An adaptive mesh rezoning scheme for moving boundary ﬂows …amasud.web.engr.illinois.edu/Papers/Masud-Manish-Khurram... · 2008. 5. 30. · An adaptive mesh rezoning scheme for

Computers & Fluids 36 (2007) 77–91

www.elsevier.com/locate/compfluid

An adaptive mesh rezoning scheme for moving boundary flowsand fluid–structure interaction

Arif Masud *, Manish Bhanabhagvanwala, Rooh A. Khurram

Department of Civil and Materials Engineering, University of Illinois at Chicago, (M/C 246), 842 West Taylor Street,

2095 Engineering Research Facility, Chicago, IL 60607-7023, USA

Received 19 May 2005; accepted 20 July 2005Available online 9 December 2005

Abstract

Arbitrary Lagrangian–Eulerian (ALE) techniques provide a general framework for solving moving boundary flows and fluid–structure interaction problems. ALE formulations allow freedom of prescribing the fluid mesh velocity which can be independentof the velocity of the fluid particles. A major challenge in ALE descriptions lies in developing mesh moving techniques to update thefluid mesh and map the moving domain in a rational way. Exploiting the notion of arbitrary mesh velocity for the fluid domain, wehave developed an adaptive mesh rezoning technique for structured and unstructured meshes. The method has been applied tomeshes composed of triangles, quadrilaterals, as well as an arbitrary combination of these two element types in the computationaldomain. This feature of the proposed scheme is very attractive from practical problem solving viewpoint in that it allows kinemat-ically complex problems to be handled effectively. A variety of test cases are shown that involve single and/or multiple movingobjects. Embedding the mesh rezoning scheme in our flow solver, we also present some representative simulations of flows overmoving meshes.� 2005 Elsevier Ltd. All rights reserved.

1. Introduction

Finite element method (FEM) in fluid mechanics hasalso reached the speed and versatility that has tradition-ally been enjoyed by the finite element techniques insolid mechanics. Thus FEM has emerged as the mostpowerful and sophisticated numerical technique for theanalysis of coupled multiphysics interaction problemsinvolving moving boundaries. Fluid–structure Interac-tion (FSI) is a multiphysics problem that involves fluidsand solids that are usually treated in different mathemat-ical settings. The solid/structural mechanics literatureis dominated by the Lagrangian description wherethe material particles are glued to the computationaldomain. On the other hand both Lagrangian and Eule-

0045-7930/$ - see front matter � 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.compfluid.2005.07.013

* Corresponding author. Tel.: +1 312 996 4887; fax: +1 312 9962426.

E-mail address: [email protected] (A. Masud).

rian viewpoints have been employed in the domain offluids. The Lagrangian viewpoint for fluids, where themesh nodal points sit on the fluid particles, is preferredfor contained fluids that have only small fluid motion.For general flow problems with large amplitude motion,the Lagrangian methods can lead to severely entangledmeshes, resulting in the failure of the algorithms or grossinaccuracies in the results. In such situations the Eule-rian description is preferred wherein the computationalmesh stays fixed and the fluid particles move throughthe stationary grid. However, if an Eulerian descriptionis employed to model fluid in FSI problems, then sophis-ticated mathematical mappings between the stationaryand the moving boundaries are required.

In order to deal with general flow problems withmoving boundaries, Arbitrary Lagrangian–Eulerian(ALE) descriptions are employed. See [1,4,9,12,13,15,19,20,25,29,32,33,41]. ALE approach is based on anarbitrary motion of the reference frame, which is

mailto:[email protected]

78 A. Masud et al. / Computers & Fluids 36 (2007) 77–91

continuously rezoned in order to allow a precise repre-sentation of the moving interfaces. Accordingly, adap-tive mesh moving schemes that alter the mesh inresponse to changes in the boundary description havealso been an area of active interest in FSI [5,6,11,14,22,24,25,33,34,39–41]. The two general techniques thathave been employed by various investigators are: (i)moving mesh proportional to the primary boundarymotion, or (ii) solving the mesh motion through a pro-posed differential equation together with well-arrangedboundary nodes as the boundary conditions. Tezduyarand co-workers propose solving modified elasticityequations wherein element Jacobian is excluded in thecalculations, thereby introducing variable stiffeningeffect in the computational domain [22,36–38]. Wangand McLay [39] propose solving fourth order differentialequations for mesh rezoning. Brackbill and Saltzman [6]solve Laplace equation with some inhomogeneous termsto optimize smoothness, orthogonality and the variationin cell volumes. An equipotential method proposed byWinslow [40] regards the mesh lines as two intersectingsets of equipotentials, with each set satisfying Laplace�sequation in the interior with adequate boundary condi-tions. Employing Winslow�s method, Godunov and Pro-kopov [17] devised an algorithm for generating meshesfor initial boundary value problems in which changesin the boundary data are reflected in the changes inthe mesh. For a good review of the various recentapproaches for mesh motion, see e.g., [5,11,14,24,34,35] and references therein.

