:omputational Mechanics 23 (1999) 151-157 @ Springer-Verlag 1999

S. Mittal, S. Aliabadi, T. Tezduyar

Abstract Recently, the Enhanced-Discretization Interface-Capturing Technique (EDICT) was introduced for simu-lation of unsteady flow problems with interfaces such astwo-fluid and free-surface flows. The EDICT yields in-creased accuracy in representing the interface. Here weextend the EDICT to simulation of unsteady viscouscompressible flows with boundary/shear layers and shock!expansion waves. The purpose is to increase the accuracyin selected regions of the computational domain. An errorindicator is used to identify these regions that need en-hanced discretization. Stabilized finite-element formula-tions are employed to solve the Navier-Stokes equations intheir conservation law form. The finite element functionscorresponding to enhanced discretization are designed tohave two components, with each component coming froma different level of mesh refinement over the same com-putational domain. The primary component comes from abase mesh. A subset of the elements in this base mesh areidentified for enhanced discretization by utilizing the errorindicator. A secondary, more refined, mesh is constructedby patching together the second-level meshes generatedover this subset of elements, and the second component ofthe functions comes from this mesh. The subset of ele-ments in the base mesh that form the secondary mesh maychange from one time level to other depending on thedistribution of the error in the computations.

S. MittalDepartment of Aerospace Engineering,Indian Institute of Technology, Kanpur, UP 208 016, India

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Correspondence to: T. Tezduyar

This work is sponsored by the Army High Performance Com-puting Research Center under the auspices of the Department ofthe Army, Army Research Laboratory cooperative agreementnumber DAAH04-95-2-0003/contract number DAAH04-95-C-00-08. The content does not necessarily reflect the position or thepolicy of the Government, and no official endorsement should beinferred. CRAY time was provided in part by the Minnesota Su-percomputer Institute.

S. AliabadiDepartment of Engineering and Army HPC Research Center,Clark Atlanta University, 223 James P. Brawley Dr. S.W.,Atlanta. GA 30314

T. Tezduya'Mechanical Engineering and Materials Science,Army HPC Researrh rpntpr ~;rp TTn;upr~itv-M" ~?16100 Main Street,Hml"tnn, TX 7700'-

Using a parallel implementation of this EDICT -basedmethod, we apply it to test problems with shocks andboundary layers, and demonstrate that this method can beused very effectively to increase the accuracy of the basefinite element formulation.

1IntroductionRecently, Tezduyar et al. [1] introduced the Enhanced-Discretization Interface-Capturing Technique (EDICT) forsimulation of unsteady flow problems with interfaces suchas two-fluid and free-surface flows. The starting point forthe EDICT is the volume of fluid (VOF) method [2]. In theEDICT, the Navier-Stokes equations are solved over a non-moving mesh and an interface function, with two distinctvalues that serves as a marker identifying the two fluids, istransported with a time-dependent advection equation. Toincrease the accuracy in representing the interface, func-tion spaces corresponding to enhanced discretization areused at and near the interface.

In this article we extend the EDICT to unsteady com-pressible flows with shock/expansion waves, boundary/shear layers, and their interactions. Our target is to in-crease the accuracy in selected regions of the computa-tional domain. These regions are identified by an errorindicator. The Navier-Stokes equations of compressibleflows in their conservation law form are solved using astabilized finite element formulation based on conserva-tion variables. The streamline-upwind/Petrov-Galerkin(SUPG) stabilization technique is employed to stabilize thecomputations against potential numerical oscillations inadvection dominated flows [3, 4, 5]. A shock-capturingterm is added to the formulation to provide stability to thecomputations in the presence of discontinuities and largegradients in the flow. The interaction between the shock-capturing term and the viscous effects and SUPG term wasaddressed earlier in [6]. With the error indicator employedto provide an estimate of the error in the computations,the regions identified for enhanced discretization usuallycoincide with those that are associated with large gradientsin the flow variables. The finite element functions corre-sponding to enhanced discretization are designed to havetwo components. Each component comes from a differentlevel of mesh refinement over the same computationaldomain. The first component comes from a base mesh.A subset of the elements in this base mesh are identifiedbased on the error estimated by the error indicator.A second mesh is constructed by patching together thesecond-level meshes generated over this subset of ele-

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E~:l E~:l (fn N;kU,ldO) 2

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In the variational formulation given by Eq. (20), the firsttwo terms and the right-hand-side constitute the Galerkinformulation of the problem. The first series of element-level integrals in Eq. (20) are the SUPG stabilization termsadded to the variational formulation to stabilize thecomputations against node-to-node oscillations in theadvection-dominated range. The second series of elementlevel integrals in the formulation are the shock-capturingterms that stabilize the computations in the presence ofsharp gradients. The stabilization coefficients c5 and 'r are

IVlt~ IIA-I '+IV2t~ IIA~:'L 0 "0 oJ

where c is the wave speed, h is the element length, hk arethe components of the Jacobian transformation matrixfrom physical to the local coordinates, and AOI is the in-verse of Reimannian metric tensor related to the trans-formation between the conservation and entropy variables[12]. As shown in Eq. (21), 'r" is subtracted from 'ra toaccount for the shock-capturing.

