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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 59 (1986) 327-362NORTH-HOLLAND

ADAPTIVE FINITE ELEMENT METHODSFOR THE ANALYSIS OF INVISCID COMPRESSIBLE FLOW:

PART I. FAST REFINEMENT /UNREFINEMENT AND MOVINGMESH METHODS FOR UNSTRUCTURED MESHES

J.T. ODEN, T. STROUBOULIS and P. DEVLOOTexas Institute for Computational Mechanics. Department of Aerospace Engineering and Engineering

Mechanics, The University of Texas at Austin, Austin, TX 78712-1085, U.S.A.

Received 16 June 1986

New adaptive finite element methods are presented for the analysis of unsteady inviscid compress-ible flow in arbitrary two-dimensional domains. The procedures described herein are used in conjunc-tion with a semi-explicit two-step algorithm for solving the time-dependent Euler equations in two spacedimensions. Two schemes are presented for monitoring the evolution of error, and error estimates areused as a basis for a mesh refinement strategy. The capability of unrefinement (adaptively coarseningthe mesh) is also included. The methods do not require a structured mesh and are applicable to quitegeneral geometries.

1. Introduction

Many would agree that the most fundamental and important questions facing users ofmodem computational methods for flow predictions are the following:

(I) How good are the answers?(II) How can one obtain the best possible answers for a fixed computational effort (or a

fixed computing budget, fixed manpower level, or a fixed and limited computing capability)?The first question is exceedingly difficult since it includes both the issue of the validity of the

physical and mathematical model of the flow phenomena itself as well as the issue of thequality of the numerical approximation of the equations characterizing the model. To simplifymatters for purposes of the present discussion, we shall dispense with the first issue and takefor granted that the classical Navier-Stokes or, in the present paper, the Euler equations areadequate models of nature for the applications in mind. Thus, the first question reduces to aword: accuracy-how accurate are the numerical solutions?

The second question is seldom asked. but it is intrinsically connected to the first. It iscommon practice in applications of computational fluid dynamics to complex flow domainsto generate extremely fine finite difference meshes in hopes of capturing all important featuresof the flow, even though the location of these special points of interest changes in time. Thisleads some to employ fine meshes in all positi6ns of the flow domain where some importan'taspect of the flow might possibly manifest itself. The quality of results is generally judged bythe invariance of solutions to further refinement if one can afford the cost of another

0045-7825/86/$3.50 © 1986, Elsevier Science Publishers BV. (North-Holland)

---

328 J. T. Oden et al., Adaptive FEM for inviscid compressible flow

r),

calculation. The fact that coarse-mesh solutions may be adequate in much of the domain atmost instants of time cannot be exploited in traditional fixed mcsh schemes._ After some thought about these issues, rather broad answers to the fundamental questionspresent themselves:

(I) Accuracy. To determine the accuracy of a computed solution, one can attempt todevelop reliable a-posteriori estimates of local error. In other words, one might hope to beable to develop procedures which use the evolving computed solution to determine sharpestimates of local errors in various norms over each mesh cell and at each time stcp.

(II) "Optimal" meshes. Use adaptive procedures to continually change the structure of themesh-the size of mesh cells, the location of grid points-so as to keep the local crrors withina preassigned limit. .

Obviously, the second answer assumes that one has some means to measure the localquality of the numerical solution and, therefore, presumes the availability of some type of aposteriori error estimate.

We describe, in this paper, algorithms and results developed in an attempt to more sharplyresolve these answers, particularly that to question II, for a class of problems in compressibleflow. More specifically, we describe here a class of very effective adaptive schemes fortime-dependent Euler equations in two dimcnsions which employ both mesh refinement (whenthe local error is large) and mesh "unrefinement" (when thc local crror is small) and whichgenerate the appropriate mesh changes as the solution evolves in time. This requires that weestimate the local approximation errors at each time step. However, only an indication of therelative error between successive meshes is esscntial in our methods; the issue of very sharp aposteriori estimates of local error (our answer to question I) is one of great concern to us andis the subject of other papers [8, 18, 19].

in designing an adaptive scheme for Euler equations, we keep the following guidelines inmind:

(1) Unstructured grids. The method must be virtually grid-independent and global-coord i-nate-free. While an initial mesh can be defined to model· the basic geometry of the flowdomain and the initial data, thereafter it musr be possible to automatically add or eliminatecells and grid points as needed to monitor local accuracy levels. This requirement considerablylessens the attractiveness of body-fitted coordinates, many elliptic/algebraic mesh generators,and various factorization algorithms which exploit such regular mesh topologies.

(2) General geometries and boundary conditions. The method must be applicable toarbitrary flow domains with virtually arbitrary geometry an~ general boundary conditions.

(3) Solid mathematical basis. Since, by its nature, any sound adaptive method must employsome type of local error estimator, it is important that the methods employed have areasonably firm mathematical basis, e.g., that a priori or a posteriori error estimates exist andthat the convergence characteristics of the method are acceptable.

(4) High accuracy. The method should be capable of delivering high-order accuracy.(5) Robustness. The method must be numerically stable and not sensitive to singularities,

distortions in the mesh, or irregularities in the data.(6) Supercomputing. The method should lend itself to modern supercomputing mcthods for

accelerating computational speed, such as easy vectorization or implementation on parallelprocessors, etc.

-..-

J. T. Oden et al., Adaptive FEM for inviscid compressible flow ~29

(2.1)

(7) Computational efficiency. The method and the supporting algorithms and data struc-tures must be computationally efficient.

We feel that these criteria can be best met by finite element methods. In the present work,we use as the basis of our adaptive schemes a semi-explicit method used by several otherauthors (e.g., [4, 15, 16, 21]): a two-step Lax-WendrofflTaylor-Galerkin scheme. It is farfrom optimal (and does not satisfy all of our criteria), but is perfectly adequate to use inconjunction with our adaptive scheme. Schemes which fulfill all of these criteria are underdevelopment and will be reported in subsequent papers.

We remark that there is a growing literature on adaptive methods in computational fluidmechanics. Adaptive procedures for incompressible viscous flow problems were developed bythe authors in a series of recent papers (see, e.g., [8,18, 19]). These methods employed avariety of different adaptive strategies, but did not come as close to satisfying the above.criteria as the methods discussed in the present work. The general subject of adaptive finite

. element methods is dealt with in a volume of collected papers edited by Babuska et al. [2]. Fora survey of adaptive finite difference schemes, see the works of Anderson [1]. Also, Bergerand Oliger [5] and Berger and Jameson [4] have recently developed adaptive finite differencemethods for hyperbolic conservation laws. Still other types of adaptive methods for hyperbolicproblems have been recently proposed by Demkowicz and Oden [10,11].

