applied - university of texas at austinoden/dr._oden_reprints/... · 2008-04-09 · adaptive...

COMPUTINGMETHODS INAPPLIEDSCIENCES ANDENGINEERINGEdited by Roland Glowinski

University of Houston and INRIA

Alain LichnewskyINRIA

New Developments in Adaptive Methods forComputational Fluid Dynamics

J. Tinsley OdenJon M. Bass

3 339

Philadelphia

Society for Industrial and Applied MaChematics

CHAPTER 9

New Developments in Adaptive Methods forComputational Fluid Dynamics

J. Tinsley Oden*Jon M. Basst

Abstract

New developments in a posteriori error estimates, smart algorithms, and h- and h-p adap-tive finite element methods are discussed in the context of two- and three-dimensionalcompressible and incompressible flow simulations. Applications to rotor-stator interaction,rotorcraft aerodynamics, shock and viscous boundary layer interaction and fluid-structureinteraction problems are discussed.

1 Introduction

A survey of recent literature in computational fluid dynamics (CFD) will reveal that conven-tional methods of flow simulation are gradually being displaced or augmented by so-calledadaptive schemes; i.e., schemes which automatically modify themselves during the courseof a calculation to accommodate changing properties of the numerical solution. Adap-tive methods represent methodologies designed to optimize and control the computationalprocess-to produce the best results, in some sense, for the least computational effort andto control the quality and stability of the evolving solution. The control parameters arethe accuracy of the solution-as measured by the error in some appropriate norm-andthe stability or robustness of the scheme-as determined by the time step, the number ofiterations, the artificial viscosity, or some similar parameter. Successful adaptive schemesmust, therefore, embody a technique for a posteriori error estimation and control. Oncean estimate of the control variables are in hand, the control itself is imposed by changingthe structure of the numerical scheme, and, typically, this is accomplished by one or moreof the following approaches:

• h-methods-in which the local mesh size h is changed by refining or regenerating themesh.

• p-methods-in which the local spectral order of the approximation is changed.• r-method-in which grid points are relocated to increase or decrease mesh density.

'Texas Institute for Computational Mecha.nics, The University of Texas, Austin, Texas, U.S.A.'The Computational Mechanics Co., Inc., Austin, Texas, U.S.A.

180

ADAPTIVE METHODS FOR COMPUTATIONAL FLUID DYNAMICS 181

The control of these parameters, and the time step At, is accomplished using smart algo-rithms, which are designed to function on unstructured meshes and to adjust themselves tochanging requirements of accuracy and stability.

A summary of the status of adaptive methods up to 1986 is found in the volume editedby Babuska, et al. (1] and a survey of later literature on the subject is given in Oden andDemkowicz (2]. More recent extensions of these types of numerical methods to problems inCFD are summarized in the papers of Oden (3, 4, 5, 6], see also (7-14].

In the present paper, we describe several new developments in adaptive methods forproblems in CFD. Some of these techniques have been incorporated into a production CFD-code, ADAP'J0, which uses h-method schemes to treat broad classes of flow problems, andin a companion code, PHASER®, which is based on a new combined h-p methodology toresolve broad classes of boundary- and initial-value problems. Details of the ADAP'J0code as it applies to chemically-reacting viscous, compressible flows and fluid-structureinteraction problems are given elsewhere (8, 9] and are not repeated here; however, someresults obtained with h-adaptivity are given in the closing section of this work. Extensionsof the hop strategies to compressible and incompressible flows are described herein togetherwith a new technique for adaptively controlling the local implicit or explicit form of a CFDalgorithm.

2 An Explicit-Implicit Algorithm for the CompressibleNavier-Stokes Equations

Computational techniques for modeling complex viscous and inviscid flows have de-veloped at an accelerated rate over the past few years. The approaches described in theliterature vary from full explicit algorithms, which are computationally inexpensive butoften severely limited by stability restrictions, to fully implicit algorithms which are uncon-ditionally stable but are much more expensive per time step. The selection of which typeof algorithm is optimal for a given application is generally not known a priori and may infact change as the features of the flowfield develop. As a result, a domain decompositionapproach is gaining popularity in the CFD community which employs an explicit formula-tion in one region of the mesh and a fully implicit formulation in another. Tezduyar, et al.(15, 16, 17, 18], for example, has demonstrated the effectiveness of methods of this type fortwo-dimensional incompressible flow simulations.

In this section, an implicit-explicit Taylor-Galerkin method for viscous compressibleflow is presented. The development consists of three basic steps. First, a general family ofimplicit Taylor-Galerkin methods are formulated which includes three implicitness param-eters. Depending on the selection of these parameters, a fully explicit scheme or a varietyof implicit schemes may be obtained. Next, criteria for automatically selecting implicitand explicit computational zones are reviewed. These criteria are based on stability andcost consideration and provide a reliable approach for selecting implicit and explicit zones.Finally, two different methods for combining implicit and explicit calculations are outlined.Full details on each of these approaches are given elsewhere (9, 20].2,1 Governing Equations

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

Ul + F;,i = Fr.i (2.1)

where U is the vector of conservation variables and Fj and FY are the inviscid and viscousfluxes, respectively. The indices i in this formula refer to the axis of a Cartesian coordinatesystem, a comma denotes partial differentiation, and the summation convention is applied.The components of these vectors in the two-dimensional case are given by

182 Oden and Bass

U = { :Vl } F i = { :~:Vi + pOl i } F Y = { ~l i }PV2 pV2Vi +p02i I CT2i

pe (pe + p)Vi CTmiVm + qi

where p is the fluid density, p is the pressure, Vi are the velocity components, e is the totalenergy per unit mass, CTij are the stress components, and qi are the components of the heatflux vector.

2.2 Implicit Taylor-Galerkin Methods

As a starting point, let us assume that the solution Un at time step tn is known and thesolution Un+1 at time tn+1 is to be calculated. Formally, the values of the solution at timestn and time tn+1 can be expressed by the second-order Taylor series expansion about anarbitrary time tn+lr where a is an implicitness parameter with values between zero and one:

(2.2)

By subtracting these two formulas a formula is obtained for an increment of the solutionbetween steps nand n + 1:

(2.3)

Observing that:

a second implicitness parameter can be introduced into equation (2.3) while still preservingthe second-order accuracy,

(2.4)

(2.6)

The next step is to express the quantities evaluated at times tn+lr and tn+1J by quantitiesevaluated at the basic steps tn and tn+1,

U~+lr = Uf + aAUt + o(At2)

U~+IJ = U~ + (JAUtt + o(At2).

