An adaptive remeshing strategy for flows with moving ...frey/papers/meshing/... · The primary objective of this work is to extend the capability of the arbitrary Lagrangian–Eulerian

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2007; 71:1009–1050Published online 5 February 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1971

An adaptive remeshing strategy for flows with moving boundariesand fluid–structure interaction

P. H. Saksono, W. G. Dettmer and D. Peric!,†

Civil and Computational Engineering Centre, School of Engineering, University of Wales Swansea,Swansea, SA2 8PP, U.K.

SUMMARY

The primary objective of this work is to extend the capability of the arbitrary Lagrangian–Eulerian (ALE)-based strategy for solving fluid–structure interaction problems. This is driven by the fact that the ALEmesh movement techniques will not be able to treat problems in which fluid–structure interface experienceslarge motion. In addition, for certain problems the need arises to capture accurately flow features, such asa region with high gradients of the solution variables. This can be achieved by incorporating an adaptiveremeshing procedure into the solution strategy.

As in our previous works (Comput. Methods Appl. Mech. Eng. 2006; 195:1633–1666; Comput. MethodsAppl. Mech. Eng. 2006; 195:5754–5779), here, the fluid flow is governed by the incompressible Navier–Stokes equations and modelled by using stabilized low order velocity–pressure finite elements. The motionof the fluid domain is accounted for by an arbitrary Lagrangian–Eulerian (ALE) strategy. The flexiblestructure is represented by means of appropriate standard finite element formulations while the motion ofthe rigid body is described by rigid body dynamics. For temporal discretization of both fluid and solidbodies, the discrete implicit generalized-! method is employed. The resulting strongly coupled set ofnon-linear equations is then solved by means of a novel partitioned solution procedure, which is based onthe Newton–Raphson methodology and incorporates full linearization of the overall incremental problem.

Within the adaptive solution strategy, the quality of fluid mesh and the solution quality indicator areevaluated regularly and compared against the appropriate remeshing criteria to decide whether a remeshingstep is required. The adaptive remeshing procedure follows closely the standard computational procedurein which the adaptive remeshing process produces a mesh that can capture salient features of the flowfield. For the problems under consideration in this work the motion of the fluid boundary very often resultsin boundaries with very high curvatures and a fluid domain that contain areas with small cross-sections.To be able to generate meshes that give result with acceptable accuracy these local geometrical featuresneed to be included in determining the element density distribution. The numerical examples demonstratethe robustness and efficiency of the methodology. Copyright 2007 John Wiley & Sons, Ltd.

Received 26 May 2006; Revised 4 October 2006; Accepted 26 October 2006

KEY WORDS: finite element; fluid–structure interaction; partitioned approach; Newton–Raphson solutionmethod; adaptive strategy

!Correspondence to: Djordje Peric, Civil and Computational Engineering Centre, School of Engineering, Universityof Wales Swansea, Swansea, SA2 8PP, U.K.

†E-mail: [email protected]

Contract/grant sponsor: EPSRC of United Kingdom; contract/grant number: GR/R92318

Copyright 2007 John Wiley & Sons, Ltd.

1010 P. H. SAKSONO, W. G. DETTMER AND D. PERIC

1. INTRODUCTION

In recent years, numerical solution of fluid–structure interaction problems has become a focus ofmajor research activity (see. e.g. [1, 2]). Generally, these problems are solved using some form ofthe mesh moving strategy, such as Lagrangian or arbitrary Lagrangian–Eulerian (ALE) techniques(see. e.g. [3–9] and the references therein). Due to possibility of mesh distortion such strategiesoften need to be complemented by a remeshing strategy. For certain classes of problems techniquesthat use fixed fluid mesh are employed. These are, however, supplemented by an appropriate strategyfor tracking the fluid–solid interfaces. In recent years strategies such as fictitious domain method[10–12] or immersed boundary methods [13, 14] have become popular.

In this work, fluid–structure interaction problems in which the spatial configuration of the fluidand solid domains experiences large motion and/or deformation are considered. In order to obtainsolutions of such problems we employ a specific combination of discretization strategies, a recentlyproposed solution procedure for the coupled sets of discrete non-linear equations and an adaptiveremeshing procedure. In particular, we employ a finite difference type time integration scheme asopposed to the space–time strategies [5, 7, 8].

The computational ingredients of the adopted strategy are as follows:For the modelling of the incompressible Newtonian fluid flow we employ the SUPG/PSPG

finite element formulation adapted to a moving domain, which allows for computationally conve-nient linear equal order velocity–pressure interpolations. This stabilized methodology is based onthe extensive work by Hughes and co-workers and has been further developed by Tezduyar andothers [15].

An ALE description is used to account for the deformation of the fluid domain which arisesfrom the motion and deformation of the solid structure.

The solid structure, depending on the physical problem under consideration, may be modelledby an appropriate standard finite element technique, involving membrane, beam, shell and/orcontinuum elements. In some cases it is sufficient to consider the solid structure as a rigid body.

The strategy adopted in this work allows the employment of non-matching fluid and structuralfinite element meshes. Hence, the fluid and the structural domains can be discretized independently.A simple interpolation strategy based on a finite element-type discretization of the interface isemployed to strongly enforce the kinematic constraints between fluid and solid. The incorporationof these constraints into the weak form of the overall problem leads to a straightforward transferof the traction fields in terms of the nodal finite element forces.

The generalized-! method is employed for the integration in time [16, 17]. This method belongsto the class of discrete and implicit single step integration schemes. For linear problems the schemecan be shown to be second order accurate and unconditionally stable. It allows user controlleddamping of high unresolved frequencies.

The fully discretized model consists of coupled sets of non-linear equations. The coupling canbe described briefly as follows: the deformation of the structure is driven by the traction forcesexerted by the fluid at the fluid–solid interface. The structural displacements, on the other hand,define the geometry and the geometry changes of the fluid domain. In a discrete FE setting thissituation may be regarded as a coupled three field problem, involving the fluid flow, the motionof the fluid mesh and the structural dynamics.