Following this introduction, we develop weak formulations of a class of problems incompressible gas dynamics. These space-time formulations are shown to be the basis of a classof Lax-WendrofflTaylor-Galerkin schemes. Our derivation of this family of algorithms isnonstandard, in that we show that a two-step scheme follows easily from the use of anumerical quadrature scheme for evaluating appropriate flux integrals. In Section 3, finiteelement models of the space-time formulation are introduced, and in Section 4 we discuss theimportant issue of a posteriori error estimation. Section 5 is devoted to a detailed discussion ofadaptive strategies. These include an h-method, wherein the mesh is refined or unrefinedwhen local errors fall outside a preassigned upper and lower bound, and an r-method, inwhich the mesh is automatically distorted.:o ~quidistribute error. In Section 6 we present there~;ultsof several numerical experiments on two-dimensional problems. These results illustratethat the performances of the adaptive schemes are quite acceptable for a class of complex flowproblems.

2. Preliminaries

We consider the motion of a perfect gas flowing through a domain n over a time interval[0, T]. We shall confine our attention to two-dimensional cases, n C !R2

; we denote by D thespace-time domain, D = n x (0, T),and by'an ·the boundary of n. The motion of the gas isgoverned by the global balance laws of physics and the second law of thermodynamics. Thus,if U = U(x, t), (x, t) E D, is the 4-vector of conservation variables, U = {p, m, E} \ with p themass density, m the linear momentum, and E the total energy, and if dn and dS denoteLebesgue measures of area (volume) and length (area) o(n and an, respectively, then wedemand that U satisfy the following system of conservation laws:

dd ( U dn = - f Q(U)n dS .t J n an

330 J.T. Oden et al., Adaptive FEM for inviscid compressible flow

Here, Q(U) is the flux and n is the unit outward normal to an. If mp m2 denote Cartesiancomponents of m, then---

Q(U) = (2.2)

p(U) = (y -1)(£ - p-Im· tm).

In (2.2), P is the thermodynamic pressure and 'Y is the ratio of specific heats. assumed here tobe constant. In addition to (2.1), V must satisfy the entropy production inequality

dd f P1](U) dn + J n' (mT/(U) + O-Iq) dS ~ 0,t n an (2.3)

with 1](V) the entropy density of the gas, 0 the absolute tempcrature, and q thc hcat flux, aswell as an initial condition,

V(x, 0) = Vo(x), x En, (2.4)

whcrc Vo is given.It is of fundamental importance to note the smoothness requirements on V in order that

(2.1) makes sense mathematically. Conservation laws (2.1) hold when the components of Vare bounded measurable (with respect to Lebesque measure in x) functions on D. Thus, wemay seek solutions in the function space

V= {V= {VI' V2, V3, V4} t I V; = V;(x, t) E L "'(0, T; L I(n)), i = 1, ... ,4} . (2.5)

In particular, (2.1) is not equivalent to the classical Euler equations,

V,+divQ(V)=O, (2.6)

(with V, = au/at and div Q = Ej iJQaJiJx/) since solutions of (2.1) may not possess derivativesacross surfaces in D. However, the conservation laws, initial and jump conditions are fullyequivalent to the following weak boundary-initial value problem:

Find U E V such that

J (V lip, + Q(V): Vip) dn dt +f V~ ip(- ,0) dn = (T,( Flip dS dto n Jo Jan

for allip E W ,

(2.7)

J. T. Oden et al.• Adaptive FEM for inviscid compressible flow 331

."P'"./

where F is the actual prescribed flux through an is W a suitable space of test functions,e.g.,

----- til - .W= {cp = {cpp CP2'CP3'CP4} CPj = CPj(x, t), CPjE C (D), lpj(x, T) = 0, l = 1, ... , 4} .(2.8)

In (2.7), we use the notation

4

U1CPt = 2: Va acpa/at,a~1

2 4

Q :Vcp= 2: 2: Qaj acpa/axj.j~1 a=1

(2.9)

On the other hand, if U is known to have integrable derivatives in D everywhere except ona family of surfaces {rk}:=p then we may consider the problem:

Find V E V such that

'.. ~'. . ... 4; 1:."; ;"~'. .) l' j! • . . •...Iv (U; cP + (div Q(U))'cp) dfl dt- In'U1(. ,0) cp(. ,0) dn

-±J cp'{skl!ul!-I!QIlI!}dSdl+J u~cpdn+(J (Qn)'ipdSdtk=1 r. n Jo an

=fTJ F'tpdSdt.foralltpEW.Jo an(2.10)

Here, Sk are the speeds of propagation of discontinuities across ~ and V, Wareappropriately redefined, e.g.,

V=, {V= (VI' V2, V3'V4)IV; EL,"'(D), V; E H1(DkJ, Dkj = D - rkJ ,o -W= {cp I CPjE C (D), tpj(x, T) = O} ,

where ~j are the surfaces on which V; suffers a jump. If F is not a prescribed flux but is merelya notation for Qn, then these flux terms cancel and do hot appear in the formulation.

Consider an arbitrary time interval ['T1, 'T2] C [0, T] and include in W functions tp(x, 'T2) ¥: o.Let w be a subset of n such that w n ukrk = 0,.and let F == Qn. Then another weak statementof the system conservation laws over w x ['Tp 'T2] is: '

Find U E VW•T such that

,. for all tp E WW•T

,

with VW•T and WW

•T appropriate spaces of trial and test functions.

(2.11)


REMARK 2.1. It is well known that (2,7), (2.10), and (2.11) may all possess nonphysicalsolutions since none of these formulations involve the entropy inequality (2.3). Thus, ingeneral, we seek solutions to (2.7), (2.10), or (2.11) in the subset KeV; K={vEVlvsatisfies (2.3) for appropriate e, q(Ve)}.

3. Finite element approximations

.. Finite element approximations of the gas dynamics problem are obtained by a directapproximation of (2.10) or (2.11) on finite-dimensional spaces approximating the spaces Yand W. The spatial domain n is partitioned into a collection fih of finite elements n. overwhich the components of trial functions V are approximated by polynomials of degree k. Inthis way, we construct a family {Vh} of finite-dimensional spaces of the type

\ ....

(3.1)

where Pk(n.) is the space of polynomials of degree k defined over n •. Alternatively, wecan use V~ln E Qk(n.), where Q/c(n.) is the space of tensor products of polynomials of,degree k on n. (e.g., QI(n.) is spanned by bilinear functions, Qz(n.) by biquadratics, etc.).In addition, a family {Wh} of finite-dimensional spaces of test functions is also constructed.We then consider Galerkin approximations of (2.7), (2.10), or (2.11) by seeking solutions tothese equalities in Vh, with V and W replaced by yh and Wh

, respectively.

