3Sd..-

Smart Algorithms and Adaptive Finite ElementMethods in CFD: Their Status and Potential

J. Tinsley Oden

Texas Institute for Computational MechanicsThe University of Texas at Austin

Austin, Texas U.S.A.

Second South African CFD SymposiumJune 25-26, 1991

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SMART ALGORITHMS AND ADAPTIVE FINITEELEMENT METHODS IN CFD:

THEIR STATUS AND POTENTIAL

J. Tinsley adenTexas Institute for Computational Mechanics

The University of TexasAustin, Texas 78712, USA

SUMMARY: This paper surveys new developments on smart algorithms and, par-ticularly, adaptive finite element methods. The idea behind these new techniques is tovary mesh parameters and the structure of algorithms so as to optimizc the computationalprocess. This involves estimation of the numerical error and then changing numerical pa-rameters, such as mesh size, node location, and spectral order, so that thc error is controlled.This approach can result in highly accurate and efficient techniques for computational fluiddynamics calculations.

1 INTRODUCTION

Finite element methods were first applied to the Navier-Stokes equations over two decadesago [1-6]. Yet, for much of the intervening twenty years, their impact on practical CFD cal-culations has been small in comparison with conventional finite difference schemes, especiallyin the calculation of high-speed flows.

Over the last half-decade, however, this situation has dramatically changed. Duringrecent years, new finite elcment techniques have appeared which are finding applicationto an increasing list of flow simulations. The versatility, reliability, and generality of thesetechniques suggest that the technology for developing general-purpose CFD codes is at hand,and that these new tools will provide a new dimension to engincering analysis and design.

At the heart of these new finite element methods is the notion of smart algorithms andadaptivity: the capability of numerical algorithms to modify themselves during a calculationto accommodate changing properties of numerical solutions. These modifications can bemanifested in changes in the structure of the mesh, the ordcr of the polynomial shapefunctions, the location of nodes, the form of the equation solver, the cxplicit or implicitcharactcr of time marching schemes, or in combinations of all of these features. vVhen addedto the well-known ability of finite elcments to handle complex geometries and boundaryconditions, modeling techniques with truly remarkable and powerful capabilities emerge.

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This paper presents a brief survey of some of the progress in adaptive mcthods and smartalgorithms developed at TICOM over the last fcw years, and it attempts to underline thestatus and potential of these methods in CFD calculations.

The goal of adaptive mcthods is the optimal control of the computational process: tocontrol the computation so as to produce the "best" results for the least effort. Differences invarious adaptivc schemes hinge on how this optimal is defincd, on how the control parametersare selected and measured, and on how the "effort" is defined. In most of the results describedhere, the control parameter is the numerical error of the solution in each finite element, sothat one attempts to control the accuracy of the computation. vVe also mention somestrategies in which numerical stability is also a factor, so that the time step, for example, isalso controlled to maintain stability at minimum computational cost.

There is an important byproduct of such a control strategy: if the error in a computationcan indeed be estimated, then the user has a ready-made mcasure of the reliability of hiscalculation. A good adaptive scheme employs ·sound, efficient, and reliable a posteriori errorestimators, to estimate the evolution of error during a computational process. In principle,such error estimators provide the user with a quantitative answer to a question asked withincreasing frequency in computer analyses; How good are the answers?

The best adaptive procedures function independently of the user, who merely prescribesa reasonable initial mcsh and a level of error he can tolerate or a dollar-value (the cost) heis willing to endure to complete a flow simulation. Thereafter, the adaptive code makes thedecisions necessary to produce solutions within the user-specified limits. Once the controlparameters (thc elemcnt errors or some reasonablc approximation of them) are available, thecode attempts to adjust them to meet control objectives-to minimize the error. Typically,the control of crror is achieved by refining the mcsh in arcas of the solution domain whcrecrrors are too large and by coarsening thc mesh (using larger elements) wherc the error issmall, or relocating nodes to increase nodal densities Ileal' regions of high crror. Also, onecould increase the degree of the local shapc functions in an element of fixed size and expectthc accuracy of the approximation to be increased. Thus, to summarize, there are severalbroad types of mesh adaptivity that can be used to control error:

h-methods-in which the element size h is used to control error. Error is reduced byrefining the mesh or by regenerating a new finer mesh. To obtain an optimal h-mesh(one with the least error possible for a fixed number of refincments), provisions for alsocoarsening a mesh should be included in the adaptive strategy.

