a new stabilized formulation for convective...

A NEW STABILIZED FORMULATION FORCONVECTIVE-DIFFUSIVE HEAT TRANSFER

M. AyubDepartment of Mathematics, Quaid-e-Azam University,Islamabad, Pakistan

A. MasudDepartment of Civil & Materials Engineering,University of Illinois at Chicago, Chicago, Illinois, USA

This article presents a new stabilized finite-element formulation for convective-diffusive heat

transfer. A mixed temperature and temperature-flux form is proposed that possesses better

stability properties as compared to the classical Galerkin form. The issue of arbitrary

combinations of temperature and temperature-flux interpolation functions is addressed.

Specifically, the combinations of C ˚ interpolations that are unstable according to the

Babuska–Brezzi inf-sup condition are shown to be stable and convergent within the present

framework. Based on the proposed formulation, a family of 2-D elements comprising 3- and

6-node triangles and 4- and 9-node quadrilaterals has been developed. Numerical results

show the good performance of the method and confirm convergence at optimal rates.

1. INTRODUCTION

This article presents a new stabilized finite-element formulation for heattransfer in fluids via conduction and convection. For a fluid in which the fluidparticles are at rest, the problem becomes one of simple heat conduction, which isalso encountered in solids. However, if the fluid particles are in motion, energy istransported both by potential gradients and by the movement of the fluid particlesthemselves. This transport phenomena is usually referred to as convection [1]. Finite-element, finite-difference, finite-volume, and boundary-element techniques have been

Received 19 September 2002; accepted 31 January 2003.

The authors wish to thank Prof. W. J. Minkowycz for helpful discussion. M. Ayub gratefully ac-

knowledges the Government of Pakistan, Ministry of Science and Technology, for the award of a post

doctoral fellowship for his visit to University of Illinois at Chicago. Partial support for this work was pro-

vided by the National Science Foundation grant CMS-98133860 and Office of Naval Research grant

N00014-02-1-0143. This support is gratefully acknowledged. The authors also wish to thank Georgette

Hlepas for preparing this manuscript.

M. Ayub is a Visiting Professor at the Department of Civil & Materials Engineering, University of

Illinois at Chicago.

Address correspondence to A. Masud, University of Illinois at Chicago, Department of Civil &

Materials Engineering (M/C 246), 842 W. Taylor St., Chicago, IL 60607-7023, USA. E-mail: amasud

@uic.edu

Numerical Heat Transfer, Part B, 44: 1–23, 2003

Copyright # Taylor & Francis Inc.

ISSN: 1040-7790 print/1521-0626 online

DOI: 10.1080/10407790390121989

1

employed in the literature to model steady as well as transient heat transfer phe-nomena (see, e.g., [2–8] and references therein). The necessity of having a higher-order-accurate temperature flux field in several critical applications has motivatedresearchers to investigate the mixed form of the equations in which the temperatureflux appears as an independent field variable in addition to the scalar temperaturefield. However, application of the standard Galerkin finite-element approach to themixed form of heat equation has not been easy, as it results in several numericaldeficiencies, namely: (1) equal-order interpolation functions for temperature andtemperature flux usually do not work, and (2) the convection term needs specialtreatment. Accordingly, various approaches based on (1) upwinding techniques, (2)the modified method of characteristics, (3) Petrov–Galerkin methods, (4) reducedand selective integration techniques, and (5) use of special interpolation polynomialshave been proposed. Nevertheless, development of elements that employ practicallyconvenient temperature and temperature flux interpolation functions has been aformidable task.

It is interesting to note that similar pathologies also appear in some alliedphysical problems, namely, the Stokes flow equations [9–12] and the advection-diffusion equations [13–17]. A standard Galerkin approach to Stokes problem shows