3.1. A two-step, Lax- Wendroffl Taylor-Galerkin scheme

We next derive a special semidiscrete, weak formulation from (2.11) which provides thebasis for the construction of a popular family of finite element schemes. We proceed with thefollowing steps:

'Step 1. Partition the time interval [0, T] according to 0 = to < t, < tz < ... < tN = T.Step 2. Apply the weak balance law (2.11) to a typical time interval [tn. tn+1l (with Tl = tn

and Tz = tn+1). .Step 3. Set lpt = 0 in (2.11) suggesting the ultimate use of a time-invariant grid (we relax

this assumption later).Step 4. Replace the time integrations in (2.11) by the elementary midpoint quadrature rule

Thus, with w = n, we obtain the semidiscrete approximation

(3.2)

• 1

"

-.-

J. T. Oden et al., Adaptive FEM for inviscid compressible flow 333

. where U~ = Uh(x, tn), etc., Uh being the approximation of U, and Qn+1/2 is the flux at the halfstep,

I"(3.3)

~ Step 5. To obtain an approximation U~+1/2, we use (2.11) again for time intervalt-} [tn' tn+1I2], this time replacing the time integrals by a simple strip rule,

We thus arrive at the following algorithm:

Step 1. With (UZ, Qn = Q(UZ)) known at the ~th time step, compute UZ+1/2 using (3.4);Step 2. Compute Qn+1/2 using (3.3);Step 3. Compute U~+I using (3.2);Step 4. Go to Step 1.

This algorithm is the finite-element-based two-step Lax-WendroffiTaylor-Galerkin scheme(see [7,20]). It is one of a family of methods advanced by Donea [13], studied by Baker andKim l3], and successfully refined and used by Lohner et a!. [15,16] and Bey et al. [6] in finiteelement applications in fluid dynamics. This semi-explicit method is of second order in time ..and can experience spurious oscillations near shocks and other types of irregularities in thesolution. These deficiencies must be reckoned with in implementing the method.

I 3.2. Artificial viscosity

As noted earlier, artificial viscosity terms are usually added to schemes such as the oneemployed here so as to dampen out o$cillations in the numerical solutions near shocks. Thecalculations described subsequently were performed adding a Lapidus viscosity term [14],which, at time step tn' is of the form .

where-v· (c(u)· VUn+a

) , (3.5)

C is the Lapidus constant, uj = mJp is the ith component of the flow velocity, and a is aparameter which determines whether viscosity is to be included implicitly (0: = 1) or explicitly(0: = 0). The viscosity term, written out in component form, is

Setting 0: = 1, we obtain for Step 2 of the procedure (instead of (3.2)),

'"' -

334 J. T. Odell et al., Adaptive FEM for inviscid compressible flow

(3.6)

\~

for all admissible test functions 'Ph'

3.3. Details of the finite element algorithm

The details of the implementation of the algorithm described above are crucial to successfulcomputations. In this work, we use meshes of four-node quadrilateral (Q I) elements overwhich the components Va (a = 1, ... , 4) of U are piecewise bilinear functions. Similarapproximations and algorithms are used by Bey et al. [6]. In addition, so-called groupapproximations of the flux Qai (a = 1, ... ,4, i = 1,2) are employed so that these componentsare also piecewise bilincar functions determincd by their valucs at element nodes. In gencral,this finite element approximation will be of the form,

N

V = 2: U ~(t)lpj(x) ,. ai-I

N

Qai = 2: Q~i(t)lpi(x) ,i= I

(3.7)

where N denotes the total number of nodes in the discrctization, and U~, Q~j are values ofUh, Qh at node j, and lpi are the global piecewise bilinear basis functions. .

As noted earlier, we advance the solution in time in two steps. It is important to note thatthe first step is essentially local, computed over each element, while the second is global andcontains the artificial viscosity terms:

Step 1. For each element net calculate a constant element vector V::1/2 from

(3.8)

Step 2. For each node j, calculate U~"+1 by solving the following system of equations:

(3.9)

Here, Q" denotes the elementwise averaged value of the flux. The coefficients Tp are definedto be constant over each element,

...-

J. T. Oden et al., Adaptive FEAJ for inviscid compressible flow 335

.'

Tp = cA~lau; / axp I ,_where c is a global constant (c = 1 in the examples), A c denotes the area of ne' and u~ denotethe components of the fluid velocity.

To speed up the calculation, we precalculate and store the following element integralsbefore the time stepping is started:

In l{Jj dfl. 'r al{Jj dn,In. aXfJ

in. l{Jj'Pjdn ,

i, j = 1, ... ,4, f3 = 1, ... ,4.

An element-by-element Jacobi conjugate gradient method is used to obtain the solution ofthe matrix problem in Step 2. Due to the structure of the mass matrix, the iterative solverrequires only a few iterations to converge fully.

3.4. Boundary conditions

In the finite element schemes developcd here, we implement the following three types ofboundary conditions:

(a) Supersonic inflow. On the part of the boundary with supersonic inflow, the values of allthe conservation variables are imposed.

(b) Supersonic outflow. On the outflow part of the boundary, the values of the conserva-tion variables and the normal flux are unknown. Boundary conditions of supersonic outflowarc implemented by adding the contribution of the boundary integral of -the normal flux to theright-hand side of thc equations of Stcp 2.

(c) Solid boundaries. On a solid boundary, the normal component of the velocity un = upnfJis zero. We note that, in general, the nodal directions are not uniquely defined. In suchcalculations, we compute the normal directions at the nOQes which satisfy global massconservation at the steady state, namely,

1 a /( 2 (J a )2) 1/2nj- l{Jj dn 2: 'Pj dn

fJ - n aXfJ fJ=1 n aXfJ

3.5. Hourglass instabilities

We now show that the Taylor-Galerkin scheme presented above can propagate undetectedspurious solutions. ,To dcmonstrate this, let us consider the scheme applied on the 2-DBurgers equations on. a mesh of rectangular elements. Burgers equations may be obtainedfrom the above formulation by redefining the flux as follows:

Consider a rectangular element with the following nodal solution at time tn:

...-


{U~r = [0,1,1, Or,~l!~r = [0,1,1, or,

{U~r = [0,1, -1, or,{U:}n = [0, 1, -1,0]' .

Then the scheme gives

{u:r+1/2 = [0,1,0, Or ,and, by letting c = 0 (no artificial viscosity), we get:

This means that the scheme propagates "hourglass" solutions undetected. This fact explainswhy in thc numerical examples the method produced oscillations ncar the outflow boundaries.This hourglassing phenomenon can be eliminated by considering each quadrilateral as a patchof two triangular elements joined along one ~iagonal of a quadrilateral.

4. Errors

The adaptive finitc element methods dcscribed here involve two basic components:(1) Error estimation-the determination of a posteriori cstimates of the evolution of error·

in thc numerical solution.,(2) Adaptation-the automatic restructuring of the approximation so as to rcduce local

clemcnt errors and the computational effort.In this paper, adaptive procedures are based on estimates of error in a single principal

dependent variable, such as the density, pressure, or the entropy. We shall choose the densityp as the driving factor in adaptivity, although other c'loices could be uscd in the algorithmsdeveloped here. Two basic procedures are used to ~st.imate local element errors.

4.1. Evolution equation for error

Consider the continuity equation for the evolution of mass density through a domain n withknown flow velocity u. A weak form of the continuity equation is .