r-methods-in which a fixed number of elements of a given order are used in a FE mesh,but in which nodes are relocated to reduce error in ccrtain regions. Thc r-methods arethus moving grid methods in which the number of degrees of freedom is fixed, but thenode locations are adapted to control error.

p-methods--in which the polynomial degree p of the element shape functions is raisedor lowered to control error. These p-refinement methods are mathematically akin tospectral methods used, for instance, in turbulence simulations, and some prefer to callthese approaches spectral adaptive methods.

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combined methods-in which combinations of h-, r-, and/or p-adaptivity are used.Some surprising results have been obtained using such combined adaptive techniques:exponential convergence, in which the error is reduced rapidly by selective adaptationof the mesh parameters.

There is a growing literature on adaptive finite element methods for broad classes ofproblems. Surveys of the state of thc subject primarily as it applies to elliptic problems canbe found in [7,8]; surveys of recent advances in adaptive methods for CFD can be found in[9-15] and other work on the subject is given in [16-21].

This note describes representative applications of smart algorithms and adaptive finiteelcment methods to sevcral classes of problems in CFD.

2 h, p, AND hp-ADAPTIVE METHODS

The adaptive methods of interest here are based on the automatic control of error by ad-justing the mesh parameters h (the mesh size) and p (the spectral order) or combinations ofhand p. Special features of the h- or h-p strategy are listed as follows:

1. For a flow domain n c lRn (n = 2 or 3), an initial mesh nh is generated. Inthe present work, this mesh can be completely unstructured and it cOllsists of quadri-lateral (n = 2) or hexahedral (n = 3) elements over which the conservation variables,U = {P,PUI,pu2,pe}T(n = 2) or, for incompressible flows, the primitive variables, (u,p) =(ttl' U2, P) are approximated by polynomials of degree p, and p can be differcnt in each clc-ment. I-Jere

P = mass densitytti = velocity componentse = total energyP = pressures

For simplicity in notation, we shall restrict the present discussion to two-dimensional prob-lems.

Each element ne in the initial mesh nh is initially regarded as the image of a masterelement fi which is mapped into its location in the grid by a smooth map Fe:

2. Let u denote a generic flow variable; u = tt (Xli X2, t), t ~ O. Let

Over fi, u is approximated by a function u* which is a tensor product of Legendre polynomialsof the type,

4 P

u· = L UA(t)~L\(~' 11) + L (lijCPi(OCPj(T/)L\=l k=2

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The functions .(fi1:J. are the usual bilinear shape functions; for fi = [-1,1] x [-1,1],

~ 1'1/;1 = - (1 +0 (1 + 11)4

1'1/;2 = - (1 - 0 (1 - 1/)4

The functions <Pi are given by

14"(1 - 0(1 + 1/)

1-(1 + 0(1-11)4

where Pi is the Legendre polynomial of degree i. The degrees of freedom aij are tangentialderivatives at the midsides of the master element, e.g.,

and mixed derivatives at thc centroid,

ai+iii(O,O) i+ j = 1,2,· .. ,p. ,earl)Thc approximation ii, mapped to elcment !le in the mesh and rcwrittcn in XI' X2 coordinatcs,is denoted

N.uh (x), X2, t) = I:adt)xdx1, X2) (XI! X2) E De

k=l

where Ne = the number of degrees of frcedom of element De = (p+ 1)2, Xk are the transformed(and relabeled) shape functions defined on the mesh, and ak(t) arc the time-dependentdegrees of freedom.

