Chapter 17 Finite Element Methods for Fluid Dynamics with Moving Boundaries and · PDF...

Chapter 17

Finite Element Methods for Fluid Dynamics withMoving Boundaries and Interfaces

Tayfun E. Tezduyar

in

Encyclopedia Of Computational Mechanics

Editors Erwin Stein, Rene de Borst and Thomas J.R. Hughes

Volume 3 Fluids

pp. 545–577

John Wiley & Sons, Ltd, Chichester, 2004

Chapter 17Finite Element Methods for Fluid Dynamics withMoving Boundaries and Interfaces

Tayfun E. TezduyarRice University, Houston, TX, USA

1 Introduction 545

2 Governing Equations 547

3 Stabilized Formulations 548

4 DSD/SST Finite Element Formulation 5495 Calculation of the Stabilization Parameters

for Incompressible Flows 5496 Discontinuity-Capturing Directional

Dissipation (DCDD) 5507 Calculation of the Stabilization Parameters

for Compressible Flows and Shock-Capturing 551

8 Mesh Update Methods 553

9 Shear–Slip Mesh Update Method (SSMUM) 55510 DSD/SST Formulation for Fluid–Object

Interactions in Spatially Periodic Flows 555

11 Space–Time Contact Technique (STCT) 55712 Fluid–Object Interactions Subcomputation

Technique (FOIST) 55813 Enhanced-Discretization Interface-Capturing

Technique (EDICT) 560

14 Extensions and Offshoots of EDICT 56015 Mixed Interface-Tracking/Interface-Capturing

Technique (MITICT) 56116 Edge-Tracked Interface Locator Technique

(ETILT) 56217 Line-Tracked Interface Update Technique

(LTIUT) 564

Encyclopedia of Computational Mechanics, Edited by ErwinStein, Rene de Borst and Thomas J.R. Hughes. Volume 3: Fluids. 2004 John Wiley & Sons, Ltd. ISBN: 0-470-84699-2.

18 Iterative Solution Methods 565

19 Enhanced Solution Techniques 56620 Mixed Element-Matrix-Based/Element-

Vector-Based Computation Technique(MMVCT) 567

21 Enhanced-Discretization Successive UpdateMethod (EDSUM) 568

22 Examples of Flow Simulations 570

23 Concluding Remarks 574

24 Related Chapters 574

Acknowledgment 574

References 574

1 INTRODUCTION

In computation of flow problems with moving bound-aries and interfaces, depending on the complexity of theinterface and other aspects of the problem, we can usean interface-tracking or interface-capturing technique. Aninterface-tracking technique requires meshes that ‘track’the interfaces. The mesh needs to be updated as the flowevolves. In an interface-capturing technique for two-fluidflows, the computations are based on fixed spatial domains,where an interface function, marking the location of theinterface, needs to be computed to ‘capture’ the inter-face. The interface is captured within the resolution ofthe finite element mesh covering the area where the inter-face is. This approach can be seen as a special case ofinterface representation techniques in which the interfaceis somehow represented over a nonmoving fluid mesh,

546 Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces

the main point being that the fluid mesh does not moveto track the interfaces. A consequence of the mesh notmoving to track the interface is that for fluid–solid inter-faces, independent of how well the interface geometry isrepresented, the resolution of the boundary layer will belimited by the resolution of the fluid mesh where the inter-face is.

The interface-tracking and interface-capturing tech-niques we have developed in recent years (see Tez-duyar, 1991, 2001b,c, 2002b,c,d, 2003) are based onstabilized formulations. The stabilized methods are thestreamline-upwind/Petrov-Galerkin (SUPG) (Hughes andBrooks, 1979; Brooks and Hughes, 1982; Tezduyarand Hughes, 1982, 1983) and pressure-stabilizing/Petrov-Galerkin (PSPG) (Tezduyar, 1991) formulations. An earlierversion of the pressure-stabilizing formulation for Stokesflows was reported in Hughes, Franca and Balestra (1986).These stabilized formulations prevent numerical oscilla-tions and other instabilities in solving problems with highReynolds and/or Mach numbers and shocks and strongboundary layers, as well as when using equal-order inter-polation functions for velocity and pressure and otherunknowns. Furthermore, this class of stabilized formula-tions substantially improves the convergence rate in iter-ative solution of the large, coupled nonlinear equationsystem that needs to be solved at every time step ofa flow computation. Such nonlinear systems are typi-cally solved with the Newton–Raphson method, whichinvolves, at its every iteration step, solution of a large,coupled linear equation system. It is in iterative solu-tion of such linear equation systems that using a goodstabilized method makes a substantial difference in con-vergence, and this was pointed out in Tezduyar et al.(1993).

In these stabilized formulations, judicious selection ofthe stabilization parameter, which is almost always knownas ‘τ’, plays an important role in determining the accuracyof the formulation. This stabilization parameter involvesa measure of the local length scale (also known as ‘ele-ment length’) and other parameters such as the localReynolds and Courant numbers. Various element lengthsand τs were proposed starting with those in Hughes andBrooks (1979), Tezduyar and Hughes (1982), Tezduyarand Hughes (1983), followed by the one introduced inTezduyar and Park (1986), and those proposed in the sub-sequently reported SUPG and PSPG methods. A numberof τs, dependent upon spatial and temporal discretiza-tions, were introduced and tested in Tezduyar and Ganjoo(1986). More recently, τs which are applicable to higher-order elements were proposed in Franca, Frey and Hughes(1992). Ways to calculate τs from the element-level matri-ces and vectors were first introduced in Tezduyar and

Osawa (2000). These new definitions are expressed interms of the ratios of the norms of the relevant matri-ces or vectors. They take into account the local lengthscales, advection field, and the element-level Reynoldsnumber. On the basis of these definitions, a τ can be cal-culated for each element, or even for each element nodeor degree of freedom or element equation. Certain vari-ations and complements of these new τs were describedin Tezduyar (2001a, 2002a,e,f, 2003). In later sections,we will describe, for the semidiscrete and space–timeformulations of the advection–diffusion equation and theNavier–Stokes equations of incompressible flows, some ofthese new ways of calculating the stabilization parame-ters. These stabilization parameters are based on the locallength scales for the advection- and diffusion-dominatedlimits.

The deforming-spatial-domain/stabilized space–time(DSD/SST) formulation (Tezduyar, 1991), developed formoving boundaries and interfaces, is an interface-trackingtechnique, where the finite element formulation of the prob-lem is written over its space–time domain. At each timestep, the locations of the interfaces are calculated as partof the overall solution. As the spatial domain occupiedby the fluid changes its shape in time, the mesh needsto be updated. In general, this is accomplished by mov-ing the mesh with the motion of the nodes governed by theequations of elasticity, and full or partial remeshing (i.e.generating a new set of elements, and sometimes also anew set of nodes) as needed. It needs to be pointed out thatthe stabilized space–time formulations were used earlierby other researchers to solve problems with fixed spatialdomains; see, for example, Hughes and Hulbert (1988).

In computation of two-fluid flows (we mean this cate-gory to include free-surface flows) with interface-trackingtechniques, sometimes the interface might be too com-plex or unsteady to track while keeping the frequencyof remeshing at an acceptable level. Not being able toreduce the frequency of remeshing in 3D might introduceoverwhelming mesh generation and projection costs, mak-ing the computations with the interface-tracking techniqueno longer feasible. In such cases, interface-capturing tech-niques, which do not normally require costly mesh updatesteps, could be used with the understanding that the inter-face will not be represented as accurately as we wouldhave with an interface-tracking technique. Because they donot require mesh update, the interface-capturing techniquesare more flexible than the interface-tracking techniques.However, for comparable levels of spatial discretization,interface-capturing methods yield less accurate represen-tation of the interface. These methods can be used aspractical alternatives in carrying out the simulations whencompromising the accurate representation of the interfaces

Finite Element Methods for Fluid Dynamics with Moving Boundaries and Interfaces 547

becomes less of a concern than facing major difficultiesin updating the mesh to track such interfaces. The needto increase the accuracy of our interface-capturing tech-niques without adding a major computational cost lead usto seeking techniques with a different kind of ‘tracking’.The enhanced-discretization interface-capturing technique(EDICT) was first introduced in Tezduyar, Aliabadi andBehr (1997, 1998) to increase accuracy in representing aninterface.

In more recent years, research efforts in flows withmoving boundaries and interfaces intensified significantly(see e.g. Farhat, Lesoinne and Maman, 1995; Lesoinneand Farhat, 1996; Lohner, Yang and Baum, 1996; Steinet al., 1998; Garcia, Onate and Sierra, 1998; Onate et al.,1998; Onate and Garcia, 1999; Cruchaga and Onate, 1999).A discussion on the geometric conservation properties ofvarious methods developed for moving boundaries andinterfaces can be found in Lesoinne and Farhat (1996),which includes a conclusion that the space–time formu-lation leads to solution techniques that inherently satisfythe geometric conservation law.

In Section 2, we describe the governing equations, andin Section 3, we summarize the stabilized semidiscrete for-mulations for the advection–diffusion equation and theNavier–Stokes equations of incompressible flows. TheDSD/SST formulation is briefly described in Section 4.Some of the ways for defining the stabilization param-eters and discontinuity-capturing terms are described inSections 5, 6, and 7. Mesh update techniques to be used inconjunction with the DSD/SST formulation are reviewed inSection 8. The shear–slip mesh update method (SSMUM),developed for objects in fast, linear or rotational relativemotion, is described in Section 9. A special DSD/SSTformulation for fluid–object interactions in spatially peri-odic flows is described in Section 10. In Section 11, wepropose the space–time contact technique (STCT) for com-putation of fluid–solid contact problems based on theDSD/SST formulation. The fluid–object interactions sub-computation technique (FOIST), which was formulated forefficient computation of some special cases of fluid–objectinteractions, is described in Section 12. The EDICT andits extensions and offshoots are described in Sections 13and 14. The extensions and offshoots of EDICT includethe enhanced-discretization mesh refinement technique(EDMRT) and enhanced-discretization space–time tech-nique (EDSTT). In Section 15 we describe the mixedinterface-tracking/interface-capturing technique (MITICT),which we propose for computation of flow problems thatinvolve both interfaces that can be accurately tracked witha moving mesh method, and interfaces that are too com-plex or unsteady to be tracked and therefore require aninterface-capturing technique. The edge-tracked interface

locator technique (ETILT), which was proposed to enableinterface-capturing techniques to have better volume con-servation and yield sharper representation of the interfaces,is described in Section 16. In Section 17, we describe theline-tracked interface update technique (LTIUT), whichwas proposed as a stabilized formulation for the time-integration of the interface update equation in conjunctionwith the DSD/SST formulation. Section 18 consists of asummary of the basic iterative solution techniques thatwere developed for solving the large, coupled nonlinearequation systems that need to be solved at every timestep of a computation. In Section 19, we describe theenhanced-iteration nonlinear solution technique (EINST)and the enhanced-approximation linear solution technique(EALST). These were developed to increase the perfor-mance of the iterative techniques used in solution of thenonlinear and linear equation systems when some parts ofthe computational domain may offer more of a challengefor the iterative method than the others. The mixed element-matrix-based/element-vector-based computation technique(MMVCT), which was proposed to improve the effec-tiveness of the iterative solution techniques for coupledproblems (such as fluid–structure interactions), is describedin Section 20. In Section 21, we describe the enhanced-discretization successive update method (EDSUM), whichwas proposed as an efficient iterative technique for solu-tion of linear equation systems in multiscale computations.The numerical examples are given in Section 22 and theconcluding remarks in Section 23.

2 GOVERNING EQUATIONS

Let t ⊂ IRnsd be the spatial fluid mechanics domain withboundary t at time t ∈ (0, T ), where the subscript t

indicates the time-dependence of the spatial domain. TheNavier–Stokes equations of incompressible flows can bewritten on t and ∀t ∈ (0, T ) as

ρ

(∂u∂t

+ u · ∇∇∇u − f)

− ∇∇∇ · σσσ = 0 (1)

∇∇∇ · u = 0 (2)

where ρ, u, and f are the density, velocity, and the externalforce, respectively. The stress tensor σσσ is defined as

σσσ(p, u) = −pI + 2µεεε(u) (3)

Here p is the pressure, I is the identity tensor, µ = ρν isthe viscosity, ν is the kinematic viscosity, and εεε(u) is thestrain-rate tensor:

εεε(u) = 12 ((∇∇∇u) + (∇∇∇u)T) (4)


The essential and natural boundary conditions for equa-tion (1) are represented as

u = g on (t )g, n · σσσ = h on (t )h (5)

where (t )g and (t )h are complementary subsets of theboundary t , n is the unit normal vector, and g and h aregiven functions. A divergence-free velocity field u0(x) isspecified as the initial condition.

If there are no moving boundaries or interfaces, thespatial domain does not need to change with respect to time,and the subscript t can be dropped from t and t . Thismight be the case even for flows with moving boundariesand interfaces if the formulation is not based on defining thespatial domain to be the part of the space occupied by thefluid(s). For example, fluid–fluid interfaces can be modeledover a fixed spatial domain by assuming that the domain isoccupied by two immiscible fluids, A and B, with densitiesρA and ρB and viscosities µA and µB. In this approach,a free-surface problem can be modeled as a special casewhere Fluid B is irrelevant and assigned a sufficientlylow density. An interface function φ serves as the markeridentifying Fluid A and B with the definition φ = 1 forFluid A and 0 for Fluid B. The interface between the twofluids is approximated to be at φ = 0.5. In this context, ρ

and µ are defined as

ρ = φρA + (1 − φ)ρB, µ = φµA + (1 − φ)µB (6)

The evolution of the interface function φ, and consequentlythe motion of the interface, is governed by a time-dependentadvection equation, written on and ∀t ∈ (0, T ) as

∂φ

∂t+ u · ∇∇∇φ = 0 (7)

To generalize equation (7), let us consider the followingtime-dependent advection–diffusion equation, written on

and ∀t ∈ (0, T ) as

∂φ

∂t+ u · ∇∇∇φ − ∇∇∇ · (ν∇∇∇φ) = 0 (8)

where φ represents the quantity being transported (e.g.temperature, concentration), and ν is the diffusivity, whichis separate from (but in mathematical significance verycomparable to) the ν representing the kinematic viscosity.The essential and natural boundary conditions associatedwith equation (8) are represented as

φ = g on g, n · ν∇∇∇φ = h on h (9)

A function φ0(x) is specified as the initial condition.

3 STABILIZED FORMULATIONS

3.1 Advection–diffusion equation

Let us assume that we have constructed some suitablydefined finite-dimensional trial solution and test functionspaces Sh

φ and V hφ . The stabilized finite element formulation

of equation (8) can then be written as follows: find φh ∈ Shφ

such that ∀wh ∈ V hφ :

∫

wh

(∂φh

∂t+ uh · ∇∇∇φh

)d +

∫

∇∇∇wh · ν∇∇∇φh d

−∫

h

whhh d +nel∑e=1

∫e

τSUPGuh · ∇∇∇wh

(∂φh

∂t+ uh · ∇∇∇φh − ∇∇∇ · (

ν∇∇∇φh))

d = 0 (10)

Here nel is the number of elements, e is the domain forelement e, and τSUPG is the SUPG stabilization parame-ter. For various ways of calculating τSUPG see Tezduyar(2002a,f).

3.2 Navier–Stokes equations of incompressibleflows

Given equations (1)–(2), let us assume that we have somesuitably defined finite-dimensional trial solution and testfunction spaces for velocity and pressure: Sh

u , V hu , Sh

p

and V hp = Sh

p . The stabilized finite element formulationof equations (1)–(2) can then be written as follows: finduh ∈ Sh

u and ph ∈ Shp such that ∀wh ∈ V h

u and qh ∈ V hp :

∫

wh · ρ(

∂uh

∂t+ uh · ∇∇∇uh − fh

)d

+∫

εεε(wh) : σσσ(ph, uh) d

−∫

h

wh · hh d +∫

qh∇∇∇ · uh d

+nel∑e=1

∫e

1

ρ[τSUPGρuh · ∇wh + τPSPG∇qh]

· [Ł(ph, uh) − ρfh

]d

+nel∑e=1

∫e

νLSIC∇∇∇ · whρ∇∇∇ · uh d = 0 (11)

where

Ł(qh, wh) = ρ

(∂wh

∂t+ uh · ∇∇∇wh

)− ∇∇∇ · σσσ(qh, wh) (12)


Here τPSPG and νLSIC are the PSPG and LSIC (least squareson incompressibility constraint) stabilization parameters.For various ways of calculating τPSPG and νLSIC, see Tez-duyar (2002a,f).