Substituting these formulas into equation (2.4) yields a two-parameter expansion:

At2

AU = At(U~ + aAUc) + (1 - 2a)2(U~ + PAUll) + 0(At3) (2.5)

Now, following the procedure outlined by Lax and Wendroff, the governing equations can beused to replace time derivatives with spatial derivatives. This substitution yields a formulafor the first derivatives:

and for the second-order derivatives:

Utt = [R;j (Fr..Io - F.\:,.\:).j + Pi (Fr." - F.\:..\:)L- [Ai (Fr..\: - Fu) L (2.7)


whereAi, Pi and R;j are the Jacobian fluxes

aFiAi = au

The convective terms in equation (2.7) involve spatial derivatives up to fourth order. Lim-iting this formula to terms with second-order derivatives, which can be effectively handledby CO continuous finite elements, yields the following approximation for the second-ordertime derivative

(2.8)

where o(µ, Ie) represents a quantity of the order of the viscosity parameters in the Navier-Stokes equations. Substitution of formulas (2.6) and (2.8) into the incremental equations(2.5) gives the implicit formula:

AU(2.9)

For the sake of maximum generality, a third implicitness can be introduced by observingthat

aAFY,i - aAFi,i = "'{~FY,i - aAFi,i + (a - "'{)o(µ, k)o(At)

Substituting this expression into (2.9) yields the three-parameter implicit form for theincrements of the conservation vector

~u(2.10)

Equation (2.10) represents a nonlinear formula for increments of the solution U at agiven time step. This formula is nonlinear due to nonlinear dependence of the fluxes andJacobians on the solution U. This equation can be linearized, while preserving the accuracy,with the resulting incremental formula:

(2.11)

2,3 Selection of Implicit and Explicit Zones

The second step in the development of an implicit/explicit algorithm is the selection ofimplicit and explicit zones. The basic strategy is that for a given time step all nodes whichviolate the stability criterion for an explicit scheme should be treated with an implicitscheme. According to this strategy, several options for an automatic adaptive selection ofimplicit/explicit zones are possible:

• User-prescribed time step At: With this option the user prescribes the time step.All nodes satisfying the stability criterion for the explicit scheme (with some safetyfactor) are explicit. This means that all the elements connected to these nodes aretreated with the explicit scheme. For all other elements an implicit scheme is applied.

184 aden and Bass

• Prescribed maximum CFL number: In this option the user prescribes the maximumCFL number that can occur for any element in n. The time step is automaticallyselected as the maximum step satisfying this condition. The choice of maximum CFLnumber may be suggested by the time accuracy arguments or the quality of results(for a Taylor-Galerkin scheme, too large a CFL number tends to smear shocks) .

• Minimization of computational cost: Obviously, both of the above approaches pro-vide a means for determining implicit and explicit zones of the computational mesh.However, one of the goals in using implicit/explicit methods is to reduce the computa-tional cost of integrating the governing equations forward in time. It therefore makessense to use the relative cost of the numerical procedure to select which zones of themesh should be treated implicitly and explicitly. The approach incorporated into theADAP':[0 code is based on the fact that for an increased time step an increasing num-ber of elements must be analyzed with the (expensive) implicit algorithm. A typicalsituation is presented in Fig. 2.1, which shows for different time steps the relativenumber of nodes that must be treated with the implicit scheme to preserve stabil-ity. On the abcissa ATE denotes the longest time step allowable for the fully explicitscheme, and AT] denotes the shortest time step for the fully implicit scheme. Therelative number of implicit nodes increases as a step function from 0 for At < AtE to1 for At > At/.Now assume that the ratio r of the computational cost of processing one implicitnode to the cost of processing an explicit node is known. Then the relative cost ofadvancing the solution in time with the implicit/explicit scheme, as compared to thefully explicit scheme, is given by the formula

D.tE E IR(At) = -(n + rn )

6.t

where nE and nI are the relative number of explicit and implicit nodes, respectively.

tit,(fully implicit)

t1TME(fully explicit)

o

~TE- fully explicit time step

~T.- fully implicit time stepN. - number of implicit nodes

N - total number of nodes

Figure 2.1: Relative number of implicit nodes for increasing time step.


Typical plots of R(At) are shown in Fig. 2.2 for the cases of a small difference anda large difference between AtE and At I. From these plots the following observationcan be made:

1. for an almost uniform mesh the mixed implicit/explicit procedure does not pro-vide computational savings~ither a fully implicit or fully explicit scheme is thecheapest depending on the time step restriction;

2. for very diverse meshes the mixed method provides considerable savings.

1

R R

~tICFLm .. )

(a) (b)

Figure 2.2: Reduction of the cost of computations due to implicit/explicit procedure.

2.4 Formulation of Implicit/Explicit Schemes

Assuming that the implicit and explicit subregions of the mesh have been properly defined,then the question arises as to what is a consistent procedure for combining the implicit andexplicit computations. One possibility, proposed by Hassan, Morgan, and Peraire [21], isbased on the following steps:

1. Perform the explicit step computations on all nodes in the mesh.

2. Perform the implicit computations in the subregions, where the stability criterion forthe explicit scheme is violated. The solution in remaining nodes is "frozen" at thisstep.

This procedure is quite simple to employ, however, it appears to be nonconservative andmay disturb the regularity of the solution in the transition zone. This is caused by the factthat during step two the "frozen" explicit nodes impose Dirichlet bounding conditions onthe edge of the implicit zone. Prescribing these conditions implies that there must existan external source of fluxes to support the prescribed state of the solution. Since no suchsource exists within the domain, the solution will not be conservative across this line.

A second approach, still under development, is based on a generalization of equation(2.11) where the implicitness parameters are not assumed constant but are continuum func-tions of the position z. With this assumption additional terms are introduced into the weakformulation when the incremental equations are integrated by parts. This approach appearsto be the most general requiring no additional interface conditions or loss of continuity acrossthe implicit/explicit line. Specific details on this approach can be found in [20].

186 aden and Bass

3 An Operator Splitting Technique

We shall make use of an operator splitting technique based on the splitting theorem ofStrang [22] and introduced recently by Demkowicz, Oden, and Rachowicz [23]. We recallthat Strang-splitting is a general technique for decomposing arbitrary evolution problemsinto a series of fractional steps so as to result in an algorithm "second-order accurate intime" (i.e., with truncation error o(At3)). Thus, if

Ut = c (D"'u, z, t) , u(z,O) = uo(z) (3.1)

is a nonlinear system of evolution equations with right-hand side dependent on variouspartial derivatives D"'u of u = u(z, t). The solution of (3.1) at time r is of the form

u(z, r) = (Gt(r)uo)(z, t) (3.2)

where Gt is an element in a semigroup G defined by T and its inverse. Next, we arbitrarilysplit c to form

and consider two fractional evolution problems of the form

Vt = a(D"'u, z, t) }

v(z,O) = uo(z)

WI == b(D"'u,:I:, t) }

W(:I:, 0) = uo(z)