NOMENCLATURE

be bubble shape functions over element

domains

B abstract bilinear form for the Galer-

kin formulation

J temperature flux field�JJ coarse-scale temperature flux field

J 0 fine-scale temperature flux field

L abstract linear form for the Galerkin

formulation

n unit normal to the boundary

P space of trial solutions and weighting

functions for the temperature field

q arbitrary weighting function for the

temperature field

s heat flow rate of source or sink

V space of trial solution and weighting

functions for the temperature flux

field

V0 space of trial solutions and weighting

functions for the fine scale tempera-

ture flux field

w arbitrary weighting function for the

temperature flux field

�ww arbitrary weighting function for the

coarse-scale temperature flux field

w0 arbitrary weighting function for the

fine-scale temperature flux field�WW the coarse-scale subproblem

W0 the fine-scale subproblem

a the given velocity field, m=s

bi coefficients for the fine-scale trial

solutions

gi coefficients for the fine-scale

weighting functions

G piecewise smooth boundary

Ge element boundaries

Gg partition of the boundary with

prescribed essential boundary

condition

Gh partition of the boundary with

prescribed natural boundary

condition

G0 union of element boundaries

k thermal conductivity, w=m ˚C

y prescribed boundary temperature

t stabilization function

j scalar temperature field, ˚C

c prescribed boundary flux

O an open bounded region

Oe element domain

O0 union of element interiors

Subscripts

nsd number of spatial dimensions

numel number of elements in the mesh

2 M. AYUB AND A. MASUD

wildly oscillating pressure field if arbitrary combinations of interpolation functionsfor pressure and velocity fields are employed. This anomalous behavior is attributedto the lack of stability in the pressure field. Mathematically, this behavior isexplained by the celebrated Babuska–Brezzi (BB) inf-sup conditions [9, 11, 12, 18,19] that govern the selection of finite-element interpolations. In particular, the BBconditions render the convenient equal-order interpolations for the velocity andpressure fields ineffective.

In advection-dominated diffusion phenomenon, the standard Galerkin finite-element method again results in nonconvergent elements. In order to correct thedeficiencies in the standard Galerkin approach, Hughes and colleagues introducedthe streamline-upwind-Petrov–Galerkin (SUPG) technique [12, 14, 16]. This tech-nique turned out to be the forerunner of a new class of stabilization schemes,called Galerkin=least-square (GLS) stabilization methods [11, 15, 17]. The essentialfeatures of these stabilized methods are: (1) circumvention of the BB (inf-sup)condition that restricts the use of many convenient interpolations, and simulta-neously, (2) stabilization of the advection operator. The key idea in the GLSstabilization approach is to augment the Galerkin finite-element formulation with aleast-squares form of the residuals that are based on the corresponding Euler–Lagrange equations. These least-squares integrals contain stabilization parametersthat are designed so that the method achieves exact solution in the case of one-dimensional model problems.

In [20, 21], Hughes revisited the origins of the stabilization schemes from avariational multiscale approach and crystallized the definition of the stabilizationparameters. In Hughes’s variational multiscale (HVM) method, different stabiliza-tion techniques appear as special cases of the underlying subgrid-scale modelingconcept. Extending this idea further, Masud [22] developed a stabilized formulationfor the incompressible Navier–Stokes equations that is based on the notion of theexistence of fine scales in the problem. The beauty of the formulation is that it is freeof any mesh-dependent parameter that needs to be designed to ensure the stability ofthe formulation. The present article follows along the lines of Masud [22], andpresents a stabilized finite-element formulation for convection-diffusion heattransfer equations. The resulting stabilized mixed finite-element formulationaccommodates equal-order interpolations for both the temperature and the tem-perature flux fields. The new formulation, hereafter termed the HVM formulation,has improved stability compared to the Galerkin formulation for the problem.

An outline of the article is as follows: Section 2 presents the governing equa-tions and the standard Galerkin form. Emphasis in the article is on the variationalmultiscale approach to the development of a new stabilized form, which is presentedin Section 3. The HVM stabilized form is presented in Section 4. Section 5 presentsnumerical results, and conclusions are drawn in Section 6.

2. A MIXED FORMULATION FOR CONVECTIVE-DIFFUSIVEHEAT TRANSFER

Let O � Rnsd be an open bounded region with piecewise smooth boundary G.We assume that nsd � 2. We consider the steady-state convective-diffusive problemwhich consists of finding the scalar temperature field j and its flux J. The strong

A NEW STABILIZED FORMULATION 3

form of the problem can be stated as follows: Given the velocity field a and con-ductivity k > 0, find j and J such that

J ¼ aj� kHj in O ð1Þ

div J ¼ s in O ð2Þ

J � n ¼ c on G ð3Þ

j ¼ y on G ð4Þ

where s is the heat flow rate of source or sink and n is the unit normal to theboundary G.

