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A new finite element procedure for the solution of the incompressible Navier–Stokes equations is presented. In the Petrov–Galerkin formulation employed, the velocities are interpolated using the flow conditions over the elements and the pressure is

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A ﬂow-condition-based interpolation ﬁnite elementprocedure for incompressible ﬂuid ﬂows
Klaus-J
€
uurgen Bathe
a,*
, Hou Zhang
b
a
Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Room 3-356,Cambridge, MA 02139, USA
b
ADINA R&D, Inc., 71 Elton Ave, Watertown, MA 02472, USA
Received 11 December 2001; accepted 18 March 2002
Abstract
A new ﬁnite element procedure for the solution of the incompressible Navier–Stokes equations is presented. In thePetrov–Galerkin formulation employed, the velocities are interpolated using the ﬂow conditions over the elements andthe pressure is interpolated to satisfy the inf–sup condition for incompressible analysis. Element control volumes areemployed to satisfy local mass and momentum conservation (as in ﬁnite volume methods), which corresponds to usingstep functions as weight functions in the ﬁnite element method. An important achievement of the discretization schemeis that no artiﬁcial parameters are set in the scheme to reach stability for low and high Reynolds (and P
eeclet) numberﬂows. The solutions of nontrivial test problems are presented to demonstrate the capability and potential of thescheme.
2002 Elsevier Science Ltd. All rights reserved.
Keywords:
Finite elements; Incompressible ﬂuid ﬂows; High Reynolds numbers
1. Introduction
Finite element methods are now abundantly used inthe analysis of solids and structures. The methods areemployed in research and commercial computer pro-grams for static and dynamic, linear and nonlinear an-alyses. Many high technology companies rely extensivelyon ﬁnite element analyses of their structural designs inorder to reach optimum functionality and cost-eﬀec-tiveness.However, considering ﬂuid ﬂow analysis, the situa-tion is quite diﬀerent. While much research eﬀort hasbeen expended over the last three decades resulting innumerous publications on ﬁnite element methods forﬂuid ﬂows, by far most ﬂuid ﬂow solutions in industryare still obtained using ﬁnite volume methods, see e.g.[1,2]. Of course, researchers in ﬁnite element analysismay––and can––claim that ﬁnite volume methods are just special ﬁnite element procedures, if ﬁnite elementmethods are interpreted as discretization techniques in abroad sense [3].Major reasons why industrial computer programs forﬂuid ﬂow analysis are not based on the classical ﬁniteelement methods, after all the research expended, arethat the ‘‘traditional ﬁnite element methods’’ do notsatisfy directly local conservation in the traditional senseand have diﬃculty to converge for high Reynolds num-ber ﬂows.There are of course also shortcomings in the ﬁnitevolume methods. Compared with ﬁnite element meth-ods, the mathematical theory for incompressible ﬂuidﬂows is less strong. However, a strong mathematicaltheory is needed to reach an optimal solution scheme.Optimal convergence, as the mesh is reﬁned, is a majoraim in ﬂuid ﬂow analysis because of the ﬁne discretiza-tions that frequently need be used and which lead tohigh cost of solutions. This shortcoming arises in part
Computers and Structures 80 (2002) 1267–1277www.elsevier.com/locate/compstruc
*
Corresponding author. Tel.: +1-617-253-6645; fax: +1-617-253-2275.