The solutions of (3.3) and (3.4) are of the form,

(3.3)

(3.4)

(3.5)

v(z, r)

w(z, r)

== (Mt(r)uo)(z, t) }

= (Nt(r)uo) (z, t)(3.6)

Under suitable smoothness and stability assumptions, it can be shown that

where 0 < Co ~ c(z) < Cl < 00 and :I: En, the spatial domain of the solution u. Thus, towithin terms of order r3,

(3.8)

Turning now to the compressible Navier-Stokes equation, we write these in the form

U t - c(U, U J , z, t) = 0

where, for two-dimensional cases,

- c(U,U J,z,t) = F;" - Fr,We now split the operator c of (3.10) into convective and diffusive parts:

(3.9)

(3.10)


(E(t)Uo) (x) = UC(x, t)

uf + Fi.i = 0

(H(t)Uo) (x) = UD(X, t)

UD _ FV. = 0t 1,1

Our version of a three-step Strang procedure consists of the following algorithm:

(3.11)

1. Use a third-order explicit-implicit Taylor-Galerkin method (obtained by setting Q = 0in (2.11» to define the convection step operator E(t):

Find UnH such that

(3.12)

t.t21 2- (1 - R)_ '" W7."A-A-Un·dxJJ 2 n.~ ,I') ,)

1,)=1

_ t.t r WT Fi(Un)nidx - (1 _ (3) A2t2 r WT AnU~dx

Jan Jan

for every W

Thus, E(t) of (3.11) is defined by a weak form of the problem,

2_ ~ i ide! i

U I - - L- A U ,i I A = F Ui=1

(3.13)

2. Use a third-order explicit-implicit Taylor-Galerkin method to define the viscous stepoperator H(t):

U(t + At) - >.t.tU,I(t + At) = U(t) + (1 - >')AtUAt) + O(At2) (3.14)

This results in the system

2 1

m~+l _ >.t.t '" q!'.-I:1 = mry + (1 - >')At '" r!l··) L- '),1) ~ '),1~1 ~1

enH _ >'At ~ (~ q!,H U!'+1 + 1I:8~+1)L- L- IJ) ,Ii=1 j=1 .

,I

= en + (1 - >.)At t (t qijUj + 11:81)•=1 )=1 ,i

(3.15)

(3.16)

(3.17)

188 aden and Bass

where mj are momentum components and 0 is the temperature. An analogous varia-tional statement to that of (3.12) is easily obtained and is omitted here for brevity.

3. Combining solutions to (3.13) and (3.14) to form the three-step operator,

S(t) = H (~) E(t)H (~)

which has the property that

IS(t)Uo(z) - U(z, t)1 ~ c(z)t3 (3.18)

4. The implicitness parameters {3 and ..\ are selected throughout the mesh according tostrategy described in the previous sections.

There remains the critical issue of boundary conditions, since the boundary conditionsof (3.12) are different and, indeed, conflict with those of (3.14). This issue discussed in acompanion paper by Demkowicz, Oden, and Rachowicz [23]. The objective is to producea set of consistent and stable boundary conditions following the theories of Strikwerda [24]and Gustafsson and Sudstrom [25] for incomplete parabolic systems. Accordingly, if 6U is aperturbation of U about a uniform flowfield, then the (linearized) Navier-Stokes equationsare stable in the sense of linearized entropy, i.e.,

dd f 6UT Ao6Udx < 0tin -

whenever on the boundary an of n,(3.19)

where

1 u v 1_[tJl2 + 1] -tJl- -t/J- -(1/1 - 1)p £ £ £

1 ( U2)

uv U-;: l+p"2 p- -p-

£2 £2

Ao= ISym. 1 ( v

2)

pv-;: l+p--;:- -J"

p£2

(3.20)

A ¥Ai.n - n,

and i = e - p-1(m? + m~)/2, and 6'tn is the perturbation in viscous flux. By representingthe normal Jacobian matrix

in terms of its left- and right-eigenvalues bi and C;, it is possible to reduce (3.18) to theform,

(3.21)

where ..\i are eigenvalues of the Euler-Jacobian matrices (..\1 = Un - C,..\2 = ..\3 = Un,..\4 =Un + C, Un = U . n, c = speed of sound). Boundary conditions for the splitting scheme arederived directly from (3.21) for all steps in the three-step algorithm so that the resultingstep is stable in the sense described above. See [23] for more details.


4 A Posteriori Error Estimates for the Navier-Stokes Equa-tions

In [1], the chapter by Oden, et a\. [13] describes an error estimation technique referredto as the error residual method (ERM) which has proved to be effective for linear ellipticproblems. Bank and Weiser [26, 27] have developed the technique independently and provedthat the technique produces a global error indicator that is bounded above and below bythe actual global HI-error for a linear second-order elliptic problem in two dimensions.Extensions of this theory to h-p finite element approximations of linear elliptic problemswere contributed by Oden, et al. [28,29,30). Demkowicz, Oden, and Rachowicz [23]recentlyextended the method to Navier-Stokes equations by applying the method to convective anddiffusive steps in an operator splitting scheme of the type described in the previous section.

4,1 Elliptic BVP's for the Model Two-Dimensional Problem

Find u E V = {v E Hl(n):v = 0 on ru c en} such that

B(u, v) = L(v) 'v'v E V (4.1)

B(u,v) = k(o'Vv+buv)dx }

L(v) = r Jvdx + r gvdsIn Jr.(4.2)

with usual assumptions on the data a,b,f,g, and dx = dXldx2' We consider an h-p finiteelement approximation of (4.1) in a space

(4.3)

where Qilp is a partition of (n),

dia(K)

PpK(K) + spaces of polynomials of degree PKdefined over element K

For K = [-1,1] x [-1,1] a master element, we define the p-interpolation operator,

by

fipuUl

U2

Ue

Then

fip: HT (K) _ Pi' (K)

Ul + U2 + U3bilinear interpolant of uedge interpolant of U, for edge E of K,

u21£ = Pp(u - u)l£Ho - projection of (u - Ul - U2) on thebubble functions of degree h-p I (vanishing on 8/( = UE)

(4.4)

(4.5)

(4.6)

190 aden and Bass

and, by standard scaling arguments, Xh.p{O) has the interpolation properties,

'r/u E H'(K),3 C independent ofu,hK,PK such that

(4.7)

Consider a coarse mesh QH, H> h, and denote

1H: H'(O) - Xhp(O)

(Ihu)IK = IIH(uIK)k E QH

The space Mil. of bubble functions is defined as

Denoting the local and global energy norms by

IIIvll12 = B(v, v) = L BK(V, v) = L IlIvlllkKeQH KeQH

we set out to derive an estimate of the error,

between the fine-mesh solution Uh and the coarse mesh approximation UH under the as-sumption that Illu - u"lll is of higher order than IIlu" - uHIII. The result is that (withstandard saturation properites in force) to within higher order terms in h and P,

Cd11eH111~ {2:IIllpKlllk}t ~ C2111eHIII.K

where 'PK are local error indicators obtained as solutions to the local problems,

Find 'PK E M,,(K) such that

Bk('PK, v,,) = Rk(V,,) 'r/ vII. E M,,(I<)

with

+ { (g - UH )v"dsJaKnr.