\,

In cpp, dn = - In v· (up)cp dn for all cp E W . (4.1)

A semidiscrete Galerkin approximation of (4.1) consists of seeking an approximate density phsuch that, over some suitable finite-dimensional space of test functions Wh,

(4.2)

If Wh C W, we may choose cp = CPh in (4.1), subtract (4.2), and obtain the following evolutionequation for the error eh(x, t) = p(x, t) - p"(X, t):

J. T. Oden et al., Adaptive FEM for inviscid compressible flow

.' The exact and approximate solutions are related according to

p = ph + he _,".f

where i is the approximation error. Thus, the error satisfies the evolution equation

In (cpe~ + V' (ui)cp) dn = (r", cp} for all cp E W ,

.where ( , h' cp) is the residual functional,

337

(4.3)

(4.4)

(4.5)

(4.6)

If we replace cp by CPh' ('h' CPh) = a by (4.3), and the evolution equation reduces to merely theorthogonaiity condition (4.3), which is automatically satisfied by error.

We obtain an approximate evolution equation for thc error as follows. Let Eh denote afine-grid approximation of eh

; Le.,

eh(x, t) = E"{x, t) = 2: E"(t)rpN(x) ,N

.. ~~.

(4.7)'

where rpN(X) denotes a polynomial basis function defined on a subgrid of finer mesh sizc thanthat used to calculate ph. Then, introduction of (4.7) into (4.5) and rcplacing cp by rpN gives

where(4.8)

(4.9)

Many possible ways for implementing (4.9) present themselves'- These equations, forexample, need not be global in the sense that an element-by-element or patch of elements in afine mesh obtained through a mesh refinement may produce sufficient accuracy to allow for anadequate indication of the evolution of error. The local velocities u and residual rh can beinterpolated using Q I-approximations on a fine-mesh level. Several of these alternatives areunder study and are to be the subject of ~ fort~coming report.' ,..

4.2. Interpolation errors

. • Let Ii be a smooth function defined over a regular domain n. The Wr,p(n) seminorm of u is( defined by

{f I ai+j IP } IIplulwT.P(fl) = fl'~ .:\ i.:\lij dn I (4.10)

I+}-r uXI uX2i. j;a.o

338 J. T. Oden et al., Adaptive FE,'rf for inviscid compressible flow

where 1~ P ~ 00 and r is a nonnegative integer. The Sobolev norm of u is

(4.11)

Let G be an arbitrary convex subdomain (a finite element) of n over which u is interpolatedby a function uh which contains complete piecewise polynomials of degree k. Then, it can beshown [17] that the local interpolation error in the Wm,p(G) semi norm is

\

hk+l

I- I C h"'P'-nlpl 1u - uh wm,p(G):S;; -,;;- • U Wk+l,p(G) ,

P(4.12)

where h is the diameter of the domain G, p is the diameter of the largest sphere that can beinscribed inside G, n is the dimension of the domain n, pi is p/(p -1), and C is a constantindependent of h, p, and u. If p is proportional to h and if it remains proportional inrefinements of G defined by parametrically reducing h, we have -

IE"I ~ Chnlp'-nlp+k+l-ml ,m.p.G U k+1.p , (4.13)

with /'lm,p,G = ,./ Wm,P(G)' etc., and Ell = U - uh •

Such estimates can be used to devise crude adaptive schemes. Suppose that u on the rightside of (4.13) is replaced by a finite element approximation uh and that lu"lk+l. =lulk+l p + O(h). Then (4.13) indicates that the local error in the Wm'P(G) seminorm" isproportional to the error indicator, hr./p·-nIP+k+l-

mlu!k+l.P' Some choices are:(i) n = 2, m = 0, k = 1, P = pi = 2:

In this case, om: must approximate the W2•2 seminorr.1 of u over G; i.e., the L 2 norm of

second partial derivatives of u.(ii) n = 2, p = 00, pi = 1, k = 0, m =0:

IEhILl(G) = Ch2lE:veragel ~ Ch3IuI1.CI>.G = Ch3 Teal!V, u(x)\.

Such estimates can give only rough indications of local errors in sufficiently fine meshes.However, they are usually easy to implement and our experience is that they can provide avery effective basis for mesh refinement strategies.

5. Adaptive mesh strategies

Let us suppose that we can calculate an error indicator O~for each finite element ne in agiven mesh at a time t. This indicator is, in general, a real number representing the local errorin a suitable norm, and it is computed using one of the procedures described in Section 4. Thedecision to adapt the numerical procedure (to refine the mesh or to move nodal points) is

t


based on whether or not local error indicators exceed preassigned tolerances. We shalldescribe two adaptive procedures in this section.

:--~--S.1. An h-refill em ell tI unrefl1lement method

Our h-procedure involves the following steps:

Step 1. For a given domain n, such as the one shown in Fig. l(a), a coarse finite elementmesh is constructed which contains only a number of elements sufficient to model basicgeometrical features of the flow domain.'

Step 2. As our adaptive process will be designed to handle groups of four elements at atime, we generate a finer starting grid by a bisection process, indicated in Fig. l(b), to obtainan initial set of element groups.

Step 3. We initiate the numerical solution procedures on this initial coarse grid, andcompute error indicators 0e over all M elements in the grid. Let

Step 4. Next, wc scan groups of a fixed numbcr P of elements and compute

where ek is the element number for group m. We take P = 4 in our current codes .

i , •········1····.······:········----:··.· ....·..--..·---. , .. , ......·..t······ ..\ ..:..\~: ...::::;.:.:::~:::::::::::::::::.::::..

: \: \.

/

( a)

(b)

Fig. 1. (a) A coarse initial mesh consisting of four-element groups and (b) the refinement and unrefinemcnt of agroup of elements.

.... -

340 J.T. Oden et al., Adaptive FEM for inviscid compressible flow

Step 5. Error tolerances are defined by two real numbers, 0 < a, {3 < 1. If Be ~ {3Bmax' werefine element ()e' This is done by bisecting ()e into four new subelements. If ():roup ~ aBmax' weunrefille group k by replacing this group with a single new element with nodes coincident withthe corncr nodes of the group. This is always possible because each group is itself the result of aninitial bisectioning.

This general process can be followed for any choice of error indicator. Moreover, it can alsobe implemented at each time step in the numerical schemes discussed in Section 3.

5.2. Data structures

An important consideration in all adaptive schemes is the data structure and associatedalgorithms needed to handle the changing number of elements, their node locations andnumbers, and the element labels.

As noted in the preceding paragraphs, the algorithm is designed to process (refine orunrefine) in groups of four elements at each local refinement/unrefinemcnt stcp. Consider, forexample, the case of an initial mesh of 20 square elements shown in Fig. 2. We assign to eachelement in this mesh an element number, NEL = 1,2, ... , NELEM, and to cach global nodca label NODE. The array NODES(J, NEL) relates the local node number J(J = 1, ... ,4) ofelement NEL to the global node number NODES. In addition, the coordinates XJ, VJ of eachnode are also providcd relative to' a fixed global coordinate system. We file these numb~rs intwo arrays:

NODES(J, NEL) is the array of global node numbers assigncd to node J of elemcnt NEL,XCO(JCO, NODE) is the array of JCO-coordinates of global node NODE (JCO = 1 or

2).Suppose that an error indicator is computed that signals that an element should be refined,.

say element 11 in the example. We must have some system for assigning appropriate labels tothe new elements and nodes. Toward this end, we can establish a convention that defines theconnectivity of the specified elemcnt with its neighbors in the mesh. This information isprovided by a third connectivity array:

NELCON(NC, NEL) is the NCth connection of element NEL, NC = 1,2, ... ,8.As seen in Fig. 2, each side of an element may be connected to two other elements so thatDimension NELCON = (8, MAXEL), with MAXEL an appropriately large number.