3. Suppose, for the moment, that the orders of approximation (the polynomial degreesp) of shape functions and the degrees of freedom are such that the global approximation Uh

(or Uh of U) is continuous over Dh' Then the discretized Navicr-Stokcs equations are solvedfor Uh and the computed numerical solution is used to compute a local error indicator, 1/e:

1/e = estimate of IIU - Uhlle

where II . lie is a suitable norm of the error U - Uh defined on element De. For detaileddiscussions on various methods for computing a posteriori error estimates, see [21]. Inaddition, a quantity ~1/= can be computed which indicates the change in error due to aunit increase in the s-th degree of freedom associated with elcmellt !le. The error indicator"'e is used to determine which elements arc to be refined or enriched; the error gradients/:1.,,; = max /:11/; are used to dcterminc if an h-refinement or a p-enrichrncnt is to be used, the

"idea being that the local adaptation is made which results in the largest dccreasc in crror

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2

3 7 4

Figure 1: A nonuniform hp-mesh generated by a sequence of maps of a master element nover which shape functions a.re defined which are tensor products of polynomiaJs of degreep.

(sec [17]). In Navier-Stokes calculations, the norm II . I/e is often selected to represent thechange in entropy over ne.

4. The computed error indicators are compared to some global error tolerance such as

{:S Ct1}l\'IAX a- E (0, 1)

T}e> Ct1}MAX

and those elements for which T}e exceeds the tolerance are listed, together with their neighbors,for refinement. For these listed elements, !:iT}; is also computed, and a mesh refinement/enrichment decision is made.

5. The data structure employed in the results given later in this paper makes use ofinterelement nodal constraints so as to maintain full continuity of the approximate flowvariables across element boundaries. If an element is h-refined (subdivided) the midsidenode becomes an inactive, constmined, or hanging node and is assigned degrees-of-freedomwhich are interpolated values at the a.ctive vertices OIl either side of the midside node (seeFig. 1a). Likewise, if polynomials of different degree occur on interelement boundaries,the lower-degree edge shape functions are enriched so that full continuity is maintained.Thus, after an h-p refinement, an element ne may be the domain of shape functions whichare tensor products of polynomials of degree $ ]1, plus other edge polynomials added tomaintain interelement continuity, and the support of these edge functions spills over intoadjacent clements (see Fig. 1b). Detailed discussion of this data structure is given ill [19].

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3 IMPLICIT jEXPLICIT FORMULATIONSTHE NAVIER-STOI(ES EQUATIONS

OF

The compressible viscous flow of a calorically perfect gas is governed by thc Navier-Stokesequations in the form

Ut + Fj,i = FY,j (1)

where U is the vector of conservation variables and Fi and FY are the inviscid and viscousfluxes, respectively, given by

Fi = {

PVj

PVl Vj + p81i

PV2Vi + p82i

(pe + p)Vj

whcre Ujj are the stress components, and qj are the components of the heat flux vector. HereUt = au lat and Fi•i = LiaFi/axj.

vVe shall outline one type of smart algorithm useful in adaptive schemes for time-dependent (unsteady) Navier-Stokes calculations; the use of explicit or implicit flow solvers.The scheme described here is based on a Taylor-Galerkin formulation, although the ideascan be used with othcr flow solvers. The idea is that the mesh size is determincd by theadaptive scheme so as to fulfill accuracy requirements while the timc step is either fixed ofselected on the ba.sis of other criteria. Clearly, to satisfy numerical stability requirements,an implicit scheme is used for any h/ tit combination that violates the local CFL conditionwhen h is large enough that the CFL condition holds, an efficient cxplicit schemc can beused. The result is an explicit/implicit scheme in which, at any time n6.t, explicit or implicittime marching schemes may be used on different portions of thc mesh.

