0(minus3)Num Res 0(minus2) Num Res 0(minus1)Num Res 0(0) Num Res 0(1)Num Res 0(2) Num Res 0(3)Num Res 1(minus3) Num Res 1(minus2)Num Res 1(minus1) Num Res 1(0)Num Res 1(1) Num Res 1(2)Num Res 1(3)

x3b

u3Uminus04 minus02

Figure 7 Numerical results versus exact solutions

Navier-Stokes problems are in Sections 421 422 and 423and flow-thermal coupled problems are in Sections 424 and425

421 A Plane Couette Flow The solver for Navier-Stokesequations in (1) was first tested with a 3D plane Couetteflow Under ideal conditions the model is of infinite lengththerefore 4 times of the height is used as the length of themodel see Figure 6 A constant velocity ( 0 0) is appliedon the upper horizontal face and no-slip conditions are seton the lower horizontal face A pressure gradient is imposedalong 119909

1for all the faces as essential boundary conditions

An unstructured 3D mesh was generated by ADVEN-TURE TetMesh [25] and the local density around the planeof 1199091= 2 where the data was picked from was enriched

The total DOF is around 1024000 The so-called Brinkmannumber [26] is believed to be the dominating parameter ofthe flow and a serious of numerical experiments is done atvarious Brinkman number To simulate the infinity lengthbetter the exact solution is enforced on both the left face(1199091

= 0) and the right face (1199091

= 4) as Dirichlet bound-ary conditions The comparisons between the computationresults and exact solutions are given by Figure 7 Dotted

0 1

1

1

x1

x2

x3

Figure 8 A lid-driven cavity model

lines are used to present the results and they are named asldquoNum Res 1(119861)rdquo where 119861 stands for the Brinkman numberCrossed lines in Figure 7 present the computation results withno exact solutions setting on the left and right faces and theyare named as ldquoNum Res 0(119861)rdquo

It can be seen form Figure 7 that both of these two setsof computation results show good agreement with the exactsolution and dotted lines are closer to the exact solutionrepresenting a better simulation to the ideal condition (cf[27 28])

422 A Lid-Driven Cavity The Navier-Stokes problemssolver was then verified by a lid-driven cavity flow Theideal gas flows over the upper face of the cube and no-slipconditions are applied to all other faces as in Figure 8

All the faces of the cube were set with Dirichlet boundaryconditions and a zero reference pressure was at the centre ofthe cube to keep the simulation stableThe pressure profiles ofthe scheme using localized stabilization parameter in (9) andthe scheme using constant (120575 = 1) parameter are comparedand the results are show in Figure 9

Figure 9(a) shows the pressure counters of the schemewith the localized stabilization parameter in (9) and theFigure 9(b) shows scheme with a constant parameter Themodel was run at Re = 10

4 and oscillations are viewedin Figure 9(b) however the isolines in Figure 9(a) is quitesmooth showing that the pressure-stabilization term has abetter control on the pressure field at high Reynolds number

The model was run at different Reynolds numbers with a128times128times128mesh to test the solvability of the new schemeAs shown in Figure 10 when the Reynolds number increasesthe eddy at right bottom of plane 119909

11199093vanishes while the

eddy at the left bottom appears due to the increasing in thespeed and the flow goes more likely around the wall Theprimary eddy goes lower and lower when Reynolds numberbecomes higher and the particle is no longer limited to asingle side of the cavity it can pass from one side to the otherand back again violating themirror symmetry as is seen fromother planes of Figure 10 Similar 3D results for highReynoldsnumber were reported by [29] and the solvability of the newsolver for high Reynolds number was confirmed


015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)

Figure 9 Pressure counters (Re = 104)

Figure 10 Velocity and pressure profiles for different Reynolds number Re = 1000 (top) Re = 3200 (middle) and Re = 12000 (bottom)along different middle planes plane 119909

11199092(left) plane 119909

11199093(middle) and plane 119909

21199093(right)

423 Backward Facing Step The solver for Navier-Stokesequations was then tested with backward facing step thefluid considered was air The problem definition is shown inFigure 11 and the height of the step ℎ is the characteristiclength An unstructured 3Dmesh was generated with 419415

nodal points and 2417575 tetrahedral elements and the localdensity of mesh was increased around the step

A laminar flow is considered to enter the domain atinlet section the inlet velocity profile is parabolic and theReynolds number is based on the average velocity at the inlet


H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3

Figure 11 Backward facing steps

(a)

r

(b)

Figure 12 Pressure counters (a) and velocity vectors (b) (Re = 200)

02468

10121416

0 100 200 300 400 500 600 700 800 900

Exp (Armaly et al)Jiang et alKu et al

Williams et alCurrent

Re

Reat

tach

men

t len

gth

(rS

)

Figure 13 Primary reattachment lengths

The total length of the domain is 30 times the step height sothat the zero pressure is set at the outlet A full 3D simulationof the step geometry for 100 le Re le 800 is present in thispaper and the primary reattachment lengths are predicted