A robust and efficient solution procedure is required to compute the complete set of unknownsat the current time instant. The partitioned solution scheme used in this work relies on the Newton–Raphson procedure, which incorporates the full linearization of the incremental problem and hence

Copyright 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2007; 71:1009–1050DOI: 10.1002/nme

ADAPTIVE STRATEGY FOR FLUID–STRUCTURE INTERACTION 1011

exhibits asymptotically quadratic convergence of the solution for all problem unknowns. Alternativestrategies, some of which resemble Quasi-Newton procedures or employ linearizations obtainedby numerical differentiation, have been suggested by Tezduyar and co-workers [18, 19], Matthiesand Steindorf [20, 21] and others [8, 22, 23].

The aforementioned framework has been implemented and tested extensively on standard bench-marks and some specific applications with a very encouraging results [3, 4, 24, 25].

For problems that involve very large motion of fluid boundaries the use of ALE strategy toaccount for the motion of fluid domain soon results in a finite element mesh with over-distortedelements which prevent the solution process to proceed. Therefore, additional numerical ingredientis required, namely, the adaptive remeshing procedure. In practice, by recognizing the fact that theappropriate mathematical framework for these complex problems is missing different criteria canbe used to automatically trigger the remeshing process (e.g. element aspect ratio [26], conditionnumber [27] or the quality of the solution [28]).

The key components of the adaptive remeshing strategy are the estimation of the solution error,the determination of element density distribution, mesh regeneration and methods for the transferof the solution variables.

Over the years different error indicators have been proposed for viscous incompressible flow(see. e.g. [29–32]). In this works, we use a norm of the velocity gradient as an indicator todetermine which part of the fluid domain requires refinement or coarsening. The value of thisindicator is then used to calculate the element size of the new mesh.

For problem where the fluid domain is fixed throughout the computation, the element size ofthe new mesh is calculated from the finite element asymptotic rate of convergence which relatesthe error on the element to some power of the element size [29, 30, 32]. In addition, the changeof spatial configuration of the domains required local geometrical features to be included in thecalculation of element density distribution [33, 34].

The generation of the new fluid mesh can be performed by using an unstructured mesh generatorbased on any standard technique such as advancing front or Delauney triangulation. Finally at somestage, the converged solution and the boundary conditions at time tn on the old mesh need to betransferred to the new mesh. The transfer is carried out by interpolating both the solution variablesand boundary conditions from the old mesh to the new one and borrows from previously designedprocedures for solid mechanics [35–38].

The paper is organized as follows; the governing equations for the motion of fluid, rigid bodyand solid are given in Section 2. Section 3 briefly describes the stabilized finite element formulationused for spatial discretization of the fluid domain. Techniques employed to move fluid mesh arepresented in Section 4 followed by a section on finite element formulation for the solid structure.In Section 6 the time integration scheme is given. Discrete overall problem and solution algorithmare presented in Sections 7 and 8, respectively. Details of the adaptive remeshing strategy for fluiddomain are outlined in Section 9. Finally, numerical examples involving interaction between fluidflow and rigid body or flexible structure are discussed in Section 10.

2. GOVERNING EQUATIONS

2.1. Fluid mechanics on a moving domain

An essential feature of the fluid–structure interaction is the motion of the boundary of the fluiddomain. The geometry of the fluid domain may change substantially during the time domain of



interest. Therefore, it is convenient to formulate the problem in the ALE description relying on amoving reference frame, in which the conservation laws are expressed.

In this context it is well-established (see e.g. [39, 40]) that the time derivative of the velocity uof the fluid particle which traverses through the co-ordinate x of the reference frame at a specifictime instant can be written as

DuDt

= "xu (u # v) + u (1)

where v= !x/!t is the velocity of the reference point. The operator "x(•) denotes the derivativeswith respect to the current referential co-ordinates x. The expression u corresponds to the changeof the material particle velocity, which is noted by an observer travelling with a point on thereference frame. The velocity u # v is denoted as the convective velocity. In the framework ofthe finite element method, the moving reference frame is identified with the finite element mesh.The Eulerian or Lagrangian representations of the material time derivative of u are easily recoveredfrom Equation (1) by setting v= 0 or v=u, respectively.

Let !f denotes the current fluid referential domain, the momentum conservation law and thecontinuity equation for incompressible flow are formulated in the referential description as

"f(u + ("xu)(u # v) # ff) # "x · rf = 0 in !f $ I (2)

"x · u= 0 in !f $ I (3)

where "f, ff and rf represent, respectively, the fluid density, the volume force vector and the Cauchystress tensor acting on the fluid. The constitutive equation for the Newtonian fluid is given in astandard way as

rf = # pI + 2#"sxu (4)

where p represents the pressure, I is the second order identity tensor, # denotes the fluid viscosityand "s

x(•) is the symmetric gradient operator with respect to the current referential co-ordinates x.

2.2. Solid mechanics

In a standard Lagrangian description, the conservation of momentum of a solid continuum maybe expressed in spatial description as

"s(d # bs) # " · rs = 0 in !s $ I (5)

where "s is the current density of the deformed solid and the vector d represents the displacementfield, whereas the body forces are given by the vector bs. The symmetric second order rs denotesthe Cauchy stress acting in the solid that is related to appropriate local strain measures by aconstitutive relation suitable to model the behaviour of the solid material.

If the solid body is considered as rigid then Equation (5), in two dimensions, is simply re-placed by

Md + Cd + Kd= fr (6)

I$$ + C$$ + K$$ = T (7)



where d= [dx , dy]T represents the rigid body translation, $ describes the rigid body rotation withrespect to the centre of gravity cg of the rigid body. The quantities M, C, K, I$, C$ and K$ denotethe mass, the damping and the stiffness moduli for translational and rotational degrees of freedom,respectively, whereas fr is the force vector acting on the rigid body and T the component alongthe out-of-plane basis vector ez of the moment T.

2.3. Boundary conditions

Let the boundary of the fluid domain be denoted by "f = "gf %"h

f %"i and the boundary of thesolid domain be represented by "s = "g

s % "hs %"i . Then the following boundary conditions have

to be satisfied

u # gf = 0 on "gf $ I (8)

rfn # hf = 0 on "hf $ I (9)

d # gs = 0 on "gs $ I (10)

rsn # hs = 0 on "hs $ I (11)

u # d= 0 on "i $ I (12)

(u # v)n= 0 on "i $ I (13)

rfn # rsn= 0 on "i $ I (14)

where "g and "h denote the part of the boundary where the velocity vector and traction vectorare prescribed, respectively, and "i is a part of boundary where the interaction between the fluidand the solid bodies is taking place. The quantities g, h, and n denote, respectively, the prescribedvelocity and traction vectors and the current outward normal unit vector on the boundary. Herethe subscripts f and s denote the association with fluid or solid body.