To outline briefly one formulation that admits such smart algorithmic structure, assumethat the solution un at time step tn is known and the solution Un+1 at time tn+1 is to becalculated. An increment of the solution between steps nand n + 1 is given by:

(2)

Observing thatur:/o = U~/{J + 0((0 - {3)6.t)

a second implicitness parameter can be introduccd while still preserving the second-orderaccuracy. The first derivatives are obtained from the Navier-Stokes equations while the

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second are given by

whereAi, Pi and Rj are the Jacobian fluxes

aFj

Ai= au Pj = aFrau

aFrRj = aU

J

The convective terms involve spatial derivatives up to fourth order. Limiting this formula toterms with second-order derivatives, which can be effectively handled by Co continuous finiteelements, can be linearized, while preserving the accuracy, with the resulting incrementalformula:

~U + Q~t(A~~U),i -/~t [(R0~U,j),i + (P~~U),j]

(1 - 2Q)/3~;2 (A? AjL\u,j),j

= ~t (F~'!l - FT!.) + (1 - 2Q)~t2 (AT! Fk~k) .It' ',' 2' t "

(4)

To select the implicit and explicit zones, all nodes which violate the stability criterion foran explicit scheme are treated with an implicit scheme. According to this strategy, severaloptions for an automatic adaptive selection of implicit/cxplicit zones are possible:

• User-prescribed time step ~t: \Vith this option the user prescribes the time step. Allnodes satisfying the stability criterion for the explicit scheme (with some safety factor)are explicit.

• Prescribed maximum CPL number: In this option the user prescribes the maximumCFL number that can occur for any elemcnt in n. The time step is automaticallyselected as the maximum step satisfying this condition. The choice of maximum OFLnumber may be suggested by the timc accuracy argumcnts or the quality of results.

• Minimization of computational cost. For additional details, see [22].

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4 INCOMPRESSIBLE NAVIER-STOKES CALCU-LATIONS

We use a penalty approximation for the incomprcssibility constraint and the method ofcharacteristics to resolve the Navier-Stokes equations for incompressible flow:

Du 1Dt == Re ~u - \7p + f in n x (0, T)

div u == 0

where Dul Dt is the material time derivative. The following two-step, sccond-order, implicitalgorithm, which is a variant of that of Pironneau [23], may be used to advance the solutionin time;

Step 1n+~

D~==ODt

Step 2

for all Wh in Wh

HereVh == Vh(:V, t) == the finite element approximation of the velocity field at point :v

in n at time t, n being the flow domain

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Uh(X, n.6.t), n = 1,2,3 ...

an arbitrary test function

a (Uh+t,Wh)

the work done by the viscous stress on the velocity gradients\7wh

I( v, w) = a numerical quadrature approximating the scalar product,No

(v, w) = 1v, wdx = Lw,v(e,) . w(e,),n '=1w, = quadrature weights,

e, = Gauss or Gauss-Lobatto itegration points.

(f, Wh) = cxternal force terms

e = the penalty parametcr, c > o.

}Vh = the space of h-p -finite element approximationsof the velocity field.

The corresponding pressure approximation is

at integration point e,.In the method of characteristics, it is understood that

wherein uh-l and ui: are connected along the characteristic line, X(:1:, tn; tn-t), drawn back-

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ward in time from point (;v, n~t); i.e., X is the solution of

dX = u(X,r) ,dt

X(t) = ;v

Some applications of these techniques are discussed in the ncxt section. Note that if

!!div uhll > co = tolcrance£2(0,) -

then a pressure correction, p; can be computed by solving

5 SOME APPLICATIONS

In concluding this note, we list a fcw representative results of numerical experimcnts withadaptive h- and hp-algorithms applied to benchmark problems in compressible and incom-pressible flow.

1. Hypersonic Flow Over a Ramp. Figure 2 shows an adapted mesh for the so-calledHolden problem of Mach 14.2, Re = 72 x 106 compressible flow over an inclined ramp.As indicated, the optimal h-adaptcd mesh contains only 5545 degrccs of freedom.Computed density contours, which are in good agrcement with experimental data, areshown in Fig. 3.

2. Oblique Shock on a Blunt Body, Mach Number = 8.03. An h-adapted mesh for inviscidflow at 13°-angle of attack is ShOWIlin Fig. 4. The 3830 degrees of freedom yieldedthe results in Fig. 5 where computed density distributions for the adapted mesh areshown.