E-mail address:
kjb@mit.edu (K.J. Bathe).0045-7949/02/$ - see front matter
2002 Elsevier Science Ltd. All rights reserved.PII: S0045-7949(02)00077-9
because interpolation functions of the ﬁeld variables arenot explicitly employed. The lack of these functions alsomeans that required derivatives cannot be evaluateddirectly for the evaluation of the viscous terms and to setup Jacobians for the Newton–Raphson iterations. Fur-thermore, it is unsatisfactory that––as in the ﬁnite ele-ment method––artiﬁcial parameters to reach stability areused.Hendriana and Bathe presented some thoughts inRef. [4] as to what might be an ‘‘ideal’’ solution proce-dure for ﬂuid ﬂows. This ideal solution scheme wouldalways give a reasonable solution to a well-posed ﬂuidﬂow problem provided a reasonable mesh is used. Thescheme would always converge fast in the Newton– Raphson iterations. For a coarse mesh, the solutionwould of course not be able to show some ﬂow detailsthat cannot be represented, but with reﬁnements of themesh, more of these details would be displayed withoptimal convergence. This quality of prediction of ﬂuidﬂow would hold even for ﬂows of Reynolds numbersthat would imply turbulence. Hendriana and Bathetested various widely published ﬁnite element solutionschemes and found that all of them were quite deﬁcientwhen measured on this ideal solution scheme.The objective of this paper is to present a ﬁnite ele-ment solution procedure that we developed in order toreach a scheme that is closer to this ideal solutionscheme. In Ref. [4], we considered 9-node elements. Wenow focus on the development of a new 9-node elementfor two-dimensional ﬂuid ﬂows, but the same approachcan also directly be employed for three-dimensionalsolutions. We regard the proposed procedure to be still aﬁnite element discretization scheme although features of ﬁnite volume methods are employed. In the procedure,the velocities are interpolated using trial functions eval-uated to incorporate the ﬂow conditions, in the spirit of Ref. [5], which provides the ‘‘upwinding eﬀect’’ in a verynatural way [6,7]. Hence no artiﬁcial upwind parametersare used. The pressure is interpolated with the aim tosatisfy the inf–sup condition for incompressible analysis,and hence here too no artiﬁcial stability constants areintroduced [6,7]. For the formulation of the ﬁnite ele-ment equations, the Petrov–Galerkin method is usedwith step functions for the weight (test) functions overcontrol volumes, and this results in satisfying locally theconditions of mass and momentum conservation (as inﬁnite volume methods).In Section 2, we ﬁrst present the ﬁnite element tech-nique, which we refer to as ‘‘a ﬂow-condition-based in-terpolation’’, or in brief FCBI, ﬁnite element procedure.We then illustrate the performance of the procedure inthe solution of some test cases with the objective toevaluate the scheme measured on the desirable charac-teristics of the ideal procedure mentioned above. Whileno detailed mathematical analysis is as yet available,based on the studies given, we can conclude already thatthe formulation approach is very valuable and that thereis much potential in the procedure.
2. The ﬁnite element procedure
In this section we present the FCBI ﬁnite elementprocedure for the solution of the Navier–Stokes equa-tions. We ﬁrst give the mathematical model consideredand then present the interpolations used. Some emphasisis given to the fact that no artiﬁcial stability constantsare employed and that the conservation of mass andmomentum is satisﬁed locally in the traditional ﬂuidﬂow sense.Throughout the paper the usual notation for Sobolevspaces is used, see e.g. [8].
2.1. Problem formulation and discretization
We consider a two-dimensional domain of an in-compressible Navier–Stokes ﬂuid subjected to essentialand natural boundary conditions (see Fig. 1). We as-sume that the ﬂuid ﬂow problem is well-posed in theHilbert spaces
V
and
P
.The diﬀerential formulation of the problem we con-sider is:Find the velocity