Equations (8), (10) and (13) determine the motion of the part of the boundary "h for both fluidand solid and the interface boundary "i . The movement of the internal finite element nodes of thefluid domain should be chosen such that the mesh quality does not deteriorate as the displacementsof the solid body become large. For this purpose several algorithms have been suggested in theliterature (see e.g. [24, 41–44]). In this work we use either the pseudo-elastic technique or atechnique based on the optimization of mesh quality as described in more detail in Section 4.

3. STABILIZED FINITE ELEMENT FORMULATION FOR THE FLUID FLOW

Let Sh , Vh , Ph be the appropriate finite element spaces of continuous piecewise linear functionson !h , where !h = !nel

e=1 !e is a standard discretization of the fluid domain ! with nel finiteelements. A stabilized velocity–pressure finite element formulation of the fluid flow then reads:

For any t & I , find uh &Sh and ph &Ph such that the following weak form is satisfied for anyadmissible %uh &Vh and %ph &Ph

Gf(uh, ph; %uh, %ph) =GGal(uh, ph; %uh, %ph) + Gstab(uh, ph; %uh, %ph) = 0 (15)



The variational form (15) consists of the standard Galerkin terms summarized in GGal, to whicha stabilization term Gstab of the momentum equation has been added.

In order to simplify the notation in the remainder of this section, which is dedicated to thedetailed presentation of GGal and Gstab, the coupling of Equation (15) with the deformation ofthe solid structure is not explicitly included. This issue, together with the appropriate notation, iselaborated in detail in Section 7.

The Galerkin terms, which can be obtained from Equations (2)–(9) by the standard procedure,read

GGal(uh, ph; %uh, %ph) ="

!h("%uh · (uh + ("xh u

h)(uh # vh) # ff)

+ "xh%uh : r(uh, ph)+%ph("xh · uh)) dv#

"

"hh

%uh · hh da (16)

where the vector field vh denotes the mesh motion, which is based on the same finite elementinterpolation as uh .

The stabilization term used in this work is similar to the one employed by the authors in[15], but has been extended to incorporate modifications required in the ALE framework. In [15],the stabilization technique is referred to as a combination of the streamline-upwind/- and thepressure-stabilizing/Petrov–Galerkin schemes (SUPG/PSPG).

The stabilization serves two purposes: first, it provides stability to the velocity field uh inconvection dominated regions of the domain. Second, it circumvents the Babuska–Brezzi condition,which standard mixed Galerkin methods are required to satisfy. Thus, it effectively renders a smoothpressure field without jeopardising the weak enforcement of the continuity condition.

The stabilization term employed here reads

Gstab(uh, ph; %uh, %ph) =nel#e=1

"

!e[&u"("xh%u

h)(uh # vh) + &p"xh %ph]

$["(uh + ("xhuh)(uh # vh) # ff) + "xh p

h] dv (17)

We note that due to the absence of the viscous term in the second pair of brackets in (17), thestabilization term does not vanish as the spatial discretization is refined. Consequently, the choice ofthe weighting parameter & and its limit behaviour are essential for the success of the methodology.

In the experience of the authors, it has proved useful (see also e.g. [15]) to employ twoparameters, here denoted as &u and &p, and hence treat the two stabilization purposes separately.Both stabilization parameters &u and &p are defined as follows

& = he

2'ue # ve'" z, z = '1$1 + ('1/('2 Ree))2

, Ree = 'ue # ve'he"2#

(18)

but they have different scaling parameters '1 and '2, which may be set independently. Thecharacteristic element size, the convective velocity in the element centroid and the element Reynoldsnumber are represented by h, ue # ve, and Ree, respectively. Thus, &u and &p are constantwithin every element and, hence, the stabilization terms are discontinuous across the inter element



boundaries, which explains the summation of integrals in (17). The parameters '1 and '2 definethe limit of z as Ree ( ) and the derivative dz/dRee at Ree = 0, respectively. The examplesin Section 10 have all been obtained with ('1 = 1, '2 = 1

3 ) for &u and ('1 = 30, '2 = 110 ) for &p.

This specific choice makes the parameter &u identical to the expression which yields nearly exactsolutions for the one-dimensional advection–diffusion equation (see [45] and references therein).In this work, the characteristic element size he is defined as the diameter of the circle, the area ofwhich corresponds to the finite element e.

The choice of the formula in (18) is rather heuristic, and many different expressions havebeen introduced in literature (see e.g. [15, 17, 46, 47]). In some early publications (e.g. [15]) itwas suggested that & should vanish as the discretization in time is refined. In the context of afinite difference time integration scheme, the expressions (18), which do not depend on the timeincrement, have been shown to yield a robust method, allowing, to a wide extent, independentrefinement of the discretizations of space and time (see [17] for a more detailed discussion andnumerical verification).

4. MOTION OF THE FLUID FINITE ELEMENT MESH

At this stage, the motion of the fluid mesh is arbitrary except for its outline: on the interfaceboundary "h

f–s, Equation (13) has to be satisfied. On the other parts of "h , the user prescribesat least the motion of the nodes normal to the current configuration of the boundary. Note that,eventually, there may be regions of !h , which are not required to adapt to a new geometry, sincethey are far away from the moving rigid body. In such regions we set vh = 0, and the flow problembecomes purely Eulerian.

4.1. Motion of internal nodes

The movement of the internal finite element nodes should be chosen such that the mesh qualitydoes not deteriorate as the displacements of the solid structure become large. For this purpose,many different algorithms have been suggested in literature (see e.g. [24, 41–44] and referencestherein).

In this work the following techniques are used:

• Pseudo-elastic technique. In this approach, the mesh is simply assumed to represent anelastic solid body. A standard Lagrangian finite element technique typically employed insolid mechanics can then be used to adapt the mesh to the new geometry of the domain.For small distortions of the geometry the linear elastic model is sufficient. In the presence

of large deformations of the fluid domain a hyperelastic model may be more suitable. Notethat the mesh need not necessarily represent an elastic continuum. In literature, alternativemethodologies have been suggested in which the mesh is, for example, assumed to be a networkof elastic springs (see e.g. [44] and references therein). In the pseudo-elastic approach usedin this work, the mesh is treated as a simple hyperelastic Neo-Hookean continuum, withtwo parameters #m and Km, representing the shear and bulk modulus, respectively. For thetwo-dimensional situation the plane strain condition is employed.