In transient fluid–structure interaction problems,monitoring the changes in the kinematic description ofthe solid and fluid continua becomes a delicate matter.A Lagrangian mesh for the structure deforms with thestructure and maintains a sharp definition of the movingboundary. On the other hand, mesh moving and meshregeneration techniques are required to accommodatethe changing geometric description of the fluid domain.For computational efficiency, a mesh update techniquethat minimizes the frequency of remeshing is attractive.This is facilitated by the adaptive mesh rezoning tech-niques (also termed as r-refinement) wherein inter-ele-ment connectivity in the mesh stays unchanged, whilethe nodal points are relocated to accommodate the spa-tial deformation imposed by the moving boundaries.This process is continued until the condition numberof the elements in the current mesh starts deteriorating.At this point, a new mesh is constructed by freezing thecalculations in time, and information is transferred fromthe previous mesh onto the new mesh using a projectionalgorithm.

An outline of the paper is as follows. Section 2 pre-sents the boundary value problem for mesh motion. Sec-tion 3 presents a modified variational form of theproblem that prevents the inversion of smaller elements

in the computational domain. An augmented Lagrang-ian formulation that results in an optimal enforcementof moving boundary constraints is presented next.Section 4 presents a conjugate gradient algorithm withdiagonal preconditioning to enhance the computationalefficiency of the proposed augmented Lagrangianmethod. Numerical results are shown in Section 5 andthe concluding remarks are presented in Section 6.

2. The boundary value problem for mesh motion

Let X � Rnsd be a bounded open set with piecewisesmooth boundary C; nsd P 2 denotes the number ofspatial dimensions. We assume that C admits thedecomposition

C ¼ Cm [ Cf ð1Þand

U ¼ Cm \ Cf ð2Þwhere Cm and Cf are the moving and the fixed portionsof the boundary respectively.

The formal statement of the boundary value problemis: Given g, the prescribed mesh displacement at themoving boundary, find the mesh displacement fieldu : X! Rnsd , such thatDu ¼ 0 in X ð3Þu ¼ g on Cm ð4Þu ¼ 0 on Cf ð5Þ

Eqs. (3)–(5) are the governing equation, the moving andthe fixed boundary conditions, respectively. The equiva-lent minimization problem can be formally written as

Find u 2 S such thatpðuÞ 6 pðvÞ 8v 2 S ð6Þwhere

pðuÞ ¼ 12ðru;ruÞ ð7Þ

Spaces relevant to the boundary value problem are

S ¼ fuju 2 ðH 1ðXÞÞnsd ; u ¼ g on Cm and u ¼ 0 on Cfgð8Þ

V ¼ fwjw 2 ðH 10ðXÞÞnsdg ð9Þ

where H1(X) denotes the space of functions in L2(X)with generalized derivatives also in L2(X). L2(X) denotesthe space of square-integrable functions on X. H 10ðXÞ is asubset of H1(X), whose members satisfy zero boundaryconditions [10].

The stationarity condition and integration by partsreveal that the Euler–Lagrange equations emanatingfrom p(u) correspond to the equations of the boundaryvalue problem (i.e. (3)–(5)).

A. Masud et al. / Computers & Fluids 36 (2007) 77–91 79

Remark. Eq. 3 works well for problems where (i) themeshes are composed of approximately equal-sizedelements, and (ii) the motion of the interface boundaryCm is of the order of the size of the elements. If themotion of the interface boundary is larger than the sizeof the elements adjacent to the moving boundary thenemploying (3) results in overturning of the elementswhich results in algorithm breakdown.

3. A modified discrete variational form of the boundary

value problem

For viscous flow calculations, the fluid mesh is locallyrefined in the areas where the small-scale effects of theboundary layers are of interest. Consequently, the fluidmeshes invariably have higher resolution close to themoving boundaries than in the far field. Our objectiveis that the smaller elements close to the moving bound-aries should translate together with the moving inter-faces with the least amount of distortion, and thelarger elements in the far field should absorb most ofthis distortion. For a given change in the condition num-ber of two elements having identical angles at the verti-ces but different element characteristic length parameterh, the nodes of the element with larger h can move moreas compared to that of the element with smaller h. Con-sequently, for a uniform change in the condition numberof elements in a mesh, the larger elements can be madeto absorb more of the interface boundary movement.Since larger elements are usually away from the movinginterfaces, so we need a mechanism to make the smallerelements stiffer as compared to the larger elements. Thiscan help move the finer boundary layer regions with theleast amount of element distortion, while translating thedeformation onto the softer larger elements that are inthe far field. This scheme results in a well-conditionedmesh for the subsequent time step calculations.