4 DSD/SST FINITE ELEMENTFORMULATION

In the DSD/SST method (Tezduyar, 1991), the finite ele-ment formulation of the governing equations is written overa sequence of N space–time slabs Qn, where Qn is theslice of the space–time domain between the time levels tnand tn+1. At each time step, the integrations involved inthe finite element formulation are performed over Qn. Thespace–time finite element interpolation functions are con-tinuous within a space–time slab, but discontinuous fromone space–time slab to another. The notation (·)−n and (·)+ndenotes the function values at tn as approached from belowand above. Each Qn is decomposed into elements Qe

n,where e = 1, 2, . . . , (nel)n. The subscript n used with nel

is for the general case in which the number of space–timeelements may change from one space–time slab to another.The Dirichlet- and Neumann-type boundary conditions areenforced over (Pn)g and (Pn)h, the complementary sub-sets of the lateral boundary of the space–time slab. Thefinite element trial function spaces (Sh

u )n for velocity and(Sh

p )n for pressure, and the test function spaces (V hu )n and

(V hp )n = (Sh

p )n are defined by using, over Qn, first-orderpolynomials in both space and time. The DSD/SST formu-lation is written as follows: given (uh)−n , find uh ∈ (Sh

u )nand ph ∈ (Sh

p )n such that ∀wh ∈ (V hu )n and ∀qh ∈ (V h

p )n:

∫Qn

wh · ρ(

∂uh

∂t+ uh · ∇∇∇uh − fh

)dQ

+∫

Qn

εεε(wh) : σσσ(ph, uh) dQ −∫

(Pn)h

wh · hh dP

+∫

Qn

qh∇∇∇ · uh dQ +∫

n

(wh)+n · ρ ((uh)+n − (uh)−n

)d

+(nel)n∑e=1

∫Qe

n

1

ρ

[τSUPGρ

(∂wh

∂t+ uh · ∇wh

)+ τPSPG∇qh

]


]dQ

+nel∑e=1

∫Qe

n

νLSIC∇∇∇ · whρ∇∇∇ · uh dQ = 0 (13)

This formulation is applied to all space–time slabs Q0,Q1,Q2, . . . , QN−1, starting with (uh)−0 = u0. For an earlier,detailed reference on the DSD/SST formulation (see Tez-duyar, 1991).

Similarly, the DSD/SST formulation of equation (8) canbe written as follows:∫

Qn

wh

(∂φh

∂t+ uh ·∇∇∇φh

)dQ +

∫Qn

∇∇∇wh · ν∇∇∇φh dQ

−∫

(Pn)h

whhh dP +∫

n

(wh)+n((φh)+n − (φh)−n

)d

+(nel)n∑e=1

∫Qe

n

τSUPG

(∂wh

∂t+ uh ·∇∇∇wh

)(

∂φh

∂t+ uh · ∇∇∇φh − ∇∇∇ · (

ν∇∇∇φh))

dQ = 0 (14)

5 CALCULATION OF THESTABILIZATION PARAMETERSFOR INCOMPRESSIBLE FLOWS

Various ways of calculating the stabilization parametersfor incompressible flows were covered earlier in detailin Tezduyar (2002a,f). In this section, we focus on theversions of the stabilization parameters (τs) denoted by thesubscript UGN, namely, the UGN/RGN-based stabilizationparameters. For this purpose, we first define the unit vectorss and r:

s = uh

‖uh‖ (15)

r = ∇∇∇‖uh‖‖∇∇∇‖uh‖ ‖ (16)

where, for the advection–diffusion equation, in equa-tion (16) we replace ‖uh‖ with |φh|.

We define the components of (τSUPG)UGN correspondingto the advection-, transient- and diffusion-dominated limitsas follows:

τSUGN1 =(

nen∑a=1

|uh · ∇∇∇Na|)−1

(17)

τSUGN2 = t

2(18)

τSUGN3 = h2RGN

4ν(19)

where nen is the number of element nodes and Na isthe interpolation function associated with node a, and the‘element length’ hRGN is defined as

hRGN = 2

(nen∑a=1

|r · ∇∇∇Na|)−1

(20)


On the basis of equation (17), we define the ‘elementlength’ hUGN as

hUGN = 2‖uh‖ τSUGN1 (21)

Although writing a direct expression for τSUGN1 as givenby equation (17) was pointed out in Tezduyar (2002a,e,f),the element length definition one obtains by combiningequations (17) and (21) was first introduced (as a directexpression for hUGN) in Tezduyar and Park (1986). Theexpression for hRGN as given by equation (20) was firstintroduced in Tezduyar (2001a). We note that hUGN andhRGN can be viewed as the local length scales correspondingto the advection- and diffusion-dominated limits, respec-tively.

We now define (τSUPG)UGN, (τPSPG)UGN, and (νLSIC)UGNas follows:

(τSUPG)UGN =(

1

τrSUGN1

+ 1

τrSUGN2

+ 1

τrSUGN3

)−1/r

(22)

(τPSPG)UGN = (τSUPG)UGN (23)

(νLSIC)UGN, = (τSUPG)UGN ‖uh‖2(24)

Equation (22) is based on the inverse of (τSUPG)UGN beingdefined as the r-norm of the vector with components1/τSUGN1, 1/τSUGN2, and 1/τSUGN3. We note that the higherthe integer r is, the sharper the switching between τSUGN1,τSUGN2, and τSUGN3 becomes. This ‘r-switch’ was intro-duced in Tezduyar and Osawa (2000). Typically, we setr = 2. The expressions for τSUGN3 and (νLSIC)UGN, givenrespectively by equations (19) and (24), were proposedin Tezduyar (2002a,e,f). We define the ‘SUPG viscosity’νSUPG as

νSUPG = τSUPG‖uh‖2(25)

The space–time versions of τSUGN1, τSUGN2, τSUGN3,(τSUPG)UGN, (τPSPG)UGN, and (νLSIC)UGN, given respectivelyby equations (17), (18), (19), (22), (23), and (24), weredefined in Tezduyar (2002a,e,f) as follows:

τSUGN12 =(

nen∑a=1

∣∣∣∣∂Na

∂t+ uh · ∇∇∇Na

∣∣∣∣)−1

(26)

τSUGN3 = h2RGN

4ν(27)

(τSUPG)UGN =(

1

τrSUGN12

+ 1

τrSUGN3

)−1/r

(28)

(τPSPG)UGN = (τSUPG)UGN (29)

(νLSIC)UGN = (τSUPG)UGN ‖uh‖2(30)

Here, nen is the number of nodes for the space–timeelement, and Na is the space–time interpolation functionassociated with node a.

Remark 1. It was remarked in Tezduyar and Osawa(2000) and Tezduyar (2001a, 2002a,f) that in marching fromtime level n to n + 1, there are advantages in calculatingthe τs from the flow field at time level n. That is,

τ ← τn (31)

where τ is the stabilization parameter to be used in march-ing from time level n to n + 1, and τn is the stabilizationparameter calculated from the flow field at time level n. Oneof the main advantages in doing that, as it was pointed outin Tezduyar (2001a, 2002a,f), is in avoiding another level ofnonlinearity coming from the way τs are defined. In gen-eral, we suggest making τs less dependent on short-termvariations in the flow field. For this purpose, we propose arecursive time-averaging approach in determining the τs tobe used in marching from time level n to n + 1:

τ ← z1τn + z2τn−1 + (1 − z1 − z2)τ (32)

where τn and τn−1 are the stabilization parameters calcu-lated from the flow field at time levels n and n − 1, andthe τ on the right-hand side is the stabilization param-eter that was used in marching from time level n − 1to n. The magnitudes and the number of the ‘averagingparameters’ z1, z2, . . . can be adjusted to create the desiredoutcome in terms of giving more weight to recently calcu-lated τs or making the averaging closer to being a trailingaverage.

6 DISCONTINUITY-CAPTURINGDIRECTIONAL DISSIPATION (DCDD)

As an alternative to the LSIC stabilization, we proposedin Tezduyar (2001a, 2002a,f, 2003) the discontinuity-capturing directional dissipation (DCDD) stabilization. Indescribing the DCDD stabilization, we first define the‘DCDD viscosity’ νDCDD and the DCDD stabilizationparameter τDCDD:

νDCDD = τDCDD‖uh‖2(33)

τDCDD = hDCDD

2ucha

‖∇∇∇‖uh‖ ‖hDCDD

uref(34)


where

hDCDD = hRGN (35)

Here uref is a reference velocity (such as ‖uh‖ at the inflow,or the difference between the estimated maximum and min-imum values of ‖uh‖), and ucha is a characteristic velocity(such as uref or ‖uh‖). We propose to set ucha = uref.

Then the DCDD stabilization is defined as

SDCDD =nel∑e=1

∫e

ρ∇∇∇wh :([

νDCDDrr − κκκCORR

] · ∇∇∇uh)

d

(36)

where κκκCORR was defined in Tezduyar (2001a, 2002a,f,2003) as

κκκCORR = νDCDD(r · s)2ss (37)

As a possible alternative, we propose

κκκCORR = νSUPG(r · s)2rr (38)

As two other possible alternatives, we propose

κκκCORR = switch(νSUPG, νDCDD(r · s)2) ss (39)

κκκCORR = switch(νDCDD, νSUPG(r · s)2) rr (40)

where the ‘switch’ function is defined as the ‘min’ function:

switch (α, β) = min (α, β) (41)

or as the ‘r-switch’ given in Section 5:

switch (α, β) =(

1

αr+ 1

βr

)−1/r

(42)

Remark 2. Remark 1 applies also to the calculation ofνDCDD.

7 CALCULATION OF THESTABILIZATION PARAMETERSFOR COMPRESSIBLE FLOWSAND SHOCK-CAPTURING

The SUPG formulation for compressible flows was firstintroduced, in the context of conservation variables, ina 1982 NASA Technical Report (Tezduyar and Hughes,1982) and a 1983 AIAA paper (Tezduyar and Hughes,1983). Here we will call that formulation ‘(SUPG)82’.After that, several SUPG-like methods for compressibleflows were developed. Taylor–Galerkin method (Donea,1984), for example, is very similar, and under certain

conditions is identical, to one of the versions of(SUPG)82. Another example of the subsequent SUPG-like methods for compressible flows in conservationvariables is the streamline-diffusion method described inJohnson, Navert and Pitkaranta (1984). Later, followingthe work in Tezduyar and Hughes (1982, 1983), the SUPGformulation for compressible flows was recast in entropyvariables and supplemented with a shock-capturing term(Hughes, Franca and Mallet, 1987). It was shown in a 1991ASME paper (LeBeau and Tezduyar, 1991) that (SUPG)82,when supplemented with a similar shock-capturing term, isvery comparable in accuracy to the SUPG formulation thatwas recast in entropy variables. Later, 2D test computationsfor inviscid flows reported in LeBeau et al. (1993) showedthat the SUPG formulation in conservation and entropyvariables yielded indistinguishable results.

Together with (SUPG)82, the 1982 NASA TechnicalReport (Tezduyar and Hughes, 1982) and 1983 AIAA paper(Tezduyar and Hughes, 1983) introduced a set of stabi-lization parameters (τs) to be used in conjunction withthat formulation. That set of τs will be called here as‘τ82’. The stabilized formulation introduced in Tezduyarand Park (1986) for advection–diffusion-reaction equa-tions included a shock-capturing term and a τ definitionthat takes into account the interaction between the shock-capturing term and the SUPG term. That τ definition,for example, precludes ‘compounding’ (i.e. augmentationof the SUPG effect by the shock-capturing effect whenthe advection and shock directions coincide). In the 1991ASME paper (LeBeau and Tezduyar, 1991), the τ usedwith (SUPG)82 is a slightly modified version of τ82, anda shock-capturing parameter, which we will call here ‘δ91’,is embedded in the shock-capturing term used. Subsequentminor modifications of τ82 took into account the interac-tion between the shock-capturing and the (SUPG)82 termsin a fashion similar to how it was done in Tezduyar andPark (1986) for advection–diffusion-reaction equations. Allthese slightly modified versions of τ82 have always beenused with the same δ91, and we categorize them all underthe label ‘τ82−MOD’. We should also point out that theelement-matrix-based τ definitions introduced in Tezdu-yar and Osawa (2000) were recently applied in Catabriga,Coutinho and Tezduyar (2002) to (SUPG)82, supplementedwith the shock-capturing term (with δ91) used in LeBeauand Tezduyar (1991).

In this section, in the context of the (SUPG)82 formu-lation and based on the ideas we discussed in Sections 5and 6, we propose alternative ways of calculating the stabi-lization parameters and defining the shock-capturing terms.For this purpose, we first define the conservation variablesvector as U = (ρ, ρu1, ρu2, ρu3, ρe) (where e is the total


energy per unit volume), associate to it a test vector-function W, define the acoustic speed as c, and define theunit vector j as

j = ∇∇∇ρh

‖ ∇∇∇ρh ‖ (43)

As the first alternative in computing τSUGN1 for each com-ponent of the test vector-function W, we propose to defineτ

ρ

SUGN1, τuSUGN1, and τe

SUGN1 (associated with ρ, ρu, andρe, respectively) by using the expression given by equa-tion (17):

τρ

SUGN1 = τuSUGN1 = τe

SUGN1 =(

nen∑a=1

|uh · ∇∇∇Na|)−1

(44)

As the second alternative, we propose to use the followingdefinition:

τρ

SUGN1 = τuSUGN1 = τe

SUGN1

=(

nen∑a=1

(c |j · ∇∇∇Na| + |uh · ∇∇∇Na|

))−1

(45)

In computing τSUGN2, we propose to use the expressiongiven by equation (18):

τρSUGN2 = τu

SUGN2 = τeSUGN2 = t

2(46)

In computing τSUGN3, we propose to define τuSUGN3 by using

the expression given by equation (19):

τuSUGN3 = h2

RGN

4ν(47)

We propose to define τeSUGN3 as

τeSUGN3 =

(he

RGN

)2

4νe(48)

where νe is the ‘kinematic viscosity’ for the energy equa-tion,

heRGN = 2

(nen∑a=1

|re · ∇∇∇Na|)−1

(49)

re = ∇∇∇θh

‖∇∇∇θh‖ (50)

and θ is the temperature. We define (τρSUPG)UGN,

(τuSUPG)UGN, and (τe

SUPG)UGN by using the ‘r-switch’ givenin Section 5:

(τρ

SUPG)UGN =(

1

(τρ

SUGN1)r

+ 1

(τρ

SUGN2)r

)−1/r

(51)

(τuSUPG)UGN

=(

1

(τuSUGN1)

r+ 1

(τuSUGN2)

r+ 1

(τuSUGN3)

r

)−1/r

(52)

(τeSUPG)UGN

=(

1

(τeSUGN1)

r+ 1

(τeSUGN2)

r+ 1

(τeSUGN3)

r

)−1/r

(53)

In defining the shock-capturing term, we first define the‘shock-capturing viscosity’ νSHOC:

νSHOC = τSHOC

(uint

)2(54)

where

τSHOC = hSHOC

2ucha

(‖∇∇∇ρh‖hSHOC

ρref

)β

(55)

hSHOC = hJGN (56)

hJGN = 2

(nen∑a=1

|j · ∇∇∇Na|)−1

(57)

Here ρref is a reference density (such as ρh at the inflow,or the difference between the estimated maximum andminimum values of ρh), ucha is a characteristic velocity(such as uref or ‖uh‖ or acoustic speed c), and uint is anintrinsic velocity (such as ucha or ‖uh‖ or acoustic speedc). We propose to set uint = ucha = uref. The parameter β

influences the smoothness of the shock front. We set β = 1for smoother shocks and β = 2 for sharper shocks (in returnfor tolerating possible overshoots and undershoots). As acompromise between the β = 1 and β = 2 selections, wepropose the following averaged expression for τSHOC:

τSHOC = 12

[(τSHOC

)β=1 + (

τSHOC

)β=2

](58)

As an alternate way, we also propose to calculate νSHOCby using the following expression:

νSHOC = ∥∥Y−1Z∥∥(

nsd∑i=1

∥∥∥∥Y−1 ∂Uh

∂xi

∥∥∥∥2)(β/2)−1(

hSHOC

2

)β

(59)

where Y is a diagonal scaling matrix constructed from thereference values of the components of U:

Y =

(U1

)ref 0 0 0 0

0(U2

)ref 0 0 0

0 0(U3

)ref 0 0

0 0 0(U4

)ref 0

0 0 0 0(U5

)ref

(60)


Z = ∂Uh

∂t+ Ah

i

∂Uh

∂xi

(61)

OR

Z = Ahi

∂Uh

∂xi

(62)

and we set β = 1 or β = 2. Here

Ai = ∂Fi

∂U(63)

where Fi is the Euler flux vector corresponding to theith spatial dimension. As a variation of the expressiongiven by equation (59), we propose for νSHOC the followingexpression:

νSHOC = ∥∥Y−1Z∥∥(

nsd∑i=1

∥∥∥∥Y−1 ∂Uh

∂xi

∥∥∥∥2)(β/2)−1

∥∥Y−1Uh∥∥1−β

(hSHOC

2

)β

(64)

As a compromise between the β = 1 and β = 2 selections,we propose the following averaged expression for νSHOC:

νSHOC = 12

[(νSHOC

)β=1 + (

νSHOC

)β=2

](65)

We can also calculate, on the basis of equation (59), aseparate νSHOC for each component of the test vector-function W:

(νSHOC

)I = ∣∣(Y−1Z

)I

∣∣( nsd∑i=1

∣∣∣∣(

Y−1 ∂Uh

∂xi

)I

∣∣∣∣2)(β/2)−1

(hSHOC

2

)β

, I = 1, 2, . . . , nsd + 2 (66)

Similarly, a separate νSHOC for each component of W canbe calculated on the basis of equation (64):

(νSHOC

)I = ∣∣(Y−1Z

)I

∣∣( nsd∑i=1

∣∣∣∣(

Y−1 ∂Uh

∂xi

)I

∣∣∣∣2)(β/2)−1

∣∣(Y−1Uh)

I

∣∣1−β(

hSHOC

2

)β

,

I = 1, 2, . . . , nsd + 2 (67)

Given νSHOC, the shock-capturing term is defined as

SSHOC =nel∑e=1

∫e

∇∇∇Wh:(κκκSHOC · ∇∇∇Uh

)d (68)

where κκκSHOC is defined as

κκκSHOC = νSHOCI (69)

As a possible alternative, we propose

κκκSHOC = νSHOC jj (70)

If the option given by equation (66) or equation (67) isexercised, then νSHOC becomes an (nsd + 2) × (nsd + 2)

diagonal matrix, and the matrix κκκSHOC becomes augmentedfrom an nsd × nsd matrix to an (nsd × (nsd + 2)) × ((nsd +2) × nsd) matrix.

In an attempt to preclude compounding, we propose tomodify νSHOC as follows:

νSHOC ← νSHOC

− switch(τSUPG(j · u)2, τSUPG(|j · u| − c)2, νSHOC

)(71)

where the ‘switch’ function is defined as the ‘min’ functionor as the ‘r-switch’ given in Section 5. For viscous flows,the above modification would be made separately with eachof τ

ρSUPG, τu

SUPG, and τeSUPG, and this would result in νSHOC

becoming a diagonal matrix even if the option given byequation (66) or equation (67) is not exercised.

Remark 3. Remark 1 applies also to the calculation ofτ

ρ

SUPG, τuSUPG and τe

SUPG, and νSHOC.

8 MESH UPDATE METHODS

How the mesh should be updated depends on severalfactors, such as the complexity of the interface and overallgeometry, how unsteady the interface is, and how thestarting mesh was generated. In general, the mesh updatecould have two components: moving the mesh for as long asit is possible, and full or partial remeshing (i.e. generating anew set of elements, and sometimes also a new set of nodes)when the element distortion becomes too high. In meshmoving strategies, the only rule the mesh motion needs tofollow is that at the interface the normal velocity of themesh has to match the normal velocity of the fluid. Beyondthat, the mesh can be moved in any way desired, with themain objective being to reduce the frequency of remeshing.In 3D simulations, if the remeshing requires calling anautomatic mesh generator, reducing the cost of automaticmesh generation becomes a major incentive for trying toreduce the frequency of remeshing. Furthermore, when weremesh, we need to project the solution from the old meshto the new one. This introduces projection errors. Also,in 3D, the computing time consumed by this projectionstep is not a trivial one. All these factors constitute a


strong motivation for designing mesh update strategies thatminimize the frequency of remeshing. In some cases wherethe changes in the shape of the computational domainallow it, a special-purpose mesh moving method can beused in conjunction with a special-purpose mesh generator.In such cases, simulations can be carried out withoutcalling an automatic mesh generator and without solvingany additional equations to determine the motion of themesh. One of the earliest examples of that, 2D computationof sloshing in a laterally vibrating container, can be foundin Tezduyar (1991). Extension of that concept to 3D parallelcomputation of sloshing in a vertically vibrating containercan be found in Tezduyar et al. (1993).

In general, however, we use an automatic mesh movingscheme. In the automatic mesh moving technique intro-duced in Tezduyar et al. (1992b), the motion of the internalnodes is determined by solving the equations of elastic-ity. As boundary condition, the motion of the nodes atthe interfaces is specified to match the normal velocity ofthe fluid at the interface. Similar mesh moving techniqueswere used earlier by other researchers (see e.g. Lynch,1982). In Tezduyar et al. (1992b), the mesh deformationis dealt with selectively based on the sizes of the elements,and also the deformation modes in terms of shape andvolume changes. Mesh moving techniques with compara-ble features were later introduced in Masud and Hughes(1997).

In the technique introduced in Tezduyar et al. (1992b),selective treatment of the mesh deformation based on shapeand volume changes is attained by adjusting the relativevalues of the Lame constants of the elasticity equations.The objective would be to stiffen the mesh against shapechanges more than we would stiffen it against volumechanges. Selective treatment based on element sizes, onthe other hand, is attained by altering the way we accountfor the Jacobian of the transformation from the elementdomain to the physical domain. In this case, the objec-tive is to stiffen the smaller elements, which are typ-ically placed near solid surfaces, more than the largerones.

The method described in Tezduyar et al. (1992b) wasrecently augmented in Stein, Tezduyar and Benney (2003,2002) and Stein and Tezduyar (2002) to a more extensivekind by introducing a stiffening power that determines thedegree by which the smaller elements are rendered stifferthan the larger ones. When the stiffening power is 0.0,the method reduces back to an elasticity model with noJacobian-based stiffening. When it is 1.0, the method isidentical to the one introduced in Tezduyar et al. (1992b).In Stein, Tezduyar and Benney (2003, 2002) and Stein andTezduyar (2002) we investigated the optimum values ofthe stiffening power with the objective of reducing the

deformation of the smaller elements. In that context, byvarying the stiffening power, we generated a family of meshmoving techniques, and tested those techniques on fluidmeshes where the structure underwent three different typesof prescribed motion or deformation (translation, rotation,and bending).

In the mesh moving technique introduced in Tezduyaret al. (1992b), the structured layers of elements generatedaround solid objects (to fully control the mesh resolutionnear solid objects and have more accurate representa-tion of the boundary layers) move ‘glued’ to these solidobjects, undergoing a rigid-body motion. No equations aresolved for the motion of the nodes in these layers becausethese nodal motions are not governed by the equations ofelasticity. This results in some cost reduction. But moreimportantly, the user has full control of the mesh resolu-tion in these layers. For early examples of automatic meshmoving combined with structured layers of elements under-going rigid-body motion with solid objects, see Tezduyaret al. (1993). Earlier examples of element layers under-going rigid-body motion, in combination with deformingstructured meshes, can be found in Tezduyar (1991).

In computation of flows with fluid–solid interfaces wherethe solid is deforming, the motion of the fluid mesh nearthe interface cannot be represented by a rigid-body motion.Depending on the deformation mode of the solid, we mayhave to use the automatic mesh moving technique describedabove. In such cases, presence of very thin fluid elementsnear the solid surface becomes a challenge for the auto-matic mesh moving technique. In the solid-extension meshmoving technique (SEMMT) Tezduyar (2001b, 2002c,d,2003), we proposed treating those very thin fluid ele-ments almost like an extension of the solid elements. Inthe SEMMT, in solving the equations of elasticity gov-erning the motion of the fluid nodes, we assign higherrigidity to these thin elements compared to the other fluidelements. Two ways of accomplishing this were proposedin Tezduyar (2001b, 2002c,d, 2003) solving the elasticityequations for the nodes connected to the thin elements sep-arate from the elasticity equations for the other nodes, ortogether. If we solve them separately, for the thin ele-ments, as boundary conditions at the interface with theother elements, we would use traction-free conditions. InStein, Tezduyar and Benney (2002) and Stein and Tezdu-yar (2002), we demonstrated how the SEMMT functionsas part of our mesh update method. We employed bothof the SEMMT options described above. In Stein, Tezdu-yar and Benney (2002) and Stein and Tezduyar (2002), wereferred to the separate solution option as ‘SEMMT – Mul-tiple Domain’ and the unified solution option as ‘SEMMT –Single Domain’.


9 SHEAR–SLIP MESH UPDATEMETHOD (SSMUM)

The SSMUM was first introduced for computation of flowaround two high-speed trains passing each other in a tun-nel (see Tezduyar et al., 1996a). The challenge was toaccurately and efficiently update the meshes used in com-putations based on the DSD/SST formulation and involvingtwo objects in fast, linear relative motion. In such cases, aspecial-purpose mesh moving method without remeshingwould not work, and an automatic mesh generation methodwould require remeshing too frequently to have an effec-tive mesh update technique. The idea behind the SSMUMwas to restrict the mesh moving and remeshing to a thinlayer of elements between the objects in relative motion.The mesh update at each time step can be accomplishedby a ‘shear’ deformation of the elements in this layer, fol-lowed by a ‘slip’ in node connectivities. The slip in thenode connectivities, to an extent, undoes the deformationof the elements and results in elements with better shapesthan those that were shear-deformed. Because the remesh-ing consists of simply redefining the node connectivities,both the projection errors and the mesh generation cost areminimized. The SSMUM can be seen as an alternative toChimera overset grid technique (Benek, Buning and Steger,1985), which requires projection of the solution betweenportions of the two overlapping grids.

1. The meshes outside the shear–slip layers can be struc-tured or unstructured. Those meshes most of the timesimply undergo rigid-body motion, as special cases,with some of them held fixed. One can exercise allthe freedom one would have in generating fixed struc-tured or unstructured meshes, such as generating verythin structured layers of elements around solid objects,combined with unstructured meshes further out.

2. In more general cases, the meshes outside the shear–slip layer can undergo more than just rigid-bodymotion, and can be updated with special-purpose meshmoving method without remeshing and/or an auto-matic mesh moving method with tolerable frequencyof remeshing.

3. Depending on the thickness of the shear–slip layer andthe rate of relative motion, it may not be necessary tohave each shear step followed by a slip step. In suchcases, multiple shear steps can be taken followed bya slip step, and this would reduce the cost associatedwith the slip steps.

4. In general, the shear–slip zone can be made of multiplelayers of elements or even unstructured meshes. Thiswould allow us to reduce the ratio of the number ofslip steps to the number of shear steps. In such cases,

one needs to consider the balance between the decreasein cost because of decreasing the frequency of theslip steps with the increase in cost due to increasedburden of redefining node connectivities in a morecomplex shear–slip zone. If the shear–slip zone ismade of unstructured meshes, then the shear step wouldrequire an automatic mesh moving method. Still, therelative cost associated with this and with redefiningconnectivities in a complex shear–slip zone wouldbe bounded by the size of this shear–slip mesh zonerelative to the total mesh size.

5. Furthermore, when the geometrical and implementa-tional requirements dictate, the shear–slip zones canhave shapes that are spatially nonuniform or temporallyvarying. For example, the shear–slip layer, instead hav-ing a disk shape, can have a conical shape. In such moregeneral cases, the shear–slip process can be handled inways similar to those described in item 4 above.

6. Also when the conditions dictate, the SSMUM canbe implemented in such a way that the mesh in theshear–slip zone is unstructured in both space and time.

The SSMUM was first implemented for computationof incompressible and compressible flows with objects inlinear relative motion, and the results for compressible flowaround two high-speed trains passing each other in a tunnelwere reported in Tezduyar et al. (1996a). In more recentyears, the implementation has been extended to objects inrotational relative motion (see Behr and Tezduyar, 1999;Behr and Tezduyar, 2001), and we describe some of theresults from those computations in the section on numericalexamples.

10 DSD/SST FORMULATION FORFLUID–OBJECT INTERACTIONSIN SPATIALLY PERIODIC FLOWS

In extending the DSD/SST formulation of incompressibleflows to computation of fluid–object interactions in spa-tially periodic flows (see Johnson and Tezduyar, 2001), weconsider a 2D computational domain (see Figure 1). Thisrectangular domain (width × height = L × H ) is assumedto contain N circular objects, with surfaces (inner bound-aries for the fluid) κ, where κ = 1, 2, . . . , N . The outerboundaries are denoted by I , II , III , and IV .

The formulation we would like to derive should beapplicable to uni-periodic (i.e. periodic in one direction),bi-periodic and tri-periodic flows, and where the total volu-metric flow rate in each periodic direction is prescribed.As the first step, the formulation for uni-periodic flows


ΓI

Ω

Γ1

Γ2

ΓII

ΓIII

ΓIV

ΓN

uI uIII

pIIIpI

V•(t )

L

H

y

x

Figure 1. Spatially periodic computational domain.

is derived. Then, this is extended to bi-periodic and tri-periodic cases.

First we reexamine equation (13), and consider only thestress terms and the associated natural boundary conditionsalong I and III :∫

Q

εεε(w) : σσσ(p, u) dQ =∫

PI

w · hI dP +∫

PIII

w · hIII dP

(72)

We have dropped the superscript h and the subscript n

to reduce the notational burden during the derivation. Theflow is assumed to be periodic in the x1 direction, and theprescribed total volumetric flow rate in this direction is V .We define u∗ and w∗ to be the periodic velocity field andthe associated weighting function:

u∗|PI= u∗|PIII

(73)

w∗|PI= w∗|PIII

(74)

Then, equation (72) becomes∫Q

εεε(w∗) : σσσ(p, u∗) dQ = −∫

PIII

(w∗ · e1)J dP (75)

where e1 is the unit vector in the x1 direction. The termJ (t) = pIII − pI represents the pressure jump across thedomain in the x1 direction, and this is an additionalunknown corresponding to the constraint imposed by pre-scribing the total volumetric flow rate in the x1 direction.This constraint, together with the incompressibility con-straint, can be written as∫

Q

q(∇∇∇ · u∗) dQ +∫

I

K

[V −

∫III

(u∗ · e1)d

]dI = 0

(76)

where K is the weighting function corresponding to J , andI represents the time interval (tn, tn+1).

To simplify the implementation, we introduce a spacereduction by decomposing p and q as

p = p∗ + J

Lx1 (77)

q = q∗ + K

Lx1 (78)

where p∗ and q∗ are continuous across the periodic bound-aries, and the discontinuities are represented by J and K .

With this, equation (75) becomes∫Q

εεε(w∗) : σσσ(p∗, u∗) dQ −∫

Q

(∇∇∇ · w∗)J

Lx1 dQ

= −∫

PIII

(w∗ · e1)J dP (79)

By integrating the second term by parts and further alge-braic manipulation, we obtain

∫Q

εεε(w∗) : σσσ(p∗, u∗) dQ +∫

Q

J

L(w∗ · e1) dQ = 0 (80)

Also as a consequence of this space reduction, equa-tion (76) becomes∫

Q

q∗(∇∇∇ · u∗) dQ +∫

Q

K

Lx1(∇∇∇ · u∗) dQ

+∫

I

K

[V −

∫III

(u∗ · e1)d

]dI = 0 (81)

By integrating the second term by parts and further alge-braic manipulation, we obtain for circular or sphericalparticles,

∫Q

q∗(∇∇∇ · u∗) dQ + 1

L∫I

K

[LV +

N∑κ=1

Vκ(U1)κ −∫

(u∗ · e1)d

]dI = 0

(82)

where Vκ and (U1)κ are, respectively, the volume andvelocity of sphere κ.