• Optimization of mesh quality. A simple strategy to compute the mesh motion can be definedby enforcing the condition, which requires that the mesh quality, with respect to a certain



criteria, is optimal at all times t & I . In this work, the chosen criteria is the ratio of the innerand outer circles of the triangular or tetrahedral finite element. Thus, the mesh movementsatisfies

W =nel#e=1

%reoutrein

&* MIN (19)

which is a simplified version of the expression used in [48]. The quantities rein and reout denotethe inner and the outer radii of a triangular or tetrahedral finite element. The equations, whichdetermine the nodal positions xfi , then read

!W!xfi

= 0, i = 1, 2, . . . , N f (20)

where N f denotes the number of nodes in the mesh interior. In the authors’ experience thismethodology renders acceptable meshes even for very distorted geometries. Note that alsothe initial mesh should satisfy (19).

Both strategies can be fully linearized and thus allow the employment of the Newton–Raphsonprocedure to solve for the new nodal positions. This is of particular importance with respect tothe overall solution procedure described in Section 8.

The employment of large time steps in problems involving severe deformations of the domainoften requires the adaptation of the mesh to substantial changes of the geometry within one timestep. In such cases the Newton–Raphson procedure may fail to converge. In this work, this problemis overcome by increment cutting within the mesh update procedure, i.e. the new displacement ofthe boundary is applied in increments if necessary.

4.2. Motion of nodes on the interface boundary

Provided that the initial boundary of the fluid mesh appropriately resolves the surface of the solidstructure, there is normally no need to allow for any tangential movement of the fluid nodes alongthe interface. The fluid mesh boundary, similarly to the fluid particles, can then be required to‘stick’ to the surface of the structure. Thus, we satisfy Equations (11) and (13) by employing apurely Lagrangian description of the interface, i.e.

vh =uh =I(dh) +(xh, t) & "hf–s $ I (21)

where the vector dh denotes the finite element approximation of the structural velocity field d. Inorder to allow for non-matching fluid and structural meshes, it is necessary to define an appropriateinterpolation operator I(•).

The current configuration of the interface boundary "hf–s is then described by

xh =I(xhs0 + dh) +(xh, t) & "hf–s $ I (22)

where xsh0 denotes the discretization of the initial configuration of the solid structure.



5. FINITE ELEMENT FORMULATION FOR THE STRUCTURE

The fluid–structure interaction solution methodology adopted in this work does not imposeany restriction on the specific choice of structural element to be used. Importantly, the struc-tural mesh at the interface is not required to match the fluid finite element mesh. Thus, anyappropriate standard finite element method may be used for the discretization of thestructure.

The starting point of a structural finite element method is the balance of momentum as givenby (5) or by an appropriate equivalent representation. A standard finite element formulation of (5)reads as follows: for any t & I , find dh &Sh such that for any %d&Vh

Gs(dh; %dh) ="

!h(%dh · "(dh # f) + "%dh : r(dh)) dv #

"

"hh

%dh · hh da = 0 (23)

whereSh andVh are the appropriate finite element spaces and !h is a finite element discretizationof the structural domain.

Introducing the vector of the nodal displacements d, the formulation (23) can be rewritten inan equivalent matrix form as

Md + Cd + Kd = P (24)

where the matrices M, C and K are denoted, respectively, as the mass, damping and stiffnessmatrices. It should be noted that some structural finite elements, such as beam and shell elements,also include the rotational degrees of freedom in addition to the translational displacements. Weassume such rotations to be included in the vector d.

6. INTEGRATION IN TIME

In order to complete the discretization of the problem, it remains to apply a numerical timeintegration scheme to the variational form given by (15). The most popular choices are standarddiscrete time stepping schemes and the so-called time finite element methods. Both approacheshave been extensively discussed in recent publications (see e.g. [48, 49] for discrete and [5, 50]for time finite element methods).

In [17, 24], we have provided a detailed comparison of implicit time integration schemeswith respect to incompressible Newtonian fluid flow in the Eulerian framework. As a result,the generalized-! method has been suggested as a very efficient and robust alternative to the moreexpensive time finite element methods. The generalized-! method has originally been developedin [51] for the second order differential equation arising in solid dynamics, but has been adaptedto the first order problem of fluid mechanics in [16]. For linear problems the scheme can be shownto be unconditionally stable and second order accurate (see [16, 17, 51]). Furthermore, it enablesuser-controlled high frequency damping, which is desirable especially for coarse discretizationin space and time. This is achieved by specifying the single integration parameter, which, forlinear problems, can be identified with the spectral radius ") associated with very large timesteps. In the following the application of the generalized-! method to Equations (15) and (24) isdescribed.



First, the time interval I = [0, T ] is replaced by a sequence of discrete time instants tn ,n = 0, 1, 2, . . . , N with t0 = 0 and tN = T . The time step size #t = tn+1 # tn is allowed tovary.

6.1. Fluid solver

The generalized-! method is now used to express uh and its time derivative uh in (15) in termsof uh and uh at the discrete time instants tn and tn+1. These values are henceforth denoted as uhn ,uhn+1, u

hn and uhn+1.

In [16], the generalized-! method is given as

uhn+1 = uhn + #t (1 # (f)uhn + #t(fuhn+1 (25)

uhn+!ff

= (1 # !ff )uhn + !ff u

hn+1 (26)

uhn+!fm= (1 # !fm)uhn + !fm u

hn+1 (27)

where (f, !fm and !ff are integration parameters. These equations can be rewritten as

uhn+!ff

= (1 # !ff )uhn + !ff u

hn+1 (28)

uhn+!fm=

%1 # !fm

(f

&uhn + !fm

#t(f(uhn+1 # uhn) (29)

uhn+1 = 1#t(f

(uhn+1 # uhn) # 1 # (f

(fuhn (30)

In GGal and Gstab, given, respectively, by the relations (16) and (17), the expressions uh

and uh are now replaced by uhn+!ff

and uhn+!fm

, respectively. Thus, the fluid velocity and its

time derivative in (15) are expressed exclusively in terms of the unknown uhn+1 and in termsof the quantities uhn and uhn , which are known from the solution at the previous time instant.Due to its nature as the Lagrangian multiplier, which enforces continuity of the flow, the pres-sure should not be subjected to a time integration scheme, but be computed independentlyfor each time increment. Once uhn+1 has been computed, the quantity uhn+1 can be obtainedfrom (30).

