Proceedings 21 Midwestern Mechanics ConferenceMichigan Technological University, 1989

NEW DIRECTIONS IN COMPUTATIONALFLUID DYNAMICS: SMART

ALGORITHMS AND ADAPTIVITY

J. Tinsley Oden

Texas Institute for Computational MechanicsThe University of Texas at Austin

1 Introduction

New developments in adaptive finite element methods, unstructured grid solvers, aposteriorierror estimation, and parallel computing promise to have a signficant impact on compu-tational fluid dynamics. These emerging technologies form the basis of developments inso-called "smart algorithms" which evolve in structure and form as the numerical solutionto a flow problem is being generated by the computer. The overall objective is to optimizethe calculation: to give the best possible results for a fixed computational effort. As a by-product, these methods also yield estimates of computational error and, thereby, provide animportant indication of the reliability of the computation. This lecture summarizes recentdevelopments in this area.

2 The Philosophy of Snl.art AIgorithlTIs and AdaptiveMethods

We begin with an outrageous idea: PUS:l aside the massive volume of work accumulated on. computational fluid dynamics over the last quarter-century and focus anew on the key issuesof numerical simulations:

• How good are the answers? What is the accuracy in a numerical simulation of theEuler equations of gas dynamics or of the Navier Stokes equations? How reliable arethe computed results?

• How good are the meshes? How does one know that a finite difference or finite elementmesh will produce a reasonable simulation of the events of interest?

• If the quality of results is known, how can they be improved? What can be done toenhance the accuracy and stability of a computation?

• How can the computational problem be optimized? In particular, how can the bestpossible results be obtained for the least computational effort?

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• Observing that these questions pertain to very localized conditions in a computationaldomain and, if answerable at all, have different answers at different places in themesh at different times, how can one cope with the localized character of the majorcomputational issues?

The field of adaptive methods and smart algorithms is developing in an attempt to resolvethese questions. The basic components of these approaches are as' follows:

1. Unstructured Meshes. Be prepared to introduce grid points, grid cells, finite el-ements, nodes, etc. or to increase the order of approximation wherever necessary toachieve acceptable accuracy at low cost. This signals the use of unstructured meshes,and with one stroke makes obsolete the majority of (fixed-grid) CFD algorithms in usetoday.

2. Aposteriori Estimates. Somehow use the computed flowfield to assess the qualityof the solution. This means construct efficient and reliabile estimates of local error asthe solution is evolving (in iterative steps or in time).

3. Adaptive Strategies. Use the error estimates (and, if appropriate, estimates ofnumerical stability based on, for instance, time-step control) to judge the quality ofthe solution; and, if the error is perceived to be too large, change the local structure ofthe approximators to reduce it. This means refine the mesh, move grid points, increasethe local spectral order of the approximation, change time stpes, or do combinationsof these strategies to bring the local error under control.

4. Smart Algorithms. Develop algorithms which themselves adapt to changing con-ditions in the solution. For instance, consider algorithms which are based on explicittime-integration schemes in some portion of the mesh where possible and which useimplicit methods at other places to allow the solution to advance in time for a fixedtime step. Other variants of smart algorithms control artificial viscosity, balance fluxes,clip off spurious oscillations, and increase automatically the local s:oectral order of theapproximation adaptively as the calculation of the solution progre5se~ in time.

Of course, the successful implementation of these ideas is a difficult and challengingproposition for research in CFD. One must face severl issues that to date are far from resolved.For example: the exceedingly complex data structures needed to manage an adaptive schememust be developed and streamlined so that the method does not collapse under the weightof its own overhead. Then there is the issue of robustness - can flow solvers be designedwhich function efficiently and remain stable when implemented on a dynamically-changinggrid. These and a host of other issues make the subject a fertile ground for new researchtopics in computational mechanics.

3 SOlue Adaptive Strategies

Many adaptive strategies now being tested are based ill one way or another on a few basicadaptive approaches.