We can now write the complete DSD/SST formulationfor fluid–object interactions in spatially periodic flows asfollows: given (u∗)−n , find u∗ ∈ (Su∗)n and p∗ ∈ (Sp∗)n suchthat ∀w∗ ∈ (Vu∗)n and ∀q∗ ∈ (Vp∗)n:

∫Q

w∗ · ρ(

∂u∗

∂t+ u∗ · ∇∇∇u∗ − f

)dQ

+∫

Q

εεε(w∗) : σσσ(p∗, u∗) dQ + 1

L

∫Q

(w∗ · J) dQ


−∫

(Pn)h∗w∗ · h∗ dP +

∫Qn

q∗∇∇∇ · u∗ dQ

+ 1

L

∫I

K ·[LV +

N∑κ=1

VκUκ −∫

u∗ d

]dI

+∫

n

(w∗)+n · ρ ((u∗)+n − (u∗)−n

)d

+(nel)n∑e=1

∫Qe

n

1

ρ

[τSUPGρ

(∂w∗

∂t+ u∗ · ∇w∗

)

+ τPSPG∇q∗ + τPVFRKL

]·[ρ

(∂u∗

∂t+ u∗ · ∇∇∇u∗ − f

)

− ∇∇∇ · σσσ(p∗, u∗) + JL

]dQ = 0 (83)

where τPVFR is the stabilization parameter associated withthe constraint equation coming from the prescribed volu-metric flow rate. We can set τPVFR = τPSPG. The superscripth remains dropped so that the notation does not become toocumbersome.

This formulation is applicable to uni-, bi-, or tri-periodicflows, with the understanding that for the flow rate andpressure jump vectors V and J, the components not cor-responding to the directions of periodicity will be set tozero.

11 SPACE–TIME CONTACTTECHNIQUE (STCT)

The STCT is a special method for free-surface flows. TheDSD/SST formulation, combined with the STCT, providesa natural mechanism to handle time-dependent free-surfaceflow problems with contacting and de-contacting surfaces.Formulating this class of problems in the space–timedomain can help us develop more effective methods. TheSTCT was introduced in Tezduyar and Osawa (1999) andTezduyar (2001c).

Let us imagine a one-dimensional problem, shown inFigure 2, where there is a possibility that between thetime levels tn and tn+1 the liquid free-surface on theright might be contacting a wall. Let us first performsome intermediate calculations for this space–time slab,where we update the positions of the free-surface nodesby assuming that the motion of these free-surface nodesare not constrained by the wall. Let us say that thesecalculations show that the new position of Node-1 at tn+1is at location (x1)2, which is beyond the wall. Next, on thewall, we predict the temporal position of Node-3. Node-3represents the contact point in the space–time domain. Wecalculate this predicted value of t3 − t1 from (x1)3 − (x1)1

x1

t

tn +1

tn

Wall

1

2

3

4

Figure 2. STCT concept in one dimension. Fluid contacting wall.

and (u1)+1 . We can now redo the calculations for this

modified space–time slab. Although we will be using apredicted value for t3 − t1, we can see the calculationfor this modified space–time slab as one in which (x1)4becomes a known, and t3 − t1 becomes an unknown. This isas opposed to the intermediate calculations, in which (x1)2was an unknown, and t2 − t1 was known. We completethe calculations for this space–time slab by performing asufficient number of iterations, in which we update t3 − t1and (u1)

+1 , as well as p+

1 , p3, and p−4 .

In extending this to multidimensional cases, we can seethe picture as follows. In the intermediate calculations, wewould have t2 − t1 as known and (x1)2, (x2)2, and (x3)2 asunknowns. In the calculations for the modified space–timeslab, (x1)4 would become a known, and we would havet3 − t1 as unknown, together with (x2)4 and (x3)4. Node-4would symbolically represent more than one node. We willprovide a picture for a simple two-dimensional case later.

Let us now imagine another one-dimensional case, shownin Figure 3, where there is a possibility that between thetime levels tn and tn+1 the liquid on the right mightbe de-contacting the wall. Again, we first perform someintermediate calculations for this space–time slab. In thesecalculations, we assume that Node-1 stays on the walland maps to Node-2 at tn+1. During these intermediatecalculations, we also predict the liquid pressure on thewall, that is, p+

1 and p−2 . Next, on the wall, we predict

the temporal position of Node-3 by calculating t3 whenthe liquid pressure becomes zero (we assume here thatthe viscous stress is negligible). Node-3 represents the de-contact point in the space–time domain. We calculate thepredicted value of t3 − t1 from p+

1 and p−2 by seeking

the zero of the linear function p(t). We can now redothe calculations for the modified space–time slab. At eachiteration of the calculations for this modified space–timeslab, we update (u1)

−4 , (x1)4, p+

1 , p3, and p−4 , as well as


t

1

2

3

4

Wall

x1

tn +1

tn

Figure 3. STCT concept in one dimension. Fluid de-contactingwall.

Wall

1

4a

3

4b

2

t

x1

tn +1

tn

x2

Figure 4. STCT concept in two dimensions. Fluid contactingwall.

t3 − t1 by seeking the zero of the linear function p(t) basedon the updated values of p+

1 and p3.Figure 4 shows a simple 2D case, where we expect

that between the time levels tn and tn+1 Node-1 might becontacting the wall. In principle, the calculation process isvery similar to the 1D contact problem. In the intermediatecalculations, we have t2 − t1 as known and (x1)2, (x2)2,and (x3)2 as unknowns. In the calculations for the modifiedspace–time slab, (x1)4a and (x1)4b become knowns, and wehave t3 − t1 as unknown, together with (x2)4a and (x2)4b.We complete the calculations for this space–time slab byperforming a sufficient number of iterations, in which weupdate t3 − t1, (x2)3, (x2)4a , (x2)4b, (u1)

+1 , (u2)

+1 , (u2)3,

(u2)−4a , and (u2)

−4b, as well as p+

1 , p3, p−4a , and p−

4b.We realize that the 2D computations will require a 3D

mesh generation in the space–time domain, and the 3Dcomputations will require a 4D mesh generation. However,

we also realize that these will be only partial mesh genera-tions, limited to the contact zones.

12 FLUID–OBJECT INTERACTIONSSUBCOMPUTATION TECHNIQUE(FOIST)

The FOIST, which was introduced in Tezduyar (2001c),is an intermediate level approximation between treatingthe objects as point masses and using a fully coupledfluid–object interaction (FOI) formulation. We assume thatthe nature of the fluid–object interactions, such as thescales involved and flow patterns expected, allows us totake into account only a one-way coupling between themain flow field and the motion of the objects. In otherwords, we assume that the main flow field influences themotion of the objects, but the presence and motion of theobjects do not influence the main flow field. With thisassumption, the main flow field can be computed withouttaking into account any of the smaller-scale objects, andat the same time, the dynamics of the objects can bedetermined by carrying out flow subcomputations oversmaller-scale domains around the objects. The boundaryconditions for these domains would be extracted from themain flow field at locations corresponding to the positionsof those boundaries at that instant.

The main flow field would be computed over a fixedmesh. The subcomputation for each object would be carriedout over a fixed mesh, and in a coordinate frame attached tothat object. In the subcomputations, we take into account thegeometry of the objects, and determine the unsteady flowfield around these objects together with the resulting fluiddynamics forces and moments. These forces and momentswould be used, while taking into account the instantaneousvalues of the moments of inertia, to compute the path andorientation of the objects.

Each subcomputation can be carried out as a two-waycoupled fluid–object interaction problem without the needfor mesh moving. This is because the coordinate frame isattached to the object, and the coupling is implementedby updating the boundary conditions as a function of theorientation of the object, rather than by updating the mesh.

Because the FOIST would typically not involve meshmoving or remeshing, by eliminating the cost associatedwith those tasks, it would result in a major reduction in thecomputational cost. The FOIST can be extended to caseswhere the main flow computation or a subcomputation mayrequire mesh update. This could happen for the main flow,for example, when it involves moving objects that are toolarge to be handled with the assumptions underlying FOIST.


For a subcomputation, this could happen, for example,when the object is undergoing deformations.

Here we also describe another level of approximation –one that is beyond FOIST, but still with more realisticassumptions than those used in treating the objects aspoint masses. In this technique, which was introducedin Tezduyar (2001c) and which we call ‘Beyond-FOIST’(B-FOIST), for each object with a different shape, wewould generate a database of fluid dynamics force andmoment coefficients. This database would be generatedfrom computations for a set of Reynolds numbers withina range of interest and a set of ‘basis directions’ for theflow velocity. These would typically be unsteady flowcomputations, but the force and moment coefficients wouldbe determined on the basis of temporal averaging of theresults.

With this database, the path and orientation of the objectswould be calculated without flow subcomputations. At eachinstant of the calculation of the path and orientation of anobject, the force and moment coefficients needed wouldbe estimated by interpolation from the database of thesecoefficients. The coefficients corresponding to the Reynoldsnumber and flow velocity directions at an instant would becalculated by a linear or higher-order interpolation withrespect to the Reynolds number, and by a directionalinterpolation with respect to the flow velocity direction. Thedirectional interpolation would use from the database thebasis directions nearest to the direction of the flow velocity.

How effective B-FOIST would be for a given problemwould depend on the balance between (i) the computa-tional cost saved by not doing the flow subcomputationsand (ii) the loss of some accuracy and the increased costassociated with generating the database. For example, ifthe fluid–object interaction problem involves many objectswith identical shapes, B-FOIST might prove quite effective,because the database generation would not involve objectswith different shapes. In addition, if these objects have oneor more symmetry planes or axes, the cost of database gen-eration would further decrease, gaining additional incentivefor B-FOIST.

The starting point for the basis directions for the flowvelocity would be the six directions identified by (1,0,0),(−1,0,0), (0,1,0), (0,−1,0), (0,0,1), and (0,0,−1). For theflow direction at an instant, the directional interpolationwould use the three nearest of these basis directions. Wenow provide more explanation of how the interpolationswould be carried out.

Let us assume that we have a database of computedforces Fα

β and moments Mαβ corresponding to the

parameter space Uα × eβ, where α = 1, 2, . . . , nU,β = 1, 2, . . . , ne. Here U 1, U 2, . . . denote the nU veloc-ity magnitudes, and e1, e2, . . . denote the unit vectors in

the ne flow directions. We note that Fαβ does not need

to be in the eβ direction. Given a flow velocity magni-tude U and direction s, we need to interpolate from ourdatabase the estimated force and moment F and M. First,we find within the parameter space Uα × eβ the veloc-ity magnitudes UL and UR nearest to U (with the conditionUL ≤ U ≤ UR), and the linearly independent unit vectorseF, eS, and eT ‘closest’ to s. The velocity magnitude inter-polation parameters zL and zR can be determined from thefollowing expressions:

zL = UR − U

UR − UL, zR = U − UL

UR − UL(84)

In some cases, it might be desirable to upgrade the lin-ear interpolation given by equation (84) to a higher-orderinterpolation. The unit vector s can be written as

s = eFsF + eSsS + eTsT (85)

Here sF, sS, and sT are the components of s, which can bedetermined by solving the following equation system:

(eF · eF)sF + (eF · eS)sS + (eF · eT)sT = eF · s

(eS · eF)sF + (eS · eS)sS + (eS · eT)sT = eS · s

(eT · eF)sF + (eT · eS)sS + (eT · eT)sT = eT · s (86)

The ‘closest’ unit vectors would be determined througha search process (with the condition sF, sS, sT ≥ 0). Weinterpolate the estimated force and moment, F and M, byusing the following expressions:

F = zL (sFFL

F + sSFLS + sTFL

T

)+ zR (

sFFRF + sSFR

S + sTFRT

)(87)

M = zL (sFML

F + sSMLS + sTML

T

)+ zR (

sFMRF + sSMR

S + sTMRT

)(88)

To increase the directional resolution of the database,additional basis directions can be defined. The first set ofadditions would be (1,1,1), (−1,1,1), (1,−1,1), (1,1,−1),(−1,−1,1), (−1,1,−1), (1,−1,−1), and (−1,−1,−1). Thesecond set of additional directions would be (1,1,0),(−1,1,0), (1,−1,0), (−1,−1,0), (1,0,1), (−1,0,1), (1,0,−1),(−1,0,−1), (0,1,1), (0,−1,1), (0,1,−1), and (0,−1,−1). Ifan object has one or more symmetry planes or axes, someof the basis directions become redundant and would beeliminated.


13 ENHANCED-DISCRETIZATIONINTERFACE-CAPTURINGTECHNIQUE (EDICT)

In the EDICT (Tezduyar, Aliabadi and Behr, 1997; Tez-duyar, Aliabadi and Behr, 1998), we start with the basicapproach of an interface-capturing technique such as thevolume of fluid (VOF) method (Hirt and Nichols, 1981).The Navier–Stokes equations are solved over a nonmovingmesh together with the time-dependent advection equa-tion governing the evolution of the interface function φ.In writing the stabilized finite element formulation for theEDICT (see Tezduyar, Aliabadi and Behr, 1998), the nota-tion we use here for representing the finite-dimensionalfunction spaces is very similar to the notation we usedin the section where we described the DSD/SST formu-lation. The trial function spaces corresponding to veloc-ity, pressure, and interface function are denoted, respec-tively, by (Sh

u)n, (Shp )n, and (Sh

φ )n. The weighting functionspaces corresponding to the momentum equation, incom-pressibility constraint, and time-dependent advection equa-tion are denoted by (Vh

u)n, (V hp )n (= (Sh

p )n), and (V hφ )n.

The subscript n in this case allows us to use differ-ent spatial discretizations corresponding to different timelevels.

The stabilized formulations of the flow and advectionequations can be written as follows: given uh

n and φhn,

find uhn+1 ∈ (Sh

u)n+1, phn+1 ∈ (Sh

p )n+1, and φhn+1 ∈ (Sh

φ )n+1,such that, ∀wh

n+1 ∈ (Vhu)n+1, ∀qh

n+1 ∈ (V hp )n+1, and ∀ψh

n+1 ∈(V h

φ )n+1:

∫

whn+1 · ρ

(∂uh

∂t+ uh · ∇∇∇uh − fh

)d

+∫

εεε(whn+1) : σσσ(ph, uh) d

−∫

h

whn+1 · hh d +

∫

qhn+1∇∇∇ · uh d

+nel∑e=1

∫e

1

ρ[τSUPGρuh · ∇wh

n+1 + τPSPG∇qhn+1]


]d

+nel∑e=1

∫e

νLSIC∇∇∇ · whn+1ρ∇∇∇ · uh d = 0 (89)

∫

ψhn+1

(∂φh

∂t+ uh ·∇∇∇φh

)d

+nel∑e=1

∫e

τφuh ·∇∇∇ψhn+1

(∂φh

∂t+ uh ·∇∇∇φh

)d = 0 (90)

Here τφ is calculated by applying the definition of τSUPG toequation (90).

To increase the accuracy, we use function spaces corre-sponding to enhanced discretization at and near the inter-face. A subset of the elements in the base mesh, Mesh-1, areidentified as those at and near the interface. A more refinedmesh, Mesh-2, is constructed by patching together second-level meshes generated over each element in this subset.The interpolation functions for velocity and pressure will allhave two components each: one coming from Mesh-1 andthe second one coming from Mesh-2. To further increase theaccuracy, we construct a third-level mesh, Mesh-3, for theinterface function only. The construction of Mesh-3 fromMesh-2 is very similar to the construction of Mesh-2 fromMesh-1. The interpolation functions for the interface func-tion will have three components, each coming from one ofthese three meshes. We redefine the subsets over which webuild Mesh-2 and Mesh-3 not at every time step but withsufficient frequency to keep the interface enveloped in. Weneed to avoid this envelope being too wide or too narrow.

14 EXTENSIONS AND OFFSHOOTSOF EDICT

An offshoot of EDICT was first reported in Mittal, Aliabadiand Tezduyar (1999) for computation of compressible flowswith shocks. This extension is based on redefining the‘interface’ to mean the shock front. In this approach, atand near the shock fronts, we use enhanced discretizationto increase the accuracy in representing those shocks. Later,the EDICT was extended to computation of vortex flows.The results were first reported in Tezduyar et al. (2000,2001). In this case, the definition of the interface is extendedto mean regions where the vorticity magnitude is larger thana specified value.

