VARIATIONAL PRINCIPLE

___ ._*._., ......,..,.......,."Jt _..,j~'_~_"' __ .-... _. ~.~ .__. .__ ~_ .-yo,. .... '-r-

Bl"'cRuse of thewcll-known advantages of variational prineiplesin so ;'ar asobtaining bounds on the variational integral and in certain existence p"'oofsusing the direct method of the calculus of variations, the custcmOl.ry ~ppro::.cain derivin{; equations of motion of finite elements is to develop appropl'iace

,~.variational principles for the problem at hand. General o,,:tremal principlesciefining the spatial and temporal evolution of classical continuun". systemshave only recemly been proposed. The postulate of minimum entropy produc-tion as set forth by Glansdorff and Progogine (3) is limited to those physicalsystems for which: (1) phenomenological coefficients are const.'l.nts; (2)Onsager reci,>rocal relations are valid: and (3) convective terms are negli(,ri-hIe. Nc'''' methods of formulating variational principles, however, have re-

Sote.-This paper (" part of the copYI'ighteJ Journal of the j~ngineering :v:ech:.nlcsDivision. Proceedings of the American Society of Civil Engineer~. \'01. 95. Xo. E:\:3.Juuc. 1959. :\lanuscript was submitted for re~·iew forpossib:e publicat:on or; September25. 1958.

! Prof. of Engrg. i\'lech .. Research lns\.. Univ. oi Ala. in Huntsvilltl. Hu..tsvilie. Ala.Z Engrg. Consultant. Sperry Rand Corp .. Huntsv:lle, Ala.3 Xumerals in parentheses refer to corresponding items in the f\ppendlx. - References.

&21FI1II"ITE-ELEMENT APPLICATIO~S

INTRODUCTION

By J. Tinsley Oden,! A. M. ASCE, and D. SomogyP

FINITE-ELEMENT APPLICATIONS IN FLUID DYNAMICS

r~1 3

It has been noted by Zienkiewicz and Cheung (10)3 and a Hllll:ber of otherinvestigators (7-12), that the finite-element concept is sufficiently general tobe applied to a wide range of field problems. Indeed, in addition to the usualstructural applications, use of the method in problems of seepage flow (11/.h(>"t conduction (9), potcntial flow (7), and Poisson's equation in the plane (10)arc among those which have been presented (see Aclmowled~ments).

In this note, the application of the finite-element method to a class of pro-blems in fluid dynamics is conside:·cd. Such applications involve several pe-cula;:-itics not encountered in the lIsual structural applications. Firstly, thereis the obvious characteristic that in ki"ematical representations, it is naturalto approhimate velocity rather than displacement fields over a finite element.Secondly, the finite elC'ments repl'esent spatial rather than material suor(:-b>1onsof the continuum: Le., ir.stead of representing finite C!len·.ents of ... fluidmatel'ial, the elements 1'epresent subregions in the space through which the'fluid moves (Eulerian description of motion). Thirdly, the local clement fieldsare specified in terms of the values of the velocity at approximate nodal pointsof the element; but these values represent the velocities at the nodes ratherthan of the nodes.

522 t ~, :

cently been developed. One technique stCl11S{romthe ('ron""j·t '" 1.... ,,: ,.', _:.~as introduced by Gl:lnsclorli and Prlgogine (4), ;lnd :1.nOl!:o';·,"''J/ \ .. : :~: .. >.

pr1nciple for Inco:npressible non-~ewtonian now w:\s Int:'(t(..'un·dL:; ll!:·j \11All of these new methods fall outside the range of \'alldity :I\:p!lc.lbl(>lo l::~principle of minimum entropy production.

For the illustration presented herein, the viscous Incompressible liowequations for the minimum entropy production, or in this case, n~in!mumvis-cous dissipation, will be studied first. The variational principle used is appH-cable to laminar f)_owsin which inertia forces are negligible. and was firstpresented by Helmholtz in 1868 (5). It is stated as follows: The motion of anincompressiole fluid that satisfied the equation of continuity, the equ:1.tionsofmotion ~'.I~dthe boundary conditions of zero stress or spec Wed velocity alongthe whole boundary is such that the dissipation integral ~

J = f Fat' = ~ f (lij dij dv (1)v v

attains a minimum for steady flow. Note that cartesian coordinates are em-ployed for this presentation. If, instead, the boundary conditions specify thevelocity on one part S 1 of the boundary, and the stresses on the remainingpart S 2 of the boundary, then

1 = ~ J cijdi jdv - 2 J (lit/idS : (2)V 52

is a minimum, in whic!l 1 = the excess of the dissipation integral over twicethe rate of which work is being done by the specified surface tractions, ci;(li j = the stress tensor: and d f j = the strain rate (rate -of -de!ormatlon) ten-801' given by

~ w·di} = aXj ~ (3)

in which V { = components of velocity and Xi cartesian coordinates.