(4.8)

(4.9)

where r" = +'V . a'VuHp - / is the residual, and [UH] is the jump in the numerical fluxaouh/8nk on oK.

Estimation techniques of the type defined by (4.8) have proved to be very effectivefor linear elliptic boundary-value problems and for certain linear classes of parabolic andhyperbolic problems [31, 32].


4,2 Extensions to Navier-Stokes

The extension of our ERM to the compressible Navier-Stokes is made possible by twoconsiderations:

1. The splitting of the discrete Navier-Stokes equations into convective and diffusivesteps are described in the previous section and

2. Symmetrizing the operators in each step.

For the Euler step, we introduce the matrix Ao of (3.20) which is the Hessian of theentropy H = -pln(pp-'Y). This matrix is a symmetrizer of the Navier-Stokes equations inthe sense that

Ao = A~ > O( i.e., Ao is positive definite) }AoAi = (AoAif ~ 0AoKij = (AoKiif for i = jAoKii = (AoKiif for i :f. j

(4.10)

Thrning to the weak form of the Euler step given in (3.12), let BE(U, W) denote thebilinear form appearing on the right-hand side of that equation. Clearly, BE(U, W) isneither symmetric nor positive definite. However, we can introduce a new Euler-blinearform,

(4.11)

which is both symmetric and positive definite.In the Euler step the symmetrizer Ao is used with the provision that its derivatives are

discarded. This not only simplifies the calculation of the symmetrized bilinear form, but infact only then is the new bilinear form symmetric and positive definite.

Next we construct a similar symmetrizer for the viscous steps. The choice of a sym-metrizer for both viscous steps is very natural: in both cases the unsymmetry of the problemis a result of a discrepancy between the solution variables (m'omentum components, totalenergy) and the test functions (velocity components, temperature). As the solution vari-ables are fixed (use of the conservative variables is one of the main issues in the presentapproach), the symmetry must be recovered by changing the test variables, i.e., represent-ing the virtual velocities and temperature in terms of a prescribed density (fixed in theviscous step) and virtual momentum components or virtual energy. Formally this is doneby introducing the symmetrizer of the form

1An = -I (4.12)P

where p is the density function and I is the identity operator.Having introduced the notion of symmetrized problems, application of the ERM is

straightforward. The procedure consists of two steps. The first step is independent of asymmetrizer (which, incidently, is not unique) and consists in solving for each finite elementK a local variational problem in the form

Find If'K E X~,P+1(J() such that

(4.13)

192 aden and Bass

(4.14)

Here X~'P+l (K) is the space Mh(K) of bubble functions defined as a subspace of the spaceof element shape functions X h,p+1 (K) obtained by raising the order of approximation byone (from p to p + 1), BK and LK are identified as element contributions to the globalbilinear and linear forms Band L respectively from element [( 1 U h.p is the h-p FE solutionand the remaining interelement boundary integral includes the average fluxes correspondingto the first order derivatives.

The relatively low cost of implementing this technique should be noted. As the dimen-sion of the space of bubble functions (for a scalar problem) is 2p + 3, where p is the order ofapproximation, the time needed to solve the local problem is manageable. From the point ofview of coding, the procedure is also comparatively straightforward since the same routinesused to solve the global problem are used to construct the local problems as well.

Functions 'PK, the solutions to the local problem, are known as error indicator function,and form the basis for the calculation of the error estimate. Globally, the error bound takesthe form

IIIU h" - Uh,,+1I11 ~ [1t BK( .. K, A... K)]'

where Ao is the particular symmetrizer and the energy norm on the left-hand side corre-sponds to the symmetrized bilinear form, i.e.,

IIIUIW = B(U, AoU) = B(U, U) (4.15)

The ERM technique thus estimates the relative error between the given h-p solution U h,p

and another h-p solution U h,p+1 obtained by raising the order of approximation p in everyelement uniformly by one. Note that:

1. the difference between Uh,p+1 and the exact solution to a fractional step is of an orderless than the relative error, and

2. it makes a little sense to consider the exact solution to the fractional step problemneglecting the time discretization error.

The introduction of symmetrizers, especially in the Euler step, seems also to be a practicalanswer to the problem of defining a "natural" norm to measure the error.

The element contribution to the right-hand side of (4.13) is defined as the element errorindicator

(4.16)

•4.3 An Interpolation Error Estimate

(4.17)

1

'f]K = !:lplp,K = _ph(JK ~ (8%0:;%£)2) i'P K+L=p 1 2

As the element error indicators are used to drive the refinements, a much simpler errorindicator in the form of the classical L2-interpolation error estimate has been implementedas well:

In this equation, the density function p alone is assumed as a basis for error indication.In practice, (4.17) is much cheaper than the residual error estimate and may be used

at least to identify the elements with negligible error (for instance, in regions of uniformflow for the convective step. Obviously, it is of little use as an indicator in the viscous stepowing to boundary conditions on p near a no-flow boundary).


4.4 Examples of h-p Adaptive Strategies

Two different adaptive strategies are described here, both in the context of steady stateproblems. As the whole method and accompanying FE code can, in principle, handletransient problems, design of an efficient space-time adaptive strategy for transient problemsremains beyond the scope of this report.

The two algorithms differ in the way the p-refinements are used. In the first algorithm,an initial mesh geometrically gradated towards the solid wall boundary is implemented usingthe technique described in the previous section. Thus, both h- and p-refinements are usedto generate the initial mesh. In the mesh modifications following convergence to the steadystate solution on the current mesh, only h-refinements are used. The use of p-refinementsis therefore limited to only the initial mesh generation. This strategy is very much in thespirit of the h-p method presented in [12] based on the SUPG algorithm.

In the second algorithm, we attempt to design a fully h-p adaptive strategy. Startingwith an initial uniform mesh (both in element size and order of approximation), we convergeto the steady state solution and proceed with the error estimation. As the error estima-tion includes both Euler and viscous steps, for each element [( we can identify two errorindicators, fl~ for the Euler and 7J~ for the viscous steps. Recall that both error indicators(squared) can be interpreted as the element contributions to the total error measured inthe appropriate energy norm. As the sum of the two contributions (energies)

2 (E)2 (V)27Jr; = TJK + flK (4.18)

is used as a basis for refinements (every element with the corresponding error indicatorexceeding a certain threshold value gets refined), we make an ad hoc decision whether tochoose the h- o~p-refinements by comparing the two contributions to the final element errorindicator TJK with each other: .