Here we describe how we extend EDICT to computationof flow problems with boundary layers. In this offshoot,the ‘interface’ means solid surfaces with boundarylayers. In 3D problems with complex geometries andboundary layers, mesh generation poses a serious challenge.This is because accurate resolution of the boundarylayer requires elements that are very thin in thedirection normal to the solid surface. This needs to beaccomplished without having a major increase in meshrefinement also in the tangential directions or creating verydistorted elements. Otherwise, we might be increasing thecomputational cost excessively or decreasing the numericalaccuracy unacceptably. In the Enhanced-DiscretizationMesh Refinement Technique (EDMRT) (Tezduyar, 2001b,2002c,d, 2003), we propose two different ways of usingthe EDICT concept to increase the mesh refinementin the boundary layers in a desirable fashion. In the


EDICT-Clustered-Mesh-2 approach (Tezduyar, 2001b,c,2002c,d, 2003), Mesh-2 is constructed by patching togetherclusters of second-level meshes generated over eachelement of Mesh-1 designated to be one of the ‘boundarylayer elements’. Depending on the type of these boundarylayer elements in Mesh-1, Mesh-2 could be structuredor unstructured, with hexahedral, tetrahedral or triangle-based prismatic elements. In the EDICT-Layered-Mesh-2approach (Tezduyar, 2001b,c, 2002c,d, 2003), a thin butmultilayered and more refined Mesh-2 is ‘laid over’ thesolid surfaces. Depending on the geometric complexity ofthe solid surfaces and depending on whether we preferthe same type elements as those we used in Mesh-1,the elements in Mesh-2 could be hexahedral, tetrahedral,or triangle-based prismatic elements. The EDMRT, as anEDICT-based boundary layer mesh refinement strategy,would allow us accomplish our objective without facingthe implementational difficulties associated with elementshaving a variable number of nodes.

In the EDSTT (Tezduyar, 2001b, 2002b,c,d, 2003), wepropose to use enhanced time-discretization in the con-text of a space–time formulation. The motivation is tohave a flexible way of carrying out time-accurate com-putations of fluid–structure interactions, where we findit necessary to use smaller time steps for the struc-tural dynamics part. There would be two ways of for-mulating EDSTT. In the EDSTT-single-mesh (EDSTT-SM) approach (Tezduyar, 2001b, 2002b,c,d, 2003), a singlespace–time mesh, unstructured both in space and time,would be used to enhance the time-discretization in regionsof the fluid domain near the structure. This, in general,might require a fully unstructured 4D mesh generation. Inthe EDSTT-multi-mesh (EDSTT-MM) approach (Tezduyar,2001b, 2002b,c,d, 2003), multiple space–time meshes, allstructured in time, would be used to enhance the time-discretization in regions of the fluid domain near thestructure. In a way, this would be the space–time ver-sion of the EDMRT. This approach would not requirea fully unstructured 4D mesh generation, and thereforewould not pose a mesh generation difficulty. In gen-eral, EDSTT can be used in time-accurate computationswhere we require smaller time steps in certain parts ofthe fluid domain. For example, where the spatial elementsizes are small, we may need to use small time steps,so that the element Courant number does not becometoo large. In computation of two-fluid interface (or free-surface) flows with the DSD/SST method, time-integrationof the equation governing the evolution of the inter-face (i.e. the interface update equation) may require asmaller time step than the one used for the fluid inte-riors. This requirement might be coming from numerical

stability considerations, when time-integration of the inter-face update equation does not involve any added stabiliza-tion terms. In such cases, time-integration with sub-time-stepping on the interface update equation can be based onthe EDSTT-SM or EDSTT-MM approaches. As an alter-native or complement to these approaches, the sub-time-stepping on the interface update equation can be accom-plished with the interpolated sub-time-stepping technique(ISTST).

In the ISTST, time-integration of the interface updateequation with smaller time steps would be carried out sep-arately from the fluid interiors. The information between thetwo parts would be exchanged by interpolation. The sub-time-stepping sequence for the interface update, togetherwith the interpolations between the interface and fluid inte-riors, would be embedded in the iterative solution techniqueused for the fluid interiors, and would be repeated at everyiteration. The iterative solution technique, which is basedon the Newton–Raphson method, addresses both the non-linear and the coupled nature of the set of equations thatneeds to be solved at each time step of the time-integrationof the fluid interiors. When the ISTST is applied to com-putation of fluid-structure interactions, a separate, ‘inner’Newton–Raphson sequence would be used at each timestep of the sub-time-stepping on the structural dynamicsequations.

15 MIXED INTERFACE-TRACKING/INTERFACE-CAPTURINGTECHNIQUE (MITICT)

In computation of flow problems with fluid–solid inter-faces, an interface-tracking technique, where the fluidmesh moves to track the interface, would allow us tohave full control of the resolution of the fluid mesh inthe boundary layers. With an interface-capturing tech-nique (or an interface representation technique in themore general case), on the other hand, independent ofhow well the interface geometry is represented, the res-olution of the fluid mesh in the boundary layer willbe limited by the resolution of the fluid mesh wherethe interface is. In computations of flow problems withfluid–fluid interfaces where the interface is too complexor unsteady to track while keeping the remeshing fre-quency under control, interface-capturing techniques, withenhanced-discretization as needed, could be used as moreflexible alternatives. Sometimes we may need to solve flowproblems with both fluid–solid interfaces and complex orunsteady fluid–fluid interfaces.


MITICT was introduced in Tezduyar (2001c), primarilyfor fluid–object interactions with multiple fluids. The classof applications we were targeting were fluid–particle–gasinteractions and free-surface flow of fluid–particle mix-tures. However, the MITICT can be applied to a largerclass of problems, where it is more effective to use aninterface-tracking technique to track the solid–fluid inter-faces and an interface-capturing technique to capture thefluid–fluid interfaces. The interface-tracking technique isthe DSD/SST formulation (but could as well be the Arbi-trary Lagrangian-Eulerian method or other moving meshmethods). The interface-capturing technique rides on this,and is based on solving over a moving mesh, in addi-tion to the Navier–Stokes equations, the advection equa-tion governing the time-evolution of the interface function.The additional DSD/SST formulation is for this advectionequation:

∫Qn

ψh

(∂φh

∂t+ uh ·∇∇∇φh

)dQ

+∫

n

(ψh)+n((φh)+n − (φh)−n

)d

+(nel)n∑e=1

∫Qe

n

τφ

(∂ψh

∂t+ uh ·∇∇∇ψh

)(

∂φh

∂t+ uh ·∇∇∇φh

)dQ = 0 (91)

This equation, together with equation (13), constitute amixed interface-tracking/interface-capturing technique thatwould track the solid–fluid interfaces and capture thefluid–fluid interfaces that would be too complex or unsteadyto track with a moving mesh. The interface-capturing partof MITICT can be upgraded to the EDICT formulation fora more accurate representation of the interfaces captured.

The MITICT can also be used for computation offluid–structure interactions with multiple fluids or for flowswith mechanical components moving in a mixture of twofluids. In more general cases, the MITICT can be used forclasses of problems that involve both – interfaces that canbe accurately tracked with a moving mesh method and inter-faces that are too complex or unsteady to be tracked andtherefore require an interface-capturing technique.

16 EDGE-TRACKED INTERFACELOCATOR TECHNIQUE (ETILT)

The ETILT was introduced in Tezduyar (2001c), to have aninterface-capturing technique with better volume conserva-tion properties and sharper representation of the interfaces.

To this end, we first define a second finite-dimensionalrepresentation of the interface function, namely, φhe. Theadded superscript ‘e’ indicates that this is an edge-basedrepresentation. With φhe, interfaces are represented as col-lection of positions along element edges crossed by theinterfaces (i.e. along the ‘interface edges’). Nodes belongto ‘chunks’ of Fluid A or Fluid B. An edge either belongsto a chunk of Fluid A or Fluid B or is an interface edge.Each element is either filled fully by a chunk of Fluid A orFluid B, or is shared by a chunk of Fluid A and a chunkof Fluid B. If an element is shared like that, the sharesare determined by the position of the interface along theedges of that element. The base finite element formulationis essentially the one described by equations (89) and (90).Although the ETILT can be used in combination with theEDICT, we assume that we are working here with the plain,non-EDICT versions of equations (89) and (90).

At each time step, given uhn and φhe

n , we determine uhn+1,

phn+1, and φhe

n+1. The definitions of ρ and µ are modified touse the edge-based representation of the interface function:ρh = φheρA + (1 − φhe)ρB, µh = φheµA + (1 − φhe)µB. Inmarching from time level n to n + 1, we first calculate φh

n

from φhen by a least-squares projection:

∫

ψh(φh

n − φhen

)d = 0 (92)

To calculate φhn+1, we use equation (90). From φh

n+1,we calculate φhe

n+1 by a combination of a least-squaresprojection:

∫

(ψhen+1)P

((φhe

n+1)P − φhn+1

)d = 0 (93)

and corrections to enforce volume conservation for allchunks of Fluid A and Fluid B, taking into account themergers between the chunks and the split of chunks. Thisvolume conservation condition can symbolically be writtenas VOL (φhe

n+1) = VOL (φhen ). Here the subscript P is used

for representing the intermediate values following the pro-jection, but prior to the corrections for volume conservation.It can be shown that the projection given by equation (93)can be interpreted as locating the interface along the inter-face edges at positions where φh

n+1 = 1/2.As an alternative way for computing φh

n from φhen , we

propose to solve the following problem:

∫INT

ψhn

(φh

n − φhen

)d

+nie∑k=1

ψhn(xk)λPEN

(φh

n(xk) − 1/2) = 0 (94)


where nie is the number of interface edges, xk is thecoordinate of the interface location along the kth interfaceedge, λPEN is a large penalty parameter, and INT is thesolution domain. The solution domain is the union of all theelements containing at least one node where the value ofφh

n is unknown. We can assume φhn to be unknown only

at the nodes of the interface edges, with known valuesφh

n = 1 (for Fluid A) and φhn = 0 (for Fluid B) at all other

nodes. We can also augment the number of nodes whereφh

n is unknown and thus enlarge the solution domain. Thiscan be done all the way to the point where INT = .As another alternative, in equation (94) we can replacethe least-squares projection term with a slope-minimizationterm: ∫

INT

∇∇∇ψhn · ∇∇∇φh

n d

+nie∑k=1

ψhn(xk)λPEN

(φh

n(xk) − 1/2) = 0 (95)

A 1D version of the way of computing φhn from φhe

n canbe formulated by minimizing (φh

n − φhen )2 along ‘chains’ of

interface edges:∫SINT

ψhn

(φh

n − φhen

)ds

+nie∑k=1

ψhn(xk)λPEN

(φh

n(xk) − 1/2) = 0 (96)

where SINT is the collection of all chains of interface edges,and s is the integration coordinate along the interface edges.This is, of course, a simpler formulation, and much of theequations for the unknown nodal values will be uncoupled.

These projections and volume corrections are embeddedin the iterative solution technique, and are carried out ateach iteration. The iterative solution technique, which isbased on the Newton–Raphson method, addresses both thenonlinear and coupled nature of the set of equations thatneed to be solved at each time step. We now provide moreexplanations of how the projections and volume correctionswould be handled at an iteration step taking us fromiterative solution i to i + 1.

1. In determining (φhen+1)

i+1P from (φh

n+1)i+1, in the first

step of the projection, the position of the interface alongeach interface edge is calculated. The calculation for anedge might yield for the interface position a value thatis not within the range of values representing that edge.This would imply the following consequences.

(a) That interface edge does not remain as an inter-face edge after the projection.

(b) The node at the end of that edge (in the directionof the interface motion) changes from one fluidto another after the projection.

(c) Different edges among those connecting thatnode to other nodes might be identified as edgesexpected to be interface edges after the projec-tion. An edge connecting that node to anothernode would be identified as an interface edge ifthe other node belongs to a different fluid. If not,it means that a chunk of one of the fluids is merg-ing with another chunk of the same fluid. It mightalso mean, as a special case, that a chunk of fluidis connecting with itself at another point.

In the second step of the projection, the interface posi-tions would be calculated along the newly identifiedinterface edges and those predicted to remain as inter-face edges after the first step of the projection. Ifadditional steps of the projection are required, the sameprocedure would be followed.

2. After the projection is complete, we need to detectthe possible occurrence of mergers between chunksand split of chunks. The mergers can be detectedas described earlier when we discussed the optionsrelated to identification of interface edges following aprojection step. To detect the split of chunks, one way isto go through a sorting process. In this process, for eachchunk, we start with one of the nodes belonging to thatchunk, identify all the nodes connected to that nodewith edges belonging to that chunk, do the same forthe newly identified nodes, and continue this recursiveprocess until all the connected nodes are identified.

After this sorting is complete, if we still have somenodes left in that chunk, this would mean that the chunkwe are inspecting has been split. The recursive processneeds to be repeated for the nodes and edges remainingin that chunk, so that any additional splits that chunkmight have undergone are detected.

3. After the process of identifying all the Fluid A andFluid B chunks is complete, we need to enforce thevolume conservation. For each chunk, we compare thevolumes corresponding to interface locations denotedby (φhe

n+1)i and (φhe

n+1)i+1. In the cases of mergers and

splits, we compare the aggregate volume of a set ofchunks corresponding to (φhe

n+1)i and constituting a

merger/split group to the aggregate volume of the setof chunks constituting the related merger/split groupcorresponding to (φhe

n+1)i+1.

4. The volume conservation for a chunk or a merger/splitgroup would be enforced by inflating or deflating itsvolume. Let us suppose that multiplying the positiveand negative increments along each interface edge bya factor (1 + x) and (1 − x), respectively, results in a


volume correction by a factor (1 + y), where y and x

are of the same sign. We need to determine the value ofx, such that the corresponding value of y is sufficientlyclose to the volume correction needed. This would bedone iteratively, and the convergence of these itera-tions can be accelerated by calculating the numericalderivative of y with respect to x and using that esti-mate in updating x at every iteration. In some cases,we may not be able to represent an increment along theedge where the interface was located prior to a projec-tion or volume correction iteration. This would happenwhen an interface edge does not remain as an interfaceedge after the projection or volume correction iteration.For such cases, we propose to measure the incrementalong the ‘extended edge’, where the edge is imaginedto extend into the neighboring element. The imaginarylocation of the interface along the extension would bewhere the extension pierces the ‘interface plane’ pass-ing through the interface locations along the edges ofthe neighboring element. The increment would be mea-sured as the distance between the imaginary locationand the location prior to the projection or volume cor-rection iteration. The new, corrected interface locationsalong the edges of the neighboring element would bedetermined by shifting the interface plane parallel toitself until it is pierced by the extended edge at thecorrected imaginary location along the extended edge.If an interface location along an edge is affected bymore than one shifting plane, then the resultant correc-tion would be calculated by a weighted averaging. Ifthe extension of an edge coincides with another edge,then the increment would simply be measured alongthe ‘edge pair’, and the interface location would becorrected along the edge pair itself. Interface locationsthat are not along such edge pairs or riding on shiftinginterface planes would be corrected by projection fromthose that are.

As an alternative to using the advection equationgiven by equation (90) and the projections given byequations (92)–(93), at each time step we can calculate(φhe

n+1)P by time-integrating an equation governing thepositions of the interface along element edges crossed bythe interface. For that purpose, for each element edgecrossed by the interface, we write the following equation:

ds

dtes · n = u · n (97)

where s is the position of the interface along the edge,es is the unit vector along the edge, n is the unit vectornormal to the interface at the point it is pierced by theedge, and u is the fluid velocity at that point. In time-integration of this equation, we can use sub-time-stepping

techniques based on the EDSTT and ISTST described inSection 14. To differentiate this version of ETILT from theprevious one, we name the first one ETILT-A and thissecond one ETILT-B, which was proposed in Tezduyar(2002b,c).

17 LINE-TRACKED INTERFACEUPDATE TECHNIQUE (LTIUT)

It was mentioned in Section 14 that in computation of two-fluid interface (or free-surface) flows with the DSD/SSTmethod, time-integration of the interface update equationmay require a smaller time step than the one used for thefluid interiors. This might be avoided if the time-integrationof the interface update equation is based on a stabilizedformulation. To this end, we proposed in Tezduyar (2002c)the LTIUT.