To determine the reattachment length the position ofthe zero-mean-velocity line was measured The points ofdetachment and reattachment were taken as the extrapolatedzero-velocity line down the wall The pressure contour inFigure 12(a) confirms the success of the pressure-stabilizationmethod velocity vectors and the primary attachment aredemonstrated in Figure 12(b) similar results have been doc-umented by many like in [10 30]

4

1

1

x1

x2

x3

ThighTlow

Figure 14 The model of infinite plates

The comparison of primary reattachment length betweencurrent results and other available benchmark results areshow in Figure 13 It is seen that the agreement is excellentat different Rayleigh numbers (cf [31 32])

424 Natural Convection of Flat Plates In order to testthe coupled solver of Navier-Stokes equations and theconvection-diffusion equation the third application modelwas the natural convection between two infinite flat platesThe geometry is given in 3-dimensional by Figure 14 No-slip boundary conditions applied on the left and right verticalwallsThe temperature on the left wall is assumed to be lowerand set at 5[119870] the right wall is set at 6[119870] An unstructured3D mesh about 1 million tetrahedral elements was generatedand the local grid density around themid-planewas enriched

The model was run at the size of 20 times 20 times 80 to getthe numerical solutions and it was compared with the exactsolutions in Figure 15 To simulate the infinity length of theplate better the exact solution is enforced on both the upperface (119909

3= 4) and the lower face (119909

3= 0) as what is

done in Section 421 and the results are present by a dottedline (ldquoNum Res 1rdquo) in Figure 12 And the model withoutexact solution set as boundary is named as ldquoNum Res 0rdquo inFigure 15


0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1

x1minus05 minus04 minus03 minus02 minus01

minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow

Figure 16 A thermal-driven cavity

With the parameter setting of ] = 05 119879119903= 55 120573 = 10

and 119886 = 1 the numerical experiment was performed Resultsof the profile on 119906

3(sdot 0 2) which are believed by many to be

very sensitive are shown in Figure 15 Both ldquoNum Res 0rdquo andldquoNum Res 1rdquo are in great agreement with the exact solutionsand ldquoNum Res 1rdquo is closer to the exact solution producing abetter simulation to the ideal condition Similar results havebeen documented in [33]

425 Thermal-Driven Cavity The new solver is also appliedto a 3-dimensional nonlinear thermal driven cavity flowproblem which is cavity full of ideal gas see Figure 16

No-slip boundary conditions are assumed to prevail on allthe walls of the cavity Both the horizontal walls are assumedto be thermally insulated and the left and right sides are keptat different temperaturesThe cube is divided into 120times120times120 small cubes and each small cube contains six tetrahedralelements The time step is set to 001 s with Pr = 071 andRa = 10

4 the steady state is achieved after 039 s as inFigure 17

Figures 17(a) and 17(b) show the contour of vorticity andthe velocity vectors at the steady stage respectively from thefront viewThe temperature contour is shown in Figure 17(c)and pressure profiles are show in Figure 17(d) The previousresults convince us of the success in solving flow-thermal

coupled problems described by (1) and (4) Similar three-dimensional results can also be found in [33 34]Thepressureprofile in Figure 17(d) is smooth and symmetric implyingthat the stabilization item in (8) works well

In order to further validate the new solver a comparisonof temperature and velocity profiles of the current solver andother benchmark results was made The centreline velocityresults 119908(sdot 05 05) and the temperature results 119879(sdot 05 05)which are believed to be very sensitive in this simulationare present in diagrams (a) and (b) of Figure 18 respectivelyThe velocity results share close resemblance to that of theADV sFlow05 and they both show themore end-wall effectscompared with the results of 2D case The three temperatureresults show good agreement with each other and the linerepresenting the current results is the smoothest as themesh is the finest among the three Similar results have beenreported by other researchers (cf [10 33 34])

Thermal convection problems are believed to be domi-nated by two dimensionless numbers by many researchersthe Prandtl number and the Rayleigh number To acquaintourselves with the solvability of the new solver and tochallenge applications of higher difficulty a wide range ofRayleigh numbers from 10

3 to 107 is studied with Pr = 071

and the results for the steady-state solution are presented inFigure 19 Dimensionless length is used and the variation ofRayleigh number is determined by changing the characteris-tic length of the model

The local Nusselt number (Nu = 1205971198791205971199091) is a concern

of many researchers as they are sensitive to the mesh sizeIn Figure 19 the diagram (a) and the diagram (b) representthe local Nusselt number at the hot wall and the coldwall respectively Similar results can also be found in [1030 35 36] The capability of the solver based on domaindecomposed Lagrange-Galerkin scheme for high Rayleighnumber is also confirmed by this figure

The new solver enables the simulation of large scaleproblems thus models of Rayleigh number up to 10