3. Subsonic Flow Over a Hemisphere. An adaptive mesh, showing unstructured andstructured regions, modeling subsonic Ai a = 0.1 flow, Re = 100, around a hemisphereis shown in Fig. 6. The explicit/implicit scheme was used with around 70 percentimplicit elements. The computed velocity field is shown in Figs. 7 and 8, with thedetails of flow recirculation shown in Fig. 9. ?\1ach number, vorticity, and pressurecontours are ShOWIlin Figs. 10, 11, and 12.

4. Supersonic Flow in a Convergent-Divergent Shock Tube. As an examplc of an h-adapted, unstructured mesh, a shock-tube problem with supersonic inflow is shown inFig. 13. The mesh is automatically adaptcd along shocks and computed pressure andMach contours arc shown.

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, 2 3 4 5 6 7 8

Figure 2: The Holden problem: Mach 14.1 hypersonic flow over an inclined ramp,Re = 72 X 106. Shown here is an h-adapted mesh of p = 1 elements, showing automaticrefinement around the shocks and boundary layers. The "optimal" h-mesh contained 5545degrees of freedom.

0.25 6.25 13.25 21.25

Figure 3: Computed density contours for the It-adapted mesh generated for the IIoldenproblem of Fig. 1; maximum = 26.!J5, minimum = 0.29.

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8

7

6

5

4

3

2

Figure 4; Mach 8.03 inviscid flow; oblique shock on a blunt body; this adapted mesh wasobtained after h-refinement; 3830 degrees-of-freedom.

0:e:

Figure 5; Density contours: max = 9.707, min = 0.956 for impinging shock problems in Fig.4.

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Figure 6: Subsonic flow (Moo = 0.1, Re = 100) around a half cylinder. Examples of anunstructured axisymmetric mesh intersecting a structured mesh after adaptive refinement.The explicit/implicit solver is used.

Figure 7: Portion of the mesh of Fig. 6 over which the implicit scheme is used.

Figure 8: Computed velocity over hemisphere: explicit/implicit solvers on an adapted mesh.

Figure 9: Computed recirculation in implicit zone.

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I'lACU "".::<!6fR CeNTBUflS "IN: 0.561 ·02 "AI;: O.ID! -00 (NHRYA:..::: O.SIC ·02

Figure 10: Mach number contours on hcmisphere.

Figure 11: Computed vorticity contours for flow around a half-sphere.

PRE5St.l~: ce~'0URS "IH 0.112 -00 HJ:ll ~ 0,120 -00 :Nf[RVAl= D.::~e ·C3

Figure 12: Pressure contours on hemisphere.

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5. Turbulent, High Reynolds Number', Low Mach Number Flow Around a TurnaroundDuct. Turbulent compressible axisymmetric flow, Re = 106, !l1a = 0.1, in a long 1800

turnaround duct is modeled in Fig. 14 using 3200 elements and 3321 nodes. A k-e-turbulence model is used togethcr with the explicit/implicit scheme. A blowup of theadapted grid is shown in Fig. 15. Computed results are shown in Fig. 16, with velocityand recirculation shown in the upper left, pressure contours in the upper right, Machnumber in the lower right, and vorticity in the lower left quadrants.

6. hp- and p-Approximations. Results obtained using adaptive hp-methods are displayedin Figs. 16 and 17. An hp-mesh modeling again the turnaround duct problem forRe = 103, A! a = 0, is shown in Fig. 16. Here an hp-mesh with p varying from 5, to3, to 1 away from the boundary-layer near the recirculation zone is shown. Velocityvectors are plotted in Fig. 17 to illustrate the recirculation zone.

7. Three-Dimensional Incompressible Flow"Around a 90° Elbow. Some preliminary resultsof hp-adaptive calculations of incompressible flow, Re = 780, around a circular ductwith a square cross section are prcsented in Figs. 18 and 19. A parabolic inflow isprescribed and a very coarse mesh consisting of 36 cubic and 28 quadratic elements,with appropriate interface enrichments. Thesc are indicated by different shadings inFig. 18. Computed contours of pressure are shown in Fig. 19.