In order to prevent the inversion of the relativelysmall elements in the boundary layer region, and thusprevent the mesh breakdown, we introduce a constraintcondition over the elements. To fix ideas, consider a 2node linear element with nodes i and j and nodal dis-placements uhi and u

hj , respectively. We want the relative

difference in the value of the displacement field at thetwo nodes to be less than the element length h.

uhi � uhj�� 6 ahe ð10Þ) ruh�� 6 a ð11Þ

where he is the length of the element, and a 2 [0,1) is thetolerance parameter for element distortion. Conse-quently, the case of least distortion in smallest elementsis attained in the limit as a! 0, namely

ruh ¼ 0 ð12ÞThis condition is applied element wise. Consequently,the modified functional for mesh rezoning can be writtenas

PðuhÞ ¼ pðuhÞ þ 12

Xnele¼1

seðruh;ruhÞXe ð13Þ

where se > 0 in Xe is a bounded, non-dimensional weightfunction that is designed such that it imposes the con-straint condition strongly over the smaller elements ascompared to that over the larger elements. It thus intro-duces a stiffening effect that is inversely proportional tothe size of the elements. Consequently, the additionalterm in (13) makes the smaller elements stiffer as com-pared to the larger elements in the mesh. This spatiallyvarying stiffening effect causes the mesh to deformnon-uniformly by translating most of the deformationto the larger elements in the mesh that usually lie inthe far field.

3.1. Design of the weight function for mesh motion

We define se as the discrete weight function for theadditional term that imposes spatially varying stiffen-ing effect in the computational domain. This functionis assumed positive, non-dimensional and bounded.The key idea in the design of this function is that itshould yield a higher value for smaller elements and alower value for the larger elements in the mesh. Onesimple definition of se proposed in Masud and Hughes[27] is

se ¼ 1� Dmin=DmaxDe=Dmax

ð14Þ

where De, Dmax and Dmin represent the areas of the cur-rent, the largest and the smallest elements in a givenmesh, respectively. Because of the spatially varying stiff-ness, the fluid mesh at the moving boundary or the so-lid–fluid interface boundary moves almost like a rigidbody, and the deformation is absorbed by the large sizeelements that are usually situated in the far fields.

Remark. We can provide an automatic control on thechange in the condition number of the element bycomparing the Jacobian of the element in the current(deformed) mesh with its corresponding value in theinitial (undeformed) mesh. For example, if the currentJacobian is either smaller or larger than a specifiedpercentage of its corresponding value in the initialundeformed mesh, the calculations can be frozen in timeand a new mesh can be constructed around the currentlocation of the bodies. This procedure can help inmaintaining the quality of the mesh in successive timestep calculations.


Remark. The definition of se given in (14) provides con-trol on the stretching and shrinking of the elements. Acontrol on the change in the interior angles of the ele-ments can be provided by a tensorial se with non-zerooff-diagonal components. We are investigating thisaspect of mesh motion and will present our work in asubsequent paper.

3.2. The augmented Lagrangian formulation

In the above formulation, the moving boundary con-ditions are embedded in the admissible spaces of func-tions, i.e., Eqs. (8) and (9). In mathematical terms thisleads to a constrained minimization problem. In FSIproblems the fluid–structure interfaces deform as a func-tion of the response of the two continua, and this pro-cess evolves as a function of time. Consequently onedoes not know, a priori, the shape of the free surfacesand fluid–solid interface boundaries in this class ofproblems. Hence, objective is to seek solution to theproblem in a larger space of functions than are given

Fig. 1. Schematic diagram of the pitching airfoil.

Fig. 2. Angular motion of the airfoil as a function of time.

by (8) and (9) wherein the boundary conditions areembedded ab initio. To relax these boundary constraintswe introduce a Lagrange multiplier p that transforms

Fig. 3. Airfoil pitching motion as a function of time. (a) Initialundeformed mesh, (b) a positive angle of attack, (c) a negative angle ofattack.


Eq. (13) into an unconstrained problem. More precisely,we define

Pðuh; phÞ ¼ PðuhÞ þ ph;Quh � g� �

ð15Þ

Fig. 4. Zoomed view of the tip and tail of t

where Q :H1(X)! H1/2(C) is a linear and a continuousoperator (called trace operator) such that Qu = traceof u on C for every smooth u. The Lagrange multiplierp appears as an extra unknown, which can be obtained

he airfoil at various stages of motion.


through the solution of the saddle-point problem. Theappropriate spaces of functions for the unconstrainedproblem are

V ¼ fuju 2 ðH 1ðXÞÞnsdg ð16ÞW ¼ fpjp 2 ðH�1=2ðCÞÞnsdg ð17Þ

Remark. The stationarity conditions for {u,p} give riseto a mixed formulation. Although in the Lagrangemultiplier formulation we do not have to satisfy ab initiothe set of boundary constraints, however, the compat-ibility between the spaces of each variable is dictated bykey stability conditions established by Babuska [2] andBrezzi [7,8], and often become a major issue in devel-oping a convergent computational methods.

Remark. Barbosa and Hughes [3] have proposed amethod that circumvents the Babuska–Brezzi conditionfor Lagrange multipliers on the boundary. In theirmethod, Lagrange multipliers appear as additionalunknowns in the system.

