ADVANCES IN ADAPTIVE METHODS INCOMPUTATIONAL FLUID MECHANICS

J. Tinsley Oden

Texas Institute for Computational MechanicsThe University of Texas at AustinAustin, Texas 78712

Abstract

Recent developments in adaptive methods in computational fluid dynam-ics (CFD) give hope that one may develop "optimal" schemes for analyzingcomplex flow; i.e., schemes which deliver the best possible accuracy for afixed computational effort. This note discusses some of the basic ideas be-hind adaptive methods and cites examples of recent results obtained usingadaptive schemes for compressible flow problems.

ADAPTIVE FEM'S

Suppose that one could estimate reliably the accuracy of a numerical so-lution, e.g., suppose that it were possible to calculate a collection of numbersBe, each of which was an indication of the actual numerical error in some ap-propriate norm for cell number e in a finite difference or a finite elementmesh over a given flow domain. Then, knowing the computational error (or,at least, knowing a good indication of it), one could legitimately ask thequestion: how can the structure of the approximation be changed in orderto reduce the error below a preassigned limit? Such numbers are called localerror indicators.

This "preassigned limit" may be determined by many factors, such asthe computer budget available to the analyst, the number of man-hoursthat can be devoted to the task, the precision of the results required in agiven calculation, or the capacity of the computer being used. Once thistolerance is assigned, there must follow an adaptive process by which thestructure of the approximation is systemically adapted to reduce error, i.e.,to improve the local quality of the solution.

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There are several families of adaptive strategies that can be used to mod-ify the structure of the approximation:

a) Mesh Refinement Schemes (h-Methods). In these schemes, the meshis automatically refined when the local error indicator exceeds a preassignedtolerance. Such h-schemes present a very difficult data management prob-lem, since they involve a dynamic regeneration of the mesh, renumberingof grid points, cells or elements, and element connectivities as the mesh isrefined. However, the h-methods can be very effective in producing near-optimal meshes for given error tolerances. The author and his colleagues havedeveloped a very fast code that enables the analyst to use h-methods veryefficiently for complex flow geometries. Furthermore, the h-method strategycan also be used to coarsen a mesh (use larger mesh cells and thereby re-duce the number of unknowns) when the local error becomes lower than anassigned lower-bound tolerance.

A sample calculation obtained with our h-method is shown in Fig. 1.Shown here is a calculation of supersonic flow in a rotor-stator flow inter-action problem in which rotor blades are moving relative to the stator ina two-dimensional flow field. The procedure dynamically refines the mesh,assigning large elements where the error is small, small elements where theerror tends to be large, and simultaneously models shocks, flow through meshinterfaces and shock interaction. Computed density contours are also givenin the figure.

b) Moving Mesh Schemes. Moving mesh schemes employ a fixed numberof grid points and attempt to dynamically move the grid points to areas ofhigh error in the mesh. Moving mesh schemes can be rather easy to imple-ment, and, therefore, do not share the difficult data management problemsof h-methods. However, they suffer from several deficiencies. Without carein their implementation, moving mesh schemes can be unstable and can re-sult in mesh tangling and local degradation of the solution. They can neverreduce errors below an arbitrary limit, and these methods often fail whentime-dependent boundary conditions are enforced, as they are incapable ofhandling the migration of regions containing irregularities or singularities inthe solution as it evolves in time. Nevertheless, when combined with othertypes of adaptive strategies, these methods can provide a useful approachtoward controlling solution error.

c) Subspace Enrichment Methods (p- or spectral Method). The subspaceenrichment methods (or spectral-type methods) generally employ a fixedmesh and a fixed number of grid cells and points. Most numerical meth-ods for partial differential equations attempt to approximate the solution ina subclass of discrete functions or by functions in some finite-dimensionalsubspace of functions in which the actual solution belongs. Thus, subspaceenrichment methods attempt to enrich this subclass of functions throughthe use of higher-order differences, spectral methods, by increasing the localpolynomial degree in finite methods, etc. If the error in any cell exceeds apreassigned tolerance, the local order of the approximation is increased to re-

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duce the error. These methods are very effective in modeling thin boundary-layers around bodies moving in a flowfield, where use of very fine meshes iscostly and impractical. The problem of developing the data managementscheme required to implement these types of adaptive methods, particularlyin cases of complex geometries, is exceedingly difficult and while we haveworked on this subject for four years, we have only treated it successfully inrecent months.

d) Combined Adaptive Methods. The best adaptive schemes for inter-nal flow studies applications involve some combination of the h-methods,moving mesh methods, and subspace--enrichment methods discussed above.In recent months, a new data management scheme has been developed forimplementing new spectral methods and combined h-spectral adaptivity onunstructured meshes, and has made some preliminary applications to theNavier-Stokes equations in two dimensions. These techniques are capableof delivering incredible accuracy: exponential convergence of solutions, bya carefully applied recipe of simultaneously refining the mesh and changingthe spectral-order. Figures 2-4 show results obtained using our h-p methodfor the Carter plate problem: compressible viscous flow over a heated flatplate. Computed density contours are shown. The final mesh consisted of1,102 bilinear elements, 42 biquadratic elements, and 161 bicubic elementsand 2,831 degrees of freedom. Standard difference methods for this problemmay require an order-of-magnitude more degrees of freedom for comparableresults. For additional details, see [2].