Acknowledgements: The numerical results rcported herc werc produced in collaborationwith Drs. "V. VV. Tworzydlo, C. Y. Huang, R. Krishnan, S. R. Kennon, and ~lssrs. S. Stowcrsand S. Vadaketh in conncction with projccts supported by the U.S. National Aeronau ticsand Space Administration.

References

1. aden, J. T., "A General Theory of Finite Elements. Part 1. Topological Considera-tions," International Journal of Numerical Methods in Engineering, Vol. 1, pp. 147-159,1969 and "Part 2. Applications," Vol. 1, No.3, pp. 247-259,1969.

2. aden, J. T., and Symogyi, D., "Finite Element Applications in Fluid Dynamics,"Journal of the Engineering Afechanics Division, ASCE, Vol. 95, No. EM3, June,pp. 821-826, 1969.

3. aden, J. T., "A Finite Element Analogue of the Navier-Stokes Equations," Jou1'nal ofthe Engineering Mechanics Division, ASCE, Vol. 96, No. EM4, 1970.

4. aden, J. T., "The Finite Element t-.lethod in Fluid :Mechanics," Lectures on Fi-nite Elements in Continuum Mechanics, Editcd by J. T. aden and E. R. A.Oliveira, Proceedings, NATO Advanced Study Institute, Lisbon, 1971, and UAH Press,Huntsville, pp. 151-186, 1973.

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PRESSURE C0NT0URS MIN = O. Q66E+00 MRX = 0.382E+OI INTERVRL:: 0.168£+00

MACH C0NT0URS MIN = 0.187E+OI MAX = 0.3QQE+Ol INTERVAL= 0.785E-OI

Figure 13: Unstructured adapted mesh for shock tube calculation.

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~......-...::-..-...- .. -.;;;- __ - -_ -.- "~~~~ .. 'a.:o-:;;:....:.=;;..o....:.,..:..)];;;~+::;.~;~~._

~.'i""""~"",,"~"'~"J~;;=;=~~~~m~~~~:;==

Figure 14: An initial mesh on a 1800 turnaround duct; Re = 106, Ma = 0.1.

Figure 15: Adapted grid for turbulent flow in turnaround duct.

17

,U'lCly

c

Figure 16: Computed results for turnaround duct problem.

18

\

\.\

\\,,

-' . .' !,,', , .r....;..' ',>--- .. i:-----~-../ ;

! 17I .•I· I:!=q,1~\ ... \,.' .; . ;..i 'I! '---

III

.. tnt I4 ~ 8 7 8 J

Figure 18: Flow separation calculated on hp-adapted mesh.

19

5

4

3

2

Figure 19: A coarse hp-mesh consisting of cubic and quadratic clements.

~inflow

0.23

Figure 20: Computed pressure contours.

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5. aden, J. T., "Finite Element Formulation of Problems of Finite Deformation and Irre-versible Thermodynamjcs of Nonlinear Continua-A Survcy and Extensions of RecentDevelopments," Recent Advances in Matrix Methods of Structural Analy-sis and Design, Edited by R. H. Gallagher, Y. Yamada, and J. T. aden, Univ. ofHuntsville Press, pp. 693-724, 1971.

6. aden, J. T., and \Vellford, L. C. Jr., "Analysis of Flow of Incompressible Viscous fluidsby the Finite Element Method," AIAA Journal, Vol. 10, No. 12, pp. 1590-1599, 1972.

7. Babuska, I., Zienkiewicz, O. C., Gago, J., and Oliveira, E. R. A. (Eds.), AccuracyEstimates and Adaptive Refinements in Finite Element Computations, JohnvViley and Sons, Ltd., Chichester, 1986.

8. aden, J. T., and Demkowicz, 1., "Advances in Adaptive Improvements: A Survey ofAdaptive Ivlethods in Computational Fluid Mechanics," State of the Art Surveysin Computational Mechanics, Edited by A. K. Noor and J. T. aden, AmericanSociety of Mechanics Engineers, N.Y., 1988.