The integration parameters are reduced to one independent control variable as follows:

(f = 12

+ !fm # !ff , !fm = 123 # "f)1 + "f)

, !ff = 11 + "f)

(31)



where "f) has to be chosen such that 0!"f)!1. For "f) = 1, the method is identical to thetrapezoidal rule, whereas the numerical damping of the method increases with smaller values of"f). For a detailed study of the generalized-! method in combination with stabilized finite elementswe refer to [17].

6.2. Solid body time integration

Similarly to Section 6.1, the quantities dn , dn , dn , n = 0, 1, . . . , Ntime $n , $n , $n , are introduced.For the time integration of the momentum conservation equation of the solid body (23), we thenemploy the generalized-! method as presented in [51] for second order initial value problems. Inthe following, only the time integration strategy for %d is presented. The rotation $ is integratedanalogously.

As suggested in [51], we define

dn+1 = dn + #t dn + #t2(( 12 # 's)dn + 'sdn+1) (32)

dn+1 = dn + #t ((1 # (s) dn + (s dn+1) (33)

dn+!sf = (1 # !sf )dn + !sf dn+1 (34)

dn+!sf = (1 # !sf )dn + !sf dn+1 (35)

dn+!sm = (1 # !sm)dn + !sm dn+1 (36)

where dn+!sf , dn+!sf , dn+!sm , are the quantities to be employed in (23). Equations (32)–(36) canbe rewritten as

dn+!sf = dn +#t !sf ((

s # 's)

(sdn +

#t2!sf ((s # 2's)

2(sdn +

#t!sf 's

(sdn+1 (37)

dn+!sf = (1 # !sf )dn + !sf dn+1 (38)

dn+!sm = 1 # !sm(s

dn + !sm#t (s

(dn+1 # dn) (39)

dn+1 = dn + #t ((s # 's)(s

dn + #t2((s # 2's)2(s

dn + #t's

(sdn+1 (40)

dn+1 = #1 # (s

(sdn + 1

#t(s(dn+1 # dn) (41)



Thus, similarly to the time integration strategy applied to the fluid, dn+!sf , dn+!sf and dn+!sm are

expressed in terms of the unknown dn+1 and the solution at the previous time instant. Once dn+1has been computed, dn+1 and dn+1 are obtained from (40) and (41). For linear problems, thefollowing formula for the time integration parameters is optimal [51]:

's = 14 (1 + !sm # !sf )

2, (s = 12 + !sm # !sf

!sf = 11 + "s)

, !sm = 2 # "s)1 + "s)

(42)

Similarly to (31), the scalar "s) can be identified as the spectral radius associated with infinitelylarge time steps and has to be chosen by the user such that 0!"s)!1. Note that (42) is not identicalto (31).

6.3. Mesh update

It still remains to discretize the movement of the mesh in time. Therefore, the configuration xhnand the velocity field vhn at the discrete time instants tn , n = 0, 1, 2, . . . , N are introduced. In thiswork, xhn and vhn are related by a simple generalized midpoint scheme

vhn+1 = 1#t(m

(xhn+1 # xhn) # 1 # (m

(mvhn (43)

where (m is an integration parameter to be chosen such that 12!(m!1. In this work, we set (m = (f

(see (31)1), and thus make it dependent on "f). The following expressions are then employed inthe weak form (15):

xhn+!ff

= (1 # !ff )xhn + !ff x

hn+1 (44)

vhn+!ff

= (1 # !ff )vhn + !ff v

hn+1 (45)

where !ff is given by (31)3. The vector field xhn+!ff

defines the configuration, which is employed

in the computation of the integrals in (15).Importantly, the mesh update strategy discussed in Section 4.1 is applied to the position of the

internal nodes of the finite element mesh at tn+1. Thus, the mesh meets the quality criteria (19) atthe time instants tn, tn+1, . . . . The mesh configuration given by xh

n+!ffis therefore an interpolant

between two ‘optimal’ configurations.

Remark 6.1Equation (21) requires that, along the interface boundary, the velocity vectors vh , uh and df–scoincide. In this work, these velocities are enforced to be identical at the discrete time instantstn, tn+1, . . . . It then follows from the similarity of (26) and (45), that the interpolated values



uhn+!ff

and vhn+!ff

also coincide. Yet, due to the different parameters employed in the time integration

of the fluid and the solid body, the quantities uhn+!ff

and vhn+!ff

are not necessarily identical to the

interpolated velocity of the solid body surface, which is associated with dn+!sf and, in the case ofrigid body, $n+!rf . However, the authors have not experienced any numerical problems associatedwith this apparent inconsistency.

7. DISCRETE OVERALL PROBLEM

The degrees of freedom associated with the finite element nodes in mesh interiors are de-noted by the vectors uf = {u, p}, xf and ds which represent, respectively, the fluid velocitiesand pressures, the position of the fluid nodes and the structural displacements. Let the vec-tor ui denote the kinematical degrees of freedom of the interface. At the adjacent boundariesof the fluid and the structural finite element meshes, the nodal positions, velocities and accel-erations of both the fluid and the solid can be expressed exclusively in terms of the motionof the interface as represented by ui. This arrangement of kinematic variables is illustrated inFigure 1.

Following the standard finite element discretization procedure, a compact representation of theoverall problem may then be written as

gf(uf, xf, ui) = 0, nf equations (46)

gi(uf, xf, ui, ds) = 0, ni equations (47)

gs(ds, ui) = 0, ns equations (48)

m(xf, ui) = 0, nm equations (49)

Figure 1. Discrete overall problem: domain decomposition based onintroduction of interface d.o.f. ui.



