A Lagrangian finite element approach for the analysis of fluid-structure interaction problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2010; 84:610–630Published online 23 April 2010 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.2911

A Lagrangian finite element approach for the analysisof fluid–structure interaction problems

M. Cremonesi∗,†, A. Frangi and U. Perego

Department of Structural Engineering, Politecnico of Milano, P.za Leonardo da Vinci 32, 20133 Milan, Italy

SUMMARY

A Lagrangian finite element method for the analysis of incompressible Newtonian fluid flows, based on acontinuous re-triangulation of the domain in the spirit of the so-called Particle Finite Element Method, ishere revisited and applied to the analysis of the fluid phase in fluid–structure interaction problems. A newapproach for the tracking of the interfaces between fluids and structures is proposed. Special attention isdevoted to the mass conservation problem. It is shown that, despite its Lagrangian nature, the proposedcombined finite element-particle method is well suited for large deformation fluid–structure interactionproblems with evolving free surfaces and breaking waves. The method is validated against the availableanalytical and numerical benchmarks. Copyright q 2010 John Wiley & Sons, Ltd.

Received 16 April 2009; Revised 6 October 2009; Accepted 13 March 2010

KEY WORDS: particle methods; Lagrangian approaches; fluid–structure interaction

1. INTRODUCTION

The analysis of fluid–structure interaction problems, in which the structure undergoes large defor-mations and the fluid motion is characterized by free surfaces and breaking waves, is of greatrelevance in many areas of engineering.

Generally, the equations of motion for a Newtonian fluid are presented in Eulerian form; thisformulation allows to conveniently solve many situations with a fixed domain or control volume.Otherwise, the Arbitrary Lagrangian Eulerian method (ALE) [1], in which the movement of thefluid particles is separated from that of the mesh nodes, is often applied. Typical difficulties offree-surface flow or fluid–structure interaction problems using Eulerian or ALE formulations arethe treatment of the convective terms, the modeling of the wave splashing and the tracking of thefree-surfaces and of the interfaces between fluids and structures which require dedicated algorithms,such as the Volume of Fluid [2] or the Level Set [3] methods.

∗Correspondence to: M. Cremonesi, Department of Structural Engineering, Politecnico of Milano, P.za Leonardo daVinci 32, 20133 Milan, Italy.

†E-mail: [email protected]

Contract/grant sponsor: Tetra Pak Packaging Solutions S.p.a.

Copyright q 2010 John Wiley & Sons, Ltd.

LAGRANGIAN APPROACH FOR FLUID–STRUCTURE 611

Most of these problems are overcome if the equations of motion of the fluid are formulated ina Lagrangian framework. In the Lagrangian formulation, the Navier–Stokes equations are writtenin material coordinates which are continuously updated with iterative procedures (see e.g. [4]).Moreover, a method to track interfaces is not necessary because the interfaces are defined by thecurrent position of the material points. One of the main advantages of the Lagrangian approach isthat the convective term in the momentum conservation equation disappears. However, if a fixedfinite element mesh is used and the positions of element nodes are updated as a consequence ofthe fluid flow, very soon the element distortion becomes excessive and the finite element meshneeds to be regenerated.

Particle andMeshless methods, (see e.g. [5]) in which the motion of each fluid particle is followedin a Lagrangian framework, have been developed to exploit the potentialities of the Lagrangianapproach. Lagrangian finite element formulations without remeshing for incompressible fluid flowhave been proposed (e.g. [6]), but with an applicability range limited to only slightly distortedmeshes.

In the present work, a Lagrangian finite element method (FEM) for the analysis of incompressibleNewtonian fluid flows, based on a continuous re-triangulation of the domain, in the spirit of theso-called Particle Finite Element Method (PFEM) [7, 8], is developed. The PFEM is a method forthe solution of fluid-dynamics problems including free-surface flows and breaking waves [7], butalso fluid–structure interactions [9, 10] or fluid–object interactions [11, 12]. This method has beenapplied to different engineering problems and has been also validated against experiments [13, 14].

