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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS. VOL 7, 1211-1228 (\987)

ADAPTIVE FINITE ELEMENT METHODS FORHIGH-SPEED COMPRESSIBLE FLOWS

1. T. ODEN, T. STROUBOULIS AND PH, DEVLOO

Texas Institute for Comprllll/ional Mechanics. Departmelll of Aerospace Engineering and Engineering Mechanics. TheUnirersit,\' of Texas al Austin, Allstin. TX 78712·1085, U.S.A.

SUMMARY

We describe an adaptive finite element algorithm for solving the unsteady Euler equations. The finite elementalgorithm is based on a Taylor/Galerkin formulation and lIsesa very fast and efficient data structure to refineand unrefine the grid in order to optimize the approximation. We give a general version of the method whichcan be applied to moving grids with sliding interfaces and we present the results for a transient supersoniccalculation of rotor-stator interaction.

KEY WORDS Adaptive Melhods Finite Elements Compressible Flow Turbomachinery

INTRODUCTION

In 1946, von Neumann. realizing the limited capabilities of analytical methods in fluid dynamics,envisioned the use of numerical simulation as a powerful tool for solving complicated problems influid dynamics. Since then tremendous progress has been made in the production of very fastcomputing devices for such calculations. But it is a debatable issue as to whether or not similarprogress has been made in the production of general and fully reliable numerical algorithms forfluid mechanics simulations.

Most of the popular methods in use in fluid dynamics calculations employ a localtinite differenceapproximation of the flow equations on a structured grid. The selection of an appropriate tinitedifference s~heme for the problems of interest is typically based on an analysis of the performance ofthe scheme for a one-dimensional problem. The selected tinite-difference scheme is then applied tothe multidimensional case by using dimensional splitting. Such procedures are based on theconstruction of smooth, orthogonal. well-structured meshes along the directions of a system ofgeneralized co-ordinates. The construction of such meshes for complicated flow domains oftenpresents a formidable task.

In this paper, we present a finite element formulation which generalizes Richtmeyer's two-stepLax-Wendroffmethod to multidimensional problems and arbitrary cell geometries. This methodwas used under different forms by several authors.1-3 and is based on the Taylor-Galerkin ideas ofDonea4•5 and Oden.I3 In contrast with most tinite difference methods. which require an elaboratestructured grid generation. our method requires a coarse initial mesh which models only thebasic geometrical features of the flow domain. Our algorithm estimates the local error and has

Based on an invited LeclUre

0271-2091/87/111211-18$09.00© 1987 by John Wiley & Sons. Ltd.

FINITE ELEMENT ALGORITHMS

where uG denotes the grid velocity.

(4)

(3)

(I)

(2)

4UTcp,= L U~iJ,pJot,

2;1

2 4

Q:\7cp= L L Q2i iJ,pJcJXj.j; 12= I

1. T. ODEN. T. STROUBOULIS AND PH. DEVLOO

We consider the motion of a perfect gas through a time-dependent two-dimensional domainn(t)cIR2, te[O,T]. If U=U(x,t) is the vector of conservation variables with p the massdensity, m the linear momentum and e the total energy. it satisiies the following space-time weakformulation.

find U = U(x, t)e V such that

- f'2 f UTcp,dndl + f UT(x, t2)CP(X, r2)dn - f UT(x, T1)cp(X,r1)dn" 010 n" n"

= f<2 f Q(U):\7cpdndl - f'2f cpT[Q(U) - U(uG'n)] dsdt. V,peW." 01" ." clOlIl

Here the sets V and Ware appropriately detined,.! cp,::::iJCP/OI. Q(U) is the Euler flux tensor.

· [1111

1112 Jp-Iml +p(U) p-lm1m2

Q(U)= P-111l111l2 p-II1I~+p(U)'p-lmI [e + p(U)] p-1m2[e + p(U)]

and p(U) is the thermodynamic pressure

p(U) = (y - l)(e- p-lm·m/2).