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• h-meihods (Mesh Refinement Schemes). In these schemes, the mesh is automaticallyrefined when a local indicator Oe of the error in grid cell e exceeds a preassigned tol-erance. Such h-schemes present a very difficult data management problem since theyinvolve a dynamic regneration of the mesh, renumbering of grid points, cells or ele-ments, and element connectivities as the mesh is refined. However, the h-methods canbe very effective in producing near-optimal meshes for given error tolerances. More-over, we have developed a very fast data mangement scheme that enables the analystto use h-methods very efficiently for complex flow geometries. Furthermore, the h-method strategy can also be used to unrefine a mesh (produce large mesh cells andthereby reduce the number of unknowns) when the local error becomes lower than anassigned lower-bound tolerance. .

• r-methods (Node Relocation or Moving Mesh Methods). These techniques employ afixed number of grid cells (elements) and grid points (nodes) and attempt to reducelocal errors by increasing the grid point density at appropriate points in the mesh. Theadvantage of these methods is their relatively simple data structure. Their disadvantageis that they involve an intrinsic error which cannot be reduced below a fixed threshold.

• p-methods (Subspace Enrichment Methods). Most numerical methods for partial differ-ential equations attempt to approximate the solution in a subclass of discrete functionsor by functions in some finite-dimensional subspace of functions in which the actual so-lution belongs. Thus, subspace enrichment methods attempt to enrich this subclass offunctions through the use of higher-order differences, spectral methods, by increasingthe local polynomial degree in finite element methods, etc. The subspace enrichmentmethods generally employ a fixed mesh and a fixed number of grid cells and points. Ifthe error in any cell exceeds a preassigned tolerance, the local order of the approxima-tion is increased to reduce the error. These methods could prove to be very effective inmodeling thin boundary layers around bodies moving in a flowfield, where use of veryfine meshes is costly and impractical. On the other hand, the problem of developingthe data management scheme required to implement these types of adaptive methods,particularly in cases of complex geometries, can be exceedingly difficult.

• Combined Adaptive Methods. It is likely (indeed, prova1:.le in some cases) that the bestadaptive schemes for applications will involve some combination of the h-methods,p-methods, or r-methods discussed above.

The question of which is the most effective adaptive scheme is roughly addressed in thegraphs of Fig. 1. There one sees a rendering intended to represent typical computationalresults obtained in representative steady flow simulations produced by robust, convergentalgorithms: the global error in some appropriate norm is plotted versus the total numberof unknowns N on a log-log scale. As is well known, standard fixed-grid finite differencemethods (and finite-volume and finite element methods) will exhibit an (asymptotic) rate-of-convergence indicated by the upper straight line in the figure. Numerous numerical exper-iments show that indeed this theoretical result holds in practice and that, even for relativelycoarse meshes, there is essentially a linear relationship between log II e II and log N, theslope of which is the rate-of-covergence of the method. The figure also demonstrates thatspectral and pseudo-spectral methods can deliver somewhat higher rates-of-covergence when

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they are applicable to the problem at hand. Unfortunately, such methods are best suited forgeometrically-simple flow domains and simple (e.g., periodic) boundary conditions. For prob-lems with singularities (or with singular-like behavior on coarse meshes) somewhat improvedrates of convergence can be obtained using optimally-graded meshes or unstructured mesheswith automatic mesh refinement. But even then the convergence is algebraic, representedby a linear decay of error on a log-log plot.

The significance of this observation is this: the analyst employing fixed-grid, fixed-ordermethods can always demand a level of accuracy which can be obtained only by using a num-ber of unknowns far in excess of the storage capabilities of the computing device on hand, nomatter how large the device is. This fact raises serious questions about the prevailing philos-ophy in large-scale calculations: use low-order fixed-grid difference schemes and increase thedensity of grid points when high resolution of primitive variables is sought. Such a grosslyinefficient approach has led some to employ FD-models with 200,000-500,000 grid points(N = 106 to 2.5 X 106) for a calculation. It is clear that to produce accurate computer simu-lations one must seek numerical schemes that exhibit non - algebraic rates of convergence,i.e., one must obtain higher accuracies per unknown.