7 can berun on small PC cluster In this simulation an unstructuredmesh of 30099775 DOF is generated the time step is 001 sand it takes about 24 hours to finish using the a small Linuxcluster of 64 PEs (64 cores266GHz)

5 Conclusions

A pressure-stabilized Lagrange-Galerkin method is imple-mented in a domain decomposition system in this researchBy using localized stabilization parameter the new schemeshows better control in the pressure field than constant sta-bilization parameter therefore it has good solvability at highReynolds number and high Rayleigh number The reliabilityand accuracy of the present numerical results are validated bycomparingwith the exact solutions and recognizednumericalresults Based on a domain decomposition method the ele-ment searching algorithm shows good numerical scalabilityand parallel efficiency The new solver reduces the memoryconsumption and is faster than classical product-type solversIt is able to solve large scale problems of over 30 milliondegrees of freedom within one day by a small PC cluster


369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)

Figure 17 Steady state of the thermal-driven cavity (Ra = 104)

0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1

CurrentADV sFlow 05 (3D)

D C Wan (2D)N Massarotti (2D)

x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture

ADV sFlow 05 (3D)K L Wong (3D)

x1

(b)

Figure 18 Centerline temperature velocity profiles of the symmetry plane (Ra = 104)


0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)

Figure 19 Local Nusselt number along the hot wall (a) and the cold wall (b)

Acknowledgments

This work was supported by the National Science Foun-dation of China (NSFC) Grants 11202248 91230114 and11072272 the China Postdoctoral Science Foundation Grant2012M521646 and the Guangdong National Science Founda-tion Grant S2012040007687

References

[1] M Bercovier O Pironneau and V Sastri ldquoFinite elementsand characteristics for some parabolic-hyperbolic problemsrdquoApplied Mathematical Modelling vol 7 no 2 pp 89ndash96 1983

[2] J Douglas Jr and T F Russell ldquoNumerical methods for con-vection-dominated diffusion problems based on combining themethod of characteristics with finite element or finite differenceproceduresrdquo SIAM Journal on Numerical Analysis vol 19 no 5pp 871ndash885 1982

[3] O Pironneau ldquoOn the transport-diffusion algorithm and itsapplications to the Navier-Stokes equationsrdquoNumerischeMath-ematik vol 38 no 3 pp 309ndash332 1982

[4] T F Russell ldquoTime stepping along characteristics with incom-plete iteration for a Galerkin approximation of miscible dis-placement in porous mediardquo SIAM Journal on NumericalAnalysis vol 22 no 5 pp 970ndash1013 1985

[5] O Pironneau Finite Element Methods for Fluids John Wiley ampSons Chichester UK 1989

[6] H Notsu ldquoNumerical computations of cavity flow problemsby a pressure stabilized characteristic-curve finite elementschemerdquo in Japan Society for Computational Engineering andScience 2008

[7] H Notsu and M Tabata ldquoA combined finite element schemewith a pressure stabilization and a characteristic-curve method

for the Navier-Stokes equationsrdquo Transactions of the JapanSociety for Industrial and AppliedMathematics vol 18 no 3 pp427ndash445 2008

[8] M Tabata and S Fujima ldquoRobustness of a characteristic finiteelement scheme of second order in time incrementrdquo in Compu-tational Fluid Dynamics 2004 pp 177ndash182 2006

[9] H Rui and M Tabata ldquoA second order characteristic finite ele-ment scheme for convection-diffusion problemsrdquo NumerischeMathematik vol 92 no 1 pp 161ndash177 2002

[10] N Massarotti P Nithiarasu and O C Zienkiewicz ldquoChar-acteristic-based split (CBS) algorithm for incompressible flowproblems with heat transferrdquo International Journal of NumericalMethods for Heat and Fluid Flow vol 11 no 3 p 278 2001

[11] F Brezzi andM FortinMixed and Hybrid Finite ElementMeth-ods vol 15 of Springer Series in Computational MathematicsSpringer New York NY USA 1991

[12] F Brezzi and J Douglas Jr ldquoStabilized mixed methods for theStokes problemrdquo Numerische Mathematik vol 53 no 1-2 pp225ndash235 1988

[13] H Jia D Liu and K Li ldquoA characteristic stabilized finite ele-ment method for the non-stationary Navier-Stokes equationsrdquoComputing vol 93 no 1 pp 65ndash87 2011

[14] H Kanayama D Tagami T Araki and H Kume ldquoA stabiliza-tion technique for steady flow problemsrdquo International Journalof Computational Fluid Dynamics vol 18 no 4 pp 297ndash3012004

[15] C R Dohrmann and P B Bochev ldquoA stabilized finite elementmethod for the Stokes problem based on polynomial pressureprojectionsrdquo International Journal for Numerical Methods inFluids vol 46 no 2 pp 183ndash201 2004

[16] H M Park and M C Sung ldquoStabilization of two-dimensionalRayleigh-Benard convection by means of optimal feedbackcontrolrdquo Physica D vol 186 no 3-4 pp 185ndash204 2003