The entire refinement (or its inverse-the unrefinement process) just described is accom-plished by specifying a series of element levels. For example, the initial coarse mesh could beassigned level O. When an element is refined, its subelements belong to a highcr level, levell,and when these subelcments are refined, elements of level 2 result, and so on. In this way, ifthe maximum level any element in the mesh can achieve is limited, then the maximum numberof elements the mesh can contain is also limited. In general, no such limit need be set.

Thus, the bookkeeping of element and node numbers evolving in a refinement process ismonitored by the arrays NODES(·,·), XCO(· ,.), NELCON(' ,.), and an array LEVEL(NEL) which assigns a level number to element NEL. Initially, the same level can be assignedto all elements, and this level is an arbitrary parameter, prescribed in advance by the user.Thus, provisions are now in hand for an arbitrary, dynamic renumbering of elements andnodes. If, for example for the mesh in Fig. 2, element 11 is to be refined, we proceed throughthe following steps:

.. .


S

1

6

rs r 6 13

12S /26

- 4 S 12 20- -

.\. -t

1

4

1

7 r4 r 14

124 127

3 6 11 I 19,r,/ X2=Y

.- T 3 8 13 18 23 28

2 7 0 lS 18

2 9 12 19 22 29

8 9 16 17

10 11 20 21 30

X1=X

7 , 3

4

8

5

6

2

. ,

Fig. 2. Mesh, node, and connectivity numbering in a model problem .

Step 1. Loop over the neighbors of element 11 (which is made possible with the NELCONarray) and check the level of the neighboring clements relative to the level of element 11.

Step 2. If any neighboring element has a level lower than element 11, then the elementcannnot be refined at this stage.

Step 3. If element 11 can be refined (as is the case in Fig. 2), we generate new elementnumbers (thus changing NELEM and new node numbers for unconstrained nodes).

Step 4. Compute the connectivity matrix NELCON for the new elements.Step 5. Adapt the connectivity matrices for the neighboring elements (since the refinement

of element 11 has now changed this connectivity).Step 6. Interpolate the solution between the unconstrained nodes.

It is clear that some strategy is needed to test if a designated element is appropriatelyconnected for a refinement to take place.

Consider, for example, the uniform grid of four clements shown in Fig. 3(a) and supposethat the error estimators dictate that element A is to be refined. Thus, A is divided into four

--

342 J.T. Oden et al., Adaptive FEAt for i/lviscid compressible flow

----- A I B

C I D

IV IIIB

I II

C D

'.( a)

IV IIIB

I II

C D

(b)

IV III VIII VIIB

I II V VI

Cl

C D

IV III VIII VII

I II V VI

C D

( c)

IV VII

IIIVIVI

C I D

Fig. 3. Sequence of refinements of a uniform mesh.

elements, I, II, III, IV, as shown, and the solution values at thc junction nodes, shown circledin the figure, are constrained to coincide with the averaged values between those marked x.Note that the connectivities change in this process, e.g., the connectivities 4 and 8 of elementB are different.

Next, assume that an additional refinement is required, and that we must next refineelement III. We impose the restriction that each element side can have no more than twoelements connected to it. Thus, before III can be refined, element B must first be refined, asindicated in Fig. 3(b). The constrained node B in Fig. 3(a) now becomes activc, while nodeCl remains a constrained node. With B bisccted, we proceed to refinc III into subelemcntsa, ~,-y, &, and new constrained nodes, again circled in Fig. 3(c), are produced. In this case,only element B had to be refined first in order to refine III, but, in general, the number ofelements that must be refined in order to refine a particular element cannot be specified. The


."

following subroutine determines the necessary refinements prerequisite to refining an elementNELl:

SUBROUTINE DIVIDE(NELl. NEL2)

NELl = the input clemcnt that needs to be refined

NELl if NELl hasbeen divided

NEL2 = output element =NELD = elcment that needsto be divided prior to NELl

Then, symbolically, we have the algorithm (for the example in Fig. 3),

RcpeatNEll = IIICALL DIVIDE (NELl, NELZ)

[

WHILE (NEL2.NE.NELl)NELl = NEL2CALL DIVIDE (NELl, NEL2)END WHILE ..UNTIL (NEL2.EQ.II1)

5.3. Moving mesh (node redistribution) methods

Another family of adaptive schemes we have considered is a node redistribution schemewhich progressively moves a fixed number of nodes as to reduce local error. One basis forsuch schemes is to equidistribute error at each time step.

For example, let 8~be an error indicator for element il~ in a !Oesh containing a fixednumber M of elements in a two-dimensional mesh. Let h = h(xl' x2) be a mesh function suchthat

and note that, approximately,

(5.1)

with dil = dX1 dX2 (this being exact for domains which are unions of square elements). Let8 = 8(xl' x2) be the mesh function which gives the local error indicator when evaluated at apoint (0 = O~for x E ilJ. We wish to minimize the total error indicator functional,

(5.2)

subject to the constraint (5.1). Using Lagrange multipliers, this leads to the optimalitycondition,

--

344

or

or

J. T. Odell et al .• Adaptive FEM for illviscid compressible flow

Suppose that meas(fle) = O'oh; and that 0e is of the form 0e = h; f(ll). Then, integrating thislast result over a typical element gives

Hence, the optimal mesh sizc distribution results when

f 0; dfl = AO'o/0' = CONST .fl.

(5.3)

In othcr words, to obtain thc optimal mcsh, we must equidislribwe the indicators JO~To usc this result to redistribute nodcs, we proceed as follows (d. [12]):Step 1. Generate an initial (generally regular) mesh with a fixed number M of clements and

computc a trial solution on this mcsh at one time step.Step 2. Computc the corresponding error indicators 0e.Step 3. For a group k of P elements (with P always 4 in this work), let Ae. denote the arca

of element i in the group. The ar~a-weighted indicators for group k are thc P numbers O)A .. ., e~

Fig. 4. Calculation of area center-of-error ;rN to cquidistribute element error indicators in a cluster of fourelements.

l

Step 4. Let y~. denote a vector from the origin of a global coordinate system to the centroidof element ej oc'group k. Then the center of error of group k is defined as the vector

Step 5. Relocate the node at the center of group k to lie at the vertex of Xk, as shown in Fig. 4.Step 6. Continue this sequence of operations over each group k of four elements until the

new location of each node does not change more than a preassigned tolerance.