For certain classes of problems, it can be proved that by simultaneously refining themesh and increasing the local spectral order of the approximation, it is possible to obtainexponential rates-of-convergence, even on meshes designed to fit very complex flow domains.The convergence characteristics of these h - p methods are illustrated by the curved linesin Fig. 1. The significance of this latter observation is that it suggests that for any given(inevitably limited)· computational resource, it may be possible to identify levels of accuracyunattainable by traditional finite difference methods but that can be attained by specialh - p adaptive schemes.

4 Sanl.ple Results

We present here some representative numerical results obtained using various adaptive strate-gies in the soluti'Jn of two- and three-dimensional compressible flow and transport problems.

1. h-Adaptive Scheme for 3D-Scalar Convection Problem. A calculaton of the rotation ofa species concentration in a three-dimensional domain is shown in Fig. 2a-2x. Thereone sees a concentration cp varying as a cosine held over a plane passing through athree-dimensional domain, and rotated several time steps. Even though the initialgrid is coarse, the mesh is automatically refined in regions of high gradient in order tomaintain the integrity of the solution through the time interval of interest. The figuresshow the dynamically adapted mesh at various times during the simulation.

2. 3D-Shock Tube. Figure 3 shows an adaptive calculation of the propagation of a planeshock wave down a shock tubes with a rectangular cross section. The h-method solutionrefines the grid ahead of the shock as it propagates.

3. h-Adaptive Schemes for Unsteady Compressible Navier-Stokes Codes for Rotor-StatorInteraction. Typical results of a rotor-stator calculation are shown in Figs. 4-7. Thereone sees a dynamically changing mesh generated after each of a specified number of

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- ..

time steps in such a way as to reduce computed errors below a preset error tolerance.Note the continuity of density (and pressure) contours across mesh interfaces and theinteraction of shocks on the moving turbine blades. The code also computes the timehistory of stresses in the blade due to fluid pressure and shear. It is also interestingto note that flows during multiple cycles of the blade rows have been computed froma start-up uniform flow to flow with periodic structure which exists after the turbineis in operation. Such calculations simulate as many as 21 complete blade revolutionsand 100,000 time steps. Note also that the adaptive process, which consumes only1 percent of the total calculation, uses around 3,500 elements to deliver the desiredaccuracy at any particular time during the computation, while a uniform mesh neededfor the same accuracy consists of around 14,000 elements.

4. Low-iV/ach Number Flow Around a Cylinder. Figures 8 and 9 show the versatility ofthe 2D h-adaptive strategy for subsonic flow around a cylinder. Note the dynamicallychanging mesh and the resolution of vortices spinning off the cylinder at !l1 = 0.65.Fully implicit schemes which function on unstructured meshes are being developed forthese problems, but the results shown were obtained with- an explicit flow solver, theeffectiveness of which was made possible by the use of a near-optimal mesh at the endof each of a designated collection of time steps.

5. Fluid-Structure Interaction: An h-p Adaptive Scheme for Viscous Compressible FlowOver a Flexible Plate. Figures 10 and 11 show flow data and an h - p mesh and densi tycontours over a flexible elastic plate deformed by the action of a viscous compressiblefluid. Low-order elements are used to capture the shock while higher-order elementsare used to model the viscous boundary layer. The flow is quasi-steady and a fullyimplicit solver with a multigrid iterative acceleration is used to compute successivemeshes as the plate deforms.

6. Chemically Reacting Viscous Compressible Flow. Two- and three-dimensional adaptivecodes for the analysis of chemically reacting flows are under development. Figures 12and 13 show computed distributions of chemical species on an h-adpated mesh. Theflow is viscous, transient, and compressible.

7. An h-r Adaptive Calculation of Shocks Structure on a Blunt Body. A moving nodetechnique (an r-method) is used to pre-condition mesh structures prior to an h-adaptivecalculation. Figures 14 and 15 show a typical calculation. There we observe an h-radaptive mesh and density contours of an inviscid gas impinging on a blunt body. Ourresults indicate that r-method preprocessing can be beneficial in aligning the initialmesh with shocks in steady supersonic flow problems with the result that a given levelof h-refinements produces better solutions than a pure h-process which is initiated onan unaligned mesh.