[17] T Barth P Bochev M Gunzburger and J Shadid ldquoA taxonomyof consistently stabilized finite element methods for the Stokesproblemrdquo SIAM Journal on Scientific Computing vol 25 no 5pp 1585ndash1607 2004

[18] P B Bochev M D Gunzburger and R B Lehoucq ldquoOnstabilized finite element methods for the Stokes problem inthe small time step limitrdquo International Journal for NumericalMethods in Fluids vol 53 no 4 pp 573ndash597 2007

[19] G C Buscaglia and E A Dari ldquoImplementation of theLagrange-Galerkin method for the incompressible Navier-Stokes equationsrdquo International Journal for Numerical Methodsin Fluids vol 15 no 1 pp 23ndash26 1992

[20] H A van der Vorst Iterative Krylov Methods for Large Lin-ear Systems vol 13 of Cambridge Monographs on Appliedand Computational Mathematics Cambridge University PressCambridge UK 2003

[21] T J R Hughes L P Franca and M Balestra ldquoA new finiteelement formulation for computational fluid dynamics VCircumventing the Babuska-Brezzi condition a stable Petrov-Galerkin formulation of the Stokes problem accommodat-ing equal-order interpolationsrdquo Computer Methods in AppliedMechanics and Engineering vol 59 no 1 pp 85ndash99 1986

[22] A Toselli and O Widlund Domain Decomposition MethodsmdashAlgorithms and Theory vol 34 of Springer Series in Computa-tional Mathematics Springer Berlin Germany 2005

[23] Q Yao H Kanayama H Notsu and M Ogino ldquoBalancingdomain decomposition for non-stationary incompressible flowproblems using a characteristic-curve methodrdquo Journal ofComputational Science and Technology vol 4 pp 121ndash135 2010

[24] Q H Yao and Q Y Zhu ldquoInvestigation of the contaminationcontrol in a cleaning roomwith amoving Agv by 3D large-scalesimulationrdquo Journal of Applied Mathematics vol 2013 ArticleID 570237 10 pages 2013

[25] ldquoAdventure Projectrdquo httpadventuresystu-tokyoacjp[26] CW Hall Laws andModels Science Engineering and Technol-

ogy CRC Press 2000[27] R S Brodkey and H C Hershey Basic Concepts in Transport

Phenomena Brodkey 2001[28] S M Richardson Fluid Mechanics Hemisphere 1989[29] E Hachem B Rivaux T Kloczko H Digonnet and T Coupez

ldquoStabilized finite element method for incompressible flows withhigh Reynolds numberrdquo Journal of Computational Physics vol229 no 23 pp 8643ndash8665 2010

[30] M T Manzari ldquoAn explicit finite element algorithm for convec-tion heat transfer problemsrdquo International Journal of NumericalMethods for Heat and Fluid Flow vol 9 no 8 pp 860ndash877 1999

[31] N A Malamataris ldquoA numerical investigation of wall effectsin three-dimensional laminar flow over a backward facingstep with a constant aspect and expansion ratiordquo InternationalJournal for Numerical Methods in Fluids vol 71 pp 1073ndash11022013

[32] P T Williams and A J Baker ldquoNumerical simulations oflaminar flow over a 3D backward-facing steprdquo InternationalJournal for NumericalMethods in Fluids vol 24 no 11 pp 1159ndash1183 1997

[33] H Kanayama K Komori and D Sato ldquoDevelopment of aThermal Convection Solver with Hierarchical Domain Decom-position Methodrdquo in Proceedings of the 8th World Congress onComputational Mechanics and the 5th European Congress onComputational Methods in Applied Sciences and EngineeringVenice Italy 2008

[34] K L Wong and A J Baker ldquoA 3D incompressible Navier-Stokes velocity-vorticity weak form finite element algorithmrdquoInternational Journal for Numerical Methods in Fluids vol 38no 2 pp 99ndash123 2002

[35] G de Vahl Davis ldquoNatural convection of air in a square cavitya bench mark numerical solutionrdquo International Journal forNumerical Methods in Fluids vol 3 no 3 pp 249ndash264 1983

[36] G de Vahl Davis and I P Jones ldquoNatural convection in squarecavity a comparison exerciserdquo International Journal for Numer-ical Methods in Fluids vol 3 no 3 pp 227ndash248 1983

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the Lagrange-Galerkin method in a domain decompositionsystem

The element searching algorithm in a domain decompo-sition system using unstructured grids is the second difficultyto implement the Lagrange-Galerkin method in a domaindecomposition system (cf [5 19]) Minev et al reported anoptimized binary searching algorithm for single PE by storingthe necessary data structures in away similar to theCSR com-pact storage format however the element information datais stored distributedly in the domain decomposition systemby the skyline format and a different way needs to be foundto overcome the extra difficulty caused by the parallel com-puting algorithm This step is critical in the sense that it canbe very computationally expensive and can thus make theentire algorithm impractical