2.1. Standard Galerkin Form

Consider the bounded domain O discretized into nonoverlapping regions Oe

(element domains) with boundaries Ge; e ¼ 1; 2; . . . ; numel, such that

O ¼[numel

e¼1

Oe ð5Þ

Let Gg and Gh be subsets of the smooth boundary G which satisfy the followingconditions:

Gg [ Gh ¼ G ð6Þ

Gg \ Gh ¼ F ð7Þ

where the superposed bar in Eq. (6) represents set closure, and F in Eq. (7) denotesthe empty set. Further, Gg and Gh represent the partitions of the boundary withprescribed essential and natural boundary conditions, respectively.

Let V � H1ðdiv;OÞ \ C�ðOÞ denote the space of trial solution and weightingfunctions for the flux J and let P � L2ðOÞ be the space of temperature trial solutionsand weighting functions, respectively. We define

H1ðdiv;OÞ ¼ fJjJ 2 ðL2ðOÞÞnsd ; divJ 2 L2ðOÞ; traceðJ � nÞ ¼ c 2 H1=2ðGÞg ð8Þ

For further elaboration on these spaces, see Brezzi–Fortin [18], Sec. 1.4.We assume that k;a, and s are given. The mixed variational form of the

problem is: Find the solution fJ;jg 2 V � P such that for all admissible variationsfw; qg 2 V � P the following holds:

ðw; JÞ � ðw;ajÞ � ðdivw; kjÞ þ ðq; div JÞ ¼ ðq; sÞ ð9Þ

where ð�; �Þ ¼ROð�Þ dO is the L2ðOÞ inner product; w and q are the weighting func-

tions for J and j, respectively. In the derivation of Eq. (9) we have applied inte-gration by parts to ðw; k HjÞ to transfer the operator from the temperature to thetemperature flux field.


We can write the Galerkin-form Eq. (9) in abstract notation as

B w; q; J;jð Þ ¼ L w; qð Þ ð10Þ

where

Bðw; q; J;jÞ ¼ ðw; JÞ � ðw;ajÞ � ðdivw; kjÞ þ ðq; div JÞ ð11Þ

Lðw; qÞ ¼ ðq; sÞ ð12Þ

Note that B : ðV � PÞ � ðV � PÞ ! R is a bilinear form and L : V � P ! R is alinear form defined on the product space W ¼ V � P.

Remark 1. The Galerkin-form Eqs. (10)–(12) suffer from pathologies enumeratedearlier.

3. HUGHES VARIATIONAL MULTISCALE (HVM) METHOD

3.1. Multiscale Decomposition

Following along the lines of Masud [22], we now present the derivation of astabilized formulation for convective heat transfer. We denote the union of elementinteriors and element boundaries by O0 and G0, respectively.

O0 ¼[numel

e¼1

ðintÞOe (element interiors) ð13Þ

G0 ¼[numel

e¼1

Ge (element boundaries) ð14Þ

We assume an overlapping sum decomposition of the temperature flux fieldinto coarse scales or resolvable scales and fine scales or subgrid scales.

JðxÞ ¼ �JJðxÞ|ffl{zffl}coarse scale

þ J0ðxÞ|ffl{zffl}fine scale

ð15Þ

Likewise, we assume an overlapping sum decomposition of the weighting functioninto coarse- and fine-scale components indicated as �ww and w0, respectively.

wðxÞ ¼ �wwðxÞ|ffl{zffl}coarse scale

þ w0ðxÞ|fflffl{zfflffl}fine scale

ð16Þ

We further make an assumption that the subgrid scales, although nonzero within theelements, vanish identically over the element boundaries.

J0 ¼ w0 ¼ 0 on G0 ð17Þ

We now introduce the appropriate spaces of functions for the coarse- and fine-scale fields and specify a direct-sum decomposition on these spaces.

V ¼ �VV�V0 ð18Þ


where �VV in Eq. (18) is the space of trial solutions and weighting functions for thecoarse-scale temperature flux field and is identified with the standard finite elementspace defined in Eq. (8),

�JJ 2 �VV � C 0ðOÞ \ V; �VVðOeÞ ¼ PkðOeÞ ð19Þ

where PkðOeÞ denotes complete polynomials of order k over Oe.On the other hand, various characterizations of V0 are possible, subject to the

restriction imposed by the stability of the formulation that requires �VV and V0 to belinearly independent. Consequently, in the discrete case V0 can contain various finite-dimensional approximations, e.g., bubble functions or p-refinements, that satisfyEq. (17).

