CILAMCE2014-0618_15867

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ALE FLUID STRUCTURE COUPLING WITH MESH MOVINGBASED ON A COARSE HIGH ORDER AUXILIARY MESH

Jeferson Wilian Dossa Fernandes

Sergio Andrés Pardo Suárez

[email protected]

[email protected]

Escola de Engenharia de São Carlos da Universidade de São Paulo - EESC/USP

Rogério Carrazedo

[email protected]

Universidade Tecnológica Federal do Paraná - UTFPR, Campus de Pato Branco

Via do conhecimento km 01, CEP 85503-390, Pato Branco, Paraná, Brazil

Rodolfo André Kuche Sanches

[email protected]

Escola de Engenharia de São Carlos da Universidade de São Paulo - EESC/USP

Av. do Trabalhador Sãocarlense, 400, CEP 13566-590, São Carlos, SP, Brazil

Abstract. The most common approach for fluid-structure interaction simulation is to employan arbitrary Lagrangian Eulerian (ALE) description for the fluid and Lagrangian descriptionfor the structure. In this case the fluid mesh need to be deformed along time to accommodate thestructural displacements, splitting the coupled problem into three subjects: the fluid dynamics,the shell dynamics and the fluid mesh dynamics. In this work we use a partitioned technique tocouple an explicit high velocity fluid solver to a nonlinear shell solver without rotation degreesof freedom. In order to move and deform the fluid mesh we develop a technique where a coarsehigh order finite element mesh, called space mesh, is immersed in the fluid mesh and fitted tothe boundaries. In the preprocessing stage, the parametric local position of each fluid meshnode is identified into the coarse mesh, as well as the space mesh element which contains thisnode. Along time, the space mesh is deformed and the new position of the fluid mesh nodes isevaluated, saving computational time. This procedure is tested by numerical examples.

Keywords: Fluid-shell interaction, ALE, Partitioned coupling

CILAMCE 2014Proceedings of the XXXV Iberian Latin-American Congress on Computational Methods in Engineering

Evandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil, November 23-26, 2014

ALE fluid structure coupling with mesh moving based on a coarse high order auxiliary mesh

1 INTRODUCTION

Fluid-structure interaction (FSI) problems simulation is a challenging and very relevantsubject for many areas of engineering. Experimental FSI analysis are, in general, very expensiveand require considerable time, leading to the search of computational tools for simulating asprecisely as possible such problems.

Although researches on FSI have already reach some maturity, there are still several is-sues which need to be studied or optimized, so that some of the best computational mechanicsresearchers are currently studying this subject.

Numerical fluid structure interaction (FSI) modelling includes three main subjects: thecomputational fluid dynamics (CFD), the computational solid dynamics (CSD) and the interac-tion problem (IP) (Sanches and Coda, 2013, 2014). The strategy used for solving the IP shouldbe able to adequately combine CSD and CFD in spite of the different characteristics of eachfield.

Mathematical modeling for physics problems is traditionally done by Eulerian or Lagrangiandescriptions. The Lagrangian description expresses the continuum medium movement in termsof the initial configuration and time, being very efficient for problems where finite displace-ments are the main variables, such as in solid mechanics (CSD), with deforming meshes. Onthe other hand, Eulerian description is defined in terms of final configuration and time, beingadequate for problems where main variables are velocities instead of displacements, such as influid flows (CFD), resulting meshes fixed in space.

Therefore, the main issue for the IP is how to combine this two different descriptions.One widely used methodology to allow fluid mesh to be deformed to accommodate structuraldisplacements is to use the arbitrary Lagrangian-Eulerian ALE description for Navier-Stokesequations, allowing to enforce arbitrary deformation to the mesh (Soria and Casadei, 1997;Donea et al., 1982; Sawada and Hisada, 2007). In this methodology, the fluid mesh needs to bedynamically deformed in order to accommodate structural displacements or vibrations and atsame time keep the fixed part of fluid flow boundaries unchanged.

In the ALE context, the coupled problem can be formulated as a three- rather than two-field problem (fluid dynamics, mesh dynamics and structure dynamics) Lesoinne and Farhat(1996). The weak form of fluid equations involves both the position and velocity of the mesh,so that the good representation of mesh motion is very important. Some good mesh movingtechniques are shown in Kanchi and Masud (2007), where the authors present an adaptive meshmoving scheme for 3D grids composed of linear tetrahedral and hexahedral elements, Degandand Farhat (2002) and Farhat et al. (1998) where the authors deal with the mesh as an structuralproblem using a 3D torsional spring analogy and Stein et al. (2003) where mesh moving andremeshing techniques are combined. It is important to mention that problems, with large scaleof displacement will require also a remeshing technique Saksono et al. (2007).

Due to its high computational cost, it is important to have efficient algorithms for all of thethree fields mentioned above, including mesh dynamics. Therefore, in this paper we develop atechnique where a coarse high order finite element mesh, called space mesh, is immersed in thefluid mesh and fitted to the boundaries. In the preprocessing stage, the parametric local positionof each fluid mesh node is identified into the coarse mesh, as well as the space mesh elementwhich contains this node. Along time, the space mesh is deformed and the new position of the

CILAMCE 2014Proceedings of the XXXIV Iberian Latin-American Congress on Computational Methods in EngineeringEvandro Parente Jr (Editor), ABMEC, Fortaleza, CE, Brazil, November 23-26, 2014

J. W. D. Fernandes, S. A. P. Suárez, R. Carrazedo, R. A. K. Sanches

fluid mesh nodes is evaluated, saving computational time. This procedure is tested by numericalexamples.