Let us assume that as we march from time level n ton + 1, each interface node A traces a line identified withunit vector eA. Typically we would select eA = nA, wherenA is the unit vector normal to the interface at node A. Welet sA denote the position of node A along that line. Theinterface update task would then consist of calculating, foreach node A, the unknown value (sA)n+1. We define a localcoordinate system (x, y, z) associated with the interfacenode A, where ez is the unit vector along the coordinateaxis z, and ez = eA. We define the 2D spatial domain A

to be the projection of the cluster of 2D interface elementssharing the node A on to the xy-plane. Limited to A

and the interval from time level n to n + 1, we write thefollowing equation:

∂z

∂t+ uxy ·

(∇∇∇xy

)z − uz = 0 (98)

where

uz = ez · u (99)

uxy = u − uzez (100)

∇∇∇xy = ∇∇∇ − ez(ez · ∇∇∇) (101)

We note that while the projected position of node A remainsfixed in the xy-plane, in general we cannot say the samething for any other node B in the cluster of 2D interfaceelements sharing node A because, as we march from timelevel n to n + 1, node B traces its own line, with unitvector eB . Therefore, unless ez · eB = 1, the position ofnode B in the xy-plane changes. Consequently, A changesits shape (i.e. deforms). With that in mind, we can solveequation (98) with the DSD/SST formulation, using the


concepts and approaches we used in Sections 4 and 15.The DSD/SST formulation of equation (98) can be writtenas

∫(QA)n

NA

(∂zh

∂t+ uh

xy ·(∇∇∇xy)zh − uh

z

)dQ

+∫

(A)n

(NA)+n((zh)+n − (zh)−n

)d

+((nel)A)n∑

e=1

∫(QA)en

τz

(∂NA

∂t+ uh

xy ·(∇∇∇xy)NA

)(

∂zh

∂t+ uh

xy ·(∇∇∇xy)zh − uh

z

)dQ = 0 (102)

Here (QA)n is the space–time slab associated with A,NA is the space–time finite element interpolation func-tion corresponding to node A, ((nel)A)n is the number ofspace–time elements in (QA)n, (QA)en is a space–time ele-ment, and τz is the SUPG stabilization parameter.

Equation (102) would be used for generating a fullydiscrete equation associated with each interface node A. Wenote that the variables (zA)+n , (zA)−n+1, (zB)+n , and (zB)−n+1are dummy unknowns. The real unknowns we are trackingare (sA)n+1 and (sB)n+1, with the following relationship tothe dummy unknowns:

(zB)−n+1 − (zB)−n = (ez · eA)((sB)n+1 − (sB)n

)(103)

18 ITERATIVE SOLUTION METHODS

The finite element formulations reviewed in the earliersections fall into two categories: a space–time formulationwith moving meshes or a semidiscrete formulation withnonmoving meshes. Full discretizations of these formula-tions lead to coupled, nonlinear equation systems that needto be solved at every time step of the simulation. Whetherwe are using a space–time formulation or a semidiscreteformulation, we can represent the equation system thatneeds to be solved as follows:

N(dn+1

) = F (104)

Here dn+1 is the vector of nodal unknowns. In a semidis-crete formulation, this vector contains the unknowns asso-ciated with marching from time level n to n + 1. In aspace–time formulation, it contains the unknowns asso-ciated with the finite element formulation written forthe space–time slab Qn. The time-marching formulationsdescribed earlier can also be used for computing a steady-state flow. In such cases, time does not have a physical

significance, but is only used in time-marching to thesteady-state solution.

We solve equation (104) with the Newton–Raphsonmethod:

∂N∂d

∣∣∣∣di

n+1

(di

n+1

) = F − N(di

n+1

)(105)

where i is the step counter for the Newton–Raphsonsequence, and di

n+1 is the increment computed for din+1.

The linear equation system represented by equation (105)needs to be solved at every step of the Newton–Raphsonsequence. We can represent equation (105) as a linear equa-tion system of the form

Ax = b (106)

In the class of computations we typically carry out, thisequation system would be too large to solve with a directmethod. Therefore we solve it iteratively. At each iteration,we need to compute the residual of this system:

r = b − Ax (107)

This can be achieved in several different ways. The compu-tation can be based on a sparse-matrix storage of A. It canalso be based on the storage of just element-level matrices(element-matrix-based), or even just element-level vectors(element-vector-based). This last strategy is also called thematrix-free technique. After the residual computation, wecompute a candidate correction to x as given by the expres-sion

y = P−1r (108)

where P, the preconditioning matrix, is an approximationto A. P has to be simple enough to form and factorizeefficiently. However, it also has to be sophisticated enoughto yield a desirable convergence rate. How to update thesolution vector x by using y is also a major subjectin iterative solution techniques. Several update methodsare available, and we use the GMRES (Saad and Schultz,1986) method. We have been focusing our research relatedto iterative methods mainly on computing the residual refficiently and selecting a good preconditioner P. Whilemoving in this direction, we have always been keeping inmind that the iterative solution methods we develop need tobe efficiently implemented on parallel computing platforms.For example, the ‘parallel-ready’ methods we designed forthe residual computations include those that are element-matrix-based (Tezduyar et al., 1997), element-vector-based(Tezduyar et al., 1997), and sparse-matrix-based (Kalro andTezduyar, 1998). The element-vector-based methods were


successfully used also by other researchers in the contextof parallel computations; see, for example, Johan, Hughesand Shakib (1991) and Johan et al. (1995).

In preconditioning design, we developed some advancedpreconditioners such as the clustered-element-by-element(CEBE) preconditioner (Tezduyar et al., 1992a) and themixed CEBE and cluster companion (CC) preconditioner(Tezduyar et al., 1992a). We have implemented, with quitesatisfactory results, the CEBE preconditioner in conjunc-tion with an ILU approximation (Kalro and Tezduyar,1998). However, our typical computations are based ondiagonal and nodal-block-diagonal preconditioners. Theseare very simple preconditioners, but are also very simpleto implement on parallel platforms. More on our parallelimplementations can be found in Tezduyar et al. (1997).

19 ENHANCED SOLUTIONTECHNIQUES

Sometimes, some parts of the computational domain mayoffer more of a challenge for the Newton–Raphson methodthan the others. This might happen, for example, at thefluid–solid interface in a fluid–structure interaction prob-lem, and in such cases, the nonlinear convergence mightbecome even a bigger challenge if the structure is goingthrough buckling or wrinkling. It might also happen ata fluid–fluid interface, for example, if the interface isvery unsteady. In the EINST (Tezduyar, 2001b, 2002b,c,d,2003), as a variation of the Newton–Raphson method, wepropose to use sub-iterations in the parts of the domainwhere we are facing a nonlinear convergence challenge.This could be implemented, for example, by identifying thenodes of the zones where we need enhanced iterations, andperforming multiple iterations for those nodes for each iter-ation we perform for all other nodes; or, at every time step,we can let those nodes have a ‘head start’ of several iter-ations prior to commencing iterations for all other nodes.In time-accurate computations of fluid–structure interac-tions with the EDSTT-SM or EDSTT-MM approaches, theEINST can be used to allow for a larger number of nonlin-ear iterations for the structure.

In some challenging cases, using a diagonal or nodal-block-diagonal preconditioner might not lead to a satis-factory level of convergence at some locations, in theparts of the domain posing the challenge. This mighthappen, for example, in a fluid–structure interaction prob-lem, where the structure or the fluid zones near thestructure might be suffering from convergence problems.The situation might become worse if the structure isgoing through buckling or wrinkling. It might also hap-pen at a fluid–fluid interface. We might also face this

difficulty in the SEMMT described in the section onmesh update methods if the elasticity equations for thenodes connected to the thin elements are solved togetherwith the elasticity equations for the other nodes. In theEALST (Tezduyar, 2001b, 2002c,d, 2003), we propose touse stronger approximations for the parts of the domainwhere we are facing convergence challenges. This couldbe implemented, for example, by identifying the elementscovering the zones where we need enhanced approximation,and reflecting this in defining the element-level constituentsof the approximation matrix. For example, for the ele-ments that need stronger approximations, we can use asthe element-level approximation matrix the full element-level matrix, while for all other elements we use a diagonalelement-level matrix. This particular EALST can be sum-marized by first expressing the assembly process for A andP as

A =nel

Ae=1

Ae (109)

P =nel

Ae=1

Pe (110)

where A is the finite element assembly operator, and thendefining Pe for the elements belonging to the enhanced-approximation and diagonal-approximation groups:

Pe =

Ae for Enhanced Approximation GroupDIAG(Ae) for Diagonal Approximation Group

(111)

Here DIAG represents a diagonal or nodal-block-diagonalapproximation operator. We note that in factorizing the sub-matrices of P corresponding to the enhanced-approximationgroup, we can use a direct solution method, or, as analternative, a second-level iteration sequence. This second-level iteration sequence would have its own precondi-tioner (possibly a diagonal or nodal-block-diagonal pre-conditioner) and its own GMRES vector space (possiblyshorter than the GMRES vector space used in the first-level iterations). To differentiate between these two ver-sions of the EALST, we will call them EALST-D andEALST-I.

In EINST and EALST, elements can be selected to theenhanced group (enhanced-iteration group in EINST andenhanced-approximation group in EALST) in a static ordynamic way. In the static way, the elements would beselected to the enhanced group on the basis of what weknow in advance about the flow problem. Elements in theparts of the mesh that are expected to offer more of a chal-lenge during a computation would belong to the enhancedgroup. For example, in a fluid–structure interaction com-putation, a thin layer of fluid elements around the structure


could be defined as the enhanced group. In the dynamicway, elements would be selected to the enhanced groupon the basis of identifying the nodes with the highest nor-malized residuals or lowest residual reduction rates. Forexample, elements connected to the nodes with the low-est 10% residual reduction rates could be defined as theenhanced group. The residuals being examined are thosefor the nonlinear equation system in EINST and the lin-ear equation system in EALST. In the dynamic way, theenhanced group would not be redefined every time step.They would be redefined frequently enough so that inbetween redefining the enhanced group we can expect tomaintain substantial overlap between the elements in theenhanced group and the elements connected to the nodeswith the lowest residual reduction rates.

20 MIXED ELEMENT-MATRIX-BASED/ELEMENT-VECTOR-BASEDCOMPUTATION TECHNIQUE(MMVCT)

The MMVCT was introduced in Tezduyar (2001c), and wasalso described in Tezduyar (2002d). Consider a nonlinearequation system of the kind given by equation (104), rewrit-ten in the following form:

N1 (d1, d2) = F1

N2 (d1, d2) = F2 (112)

where d1 and d2 are the vectors of nodal unknowns cor-responding to unknown functions u1 and u2, respectively.Similarly, we rewrite equation (106) in the form

A11x1 + A12x2 = b1

A21x1 + A22x2 = b2 (113)

where

Aβγ = ∂Nβ

∂dγ

(114)

Rewriting equations (104) and (106) in this fashionwould help us recognize or investigate the properties asso-ciated with the individual blocks of the equation system.It would also help us explore selective treatment of theseblocks during the solution process. For example, in the con-text of a coupled fluid–structure interaction problem, u1 andu2 might be representing the fluid and structure unknowns,respectively. We would then recognize that computation ofthe coupling matrices A12 and A21 might pose a significantchallenge and therefore alternative approaches should beexplored.

Iterative solution of equation (113) can be written as

P11y1 + P12y2 = b1 − (A11x1 + A12x2

)P21y1 + P22y2 = b2 − (

A21x1 + A22x2

)(115)

where Pβγ’s represent the blocks of the preconditioningmatrix P. Here we focus our attention to computation ofthe residual vectors on the right-hand side, and explorealternative ways for evaluating the matrix–vector products.

Let us suppose that we are able to compute, withouta major difficulty, the element-level matrices Ae

11 and Ae22

associated with the global matrices A11 and A22, and that weprefer to evaluate A11x1 and A22x2 by using these element-level matrices. Let us also suppose that calculation of theelement-level matrices Ae

12 and Ae21 is exceedingly difficult.

Reflecting these circumstances, the computations can becarried out by using a mixed element-matrix-based/element-vector-based computation technique (Tezduyar, 2001c,d):

(A11x1 + A12x2

) =nel

Ae=1

(Ae11x1) +

nel

Ae=1

limε1→0[

Ne

1(d1, d2 + ε1x2) − Ne

1(d1, d2)

ε1

]

(A21x1 + A22x2

) =nel

Ae=1

(Ae22x2) +

nel

Ae=1

limε2→0[

Ne

2(d1 + ε2x1, d2) − Ne

2(d1, d2)

ε2

](116)

where ε1 and ε2 are small parameters used in numericalevaluation of the directional derivatives. Here, A11x1 andA22x2 are evaluated with an element-matrix-based compu-tation technique, while A12x2 and A21x1 are evaluated withan element-vector-based computation technique.

In extending the mixed element-matrix-based/element-vector-based computation technique described above to amore general framework, evaluation of a matrix–vectorproduct Aβγxγ (for β, γ = 1, 2, . . . , N and no sum) appear-ing in a residual vector can be formulated as an intentionalchoice between the following element-matrix-based andelement-vector-based computation techniques:

Aβγxγ =nel

Ae=1

(Aeβγxγ) (117)

Aβγxγ =nel

Ae=1

limεβ→0[

Ne

β(. . . , dγ + εβxγ, . . .) − Ne

β(. . . , dγ, . . .)

εβ

](118)


Sometimes, computation of the element-level matrices Aeβγ

might not be prohibitively difficult, but we might still preferto evaluate Aβγxγ with an element-vector-based compu-tation technique. In such cases, instead of an element-vector-based computation technique requiring numericalevaluation of directional derivatives, we might want to usethe element-vector-based computation technique describedbelow.

Let us suppose that the nonlinear vector functionNβ corresponds to a finite element integral formBβ(Wβ, u1, . . . , uN). Here Wβ represents the vector ofnodal values associated with the weighting function wβ,which generates the nonlinear equation block β. Letus also suppose that we are able to, without a majordifficulty, derive the first-order terms in the expansion ofBβ(Wβ, u1, . . . , uN) in uγ. Let the finite element integralform Gβγ(Wβ, u1, . . . , uN, uγ) represent those first-orderterms in uγ. We note that this finite element integral formwill generate (∂Nβ/∂dγ). Consequently, the matrix–vectorproduct Aβγxγ can be evaluated as (Tezduyar, 2001c,d)

Aβγxγ = ∂Nβ

∂dγ

xγ =nel

Ae=1

Gβγ(Wβ, u1, . . . , uN, vγ) (119)

where, vγ is a function interpolated from xγ in the same waythat uγ is interpolated from dγ. This element-vector-basedcomputation technique allows us to evaluate matrix–vectorproducts without dealing with numerical evaluationof directional derivatives. To differentiate between theelement-vector-based computation techniques defined byequations (118) and (119), we will call them, respectively,numerical element-vector-based (NEVB) and analyticalelement-vector-based (AEVB) computation techniques.

Remark 4. In fluid–structure interaction computations,where the structure is light, in the absence of taking intoaccount the coupling blocks A12 and A21, we propose ashort cut approach for improving the convergence of thecoupling iterations. In this approach, to reduce ‘overcorrect-ing’ (i.e. ‘overincrementing’) the structural displacementsduring the coupling iterations, we artificially increase thestructural mass contribution to the matrix block correspond-ing to the structural mechanics equations and unknowns.In the context of this section, with the understanding thatsubscript 2 denotes the structure, this would be equiv-alent to artificially increasing the mass matrix contribu-tion to A22. This is achieved without altering b1 or b2(i.e. F1 − N1 (d1, d2) or F2 − N2 (d1, d2)), and thereforewhen the coupling iterations converge, they converge tothe solution of the problem with the correct structuralmass.

Remark 5. In fluid–structure interaction computationswith light and thin structures (such as membranes), it mightbe desirable to eliminate the higher spatial modes of thestructural response normal to the membrane. We propose toaccomplish that by adding to the finite element formulationof the structural mechanics problem a ‘directional-inertiastabilizing mass (DISM)’ term, which we define as

SDISM =nel∑e=1

∫(s)e

w · (η ρs nn

) ·(

d2yh

dt2

)ds (120)

where s is the membrane domain, yh is the displacement,ρs is the material density, n is the unit vector normal tothe membrane, and η is a nondimensional measure of thecurvature in (d2yh/dt2).