This process should approximately equidistribute the element error indicators.

.,,

J. T. Oden et al., Adaptive FEM for inviscid compressible flow

--- - 4 4

Xk = ~l Y~i(8~/A~)/Ll (8~/A~).1_ ,a

34S

(5.4)

6. Numerical examples

In this section, we present the results of several numerical experiments on representativetest problems. Six examples are presented, the first five involving steady-state examples, oneof the following two strategies is used.

STRATEGY AStep A.I. The numerical solution is computed on a fixed mcsh and is advanced in time until

a steady state is reached.Step A.2. After convcrgence to a steady state, error indicators 8~arc computed over each

element. In the calculations discussed below, we employ the interpolation estimates and use

(6.1)

..•

where A~ is the area of the element.Step A.3. The mesh is refined/unrefined using the criteria and algorithms discussed 10

Section 5.

STRATEGY BStep B.1. Same as Step A.1.Step B.2. After convergence to a steady state, error jndicators 8~are computed according

to

(6.2)

Step B.3. In applying the node redistribution (moving mesh) algorithm, a modified errorindicator e~is employed, which is designed to be always greater than unity even when (J~ =O.In particular, we use

In our examples a = 81, {3 = 1, and 'Y = 8.

'-


Step B.4. Nodes are redistributed a total of K times using the procedure described In

Section 5.3. In the examples, we take only two iterations (K = 2).

__ We proceed to the examples.

6.1. Shock reflection problem

We begin with a problem for which an exact solution is known and which has been used as abenchmark problem by others.

The problem involves the steady flow of a perfect gas in a rectangular duct in which density,velocity, and energy are prescribed in each of four triangular wedges in such a way that theappropriate jump conditions (the Rankine-Hugoniot conditions) are exactly satisfied. Thus, aproblem of shock reflection fOf which an exact solution is known is obtained. Dimensions anddata are given in Fig. 5. In this and all the other problems, the solution is considered to haveconverged to steady state when the magnitude of the L 2 norm of the density is reduced bythree orders of magnitude.

The time step is monitored by the fOfmula

At = min{0.50VA:/(lul + en .r

Here, C2 = yP/p and /u12 = u~ + ui, y == 1.40. The constants multiplying the artificial viscousterms were selected locally as:

where the bar denotes average element values. A Lapidus constanC of 1.0 was used in allcalculations.

The results of a uniform coarse initial mesh approximation are shown in Fig. 6. Thecomputed density contours an.~ also shown in this figure. Note that only a rough indication ofthe location of the'shock is possilJle with this mesh.

A much better resolution is given in Fig. 7, where the adaptively refined mesh shown iscomputed with refinement parameters a = 0.10, f3 = 0.50 (recall Section 5). Note that !l0

Fig. 5. A shock reflcction problcm. Inflow values of the conservation variables are prescribed as indicatcd inregions I and II, and outflow values are computed in region III to satisfy the conservation laws.

. -(a) (b)

"\ - "..

\'

•II

Fig. 6. Reflecting shock problem. (a) Initial mesh and (b) density contours ..

(a)++

I II I

I

I

I

++Fig. 7. Reflecting shock problem. (a) Mesh and (b) density contours obtained with one level of refinement(a = 0.10, {3 = 0.50).

w...-...I

--


"unrefinement" appears to have taken place with these parameter choiccs, but that the simpleerror estimation scheme is capable of detecting the general area of the shock line. The muchimproved density profiles are indicated in the figure.

-- Still better results are obtained with the same a and 13 but with two levels of refinement, asindicated in Fig. 8. Note that in this case large elements appear in the mesh, indicatingunrefinement as well as refinement of the original mesh. The corresponding density surface isgiven in Fig. 9, where quite sharp shock fronts are observed. Note some spurious oscillation isencountered near the outflow boundary, as should be expected from the deficiencies of the31gorithm noted in Section 4.

The same problem was also analyzed using the node redistribution algorithm discussed inSection 5.3 with ten node redistribution iterations. Results are shown in Fig. 10. There, theoriginal coarse initial mesh of Fig. 6 is progressively distorted to conform to the rcflectedshock locations. Corresponding density contours are also given in the figure.

(a)

Fig. 8. Reflecting shock problem. (a) Mesh and (b) density contours obtained with two levels of refinement(a = 0.10, {3 = 0.50).

,

l

of

J.T. .aden et ai., Adaptive FEM for inviscid compressible flow 349

Fig. 9. Reflccting shock problcm. 3·D view of the converged dcnsity function obtaincd with two Icvels ofrcfinement.

6.2. NACA 0012 airfoil in supersonic wind tunnel

In this examplc, the supcrsonic l10w through a narrow wind tunnel. containing a NACA0012 airfoil is studied. The inl10w Mach number was set at M"" = 2, with 'Y = 1.40, andsymmetry IS exploited to reduce the computational effort:"

The initial.coarse mesh and density computed contours are giye.n in_Fig, 11. Note that thecritical features. of th~solution-the reflfcted shock and contact discontinuity-are lost withthis coarse mesh. A refined/unrefined mesh: obtained with parameters a = 0.10, f3 = 0.10 isshown in Fig. 12 together with a greatly improved density approximation. In these andsubsequent calculations, a CFL number of 0.5 and a Lapidus constant of 1.0 were employed.Results of a node redistribution scheme for the coarsemesh"are shown in Fig. 13. In theseresults, ten iterations of the node redistribution algorithm were used.

6.3. Supersonic flow in a wind tunnel with a step

The steady-state solution of the problem of a wind tunnel with a step introduced into theflow is next considered. The inflow Mach number was·selected oM"" =='3.0and 'Y = 1.40. Theinitial coarse mesh is shown in Fig. 14 with the corresponding density profiles, and results ofthe adaptive refinement/unrefinement scheme with a = 0.15 and {3 = 0.20 are shown in Fig.15. The mesh refinement algorithm was also used, with the mesh and density profiles obtainedafter 10 iterations shown in Fig. 16. We see that the adaptive scheme captures well thefeatures of the flow including tbe contact discontinuity at the top near the point of reflection ofthe bow'shock. However, some oscillations are present do\vnstream: and they are believcd tobe due to the nonmonotonicity of the solution algorithm. The results presented for therefinement/unrefinement procedure have been constrained by a maximum number of 2000

JI

t..>lJlo

Fig. 10. Reflecting shock problem. (a) Mesh and (b) density contours obtained after 10 applications of the meshredistribution algorithm .

(b)

I flIl:.J..J-

~~).l}~~."'~~:::'0'.,S·<::1:;'....~.....(:)

~.,:Jl...0:

"'~(:)

<:

~:-iC~::--l-

\~

-

"\.,

~

• I I l

~

(a)

~

. (a)

'.

Fig. 11. NACA 0012 airfoil in supersonic wind tunnel. (a) Initial mesh and (b) density contours.

~ -- ..... ..-. ..