When remeshing is performed, data have to be transmitted from the old mesh to the new one.In the present approach, a discretization of a 2D domain with three node triangles is used, so thatthe remeshing consists of a fast Delaunay triangulation with prescribed node positions. In viewof the assumed incompressibility constraint, both velocity and pressure fields are independentlydiscretized. To avoid interpolations from the old to the new mesh, only degrees of freedom ofparticles located at the vertices of triangles are used for the discretization, so that linear shapefunctions have to be used for both velocity and pressure. However, it is well known that this typeof discretization does not satisfy the LBB inf-sup compatibility condition [15] and a stabilizationmethod is required to avoid spurious oscillations in the pressure field. One way to alleviate thesedifficulties is to adopt iterative procedures where velocities and pressures are estimated separately,solving more stable problems of reduced size. A fractional step method [16] is used to thispurpose in the original implementation of the PFEM coupled with Finite Calculus [17] or withthe Orthogonal Subscale stabilization [18]. In the implementation proposed here the Lagrangianversion of a pressure-stabilizing/Petrov-Galerkin (PSPG) stabilization technique [19] has beenderived and adopted. This stabilization allows for an accurate enforcement of the incompressibilityconstraint. The numerical tests in Section 5 show that mass conservation can be achieved with areasonable accuracy and that the ‘mass loss’ problem reported in [20] is avoided. Other sourcesof possible mass variation, intrinsic to the formulation of the PFEM, are discussed in Section 4.3.

The fluid–structure interaction problem is solved using a staggered iterative scheme in whichthe solid and the fluid parts are solved separately. A new methodology, based on the definitionof ‘ghost’ fluid particles on the solid surface, is proposed for the identification of the evolvingfluid–structure interface.

Typically, PFEM formulations addressing fluid–structure interaction problems also employPFEM for the modeling of the solid part [9, 10]. A different approach is investigated here, where thesolid part is treated with a classical finite element scheme. This choice has several potential advan-tages in view of future generalizations. When PFEM is employed everywhere, the same elements

Copyright q 2010 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2010; 84:610–630DOI: 10.1002/nme

612 M. CREMONESI, A. FRANGI AND U. PEREGO

(triangles and tetrahedra) are utilized also for the solid, and this might be a serious drawback inapplications where e.g. quadrangles/hexahedra are known to perform better or possibly when theapplication of structural elements such as beams and shells is envisaged. More importantly, thedifferent discretizations of the fluid and structural domains should prove beneficial when the solidpart behaves according to a non-linear constitutive law with internal variables, which are usuallydefined at element Gauss points. As the only entity which is tracked by the PFEM are nodes,to which all the pieces of information are attached, this would require a mapping of the internalvariables between different meshes with consequent deterioration of the intrinsic accuracy of theresults and of the computing time.

The current implementation of the method is limited to 2D problems. The extension to 3Dproblems does not present conceptual difficulties and is currently in progress.

The outline of this work is as follows. In the following section the basic ideas of the PFEM areillustrated. In Section 3 the fluid-dynamics equation in a Lagrangian framework are introduced, aspace-time finite element semi-discretization is presented and the stabilization of the incompress-ibility condition is described. In Section 4 the fluid–structure interaction algorithm is explained,and finally a variety of fluid–structure interaction problems are solved in Section 5 to show theefficiency of the proposed method.

2. PARTICLE FINITE ELEMENT METHOD

2.1. Summary of the steps

The PFEM is conceived for the solution of fluid-dynamics problems using a Lagrangian approachbased on the formulation of the Navier–Stokes equations in material coordinates. The PFEM, in theform implemented in the present work, consists of the following steps (see also [8] and referencetherein).

1. Fill the fluid domain with a set of points referred to as ‘particles’. The accuracy of thenumerical solution is clearly dependent on the considered number of particles.

2. Generate a finite element mesh using the particles as nodes. This is achieved using a Delaunaytriangulation.

3. Identify the external boundaries to impose the boundary conditions and to compute thedomain integrals.

4. Solve the non-linear Lagrangian form of the governing equations finding velocity and pressureat every node of the mesh.

5. Update the particle positions using the computed values of velocity and pressure.6. Go back to step 2 and repeat for the next time step.

In this solution scheme, not only is the numerical solution of the equations critical from thecomputational point of view, but so are the generation of a new mesh and the identification ofthe boundaries. To this purpose a Delaunay triangulation scheme is adopted together with theboundary identification method presented in the following section.

2.2. Identification of the boundaries

In a Lagrangian framework the external boundary �� and the reference volume � are defined bythe position of the material particles. Everytime the mesh is regenerated, the particles belonging



Figure 1. (a) Distribution of points; (b) Delaunay triangulation; and(c) Delaunay triangulation with �-shape.

to the boundary may change and the new boundary nodes (and therefore the particles) have tobe identified. The Delaunay triangulation generates the convex hull of the set of particles, i.e. theconvex figure with the minimum area that encloses all the particles belonging to the set. Moreover,the convex hull may not be conformal with the external boundaries. To clarify this problem, a setof points is shown in Figure 1(a) and its Delaunay triangulation in Figure 1(b). It is clear thatthe Delaunay triangulation does not match the real boundaries. A possibility to overcome thisproblem is to correct the generated mesh using the so-called �-shape method [21]. The key idea isto remove the unnecessary triangles from the mesh using a criterion based on the mesh distortion.For each triangle e of the mesh, the minimal distance he between two nodes in the element andthe radius Re of the circumcircle of the element are defined. If h is computed as the mean valueof all the he, the shape factor

�e= Re

h� 1√

3(1)

is an index of the element distortion. It is worth recalling that Re/he=1/√3 is the ratio for an

equilateral triangle. All the elements that do not satisfy the condition �e��, where usually ��1 isassumed, are removed from the mesh. Increasing the value of �, fewer triangles are removed fromthe original mesh and, for �→∞, the original Delaunay triangulation is always recovered.