Time discretization of the variational formulation givenby Eq. (20) is achieved with the generalized trapezoidalrule. For unsteady computations, temporally we employ asecond-order accurate procedure.

5Construction of function spaces - enhanced discretizationWe begin with a base mesh which we call Mesh-I. The setof elements and nodal points in Mesh-I are EI and 1]1,respectively. At a time level n, a subset (EI)~ from this setof. elements is identified, based on an error indicator, re-quiring enhanced discretization for higher accuracy. Amore refined mesh, Mesh-2, is constructed over the set ofelements (EI)~ by subdividing each of these elements into

Fig. 3. Shock-reflection prob-lem: Mesh-2 shown on top ofMesh-l and density att=O.5,1.O,1.5,2.0,2.5,3.0

and shock/expansion waves. The Navier-Stokes equationsare solved using stabilized finite element formulations thatare known to possess good stability and accuracy prop-erties. To further increase the accuracy of the computa-tions an error indicator is employed to identify the regionsin the computational domain that need enhanced dis-cretization. The finite element functions corresponding toenhanced discretization are designed to have two com-ponents, with each component coming from a differentlevel of mesh refinement over the same computationaldomain. The first component of the functions for the flowvariables comes from the base mesh. A subset of the ele-ments in this mesh, for which the error indicator showsthe error larger than a certain threshold value, are iden-tified for enhanced discretization. A more refined mesh isconstructed by patching together the second-level meshesgenerated over this subset of elements. This subset of el-

, ement may change from one time level to other dependingon the error distribution in the computations. The secondcomponent of the functions for the flow variables comes

~ from this more refined mesh. However, the second-levelmesh is not redefined at every time level, but frequentlyenough to keep the zones, with error larger than a pre-

Fig. 4. Mach 3, Re 105 flow past a circular cylinder: Mesh-l (2396 determined threshold, covered with these higher levelnodes and 4694 triangular elements) meshes. The methodology has been implemented on a

shared-memory parallel computer and results from test7 computations for flows involving shock waves and boun-Concluding remarks dary layers demonstrate that the EDICT can be utilized toIn this paper we have extended the EDICT to unsteady increase accuracy of solutions that involve crisp shocksviscous compressible flows with boundary/shear layers and layers.

,...,<.

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~,\~

'I< .../I

[7]

[8]

[9]

[10]

.l11J

[12]

Lohner R (1987) An adaptive finite element scheme fortransient problems in CFD. Compo Meth. Appl. Mech. Eng.61:323-338Oden JT, Demkowicz L, Racowicz W, Wastermann TA(1990) A posteriori error analysis in finite elements: Theelement residual method for symmetrizable problems withapplications to compressible Euler and Navier-Stokesequations. Compo Meth. Appl. Mech. Eng. 82:183-203Johan Z, Hughes TJR, Shakib F (1991) A globally con-vergent matrix-free algorithm for implicit time-marchingschemes arising in finite element analysis in fluids. CompoMeth. Appl. Mech. Eng. 87:281-304Aliabadi SK, Tezduyar TE (1995) Parallel fluid dynamicscomputations in aerospace applications. Int. J. Num. Meth.Fluids 21: 783-805Mittal S (1997) Finite element computation of unsteadyviscous compressible flows past stationary airfoils. Toappear in Computat. Mech.Hughes TJR, Mallet M (1986) A new finite element for-mulation for computational fluid dynamics: IV. A dis-continuity-capturing operator for multidimensionaladvective-diffusive systems. Compo Meth. Appl. Mech,Eng. 58:329-339

References[1] Tezduyar TE, Aliabadi S, Behr M (1997) Enhanced-dis-

cretization interface-capturing technique. In Proceedingsof the ISAC' 97 High Performance Computing on Multi-phase Flows, Tokyo, Japan

[2] Hirt CW, Nichols BD (1981) Volume of fluid (VOF)method for the dynamics of free boundaries. J. Comput.Physics 39:201-225

[3] Tezduyar TE, Hughes TJR (1983) Finite element formula-tions for convection dominated flows with particular em-phasis on the compressible Euler equations. InProceedings of AIAA 21st Aerospace Sciences Meeting,AIAA Paper 83-0125, Reno, Nevada

[4] Hughes TJR, Tezduyar TE (1984) Finite element methodsfor first-order hyperbolic systems with particular emphasison the compressible Euler equations. Compo Meth. Appl.Mech. Eng. 45:217-284

[5] Mittal S (1997) Finite element computation of unsteadyviscous compressible flows. To appear in Compo Meth.Appl. Mech. Eng.

[6] Aliabadi S, Tezduyar TE (1993) Space-time finite elementcomputation of compressible flows involving movingboundaries and interfaces. Compo Meth. Appl. Mech. Eng.107:209-223

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