Figure 2. General interface modelling, transfer based on finite element typeinterpolation of the interface domain.

where the numbers of scalar equations nf corresponds to the number of velocity and pressuredegrees of freedom of the fluid mesh reduced by the fluid velocities on the interface boundary.The number of interface degrees of freedom is denoted by ni, whereas ns represents the numberof structural degrees of freedom not including those on the interface boundary of the structuralmesh. Finally, the number of degrees of freedom which describe the motion of the fluid mesh isgiven as nm. The systems (46)–(49) are strongly coupled and highly non-linear.

We note that the introduction of the independent interface degrees of freedom, as schematicallyshown in Figure 1, allows a modular computer implementation and thus facilitates future extensionsof the methodology. Such extensions may include, for instance, independent remeshing of the fluidor the solid domains or the employment of a ‘mortar’ strategy for the data transfer between theadjacent phases. Furthermore, it is pointed out that the computational cost of solving (46)–(49)depends strongly on the number ni of interface degrees of freedom. In many cases, the choiceof a coarse interface discretization may allow a good computational model of the fluid–structureinteraction at low computational cost. Finally, the introduction of independent interface degrees offreedom facilitates the application of the computational framework to different physical problemssuch as fluid–rigid body interaction or free surface flow (see references in [3]).

If the structures under considerations can be regarded as rigid bodies, Equation (47) becomesgi(uf, xf, ui) = 0 and the set of Equations (48) can be ignored. Furthermore, the interface veloc-ity ui becomes ui = {d, $} where d denotes the rigid body translations and $ is the rigid bodyrotation.

Remark 7.1At the outset, the derivation of system (46)–(49) was based on identifying the kinematics of theinterface with the discretization of structural surface. However, the coupled system (46)–(49) mayalso be obtained from a more generic framework based on the introduction of an independentdiscretization of the interface, as shown schematically in Figure 2.



8. SOLUTION ALGORITHM

Given the solution at the time instant tn , the non-linear coupled system (46)–(49) has to be solvedfor uf, xf, ds and ui at the next time instant tn+1. The solution strategy adopted in this work isessentially based on the Newton–Raphson method and is summarized in Box 1.

Box 1. Solution algorithm for coupled system of Equations (46)–(49).

1. estimate uf, ui

2. fluid mesh solver m(xf, ui) = 0 * xf,!xf

!ui

3. solid solver gs(ds, ui) = 0 * ds,!ds

!ui

4. if gf(uf, xf, ui), gi(uf, xf, ui, ds)<tol, then exit

5. compute derivatives

A = !gf

!uf, B = !gf

!ui+ !gf

!xf!xf

!ui, C = !gi

!uf, D = !gi

!ui+ !gi

!xf!xf

!ui+ !gi

!ds!ds

!ui

6. combined fluid + interface solver'

A B

C D

( )#uf

#ui

*

=#)

gf

gi

*

,

)uf

ui

*

,)

uf

ui

*

+)

#uf

#ui

*

7. goto 2.

It should be noted that the derivative matrices !xf/!ui and !ds/!ui required in step (5) maybe obtained from solving two systems of linear equations for ni different right hand sides.The coefficient matrices to be employed are the stiffness matrices of the converged fluid meshsolver and the solid solver, respectively. The implementation of the derivatives of the fluidnodal forces with respect to the fluid mesh configuration, as required in step 5 of the algo-rithm, is tedious but straightforward. We note that the computational cost of calculating thesubmatrices B, C and D is essentially determined by the number of interface degrees of free-dom ni. Similarly, the number of non-zero coefficients in the system matrix of the combinedfluid+ interface solver depends largely on ni. We should also note that no conceptual differencesexist in using the described strategy for either two- or three-dimensional fluid–structure interactionproblems.

Remark 8.1A number of different solution strategies for fluid–structure interaction problems has recentlybeen discussed by Tezduyar and co-workers [52, 53]. The described strategies range from a sim-ple ‘block-iterative’ coupling, in which cross-coupling matrix blocks representing the coupling



between the different sets of discretized equations are not included in computation, to the so-called ‘direct’ coupling procedures, which incorporate all matrix blocks representing the couplingbetween different sets of discretized equations. We note that, in contrast to the strategy de-scribed in [53], our solution strategy given in Box 1 also includes evaluation of all cross-couplingblocks related to the mesh movement. In addition, all directional derivatives representing thecross-coupling terms in Box 1 are evaluated in analytical form rather than by resorting to nu-merical differentiation as reported in [53]. Although we believe that further numerical studiesare required to establish the importance of some of the coupling terms, some very useful guid-ance notes regarding possible range of applications of different solution procedures is providedin [53].

9. ADAPTIVE REMESHING OF THE FLUID DOMAIN

The ALE formulation has been widely used in solving fluid–structure interaction problems (see e.g.[3–5, 7, 8]). However, for problems where the motion of the fluid–structure interface is large themesh movement algorithm will either diverge or produce a mesh with over-stretched elements thatare unsuitable for computation. In such a case, the fluid domain must be remeshed, either locallyor globally in order for the computation to proceed. Furthermore, the motion of fluid–structureinterface cause the relevant flow features to change and an adaptive remeshing procedure willenable these features to be captured accurately.

9.1. Remeshing criteria

After the computation of each set of solution variables associated with the time steps tn, tn+1, . . .the quality of the fluid mesh should be measured with an appropriate criteria such as, for instance,the element aspect ratio. If the mesh quality is found to have deteriorated substantially dueto the deformation of the fluid domain or the ratio between target element size and current elementsize exceeds a preset value !size, i.e. hnew/hn>!size or hnew/hn<1/!size, a new fluid mesh isgenerated. Here, the norm of the velocity gradient '"u' is used to indicate area that requiresrefining or coarsening. The norm '"u' is also used to determine the appropriate new elementsizes through the following expression:

hnew =

+,,-

,,.

hmin +%'"u'hmin # '"u'

'"u'hmin

&2

(hmax # hmin), '"u'!'"u'hmin

hmin, '"u'>'"u'hmin

(50)

where '"u'hmin is the norm of velocity gradient associated with hmin, and hmax and hmin are themaximum and minimum allowable element size, respectively.