Once the unnecessary triangles are removed from the Delaunay triangulation, the particles whichactually belong to the boundary can be identified. Figure 1(c) shows the boundary identificationfor the example introduced before.

The �-shape method can also be used for the identification of the fluid particles which separatefrom the rest of the domain. When a particle on the boundary is a vertex of only one triangleand the triangle is overly distorted, the particle is separated from the domain and the entire massof the triangle is assigned to this particle in order to preserve the total fluid mass. After separation,the motion of these particles is governed by the body force and the initial velocity which they aresubjected to. At each new Delaunay triangulation, the separated particles become the vertices oftriangles. In the case that a separated particle has approached the fluid free boundary adequately,the triangle is not eliminated by the �-shape method check and the particle is again incorporatedin the fluid mass.



3. FLUID-DYNAMICS PROBLEM

3.1. Governing equations

Let x denote the current position of a fluid particle in the current configuration at time t , u=u(x, t)its velocity and �=�(x, t) the Cauchy stress tensor at time t ∈[0,T ]. For an incompressible fluid,momentum and mass conservation write as:

�DuDt

= div �+�b in �, (2)

divu= 0 in �, (3)

where � is the volume occupied by the fluid mass in the current configuration, �(x) is the density,b(x, t) the external body forces and D/(Dt) denotes the material time derivative. The initialconditions are prescribed assigning the initial velocity u0(x) at t=0 whereas both Dirichlet andNeumann type boundary conditions are imposed as:

u(x, t) = u(x, t) ∀x∈�D, (4)

�(x, t) ·n= h(x, t) ∀x∈�N , (5)

where u(x, t) and h(x, t) are assigned functions, n denotes the outward normal to the boundaryand �D∪�N =��.

Defining X as the position of a particle in a chosen reference configuration, the momentumconservation can be written in the Lagrangian framework as follows:

�DUDt

= 1

JDivP+�b in �0, (6)

where �0 is the volume occupied by the fluid mass in the reference configuration, U=U(X, t)denotes the evolution of the velocity field referred to in the original configuration, J = det Fand P= J�F−T are the first Piola-Kirchhoff stress tensors, with F deformation gradient. For aNewtonian incompressible fluid the Cauchy stress tensor � is related to the velocity u and thepressure p by:

�=−pI+2�D(u), (7)

where � is the dynamic viscosity and D(u)= 12 (gradu+graduT) is the symmetric part of the

gradient of the velocity. Using the constitutive equation (7), and the fact that gradu=(GradU)F−1,the Lagrangian momentum conservation for a Newtonian fluid writes as:

�DUDt

=− 1

JDiv(J pF−T)+ 1

J�Div[J (GradU)F−1F−T]+�b in �0, (8)

Similarly, noting that for an incompressible fluid, mass conservation implies volume conservation,one has:

D(�d�)

D t= DJ

Dt= JF−T :GradU=0, (9)

Finally, since

JF−T :GradU=Div(JF−1U)−Div(JF−T)U (10)



and Div(JF−T)=0, one can write the Lagrangian version of the mass conservation equation inthe form:

Div(JF−1U)=0 in �0. (11)

Equations (8) and (11) represent the Lagrangian form of the Navier–Stokes equations. In Equa-tion (8), the convective term, which represents the source of non-linearity of the problem in theEulerian approach, does not appear. In the Lagrangian approach, the non-linearity is due to the factthat the current configuration differs by large displacements from the original one; this non-linearityappears in the equations through the deformation gradient F.