IF 7J~ > flk THENuse h-refinement

ELSEuse p-refinement

ENDlFThis very arbitrary decision is motivated by standard observations concerning the typeof irregularities associated with both fractional steps; shocks are better handled by theh-refinements, p guarantees an exponential rate of convergence for elliptic problems. Ofcourse, a more rigorous mathematically based criterion would be preferable.

Formally, our adaptive algorithm is structured as follows:Step 1. Input data and generate an initial mesh with a corresponding solution UO.Step 2. Iterate towards the steady state solution on the current mesh

FOR n = 1, ... , maJCn• determine the next time step solution Un solving both Euler and

viscous steps according to the operator splitting procedure beingused

• IF IUn - un-II>' THEN• proceed with iterationsELSE• GO TO Step 3ENDlF

END FORSTOP and output a warning (no convergence to the steady state solutionhas been achieved)

194 aden and Bass

Step 3. Calculate the global error estimate err-estIF err-est < TOL THEN

• STOPELSE

• adapt the mesh• set UO to the current solution• GO TO Step 2

ENDIF

5 Review of an h-p Data Structure

Details of a general finite element data structure for adaptive h-p methods can be foundin [29]. For present purposes, it suffices to only outline its major features. The principalfeatures of the data structure are listed as follows:

1. Element shape functions are tensor products of integrated Legendre polynomials Xiof degree PK:

PK

uhPIK = L aii(t)Xi(xdxi(X2)i..1=O

2. Degrees of freedom are nodal values Ui, i = 1,2,3,4, tangential derivatives at midsides,and mixed partial derivatives at the centroids of an element.

3. Only I-irregular h-refinements are allowed, meaning that for two-dimensional prob-lems, only one node per element edge is permitted that is not a vertex.

4. The maximum rule is applied at element interfaces, i.e., if adjoining elements K, J EQH have a common side rKJ = oJ( noJ, and if PK > PJ, then the polynomial basisis enriched by the addition of polynomials up to degree PK to the element shapefunctions of element J along the common edge rK J. These augmented functions forJ consist of tensor products of polynomials of degree ~ PJ plus a set of polynomialsof degree PK on rKJ n J which vanish on 8J - rKJ·

5. Global continuity of the FE basis functions is enforced by the imposition of interele-ment continuity constraints. The transformation of unconstrained (globally dis-continuous) basis functions to continuous basis functions is defined by a collection ofBoolean maps which are also used to reduce element stiffness, mass, and load matricesto the appropriate forms. Thus the h-p finite element space is indeed a conformingspace of FE approximations.

6. Provisions for "unrefinement" and "unenrichment" throughout the mesh are provided.In other words, if error indicators reveal that the local error 11K is below a presettolerance, then a cluster of elements around J< can be collapsed into a larger elementor the local spectral order PK can be decreased.

More recently, an anisotropic p-method has also been implemented [23] which pro-vides for a special h-p treatment of boundary layers. In these strategies, low-order P andlarge h (then, flat elements) are used to approximate the variation in flow velocities tangentto walls (no-flow boundaries), while larger order polynomials are used to simulate the vari-ation in directions normal to the wall. The result is a thin boundary-layer approximationnot unlike PNS methods proposed in the CFD literature.


6 Moving Mesh Methods for Fluid-Structure Interactionand Adaptivity

Among the adaptive methods employed in the ADAPT0 code are those designed to ac-comodate motion of bodies through a flowneld or deforming or receding boundaries, orgeneral f1uid-structure interaction problems. These are handled by a special formulation ofthe Navier-Stokes equations in referential coordinates defined as follows:

z = spatial coordinates of particle z at time t

x(X,t), X = (X1,X2) = material coordinate

¢(y, t), Y = referential or grid coordinates

Thus,

vDzDt

a¢at

particle velocity

'd I' de!gn ve OClty, VD = va - v

vD = 0 ~ Lagrangian description of motions

va = 0 ~ Eulerian descriptions of motion

va :j:. 0 ~ Quasi-Eulerian descriptions of motion

A weak bilinear FE approximation of the Navier-Stokes equations in referential coor-dinates for a moving quadrilateral element ]{(t) over a time interval [ti, t2] is then givenby

(6.1)

_ rc' [ (U°,pj(v~nk)-FOk,pjnk)dsJtl JoK(t)

where ,pj,} = 1,2,3,4 are the shape functions for node} in element I<,a = 1,2,3,4 for.two-dimensional problems, and Fok are the fluxes (k = 1,2). This formulation can beapplied to a variety of moving mesh simulations by appropriately defining v~. Examplesinclude:

rotor-stator interactions: v~ is assigned the difference velocity between the rotor-stator blades [14].

fluid-structure interaction: v~ is prescribed to define a smooth mesh moving betweena fixed Eulerian grid and a deforming Lagrangian grid [33].

A sample calculation is provided in Section 8.

196 aden and Bass

7 Incompressible Navier-Stokes Calculations Using h-p Fi-nite Element Methods.

This section presents a general approach for solving the incompressible Navier-Stokes equa-tions using the general h-p finite element adaptive system outlined in Section 5. The basicapproach employed here is to use a penalty approximation for the incompressibility con-straint and the method of characteristics to handle the convective terms. Writing theNavier-Stokes equations in the form,

Dv 1Dt == Re ~v - V'p + f in n x (0, T)

div v == 0

where Dv / Dt is the material time derivative, we again use operator splitting. The followingtwo-step, second-order, implicit algorithm, which is a variant of that of Pironneau [31], maybe used to advance the solution in time:

Step 1

Step 2

HereVh == Vh(X, t) ==

for all Wh in Wh

the finite element approximation of the velocity field at point xin n at time t, n being the flow domain

vi: == vh(X,nAt). n=1,2,3 ...

Wh == an arbitrary test function

the work done by the viscous stress on the velocity gradientsV'wh


I( v, w) = a numerical quadrature approximating the scalar product,Na

(v, w) = 1v, wdx = LWIV«I)' W(~I),o 1=1

WI = quadrature weights,

el = Gauss or Gauss-Lobatto itegration points.

(f, Wh) = external force terms

t = the penalty parameter, t > O.