The remainder of this paper is organized as fol-lows in Section 2 the formula of the governing equationand the pressure-stabilization Lagrange-Galerkin method isdescribed Section 3 focuses on the parallel implementationof this scheme Numerical results and comparisons withclassical asymmetric product typemethods in [20] are shownin Section 4 Conclusions are drawn in Section 5

2 Formulation

21 The Governing Equations Let Ω be a three-dimensionalpolyhedral domain with the boundary 120597Ω The conservationequations of mass and momentum are governed by

120597119906

120597119905+ (119906 sdot nabla) 119906 minus 2]nabla sdot 119863 (119906) + nabla119901 = 119891

buoyancy

in Ω times (0 119905)

nabla sdot 119906 = 0 in Ω times (0 119905)

119906 = on Γ1times (0 119905)

3

sum

119895=0

120590119894119895119899119895= 0 on 120597Ω

Γ1times (0 119905)

119906 = 1199060

in Ω at 119905 = 0

(1)

where Γ1sub 120597Ω and

119891buoyancy

= 120573 (119879119903minus 119879) 119892 (2)

is the gravity force per unit mass derived on the basis ofBoussinesq approximation 119892 is the gravity [ms2] 120573 119879and 119879

119903are the thermal expansion coefficient [1119870] the

temperature [119870] and the reference temperature [119870] and119906 119905 ] and 119901 are velocity vector [ms] time [s] kinematicviscosity coefficient [m2s] and kinematic pressure [m2s2]respectively 120590

119894119895is the stress tensor [Nm2] defined by

120590119894119895(119906 119901) equiv minus119901120575

119894119895+ 2]119863

119894119895(119906)

119863119894119895(119906) equiv

1

2(120597119906119894

120597119909119895

+

120597119906119895

120597119909119894

) 119894 119895 = 1 2 3

(3)

with the Kronecker delta 120575119894119895

The fluid is assumed to be incompressible according toBoussinesq approximation and the density is assumed to beconstant except in the gravity force term where it depends ontemperature according to the indicated linear law see (2)Theenergy equation is

120597119879

120597119905+ 119906 sdot nabla119879 minus 119886Δ119879 = 119878 in Ω times (0 119905)

119879 = on Γ2times (0 119905)

119886120597119879

120597119899= 0 on 120597Ω

Γ2times (0 119905)

119879 = 1198790

in Ω at 119905 = 0

(4)

where Γ2sub 120597Ω 119886 is the thermal diffusion coefficient [m2s]

and 119878 is the source term with the unit of [119870s]

22 The Lagrange-Galerkin Finite-Element Method Somepreliminaries are arranged for the derivation of a finiteelement scheme of (1) and (4) Let the subscript ℎ denotethe representative length of the triangulation and let I

ℎequiv

119870 denote a triangulation of Ω consisting of tetrahedralelements Given that 119892 is a vector valued function on Γ

1 the

finite element spaces are as follows

119883ℎequiv Vℎisin 1198620(Ω)3

Vℎ

1003816100381610038161003816119870isin 1198751(119870)3 forall119870 isin I

ℎ

119872ℎequiv 119902ℎisin 1198620(Ω) 119902

ℎ

1003816100381610038161003816119870isin 1198751(119870) forall119870 isin I

ℎ

119881ℎ(119892) equiv V

ℎisin 119883ℎ Vℎ(119875) = 119892 (119875) forall119875 isin Γ

1

Θℎ(119887) equiv 120579

ℎisin 119872ℎ 120579ℎ(119875) = 119887 (119875) forall119875 isin Γ

2

119881ℎequiv 119881ℎ(0) Θ

ℎequiv Θℎ(0) 119876

ℎ= 119872ℎ

(5)

Let (sdot sdot) defines the 1198712inner product the continuous

bilinear forms 119886 and 119887 are introduced by

119886 (119906 V) equiv 2] (119863 (119906) 119863 (V))

119887 (119906 V) equiv minus (nabla sdot 119906 119902)

(6)

respectivelyLet Δ119905 be the time increment and let 119873

119905equiv [119905Δ119905] be the

total step number Let the superscript 119899 denote the time stepa finite element approximation of (1) is described as followsfind (119906119899

ℎ 119901119899

ℎ)119873119905

119899=1isin 119881ℎ(119892)times119876

ℎ such that for (V

ℎ 119902ℎ) isin 119881ℎtimes119876ℎ

(

119906119899


ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 Vℎ) + 119886 (119906

119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) = (119891

119899 Vℎ)

119887 (119906119899

ℎ 119902ℎ) = 0

(7)

where 1198831(sdot sdot) denotes a first-order approximation of a parti-

clersquos position [5] and the notation ∘ denotes the compositionof functions



ℎand 119876

ℎ A


sum

119870isinIℎ

120575119870ℎ2

119870(nabla119901119899




120575119870=

120572

for log10[Max 10038171003817100381710038171003817nabla119901

119899minus1

ℎ

1003817100381710038171003817100381724

119894=1] le 1


119899minus1

ℎ

1003817100381710038171003817100381724

119894=1]

otherwise

(9)