FINITE ELEMENTS FOR VISCOUS INcOMPRESSIDLE FLUIDS

Consideration is now given to a typical finite element of a finite-elementmodel of a general, three-dimensional, \1scous incompl'essible flow field.The integral I of Eq. 2 must be minimized. For ~ewtonian fluids. the rate ofdissipation of energy per unit time may be written

1 1Q = '2(lij dji = 2µd{jd.U (4)

in which µ = the viscosity. Thus, according to Eqs. 2, 3, and 4, for !inite el-ement e

~in which commas denote diIferentlation with respect to ,; (t. /.' " " .'. ,

. (5)

- ..-----.- .. _. .. _._.- .~-_.... _ ....... ~ .. ."-_.....:r,.:.. ~ ... _ -. _ - __ .... •.• ---.----

alW~ li l' Nt = 0 (7)NI

In a ma:1nel'similar to the usual displacement approximations, the velocitycomponents Fi are approximated ovcr a finite element by functions of theform

I

~~f'K

t823FINITE-ELEMENT APPLICA TIO~S

G

Vi ~ 2: I!.'N(Xk) l'Ni (6)10.'01

in which Ill,vo; j) = ge:1eralized Lagrange interpolation fur.ctions [i.e., 1P,.,.(xUi)= IiNM1:V'Xi :: the velocity components at node N of the element: and G :: thetotal number of nodes of the!element. Upon introducing Eq. 6 into Eq. 5, func-tionalle reduces to an ordinary function of generalized nodal velocities VNi .Because Ie must be a minimum locally for element e, it is seen that

G

(j I = re .....,

for arbitrary variations li VN/' Thus, the following system of equations areobtained for a typical element

G\'L.J kRNilil ~'Rm - PNi = 0 , (8)

Ral

in which kRNim:: f µ ~IR ,i I/;,\" ,II' dv (9a)Ve

and PN j = f a i l/WciS (9i1)52

The array l( RNi III may be ir,tcrpreted as a "stiffness" array for the finite el-ement and PSj are components of "consistent" gencralized force at node N.It is not difficult to recast these equations in matrix form if desired. The pro-cess of assembling elemen~s together into a particular model is not presentedherein: the connection process depends upon purely topological features of themodel and is identical to the well-known procedures used in structuralaptllications.

Note that the tractions in Eq. 10 are given by

~ ai = a{j IIj = (- P liij + Tjj)lIj - (10)

in which llj :: components of a unit exterior normal to the boundary of the el-ement; p = the hydl·ostatic prcssure; and Tjj = the so-called viscous stresstensor. Thus, the nodal forces P]..·j may be represented as the sum of twoparts: one part due to hyd.ostatic pressure and one due to contact forces. Theformer is oftcn prescribed in practical applications.

UNSTEADY FLOW

Thc more general problem of unsteady flow of an arbitrary fluid is nowexamined, withoat resorting to specific variational principles. The globalform of the conservations of energy for a finite element is (.2)

"- 824 Junc·. UlI39 ..

+ f (1i vi ds + Qe .se

. . . . . . . . . (11)

in which the first integral represents the rate-of-change of kinetic eaerg)':= the specific internal energy: F i = the body force~: and Q (' = the heat sup-plied to the element. If mass and linear momentum are conserved

.. (12)

~so that

+ f (1 i vi ds (13)Se

Introducing Eq. 6 into Eq. 13, it is found that for a typical element

+ f (1ij r/iN.j dv - PNi] "11'; = 0Ve

. . . . . . . . . . . . . . . . . . .. (14)

in which mNM f I/'N1/JM pdt' - (15)I' e

b.1NJHR

.................. (16)

PN( = f pF( WN dv +l'e

f (1i ~JNds'"se

. . . . . . . . . . . . . . . .. (17)

and mXM = the consistent mass matrix for the element: and l'Ali representthe components of eVi let at node M. Because Eq. 14 must hold for arbitraryVNi' the general equations of motion of a finite element. including the non-linear convective terms. is

G

L: (11INM l'MiM\

which is valid for arbitrary continuous fluids.

(lB)

~ ....- -.......--.......

E~l 3 FINITE-ELEMENT APPLICATIONS 825

If, as a special case, laminal' flow of a ViSCOllSincompressible fluid isconsidered, then the acceleration terms 11INMV,.1i + b/'MR V,\fj l'ni in Eq.18are dropped and (Jij = (µ dij /2) - P Gij is introduced iato the remainingintegral. Upon simplifying, the result reduces to Eq. 8,

BRIEF EXAMPLE

.For two-dimensional problems, the simplest finite elements are the tri-angular elements with three nodes as indicated in Fig ..1. In this case, functionsWN(X, y) acquire the familiar linear form l/iN = aN + bNI x + bNa .v in whichaN' bN1, and b'\'2 are known in terms of element geometry and N = 1, 2, 3.Such local interpolation funcqons are complete and, on cOimecting elements,they insure complete continuity of the velocitY fields across intel'elementboundaries. Thus, convergence of the results to the exact solution is guaran-

FlG. l.-FT!\ITF. ELE;\lDIT fiEP-RESENTATTO~ OT-' ilUlEGULARGED:\IETRY

FIG. 2.-CilOSS SECTION OF SQUARECHANl\EL

teed ii the finite element network is properly refined (6). By connectiilg such~'triangular elements together, any irregular geometry can be app1'Oximated

and element size_s can bc graded to allow for rapid variatio:l oi the velocityfield. Moreovel', nonho:nogeneous and anisotropic fluids are easily handledby assigning different constitutive equations to each clement. Boundary con-ditions are applied by merely specifying nodal velocities or generalized nodalpressures /lNi at boundary nodes. Convergence rates, though reasonably highfor triangular element models, Can be Significantly improved by using higher-degree polynomial interpolation functions and, correspondingly, more nodalpoints per element.