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

The corresponding pressure approximation is

at integration point el'In the method of characteristics, it is understood that

wherein vi:-1 and vi: are connected along the characteristic line, X(z,tn;tn_d, drawnbackward in time from point (z, nAt); i.e., X is the solution of

dX = V(X,T),dt

X(t) = z

8 Numerical Examples

8.1 Convection-Diffusion Error Estimation

To demonstrate the robustness of the error estimation procedure developed in Section 4 alinear convection-diffusion problem was solved:

au au au~ + a-;:;-- + a-a - t~u = f in n = (0 64)2v< VXl X2 '

with boundary, initial conditions, and the source term f corresponding to the exact solution

where A = 100, k = 0.03 and x~ = xg = 16. Several values of the parameters t and a wereused in the investigation in order to check whether the accuracy of the error estimate wasaffected by the Peclet number

198 aden and Bass

lalhp = 2 max laijl

ij1

where lal = (o:~+ o:n~ (so that P = o:h/2! here), where h is the mesh size of the com-putational cell. Large (resp. small) values of the Peclet number characterize convection-dominated (resp diffusion-dominated flows.

The a.pproximate solutions were calculated on an adaptive grid with three levels of h-refinement and minimum mesh-size h = 1. Four cases were in this study: (a = I, ! =0.01), (0: = 1, ! = 0.1), (a = 1, ! = 1), and (0: = 0.1, ! = 1) corresponding to thevalues P = 70.71, P = 7.071, P = 0.7071 and P = 0.0707, respectively. In each case, weintegrated the solution 136 time steps. Figure 8.1 show the evolution of the global errornorms and its estimates. It is clear from these results that the error estimate remains sharpfor all the values of the Peclet number tested.

A

B

co

Set C: a = 1,1), E = 1.Set D: a = (0.1,0.1), E = 1.

Figure 8.1: Forced convection-djffusion of a Gaussian-Hill.Time history of the global IlIlellll-norm of the error and its estimate for 4 different sets ofparameters.Set A: a = (1,1), E = 0.01.Set B: a = (1,1), E = 0.1.

8.2 Rotor-Stator Interaction

The ADAP10 code has several unique options for modeling multi-blade and/or multi-stage compressor or turbine geometries under subsonic or supersonic operating conditions.A rather complex example of subsonic inviscid flow through a compressor is shown in Fig.8.2. Here a uniform inflow of Mach = 0.0675 is initially specified along the left boundaryat an angle of zero degrees. No-flow conditions are enforced along the surface of each bladewith cyclic boundaries along the top and bottom. There are three grid velocities specifiedhere, one for each row of blades, with values of zero for the left and right blade rows andVG - 0.08654 for the center row of blades. The adaptive package was set up to use only h-refinements with the grid being dynamically adapted along one level each time the elementsalong the interface become aligned.

The plots in Fig. 8.2 show the pressure, velocity vectors, and adapted grid after 1100time steps. Note the continuity of pressure contour at the sliding interface and the intersec-tion of the pressure contour values with the inflow boundary. The continuity of the contoursis a direct result of the conservative algorithm used along the interface and the contours


intersecting the inflow boundary are a result of the extrapolation procedure for the subsonicinflow condition. Also note the velocity vectors in this figure are absolute velocity vectorsand thus the center row of blades shows a downward velocity component on the blade. Asrequired from the no-flow conditions on the other rows of blades, the velocity vectors hereare tangential to the boundary surface.

During the course of this calculation the maximum adaptive level was set to only one.With an initial number of elements in the grid of ....4400 (at level zero) this corresponds to....22,000 elements if every element is h-adapted. Monitoring the adaptive procedure duringthis calculation a maximum of .... 10,000 elements were used which represents a savings ofmore than 50 percent over a uniform global refinement.

I

l:Figure 8.2: Multiblade, multistage rotor/stator interaction. Adapted grid, pressure con-tours, and velocity vectors after 1100 time steps (Mach = 0.0675).

8.3 Erosion of Propellant Boundary

Since the space shuttle disaster, there has been an increased interest in accurately modelingthe flow conditions present in the solid rocket boosters. The question of central importanceis: what happens inside the motor casing if a flaw exists in the solid propellant, and willsuch a flow ultimately cause the booster to fail? This problem is truly a fluid-structureinteraction problem where the interface between the solid and fluid continually change asthe propellant is burned away.

A typical situation of an initial flaw in the solid propellant is shown in Fig. 8.3b. As weare concerned mostly with the effects of this flaw on the performance, let us for now onlyconsider a small region around the erosion pocket as shown in Fig. 8.3a. To simplify theanalysis further, let us also assume that the flow is inviscid and the interface between thefluid and solid erodes at a uniform rate independent of the temperature or pressure. With

200 aden and Bass

Figure 8.3: Solid rocket motor casing with an initial flaw and an enlarged view of thecomputational domain in the region of the flaw.

the absence of viscous effects we will also restrict an adaptive package to h-refinements orun refinements.

The farfield flow conditions prescribed are Mach = 0.65 at a = 0.0 degrees and the gridvelocity va = O.ln where n is the unit normal to the boundary. The adaptive package wasset to adapt the solution every 25 steps with a maximum level of adaptation of 2.

The results after 400 time steps are shown in Fig. 8.4. Figure B.4a shows the fullyunstructured adapted grid with 1800 degrees of freedom, most of which have been placedin the region of the shock. Figure 8.4b shows the corresponding density contours for thissame instant in time. As anticipated from the initial configuration of the flow, we see ashock wave being reflected upstream and recirculation has developed in the eroding pocket.

8.4 Flow Around a Cylinder

Using the two-step operator splitting method outlined in Section 7, the subsonic, incom-pressible flow around a cylinder was solved with uniform p-approximations of order p = 3,and underintegration of order 1. The computational domain selected was 16 cylinder di-ameters wide and 32 diameters long. A constant horizontal velocity was prescribed onthe inlet and both sides of the domain, and a zero velocity was prescribed on the cylinderboundary. For low Re (up to Re = 140), symmetrical solutions were obtained and forlarger Re, unsymmetrical solutions (vortex sheets) were obtained, without introducing anyartificial unsymmetry for larger time values. Figure 8.5 shows the horizontal velocity U:r

For Re = 1000, where unsymmetrical solutions developed near 30-40 sec. Figure 8.6 showsthe corresponding velocity vectors at time = 102 sec.

Note the adapted grid in Fig. 8.5 is symmetric even though the solution is nonsymmetric.This is a result of turning off the unrefinement capability in the adaptive package. Thus,


Figure 8.4: Subsonic flow past an eroding cavity. Dynamically adapted grid and densitycontours at 400 time steps. (Mach = 0.65, burning velocity = 0.01.)

once an element is h-refined it cannot be unrefined and since the solution is oscillating wesee a symmetric result. Also note the large amount of detail available in the velocity vectorplot. This is a result of using third-order elements in the wake region of the cylinder. Ifbilinear elements had been used instead, a much finer mesh would have been needed toprovide a similar level of detail.

