LEXTENS IN H E A T A N D M A S S
TRANSFER
Vol. 7, pp. 227-233, 1980
©Pergamon
A
STRESS
0094-4548/80/0501-0227502.00/0
Press Ltd. Printed in the united States
HYBRID FINITE ELEMENT METHOD
-
FOR STOKES' FLOW
C. Bratianu* and S.N. Atluri**
Center for the Advancement
of Computational Mechanics
Georgia Institute of Technology,
(Co~cated
Atlanta, Ga.
30332
by J.P. Hartnett and W.J. Minkowvcz)
Introduction
Computational
fluid mechanics based on the finite element method has
been developed so far following two distinct approaches
function,
or stream functlon-vortlcity formulation
primitive variables
formulation
[3,4,5].
(a) using a stream
[1,2], and (b) using a
Within the first approach a stream
function is defined such that the continuity equation is identically satisfied and the momentum equations are combined into a single equation expressed
in terms of stream function,
or stream function and vorticity.
In the sec-
ond approach velocities and pressure become the main variables of the working equation system.
variational,
The finite element method is then implemented via a
quasi-variational
approach or the method of weighted residuals
[6,7,8,9].
The purpose of this paper is to present a new approach based on an assumed stress distribution and a variational
sure distributions
formulation.
Stress and pres-
are assumed such that the equilibrium equations are satis-
fied in the interior of each discrete element while boundary velocities are
assumed such that the compatibility with the neighboring elements is maintained.
*Doctoral Candidate, School of Nuclear Engineering;
Mechanics, School of Civil Engineering.
227
** Regents' Professor of
228
C. Bratianu and S.N. Atluri
Vol. 7, NO. 3
A Variational Functional Formulation for Stokes' Flow
Let us consider a discrete domain formed by assembling a finite number
of finite sized elements which have piecewise continuous boundaries at which
neighboring elements adjoin.
We denote the sub-domain of each element by
V
and its boundary by 8V ; that portion of 8V which represents the intern
n
n
element boundary by S , the traction boundary region by Son , and the velocn
ity boundary region by S . With these above notations, it is obvious that
un
the field variables taken into consideration must satisfy the following
necessary conditions in order that the solution generated for the finite
element assembly would approach that for an otherwise continuous domain.
oij,j + 0F.i = 0
in Vn
Vij = ½(ui, j + uj, i)
3A
°ij = ~Vij
u.
i
= u.
i
in V
on S
°ijni = Ti
(i)
in V n
(2)
(3)
n
(4)
un
on Son
(5)
In addition, at the inter-element boundary, one has to satisfy [i0]:
(a)
the velocity continuity condition u~ = ui at Sn, and (b) the traction con+ tinuity condition T.+~.=0 at S . The following notations have been used
1
so far:
l
n
oij for denoting stress, p - density, F i - specific body forces,
velocity, Vii~ - rate of strain tensor , nj - direction cosines of surui
face normal, T. - surface traction, A - strain energy density function given
1
by
A(p,Vij) = -PVkk + ~VijVij
where p is viscosity.
(6)
Then the total potential energy functional [ii, 12]
can be written as follows
~p = ~ I f [ A ( p , V i j ) - 0 F i u i ] d v - f T i u i d s I
(7)
Vn
Son
This variational functional can be generalized by including the compatibility eq. (2) as a constraint via the Lagrange multipliers o...
m3
~G = ~
A(PiVij) - °ijVij - (°ij'j
+
PFl)Ui]dv
<
V
n
js
on
(8)
For practical applications this functional involves too may variables
and too many constraints imposed upon the admissible functions.
One way
Vol. 7, NO. 3
FINITE ~ q T
FOR STOKES' FLOW
229
of simplifying this formulation is to consider some of the conditions (I)
through (5) satisfied a priori.
The modified functional used in present re-
search has been obtained making the following considerations:
(a)
V
assume stress and pressure distributions within each discrete element
such that the equilibrium equation (i) is satisfied a priori;
n
(b) assume boundary velocities ui such that the continuity requirement
~+
u. = ~_ at S and the velocity boundary condition (4) are satisfied a priori;
1
l
n
(c) assume that relations of eq. (3) are invertible for V.. in terms of
13
~ij [i0], and that the contact transformations exist.