· ~

(a)

(a)

" .... ~

(b)

,

Fig. 12. NACA 0012 airfoil in supersonic wind tunnel. (a) Mesh and (b) density contours obtained with one levelof refinement (a = 0.10, f3 = 0.10).

(b)

Fig. 13. NACA 0012 airfoil in supersonic wind tunnel. (a) Mesh and (b) density contours obtained after 10applications of the mesh redistribution algorithm.

~~o~:3

~s:,:-

~§-~~..,.~:;......

Ci>..,S·0::£:;.l">is.:8~..,~'"s:;;;-

'::bc·~

l;o.lVI.....

.'I

I

I

I

I

I

I

".

(a) (b)••• •. .••••

I

I

Fig. 14. Supersonic flow in a wind tunnel with a step. (a) Initial mesh and (b) density contours.

(b)

Fig. 15. Supersonic flow in a wind tunnel with a step. (a) Mesh and (b) density contours obtained with one level ofrefinement (a = 0.15, P = 0.20).

\,

wVIN

:--..:-;Ci}::~I:l:-

~~~;,:."';;;3::

C;>.,S·'<:!:j'

"~::;~~i:j.;~~~

JI

'.-. ~ :'-' , - ,

· "-

(a)

~ ... \,0

",

(a)

Fig. 16. Supersonic flow in a wind tunnel with a step. (a) Mesh and (b) density contours obtained after 10applications of the mesh redistribution algorithm.

(b)

-..... --- -i---~_--:::-~::-C:::;'-~- - -~ ..... - ---~:- '- ~ ~--~-: ~:~~~-~~~:-~-~~~

i- ~~_- __ ~ __ -~--~-~~~

:-~~::~~~~~-~~-~---~i-----~---~....:::t:::i-::::~:::t:::- ...._ i- .....'"""-~~--~....

Fig. 17. Supersonic flow over a 200 ramp. (a) Initial mesh and (b) density contours.

wVIW


nodes or 2000 elements that can be allowed. In the refined mesh shown, this constraint hasbeen achieved.

~---0.4. Supersonic flow over a 20° ramp

We next consider the steady supersonic flow through a conduit with a 20° ramp. The gas(with y:::: 1.4) enters as a uniform M:::: 3.0 flow through the left side of the ramp and a shock '"develops at the ramp root. A coarse initial mesh and the computed density contours areillustratcd in Fig. 17. For this problem, a reasonably good indication of the orientation of theshock is obtained.

Adaptive mesh results are shown in Figs. 18 and 19 for choices of the parameters ofa ::::0.20 and {3:::: 0.50 with one and two levels of refinement, respectively. Notice thatspurious oscillations at the outflow boundary above the ramp root, due to the hourglassoscillations described in Section 3, cause unneccssary refinements in this region. Similarly, inregions between the shock and the ramp, some unnecessary refinement results fromoscillations in the numerical solution. Nevertheless, striking improvement in the quality of thesolution is seen to result from the refinement procedure. .

In this particular problem, the node redistribution algorithm works remarkably well. Acomputed distorted coarse mesh, obtained after ten applications of the node redistributionalgorithms, is shown in Fig. 20 with the resulting density contours.

6.5. BlUlIlleading edge of 8' HTT panel holder in hypersonic flow

The problem of the blunt leading edge of the 8' HTT panel holder in a supersonic flow fieldwith free-stream Mach number M",,:::: 6.57, y:::: 1.38, and 0° angle of attack was solved toobtain the steady-state solution. This problem has also been studied by Bey et al. [6].

A coarse mesh solution is indicated in Fig. 21 and an adaptively refined/unrefined mesh andsolution, obtained for a = 0.05 and {3 ::::0.15, are shown in Fig. 22. A distorted mesh andcorresponding density map are indicatec.' in Fig. 23. In this particular problem, neither theh-melhod nor the r-method gave particularly good results, as a poor approximation of thesolution between the shock and blunt body results from spurious oscillations in the basictim~marching algorithm. In the case of mesh adaptation using redistribution, the solutionactually diverges afteI: four passes through the adaptive scheme due to the badly graded(hourglassed) mesh produced from the oscillations of the adaptive scheme downstream of theshock.

6.6. Transient adaptive ~olution for supersonic flow over a 20° ramp.

In all the examples presented above, a time-accurate time stepping scheme is used, but theadaptive scheme was not used stepwise for the transient solution since our primary interestwas to increase accuracy in the steady-state solution. The adaptive method used to tracktransient fronts is described as follows:

Step 1. Choose a structured mesh with the finest mesh size to be allowed in the calculationto be the initial mesh. This is done to avoid large variations of the time step during the timestepping.

l.."

,.' .,. ..... ~ ..

44

I

(a) (b)

Fig. 18. Supersonic flow over a 20° ramp. (a) Mesh and (b) density contours obtained with one level of refinement(a = 0.20, fJ = 0.50).

(a) (b)

Fig. 19. Supersonic flow over a 20° ramp. (a) Mesh and (b) density contours obtained with two levels of refinement(a = 0.20, fJ = 0.50).

U>IJ\IJ\

II

Fig. 20. Supersonic flow over a 20" ramp. (a) Mesh and (b) density contours obtained after 10 applications of themesh redistribution algorithm.

:--~0l}:::~l:l:--.):..§-~~..,.~3::'C''"S·'<:Z::;.05:00

~'"D:....5.:~

':::1:..0~

~CJ\0-

(b)

--~~--,~-...........-•

(b)

(a)

(a)

".

Fig. 21. Blunt leading ~~ge In hypersonic flow. (a) Initial mesh and (b) density contours.

'- .l... _i ....... ~

-.-

.J.T. Oden et al., Adaptive FEM for inviscid compressible flow 357

"

(a)

Fig. 22. Blunt leading edge in hypersonic flow field. (n) Mesh and (b) density contours obtained with one lcvel ofrcfinement (a = 0.05, f3 :; 0.15).

.,'II

(a)

Fig, 23. Blunt leading edge in hypersonic flow. (a) Mesh and (b) density contours obtained after 4 applications ofthe mesh redistribution.


Fig. 24. Transient supersonic flow over a 200 ramp. Initial mesh.

)..

Step 2. Every N time stcps (N = 50 in the prcsent problem) go through the refinemcntiunrefinemcnt process (only unrefincment after the first N time steps).

The above adaptive stratcgy was employed to solve the problem of a 20° ramp which issuddcnly introduced in a supersonic flow field with M"" = 3.0, 'Y = 1.40. The solution wasintegrated to a steady state, and it is demonstrated that the mesh adapts to the shock front asthe shock front moves from its initial to its stcady-state position. .

The initial coarse mesh is shown in Fig. 24 and the evolution of a refined/unrefined meshfor various time intervals is illustrated in Figs. 25-29. The refinement parameters used werea = 0.5 and f3 = 0.25, and a total of 250 time steps were used to track the solution froID itsinitial to the final steady state. The final steady result is ;;imilar to that obtained earlier andshown in Figs. 18 and 19.

Acknowledgment

The bulk of the work reported here is a component of a program for the development offinite element methodologies for aerothermal heating analysis and related computationalfluid/structural dynamics problems supported by the NASA Langley Research Center. Thesupport of Mr. A. Weiting and Dr. G. Ohlsen of NASA is gratefully acknowledged. Much ofthe preliminary work on this effort was done in collaboration with Lawrence W. Spradley ofthe Lockheed Missile and Space Co., Huntsville Research and Engineering Center. Theadaptive data structures discussed in Section 5 were developed in a study of adaptive finiteelement._ m.etho~~ .i!!. _~Ql_id,!TIechanicssponsored by the Office of Naval Research underContract N00014-84-K-0409. The support of some of the effort of P. Devloo and T.Strouboulis through Contract NAS8-36647 with the NASA Marshall Space Flight Center isalso acknowledged.

·- .

(a) (b)

" --,..

JI

(a)

Fig. 25. Transient transonic flow over a 20° ramp. (a) Mesh and (b) density contours after 50 time steps (a = 0.05,J3 = 0.25).

(b)

Fig. 26. Transient supersonic flow over a 20° ramp. (a) Mesh and (b) density contours after 100 time steps{a = 0.05, {3 = 0.25).

.~

:--~o~;:~;,:-).

~~.'"~~ti'...S·'=!:j'l"l~l"lC