v
ð
x
;
t
Þ 2
V
and pressure
p
ð
x
;
t
Þ 2
P
such that
rð
q
v
Þ ¼
0
ð
x
;
t
Þ 2
X
½
0
;
T
ð
1
Þ
o
q
v
o
t
þrð
q
vv
s
Þ ¼
0
ð
x
;
t
Þ 2
X
½
0
;
T
ð
2
Þ
subject to the (suﬃciently smooth) initial and boundaryconditions
v
ð
x
;
0
Þ ¼
0
v
ð
x
Þ
;
p
ð
x
;
0
Þ ¼
0
p
ð
x
Þ
x
2
X
ð
3
Þ
v
¼
v
s
ð
x
;
t
Þ 2
S S
v
ð
0
;
T
ð
4
Þ
s
n
¼
f
s
ð
x
;
t
Þ 2
S
f
ð
0
;
T
ð
5
Þ
Ω
Fig. 1. Schematic of ﬂuid ﬂow problems considered.1268
K.J. Bathe, H. Zhang / Computers and Structures 80 (2002) 1267–1277
where
s
¼
s
ð
v
;
p
Þ ¼
p
I
þ
l
r
v
h
þðr
v
Þ
T
i
ð
6
Þ
q
is the density,
l
is the viscosity,
X
2
R
2
is a domainwith the boundary
S
¼
S S
v
[
S
f
ð
S
v
\
S
f
¼ ;Þ
,
T
is thetime span considered,
v
s
are the prescribed velocities onthe boundary
S S
v
,
f
s
are the prescribed tractions on theboundary
S
f
and
n
is the unit normal to the boundary.We note that we set out to solve the ‘‘conservativeform’’ of the Navier–Stokes equations [7]. The reason isthat we want to satisfy the local conservation of massand momentum in the classical sense.Our objective is to develop a solution procedure forthe Navier–Stokes equations that is close to the idealscheme mentioned in Section 1, see also Ref. [4]. As iswell known, the Navier–Stokes equations are self-con-sistent up to very high Reynolds numbers and weaksolutions exist, provided of course transient analysisconditions are considered and the boundary and initialconditions are suﬃciently smooth [9,10]. Hence, it isreasonable to require that a numerical solution schemeshould be able to solve ﬂow conditions at very highReynolds numbers.For the ﬁnite element solution, we use a Petrov– Galerkin variational formulation with subspaces
V
h
,
U
h
and
W
h
of
V
, and
P
h
and
Q
h
of
P
of the problem in Eqs.(1)–(6). The formulation used is:Find
v
2
V
h
,
u
2
U
h
,
p
2
P
h
such that for all
w
2
W
h
and
q
2
Q
h
Z
X
w
o
q
u
o
t
þrð
q
uv
s
ð
u
;
p
ÞÞ
d
X
¼
0
ð
7
Þ
Z
X
q
rð
q
u
Þ
d
X
¼
0
ð
8
Þ
To deﬁne the spaces used in the formulation, considerFig. 2, where we show a mesh of elements in their naturalcoordinate systems. To obtain the matrices correspond-ing to a general two-dimensional geometry, the usualisoparametric transformations are used [7]. The ﬁgureshows a patch of typical 9-node elements, see Fig. 2(a),and a ‘‘sub-element’’, see Fig. 2(b). This sub-element isdeﬁned by four nodes of the 9-node element and is usedfor the interpolation of velocities. Each 9-node element isthought of to consist of four 4-node sub-elements.For the deﬁnition of the space
U
h
, we refer to the sub-element. The trial functions in
U
h
are deﬁned as
h
u
1
h
u
4
h
u
2
h
u
3
¼
h
ð
n
Þ
h
T
ð
g
Þ ð
9
Þ
where
h
T
ð
y
Þ ¼ ½
1
y
;
y
(
y
¼
n
;
g
with 0
6
n
;
g
6
1).Similarly, an element in the space
P
h
is given by (refer toFig. 2(a))
h
p
1
h
p
4
h
p
2
h
p
3
¼
h
ð
r
Þ
h
T
ð
s
Þ ð
10
Þ
with 0
6
r
;
s
6
1.The trial functions in
V
h
are deﬁned using the ﬂowconditions along each side of the sub-element. Thefunctions are, for the ﬂux through
ab
,
h
v
1
h
v
4
h
v
2
h
v
3
¼
h
ð
x
1
Þ
;
h
ð
x
2
Þ
h
ð
g
Þ
h
T
ð
g
Þ ð
11
Þ
with
x
k
¼
e
q
k
n
1e
q
k
1
;
q
k
¼
q
uu
k
D
x
k
l
ð
12
Þ
where
uu
k
2
U
h
and is the velocity at the center of thesides considered (
n
¼
1
=
2 and
g
¼
0, 1 for
k
¼
1, 2 re-spectively). Analogously, the functions are constructedfor the ﬂux through
bc
, and so on.Note that these functions satisfy the requirement
P
h
i
¼
1, although diﬀerent ﬂow conditions may bepresent at the element edges [7].The elements in the space
Q
h
are step functions.Referring to Fig. 2(a), we have, at node 2, for example,
Fig. 2. 9-Node elements and a sub-element in isoparametric coordinates. Domains over which the (constant) weight functions aredeﬁned.
K.J. Bathe, H. Zhang / Computers and Structures 80 (2002) 1267–1277
1269
h
q
2
¼
1
ð
r
;
s
Þ 2
12
;
1
0
;
12
0 else
ð
13
Þ
Similarly, the weight functions in the space
W
h
are alsostep functions. Considering the sub-element we have atnode 1, for example,
h
w
1
¼
1
ð
n
;
g
Þ 2
0
;
12
0
;
12
0 else
ð
14
Þ
Remark 1.