3.2. Space discretization and pressure stabilization

To discretize the Lagrangian Navier–Stokes equations (8)–(11), a weak form is required. Introducingthe spaces of trial and test functions (see e.g. [22]), defined as

S = {u∈H1(�)|u= u on �D}, (12)

S0 = {w∈H1(�) |w=0 on �D}, (13)

Q = L2(�), (14)

the variational form is obtained multiplying equation (8) by a vector of test functions W∈ S0 andEquation (11) by a scalar test function q∈Q. Then, integrating over the domain �0 and applyingthe Green formula, the weak form reads asfind U∈ S×[0,T ] and p∈Q×[0,T ] such that∫

�0

�0DUDt

·Wd�0 =∫

�0

J pF−T :Grad(W)d�0−∫

�0

�JGrad(U)F−1F−T :Grad(W)d�0

+∫

�0

�0bWd�0+∫

�N

W·h ∀W∈ S0, (15)

∫�0

Div(JF−1U)q d�0 = 0 ∀q∈Q. (16)

The selected discretization (linear elements for velocity and pressure) is known to lead tospurious oscillations in the pressure field and a stabilization procedure is required. The fractionalstep method, adopted by various authors, has been shown in [20] to be responsible for the massloss due to the strong enforcement of the pressure boundary conditions at the free surface. Indeed,if the three nodes of a triangular element belong to the free surface, the pressure turns out to beassigned completely on this element and consequently the velocity computed with the fractionalstep does not respect the incompressibility constraint. For this reason, a monolithic approach witha PSPG stabilization is adopted herein.

This stabilization, proposed by Tezduyar et al. [19] as a generalization to finite Reynolds numberflows of the Petrov-Galerkin stabilization originally proposed by Hughes et al. [23], is achievedby adding to the Galerkin formulation (15)–(16) a series of integral terms. These integrals involvethe product of the stabilizing term times the residual of the momentum equation, so that theformulation is consistent (i.e. the exact solution still satisfies the stabilized formulation).



For the present purpose the stabilized formulation (presented in [19] for the Eulerian problem)has to be transformed into its Lagrangian counterpart. As already mentioned, in a Lagrangianapproach the convective terms disappear from the equation as well as the associated stabilizingterms. As a consequence of these remarks, in the stabilized formulation Equation (15) remainsunchanged, whereas Equation (16) has to be replaced by

∫�0

q Div (JF−1U)d�0+Nel∑e=1

∫�e0

�epspg1

�0Gradq

(�0

DUDt

+ 1

JDiv(J pF−T)

− 1

J�Div(J GradUF−1F−T)−�0b

)d�0=0 (17)

∀q∈Q, where the stabilization parameter is defined as

�epspg= ze2‖u‖ (18)

and ze is the ‘element length’ defined to be equal to the diameter of the circle which is area-equivalent to the element e.

Introducing a Galerkin isoparametric finite element discretization, the semi-discrete form ofequations (15)–(17) is obtained as

M(x)DVDt

+K(x)V+DT(x)P=B, (19)

C(x)DVDt

+D(x)V+L(x)P=H(x), (20)

where V is the vector of nodal velocity, M is the mass matrix, K is the fluid stiffness matrix,D is the discretization of the divergence operator, B is the vector of external forces and the matricesC, L and H stem from the discretization of Equation (17).

The second-order term Div[J Grad UF−1F−T] in the stabilized Equation (17) vanishes for linear(triangular and tetrahedral) elements and is often neglected also for higher-order elements [24].

Equations (19)–(20) govern the motion of the particles belonging to the fluid mass. The motionof the isolated particles, separated from the bulk, is governed by the equation of a particle massin motion under the effect of its gravity load and initial velocity.

3.3. Time integration

For simplicity, a partition of the time domain [0,T ] into N time steps, of the same length�t is considered. Let us focus on the time step tn → tn+1 and enforce equilibrium at tn+1.The acceleration and the current configuration are expressed according to the following implicitbackward-difference integration scheme:

DVDt

= Vn+1−Vn

�t,

xn+1 = xn+Vn+1�t.

(21)



Choosing as reference configuration the known configuration at tn , in the final discretized system

1

�tM(Vn+1−Vn)+K(Vn+1)Vn+1+DT(Vn+1)Pn+1 =Bn+1, (22)

1

�tC(Vn+1)(Vn+1−Vn)+D(Vn+1)Vn+1+L(Vn+1)Pn+1 =H(Vn+1) (23)

all the matrix and vector operators generally depend non-linearly on the unknown vector Vn+1.The pressure stabilization introduces a pressure term in the mass conservation equation and adds

a term depending on the acceleration to the matrix D, destroying the formal symmetry of the system.Following [9], a Picard algorithm is here privileged over a full Newton–Raphson scheme for

the solution of the non-linear system. This is a simple fixed point method in which, at iterationk, the matrices in equations (22) and (23) are evaluated at the configuration estimated at iterationk−1. As shown in [25] the Picard algorithm implies less extra storage than the Newton–Raphsonalgorithm for a Newtonian fluid. Iterations are repeated until the correct geometry at tn+1 is found,together with Vn+1 and Pn+1.