21 ENHANCED-DISCRETIZATIONSUCCESSIVE UPDATE METHOD(EDSUM)

In this section, we describe a multilevel iteration methodfor computation of flow behavior at small scales. TheEDSUM (Tezduyar, 2001c,d) is based on the EDICT.Although it might be possible to identify zones where theenhanced discretization could be limited to, we need tothink about and develop methods required for cases wherethe enhanced discretization is needed everywhere in theproblem domain to accurately compute flows at smallerscales. In that case, the enhanced-discretization would bemore widespread than before, and possibly required forthe entire domain. Therefore an efficient solution approachwould be needed to solve, at every time step, a very large,coupled nonlinear equation system generated by the multi-level discretization approach.

Such large, coupled nonlinear equation systems involvefour classes of nodes. Class-1 consists of all the Mesh-1 nodes. These nodes are connected to each other throughthe Mesh-1 elements. Class-2E consists of the Mesh-2 edgenodes (but excluding those coinciding with the Mesh-1nodes). The edge nodes associated with different edges arenot connected (except those at each side of an edge, but wecould possibly neglect that a side node might be connectedto the side nodes of the adjacent edges). Nodes withinan edge are connected through Mesh-2 elements. Class-2F contains the Mesh-2 face nodes (but excluding thoseon the edges). The face nodes associated with differentfaces are not connected (except those at sides of a face, butwe could possibly neglect that those side nodes might beconnected to the side nodes of the adjacent face). Nodeswithin a face are connected through Mesh-2 elements.Class-2I nodes are the Mesh-2 interior nodes. The interior


nodes associated with different clusters of Mesh-2 elementsare not connected. Nodes within a cluster are connectedthrough Mesh-2 elements.

On the basis of this multilevel decomposition concept,a nonlinear equation system of the kind given by equa-tion (104) can be rewritten as follows:

N1 (d1, d2E, d2F , d2I ) = F1

N2E (d1, d2E, d2F , d2I ) = F2E

N2F (d1, d2E, d2F , d2I ) = F2F

N2I (d1, d2E, d2F , d2I ) = F2I (121)

where the subscript ‘n + 1’ has been dropped to simplifythe notation.

This equation system would be solved with an approxi-mate Newton–Raphson method. At each nonlinear iterationstep, we would successively update the solution vectorscorresponding to each class. While updating each class, wewould use the most recent values of the solution vectorsin calculating the vectors N1, N2E , N2F , and N2I and theirderivatives with respect to the solution vectors. We wouldstart with updating the Class-1 nodes, then update the Class-2E, Class-2F, and Class-2I nodes, respectively. The processis shown below, where each class of equations are solvedin the order they are written.

∂N1

∂d1

∣∣∣∣(di

1, di2E

, di2F

, di2I )

(di

1

)= F1 − N1

(di

1, di2E, di

2F , di2I

)∂N2E

∂d2E

∣∣∣∣(di+1

1 , di2E

, di2F

, di2I )

(di

2E

)

= F2E − N2E

(di+1

1 , di2E, di

2F , di2I

)∂N2F

∂d2F

∣∣∣∣(di+1

1 , di+12E

, di2F

, di2I )

(di

2F

)

= F2F − N2F

(di+1

1 , di+12E , di

2F , di2I

)∂N2I

∂d2I

∣∣∣∣(di+1

1 , di+12E

, di+12F

, di2I )

(di

2I

)

= F2I − N2I

(di+1

1 , di+12E , di+1

2F , di2I

)(122)

This sequence would be repeated as many times as needed,and, as an option, we could alternate between this sequenceand its reverse sequence:

∂N2I

∂d2I

∣∣∣∣(di

1, di2E

, di2F

, di2I )

(di

2I

)

= F2I − N2I

(di

1, di2E, di

2F , di2I

)∂N2F

∂d2F

∣∣∣∣(di

1, di2E

, di2F

, di+12I )

(di

2F

)

= F2F − N2F

(di

1, di2E, di

2F , di+12I

)∂N2E

∂d2E

∣∣∣∣(di

1, di2E

, di+12F

, di+12I )

(di

2E

)

= F2E − N2E

(di

1, di2E, di+1

2F , di+12I

)∂N1

∂d1

∣∣∣∣(di

1, di+12E

, di+12F

, di+12I )

(di

1

)

= F1 − N1

(di

1, di+12E , di+1

2F , di+12I

)(123)

Updating the solution vector corresponding to each classwould also require the solution of a large equation system.These equation systems would each be solved iteratively,with an effective preconditioner, a reliable search technique,and parallel implementation. It is important to note that thebulk of the computational cost would be for Class-1 andClass-2I. While the Class-1 nodes would be partitionedto different processors of the parallel computer, for theremaining classes, nodes in each edge, face, or interiorcluster would be assigned to the same processor. Therefore,solution of each edge, face, or interior cluster would belocal. If the size of each interior cluster becomes too large,then nodes for a given cluster can also be distributed acrossdifferent processors, or a third level of mesh refinement canbe introduced to make the enhanced discretization a tri-levelkind.

A variation of the EDSUM could be used for the iterativesolution of the linear equation system that needs to besolved at every step of a (full) Newton–Raphson methodapplied to equation (121). To describe this variation, wefirst write, similar to the way we wrote equations (113)and (114), the linear equation system that needs to besolved:

A11x1 + A12Ex2E + A12F x2F + A12I x2I = b1

A2E1x1 + A2E2Ex2E + A2E2F x2F + A2E2I x2I = b2E

A2F1x1 + A2F2Ex2E + A2F2F x2F + A2F2I x2I = b2F

A2I1x1 + A2I2Ex2E + A2I2F x2F + A2I2I x2I = b2I

(124)

where

Aβγ = ∂Nβ

∂dγ

(125)


with β, γ = 1, 2E, 2F , 2I . Then, for the iterative solutionof equation (124), we define two preconditioners:

PLTOS =

A11 0 0 0A2E1 A2E2E 0 0A2F1 A2F2E A2F2F 0A2I1 A2I2E A2I2F A2I2I

(126)

PSTOL =

A11 A12E A12F A12I

0 A2E2E A2E2F A2E2I

0 0 A2F2F A2F2I

0 0 0 A2I2I

(127)

We propose that these two preconditioners be used alternat-ingly during the inner iterations of the GMRES search. Wenote that this mixed preconditioning technique with mul-tilevel discretization is closely related to the mixed CEBEand CC preconditioning technique (Tezduyar et al., 1992a)we referred to in Section 18. Along these lines, as a mixedpreconditioning technique that is more closely related tothe mixed CEBE and CC technique, we propose that thefollowing three preconditioners be used in sequence duringthe inner iterations of the GMRES search:

PL =

A11 0 0 00 DIAG 0 0(

A2E2E

)0 0 DIAG 0(

A2F2F

)0 0 0 DIAG(

A2I2I

)

(128)

PSETOI =

DIAG(A11

)0 0 0

0 A2E2E 0 00 A2F2E A2F2F 00 A2I2E A2I2F A2I2I

(129)

PSITOE =

DIAG(A11

)0 0 0

0 A2E2E A2E2F A2E2I

0 0 A2F2F A2F2I

0 0 0 A2I2I

(130)

As possible sequences, we propose (PL, PSETOI ,PSITOE , . . . , PL, PSETOI , PSITOE ), as well as (PL,PSETOI , . . . , PL, PSETOI ) and (PL, PSITOE , . . . , PL, PSITOE ).As a somewhat downgraded version of PL, we can use apreconditioner that is equivalent to not updating x2E , x2F ,and x2I , instead of updating them by using DIAG

(A2E2E

),

DIAG(A2F2F

), and DIAG

(A2I2I

). Similarly, as down-

graded versions of PSETOI and PSITOE , we can use precon-ditioners that are equivalent to not updating x1, instead ofupdating it by using DIAG

(A11

).

To differentiate between the two variations of theEDSUM we described in this section, we call the

nonlinear version, described by equations (122) and(123), EDSUM-N, and the linear version, described byequations (124)–(130), EDSUM-L.

22 EXAMPLES OF FLOW SIMULATIONS

In this section, we present examples of flow simulationscarried out by using some of the methods described in theearlier sections. In some cases, at high Reynolds numbers,we use a simple turbulence model, where the physical vis-cosity is augmented by an eddy viscosity as proposed bySmagorinsky (see Smagorinsky, 1963; Kato and Ikegawa,1991). All results were obtained by computations on par-allel computing platforms. All computations were carriedout in 3D. Each numerical example is described briefly andreferences are given to our earlier publications for moreinformation.

22.1 Flow around two high-speed trains in atunnel

Two high-speed trains are passing each other in a tunnel.Each train has a speed of 100 m/s. The Reynolds num-ber based on the train length is around 67 million. Themesh consists of 101 888 hexahedral elements and 230 982space–time nodes, and leads to 900 274 coupled, nonlinearequations that need to be solved at every time step. Herethe compressible flow version of the DSD/SST formula-tion is used together with the SSMUM to handle the meshwhile the trains are in rapid, relative motion. Figure 5 shows

Figure 5. Flow around two high-speed trains in a tunnel. Thetrains and pressure distribution at different instants. A colorversion of this image is available at http://www.mrw.interscience.wiley.com/ecm


the trains and pressure distribution at different instants. Formore on this simulation see Tezduyar et al. (1996a).

22.2 Flow past a rotating propeller

In this computation, the DSD/SST formulation and SSMUMare applied to simulation of flow past a propeller in alarge box. The Reynolds number, based on the upstreamflow velocity and the propeller diameter, is approximately1 million. The rotational speed of the propeller, scaledwith the upstream flow velocity and the propeller diam-eter, is 9.11π. The mesh consists of 305 786 space–timenodes and 939 213 tetrahedral elements. Figure 6 showsthe mesh at the outer boundaries and on the outer surfaceof the shear–slip layer, as well as the mesh on the pro-peller surface and on the inner surface of the shear–sliplayer. Figure 7 shows cross-sectional views of the meshand the shear–slip layer. Figure 8 shows the pressure onthe propeller surface and the isosurface of −0.025 valueof pressure, as well as the helicity on the propeller surfaceand the isosurface of 0.2 value of helicity. For more on thissimulation, see Behr and Tezduyar (1999).

22.3 Flow past a helicopter

Here we apply the DSD/SST formulation and SSMUM forcomputation of flow past a helicopter with its main rotorin motion. In this computation, the tail rotor is excludedfrom the model, although the SSMUM approach couldhave been used to model it too. The mesh consists of361 434 space–time nodes and 1 096 225 tetrahedral ele-ments. Figure 9 shows the surface meshes for the fuse-lage, rotor, and the top portion of the inner surface of

the shear–slip layer. Figure 10 shows the cross-section ofthe mesh through the center of the rotor hub. The heli-copter is assumed to be in a forward horizontal flight at aspeed of 10.0 m s−1, with a rotor tip velocity of 200 m s−1.Figure 11 shows the air pressure distribution on the surfacesof the fuselage and rotor. For more on this simulation, seeBehr and Tezduyar (2001).

22.4 Fluid–object interactions with 1000 spheresfalling in a liquid-filled tube

The core method for this simulation is the DSD/SST for-mulation. The methods layered around this include thefollowing: an efficient distributed-memory implementationof the formulation; fast automatic mesh generation; a meshupdate method based on automatic mesh moving withremeshing only as needed; an efficient method for pro-jecting the solution after each remesh; and multi-platform(heterogeneous) computing. Here, while mesh partitioning,flow computations, and mesh movements were performedon a 512-node Thinking Machines CM-5, automatic meshgeneration and projection of the solution were accomplishedon a 2-processor SGI ONYX2. The two systems communi-cated over a high-speed network as often as the computationrequired remeshing. In more recent simulations of this classof problems (see the next numerical example), the CM-5has been replaced by a CRAY T3E-1200. The spheres,in addition to interacting with the fluid, interact and col-lide with each other and with the tube wall. The averageReynolds number is around 8. The mesh size is approxi-mately 2.5 million tetrahedral elements, resulting in about5.5 million coupled, nonlinear equations to be solved everytime step. The number of time steps is around 1100 in the

(a) (b)

Figure 6. Flow past a rotating propeller. (a) Mesh at the outer boundaries and on the outer surface of the shear–slip layer. (b) Meshon the propeller surface and on the inner surface of the shear–slip layer.


(a) (b)

Figure 7. Flow past a rotating propeller. Cross-sectional views of the mesh and the shear–slip layer. (a) Side view and (b) Top view.

(a) (b)

Figure 8. Flow past a rotating propeller. (a) Pressure on thepropeller surface and the isosurface of −0.025 value of pressure.(b) Helicity on the propeller surface and the isosurface of 0.2value of helicity. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

Figure 9. Flow past a helicopter. Surface meshes for the fuselage,rotor, and the top portion of the inner surface of the shear–sliplayer.

Figure 10. Flow past a helicopter. Cross-section of the meshthrough the center of the rotor hub.

(a) (b)

Figure 11. Flow past a helicopter. Air pressure on the surfacesof the fuselage (a) and rotor (b). A color version of this image isavailable at http://www.mrw.interscience.wiley.com/ecm

simulation. Figure 12 shows the spheres at four differentinstants during the simulation. The first picture shows theinitial distribution. The colors are for identification pur-pose only. For more on this simulation, see Johnson andTezduyar (1999).


Figure 12. Fluid–object interactions with 1000 spheres falling ina liquid-filled tube. Pictures show, at four instants, the spheresas they fall. A color version of this image is available athttp://www.mrw.interscience.wiley.com/ecm

22.5 Fluid–object interactions in spatiallyperiodic flows

Here we carry out simulation of fluid–object interactionsin spatially tri-periodic flows, where the solid objects areequal-sized spheres and fall under the action of gravity. Inour study, we keep the volume fraction of the spheres in thefluid–particle mixture constant at 26.8%, and investigatehow the mixture behavior varies as the size of the periodiccell is varied to contain more and more spheres. The flowparameters are defined in such a way that a single sphere,at the solid volume fraction of 26.8%, falls at a terminalReynolds number of approximately 10. The total volumetricflow rate is set to zero in all three directions, and thisis what we expect to be the case in sedimentation. Wecomputed five cases, where the number of particles in theperiodic cell were 1, 8, 27, 64, and 125. The mesh sizes forthese five cases were approximately 0.012, 0.120, 0.400,0.930, and 1.8 million tetrahedral elements, respectively.Here, while the projection of the solution after a remesh,mesh partitioning, flow computations, and mesh movements

Figure 13. Fluid–object interactions in spatially periodic flows.For the five cases, the velocity vectors at a cross-section. Fromleft to right and top to bottom: 1, 8, 27, 64, and 125 particles/cell.

were performed on a CRAY T3E-1200, the automaticmesh generation was carried out on a PentiumII-basedPC. The two systems communicated over a high-speednetwork as often as the computation required remeshing.The results from these five cases are very similar, evenquantitatively, observed in terms of average settling speed,volumetric fluid flow rate, particle drag, and pressure jumpvalues. Figure 13 shows, at a vertical cross-section, thevelocity vectors. To make the comparison between theresults from these five cases easier, except for the lastcase, we show the velocity vectors over multiple periodiccells. The number of periodic cells assembled together forthe cases of 1, 8, 27, 64, and 125 particles in a periodiccell are 5 × 5, 3 × 3, 2 × 2, 2 × 2, and 1 × 1, respectively.For more on this simulation, see Johnson and Tezduyar(2001).


Figure 14. Free-surface flow past a bridge support. The bridge-support and the free-surface color-coded with the velocity mag-nitude. A color version of this image is available at http://www.mrw.interscience.wiley.com/ecm

22.6 Free-surface flow past a circular cylinder

The Reynolds number based on the upstream velocity is10 million. The upstream Froude number is 0.564. Themesh consists of 230 480 prism-based space–time elementsand 129 382 nodes. The DSD/SST formulation is used withan algebraic mesh update method. The free-surface heightis governed by an advection equation and solved with astabilized formulation. Figure 14 shows, at an instant, thecylinder together with the free-surface color-coded with thevelocity magnitude. For more on this simulation, see Guler,Behr and Tezduyar (1999).