~~~5:<b~~

wVI\0

".

(a)

(b)

(b)

Fig. 27. Transient supe~nic flow over a 20° ramp. (a) Mesh and (b) density contours after 150 time steps(a = 0.05, P = 0.25).

(a)

Fig. 28. Transient supersonic flow over a 20° ramp. (a) Mesh and (b) density contours after 200 time steps(a = 0.05, P = 0.25) .

'.'.

to»g

I-,

III~

~~0~:::~~:-~f}~~,

'"'"~E::

'0>...5''"1:;'

'"~'"0::;~~t:l5:~~'"

II

.-. _J1' ...- - ... ..It -..,..,

J.T. Oden et al.• Adaptive FEM for inviscid compressible flow 361

---- --

(a)

IFig. 29. Transient supersonic flow over a 20" ramp. (a) Mesh and (b) density contours after convergence to steadystate (a = 0.05, {3 = 0.25). .

References

[1] D.A. Anderson, Adaptive mesh schemes based on grid speeds, in: K. Ghia and U. Ghia. eds .• Advances inGrid Generation FED-5 (ASME, New York, 1983) 311-318. . -

[2] I. Babuska, O.C. Zienkiewic, J.P. de S.R. Gago and A. de Oliveira, eds., Adaptive Methods and ErrorRefinement in Finite Element Computation (Wiley, London, 1986).

[3] A.J. Baker and J.W. Kim, Analyses on a Taylor weak-statement algorithm for hyperbolic conservation laws,Tech. Rep!. CFDL/86-1, Computer Fluid Dynamics Laboratory, University of Tennessee, Knoxville, TN,1986.

(4} M. Berger and A. Jameson, Automatic adaptive grid refinements_ for .the,Euler equations, MAE Rep!. 1633,Princeton University, Princeton, NJ, 1983.

[5} MJ. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations, J. Compu!.Phys. 53 (3) (1984) 484-512.

[6} K.S. Bey, E.A. Thornton, P. Dechaumpai and R. Ramakrishnan, A new finite element approach forprediction of aerothermal loads-progress in inviscid now computations, AIAA Paper No. 85-1533-CP.

[7] S.Z. Burstein. Finite difference calculations for hydrodynamic flows containing discontinuities, J. Comput.Phys. 2 (1967) 198-222.

[8] L. Demkowicz. J.T. Oden and T. Strouboulis, Adaptive finite elements for flow problems with movingboundaries. Part I: Variational principles and a posteriori estimates, Compu!. Meths. Appl. Mech. Engrg. 46(1984) 217-251.

....-


I

~:I.J

I

L-1\'I •

[9] L. Demkowicz, J.T. aden and T. Strouboulis, An adaptive p-version finite element method for transient flowproblems with moving boundaries, in: R.H. Gallagher, G.F. Carey, J.T. Oden and O.C. Zienkiewicz, cds.,Finite J~lements in Auids VI (Wiley, London, 1985) 291-305.

[10] L. Demkowicz and J.T. Oden, An adaptive characteristic Petrov-Galerkin finite element method forconvection dominated linear and non-linear parabolic problems in one space variable, TICOM Rept. 85-3,The University of Texas at Austin, Austin, TX, 1985.

[11] L. Demkowicz and J.T. Oden, An adaptive characteristic Petrov-Galerkin finte element method forconvection-dominated linear and nonlinear parabolic problems in two space variables, Comput. Meths. Appl.Mech. Engrg. 55 (1986) 63-87.

(12] A.R. Diaz, N. Kikuchi and J.E. Taylor, A method of grid optimization for finite element methods, Comput.Meths. Appl. Mech. Engrg. 41 (1983) 29-45.

[13] J. Donea, A Taylor Galerkin method for convec~ive transport problems, Internat. J. Numer. Meths. Engrg.20 (1984) 101-119.

(14] A. Lapidus, A detached shock calculation by second-order finite differences, J. Comput. Phys. 2 (1967)154-177.

(15] R. LOhner, K. Morgan and O.c. Zienkiewicz, Adaptive grid refinement for the Euler and compressibleNavier-Stokes equations, in: 1. Babuska, O.C. Zienkiewicz, J.P. de S.R. Gago and A. de Oliveira, eds.,Adaptive Methods and Error Refinement in Finite Element Computation (Wiley, London, 1986) 281-297.

(16] R. Lohner, K. Morgan and O.C. Zienkiewicz, An adaptive finite element procedure for compressible highspeed flows, Comput. Meths. Appl. Mech. Engrg. 51 (1985) 441-465.

(17] J.T. Oden and G.F. Carey, Finite Elements: Mathematical Aspects, Vol. IV (Prentice-Hall, Englewood Cliffs,NJ, 1981).

[18] J.T. aden, L. Demkowicz, T. Strouboulis and P. Devloo, Adaptive methods for problems in solid and fluidmechanics, in: 1. Babuska, O.C. Zienkiewicz, J.P. de S.R. Gago and A. de Oliveira, eds., Adaptive Methodsand Error Refinement in Finite Element Computation (Wiley, London, 1986).

(19] T. Strouboulis, J.T. Oden and L. Demkowicz, A-posteriori error estimates for some Galerkin space-timeapproximations of the unsteady Navier-Stokes equations, TICOM Rep!. 85-5, The University of Texas atAustin, Austin, TX.

[20] J. von Neumann and R.D. Richtmeyer, A method for numerical calculation of hydrodynamic shocks, J. Appl.Phys. 21 (1950) 232-236.

[21] O.C. Zienkiewicz, R. LOhner and K. Morgan, High speed inviscid compressible flow by the finite elementmethod, in: J.R. Whiteman, ed., The Mathematics of Finite Elements and Applications V-MAFELAP 1984(Academic Press, London, 1985).

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