8.5 Carter Flat Plate Problem

To demonstrate the enormous potential associated with h-p adaptive methods, the super-sonic flow over a flat plate was solved. This problem has been solved by several investigatorsstarting with Carter who obtained numerical solutions using the Brailovskaya scheme. Themain features present in this problem which are of interest include the shock emanatingfrom the front edge of the plate and the viscous boundary layer region. Of particular in-terest in the determination of highly accurate heating rates and the prediction of the skinfriction along the surface of the plate and at the exit boundary.

The computational domain for this problem is shown schematically in Fig. 8.7. Thesupersonic free stream condition (Mach = 3.0, 0 = 0.0, "I = 1.4, and Too = 3900 R) are

202 aden and Bass

(;-1 2)

Too l+~Moo

Figure 8.5: Horizontal velocity component Ux at 112 seconds for Reynolds number = lOuu,and and h-adapted third order grid.

specified for the Kind = 1 boundary, while outflow symmetry conditions are applied for allKind = 5 boundaries. The flat plate is represented by an isothermal no-slip boundary withthe temperature set equal to

This remaining boundary segment Kind = 4 was treated as a no-flow symmetry boundary.


Figure 8.6: Velocity vectors at 112 seconds for Reynolds number = 1000.

KlND=6 (Symmetry III01' KIND=l (Subsonic inflow/outflow)

--..VKIND=5 (S......-------! Ymmet r y IIl

KIND=l {Supersonic inflow) 0.76

// 1.0 _II

KIND=2 (Temperature prescribed)

or KIND=3 (Heat Clux prescribedl

KlND=4 (Symmetry I)

~

HFigure 8.7: Schematic of Carter flat plate problem showing domain dimensions and bound-ary types.

204 aden and Bass

MESH

t ! I I I

(a)

1.1

2

I I':: •

4

PR.E.SSURE

6 7 D.O.F= 3252

1.9

I.J ..J t r., ::

:: .,~ :t.

1.1

(b)o 0.1 0.6 Q.7

M1N=1.0286984MAX=I.9371926PROflLEaoEXrr

Figure 8.8: Carter flat plate problem. Mach = 3.0, Re = 104. a) hop -adapted grid; b)

pressure distribution at the e..xitboundary.


DeNSITY

1-'

0.7

1.1

1..

1-'

o

0.1

0.1

OJ

. OJ

n:MPI!RATURE

0.6

0.6

0.7

0.7

MlN-o.s I 02617MAX·I.saSa~2PROFII...E-EXIT

MlN=1.01797S9MAX.2.1PROFILE-EXIT

Figure 8.9: Carter flat plate problem. Mach = 3.0, Re = 104. a) Density profile at the exitplane; b) temperature distribution at the exit plane.

206 aden and Bass

Pl£SStIlUl

M!H·I.0216ncMAJ(.4.14S1$39PROFlLSaSOOY

H.EAT FLUX COEFFICIENT

o.m

0.13

0.011

.0011 I ~l1/'l·"().02U9J, I , I , I I MAJ(oOJU1U10..2 OJ 0.1 I PROFl1.&BOOY

SKIN FRICTION COEFFICll!NT

I1..1

I.JU

011

OJ"

~~M1Ho00272IUMAX.I..,S3W

0..2 .u 0.1 I PllOFlUl-BOOY

Figure 8.10: Carter flat plate problem. Mach = 3.0, Re = 1Q4. a) Pressure distributionalong the plate; b) heat flux coefficient along the plate; c) skin friction along the plate.


The Reynolds number, Re = 104, was based on the length of the plate the the Prandtlnumber for air was taken as 0.72.

Solutions for the Carter problem were obtained using three successively finer mesheswith a maximum order of p = 4 in the boundary layer region. (Note in this examplewe have also used the anisotropic p elements which are linear in the streamwise directionand higher order p normal to the plate to capture the boundary layer effects and reducethe computational effort.) Figures 8.8 and 8.9 shows the final adapted hop mesh with lessthan 3300 degrees of freedom and the density, temperature, and pressure profile at the exitboundary for each of the three successively finer meshes. Also included in these profileplots are the results obtained by Carter on a grid with approximately 14,000 grid points.Comparing the results obtained here with those of Carter, we see a very good agreementbetween the density and temperature distributions while our pressure values appear muchmore stable, particularly in the boundary layer region. Ref,ults for the pressure, skin frictioncoefficient, and heat flux coefficient along the plate are also shown in Fig. 8.10. In each casethe results obtained with the h-p methodology are comparable to other existing solutionswith a significantly reduced number of degrees of freedom and without the oscillation in theboundary layer region.

A cknow ledgements

Our work on error estimation and h-p adaptive procedures was initiated under a contractwith the U.S. Office of Naval Research. Extensions to aerodynamic heating in supersonicflow simulations was done through support of the Aerothermal Loads branch of NASALangley Research Center. The new operator-splitting techniques, error estimation methods,:\nd anisotropic p-refinement methods were developed during a study supported by NASAAmes Research Center.

References

1. BABUSKA, 1., ZfENKIEWICZ, O. C., GAGO, J., and OLIVEIRA, E. R. A. (Eds.),Accuracy Estimates and Adaptive Refinements in Finite Element Compu-tations, John Wiley and Sons, Ltd., Chichester, 1986.

2. ODEN, J. T., and DEMKOWICZ, L., "Advances in Adaptive Improvements: A Sur-vey of Adaptive Methods in Computational Fluid Mechanics," State of the ArtSurveys in Computational Mechanics, Edited by A. K. Noor and J. T. Oden,American Society of Mechanical Engineers, N.Y., 1988.

3. ODEN, J. T., "Adaptive Finite Element Methods for Problems in Solid and FluidMechanics," Finite Element Theory and Application Overview, Edited by R.Voight, Springer-Verlag, NY., 1988.

4. ODEN, J. T., "Adaptive FEM in Complex Flow Problems," The Mathematics ofFinite Elements with Applications, Edited by J. R. Whiteman, London AcademicPress, Lt., Vol. 6, pp. 1-29, 1988.

5. ODEN, J. T., "Smart Algorithms and Adaptive Methods in Computational FluidDynamics," Proceedings CANCAM (Canadian Congress on Applied Me-chanics), Ottowa, Canada, 1989.

208 aden and Bass

6. ODEN, J. T., "Progress in Adaptive Methods in Computational Fluid Dynamics,"Adaptive Methods for Partial Differential Equations, Ed. by J. Flaherty, etaI., SIAM Publications, Philadelphia, 1989.

7. ODEN, J. T., DEMKOWICZ, L., STROUBOULIS, T., and DEVLOO, P., "Adap-tive Methods for Problems in Solid and Fluid Mechanics," Accuracy Estimatesand Adaptive Refinements in Finite Element Computations, Edited by I.Babuskall O. C. Zienkiewicz, J. Gago, and E. R. A. de Oliveira, John Wiley and Sons,Ltd., London, 1986.