Ideally, in this solution scheme, the triangulation should be performed at every time step.However, to reduce the computing time of the method, the mesh is regenerated only when it isglobally too distorted. For this purpose an index of mesh distortion is necessary. Starting fromthe shape factor �e, defined in Equation (1), a simple measure of mesh quality can be obtaineddefining for each element a distortion factor �e:

�e=√3�e=√

3Re

he�1, (24)

where the equilateral triangle (�e=1/√3) has been considered as the best possible element.

The quality of the entire mesh is then evaluated by an arithmetic mean

�= 1

Nel

Nel∑e=1

�e, (25)

where Nel is the number of elements. The mesh is regenerated only if �>�, where �>1 is a fixedparameter. Increasing the value of �, more and more distorted meshes are accepted.

4. FLUID–STRUCTURE INTERACTION

As recalled in the Introduction, the available PFEM formulations addressing fluid–structure inter-action problems also employ PFEM for the modeling of the solid part. On the contrary, a differentapproach is investigated here, where a classical FEM is employed for the solid part.

4.1. Definition of the problem

A fluid occupying the domain � f and a solid structure occupying the domain �s at time t ∈[0,T ]are considered. The fluid–structure interface � is the common boundary between � f and �s ,�=� f ∩�s . The fluid problem is governed by the Lagrangian Navier–Stokes equations introducedin Section 3, whereas the structural problem is governed by the classical Lagrangian elastodynamicsequations. In particular, defining d=d(X, t) the displacement field, the motion of the structure in



Lagrangian coordinates is governed by

�sD2dDt2

− 1

JDivP(d)=�sb in �s, (26)

where �s denotes the material density, b the body forces and P(d) is the first Piola-Kirchhoffstress tensor. This equation has to be completed with a constitutive law; here a linear elastic modelis considered under the assumption of small strains of the solid phase (whereas displacements maybe large). The numerical solution of the elastodynamics problem is achieved with a classical FEMwith quadrilateral four node elements and bilinear displacement interpolation. The time integrationis performed with the Newmark scheme and the non-linearity due to the large displacement issolved with a Newton–Raphson scheme (see e.g. [26, 27] for details).

The fluid and the structural problems are coupled on the interface � by continuity conditions

u(t) = d(t) on �, (27)

� f n f +�sns = 0 on �, (28)

where both normal and tangential continuity of the velocities is required, and ns =−n f are theoutward normals on � to �s and � f , respectively, and �s and � f are the corresponding Cauchystresses at the interface.

4.2. Fluid–structure interaction algorithm

The proposed interaction algorithm is illustrated below, assuming a discretization of the soliddomain by four node quadrilateral elements.

• At the start of the analysis, a set of fictitious fluid particles, hereafter referred to as ‘ghostparticles’, is artificially created. A ghost particle is geometrically superposed to each of thenodes of all the solid surfaces that can possibly come into contact with the fluid domain(Figure 2(a)). It is worth stressing that different options could be in principle adopted, byplacing a different number of ghost particles on the contact surfaces. As the density of theghost particles governs the size of the fluid mesh at the contact surface (see the followingsteps), the choice adopted herein seems reasonable if the initial solid and fluid meshes exhibitsimilar refinement.

• Perform the Delaunay triangulation (Figure 2(b)).• Apply the �-shape method to remove the unnecessary triangles (Figure 2(c)).• Check whether the fluid is in contact with the solid part. The fluid element is assumed to be

in contact with the solid if at least one of its nodes is a ghost particle.

◦ If the two discretized domains are not in contact (Figure 2(c)), the fluid and the solidanalyses are performed separately without any interaction.

◦ If the two discretized domains are in contact (Figure 2(e)), a coupled analysis is necessary.

The coupled analysis is performed with a staggered scheme (Figure 3), based on the Dirichlet–Neumann approach [28]. For the assigned increment of loading:

(a) Solve the structural-dynamics problem considering the fluid–solid interface as a Neumannboundary for the structure.

(b) Compute displacements on the current fluid–solid interface.



Figure 2. Sketch of fluid structure interaction: (a) the two domains with particle (fluid) and quadrilateralelements (solid). Ghost fluid particles are superposed to the nodes of the solid boundary; (b) Delaunaytriangulation; (c) Delaunay triangulation after �-shape correction; (d) same as (a) but with the two domains

in contact; and (e) same as (c) but with the two domains in contact.

Figure 3. Fluid–structure interaction algorithm.

(c) Update the discretization of the fluid domain according to the new position of the boundary.(d) From the displacements compute the velocities to be assigned to the ghost fluid particles

superposed to the nodes of the solid mesh at the current fluid–solid interface (u=(dn+1−dn)/�t).

(e) Solve the fluid-dynamics problem under the assigned increment of velocities. Consider thefluid–solid interface as a Dirichlet boundary for the fluid.