Moreover. the following notation is used:

We obtain a tinite element approximation of (I) by partitIOning the space-time domainD=UO"'~Tn(l) into subdomains D.=U,.", .. ,•• ,n(t) with 0=IO<tl<· .. <t.<t.+1<· .. <tN = T, by discretizing each subdomain and by using the discrete spaces of test and trial functionsdetined by the discretization. In order to discretize (I) in the case of variable domains we note that6

oCPJot = - UG'\7CP2

the ability to adapt the grid in order to produce optimal approximations.The grid is dynamically adapted during thc calculation by locally rctining elements with large

error and by unretining 'groups' of elements with small 'group error'. When changing the structureof the grid a major problem of bookkeeping arises. We solve the data management problem byusing a 'fast' data structure which permits local grid changes using a local modification of the datastructure. The fact that we do not have to search through the whole list of elements in order tomodify one element makes the data structure very attractive for transient adaptive calculations.

Following this Introduction, we give a brief description of the solution algorithm, we present theadaptive tinite clement pocedures for steady-state and transient calculation and we close the paperwith the presentation of some numerical examples including the adaptive calculation of rotor-stator interaction.

1212

Using numerical quadrature to approximate the space-time integrals in (1),3 we obtain thefollowing two-step scheme.

Second step

For each node calculate V~+ I by solving the following system of equations:

f (f 4J14JjdO)V~."+l = f (f 4J14Jjdo)v~·"+6tf Q"+1/204JidOj=! 0"+1 j=l !l" 0"" oXII

- 6t f IlIl(Q;; 1/2 - Q;/I)4Jids - 6t f IlIlQ;/l4Jids·. (6)~'-J ,'ft"

II

I

'IIIII

(5)

(8)

1213

M{U}"+l = {R}.

HIGH-SPEED COMPRESSIBLE FLOWS

First step

For each elcment O. calculate a constant element vector V:,~1/2 from

A"+ 1 V"+ 1/2 = ~ (f A..dO) Vi." _ 6t A:+ I (f 04J1dO)QI."• a.. L... 'f', 2 A" + 1/2 /2 ~X •• /1i = 1 0: e Cl~~ 1 V P

4 4 f 0+ L L 6t -(4JI4Jj)U~G V~·"dO.1=1j=l n;-1/10X/I

Here <1:/1 denotes an elementwise averaged value of thc nux and Q./I is defined by- GQ'!1 = Q.P - Val/p" (7)

Equations (6) ana (7) define a two-step TG/FEL W (Taylor-Galerkin/finite-element Lax-Wendroff) method which was introduced in thc finite difference literature by Burstein 7 used forthe first time in a finite element context by Odcn,13 and is one of the Taylor-Galerkin methodsintroduced by Donea,4 studied by Baker and Kim5 and employed by Lohner et at.! and others2

.3

to the solution of compressible flow equations. The second step of the scheme involves a globalcalculation of the form

Here M denotes the consistent mass matrix, {R} the load vector whose definition can be easilydeduced from (6) and {U} = [U l' U2, U3, ... , U M]T is the global vector of nodal unknowns. Theinversion of the mass matrix can be performed by Jacobi iteration 1." or a preconditioncd Jacobiconjugate gradient. 3

The TG/FEL W mcthod provides us with a fast, multidimensional time-stepping algorithm withhigh 'resolution' (high order of accuracy) in smooth regions of the flow and which applies tounstructured adaptive grids. Artificial diffusion is added in order to stabilize the scheme in thepresence of discontinuities 7,OJ and to suppress hour-glassing modes,3

Artificial diffusion terms prevent the occurrence of non-linear instabilities, but do not eliminatenon-physical oscillations from the solution. In recent years. Boris. Book and others8

.10

,ll havedeveloped the theory of flux-corrected transport (FCT) in an attempt to correct finite-differencetransport schemes in order to avoid non-physical oscillations in the solution. Fully multi-dimensional FCT schemes were presented by Zalesak,'! and recently Lohner et at.8 prcsented aflux-corrected procedure for the TG/FELW. We now give a short exposition of the FCT-TG/FEL W algorithm which we employed in some of our examples.

ADAPTIVE PROCEDURES

Here ML denotes the lumped mass matrix and {V} denotes the vector of added diffusion, withnodal contributions of the form

(9)

(10)

IV1dU7r} = {R} + {V}.