8. A New h-p Scheme for Optimal Computations. Both two- and three-dimensional h-p adaptive codes are operational for the analysis of general linear boundary-valueproblems. Extensions to steady-state Euler equations are under study. These codesemploy an optimization algorithm which chooses the optimal distribution of h (meshsize) and p (polynomial degree/spectral order) to produce a solution with a givenlevel of local accuracy with a minimum number of unknowns. Potential extensions of

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these techniques to the Navier-Stokes equations are in early stages of development andemploy data structures quite different from the h-p schemes used in example 5.

Figure 16 contains a typical computer-generated h - p mesh for a model problem of \potential flow over a cubic domain. Some elements are removed to expose differentshades corresponding to different polynomial degrees. The scheme generated the op-timal h - p mesh shown, with polynomials of up to seventh degree in areas of rapidchange of the solutions, and with larger elements and lower polynomial degrees in areasof the domain wherein the solution is roughly piecewise constant.

Acknowledgments

The work reported here is joint work done with the help of several colleagues and studentsover the last several years under support from several sources. The early work on h - Pschemes for fluid-structure interaction was completed under a project supported by ONRon algorithms for accurate flux and stress prediction and fluid-structure interaction. Theadaptive techniques for rotor-stator interaction are under support of NASA Lewis ResearchCenter. The model problem illustrated in Figures 4-7 was based on the Supersonic Flow-Through Fan developed at NASA Lewis Research Center. The preliminary work on h - Pmethods and subsonic flows was completed in a feasibilitiy study for NASA Marshall SpaceFlight Center. Our studies of chemically reacting flows are supported by the Air ForceWeapons Lab, Kirtland AFE. Further work on error estimation and h - p adaptive schemesis the subject of ongoing work which forms a part of several projects including a project topredict aerothermalloads under contract with NASA Langley Research Center.

The help of J.M. Bass, T. Strouboulis, C.Y. Huang, and C.W. Berry on theADAPT/2DTM and ADAPT/3DTM codes, of R.Chen, P. Pattani, and P. Devloo on turbu-lence modeling and on the h - p fluid-structure interaction work, and of M. Edwards on theh - r schemes is gratefully acknowledged. Special thanks are given to L. Demkowicz whoparticipated in many of the adaptive projects over the last four years and to W. Rachowiczwho continues to contribute many original ideas to our work on h - p schemes.

Iteferences

The following are representative references where more details on adaptive methods in CFDcan be found.

1. Oden, J. T. and Demkowicz, L., "Advances in Adaptive Improvements: A Survey ofAdaptive Methods in Computational Fluid Mechanics," State of the Art Surveys inComputationallvfechanics, Edited by A. K. Noor and J. T. Oden, American Society ofMechanical Engineers, N.Y., 1988.

2. Demkowicz, L., Oden, J. T., and Strouboulis, T., "Adaptive Finite Element Methodsfor Flow Problems with Moving Boundaries. I: Variational Principles and AposterioriEstimates," Computer i\1ethods in Applied lvlechanics and Engineering, Vol. 46, pp.217-251,1984.

3. Oden, J. T., Demkowicz, L., Strouboulis, T., and Devloo, P., "Adaptive Methods forProblems in Solid and Fluid Mechanics," Accuracy Estimates and Adaptive Re-

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finements in Finite Element Computations, John vViley and Sons Ltd., London,1986.

4. Demkowicz, L. and Oden, J. T., "An Adaptive Characteristic Petrov-Galerkin FiniteElement Ivlethod for Convection-Dominated Linear and Nonlinear Parabolic Problemsin One Space Variable," Journal of Computational Physics, Vol. 68, No.1., pp. 188-273, 1986.

5. Oden, J. T., and Bass, J. lVI., "Adaptive Finite Element Methods for a Class of Evolu-tion Problems in Viscoplasticity," International Journal of Engineering Science, Vol.25, No.6, pp. 623-653, 1987.