In (15), Lagrange multipliers are the additionalunknowns that need to be solved for. In order to retainthe size of the system to that of the primal variables, wepropose an augmented Lagrangian formulation.

Peðuh; phÞ ¼ Pðuh; phÞ þe2

Quh � g�� 2 ð18Þ

where e is the user specified penalty parameter and j Æ jdenotes norm on W. The augmented Lagrangian formu-lation can be viewed as a combination of the penaltyfunction and the Lagrange multiplier method. This for-mulation combines the two concepts to eliminate manyof the disadvantages associated with either methodalone (see e.g. [18]). It can easily be proved that any sad-

Fig. 5. Schematic diagram of store separation.

dle point of Pe is a saddle point of P and that the con-verse also holds. This is due to the fact that ejQuh � gj2vanishes when the constraint Quh = g is identicallysatisfied.

Fig. 6. Store separation problem. Spatial configuration at time (a)t = 0.01, (b) t = 0.15, (c) t = 0.45.


Remark. It is important to note that for p = 0, we have

Peðu; 0Þ ¼ PðuÞ þe2

Qu� gj j2 ð19Þ

This is the classical penalty function formulation for theconstraint Qu = g. The advantage of the augmentedLagrangian formulation is that due to the termhp,Qu � gi, the exact solution of the problem (13) canbe determined without making e tend to infinity, which,using ordinary penalization methods would have the ef-fect of causing deterioration in the conditioning of thesystem to be solved.

The variational equation emanating from (18) is

0 ¼ dde

Peðuþ ew; p þ eqÞ� ��

e¼0

¼ ðrw;ruÞ þ seðrw;ruÞ þ q;Qu� gh iþ Qw; ph i þ e Qw;Qu� gh i ð20Þ

3.3. The finite element form

Let Vh and Wh represent the finite-dimensional sub-spaces of V and W, respectively. We think of Vh andWh as typical finite element spaces involving piecewisepolynomial interpolations. The finite element formemanating from the variational problem (20) can beexpressed as: find {uh,ph} 2 Vh · Wh such thatBeðwh; qh; uh; phÞ ¼ Lðfwh; qhgÞ 8fwh; qhg 2 V h � W h

ð21Þwhere

Beðwh; qh; uh; phÞ ¼ ðrwh;ruhÞ þ e Qwh;Quh� �

þ Qwh; p� �

þ q;Quh� �

þXnele¼1

seðrwh;ruhÞXe

ð22Þ

Fig. 7. A schematic diagram for the moving shock wave problem.

Lðfwh; qgÞ ¼ e Qwh; g� �

þ q; gh i ð23Þ

Fig. 8. High resolution region of the mesh following the evolution ofthe interval layers in the fluid domain. (a) Initial undeformed mesh attime t = 0.0. An intermediate deformed mesh at time (b) t = 0.25, (c)t = 0.75.


4. A preconditioned conjugate gradient method for

augmented Lagrangian formulation

This section presents a preconditioned conjugategradient algorithm for the augmented Lagrangian for-mulation. This algorithm is a modification of the pre-conditioned conjugate gradient algorithm presented in[16]. For a detailed account of conjugate gradient algo-rithms see e.g., [16,18] and references therein.

Remark. For 0 < qj 6 2e, and for all p0 2W, thesequence uh defined by the algorithm converges to thesolution u of p(u) (see, e.g., [18]). We takeqj ¼ e ð24Þ

Box 1. Given the linear system Av = b with aconstraint Qv � g = 0 and the preconditioner P,where A and P are symmetric, positive definite.

Step 1. Initialize and solve uncoupled equations:

r0 ¼ b ð25Þv0 ¼ 0 ð26ÞP0 ¼ 0 ð27Þ

for l ¼ 1; 2; . . . ;N eqif Akl ¼ 0 for all k < l then

vl ¼ rl=All ð28Þrl ¼ 0 ð29Þ

endif

continue

q1 ¼ z1 ¼ P�1r0 ð30Þ

Step 2. Iterate for j = 1,2, . . . , jmax

Perform line search to update solution and residual:

aj ¼rj�1 � zjqj � Aqj

ð31Þ

vj ¼ vj�1 þ ajqj ð32Þrj ¼ rj�1 � ajAqj ð33Þ

Check convergence (d is a user-defined tolerance):

if krjk 6 dkr0k return ð34ÞUpdate the multiplier:

pjþ1 ¼ pj þ qjQvj � gj ðq > 0Þ ð35ÞrjjC ¼ rjjC þ pjþ1 ð36ÞCompute new conjugate search direction:

zjþ1 ¼ P�1rj ð37Þ

bjþ1 ¼rj � zjþ1rj�1 � zj

ð38Þ

qjþ1 ¼ zjþ1 þ bjþ1qj ð39Þ

5. Numerical simulations

The finite element formulation presented in (21)–(23)has been implemented for 4-node quadrilaterals and 3-node triangles. It has been applied to meshes that arecomposed of bilinear quadrilaterals and linear triangles,and has also been applied to composite meshes that arecomposed of a combination of these two element typesin the same domain. In this section we present variousproblems from different fields of engineering thatrequire a mesh moving technique that is embedded inthe solution procedure.