Then the modified variational formulation is given by
~M(Oij'Oi)
= Znl~Vn
ij
n
on
where B(o~j) is the complementary energy function
A Finite Element Scheme.
Let us express stresses and boundary velocities in a matrix form:
{o'} = [P] { 8 }
{o}
=
[R]
{8}
-
{6}a I
{0} = [Q] { q }
(11)
Matrices [P], [R] and [Q] are expressed in terms of polynomials, and {8},
{q}, and ~i represent unknown parameters.
The task now is to express the
unknown parameters 8 in terms of q's via the variational functional ~M of
eq. (9), and to generate the wroking equation systems [13, 14].
Intro-
ducing eqs. (ii) into eq. (9) and neglecting the body forces Fi' one obtains
=~q}T[A]{8}
- {q}T{B}~ I - {q}T{So} - ½{8}T[H]{8}]
(12)
where:
[A] = ~
[Q]T[N][R] ds, {B} = ~
- ~V n
{So} =/S
[Q]T[N]{Co}dS, [H] <
=Fnl n 2 0 ],
IN]
[0
[Q]T[N]{~}ds
Vn
[c] =
nl n2
i 0 0
2
0
~[P]T[c][P] dv
(13)
Taking the first variation of the functional ~ with respect to {8} and putting
230
C. Bratianu and S.N. Atluri
Vol. 7, No. 3
6z=0, one obtains
{8} = [H-I]T[A]T{q}
(14)
which introduced into eq. (12) yields
= ~[½{q}T[K]{q}
with [K] = [A][H-I]T[A] T.
- {q}r{B}~ I - {q}T{So} ]
(15)
Let us now expand all the element matrices of eq.
(15) into system matrices and denote these new matrices by (*).
~* = ½{q*}T[K*]{q*}
- {q*}T[B*]{~}
- {q*}{S~}
(16)
Taking the first variation of this above functional with respect to all q*'s
*w
and ~i s, which are independent at the system level, one obtains finally the
following system of equations
[K*]{q*} - [B*]{a~} - {S~} : 0
[B*]{q*} = 0
(17)
Solving for q*'s and ~l'S and using eqs. (Ii) and (14) one obtains the problem solution in terms of velocities,
pressure,
and stresses.
Example
To illustrate the present method the steady, laminar fluid flow through
a parallel channel is considered
tion expressed by polynomials
tribution
(FIG. i).
Let us assume a stress distribu-
of second order, and a constant pressure
(i.e. p=~l ) within each element.
dis-
Applying the equilibrium equ-
ations, one obtains the following final expressions
for the deviatoric
stress
~II = E1 + 82x + 83Y + 84xY + 85 x2 + 86 y2
o~2 = 87 + 88x - 82y - 2B5xY + ~9 x2 - ½84 y2
°22 = ~i0 + 811 x - 88Y - 2B9xY + B12 x2 + 85 y2
Velocity expressions
are obtained from the conventional
For the present example isoparametric
nodes have been used.
the system equations
Introducing
quadrilateral
(18)
finite element method.
elements with eight
eqs. (18) and velocity expressions
into
(17) one obtains a system of linear equations which can
be solved by using a standard solution procedure.
in this example are shown in FIG. i.
Boundary conditions used
Figures 2 and 3 present computed re-
sults in comparison with the analytical solutions.
As it can be seen from
Vol. 7, NO. 3
FINITE ~ F O R S T O K E S '
FIEW
231
WALL----~
1.O
0.8
P-24
V-O
o1,
o.6
o -~, I ° 'I o
o
P-12
o $ ~ I o
o
~o
0.3
CENTER
LINE_-~._CL.
0
O. 5
1.0
1.5
X
2.0
FIG. i
Discrete domain and boundary conditions.