1. T. ODEN. T. STROUBOULIS AND PH. DEVLOO

Step I: 'diffusioll'step

Compute {U7/ I } from

f ( (hPi aU" iJ¢JiU")Vi = Dx~;;- + Dy'7'"". ~. dn,

n vX (IX CJ cy

For a mesh of quadrilaterals we let

We now present adaptive procedures for the equations of compressible flow. The organization ofthe adaptive procedure depends on the desired result. Adaptive proc.edures for steady stateproblems are different from adaptive procedures for the accurate integration of the transientrcsponse. To initiate the adaptive procedure for a given flow domain, a coarse finite elementmcsh is defined which contains only a number of elements sufficient to model the basic geometricfeatures of the flow domain (see Figure l(b)), Each element of the initial mesh is assigned a 'level'equal to zero. Then the initial mesh is bisected uniformly several times in order to construct aninitial grid which has the 'group' structure. Note that when an element is refined a group of fourelements is defined and each one of the four new elements has a Icvel one unit higher than theparent element (for more details about the data structures sce Reference 3).

An adaptivc proccdure for stcady-state solutions of the Euler equations involves the foIlowingstcps:

1. For a given finite element grid determinc thc steady-state solution.2. Compute error indicators 0.. overall elements in the grid. Let

0MAX = max 0,.I..,'" .\1

Dx = D), = cA~.

where c is a constant and A~ denotes the area of the element nt>.

Step I I: 'anridiffusioll' step

Compute {U" + I} as the limit of the sequencc of itcrates {Ulii I},i= I.2. 3.... defined by

1\1d Uri~ " J - u~/1 } = I (FIIJ) • (11 )

Flil =(ML - M)Urii I - V.

Hcre I denotes the flux limiting function of Zalesak.11

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The FCT procedure consists in solving equation (8) by using first a 'strong' diffusion step 10.11 toobtain a 'transported and diffused' solution which is free of non-physical oscillations: then anantidiffusion step with nux limiting11 in order to steepen the solution at discontinuities. Tn.particular. we have:

..;;iJ...._oc , .. IiIlIl

1215

( a.)

REFlliE~

...-: I .···r····· ..· ~ .

~UliREFINE

HIGH-SPEED COMPRESSIBLE FLOWS

3. Scan groups of four elements and compute

where Ink is the k1h element in group m.4. For given tolcrances tx, PER 0 < tx, P < 1, if Oe ~ P(}MAX reline elemen t Oe by bisecting it into

four new elements. If (}OROUP ~ aOMAX unrefine group /II by replacing the group by a singlenew element with the nodes coincident with the corner nodes of the group.

5. Go to step I.

We also present an example of an h-rclinemcntlunrelinemcnt strategy for transient calculation.The goal of this algorithm is accuratcly to follow the features of the transient response and topreserve temporal accuracy. The basic steps of the algorithm are

(a) Advance thc solution N time steps.(b) Do the following until no more elements can be relined:

(1) compute the elemcnt error indicators (}e

(2) relinc all elements with (}e ~ P(}MAX(3) integrate the last N time stcps with the updated (refined) mesh

Figure I. (a) Relinement and unrelinemenr of 3 four·demcntgroup and (bl a coarse initial mesh consisting orrour·elementgroups

t= ~_

NUMERICAL EXAMPLES

Supersonic rotor-stator interaction

We applicd the 'two-pass' version of the adaptive procedure for transient calculations to aproblem of rotor-stator interaction.6•12 We consider two rows of unsymmetric aerofoils which areshown, together with the initial discretization of the domain, in the upper part of Figure 5(a). Weassumed that the stator and the rotor have the same number of aerofoils and we perform thecomputation on domains corresponding to one rotor and one stator aerofoil, simulating the

1. T. ODEN, T. STROUBOULIS AND PH. DEVLOO1216

Supersonic flow over a sharp comer

We consider the problem of a uniform Mach 3 ()' = 1'40) flow which is deflected by a 20° ramp.The flow enters with uniform flow conditions through the left boundary and, travellinghorizontally with constant supersonic speed, arrives at the corner and turns discontinuously 20°into the direction of the other leg of the angle travclling again at constant velocity. The flow domainconsists of two constant states which are separatcd by an oblique shock front attached to thccorner tip.