23 CONCLUDING REMARKS

We provided an overview of the stabilized finite elementinterface-tracking and interface-capturing techniques wedeveloped in recent years for computation of fluid dynam-ics problems with moving boundaries and interfaces. Wealso highlighted the calculation of the stabilization param-eters used in these stabilized formulations. The interface-tracking techniques are based on the DSD/SST formula-tion, where the mesh moves to track the interface. Theinterface-capturing techniques, which were developed fortwo-fluid flows, are based on the stabilized formulation,over nonmoving meshes, of both the flow equations andan advection equation. The advection equation governs thetime-evolution of the interface function marking the inter-face location. We also described in this paper some of theadditional methods developed to increase the scope andaccuracy of the interface-tracking and interface-capturingtechniques. Among these methods are the SSMUM for fast,linear or rotational relative motions; a special DSD/SSTformulation for spatially periodic fluid–object interactions;the STCT for fluid–solid contact problems; and the FOIST,

which is a subcomputation technique for fluid–objectinteractions. Also among these methods is the EDICT,which was developed to increase the accuracy in capturingthe interface, and extensions and offshoots of the EDICT,such as the EDMRT and EDSTT, and other enhancediterative solution techniques such as the EINST, EALST,EDSUM, and MMVCT. Other methods that were developedto increase the scope and accuracy of the interface-trackingand interface-capturing techniques are the MITICT, ETILT,and LTIUT. The MITICT was developed for the classes ofproblems that involve both – interfaces that can be accu-rately tracked with a moving mesh method and interfacesthat are too complex or unsteady to be tracked and thereforerequire an interface-capturing technique. The ETILT wasdeveloped to improve the interface-capturing techniqueswith better volume conservation properties and sharperrepresentation of the interfaces. With the LTIUT, in com-putation of two-fluid interfaces with the DSD/SST method,the interface update equation can be solved with a stabilizedformulation.

24 RELATED CHAPTERS

(See also Chapter 14 of Volume 1, Chapter 21 ofVolume 2, Chapter 2, Chapter 13, Chapter 18 of thisVolume)

ACKNOWLEDGMENT

This work was supported by the US Army Natick SoldierCenter and NASA Johnson Space Center.

REFERENCES

Behr and Tezduyar T. Shear-slip mesh update method. Com-puter Methods in Applied Mechanics and Engineering 1999;174:261–274.

Behr M and Tezduyar T. Shear-slip mesh update in 3D com-putation of complex flow problems with rotating mechanicalcomponents. Computer Methods in Applied Mechanics andEngineering 2001; 190:3189–3200.

Benek J, Buning P and Steger J. A 3-D Chimera grid embeddingtechnique. Paper 85-1523, AIAA, 1985.

Brooks A and Hughes T. Streamline upwind/Petrov-Galerkin for-mulations for convection dominated flows with particularemphasis on the incompressible Navier-Stokes equations. Com-puter Methods in Applied Mechanics and Engineering 1982;32:199–259.


Catabriga L, Coutinho A and Tezduyar T. Finite element SUPGparameters computed from local matrices for compressibleflows. In Proceedings of the 9th Brazilian Congress of Engi-neering and Thermal Sciences , ENCIT Paper No. CIT02-0221,Caxambu, 2002.

Cruchaga M and Onate E. A generalized streamline finite ele-ment approach for the analysis of incompressible flow prob-lems including moving surfaces. Computer Methods in AppliedMechanics and Engineering 1999; 173:241–255.

Donea J. A Taylor-Galerkin method for convective transport prob-lems. International Journal for Numerical Methods in Engineer-ing 1984; 20:101–120.

Farhat C, Lesoinne M and Maman N. Mixed explicit/implicittime integration of coupled aeroelastic problems: Three-fieldformulation, geometric conservation and distributed solution.International Journal for Numerical Methods in Fluids 1995;21:807–835.

Franca L, Frey S and Hughes T. Stabilized finite element meth-ods: I. Application to the advective-diffusive model. Com-puter Methods in Applied Mechanics and Engineering 1992;95:253–276.

Garcia J, Onate E, Sierra H and Idelsohn S. A stabilized numeri-cal method for analysis of ship hydrodynamics. Proceedings ofECCOMAS 1998. John Wiley: Athens, 1998.

Guler I, Behr M and Tezduyar T. Parallel finite element compu-tation of free-surface flows. Computational Mechanics 1999;23:117–123.

Hirt CW and Nichols BD Volume of fluid (VOF) method for thedynamics of free boundaries. Journal of Computational Physics1981; 39:201–225.

Hughes T and Brooks A. A multi-dimensional upwind schemewith no crosswind diffusion. In Finite Element Methods forConvection Dominated Flows, AMD-Vol. 34, Hughes T (ed.).ASME: New York, 1979; 19–35.

Hughes T, Franca L and Balestra M. A new finite element for-mulation for computational fluid dynamics: V. Circumventingthe Babuska-Brezzi condition: a stable Petrov-Galerkin for-mulation of the stokes problem accommodating equal-orderinterpolations. Computer Methods in Applied Mechanics andEngineering 1986; 59:85–99.

Hughes T, Franca L and Mallet M. A new finite elementformulation for computational fluid dynamics: VI.Convergenceanalysis of the generalized SUPG formulation for lineartime-dependent multi-dimensional advective-diffusive systems.Computer Methods in Applied Mechanics and Engineering1987; 63:97–112.

Hughes T and Hulbert G. Space-time finite element methodsfor elastodynamics: formulations and error estimates. Com-puter Methods in Applied Mechanics and Engineering 1988;66:339–363.

Johan Z, Hughes T and Shakib F. A globally convergent matrix-free algorithm for implicit time-marching schemes arising infinite element analysis in fluids. Computer Methods in AppliedMechanics and Engineering 1991; 87:281–304.

Johan Z, Mathur K, Johnsson S and Hughes T. A case study inparallel computation: Viscous flow around an Onera M6 wing.International Journal for Numerical Methods in Fluids 1995;21:877–884.

Johnson A and Tezduyar T. Advanced mesh generation andupdate methods for 3D flow simulations. Computational Mech-anics 1999; 23:130–143.

Johnson A and Tezduyar T. Methods for 3D computation offluid-object interactions in spatially-periodic flows. ComputerMethods in Applied Mechanics and Engineering 2001;190:3201–3221.

Johnson C, Navert U and Pitkaranta J. Finite element methodsfor linear hyperbolic problems. Computer Methods in AppliedMechanics and Engineering 1984; 45:285–312.

Kalro V and Tezduyar T. Parallel iterative computational methodsfor 3D finite element flow simulations. Computer AssistedMechanics and Engineering Sciences 1998; 5:173–183.

Kato C and Ikegawa M. Large eddy simulation of unsteady turbu-lent wake of a circular cylinder using the finite element method.In Advances in Numerical Simulation of Turbulent Flows, FED-Vol. 117, Celik I, Kobayashi T, Ghia K and Kurokawa J (eds).ASME: New York, 1991; 49–56.

LeBeau G, Ray S, Aliabadi S and Tezduyar T. SUPG finite ele-ment computation of compressible flows with the entropyand conservation variables formulations. Computer Methods inApplied Mechanics and Engineering 1993; 104:397–422.

LeBeau G and Tezduyar T. Finite element computation of com-pressible flows with the SUPG formulation. In Advances inFinite Element Analysis in Fluid Dynamics, FED-Vol. 123,Dhaubhadel M, Engelman M and Reddy J (eds). ASME: NewYork, 1991; 21–27.

Lesoinne M and Farhat C. Geometric conservation laws for flowproblems with moving boundaries and deformable meshes, andtheir impact on aeroelastic computations. Computer Methods inApplied Mechanics and Engineering 1996; 134:71–90.

Lohner R, Yang C and Baum J. Rigid and flexible store sepa-ration simulations using dynamic adaptive unstructured gridtechnologies. In Proceedings of the First AFOSR Conferenceon Dynamic Motion CFD , New Brunswick, 1996.

Lynch D. Wakes in liquid-liquid systems. Journal of Computa-tional Physics 1982; 47:387–411.

Masud A and Hughes T. A space-time Galerkin/least-squaresfinite element formulation of the Navier-Stokes equationsfor moving domain problems. Computer Methods in AppliedMechanics and Engineering 1997; 146:91–126.

Mittal S, Aliabadi S and Tezduyar T. Parallel computation ofunsteady compressible flows with the EDICT. ComputationalMechanics 1999; 23:151–157.

Onate E and Garcia J. A methodology for analysis of fluid-structure interaction accounting for free surface waves. In Pro-ceedings of European Conference on Computational Mechan-ics , Munich, 1999.

Onate E, Idelsohn S, Sacco C and Garcia J. Stabilization of thenumerical solution for the free surface wave equation influid dynamics. Proceedings of ECCOMAS 1998. John Wiley:Athens, 1998.

Saad Y and Schultz M. GMRES: A generalized minimal residualalgorithm for solving nonsymmetric linear systems. SIAM Jour-nal of Scientific and Statistical Computing 1986; 7:856–869.

Smagorinsky J. General circulation experiments with the primitiveequations. Monthly Weather Review 1963; 91(3):99–165.


Stein K, Benney R, Kalro V, Tezduyar T, Leonard J and Acco-rsi M. Parachute fluid-structure interactions: 3-D computation.In Proceedings of the 4th Japan-US Symposium on Finite Ele-ment Methods in Large-Scale Computational Fluid Dynamics ,Tokyo, 1998.

Stein K and Tezduyar T. Advanced mesh update techniques forproblems involving large displacements. In Proceedings ofthe Fifth World Congress on Computational Mechanics , On-line Publication: http://wccm.tuwien.ac.at/, Paper-ID: 81489,Vienna, 2002.

Stein K, Tezduyar T and Benney R. Mesh update with solid-extension mesh moving technique. In Pre-Conference Proceed-ings of the Sixth Japan-US International Symposium on FlowSimulation and Modeling , Fukuoka, 2002.

Stein K, Tezduyar T and Benney R. Mesh moving techniques forfluid-structure interactions with large displacements. Journal ofApplied Mechanics 2003; 70:58–63.

Tezduyar T. Stabilized finite element formulations for incom-pressible flow computations. Advances in Applied Mechanics1991; 28:1–44.

Tezduyar T. Adaptive determination of the finite element stabi-lization parameters. In Proceedings of the ECCOMAS Computa-tional Fluid Dynamics Conference 2001 (CD-ROM), Swansea,2001a.

Tezduyar T. Finite element interface-tracking and interface-capturing techniques for flows with moving boundaries andinterfaces. Proceedings of the ASME Symposium on Fluid-Physics and Heat Transfer for Macro- and Micro-Scale Gas-Liquid and Phase-Change Flows (CD-ROM), ASME PaperIMECE2001/HTD-24206. ASME: New York, 2001b.

Tezduyar T. Finite element methods for flow problems with mov-ing boundaries and interfaces. Archives of Computational Meth-ods in Engineering 2001c; 8:83–130.

Tezduyar T. Calculation of the stabilization parameters in SUPGand PSPG formulations. In Proceedings of the First South-American Congress on Computational Mechanics (CD-ROM),Santa Fe-Parana, 2002a.

Tezduyar T. Computation of moving boundaries and interfaceswith interface-tracking and interface-capturing techniques. InPre-Conference Proceedings of the Sixth Japan-US Interna-tional Symposium on Flow Simulation and Modeling , Fukuoka,2002b.

Tezduyar T. Interface-tracking and interface-capturing techniquesfor computation of moving boundaries and interfaces. InProceedings of the Fifth World Congress on ComputationalMechanics , On-line Publication: http://wccm.tuwien.ac.at/,Paper-ID: 81513, Vienna, 2002c.

Tezduyar T. Interface-tracking, interface-capturing and enhancedsolution techniques. In Proceedings of the First South-AmericanCongress on Computational Mechanics (CD-ROM), Santa Fe-Parana, 2002d.

Tezduyar T. Stabilization parameters and element length scalesin SUPG and PSPG formulations. In Book of Abstracts ofthe 9th International Conference on Numerical Methods andComputational Mechanics , Miskolc, 2002e.

Tezduyar T. Stabilization parameters and local length scales inSUPG and PSPG formulations. In Proceedings of the Fifth

World Congress on Computational Mechanics , On-line Pub-lication: http://wccm.tuwien.ac.at/, Paper-ID: 81508, Vienna,2002f.

Tezduyar T. Stabilized finite element formulations and interface-tracking and interface-capturing techniques for incompress-ible flows. In Numerical Simulations of Incompressible Flows,Hafez M (ed.). World Scientific: New Jersey, 2003; 221–239.

Tezduyar T, Aliabadi S and Behr M. Enhanced-DiscretizationInterface-Capturing Technique. In Proceedings of the ISAC’97 High Performance Computing on Multiphase Flows, Mat-sumoto Y and Prosperetti A (eds). Japan Society of MechanicalEngineers: Tokyo, 1997; 1–6.

Tezduyar T, Aliabadi S and Behr M. Enhanced-discretizationinterface-capturing technique (EDICT) for computation ofunsteady flows with interfaces. Computer Methods in AppliedMechanics and Engineering 1998; 155:235–248.

Tezduyar T, Aliabadi S, Behr M, Johnson A, Kalro V andLitke M. Flow simulation and high performance computing.Computational Mechanics 1996a; 18:397–412.

Tezduyar T, Aliabadi S, Behr M, Johnson A, Kalro V andLitke M. High performance computing techniques for flowsimulations. In Solving Large-Scale Problems in Mechanics:Parallel Solution Methods in Computational Mechanics,Papadrakakis M (ed.). John Wiley & Sons: Chichester, 1997;363–398.

Tezduyar T, Aliabadi S, Behr M, Johnson A and Mittal S. Par-allel finite-element computation of 3D flows. IEEE Computer1993; 26(10):27–36.

Tezduyar T, Behr M, Aliabadi S, Mittal S and Ray S. A newmixed preconditioning method for finite element computa-tions. Computer Methods in Applied Mechanics and Engineer-ing 1992a; 99:27–42.

Tezduyar T, Behr M, Mittal S and Johnson A. Computationof unsteady incompressible flows with the finite elementmethods – space-time formulations, iterative strategies andmassively parallel implementations. New Methods in TransientAnalysis, PVP-Vol.246/AMD-Vol.143. ASME: New York,1992b; 7–24.

Tezduyar T and Ganjoo D. Petrov-Galerkin formulations withweighting functions dependent upon spatial and temporal dis-cretization: Applications to transient convection-diffusion prob-lems. Computer Methods in Applied Mechanics and Engineer-ing 1986; 59:49–71.

Tezduyar T and Hughes T. Development of Time-Accurate FiniteElement Techniques for First-Order Hyperbolic Systems withParticular Emphasis on the Compressible Euler Equations.NASA Technical Report NASA-CR-204772, NASA: MoffetField, 1982.

Tezduyar T and Hughes T. Finite element formulations for con-vection dominated flows with particular emphasis on thecompressible Euler equations. In Proceedings of AIAA 21stAerospace Sciences Meeting , AIAA Paper 83-0125, Reno,1983.

Tezduyar T and Osawa Y. Methods for parallel computationof complex flow problems. Parallel Computing 1999;25:2039–2066.


Tezduyar T and Osawa Y. Finite element stabilization param-eters computed from element matrices and vectors. Com-puter Methods in Applied Mechanics and Engineering 2000;190:411–430.

Tezduyar T, Osawa Y, Stein K Benney R, Kumar V and Mc-Cune J. Numerical methods for computer assisted analysisof parachute mechanics. In Proceedings of 8th Conferenceon Numerical Methods in Continuum Mechanics (CD-ROM),Liptovsky, 2000.

Tezduyar T, Osawa Y, Stein K, Benney R, Kumar V and Mc-Cune J. Computational methods for parachute aerodynam-ics. In Computational Fluid Dynamics for the 21st Century,Hafez M, Morinishi K and Periaux J (eds). Springer: Berlin,2001; 290–305.

Tezduyar T and Park Y. Discontinuity capturing finite elementformulations for nonlinear convection-diffusion-reaction prob-lems. Computer Methods in Applied Mechanics and Engineer-ing 1986; 59:307–325.

Chapter 17 Finite Element Methods for Fluid Dynamics with Moving Boundaries and · PDF...

Documents

Transcript of Chapter 17 Finite Element Methods for Fluid Dynamics with Moving Boundaries and · PDF...