We first define the various parameters that describethe geometry of the problems presented in this section.In the geometric descriptions, X specifies the fluiddomain, and Cm and Cf indicate the moving and thefixed boundaries of the fluid mesh, respectively. Thenodal displacements are specified on the moving bound-aries by functions gx(X, t), gy(X, t), and gh(X, t), whichare functions of time and the spatial coordinates. T spec-ifies the total time for the simulation.

5.1. Pitching airfoil

Analysis of a pitching airfoil is important for study-ing the aerodynamic stability as well as the dynamicbehavior of an airplane wing (see e.g., [31]). Fig. 1 pre-sents the schematic diagram of the pitching airfoilproblem.

An unstructured triangular mesh is generated aroundthe airfoil. The mesh is composed of 9177 3-node trian-gles with 4683 nodes. The airfoil is given a prescribedangular rotation (about its centroid) described by anunder damped equation given in (40). The maximumpitch angle is 30� and the various parameters in (40)are: n = 0.035, xn = 100, xd = 100, V0 = 1.0, X0 = 0.38.

ghðX ; tÞ ¼ 100e�nxnt X 0 cosðxd tÞþV 0þ nxnX 0

xdsinðxd tÞ

� �

ð40ÞThe graph shown in Fig. 2 presents the angular pitchingmotion of the airfoil. The spatial orientation at varioustime levels is shown in Fig. 3. Fig. 4 shows the close upview of the deformed mesh at various stages of deforma-tion. Maintaining the quality of the spatial mesh isimportant for a uniform spatial resolution of the solu-tion in time-dependent adaptive mesh simulations. Itcan be seen that the quality of the mesh around the air-foil, especially around the tip and the tail is comparableto the quality in these regions in the initial undeformedmesh.

5.2. Store separation

Store separation is a typical example of multi-bodymovement in aerodynamics (see Fig. 5). Such simula-

Fig. 10. Pulsatile flow of blood causing expansion in the distensibleartery wall. An intermediate deformed mesh at time (a) t = 0.01, (b)t = 0.04, (c) t = 0.36, (d) t = 0.51.


tions are carried out to study the motion of objectsdropped from flying vehicles [36]. In order to modelthe interaction of boundary layers from each of the indi-vidual bodies, the region between the bodies is usuallydiscretized with a higher density of elements, as shownin Fig. 6. As the bodies move away, the smaller elementsin this dense region are stretched, thus providing a con-tinuous variation in the spatial mesh for subsequenttime step calculations. Once the bodies are sufficientlyfurther away such that their boundary layers do notdirectly interact, then the region between the bodiescan be discretized via larger elements while keeping fineelements only in the immediate vicinity of each of thebodies to capture the effects of the individual boundarylayers. For the purpose of presenting the ideas, the com-putational domain is discretized via 21,251 3-node trian-gles with 11,012 nodal points. As can be seen inFig. 6(a)–(c), most of the elements are placed aroundand in between the two bodies. Although the elementsin between the two bodies get stretched and distortedmany times their initial size and shape, overturning ofelements does not occur. The mesh shown in Fig. 6 isonly intended to serve as a test case for the mesh movingmethod when applied to this class of problems. Anactual numerical simulation would necessitate a meshwith still higher density around the bodies.

The function that models the motion of the fallingobject is given in (41). Fig. 6 shows the two bodies at dif-ferent time levels to demonstrate the application of themesh moving scheme for the store separation problems.

gxðX ; tÞ ¼ tgyðX ; tÞ ¼ 3:5714X 2 � 4:6071X þ 0:0107

ghðX ; tÞ ¼ogyox

9>>>=>>>;

ð41Þ

where gx, gy and gh represent the x displacement, the ydisplacement and the rotation (about its centroid) ofthe falling object. In an actual simulation the motionof the falling object is dictated by the gravitationalforces and the drag forces, and the trajectory and orien-tation of the object is an outcome of the entire compu-tational process.

Fig. 9. A schematic diagram of the mo

5.3. Shock wave propagation

This problem is designed to show that the proposedmesh rezoning scheme can also be applied to move theinternal layers containing higher mesh density in a nar-row banded region. These internal mesh layers caneither be given a prescribed motion, or they can be madeto follow certain features in the computed solution,namely, traveling shock waves or evolving zones of

ving pulse in an idealized artery.

Fig. 11. Schematic diagram of the oscillating beam.

Fig. 12. Oscillating beam in fluid domain (unstructured triangularmesh). (a) Initial undeformed mesh, (b) deformed mesh showingmaximum tip amplitude.