1198831(119906119899minus1

ℎ Δ119905) equiv 119909 minus 119906

119899minus1

ℎΔ119905 (10)


Step 2 Find 119879119899

ℎby

(

119879119899


ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 120579ℎ) + (119886nabla119879

119899

ℎ nabla120579ℎ) = (119878

119899 120579ℎ)

(11)

Step 3 Find (119906119899

ℎ 119901119899

ℎ) by

(

119906119899


ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 Vℎ) + 1198860(119906119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) + 119887 (119906

119899

ℎ 119902ℎ)

+ sum

119870isinIℎ

120575119870ℎ2

119870(nabla119901119899


= (119891119899 Vℎ) + (120573 (119879

119903minus 119879119899

ℎ) 119892 Vℎ)

(12)


defined by

1003817100381710038171003817(119906 119901 119879)10038171003817100381710038171198671times1198712times1198671

equiv1

radicRe1199061198671(Ω)3

+1003817100381710038171003817119901

10038171003817100381710038171198712+ 1198791198671(Ω)

(13)


diff =

10038171003817100381710038171003817(119906119899 119901119899 119879119899) minus (119906

119899minus1 119901119899minus1

119879119899minus1

)100381710038171003817100381710038171198671times1198712times1198671


10038171003817100381710038171198671times1198712times1198671

le ErrNS

(14)



3 Implementation


119870119906 = 119891 (15)

is written as

[[[[[[[[[

[

119870(1)

1198681198680 sdot sdot sdot 0 119870

(1)

119868119861119877(1)

119861

0 d

d


119868119868119870(119873)

119868119861119877(119873)

119861

119877(1)119879

119861119870(1)119879


(119873)119879

119861119870(119873)119879

119868119861119870119861119861

]]]]]]]]]

]

[[[[[[[[[

[

119906(1)

119868

119906(119873)

119868

119906119861

]]]]]]]]]

]

=

[[[[[[[[[

[

119891(1)

119868

119891(119873)

119868

119891119861

]]]]]]]]]

]

(16)





119873

sum

119894=1

119877(119894)119879

119861(119870(119894)

119861119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119870(119894)

119868119861)119877(119894)

119861119906119861

=

119873

sum

119894=1

119877(119894)119879

119861(119891(119894)

119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119891(119894)

119868)

(17)


119906(119894)

119868= 119870(119894)minus1

119868119868(119891(119894)

119868minus 119870(119894)119879

119868119861119877(119894)

119861119906119861) 119894 = 1 119873 (18)


119861in to (18)


119868







Part interface

ei

ecurrent

R2

ne (1205821)

ne (1205822)

ne (1205823)




119894


119890119894+1

= 119899119890(Min1205821 1205822 1205823)


119894



119899minus1








0 1000 2000 3000 4000 5000Iteration times

120575 = 0001

120575 = 0005

120575 = 001

120575 = 01

120575 = 1

120575 = 10

120575 = 100


Resid

ual

(a)

0 2000 4000 6000 8000 10000Iteration times

Resid

ual

Re = 10E6 (120575 = 0005)Re = 10E5 (120575 = 0005)Re = 10E3 (120575 = 0005)Re = 10E4 (120575 = 0005)



(b)


119870


0100200300400500600700800

90 180 270 360 450 540 630 720 810 900 990 1080

NoneCurrent

Num

ber o

f ite

ratio

ns





















0

400

800

1200

1600

2000

2400

2800

5000 35000 65000 95000 125000 155000 185000

Elap

sed

time (

s)

DOF

Current


(a)

5000 35000 65000 95000 125000 155000 185000DOF

Current


0

100

200

300

400

500

Tota

l mem

ory

usag

e (G

B)(b)


12345678

8 16 24 32 40 48 56 64


Number of PEs

Spee

d-up








x1

x2

x3

0

1

1

4

b


0010203040506070809

1

0 02 04 06 08 1 12 14


x3b

u3Uminus04 minus02









0 1

1

1

x1

x2

x3












015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)



11199092(left) plane 119909


21199093(right)





H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎand 119876

ℎ A


sum

119870isinIℎ

120575119870ℎ2

119870(nabla119901119899




120575119870=

120572

for log10[Max 10038171003817100381710038171003817nabla119901

119899minus1

ℎ

1003817100381710038171003817100381724

119894=1] le 1


119899minus1

ℎ

1003817100381710038171003817100381724

119894=1]

otherwise

(9)






1198831(119906119899minus1

ℎ Δ119905) equiv 119909 minus 119906

119899minus1

ℎΔ119905 (10)


Step 2 Find 119879119899

ℎby

(

119879119899


ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 120579ℎ) + (119886nabla119879

119899

ℎ nabla120579ℎ) = (119878

119899 120579ℎ)

(11)