Asa brief example, the problem of flow of a homogeneous, viscous, incom-pressible fluid through the square channel in Fig. 2 is considered. In thiscase, Eq. 9a gives

RRNim = µ Ao bRi bNm , , (19)

~, ,.. ....826 June, 1969

. ,I

•(1\

inwhichAo = the area of the element and i,m" 1,2. T!ll' \'l'\'·(\l) ,,' L.

origin is of the form V 0 = - a (a2 I1l) dp/dz, in which C' " 0_2~'7cor:-t'"'j" 1~~3

to the exact solution. A crude model indicated in Fig. 2 (\wo pl~'nll'at" \"':quadrant) yields a s'.!rprizingly close value of a = 0.333. CO:-:-~'~;)/lt"\ll'\:stresses, however, are srf10 in error. A finer gridwork of eight elements ~~quadrant gives 0 = 0.313 and stresses less than 5%in error. Greater accu-racy, of course. is o!Jtained with still finer finite-element networks.

ACK,,'iOWLEDGMENTS

The writers have become aware recently of the unpublished work of p,Tong of MIT on the problem of sloshing of a liquid in an elastic cylinder. H.C, Martin of the Univ. of Washington (7). has recently applied the method to"'inumber of problems involving steady flow of an incompressible fluid, By pri-vate communication, B. Irons of the Univ, of Swansea. Wales, has informedthe senior writer of work in progress on the finite-elel?lcnt analysis or vis-cous flows,

APPENDcr.-REF~RENCES

I. Bird, R. n., "New Variational Principle ror Incompressible Non·Newtonian Flow," "hy.ric.I of1-1llidr, Vol. 3, No.4, July·Aug" 1960.

2. Eringen, A, CooMt'chtll/ics of COlllill/Ia, John Wiley and Sons. :-Jew York, 1967.3. GJansdorff, P., and Prigogine, I., "Sur Ic.~Proprieles Dirrcrenticllcs ue la Produclion d'Entrpie,"

"h....1/(Il. Gf:!Y. 20, 1954.4. Giansdorrr. P.. and Prigogine, I.. "On a General Evolution Criterio!' in ~lacroseopic Physics,"

"hY.lica. Gray. 20, 1964.5. Ilelmholtz, H .• "Zur Theorie det Stationaren Strome in Reibcndcn Flussigkcilen," Vcrhtllidlun,

gcn d.,s ntlltlfMrturisch,medi:in(rcilen Vnl'ins I.U Heidelberg, B,md 5, Oct., 1868.6. Key, S. W., "A Convergence Invcstigation of the Direct Slirrness Method." Thesis submitted to

the UniY. of Washington. in 1966. in p;.mial fulfillment of the requirements for the degree ofDoclor of Philosop~y,

7. Martin, H. C, "Finite Element Analysis of Fluid flows," ProCl!ed/ng ... Second ConfeT<'nCl!onMatrix Methods in Struclllral MechanicJ, \'iright,Pattcrson Air Force Base, Oct., 19611.Ohio(to appear).

8. aden, J. T .. "A Gencrali7.alion of the Finite Element Concept and Its Applic:!tion to a Class ofProblems in :-;onlinear Viscoelasticity." De"elopmems in Theort>tical alld Applied .\feclwllics.Vol. IV, (Proceedings. Fourth Southeastern Conference on Theoretical and Applied :'olechanics),Pergamon Press, London. (in press).

9. Wilson, E. L., and Nickell, R. E.... Application of the Finite Element :-'lethod to lIeat Conduc,tion Analysis." Nuclear Engineering and Design, Vol. 4, 1966. p. 276.

10.Zienkicwicz, O. C., and Cheung, Y_ K., "Finite Elements in the Solution of Field Problems,"TlIl'loirginen, Vol. 24, Sept .. 1965, pp. 501-510.

II. Zienkiewicz, O. C, Mayer, P., and Cheung, Y_ K .. "Solution of Anisotropic Sccpafe Problem,hy,Finile Elements." Journal oj the Enginrering Mechanics Division. ASeE, \'01. 92, S" I;\! I.Proc. Paper 4676. Feb., /966, pp_ 111-120.

12. Zienkiewicz, 0_ C. and Cheung, Y. K.. rhe Fit,ite Element Mt'lhod in Slm<lur,,1 ,1ft.! ('''fi:''''11//11' Me,'hanic.r, McGraw-Hili Book Co_, London, 1967, pp. 14R-16S.