J0 2 V0 ¼ J0jJ0 ¼ 0 on G0f g ð20Þ

w0 2 V0 ¼ S0 ð21Þ

We assume that the fine-scale temperature field is zero, and therefore we haveonly the coarse-scale temperature component. The appropriate spaces of functionsfor the temperature field are

j 2 P � C0ðOÞ \ P; PðOeÞ ¼ PkðOeÞ ð22Þ

Remark 2. In general, the space of weighting functions satisfies the homogeneousessential boundary conditions. However, for the case of Neumann boundary con-ditions, the space of weighting functions �VV coincides with the trial solution space �SS.Furthermore, for the fine-scale fields, both S0 and V0 coincide.

We now substitute the trial solutions Eq. (15) and the weighting functionsEq. (16) in the standard variational form Eq. (9), and this becomes the point ofdeparture from the conventional Galerkin formulations.

ð�wwþ w0; �JJþ J0Þ � ð�wwþ w0;ajÞ � ðdiv ð�wwþ w0Þ; kjÞ þ ðq; div ð�JJþ J0ÞÞ ¼ ðq; sÞð23Þ

With suitable assumptions on the fine-scale field, as stipulated in Eq. (17), andemploying the linearity of the weighting function, we can split the problem intocoarse- and fine-scale parts, indicated as �WW and W0, respectively.

The coarse-scale subproblem �WW can be written as

ð�ww; �JJþ J0Þ � ð�ww;ajÞ � ðdiv �ww; kjÞ þ ðq; div ð�JJþ J0ÞÞ ¼ ðq; sÞ ð24Þ

It is important to note that the weighting-function slot in the coarse-scale problem �WWcontains only the coarse-scale components.

The fine-scale subproblem W0 can be written as

ðw0; �JJþ J0Þ � ðw0;ajÞ � ðdivw0; kjÞ ¼ 0 ð25Þ

In W0 the weighting function slot contains only the fine-scale components. Thegeneral idea at this point is to solve the fine-scale problem, defined element-wise,to obtain the fine-scale solution J0. This solution is then substituted in the coarse-scale problem, Eq. (24), thereby eliminating the fine scales, yet retaining theireffect.


3.2. The Fine-Scale Problem (W0)

Let us now consider the fine-scale part of the weak form W0, which, because ofthe assumption on the fine-scale space, is defined over O0. Exploiting linearity of thesolution slot in the first term and employing integration by parts to the second termon the left-hand side in Eq. (25), we have

ðw0; �JJÞ þ ðw0; J0Þ � ðw0;ajÞ � ðdivw0; kjÞ ¼ 0 ð26Þ

Applying integration by parts to the last term and then rearranging, we obtain

ðw0; J0Þ ¼ �ðw0; �JJ� ajþ kHjÞ ð27Þ

Rearranging and combining terms, we get

ðw0; J0Þ ¼ �ðw0; �rrÞ ð28Þ

where

�rr ¼ �JJ� ajþ kHj ð29Þ

To crystallize ideas, and without loss of generality, we assume that the finescales are represented via bubbles over element domains, i.e.,

J0jOe ¼ beðxÞb ) J0ijOe ¼ beðxÞbi ð30Þ

w0jOe ¼ beðxÞg ) w0ijOe ¼ beðxÞgi ð31Þ

where be represents the bubble shape functions over element domains, i ¼1; 2; . . . ; nsd, and bi and gi represent the coefficients for the fine-scale trial solutionsand weighting functions, respectively. Substituting Eq. (30) and Eq. (31) in the fine-scale problem Eq. (28) defined over a typical element, we get