Fig. 13. Zoomed view of the mesh around region of maximumdeformation. (a) Tip region undeformed mesh, (b) tip region deformedmesh.



discontinuity in the computed flow field. For example,supersonic flying vehicles produce shock waves aroundtheir leading edges [21]. The angle between the shockand the body can change as a function of the changein the angle of attack of the body. To solve similar classof problems, Tezduyar and co-workers have proposedan Enhanced Discretization Interface-Capturing(EDICT) technique [30].

In this study, we consider a triangular body flying atsupersonic speed that generates a shock wave at theleading edge. Fig. 7 shows a schematic diagram of theproblem. The domain is discretized with 5777 linear tri-angles and the total number of nodes is 3087. Eq. (42)mimics the kinematics of the shock wave and therotation of the body, where gh1 mimics the orientationof the upper shock and gh2 mimics the orientation ofthe lower shock. Fig. 8 shows the graphical representa-tion of the solution adaptive mesh at various time levels.Once again, the objective of this simulation is to showthe application of the proposed method to the changein the orientation of the internal layers in the mesh.

gh1ðX ; tÞ ¼ 3:2 sinð2ptÞgh2ðX ; tÞ ¼ �3:2 sinð2ptÞ

�ð42Þ

Fig. 14. Computed pressure field for half cycle oscillation (

5.4. Pulsatile motion in distensible arteries

Modeling of blood flow through distensible arteries isan example from biofluid dynamics. The physics of theproblem involves a viscous incompressible fluid interact-ing with a compliant flexible elastic multilayered arterialwall. A simple structured mesh is generated to show theapplication of the mesh rezoning method to study thepulsatile motion of blood through a 2D idealization ofa flexible artery. Fig. 9 shows the schematic diagramof the problem. The mesh is composed of 2500 4-nodeelements.

In this study a half sine-wave travels along the artery.The pulse position at different time levels is shown inFig. 10.

5.5. Periodically oscillating beams

A beam oscillating in the fluid domain is a typicalexample of fluid–structure interaction problems. Inaddition to its application in civil and mechanicalengineering systems, such problems are also of funda-mental significance in Micro Electro-Mechanical System(MEMS) devices.

t = 0–20): (a) t = 14, (b) t = 15, (c) t = 17, (d) t = 19.


Fig. 11 shows the computational domain. The mesh iscomposed of 9509 3-node triangles with 5024 nodes. A

Fig. 15. Schematic diagram of the multiple body motion problem.

Fig. 16. Motion of multiple objects in the fluid domain. An interme-diate deformed mesh at time (a) t = 0.01, (b) t = 0.50.

flexible beam is attached to a circular base and it under-goes cyclic motion in its fundamental mode of vibrationas given by Eq. (43). The fluid is flowing from left toright with a given flow velocity. Fig. 12 shows the beamposition at different time levels during the transient anal-ysis, while Fig. 13 presents the zoomed view of the meshat the tip of the beam. The multiscale finite element for-mulation for the Navier–Stokes equations [26,28] isemployed to solve the fluid flow problem to obtain thepressure field shown in Fig. 14.

A0 ¼ð4X 2 � 4X þ 1Þ

75gyðX ; tÞ ¼ A0 sinð2ptÞ

9=; ð43Þ

Fig. 17. Zoomed view of the meshes around moving bodies.


5.6. Multiple moving cylinders

This is an example from heat transfer problemswherein a coolant fluid flows around high temperatureslender pipes that undergo large amplitude oscillationsbecause of fluid–structure interaction effects. The multi-scale/stabilized finite element method for the Navier–Stokes equations [26,28] is employed to solve the flowfield.

For this cross-sectional two-dimensional model, cir-cles of unit diameter represent the transverse cylinders.The cylinders are 0.5D apart where D is the diameterof the cylinder. Fig. 15 shows the schematic diagramof the problem. Similar problems in 2D and 3D havebeen solved by Johnson and Tezduyar in [23,24].

The multi-body motion is simulated with sine func-tions given in (44).

gyiðx; tÞ ¼ A0i sinð2ptÞ ð44Þ

where A0i is the maximum amplitude for the body �i�.Fig. 16 shows the displaced positions of the cylindersat various time levels and Fig. 17 shows the close upview of the deformed meshes at two extreme configura-tions. Fig. 18(a) presents the snapshot of the computedpressure field around the moving cylinders at time t0,and Fig. 18(b)–(c) at the beginning of the fourth quarter

Fig. 18. Zoomed view of the meshes around moving bodies. (a) Beginn

cycle, respectively. In these snapshots the odd numbercylinders (1, 3 and 5) are translating in the �y directionand the even number cylinders are translating in the +ydirection.