(f) Compute the fluid stress on the current fluid–solid interface.(g) If convergence is not reached, go back to (a) solving again the structural problem under the

updated value of boundary tractions on the fluid–solid interface.

Owing to the so-called added-mass effect [29], the convergence properties of the Dirichlet–Neumann algorithm depend heavily on the density ratio between the fluid and the solid phases.When the material density of the structure is comparable to the fluid one, this method is known to



have convergence problems [28]. To enhance convergence, a relaxation parameter can be introduced.Nevertheless, this parameter decreases as the added-mass effect increases, hence convergence couldbe very slow [29, 30]. In all the examples considered in the subsequent sections the added-masseffect is negligible and convergence problems are avoided.

4.3. Comments on mass and momentum conservation

The proposed method does not guarantee fluid mass conservation exactly, mainly due to the �-shapemethod employed for the boundary identification.

When a particle separates from the bulk of the fluid through deletion of one or more elements,a mass equal to the total mass of the deleted elements is attached to the particle. If subsequentlythe same particle merges back with the bulk, the mass added to the bulk itself is not exactly themass of the particle, but is the mass of the newly created elements that first connect the particle tothe bulk according to the �-shape algorithm. It can be reasonably argued that these mass variationscompensate each other and that the average error is negligible. This is investigated empirically inthe examples. This issue also affects the momentum equation, which is enforced separately for thebulk of the fluid and for the isolated particles. The momentum added to the bulk when an isolatedparticle is re-attached, is not exactly the momentum of the particle itself. Similar remarks alsohold for the formation and annihilation of internal cavities.

A somewhat different situation arises at the solid/fluid boundaries. Along these surfaces, a sortof fluid boundary layer is formed, consisting of the elements connecting the fluid boundary nodes tothe ghost nodes on the solid boundary. The thickness tL of this layer is affected by the � parameterselected in Equation (1) and by the typical size hS of the solid elements on the interface. Ideally,when �→1, the value of tL will be approximately equal to hS . It is quite intuitive that when contactbetween fluid and solid is lost at some interface, this will induce a mass loss equal to the massof the boundary layer, since the �-shape method simply deletes elements. The converse situationis found when a new contact surface is created: the total mass will increase. Hence, as verifiedin e.g. Example 5.4, a mass oscillation will be measured associated with the instantaneous valueof the mass of all the boundary layers in the analysis. Indeed, this oscillation will progressivelydisappear as the solid mesh is refined and, anyway, can be measured and filtered at any time stepof the time-integration procedure.

5. NUMERICAL EXAMPLES

In all the examples considered in this section the values of �=1.4 and �=1.5 of the shapeparameters are fixed unless differently stated.

5.1. Free-sloshing of an incompressible fluid in a container

In the first example the free oscillation of an incompressible fluid in a rigid and fixed container isconsidered (see e.g. [6] and [4]), as depicted in Figure 4.

The initial free surface profile is enforced as

�(x,0)=a sin(

bx1), (29)



Figure 4. Sloshing problem. Computational domain.

Figure 5. Small amplitude sloshing. Time histories of the surface height at x1=b/2 and x1=−b/2.

where b is the width of the container, � denotes the height of the wave and a is the maximumamplitude. Fictitious values of fluid viscosity �=0.01kg/ms and density � f =1.0kg/m3 areconsidered. In the calculation, an initial mesh of 1421 nodes and 2647 elements is used and a timestep of 0.005s is adopted.

A set of analyses has been performed assuming a small initial amplitude a=0.01. The timeevolution of the free surface height � at x1=±b/2 is presented in Figure 5, where circles denotethe peak values of � as extrapolated from [6] and squares from [4]. A good agreement is observed.

As mentioned in Section 4.3, particle separation and element cancellation induced by the �-shapemethod may lead to oscillations of the fluid total mass, despite the assumed incompressibility, asis well known in the literature (e.g. [20]). As empirical verification of mass preservation for theproblem at hand, the exact mass si computed theoretically and numerically; a maximum upper-bound error of 0.09% and a maximum lower-bound error of 0.14% are obtained. Only smalloscillations are evidenced and mass loss does not seem to be an issue for the proposed formulation.



Figure 6. Solitary wave propagation. Computational domain.

5.2. Solitary wave propagation

In this example, the propagation of a solitary wave is addressed. The motion of a viscous incom-pressible fluid in a fixed tank is analyzed under assigned initial conditions and the action of thegravity force (Figure 6).