ZO0ðbew0

ebeJ0eÞ dO ¼ �

ZO0

bew0e�rr

� �dO ð32Þ

Writing Eq. (28) in indicial notation,

ðw0i; J

0iÞ ¼ �ðw0

i; �rriÞ ð33Þ

Substituting Eq. (30) and Eq. (31) in Eq. (33), we obtain

ðbegi; bebiÞ ¼ �ðbegi; �rriÞ

In the integral form, this can be written as

gi

ZOeðbeÞ2 dO

� �bi ¼ �gi

ZOebe dO

� ��rri

Since gi is arbitrary, we have

bi ¼ �ROe be dO�rriROe ðbeÞ2 dO

ð34Þ


Putting bi in Eq. (30), we obtain

J0 ¼ �be

ROe be dO �rrR

Oe ðbeÞ2 dO) J0 ¼ �t �rr ð35Þ

where

t ¼ �be

ROe be dOR

Oe ðbeÞ2 dOð36Þ

3.3. The Coarse-Scale Problem ð �WWÞ

Employing linearity of the solution slot in (24), we get

ð�ww; �JJÞ þ ð�ww; J0Þ � ð�ww;ajÞ � ðdiv �ww; kjÞ þ ðq; div �JJÞ þ ðq; div J0Þ ¼ ðq; sÞ ð37Þ

Integration-by-parts of ðq; div J0Þ together with the condition Eq. (17) on finescales leads to the second and sixth terms being combined as ð�ww; J0Þþðq; divJ0Þ ¼ ð�ww� Hq; J0Þ. Consequently, we obtain

ð�ww; �JJÞ � ð�ww;ajÞ � ðdiv �ww; kjÞ þ ðq; div �JJÞ þ ð�ww� Hq; J0Þ ¼ ðq; sÞ ð38Þ

Using Eq. (35) in Eq. (38) and then substituting �rr from Eq. (29), we obtain

ð�ww; �JJÞ � ð�ww;ajÞ � ðdiv �ww; kjÞ þ ðq; div �JJÞ � ð�ww� Hq; tð�JJ� ajþ kHjÞÞ ¼ ðq; sÞð39Þ

4. THE HVM FORM

We assume the existence of a bubble function such that t ¼ 12. Therefore, Eq.

(39) becomes

ð�ww; �JJÞ � ð�ww;ajÞ � ðdiv �ww;kjÞ þ ðq; div �JJÞ þ 1

2ð��wwþ Hq; �JJ� ajþ kHjÞ ¼ ðq; sÞ

ð40Þ

Furthermore, in order to obtain convergence for the flux field in H1ðdiv ;OÞspace, we add the least-squares stabilization term for Eq. (2) to Eq. (40), pre-multiplied by t ¼ 1

2 as well.

ð�ww; �JJÞ � ð�ww;ajÞ � ðdiv �ww; kjÞ þ ðq; div �JJÞ þ 1

2ð��wwþ HqÞ; ð�JJ� ajþ kHjÞ� �

þ 1

2div �ww;

h2

kðdiv �JJ� sÞ

� �¼ ðq; sÞ ð41Þ

It is important to realize that the zeroth-order term in the weighting functionslot for the first stabilization integral in Eq. (41) has a negative sign. In thisderivation this feature appears naturally and provides better stability to the HVMformulation. Furthermore, the HVM stabilized form Eq. (41) is defined completely


in terms of the resolvable coarse scales in which the effect of the fine scales isrepresented in the form of residuals of the resolvable scales (see Hughes [20]). Inorder to keep the notation simple, we will drop the superposed bars from here on.

Simplifying the above expression, we get

1

2ðw; JÞ � 1

2ðw;ajÞ � 1

2ðdivw; kjÞ þ 1

2ðq; div JÞ � 1

2ðHq;ajÞ

þ 1

2ðHq; kHjÞ þ 1

2divw;

h2

kdiv J

� �¼ ðq; sÞ þ 1

2divw;

h2

ks

� �ð42Þ

We can write the HVM-form Eq. (42) in abstract notation as

BHVM w; q; J;jð Þ ¼ LHVM w; qð Þ ð43Þ

where

BHVMðw; q; J;jÞ ¼ 1

2B w; q; J;jð Þ � 1

2ðHq;ajÞ þ 1

2ðHq; kHjÞ þ 1

2divw;

h2

kdiv J

� �

ð44Þ

LHVMðw; qÞ ¼ Lðw; qÞ þ 1

2

�divw;

h2

ks

�ð45Þ

where B w; q; J;jð Þ and Lðw; qÞ are given in Eq. (11) and Eq. (12), respectively.

5. NUMERICAL EXAMPLES

Figure 1 shows the elements employed in the numerical studies. Dots corres-pond to the temperature nodes and circles correspond to the temperature flux nodes.

Figure 1. A family of 2-D linear and quadratic elements.