Figure 5 shows a representative log-log plot of error versus problem sizeand emphasizes the fact that h-p techniques provide the best available wayto get the most out of one's computational effort. Even more significant isthe observation that these special adaptive techniques can produce numericalsolutions to problems which are impossible to obtain by conventional finitedifference or finite element techniques on the largest existing supercomputers!Indeed, to reproduce the accuracy obtainable by h-p methods on some modelelliptic problems, a finite difference mesh consisting of over ten million gridpoints would be required.

Calculation of Error Indicators

There are several methods for calculating estimates of the solution errorin discrete approximations of boundary- or initial-value problems. In devel-oping error indicators it is always desirable if not theoretically necessary toensure that the error indicator be bounded above and below by the actualerror globally in some appropriate norm, i.e., one attempts to construct anumber 8, called the global error indicator, which has the properties

(.1)

3

(.2)

with

where CI and C2 are constants independent of the mesh size parameter h,e is the actual approximation error, and II . II is some norm appropriate forthe problem at hand. In (2), e is determined by a collection of local errorindicators calculated over each cell K in the mesh, and generally 1 :::;p :::;00.Condition (2) ensures that rate of convergence of the global error indicatoris precisely the same as that of the actual error and that by designing analgorithm that systematically reduces e we also reduce Ilell·

Except for simple one-dimensional cases, it is generally possible to deriveerror indicators that satisfy (1) only asymptotically, for sufficiently small hor large p. For example, for the model problem,

-V· aVu + bu. = f in nu = Uo on anIau

aan = 9 on an2

(n C ll~an = anI U an2), one can prove under standard hypotheses thatconstants CI and C2, independent of h and p, exist such that

{ }

1/2

e = ~ IIIlflKlilk

and III . III the energy norm,

IIIulW = LBK(u, u)K

L: { (alVul2 + bu.2)dxK JK

I:IlIulllk = { (aVu . Vu + bu2)dxK In

and CPK is a solution of the local problem (posed over each element K):

(.3)

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Here Bh is a "space of u bubble functions'~containing higher-order polyno-mials which vanish at the nodes of each element, Uh is the finite elementapproximations of U on a mesh with mesh size h and polynomial degree p,and rh is the re~idual

rh = f + V' . aV'uh - bUh

Thus, every factor on the right-hand-side of (3) is known and (3) representsa well-posed problem for the local error indicators «JK. We have developedand implemented a similar error~timation procedure for non-self-adjointproblems.

Ordinarily, one seeks a more easily calculated error indicator than (3) todrive the adaptive process and reserves a scheme such as (3) for the end ofa computation to obtain a more precise estimate of the error. For instance,one can show that [2]

where hK = dia (K), 11'llo,K denotes the L2- norm over K, ~aaUh/anK] isthe jump in the computed flux over the boundary and C is a constant. Thequantity <PK = c-I PK, where PK is the quantity on the right side of the aboveinequality, represents an easily computed error indicator, which is generallysufficiently accurate to correctly direct the adaptive process through solutionsof increasing quality. Space limitations do not permit a discussion of suchschemes here, but more details can be found in companion papers [1].

Acknowledgement The support of this work by the Office of Naval Re-search under Contract N00014-84-K-0409-MOO-P00005is gratefully acknowl-edged.

References

1. Devloo, P., Oden, J. T., and Pattani, P., "An h-p Adaptive FiniteElement Method for the Numerical Simulation of Compressible Flow,"Computer MethotU in Applied MechanicJ and Engineering (to appear).

2. Oden, J. T., "Notes on Aposteriori Error Estimates for Finite ElementApproximations of Boundary- and Initial-Value Problems," TICOMR.ept., No. 88-03, Austin, 1988.

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Figure 1. Rotor-stator flow interaction: Here one sees a computer-generated mesharound two rotor-blades (on the right) moving with respect to a stator blade(on the left) in a simulation of rotor-stator flow interaction in a turbineengine. The mesh is dynamically rermed. using at a given time only thenumber of cells needed to deliver a specified level of accuracy. Cells areremoved dynamically if they are not needed. Figure (a) also contains aninstantaneous plot of pressure contours. Figure (b) shows the solution at alater time; note an entirely new optimal mesh prevails. since the solution haschanged. In these calculations. approximately one-third the number ofunknowns for a conventional uniform mesh solution are used to obtainequivalent accuracies.

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Figu~ 1. (b)

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Supersonic Inflow (Dirichlet)Supersonic Inflow (Dirichlet)Mach.3Re.1000

Pr.O.72Y. 1.4

y

outflow 0.75

0.1 L-1.0

Figure 2. Data and geometry for the Carter plate problem.

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Figure 3. An h-p mesh for the Carter problem consisting of linear.quadratic and cubic elements in the boundary layer.

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H-f-

f-

-II

II

-f- -f-

f-

...

II

Figure 4. Computed pressure contours.

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LOG11Error11

h-method (uniform refinement)

h-method (adaptive)

an adaptiveh-p method

p-method(uniform refinement)

LOG(No. of Unknowns)

Figure 5. Plots of convergence rates. It is clear that for very high reesolution ofcomplex flow features, traditional finite difference and finite element methodsare grossly inadequate; the use of combined h-spectral methods seems to bethe most promising approach for fitting very large problems on today'smainframes, particularly if high accuracy is nequired.

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