- - ANALYTICAL SOLUTION
® COMPUTED
SOLUTION
-®
ANALYTICAL SOLUTION
COMPUTED SOLUTION
CENTROID VALUES
20
15
.:1 "
ol
0
, , u.sum,.~.,
.2 .4 .6
5,
,
.8
L-
1.0
0
o
0;5
× "-"~
%
1.g
FIG. 2
FIG. 3
Velocity solution at
station x = 1.0
Pressure solution
2:0
232
C. Bratianu and S.N. Atluri
Vol. 7, No. 3
FIG. 2, the computed velocity solution is in good agreement with the analytical solution.
Although the pressure distribution was considered constant
within each element it is obvious from FIG. 3 that the computed solution
yields correct values for pressure at the centroid of each element.
There-
fore, the system pressure distribution can be obtained fairly accurate by
interpolation,
using as data points pressure values at the center of each
discrete element.
Conclusions
Fluid flow solutions based on the finite element method have been developed so far using a stream function or stream function/vorticity formulation,
and a primitive variable approach.
The present paper describes a
new approach in solving viscous incompressible fluid flow problems based on
a hybrid finite element method.
Stress and pressure distributions are as-
sumed such that the equilibrium equations are satisfied within each discrete element while boundary velocities are chosen so that the compatibility
with neighboring elements is maintained.
A modified variational functional
is formulated and a finite element scheme developed.
The method is illus-
trated by an example of a steady, laminar channel flow.
Stress distribu-
tion is represented by polynomials of second order while pressure is assumed constant within each element.
analytical solution.
However,
Results are in good agreement with the
it is expected that improved solutions can
be obtained by considering a linear or higher order pressure distributions.
These results will be reported elsewhere.
i.
References
R.T. Cheng, "Numerical Solution of the Navier-Stokes Equations by the
Finite Element Method", The Physics of Fluids, 12, 2098~2105(1972).
2.
A.J. Baker, "Finite Element Solution Algorithm for Viscous Incompressible Fluid Dynamics", Int. J. Num. Meth. Eng., 6, 89-101 (1973).
3.
C. Taylor and P. Hood, "A Numerical Solution of the Navier-Stokes
Equations Using the Finite Element Technique", Computers & Fluids,
!, 73-100 (1973);
4.
Y. Yamada, K. Ito, Y. Yokouchi, T. Tamano and T. Ohtsubo, "Finite
Element Analysis of Steady Fluid and Metal Flow", in Gallagher et. al.
(eds.) Finite Elements in Fluids, vol. I, pp. 73-95, John Wiley & Sons,
New Yor (1975).
5.
D.K. Gartling, "Finite Element Analysis of Viscous, Incompressible
Fluid Flow", Ph.D. Thesis, University of Texas, Austin (1975).
Vol. 7, NO. 3
FINITE k-ff.~MI~]B-TFORSTOKES' FLOW
233
6.
J.T. Oden, Finite Elements of Nonlinear Continua, McGraw-Hill Book Co.,
New York (1972).
7.
B.A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York (1972).
8.
K.H. Huebner, The Finite Element Method for Engineers, John Wiley &
Sons, New York (1975).
9.
O.C. Zienkiewicz, The Finite Element Method, 3d ed., McGraw-Hill, London
(1977).
I0.
S.N. Atluri, "On 'Hybrid' Finite Element Models in Solid Mechanics",
in R. Vichnevetsky (ed.), Advances in Computer Methods for Partial
D~fferential Equations, Rutgers University Press, New Brunswick,
pp. 346-356 (1975).
ii.
K. Washizu, Variational Methods in Elasticity and Plasticity, 2nd ed.,
Pergamon Press, London (1974).
12.
P. Tong, "The Finite Element Method for Fluid Flow", Japan-U.S. Seminar
on Matrix Methods of Structural Analysis and Design, Aug., 25-30 (1969),
Tokyo, Japan.
13.
P. Tong and T.H.H. Pian, "A Variational Principle and the Convergence
of a Finite-Element Method Based on Assumed Stress Distribution",
Int. J. Solids Structures, 5, 463-472 (1969).
14.
P. Tong, "An Assumed Stress Hybrid Finite Element Method for an Incompressible and Near-Incompressible Material", Int. J. Solids Structures, 5, 455-461 (1969).