6. Conclusions

We have presented an adaptive mesh rezoningscheme for simulation and analysis of fluid dynamicsproblems that involve moving and deforming bound-aries. The method is based on a Galerkin/least-squarestype modification of the Laplace equation that intro-duces spatially varying scalable-incompressibility effectsin the computational domain. Smaller elements behavestiffer as compared to the larger elements, and thusmaintain their shape during the rezoning process. Sincesmaller elements invariably lie in the boundary layerregions, the quality of subsequent meshes is comparableto that of the original mesh. The motion of the fluid–solid interface boundaries is accommodated via anaugmented Lagrangian enforcement of the evolvingboundary conditions. It thus results in an optimalenforcement of the interface constraints that are dic-tated by the continuum requirements in the problem.The method can also be employed to make the internal

ing of the cycle, (b) intermediate step, (c) one-fourth of the cycle.


layers in the fluid mesh follow some solution features(e.g., the shock fronts) which are a function of the com-puted solution in time-dependent calculations. A varietyof test cases from various fields of engineering are pre-sented to show the range of applicability of the proposedmesh rezoning method.

Acknowledgement

Support for this work was provided by the US Officeof Naval Research under grant N00014-02-1-0143. Thissupport is gratefully acknowledged.

References

[1] Aquelet N, Souli M, Olovsson L. Euler–Lagrange coupling withdamping effects: application to slamming problems. Comput MethAppl Mech Eng 1984;29:329–49.

[2] Babuska I. Error bounds for finite element method. Numer Math1971;16:322–33.

[3] Barbosa HJC, Hughes TJR. The finite element method withLagrange multipliers on the boundary: circumventing the Bab-uska–Brezzi condition. Comput Meth Appl Mech Eng 1991;85(1):109–28.

[4] Belytschko T, Kennedy JM, Schoeberie DF. Quasi-Eulerian finiteelement formulation for fluid–structure interaction. ASME J PressVess Technol 1980;102:62–9.

[5] Bottasso CL, Detomi D, Serra R. The ball-vertex method: anew simple spring analogy method for unstructured dynamicmeshes. Comput Meth Appl Mech Eng 2005;194(39–41):4244–64.

[6] Brackbill JU, Saltzman JS. Adaptive zoning for singular problemsin two dimensions. J Comput Phys 1982;46:342–68.

[7] Brezzi F, Fortin M. Mixed and hybrid finite element meth-ods. New York–Heidelberg–Berlin: Springer-Verlag; 1991.

[8] Brezzi F. On the existence, uniqueness and approximation ofsaddle-point problems arising from Lagrange multipliers. Rev FrAutomat Inform Recher Oper. Ser Rouge Anal Numer 1974;R-2:129–51.

[9] Chen JS, Liu WK, Belytschko T. Arbitrary Lagrangian–Eulerianmethods for materials with memory and friction. In: TezduyarTE, Hughes TJR, editors. Recent developments in computationalfluid dynamics, AMD-vol. 95. 1988.

[10] Ciarlet PG. The finite element method for elliptic prob-lems. Amsterdam: North-Holland; 1978.

[11] Degand C, Farhat C. A three-dimensional torsional springanalogy method for unstructured dynamic meshes. Comput Struct2002;80:305–16.

[12] Donea J. Arbitrary Lagrangian–Eulerian finite element methods.In: Belytschko T, Hughes TJR, editors. Computational methodsfor transient analysis. Amsterdam: North-Holland; 1983. p.473–516.

[13] Donea J, Fasoli-Stella P, Giuliani S. Lagrangian and Eulerianfinite element techniques for transient fluid–structure inter-action problems. In: Transactions of the 4th internationalconference on structural mechanics in reactor technology, PaperB1/2. 1977.

[14] Farhat C, Degand C, Koobus B, Lesoinne M. Torsional springsfor two-dimensional dynamic unstructured fluid meshes. ComputMeth Appl Mech Eng 1998;163:231–45.

[15] Farhat C, Pierson K, Degand C. Multidisciplinary simulation ofthe maneuvering of an aircraft. Eng Comput 2001;17:16–27.

[16] R.M. Ferencz, Element-by-element preconditioning techniquesfor large-scale, vectorized finite element analysis in nonlinear solidand structural mechanics. Ph.D. Thesis, Division of AppliedMechanics, Stanford University, 1989.

[17] Godunov SK, Prokopov GP. The use of moving meshes in gas-dynamic calculations. USSR Comput Math Math Phys 1972;12(2):182–91.

[18] Glowinski R, Tallec PL. Augmented Lagrangian and operator-splitting methods in nonlinear mechanics. SIAM Studies inApplied Mathematics. Philadelphia, Pennsylvania: Society forIndustrial and Applied Mathematics; 1989.

[19] Hassan O, Probert EJ, Morgan K, Weatherill NP. Unsteady flowsimulation using unstructured meshes. Comput Meth Appl MechEng 2000;189:1247–75.

[20] Hughes TJR, Liu WK, Zimmerman TK. Lagrangian–Eulerianfinite element formulation for incompressible viscous flows.Comput Meth Appl Mech Eng 1984;29:329–49.