Step 3 Find (119906119899

ℎ 119901119899

ℎ) by

(

119906119899


ℎ∘ 1198831(119906119899minus1

ℎ Δ119905)

Δ119905 Vℎ) + 1198860(119906119899

ℎ Vℎ)

+ 119887 (Vℎ 119901119899

ℎ) + 119887 (119906

119899

ℎ 119902ℎ)

+ sum

119870isinIℎ

120575119870ℎ2

119870(nabla119901119899


= (119891119899 Vℎ) + (120573 (119879

119903minus 119879119899

ℎ) 119892 Vℎ)

(12)


defined by

1003817100381710038171003817(119906 119901 119879)10038171003817100381710038171198671times1198712times1198671

equiv1

radicRe1199061198671(Ω)3

+1003817100381710038171003817119901

10038171003817100381710038171198712+ 1198791198671(Ω)

(13)


diff =

10038171003817100381710038171003817(119906119899 119901119899 119879119899) minus (119906

119899minus1 119901119899minus1

119879119899minus1

)100381710038171003817100381710038171198671times1198712times1198671


10038171003817100381710038171198671times1198712times1198671

le ErrNS

(14)



3 Implementation


119870119906 = 119891 (15)

is written as

[[[[[[[[[

[

119870(1)

1198681198680 sdot sdot sdot 0 119870

(1)

119868119861119877(1)

119861

0 d

d


119868119868119870(119873)

119868119861119877(119873)

119861

119877(1)119879

119861119870(1)119879


(119873)119879

119861119870(119873)119879

119868119861119870119861119861

]]]]]]]]]

]

[[[[[[[[[

[

119906(1)

119868

119906(119873)

119868

119906119861

]]]]]]]]]

]

=

[[[[[[[[[

[

119891(1)

119868

119891(119873)

119868

119891119861

]]]]]]]]]

]

(16)





119873

sum

119894=1

119877(119894)119879

119861(119870(119894)

119861119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119870(119894)

119868119861)119877(119894)

119861119906119861

=

119873

sum

119894=1

119877(119894)119879

119861(119891(119894)

119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119891(119894)

119868)

(17)


119906(119894)

119868= 119870(119894)minus1

119868119868(119891(119894)

119868minus 119870(119894)119879

119868119861119877(119894)

119861119906119861) 119894 = 1 119873 (18)


119861in to (18)


119868







Part interface

ei

ecurrent

R2

ne (1205821)

ne (1205822)

ne (1205823)




119894


119890119894+1

= 119899119890(Min1205821 1205822 1205823)


119894



119899minus1








0 1000 2000 3000 4000 5000Iteration times

120575 = 0001

120575 = 0005

120575 = 001

120575 = 01

120575 = 1

120575 = 10

120575 = 100


Resid

ual

(a)

0 2000 4000 6000 8000 10000Iteration times

Resid

ual

Re = 10E6 (120575 = 0005)Re = 10E5 (120575 = 0005)Re = 10E3 (120575 = 0005)Re = 10E4 (120575 = 0005)



(b)


119870


0100200300400500600700800

90 180 270 360 450 540 630 720 810 900 990 1080

NoneCurrent

Num

ber o

f ite

ratio

ns





















0

400

800

1200

1600

2000

2400

2800

5000 35000 65000 95000 125000 155000 185000

Elap

sed

time (

s)

DOF

Current


(a)

5000 35000 65000 95000 125000 155000 185000DOF

Current


0

100

200

300

400

500

Tota

l mem

ory

usag

e (G

B)(b)


12345678

8 16 24 32 40 48 56 64


Number of PEs

Spee

d-up








x1

x2

x3

0

1

1

4

b


0010203040506070809

1

0 02 04 06 08 1 12 14


x3b

u3Uminus04 minus02









0 1

1

1

x1

x2

x3












015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)



11199092(left) plane 119909


21199093(right)





H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119873

sum

119894=1

119877(119894)119879

119861(119870(119894)

119861119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119870(119894)

119868119861)119877(119894)

119861119906119861

=

119873

sum

119894=1

119877(119894)119879

119861(119891(119894)

119861minus 119870(119894)119879

119868119861119870(119894)minus1

119868119868119891(119894)

119868)

(17)


119906(119894)

119868= 119870(119894)minus1

119868119868(119891(119894)

119868minus 119870(119894)119879

119868119861119877(119894)

119861119906119861) 119894 = 1 119873 (18)


119861in to (18)


119868







Part interface

ei

ecurrent

R2

ne (1205821)

ne (1205822)

ne (1205823)




119894


119890119894+1

= 119899119890(Min1205821 1205822 1205823)


119894



119899minus1








0 1000 2000 3000 4000 5000Iteration times

120575 = 0001

120575 = 0005

120575 = 001

120575 = 01

120575 = 1

120575 = 10

120575 = 100


Resid

ual

(a)