9. aden, J. T., "Adaptive Finite Elemcnt Methods for Problems in Solid and FluidMechanics," Finite Element Theory and Application Overview, Edited by R.Voight, Springer-Verlag, N.Y., 1988.

10. aden, J. T., "Adaptive FEivl in Complex Flow Problems," The Mathematics ofFinite Elements with Applications, Edited by J. R. Whiteman, London AcademicPress, Ltd., Vol. 6, pp. 1-29, 1988.

11. aden, J. T., "Smart Algorithms and Adaptive Methods in Computational Fluid Dy-namics," Proceedings of CAN CAM (Canadian Congress on Applied Mechan-ics), Ottowa, Canada, 1989.

12. aden, J. T., "Progress in Adaptive I\fethods in Computational Fluid Dynamics,"Adaptive Methods for Partial Differential Equations, Ed.by J. Flaherty, etaI., SIAM Publications, Philadelphia, 1989.

13. aden, J. T., Strouboulis, T., and Devloo, P., "Adaptive Finite Element Methods forthe Analysis of Inviscid Compressible Flow: I. Fast RefinementjUnrefinement andMoving Mesh Methods for Unstructured Meshes," Computer Methods in Appl. Mech.and Engrg., Vol. 59, No.3, 1986.

14. aden, J. T., Strouboulis, T., and Dcvloo, P., "Adaptive Finite Element j\ilethods forCompressible Flow Problems," Numerical Methods for Compressible Flows-Finite Finite Difference, Element and Volume Techniques, Edited by T. E.Tezduyar and T. J. R. Hughes, AMD-Vol. 78, ASME, New York, pp. 115-126,1987.

15. aden, J. T., Strouboulis, T., and Dcvloo, P., "Adapti\'e Finite Element \lethodsfor High-Speed Compressible Flows," International Journal for Numerical Methods inFluids, Vol. 7, pp. 1211-1228, 1987.

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16. Odcn, J. T., Bass, J. :M., Huang, C. Y., and Berry, C. \V., "Recent Results on SmartAlgorithms and Adaptive !vlethods for Two- and Three-Dimensional Problems in Com-putational Fluid Mechanics," Computers and Structures, Vol. 35, No.4, pp. 381-396,1990.

17. Rachowicz, W., Oden, J. T., and Demkowicz, L., "Toward a Universal h-p AdaptiveFinite Element Strategy, Part 3. A Study of the Design of h-p 11eshes," ComputerMethods in Applied Mechanics and Engineering, 77, pp. 181-212, 1989.

18. aden, J. T., "Smart Algorithms and Adaptive Methods in Fluid-Structure Interac-tion," AIAA Aerospace Sciences Meeting, AIAA-90-0577, Reno, NV, 1990.

19. Demkowicz, L., aden, J. T., Rachowicz, W., and Hardy, 0., "Toward a Universalh-p Adaptive Finite Element Strategy, Part 1. Constrained Approximation and DataStructures," Computer !v[ethods in Applied kfechanics and Engineering, 77, pp. 79-112, 1989.

20. Bass, J. M., and aden, J. T., "Adaptive Computational Methods for Chcmically-Reacting Radiative Flows," International Journal of Engineering Science, Vol. 26, No.9, pp. 959-992, 1988.

21. aden, J. T., Demkowicz, L., \Vestermann, T., and Rachowicz, W., "Toward a Uni-versal h-p Adaptive Finite Element Strategy, Part 2. A Poste1'iori Error Estimates,"Computer Methods in Applied Mechanics and Engineering, 77, pp. 113-180, 1989.

22. Odcn, J. T., and Bass, J. M., "New Developments in Adaptive Methods for Compu-tational Fluid Dynamics," Proceedings of the Ninth International Conferenceon Computing Methods in Applied Sciences and Engineering, Paris, France,January 29-February 2, 1990, SIA~\'I, pp. 180-210.

23. Pironneau, 0., "On the Transport-Diffusion Algorithm and its Applications to theNavier-Stokes Equations," Numerische Mathematik, 38, pp. 309-332, 1982.

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