[21] Hughes TJR, Mallet M. A new finite element formulation forcomputational fluid dynamics: IV. A discontinuity-capturingoperator for multidimensional advective–diffusive systems. Com-put Meth Appl Mech Eng 1986;58:329–39.

[22] Johnson AA, Tezduyar TE. Mesh update strategies in parallelfinite element computations of flow problems with movingboundaries and interfaces. Comput Meth Appl Mech Eng 1994;119:73–94.

[23] Johnson AA, Tezduyar TE. Simulation of multiple spheres fallingin a liquid-filled tube. Comput Meth Appl Mech Eng 1996;134:351–73.

[24] Johnson AA, Tezduyar TE. Advanced mesh generation andupdate methods for 3D flow simulations. Comput Mech 1999;23:130–43.

[25] Koobus B, Farhat C. Second-order time-accurate and geometri-cally conservative implicit schemes for flow computations onunstructured dynamic meshes. Comput Meth Appl Mech Eng1999;170:103–29.

[26] Masud A. On a stabilized formulation for incompressible Navier–Stokes equations. In: Kanayama H, editor. Proc 6th Japan–USinternational symposium on flow simulation and modeling,Fukuoka, Japan. May 2002, p. 33–8.

[27] Masud A, Hughes TJR. A space–time Galerkin/least-squaresfinite element formulation of the Navier–Stokes equations formoving domain problems. Comput Meth Appl Mech Eng 1997;146:91–126.

[28] Masud A, Khurram RA. A multiscale finite element method forthe incompressible Navier–Stokes equation, Comput Meth ApplMech Eng, in press.

[29] Masud A, Khurram R, Bhagwanwala M, A stable method forfluid–structure interaction problems. In: Yao ZH, Yuan MW,Zhong WX, editors. Proceedings CD-ROM of the sixth worldcongress on computational mechanics. ISBN 7-89494-512-9,Beijing, China, 2004.

[30] Mittal S, Aliabadi S, Tezduyar T. Parallel computation ofunsteady compressible flows with the EDICT. Comput Mech1999;23:151–7.

[31] Mittal S, Tezduyar TE. A finite element study of incompressibleflows past oscillating cylinders and airfoils. Int J Numer MethFluids 1992;15:1073–118.

[32] Lesoinne M, Farhat C. Geometric conservation laws for flowproblems boundaries and deformable meshes, and their aero-elastic computations. Comput Meth Appl Mech Eng 1996;134:71–90.

[33] Nkonga B. On the conservative and accurate CFD approxima-tions for moving meshes and moving boundaries. Comput MethAppl Mech Eng 2000;190:1801–25.

[34] Stein K, Tezduyar T, Benney R. Mesh moving techniques forfluid–structure interactions with large displacements. J Appl Mech2003;70:58–63.


[35] Stein K, Tezduyar TE, Benney R. Automatic mesh update withthe solid-extension mesh moving technique. Comput Meth ApplMech Eng 2004;193:2019–32.

[36] Tezduyar T, Aliabadi S, Behr M, Johnson A, Kalro V, Litke M.Flow simulation and high performance computing. Comput Mech1996;18:397–412.

[37] Tezduyar TE, Behr M, Liou J. A new strategy for finite elementcomputations involving moving boundaries and interfaces—thedeforming-spatial-domain/space–time procedure: I. The conceptand the preliminary tests. Comput Meth Appl Mech Eng 1992;94:339–51.

[38] Tezduyar TE, Behr M, Mittal S, Johnson AA. Computation ofunsteady incompressible flows with the stabilized finite ele-

ment methods—space–time formulations, iterative strategies andmassively parallel implementations. New methods in transientanalysis, PVP-vol. 246/AMD-vol. 143. New York: ASME; 1992.p. 7–24.

[39] Wang HP, McLay RT. Automatic remeshing scheme for model-ing hot forming process. J Fluid Eng 1986;108:465–9.

[40] Winslow AM. Equipotential zoning of two-dimensional meshes.University of California, Lawrence Radiation Laboratory Report,UCRL-7312, 1963.

[41] Zhao Y, Forhad A. A general method for simulation of fluid flowswith moving and compliant boundaries on unstructured grids.Comput Meth Appl Mech Eng 2003;192:4439–66.

An adaptive mesh rezoning scheme for moving boundary flows and fluid-structure interactionIntroductionThe boundary value problem for mesh motionA modified discrete variational form of the boundary value problemDesign of the weight function for mesh motionThe augmented Lagrangian formulationThe finite element form

A preconditioned conjugate gradient method for augmented Lagrangian formulationNumerical simulationsPitching airfoilStore separationShock wave propagationPulsatile motion in distensible arteriesPeriodically oscillating beamsMultiple moving cylinders

ConclusionsAcknowledgementReferences

Documents

An adaptive mesh rezoning scheme for moving boundary ﬂows …amasud.web.engr.illinois.edu/Papers/Masud-Manish-Khurram... · 2008. 5. 30. · An adaptive mesh rezoning scheme for