The fact that two spaces for the trial func-tions are used needs a comment. We could use only thespace
V
h
(and hence substitute
u
by
v
in Eq. (7)), but thenclearly the discrete ﬁnite element equations would be-come even more nonlinear, because the trial functions in
V
h
contain the exponential expressions. Therefore, weuse both spaces
U
h
and
V
h
, but of course the functions inthese spaces are ‘‘attached’’ to the same nodal velocityvariables. For example, the velocities
v
and
u
in Fig. 2are attached to the trial functions in
U
h
and
V
h
by
v
¼
h
vi
v
i
ð
15
Þ
u
¼
h
ui
v
i
ð
16
Þ
where
v
i
are the nodal velocity variables.The rational for proceeding in this way lies in that itis
v
in Eq. (7) which introduces the instability in thenumerical solution and which therefore needs to be inter-polated exponentially. Another reason is that we wantthe scheme to be applicable to any transport equation,for example, the advection–diﬀusion equation where, inthe convective term, the temperature would be interpo-lated in
V
h
and the velocity in
U
h
.
Remark 2.
We should note that the evaluation of Eqs.(7) and (8) with the given functions reduces to an eval-uation of the integrals around the control volumesshown in Fig. 2(a). Considering Eq. (7), we have onecontrol volume for each ﬁnite element node in the ele-ment assemblage (see control volumes
M
1
,
M
2
and
M
3
),and considering Eq. (8) we have one control volume foreach pressure node in the element assemblage (see con-trol volume
C
). The use of these control volumes en-forces the local conservation of momentum and mass,respectively.
Remark 3.
The ﬁnite element procedure given here isrelated to ﬁnite volume methods [1,11], discontinuousﬁnite element methods [12,13] and the use of bubbles[14]. Our objective in constructing the FCBI procedure isto synthesize ideas in order to obtain a solution proce-dure closer to the ideal scheme described in Ref. [4], anda scheme that mathematically can be analyzed such thatfurther ideas of improvements will come forth.
2.2. On the conservation of mass and momentum
In this section we endeavor to discuss the importantproperties of conservation of mass and momentum andin which way these conservation conditions are satisﬁed.We consider ﬁrst the traditional ﬁnite element methodsand then our procedure. We refer to ‘‘ﬂux conservation’’because the essence of the conditions is to conserve the‘‘ﬂux’’ as imposed through the divergence operator inEqs. (1) and (2) (the mass ﬂux in the ﬁrst equation andthe momentum ﬂux in the second equation).The ﬁnite element methods for ﬂuid ﬂows were de-veloped because a great success of ﬁnite element proce-dures was seen in structural analysis. For the analysis of structures, Lagrangian formulations and the principle of virtual work are used to obtain the well-known ﬁniteelement equations
F
¼
R
ð
17
Þ
where
R
is a vector of all externally applied nodal forcesand
F
is a vector equivalent (in the virtual work sense) tothe element internal stresses. The details of derivation of these equations are widely available, but we should notethat the nodal force vector
R
contains the contributionsof all externally applied forces, including the contribu-tions from surface tractions, body forces, concentratedloads, initial stresses, and in transient analysis inertiaforces. Of course, Eq. (17) is applicable in linear andnonlinear analyses. The vector
R
is assembled by sum-ming over all element contributions, and similarly, thevector
F
is obtained as
F
¼
X
m
F
ð
m
Þ
ð
18
Þ
where
F
ð
m
Þ
lists the nodal forces equivalent to the ele-ment stresses of element
m
. Using the notation of Ref.[7], we have
F
ð
m
Þ
¼
Z
V
ð
m
Þ
B
ð
m
Þ
T
s
ð
m
Þ
d
V
ð
m
Þ
ð
19
Þ
where
B
ð
m
Þ
is the strain displacement matrix of element
m
,
s
ð
m
Þ
is the stress in element
m
, and we are integratingover the element volume. While the above relations arewritten for the commonly used displacement-based ﬁniteelement methods, in order to focus on the essence of thediscussion, the same equations are also fundamentalwhen considering mixed methods [7].Considering Eqs. (17)–(19), we can directly infer(prove) that in ﬁnite element analysis the following twofundamental Properties I and II are satisﬁed:
Property I
(Nodal point equilibrium).
At each node inthe element assemblage, the sum of the element forces
F
ð
m
Þ
is equal to the externally applied forces listed in
R
.