0 2000 4000 6000 8000 10000Iteration times

Resid

ual

Re = 10E6 (120575 = 0005)Re = 10E5 (120575 = 0005)Re = 10E3 (120575 = 0005)Re = 10E4 (120575 = 0005)



(b)


119870


0100200300400500600700800

90 180 270 360 450 540 630 720 810 900 990 1080

NoneCurrent

Num

ber o

f ite

ratio

ns





















0

400

800

1200

1600

2000

2400

2800

5000 35000 65000 95000 125000 155000 185000

Elap

sed

time (

s)

DOF

Current


(a)

5000 35000 65000 95000 125000 155000 185000DOF

Current


0

100

200

300

400

500

Tota

l mem

ory

usag

e (G

B)(b)


12345678

8 16 24 32 40 48 56 64


Number of PEs

Spee

d-up








x1

x2

x3

0

1

1

4

b


0010203040506070809

1

0 02 04 06 08 1 12 14


x3b

u3Uminus04 minus02









0 1

1

1

x1

x2

x3












015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)



11199092(left) plane 119909


21199093(right)





H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1000 2000 3000 4000 5000Iteration times

120575 = 0001

120575 = 0005

120575 = 001

120575 = 01

120575 = 1

120575 = 10

120575 = 100


Resid

ual

(a)

0 2000 4000 6000 8000 10000Iteration times

Resid

ual

Re = 10E6 (120575 = 0005)Re = 10E5 (120575 = 0005)Re = 10E3 (120575 = 0005)Re = 10E4 (120575 = 0005)



(b)


119870


0100200300400500600700800

90 180 270 360 450 540 630 720 810 900 990 1080

NoneCurrent

Num

ber o

f ite

ratio

ns





















0

400

800

1200

1600

2000

2400

2800

5000 35000 65000 95000 125000 155000 185000

Elap

sed

time (

s)

DOF

Current


(a)

5000 35000 65000 95000 125000 155000 185000DOF

Current


0

100

200

300

400

500

Tota

l mem

ory

usag

e (G

B)(b)


12345678

8 16 24 32 40 48 56 64


Number of PEs

Spee

d-up








x1

x2

x3

0

1

1

4

b


0010203040506070809

1

0 02 04 06 08 1 12 14


x3b

u3Uminus04 minus02









0 1

1

1

x1

x2

x3












015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)



11199092(left) plane 119909


21199093(right)





H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

400

800

1200

1600

2000

2400

2800

5000 35000 65000 95000 125000 155000 185000

Elap

sed

time (

s)

DOF

Current


(a)

5000 35000 65000 95000 125000 155000 185000DOF

Current


0

100

200

300

400

500

Tota

l mem

ory

usag

e (G

B)(b)


12345678

8 16 24 32 40 48 56 64


Number of PEs

Spee

d-up








x1

x2

x3

0

1

1

4

b


0010203040506070809

1

0 02 04 06 08 1 12 14


x3b

u3Uminus04 minus02









0 1

1

1

x1

x2

x3












015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)



11199092(left) plane 119909


21199093(right)





H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x1

x2

x3

0

1

1

4

b


0010203040506070809

1

0 02 04 06 08 1 12 14


x3b

u3Uminus04 minus02









0 1

1

1

x1

x2

x3












015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)



11199092(left) plane 119909


21199093(right)





H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










015

012

009

006

003

0

(a)

021

016

012

007

002

minus003

(b)



11199092(left) plane 119909


21199093(right)





H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H = 0101

W=09

L = 15

S = 0049

h = 0052

l=005

x1

x2x3


(a)

r

(b)


02468

10121416

0 100 200 300 400 500 600 700 800 900



Re

Reat

tach

men

t len

gth

(rS

)




4

1

1

x1

x2

x3

ThighTlow






3= 0) as what is



0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0005

01015

02

0 01 02 03 04 05

Exact value

u3 (middot 0 2)

Num Res 0Num Res 1


minus005

minus01

minus015

minus02


1

1

1

x1

x2

x3

ThighTlow



















5 Conclusions



369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










369

271

173

75

minus24

minus122

(a)

19

15

12

8

4

0

(b)

1

08

06

04

02

0

(c)

7094

4489

1884

minus721

minus3326

minus5931

(d)


0

5

10

15

20

25

0 01 02 03 04 05 06 07 08 09 1



x1

minus5

minus10

minus15

minus20

minus25

Velo

city

(w)

(a)

0

01

02

03

04

05

06

07

08

09

1

0 01 02 03 04 05 06 07 08 09 1

Current

Tem

pera

ture


x1

(b)



0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu0 10E3

Nu0 10E4

Nu0 10E5

Nu0 10E6

Nu0 10E7

(a)

0

01

02

03

04

05

06

07

08

09

1

0 10 20 30 40Nu

x3

Nu1 10E3

Nu1 10E4

Nu1 10E5

Nu1 10E6

Nu1 10E7

(b)


Acknowledgments


References













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article A Pressure-Stabilized Lagrange-Galerkin...

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