When dealing with complex geometries some local geometrical features of the domain, such ascurvature or domain width, must be taken into account in the determination of the new elementsizes. In order to find the required minimum element size, we obtain the element size using differentexpressions that represent local geometrical requirements and compare them with the one obtainedfrom Equation (50). The expressions employed in this work to represent the local curvature and



domain width are

hnew =min%hnew,

2)N**

&(51)

hnew =min%hnew,

DND

&(52)

where * is the local curvature, D is the local width of the domain, and N* and ND are integers. Thefield of new element sizes is then smoothed to avoid problem during mesh generation procedure.Using the old mesh as the background mesh, the fluid domain is then remeshed by an advancingfront mesh generator.

9.2. Transfer operation procedure

The transfer procedure of the velocity and pressure fields from the old mesh Th onto the newmesh Th+1 (see, Figure 3 can be subdivided into three stages [28, 35–38])

(i) Construction of the background meshFor each node A of the new mesh Th+1 with known co-ordinates xh+1

n,A the so-called

background element !he is found in the old mesh Th for which xh+1

n,A & !he .

(ii) Evaluation of the local co-ordinatesThe local co-ordinates nhA of the node A of the new mesh within the background element!he can be evaluated by solving the following equation:

xh+1n,A =

3#B=1

NhB(nhA)xhn,B (53)

where N are the interpolation functions.

Figure 3. Transfer operator.



(iii) Mapping of the nodal valuesBy using the shape functions Nh

B(nhA) the nodal values Khn,B are mapped from the nodesB of the old mesh Th to the nodes A of the new mesh Th+1. This mapping may beexpressed as

Kh+1n,A =

3#B=1

NhB(nhA)Khn,B (54)

where Khn denotes the array containing all nodal information at given time step tn andmesh Th , that are being transferred. We note that in addition to the velocity field, theacceleration field must also be transferred. This is the result of the chosen discrete timeintegration procedure described in Section 6.

The new mesh is then employed as a representation of the velocity and pressure fields at timeinstant tn , and the ALE strategy is used to compute the solution at tn+1.

10. NUMERICAL EXAMPLES

In order to demonstrate the performance of the combined ALE and adaptive remeshing procedure,several numerical examples are presented in this section.

10.1. A model turbine simulation

In this example, we simulate 2D flow inside a simple model turbine. The intention here is totest the robustness of our combined ALE-adaptive remeshing procedure. We note that for thistype of problem the shear-slip mesh update strategy of Behr and Tezduyar [54–56] is expectedto provide a more efficient approach. The geometry of the turbine and initial finite elementmesh are given in Figures 4 and 5, respectively. A parabolic free-stream velocity with umax

inflow =50 is prescribed at the left end of the domain. The viscosity and the density of the fluid aretaken as # = 0.1 and " = 1.0, respectively. The Reynolds number based on maximum velocity

Figure 4. A model turbine simulation: geometry.



Figure 5. A model turbine simulation: initial finite element mesh.

and the size of the inlet pipe is Re= 200. The turbine is modelled as a rigid body with the momentof inertia with respect to its centre taken as I$ = 1.5. The severity of the mesh deformation at thetip of the blades and at the corner of the base of the blade allows only a single ALE step after eachremeshing step. Figures 6–8 show, respectively, the finite element mesh, vorticity, and pressurecontour plot of the flow at different time instants. The time history of the angular velocity of theimpeller is shown in Figure 9.

10.2. Particulate flow

A class of problems in which combined ALE-adaptive remeshing procedure is required is thedirect numerical simulation of particulate flows [57, 58]. Recently, the use of fictitious domainmethod to simulate particulate flows has gained popularity [59–61].

We consider 2D rigid particles of circular shape with radius R = 0.1 cm falling under the actionof gravitational force inside a tube of width D = 20R. The length of the tube is L = 80R andthe tube is filled with liquid. The density and viscosity of the liquid used in these examplesare "l = 0.998 g/cm3 and # = 0.0101 dyn s/cm2, respectively, and the density of the particle is"p = 1.002"l.

As shown in Figure 10, the particle is initially stationary at the distance of 7.2 cm from thebottom of the tube and is released at time t = 0.0 s. The no-slip boundary conditions are imposedon the tube wall and bottom while the pressure boundary condition is prescribed on a node at thefree-surface. This arrangement represents a tube of finite length.

The time history of the particle vertical velocity for tubes of width D/R = 20 and 100 is shownin Figure 11. The velocity curves clearly show the influence of the wall effect on the settlingvelocity. The maximum Reynolds number and the settling velocity, respectively, are Re= 7.74and Us = 0.391 cm/s for D/R = 20 tube and Re= 8.83 and Us = 0.446 cm/s for D/R = 100 tube.The analytical settling velocity for this problem can be obtained by employing the followingformula [62]:

Us =/2gmp("p # "l)

"l"pApCD(55)



Figure 6. A model turbine simulation: finite element mesh at different time instants.



Figure 7. A model turbine simulation: vorticity contour plots at differenttime instants; ! # 700.0, 0 + 700.0! .



Figure 8. A model turbine simulation: pressure contour plots at differenttime instants; ! # 20000.0, 0 + 20000.0! .



Figure 9. A model turbine simulation: time history of angular velocity of the impeller.

Figure 10. One particle sedimentation: geometry and finite element mesh.



Figure 11. One particle sedimentation: time history of vertical velocity.

where mp is the mass of the particle, Ap is the projected area and CD is the drag coefficient. For acylindrical particle and 1<Re<104 the drag coefficient CD can be estimated using the expressionproposed by Ceylan et al. [63]

CD = 1 + 4.44 Re#2/3 exp(0.684 Re#1/8) (56)

Often in practice a simpler expression is used, given by

CD = 1 + 10$ Re#2/3 (57)

Here, the Reynolds number is defined as

Re="pU2R

#

The use of Equations (56) and (57) gives Us = 0.48169 and 0.42435 cm/s, respectively. Withrespect to the first expression (56), the computational result for settling velocity undershootsthe analytical solution by 7.8% and with respect to the second one it overshootsby 5.0%.

For the problem with two particles, to prevent particle overlapping, a repulsive force hasbeen imposed when the distance between the two particles is less than a preset minimum



Figure 12. Two particles sedimentation: geometry, finite element mesh at t = 0.0and the paths of both particles.

distance dmin. This is a standard practice used in direct numerical simulation of particulate flows(see, [59, 60, 64]). The repulsive force Fp is given by

Fp =

+,-

,.