This property of course directly follows from Eq. (
17
), but we
1270
K.J. Bathe, H. Zhang / Computers and Structures 80 (2002) 1267–1277
must recall that at those nodes where displacements are prescribed, the reactions are calculated from the summa-tion of the element internal forces.
Property II
(Element equilibrium).
Each element m is inequilibrium under its forces
F
ð
m
Þ
.
This property followsdirectly from Eq. (
19
) and holds for any properly formu-lated finite element. The proof is simple and based onsubjecting the element m to rigid body translations and rotations.
These properties are illustrated in Fig. 3 and hold of course for any coarse or ﬁne mesh. For details on theseproperties and a demonstrative example, see Ref. [7, pp.177–182].The above equilibrium properties express funda-mental requirements that structural engineers are usedto work with. If only nodal point concentrated loads areapplied to a truss or beam model, the exact solutions tothe mathematical models are directly obtained (if ap-plied distributed loads are present, special techniquescan be used that then also lead to exact solutions).However, in the general ﬁnite element analysis of solids,plates or shells, of course, only approximate solutionsare obtained. Speciﬁcally, consider the widely used dis-placement-based ﬁnite element discretization. Althoughthe conditions of compatibility and stress–strain rela-tionships are fulﬁlled, and the equilibrium Properties Iand II are satisﬁed, diﬀerential equilibrium (within theelements and on the boundary) is in general not satisﬁed.This results into stress jumps between elements and thefact that the externally applied body forces and surfacetractions are not balanced by the ﬁnite element internalstresses. Of course, as is well-known, these errors di-minish suﬃciently as the mesh is reﬁned [7].Consider next the generic ﬂuid ﬂow problem, sche-matically shown in Fig. 1, and governed by the Navier– Stokes equations (1)–(6). If the standard ﬁnite elementdiscretization is carried out, algebraic equations of theforms of Eqs. (17)–(19) are also obtained but of coursefor the nodal ﬂuxes. Properties I and II are also appli-cable––but naturally for the nodal ﬂuxes [7]. Hence,‘‘nodal ﬂux equilibrium’’ as schematically shown in Fig.3 on the assemblage nodal level and element level issatisﬁed.The conditions fulﬁlled in traditional ﬁnite elementﬂuid ﬂow analyses are therefore very similar to thosein structural analysis. However, whereas in structuralanalysis, the conditions of ‘‘force equilibrium’’ expressedin Properties I and II are suﬃcient to satisfy structuralengineers, the ‘‘ﬂux equilibrium’’ alone, expressed also inProperties I and II, is generally not suﬃcient to satisfyan analyst of ﬂuid ﬂow problems. Here recall that usingthe Lagrangian formulation for structural analysis, massconservation is automatically satisﬁed. However, inEulerian formulations of ﬂuid ﬂows, ﬂux conservationfor any sub-domain (of ﬁnite elements or ﬁnite volumes)in the traditional sense (integrating around the sub-domain boundary) needs to be fulﬁlled, and this prop-erty is of course what ﬁnite volume methods are basedon and hence always directly satisfy. Of course, standardﬁnite element schemes can be extended with specialtechniques such as post-processing methods in order tomore closely satisfy ﬂux equilibrium (in the control vol-ume sense) but such methods add to the complexity andcost of solution. Also, ﬁnite element methods with dis-continuous weight functions (
w
in Eq. (7)) can be usedand this is what we have selected to pursue in our de-velopments.Considering the FCBI ﬁnite element procedure pre-sented above, since the weight functions are chosen as thegiven step functions, ﬂux conservation is satisﬁed in thecontrol volume sense over the sub-domains shown in Fig.2(a). Hence the Property I can be restated as follows:
Property I for FCBI scheme
(Control volume ‘‘ﬂuxequilibrium’’––that is, mass and momentum conserva-tion).
The FCBI finite element solution procedure satisfieslocal mass and momentum conservation for the control volumes used to construct the algebraic finite elementequations.
This property is an important characteristic of thesolution scheme because of the theoretical and practicalneed to satisfy locally conservation of mass and mo-mentum. The property is probably also, in part, thereason why reasonable solutions of ﬂuid ﬂow problemswith very high Reynolds numbers could be obtainedwith relatively coarse meshes (see Section 3).
Fig. 3. Nodal point and element equilibrium in a ﬁnite elementanalysis (taken from Ref. [7]).
K.J. Bathe, H. Zhang / Computers and Structures 80 (2002) 1267–1277
1271

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