The calculation of the steady flow was performed with the adaptive scheme described above withconstants ex = 0·05, P = 0·15. Figures 2-4 show thc computed steady-state pressure contours withzero, one and two levels of relinement, respectively. The results were obtained using the FCTversion of the algorithm with 'viscous' constant, c = 0'125 and C.F.L. number equal to 0·25.

(4) go to (I)(c) Compute the element error indicators 0. and unrefine all groups with lFoROUP ~ exOMAX'

(d) Go to (a).

We note that the 'do loop' in step (b) converges when no more elements can be refined (themaximum level of rcfinements is fixed). Although the iteration in step (b) guarantees a 'fully upated'mesh it may lead to an expensive scheme if more than a few passes are required for convergence ofthe 'do loop'. A cheaper alternative is presented by the following 'two-pass' scheme:

(a) Advance the solution N time steps.(b) Compute the element error indicators 0•.(c) Refine aU elements with O. ~ POMAX'(d) Integrate the last N time steps with thc relined mesh obtained in step (c).(e) Compute the element error indicators O. and

(1) unrefine all groups with 00ROUP ~ aOMAX

(2) reline all elements with 0.~ POMAX'

(£) Go to step (a).

The adaptive procedures presented above require the computation of reliable errorindicators Oe' In our calculation we employed thc following definition:

J A. lop IO.=-sup -.Pe i~1.2 OXi

Here P. denotes an average value of density for clement n. and A. denotcs the clement area. Thcnumerical examplcs show that our error indicator captures well the locution of variable shocks,For alternative delinitions pf error indicators see Rcfercnces 3 and 8.

iIIiiMIIIili6 --~--

Figure 2. Upper: initial coarse mesh, and lower: corresponding pressure contours for supersonic now ovcr a ramp

Figurc 3. Upper: first levcl refinement. and lower: corresponding pressure contours for supersonic now over a ramp

- -------

Figure 4. Uppcr: sccond levcl refiilemcnt, and lowcr: corresponding pressure C0l110UrSfor supersonic now over a ramp

J. T. ODEN. T. STROUBOULlS AND PH. DEVLOO'

ACKNOWLEDGEMENTS

Support of a portion of our research on adaptive methods has been provided by ONR underContract NOOOI4-84-K-0409 for error estimation and transient problcms in solid mechanics. Thedata management schcme and CFO codes described here were completed under support of theAerothermal Loads Branch of NASA Langely Research Center, Contract NAS 1-17894.

presence of the remaining aerofoils by periodic boundary conditions. Wc imposed supersonicinflow boundary conditions at the stator corresponding to a frce-stream Mach number equal tothrce (i' = lAO). Boundary conditions of supersonic outflow werc assumed on thc right boundary ofthc rotor. In the beginning. alI the aerofoils are kept lixed in order to obtain a stcady flow pattern(lower part of Figure 5(a)) which was used as an initial condition in our calculations. After thesteady state is reached we start moving the left row of acrofoils (stator) upward with a velocity equalto one-third of the inflo\\' velocity.

Figures 5(a)-5(i) show the computed adaptive linite element meshes and the correspondingdensity contours for various stages of the stator motion. Figure 5(a) shows the initial mesh and thedensity contours for the initial conditions. The remaining plots show the evolution of the mesh andthe density contours for 1/4,2/4,3/4,4/4,5/4,6/4,7/4,8/4 cycles of motion. The results show thatthe error indicator captures well discontinuities of variable strength and that the mesh isdynamically adapted to follow the prevailing features of the solution.