5.2.1. Solitary wave in a rigid tank. In order to perform a comparison with the available analyticaland numerical results (e.g. [31–34]) a rigid tank is assumed at first. In the limit case k→∞ (e.g. awave at infinite distance from limiting walls) the analytical Laitone’s approximation for a solitarytravelling wave [35] is available and is employed here as a reference solution:

u1(x, t) =√gd

(H

d

)sech2

(√3H

4d3(x1−ct)

), (30)

u2(x, t) =√3g

d

(H

d

)3/2

x2 sech2

(√3H

4d3(x1−ct)

)tanh

(√3H

4d3(x1−ct)

), (31)

�(x, t) = Hsech2(√

3H

4d3(x1−ct)

). (32)

In Equations (30)–(32)

c=√gd

(1+ H

d

)

and �, H , k and d are defined in Figure 6; g is the gravity acceleration and u1 and u2 denotevelocity components.

The initial wave height � and velocity field u(x,0) for the numerical solution are generatedaccording to Laitone’s approximation (30)–(32) by setting t=0. This formula holds only for aninfinitely long channel in which the fluid is still at an infinite distance from the wave peak. However,the fluid can be considered to be at rest if it as sufficiently distant from the wave crest. In thesolution d=10, H =2 and k=8 will be considered. A fluid viscosity �=1.0kg/ms and a density� f =1.0kg/m3 are adopted in the calculations. In the analysis an initial mesh of 8210 nodes and15 366 elements is used and a time step of 0.001s is adopted.

The run-up height R (i.e. the maximum height of the wave at the sidewall), the correspondingtime and the maximum pressure are compared with the numerical solutions of Ramaswamy



Table I. The run-up height, the corresponding time and the maximum pressure.

Theory Fluent [31] [32] [33] [34] Pres. meth.

Height 14.2 14.19 14.48 14.4 13.4 14.27 14.19Time — 7.55 7.7 7.66 7.6 7.6 7.69Pressure — — 130 132.2 — 131.66 127.9

Figure 7. Solitary wave with elastic wall. Horizontal displacement at the top of the retaining wall.

et al.[31], Hansbo [32], Navti et al.[33], and Duarte et al.[34] and against the analytical estimate ofR=14.2 obtained in [36]. A good agreement is found, as shown by the data collected in Table I.

5.2.2. Solitary wave in a tank with an elastic wall. As a first validation of the fluid–structureinteraction algorithm, the same travelling solitary wave is now assumed to collide against anelastic wall (see also [37]) defined by the following parameters: thickness s=1m, height L=16m,Young modulus E=10MPa, Poisson coefficient =0.3 and density �s =10kg/m3. The overallgeometrical dimensions and fluid properties are assumed as in the rigid-wall case.

In Figure 7 the horizontal displacement of the top of the structure is shown whereas thecomparison between the run-up height d+� obtained with the rigid and elastic walls is presentedin Figure 8. In the latter case, the maximum run-up height slightly decreases, leading to R=14.13.

5.3. Deformation of an elastic gate

A container formed by two walls is considered. One of the two walls is rigid, and the other oneconsists of an upper rigid part and of a lower deformable gate made of rubber. The rubber gate isfree at its lower end and clamped at its upper side. The rubber plate is held fixed by an externalrigid support so that the container can be filled with water. Then the rigid support is suddenlyremoved, so that the rubber gate can deform and water flows beneath it. The geometry of the



Figure 8. Solitary wave with elastic wall. Comparison with the run-up heightd+� obtained with the rigid wall.

Figure 9. Deformation of an elastic gate. Geometry of the problem.

problem is sketched in Figure 9. The geometrical dimensions of the system can be found in [38],where the experimental results are provided and the problem is simulated numerically using theSPH approach. Following [38] the density of the rubber is taken as �s =1100kg/m3, its Youngmodulus as E=12MPa and its Poisson ratio as =0.4. A viscosity �=0.001kg/ms and a density� f =1000kg/m3 are adopted for the water. For the numerical solution of the problem, an initialfluid mesh of 7470 nodes and 14 374 elements and a solid mesh of 285 nodes and 503 quadrilateralelements are adopted and a time step of 0.001s is used.

Figure 10 shows a snapshot of the deformation of the elastic gate and of the emptying of thecontainer. In Figure 11, the horizontal and vertical displacements computed at the free end of thegate are compared with those obtained numerically and experimentally in [38]. Taking into accountuncertainties in the material properties and some issues arisen in [38] concerning the experimentalresults, the agreement can be considered satisfactory.

5.4. Filling of an elastic container

The proposed particle method is particularly suited to model fluid–structure interactions involvinglarge fluid motions with large displacements of the structure. To give a qualitative example of



Figure 10. Deformation of an elastic gate. Snapshot of the deformed shape at t=0.28s.