The following quadrature rules were used throughout: linear quadrilaterals, 2� 2Gauss quadrature; quadratic quadrilaterals, 3� 3 Gauss quadrature; linear trian-gles, 4-point quadrature; quadratic triangles, 7-point quadrature (see Hughes [23],chap. 3).

5.1. Convergence Study

The first numerical simulation is a study of convergence rates. The domainunder consideration is a biunit square, and the exact temperature solution is given by

j ¼ sin2pxL

sin2pyL

ð46Þ

The temperature flux is computed from Eq. (1); s is calculated from Eq. (2) by takingdivergence of the temperature flux field. In specifying the boundary-value problem,s is integrated over O while c is prescribed nodally at the boundary.

For linear quadrilateral elements, the meshes employed consisted of 100, 400,1,600, and 6,400 elements. The linear triangular element meshes consisted of exactlytwice as many elements. For quadratic quadrilateral elements, the meshes employedconsisted of 25, 100, 400, and 1,600 elements. Again, the quadratic triangular ele-ment meshes consisted of twice as many elements. The element mesh parameter, h, istaken to be the edge length of the elements for the quadrilaterals, and the short-edgelength for triangles.

5.1.1. Continuous temperature elements. In the first case (Figures 2–5)we consider the stabilized formulation given in Eqs. (43)–(45). Figures 2 and 3 showthe L2 norm and H1 seminorm of the temperature field, and the L2 norm of thetemperature flux field for the linear quadrilaterals and triangles. The theoreticalconvergence rates for the HVM form presented in Eq. (43) (excluding theconvective terms) are presented by Masud and Hughes [24]. We see that the rates

Figure 2. Convergence rates for continuous temperature 4-node quadrilaterals.


predicted by the theory are achieved. In fact, optimal L2 rates of convergence fortemperature and temperature flux are also attained for these cases. Figures 4 and5 present the corresponding rates for the quadratic elements. We see optimal ratesfor the temperature field. The rate for the temperature flux field is less thanoptimal in the L2 norm; however, the attained rate is in accordance with thetheoretical predictions.

5.1.2. Discontinuous temperature elements. The second test caseinvestigates the discontinuous temperature field elements. The method isconvergent for discontinuous temperature interpolations as long as temperatureflux is interpolated with quadratic functions or higher (see Masud and Hughes

Figure 3. Convergence rates for continuous temperature 3-node triangles.

Figure 4. Convergence rates for continuous temperature 9-node quadrilaterals.


[24] for an equivalent field theory). The computed rates for the 9-node quadrilateralsand 6-node triangles are presented in Figures 6 and 7. Optimal rates are noted for thetemperature. The labels in the figures ‘‘H1 Dis-Temp’’ mean the temperature error inthe H1ð~OOÞ norm in the discontinuous-temperature case.

It is interesting to note that even though the theory does not predict con-vergence for the 4-node quadrilaterals with discontinuous temperature fields (seeMasud and Hughes [24]), we have obtained optimal convergence rates in all normsfor this element as well. Figure 8 shows the computed convergence rates for thediscontinuous-temperature linear quadrilaterals. However, the element-wise con-stant part of the temperature field for discontinuous temperature linear triangles is

Figure 5. Convergence rates for continuous temperature 6-node triangles.

Figure 6. Convergence rates for discontinuous temperature 9-node quadrilaterals.


not stabilized by this formulation. Numerical results (not shown) exhibitedtemperature oscillations.

Figures 9a and 9b present the elevation plots of the temperature and theabsolute value of the temperature flux fields for the continuous temperature case fora representative mesh of 4-node elements. Likewise, we have plotted the contours ofthe temperature and the temperature flux fields for the continuous case. Figures 10aand 10b present the contour plots of the exact temperature flux field for convectionvelocity a ¼ 0 and a ¼ 10, respectively. Figures 11a and 11b present the contourplots of the computed temperature flux field for 4-node quadrilaterals, for values of

Figure 7. Convergence rates for discontinuous temperature 6-node triangles.

Figure 8. Convergence rates for discontinuous temperature 4-node quadrilaterals.


Figure 9 (a). Elevation plot of the temperature field. (b) Elevation plot of the absolute value of the

temperature-flux field (a=1).


Figure 10. Absolute temperature flux contours. Exact solution. (a) Convection velocity a¼ 0. (b)

Convection velocity a=10.