6. Oden, J. T., Strouboulis, T., and Devloo, P., "Adaptive Finite Element Methods forthe Analysis of Inviscid Compressible Flow: 1. Fast Refinement / Unrefinement andMoving Mesh Methods for Unstructured Meshes," Computer !vlethods in Appl. Mech.and Engrg., Vol. 59, No.3, 1986.

7. Oden, J. T., Devloo, P., and Strouboulis, T., "Implementation of an Adaptive Refine-ment Technique for the SUPG Algorithm," Computational Alethods in Appl. !vlech.and Engrg., Vol. 61, pp. 339-358, 1987.

8. Oden, J. T., Strouboulis, T., Devloo, P., and Howe, M., "Recent Advances in ErrorEstimation and Adaptive Improvement of Finite Element Calculations," Computa-tional Mechanics Adyances and Trends, Edited by A. K. Noor, AMO - Vol. 75,ASME, pp. 369-410, 1986.

9. 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, N. Y., 1988.

10. Oden, J. T., Strouboulis, T., and Devloo, P., "Adaptive Finite Element Methods forCompressible Flow Problems," Numerical Methods for Compressible Flows -Finite Difference, Element and Volume Techniques, Edited by T. E. Tezduyarand T. J. R. Hughes, AMO - Vol. 78, ASME, New York, pp. 115-126; 1987.

11. Oden, J. T., Strouboulis, T., and Devloo, P., "Adaptive Finite Element Methodsfor High-Speed Compressible Flows," International Journal for Numerical Methods inFluids, Vol. 7., pp. 1211-1228, 1987.

12. Devloo, P., Oden, J. T., and Pattani, P., "An h-p Adaptive Finite Element Methodfor the Numerical Simulation of Compressible Flow," Computer Methods in AppliedA1echanics and Engineering, Vol. 70, pp. 203-235, 1988.

13. Bass, J. 1'.1., 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.

14. Oden, J. T., Strouboulis, T., and Bass, J. M., "Paradigmatic Error Calculations forAdaptive Finite Element Approximations of Convection Dominated Flows," RecentAdvances in Computational Fluid Dynamics, ASME Publication, (to appear).

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15. Demkowicz, L., and Oden, J. T., "An Adaptive Characteristic Petrov-Galerkin FiniteElement IVlcthods for Convection-Dominated Linear and Nonlinear Parabolic Problemsin Two Space Variables," Comput. Meth. Appl. klech. Engrg., Vol. 55, pp. 63-87,1986.

16. Demkowicz, L., and Oden, J. T., "On a Mesh Optimization Method Based on a rvlin-imization of Interpolation Error," International Journal of Engineering Science, Vol.24, No.1, p. 55-68, 1986.

17. Oden, J. T., "Adaptive FEM in Complex Flow Problems," The Mathematics ofFinite Elements with Applications, Vol. 6, pp.I-29, Edited by J.R. Whiteman,London Academic Press, Ltd., 1988.

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-~a I .. .... I --.~~w-Cla-1

Log (N)N = no. degrees-af-freedom

Figure 1. The error estimations of various computational methods.

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Figure 2: Numerical results displaying the continuous motion of the mass concentration in onedirection, and the corresponding adapted grids.(a) initial conditions, (b) 2 time steps,(c) grid adapted 2 time steps, (d) 4 time steps,(e) grid adapted 4 time steps, (e) 6 time steps.

- ..

Figure 2 continued. (g) grid adapted 6 time steps(i) grid adapted 8 time steps,(k) grid adapted 4 time steps,

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(h) 8 time steps,(j) 10 time steps,(1) 12 time steps.

(0)

(q)

Figure 2 continued. (m) grid adapted 12 time steps(0) grid adapted 14 time steps(q) grid adapted 16 time steps,

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(n) 14 time steps,(p) 16 time steps,(r) 18 time steps.

Figure 2 continued. (s) grid adapted 18 time steps(u) grid adapted 20 time steps,(w) grid adapted 22 time steps,

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(t) 20 time steps,(v) 22 time steps,(x) 24 time steps.