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· If ..r

I\

~~'// '-----I~ ----~

Figure 5(a). Upper: initial mesh for dynamic rotor-stator now interaclion problem. and lower: computed initial pressuredistribution. The next sequence of Figures shows the dynamically relined and unrelined meshes and corresponding density

contours for various time instants measured in fractions of a rotor-slalor period P

--1220 1. T. ODEN, T. STROUBOtlLlS AND PH. DEVLOO

Figure 5lb). 1/4 P

HIGH-SPEED COMPRESSIBLE FLOWS

Figure SIc). 2/4 P

1221

_t- • ----.... j

1222

-----. ,--.--

1. T. ODEN. T. STROUBOULIS AND PH, DEVLOO

Figure SId). 3/4 P

HIGH-SPEED COMPRESSIBLE FLOWS

Figure 5(e~ 4/4 P

1223

Figure S(f). 5/4 P

1. T. ODEN. T. STROUBOUL/S AND PH. DEVLOO

1224

HIGH-SPEED COMPRESSIRLE FLOWS

Figure 5(gt, 6/4 P

1225

I',

--

1226

Pigure 5/h), 7/4 p

...;.; t ',. ' M Mr. e--.

HIGH-SPEED COMPRESSIBLE FLOWS

Figure 5(i). 8/4 P

1227

I. R. Lohner, K. Morgan and O. C. Zienkiewicz, 'An adaplive finite clcment procedure for compressible high spced flows',Compllt. MClh. Appl. Mech. Eng., 51, 441-466 (1986).

2. K. S. Bcy. E. A. Thornton, P. Dcchaumpai and R. Ramakrishnan. 'A new finite element approach for prediction ofaerothermalloads-progress in inviscid flow computalions', 7th A IAA Compo H Dyn. Conr.. Cincinna ti. A I AA PaperNo, 85-1533-CP. 1985,

3. 1. T. Oden, T. Strouboulis and Ph, Devloo. 'Adaptive finite element mcthods for the analysis of inviscid compressibleflow: I. fast refinernenl/unrefincment and moving mesh melhods for unslrllcturcd meshes', Compllt. Meth. Appl. Mech.Eng., 59(3), 327-362 (1986).

4. 1. Donea, 'A Taylor-Galerkin method for convective transport problems', 11lI.j.nwner. meth. eng .•20. 101-120 (1984).5. A.1. Baker and J. W. Kim. 'Analysis of a Taylor weak-statemcnt algorithm for hyperbolic conscrvation laws', Tech.

Rept. CFDL/86-1. Computer fluid Dynamics Laboratory, Univ. of Tennessee. Knoxville. TN. 1986.6. T. Strouboulis. Ph. Devloo and 1. T Oden. 'A moving-grid finitc elcment algorithm for supcrsonic now intcraction

between moving bodies', Comput. Meth. Appl. Mech. Eng., 59(2). 235-255 (1986),7. S. Z. Burstein, 'Finite difference calculations for hydrodynamic flows containing discontinuities', J. Compwational

Ph}'sics. 20, 198-222 (1967).8. R. Lohner. K. Morgan, M. Vahdati, J. P. Boris and D, L. Book. 'FElIl-Fer: combining unstructured grids with high

rcsolution'. (to appear).9. A. Lapidus, 'A dctached shock calculation by second order finite differences', J. Computational Physics. 2, 154-177

(1967).10, 1. P. Roris and D. L. Rook, 'Flu~ correctcd transport I: SHASTA. A nuid lransporl algori1hm that works'. J. Compo

Physics. 11, 38-69 (1973).11. S. T Zalcsak, 'Fully multidimensional flux·corrected transport algorithm for fluids'. J. Computational Physics, 31,

335-362 (1979),12. 1. T. Oden. T. Strouboulis, Ph. Devloo, S. 1. Robertson and L. W. Spradley, 'Adaptive and moving mesh finile clement

methods for flow interaction problems', Proceedings Sixth Inter. Confen'nce IlII Fin;II' Eleml'llI A/I,thuds in Fillids,Antibes. France. June 1986-

13, J. T. Oden, 'Formulation and Applicalion of Certain Primal and Mi~cd Finitc Elemcnt Models of Finite ElasticDeformations'. Computer Melhor/.~ 011 Applied Science and Engineering, Part I.Edited by ItGlowinski and J. L. Lions,In!'1. Symposium al Versaille, 334-365 (1973). .

14. L. W, Spradley. I. F. Stalnaker, M, A. Robinson and K. E. Xiqlles, 'Finite clement algorithms for compreSSIble flowcompulation on a supercomputer', Finite Elements in Fluids, 6, 135-1 S6 (1985).

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1. T ODEN, T STROUROULIS AND PH. DEVLOO

REFERENCES

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