Figure 11. Absolute temperature flux contours for 4-node elements. (a) Convection velocity a¼ 0.

(b) Convection velocity a¼ 10.


a ¼ 0 and a ¼ 10, respectively. These plots present a qualitative comparison of thecomputed solution with the corresponding exact solution, and show a stableresponse of the elements. Likewise, Figures 12a–12b and 13a–13b present the com-puted temperature flux fields for the 3-node triangles and the 9-node quadrilaterals,for values of a ¼ 0 and a ¼ 10, respectively. Similar plots were obtained for the6-node triangles and are therefore not shown here.

5.2. The Convection-Dominated Problem

This section presents numerical results for a 1-D convection-dominated pro-blem for which the exact solution is known in terms of the Peclet number of theproblem (see Hughes and Brooks [16]). The element Peclet number in this simulationis 0.3. The prescribed boundary conditions are temperature j ¼ 0 at x ¼ 0, andj ¼ 1 at x ¼ 1. Figure 14 presents the performance of the linear elements, whileFigure 15 presents the results for the quadratic elements. Also plotted is the exactsolution for a quantitative comparison. Figure 16 presents the comparison with theexact solution for 4-node elements for the case where there is no div stabilization andthe case with the proposed div stabilization terms. Similar behavior was observed forthe other element types and is therefore not shown here. In all cases, we observed

Figure 12. Absolute temperature flux contours for 3-node triangles. (a) Convection velocity a¼ 0.



Figure 12. Continued.

Figure 13. Absolute temperature-flux contours for 9-node quadrilaterals. (a) Convection velocity a¼ 0.



oscillations in the absence of the div stabilizing terms. A 3-D view of the temperatureprofile for the 4-node quadrilaterals without and with div stabilization is shown inFigures 17a and 17b, respectively. Once again a stable response is observed in theentire domain with the div stabilized formulation Eq. (42).

Figure 13. Continued.

Figure 14. Temperature distribution for linear elements for the convection-dominated problem.


6. CONCLUSIONS

We have presented a new stabilized mixed finite element method for con-vective-diffusive heat transfer. The proposed method possesses better stabilityproperties compared with the classical Galerkin form. A salient feature of the for-mulation is that it allows equal-order interpolations for the temperature and thetemperature flux fields that are otherwise unstable within the classical Galerkinmethod. The proposed method works equally well for continuous and discontinuoustemperature elements. The numerically obtained convergence rates confirm thetheoretically predicted rates for equal-order linear and quadratic elements, as aregiven by Masud and Hughes [24]. In several cases, optimal rates are attained in the

Figure 15. Temperature distribution for quadratic elements for the convection-dominated problem.

Figure 16. Temperature distribution for linear elements with and without div stabilization.


norms considered. Qualitative and quantitative comparison of the computed resultswith various test problems confirm the superior performance of the proposedmethod.

Figure 17. Elevation plots of the temperature field for 4-node quadrilaterals: (a) without div stabilization;

(b) with div stabilization.


REFERENCES

1. W. M. Kays, Convective Heat and Mass Transfer, 2d ed., chap. 4, McGraw-Hill, New

York, 1966.2. B. R. Baliga, A Control-Volume Based Finite Element Method for Convective Heat and

Mass Transfer, Ph.D. thesis, University of Minnesota, Mineapolis, MN, 1978.

3. B. R. Baliga and S. V. Patankar, A New Finite Element Formulation for Convection-Diffusion Problems, Numer. Heat Transfer, vol. 3, pp. 393–409, 1980.

4. T. J. R. Hughes, I. Levit, and J. M. Winget, Element-by-Element Implicit Algorithms for

Heat Conduction, J. Eng. Mech., vol. 109(2), pp. 576–585, 1983.5. T. J. R. Hughes and T. E. Tezduyar, Analysis of Some Fully Discrete Algorithms for the

One-Dimensional Heat Equation, Int. J. Numer. Meth. Eng., vol. 21, pp. 163–168, 1985.6. K. A. R. Ismail and V. L. Scalon, A Finite Element Free Convection Model for the Side

Wall Heated Cavity, Int. J. Heat Mass Transfer, vol. 43, pp. 1373–1389, 2000.7. R. Marshall, J. Heinrich, and O. Zienkiewicz, Natural Convection in a Square Enclosure

by a Finite-Element, Penalty Function Method Using Primitive Fluid Variables, Numer.