0 d>dmin

Cp(d)ri # r j

'ri # r j'd!dmin

(58)

where d = 'ri # r j' # Ri # R j the distance between particle i and j , Cp(d) specifies the mag-nitude of the force, and ri , r j , Ri and R j are the position and the radius of particles i and j ,respectively.

The particles are arranged, initially, as shown in Figure 12 together with the finite elementmesh at time t = 0.0 s. The figure also shows the settling trajectories for each particle. Figures 13and 14 show, respectively, the finite element mesh and the vorticity contour plot at different timeinstants. These figures also show the drafting, kissing and tumbling behaviour of the two particlescommonly observed in particulate flows. The maximum Reynolds number attained by the particles



Figure 13. Two particles sedimentation: finite element mesh at different time instants.

is Re= 35.21. The time history of horizontal and vertical velocity are depicted in Figures 15and 16, respectively.

The particles arrangement and the finite element mesh at t = 0.0 for particulate flow involv-ing four particles is given in Figure 17. The properties of the particles used in this case aresimilar to the two particles flow. The maximum Reynolds number for this case is Re= 29.5132(and the simulation requires 617 remeshings). The paths of each particles are shown in



Figure 14. Two particles sedimentation: vorticity contour plot at differenttime instants; ! # 4.0, 4.0! .



Figure 15. Two particles sedimentation: time history of horizontal velocity.

Figure 16. Two particles sedimentation: time history of vertical velocity.



Figure 17. Four particles sedimentation: geometry, finite element mesh at t = 0.0and the paths of each particles.

Figure 17 while the finite element meshes and vorticity contour plots at different time instantsare depicted, respectively, in Figures 18 and 19. Figures 20 and 21 show the time history ofthe horizontal and vertical velocity, respectively. It could be observed that the initial symmetricarrangement of particles is closely preserved throughout this simulation. For such flows, it is ex-pected that small geometric perturbation will have significant effect on stability of the symmetricpattern.

10.3. Airbag deployment

In the next two examples we will consider fluid–structure interaction problems in which thefluid flow is confined inside a space bounded by a flexible structure. This example simulatesdeployment of an airbag with geometry given in Figure 22. The airbag is modelled by using axisym-metric membrane element of incompressible Neo-Hookean-type material given by the strain energyfunction with reference density "0 = 2.5 g/cm3, shear modulus of G = 4.0$ 104 dyn/cm2 andbulk modulus K = 5.0$ 105 dyn/cm2. The thickness of the membrane is taken to be T = 0.06 cm.The fluid used to fill the airbag is air with viscosity # = 1.73$ 10#4 dyn s/cm2 and density"f = 1.23$ 10#3 g/cm3. The flow at the inlet is assumed to have a parabolic velocity distribution



Figure 18. Four particles sedimentation: finite element mesh at different time instants.

and the time history of the maximum inflow velocity is given in Figure 23. The airbag geometryis designed in such a way that at the beginning of the simulation the membrane is in unstretchedstate. This is done to show that the proposed solution strategy works without the need to giveadded mass to the membrane. The finite element meshes and the vorticity contour plots at differenttime instants are presented in Figures 24 and 25, respectively.



Figure 19. Four particles sedimentation: vorticity contour plot at differenttime instants; ! # 4.0, 4.0! .



Figure 20. Four particles sedimentation: time history of horizontal velocity.

Figure 21. Four particles sedimentation: time history of vertical velocity.



Figure 22. Airbag deployment: geometry.

Figure 23. Airbag deployment: time history of maximum inflow velocity.



Figure 24. Airbag deployment: finite element mesh at different time instants.



Figure 25. Airbag deployment: vorticity contour plots at different timeinstants; ! # 180.0, 180.0! .



Figure 26. Balloon inflation: geometry.

Figure 27. Balloon inflation: time history of maximum inflow velocity, Umaxinflow.



Figure 28. Time history of volume, V .

Figure 29. Time history of vertical displacement of balloon tip, dy .

10.4. Balloon inflation

This last example deals with the inflation of a rubber balloon. The geometry of the balloon is givenin Figure 26. As in Section 10.3 the balloon is modelled using axisymmetric membrane elementand the rubber is modelled using an Ogden-type of material given by the strain energy function.The material parameters for the rubber are

#1 = 6300.0 dyn/cm2, !1 = 1.3

#2 = 12.0 dyn/cm2, !2 = 5.0

#3 = #100.0 dyn/cm2, !3 = # 2.0



Figure 30. Balloon inflation: finite element meshes at different time instants.

The bulk modulus is K = 1.0$ 105 dyn/cm2 and the reference density is "0 = 2.0 g/cm3. Mem-brane thickness used in this example is T = 0.05 cm. The mechanical properties of air are similarto the ones in Section 10.3. The flow at the inlet is assumed to have parabolic velocity distri-bution and has the average value U ave

inflow = 220.0 cm/s with the associated Reynolds’s numberRe= 1564. The evolution of the prescribed inflow velocity is shown in Figure 27. Figures 28and 29 show time history of the volume of the balloon and vertical displacement of the tip of theballoon, respectively. The finite element mesh evolutions and the vorticity distributions are given in



Figure 31. Balloon inflation: vorticity contour plots at different timeinstants; ! # 300.0, 300.0! .

Figures 30 and 31, respectively. By inspecting these figures it can be observed that the proposedadapted finite element meshes clearly capture the essential features of the fluid flow.

11. CONCLUSIONS

The capability of a fully implicit finite element method for the simulation of fluid–structureinteractions is extended by incorporating a remeshing strategy. It is demonstrated that the large



fluid–structure interface motion can be modelled easily with the combination of ALE strategy andan adaptive remeshing procedure.

The adopted remeshing procedure is triggered by a mesh quality check and a simple remeshingindicator. Different element-size control schemes that represent the salient features of the solutionand the local geometrical characteristics are used simultaneously in order to capture the relevantfeatures of the flow.

The adaptive strategy has proven to be robust and efficient on a variety of complex problemsinvolving moving boundaries and fluid–structure interaction. Our ongoing research is focused onthe development of enhanced transfer operation procedures and modelling of novel multi-physicsapplications incorporating multiple scales. The outcomes of this research will be reported in theforthcoming publications.

ACKNOWLEDGEMENTS

This work was partly funded by EPSRC of UK under grant no. GR/R92318, awarded to the Universityof Wales Swansea, and this support is gratefully acknowledged.

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