(a)

contact surface

fronc

bead of the

expansion wave

tail of the

expansion wave

(b)

Figure 3..: Solution of the Euler equations for a shock tube problem in three dimensions.Each set of figures illustrates velocity vectors on the three inside boundary sufraces, densitycontours on the center plane, and density contours on the three inside boundary surfaces.(a) Time step 25. (b) Time step 50.

, I-

Figure 4. Dynamically evolving adaptive grid at one time instant for Navier-Stokes solution ofrotor-stator interaction. Mesh for rotor-blades on right is moving relative to fixed stator blademesh along sliding interface.

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ClCl........Cl

Cl

II-lcr:>Cl:::LUI-Z

Cl...

Cl

II

xa:L

ClCl...Nr--

Cl

II

z

U"lCl:::::JIS)

I-ZIS)

u

:cuex:~

Figure 5. Computed instantaneous Mach number contours at 3.5 cycles for rotor-stator flowinteraction simulation.

16

aI

Q")

NcoaII-lcr:>0::WI-Z

a+

Q")

N

aII

xcr:z

aa+

Q")

aUl

aII

z

>-I-

(f)

zwa

Figure 6. Computed instantaneous density contours at 3.5 cycles of rotor-stator flow interactionsimulation.

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aa+

('r)

aII

....Ja:>a::::LUI-Z

a+Nco('r)

aII

xa:~

aa+

U1U1U1

aII

U1a::::::JCSJI-ZCSJU

l1Ja::::::JU1U1l1Ja::::ll..

Figure 7. Computed instantaneous pressure contours at 3.5 cycles of viscous rotor-stator flowinteraction simulation.

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- .

(a)

(b) ...,.." ~ • .,.. ...... 1... _. "In... I~ •.... ~I

Figure 8. Viscous cylinder problem with M = 0.64, flow perturbed after 2000 time steps.Vortices are generated and shedded; (a) instantaneous grid and (b) density contours.

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..c:2IlI ~'IlII''' •• 1.11S"" IU' 1.IM"l I~' 1.1n -4,

Figure 9. (a) Computed Mach number contours and (b) Velocity vectors for viscous cylinderproblem.

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•.

·Supersonic Inflow (Dirichlet)Supersonic Inflow .(Dirichlet)MachaJRe.l000

Pr.O.72

1. 1.4

y

outflow 0.75

0.1

x

.', ..~-'. L=1.0

Figure 10. Supersonic viscous flow over an elastic flat plate.

I I I I I I I I I II I I I I II I I II I I I I I I I I \ I I I I

IIIII~

I· I I I I I I il"""", , I I I I I II 'I j I I I ••. , ,

I I I IIIII~' >,r I : I : : 1 I • I .' ..

• , I I II I I I :: , ,. ',,' ":"I' • _ f" I I

I I ". "". ".-'III'~ I',A. •

j j , I' • I I

I "I I. . ...~- - .., ...

• "II'... . ' I. <j'l -'" ..

Figure 11. An instantanteous adapted h-p mesh with overlayed density contours for viscouscompressible flow over a defoffi1ing elastic plate; high-order spectral elements are used to modelthe viscous boundary layer while h-adaptive refinement is used to capture the shock.

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~_f

, .~

.1'- __

IIIII111 _

I I

11_,

L.I I I

I I , •

I I I I I I H-1

Contour Line Data: min = 0.15 E-5 max = 2.85 E-5 Interval = 0.3 E-5 (g/cm3)

--..I

Figure 12. Adaptive mesh and total mesh density contours for a computational region beginningjust downstream of the nozzle throats (see [13]).

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Figure 13. Normalized species density distribution compted on an h-adapted mesh.

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Figure 14. An h-r adaptive mesh with combined node relocation and mesh refinement.

Figure 15. Computed density contours for supersonic flow calculation of bow shocks on a bluntbody.

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""'-."-.."'.

Figure 16. An optimal h-p mesh for potential flow within a cube; different shades indicatedifferent spectral orders of various elements.

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