Heat Transfer, vol. 1, pp. 433–451, 1978.8. J. M. Winget and T. J. R. Hughes, Solution Algorithms for Non-Linear Transient Heat

Conduction Analysis Employing Element-by-Element Iterative Strategies, Comput. Meth.

Appl. Mech. Eng., vol. 52, pp. 711–815, 1985.9. F. Brezzi, J. Douglas, and L. D. Marini, Two Families of Mixed Finite Elements for

Second Order Elliptic Problems, Numer. Math., vol. 47, pp. 217–235, 1985.

10. F. Brezzi, L. P. Franca, T. J. R. Hughes, and A. Russo, b ¼Rg, Comput. Meth. Appl.

Mech. Eng., vol. 145. pp. 329–339, 1997.11. L. P. Franca and T. J. R. Hughes, Two Classes of Mixed Finite Element Methods,

Comput. Meth. Appl. Mech. Eng., vol. 69, pp. 89–129, 1988.

12. T. J. R. Hughes, L. P. Franca, and M. Balestra, A New Finite Element Formulation forComputational Fluid Dynamics: V. Circumventing the Babuska–Brezzi Condition: AStable Petrov–Galerkin Formulation of the Stokes Problem Accommodating Equal-Order

Interpolations, Comput. Meth. Appl. Mech. Eng., vol. 59, pp. 85–99, 1986.13. F. Brezzi and A. Russo, Choosing Bubbles for Advection-Diffusion Problems, Math.

Models Meth. Appl. Sci., vol. 4, pp. 571–587, 1994.

14. A. N. Brooks and T. J. R. Hughes, Streamline Upwind=Petrov–Galerkin Methods forConvection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations, Comput. Meth. Appl. Mech. Eng., vol. 32, pp. 199–259, 1982.

15. L. P. Franca, S. L. Frey, and T. J. R. Hughes, Stabilized Finite Element Methods: I.

Application to the Advective-Diffusive Model, Comput. Meth. Appl. Mech. Eng., vol. 95,pp. 253–276, 1992.

16. T. J. R. Hughes and A. N. Brooks, Streamline-Upwind=Petrov–Galerkin Methods for

Advection Dominated Flows, Proc. 3d Int. Conf. on Finite Element Methods in Fluid Flow,Banff, Canada, 1980, pp. 283–292.

17. T. J. R. Hughes, L. P. Franca, and G. M. Hulbert, A New Finite Element Formulation for

Computational FluidDynamics: VIII. TheGalerkin=Least-SquaresMethod for Advective-Diffusive Equations, Comput. Meth. Appl. Mech. Eng., vol. 73, pp. 173–189, 1989.

18. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in

Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991.19. D. S. Malkus and T. J. R. Hughes, Mixed Finite Element Methods—Reduced and

Selective Integration Techniques: A Unification of Concepts, Comput. Meth. Appl. Mech.Eng., vol. 15, pp. 63–81, 1978.

20. T. J. R. Hughes, Multiscale Phenomena: Green’s Functions, the Dirichlet-to-NeumannFormulation, Subgrid Scale Models, Bubbles and the Origins of Stabilized Methods,Comput. Meth. Appl. Mech. Eng., vol. 127, pp. 387–401, 1995.


21. T. J. R. Hughes, G. Feijoo, L. Mazzei, and J.-B. Quincy, The Variational Multiscale

Method: A Paradigm for Computational Mechanics, Comput. Meth. Appl. Mech. Eng.,vol. 166, pp. 3–24, 1998.

22. A. Masud, On a Stabilized Finite Element Formulation for Incompressible Navier-Stokes

Equations, Proc. 4th U.S.–Japan Conf. on Computational Fluid Dynamics, Tokyo, Japan,2002 pp. 33–38.

23. T. J. R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite ElementAnalysis, chap. 3, Prentice-Hall, Englewoods Cliffs, NJ, 1987; Dover, New York, 2000.

24. A. Masud and T. J. R. Hughes, A Stabilized Mixed Finite Element Method for DarcyFlow, Comput. Meth. Appl. Mech. Eng., vol. 191, no. 39–40, pp. 4341–4370, 2002.


a new stabilized formulation for convective...

Documents