8. ODEN, J. T., and BASS, J. M., "Adaptive Finite Element Methods for a Class ofEvolution Problems in Viscoplasticity," International Journal of Engineering Science,Vol. 25, No.6, pp. 623-653, 1987.

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

10. ODEN, J. T., DEMKOWICZ, L., and STROUBOULIS, T., "Adaptive Finite ElementMethods for Flow Problems with Moving Boundaries. 1: Variational Principles andA Posteriori Estimates," Computer Methods in Applied Mechanics and Engineering,Vol. 46, pp. 217-251, 1984.

11. ODEN, J. T., STROUBOULlS, T., and DEVLOO, P., "Adaptive Finite ElementMethods for the Analysis of lnviscid Compressible Flow: I. Fast RefinementjUnre-finement and Moving Mesh Methods for Unstructured Meshes," Computer Methodsin Appl. Mech. and Engrg., Vo\. 59, No.3, 1986.

12. ODEN, J. T., DEVLOO, P., and STROUBOULIS, T., "Implementation of an Adap-tive Refinement Technique for the SUPG Algorithm," Computational Methods inAppl. Mech. and Engrg., Vol. 61, pp. 339-358, 1987.

13. ODEN, J. T., STROUBOULlS, T., and DEVLOO, P., "Adaptive Finite ElementMethods for Compressible Flow Problems," Numerical Methods for Compress-ible Flows-Finite Difference, Element and Volume Techniques, Edited byT. E. Tezduyar and T. J. R. Hughes, AMD-Vo\. 78, ASME, New York, pp. 115-126,1987.

14. ODEN, J. T., STROUBOULIS, T., and DEVLOO, P., "Adaptive Finite ElementMethods for High-Speed Compressible Flows," International Journal for NumericalMethods in Fluids, Vo\. 7, pp. 1211-1228, 1987.

15. TEZDUYAR, T. E., and LIOU, J., "Element-by-Element and Implicit-Explicit FiniteElement Formulations in Computational Fluid Dynamics," Domain DecompositionMethods for Partial Differential Equations, R. Glowinski, et a\. (eds.), SIAM,pp. 281-300, 1988.


16. TEZDUYAR, T. E., and LIOU, J., "Adaptive Implicit-Explicit Finite Element Al-gorithms for Fluid Mechanics Problems," to appear in Computer Methods in AppliedMechanics and Engineering; also in Recent Developments in ComputationalFluid Dynamics, T. E. Tezduyar and T. J. R. Hughes (eds.), AMD-Vol. 95, pp.163-184, ASME, New York, 1988.

17. TEZDUYAR, T. E., LIOU, J., NGUYEN, T., and POOLE, S., "Adaptive Implicit-Explicit Parallel Element-by-Element Factorization Schemes," Chapter 34 in DomainDecomposition Methods, T. F. Chan, et al. (eds.), SIAM, pp. 443-463, 1989.

18. TEZDUYAR, T. E., and LIOU, J., "Adaptive Implicit-Explicit Methods for FlowProblems," Finite Element Analysis in Fluids, T. J. Chung and G. R. Karr(eds.), University of Alabama-Huntsville Press, Huntsville, Alabama, 1989.

19. LERAT, A., "Implicit Methods of Second-Order Accuracy for the Euler Equations,"AIAA Journal, 23, I, pp. 33-40, 1985.

20. "Analysis of Flow-, Thermal-, and Structural-Interaction of Hypersonic StructuresSubjected to Severe Aerodynamic Heating," Annual Technical Report, Bolling AirForce Base, Cont'ract No. F49620-C-88-001.

21. HASSAN, 0., MORGAN, K., and PERAIRE, J., "An Adaptive Implicit/ExplicitFinite Element Scheme for Compressible Viscous High Speed Flows," AIAA 27thAerospace Sciences Meeting, Reno, Nevada, January 9-12, 1989.

22. STRANG, G., "On the Construction and Comparison of Difference Schemes," SIAMJ. Numer. Anal., Vol. 5, No.3, pp. 506-517, September, 1968.

23. DEMKOWICZ, L., ODEN, J. T., and RACHOWICZ, W., "A New Finite ElementMethod for Solving Compressible Navier-Stokes Equations Based on an OperatorSplitting Method and h-p Adaptivity," Computer Methods in Applied Mechanics andEngineering (to appear).

24. STRIKWERDA, J. C., "Initial Boundary Value Problems for Incompletely ParabolicSystems," Ph.D. Thesis, Dept. of Math., Stanford University, Stanford, CA, 1976.

25. GUSTAFSSON, B., and SUDSTOM, A., "Incompletely Parabolic Problems in FluidDynamics," SIAM J. Appl. Math., Vo\. 35, No.2, pp. 343-357, September, 1978.

26. BANK, R. E., and WEISER, A., "Some A Posteriori Error Estimates for EllipticPartial Differential Equations," Mathematics of Computation, Vo\. 44, No. 170, pp.283-301, 1985.

27. BANK, R. E., "Analysis of Local A Posteriori Error Estimates for Elliptic Problems,"Accuracy Estimates and Adaptive Refinements in Finite Element Compu-tations, Edited by 1. Babuska, et aI., John Wiley and Sons, Ltd., Chichester, pp.119-128, 1986.

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28. ODEN, J. T., DEMKOWICZ, L., WESTERMANN, T., and RACHOWICZ, W., "To-ward a Universal h-p Adaptive Finite Element Strategy, Part 2. A Posteriori ErrorEstimates," Computer Methods in Applied Mechanics and Engineering.

29. DEMKOWICZ, L., ODEN, J. T., RACHOWICZ, W., and HARDY, 0., "Toward aUniversal h-p Adaptive Finite Element, Part 1. Constrained Approximation and DataStructures," Computer Methods in Applied Mechanics and Engineering.

30. RACHOWICZ, W., ODEN, J. T., and DEMKOWICZ, L., "Toward a Universal h-p Adaptive Finite Element Strategy, Part 3. A study of the Design of h-p Meshes,"Computer Methods in Applied Mechanics and Engineering (to appear).

31. ODEN, J. T., STROUBOULIS, T., and BASS, J. M., "Paradigmatic Error Calcula-tions for Adaptive Finite Element Approximations of Convection Dominated Flows,"Recent Advances in Computational Fluid Dynamics, ASME Publication.

32. STROUBOULIS, T., and ODEN, J. T., "A Posteriori Error Estimates for Problemsin Computational Fluid Dynamics," Computer Methods in Applied Mechanics andEngineering. (to appear).

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

34. PIRONNEAU, 0., "On the Transport-Diffusion Algorithm and its Application to theNavier-Stokes Equations," Numer. Mathematik, 38, pp. 309-332, 1982.

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