Figure 11. Deformation of an elastic gate. Horizontal (x-disp) and vertical (y-disp) displacementsof the free end of the gate. Comparisons between experimental values (exp), SPH approach

(SPH) and present approach (p.a.).

the potential of the method, the following idealized problem has been modeled. Under the actionof the gravity force a fluid (water in the following) drops down from a funnel-shaped rigidcontainer into a thin elastic container for which a linear elastic constitutive law is assumed to hold.The elastic properties of the container are given as: Young modulus E=210kPa, Poisson ratio =0.3 and density �s =2000kg/m3. A fluid viscosity �=0.001kg/ms and a density � f =1000kg/m3

are adopted for the fluid. In the solution an initial fluid mesh of 5749 nodes and 11 103 elementsand a solid mesh of 2188 nodes and 3365 quadrilateral elements are adopted and a time step of0.001s is used.

In this example all the capabilities of the described formulation are required, since a flowwith evolving free surfaces and fluid–structure interaction in which the solid undergoes largedisplacements come into play. Figure 12 shows the snapshots of the problem configuration atdifferent time steps, in which the dots represent the nodes of the fluid mesh (here identified as fluidparticles) whereas the quadrilateral finite elements represent the discretization of the structure.

This example is also used to test the mass conservation. As introduced in Section 4.3 the methoddoes not guarantee the fluid mass conservation when the problem presents evolving fluid–structure



Figure 12. Filling of an elastic container. Snapshots at different time steps.

interfaces. Here an empirical verification is performed. Figure 13 shows the evolution of the totalmass for two different values of �. It should be remarked that the total mass fluctuates duringthe analysis. As discussed in Section 4.3 these oscillations are mainly due to the generation andremoval of interface elements between solid and fluid.

On the contrary, Figure 14 illustrates the mass conservation for two different meshes (for a fixedvalue of �=1.3) and shows that a refinement of the mesh size reduces the mass loss due to theevolving interface layer, as expected.

Finally, Figure 15 shows the evolution of the mass when the mass associated with the interfaceelements is not taken into account. It can be noticed that the mass without interfaces remainsalmost constant confirming the good mass conservation properties of the method. The value ofmass for the fine mesh appears to be greater than the value for the coarse one; this is due to thedifferent dimensions of the boundary layers and it indicates that, reducing the size of the mesh,the size of the boundary layer diminishes and so does the mass loss.

6. CONCLUSIONS

A Lagrangian finite element approach, based on the Particle FEM, has been presented for thesimulation of fluid–structure interaction problems dominated by free-surface flows and possiblylarge displacements of the structure.



Figure 13. Filling of an elastic container. Verification of the mass preservation:comparison of different � values.

Figure 14. Filling of an elastic container. Verification of the mass preservation:comparison of different mesh sizes.

The enforcement of equilibrium of the fluid and solid parts has been performed at the end of eachstep of the time history following an implicit approach. The coupling has been analyzed using aclassical staggered scheme based on the Dirichlet–Neumann technique in which the fluid–structureinterface is treated as a Neumann boundary for the solid and as a Dirichlet one for the fluid.

The adopted strategy allows for the independent modeling of the fluid and structural domains.In particular, the latter can be discretized using classical finite elements enabling the coupling ofthe PFEM with existing FEM codes.

Stability of the discretized fluid problem has been achieved by a Petrov-Galerkin pressurestabilization [19] which, on the one hand, leads to a monolithic solution for pressures and velocities



Figure 15. Filling of an elastic container. Verification of the mass preservationwithout boundary layer for �=1.3.

whereas, and on the other hand, allows for an accurate enforcement of the incompressibilityconstraint which greatly contributes to guarantee mass conservation during the analysis.

A novel fluid–structure interaction strategy has been proposed, based on the definition of ghost-particles on the solid boundary. The activation of the interaction is governed by a thin layer ofinterface elements which are generated and removed on the basis of the same �-shape criterionwhich is used for the identification of the boundary of the fluid domain.

The mass conservation problem has been discussed in detail, pointing out the role of the layerof elements which are generated and removed during the evolution of the fluid–structure interface.This role has also been quantified by means of a numerical test.

A set of simple examples has been analyzed and compared, whenever possible, with the availableanalytical and numerical solutions showing the robustness of the adopted scheme. Finally, thefilling of a thin and highly deformable container has been simulated in order to put in evidencethe modeling capabilities of the technique.

Extension of the approach to three-dimensional problems and materials with non-linear consti-tutive laws are currently under investigation together with the analysis of the potential benefits ofapplying a monolithic approach to the solution of the fluid–structure interaction problem.

ACKNOWLEDGEMENTS

The financial support from Tetra Pak Packaging Solutions S.p.a. is gratefully acknowledged.

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A Lagrangian finite element approach for the analysis of fluid-structure interaction problems

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