We assume the fluid flow to be compressible, which allows us to efficiently employ anexplicit time integrator for saving computational effort. The governing equations are written inthe ALE convective form and integrated in time based on characteristics. This automaticallyintroduces diffusion in stream direction, stabilizing spurious variations due the employment ofthe standard Galerkin procedure Zienkiewicz and Taylor (2000).

For dynamics of structures, we employ a novel unconstrained geometric non-linear formu-lation for shell analysis based on positions and unconstrained vectors introduced by Coda andco-workers Coda and Paccolla (2007); Coda and Paccola (2009); Coda (2009); Coda and Pac-cola (2010); Sanches and Coda (2013). The use of positions as nodal parameters is motivatedby the work present by Bonet et al. Bonet et al. (2000). The main advantage of this formulationsare the absence of large rotation description, resulting in a constant mass matrix and thereforemaking naturally possible the use of the Newmark time integrator as an momentum conservingalgorithm and conserves energy for small strain problems (see Sanches and Coda (2013)).

In order to couple fluid and structure, we employ a partitioned scheme, which among otheradvantages, allows to use completely different solvers for solid and fluid as well as differenttime steps (Sanches and Coda, 2013, 2014; Felippa et al., 2001; Teixeira and Awruch, 2005).

This paper is organized as follows: In section 1 we present the employed shell formulation,in section 2 we present the compressible fluid dynamics formulation, in section 3 we describethe coupling scheme and the mesh moving technique, in section 4 we present numerical studiesof the proposed coupling scheme, and finally in section 5 we draw the conclusions.

2 SHELL DYNAMICS

Shell structures are solids with one dimension much larger than the others. Therefore,the mid surface of coordinates X in the initial configuration and x in the actual configuration,serves as a reference to the solid mapping. The mappings f 0 and f 1, from an auxiliary non-dimensional space respectively to the initial and current configurations may be written for anypoint out of the middle surface, its position at initial and actual configuration as:

f0i = Xi = Nj(ξ1, ξ2)X

mji +

h0

2ξ3Nj(ξ1, ξ2)e

0ij, (1)

and

f1i = xi = Nj(ξ1, ξ2)x

mji +

h0

2

ξ3 + ajNj(ξ1, ξ2)ξ

23

Nj(ξ1, ξ2) Gij, (2)

where Gij are the nodal values (unknowns) for the generalized vector at node j at final configu-ration, h0 is the initial thickness, e0i is the i− th component of the unitary vector

−→e0 , normal to

the middle surface at initial and a is the strain rate along thickness.

Finally, the positional mapping from initial to current is represented by:

f = f (X) =f1f0−1

. (3)




The gradient A of mapping function may be expressed by:

A = ∇f =A1

A0

−1. (4)

After evaluating the gradient A, the Green strain tensor and the specific strain energy maybe obtained, following Ogden (1984):

Eij =1

2[AkiAkj − δij] =

1

2[Cij − δij] . (5)

The variables Cij and δij are the right Cauchy-Green stretch tensor and the Kroeneckerdelta, respectively. The following quadratic strain energy per unit of initial volume is adopted,

ue =1

2EijCijklEkl (6)

resulting into a linear elastic constitutive law relating second Piola-Kirchhoff stress and Greenstrain, usually called Saint-Venant–Kirchhoff elastic law, i.e.:

Sij =∂ue

∂Eij= CijklEkl (7)

where The Cijkl are the components of the elastic constants tensor.

From preceding developments, one may write the equilibrium equation as the minimizationof the energy functional as:

∂Ue

∂x− F+Mx+Cx = 0, (8)

where F is the external forces vector, C is the dissipative matrix and M is the mass matrix.

Coda and Paccola proved that for a positional total Lagrangian description, the Newmark β

with γ = 1/2 presents momentum conservative properties for most of shell dynamics problemsand conserves energy for small strains if the time step is sufficiently large that the asymp-totic energy convergence dominates or small enough that a uniform bound on the energy isachieved (Coda and Paccola, 2009) - (see Hughes (1976) for more details with respect to en-ergy conservation for constant mass matrix nonlinear dynamics with the average accelerationtime integration).

From Newmark β method, the equilibrium equation for a given instant s+ 1 becomes:

∂Ue

∂x

S+1

− FS+1 +M

β∆t2xS+1 −MQS +CRS +

γC

β∆txS+1 − γ∆tCQS = 0, (9)

where QS = xSβ∆t2 +

xSβ∆t +

12β − 1

xS and RS = xS +∆t (1− γ) xS .

Equation (9) represents a nonlinear system, which we solve employing Newton-Raphsonmethod. Each node will have 7 nodal parameters: 3 position vector components xi with i = 1, 2or 3, 3 components of the generalized position vector Gi with i = 1, 2 or 3 and the strain ratioalong thickness a.



3 FEM FOR FLUID DYNAMICS WITH MOVING BOUNDARIES

The Eulerian description of fluid dynamics governing equations (Navier-Stokes) is givenby:

∂ρ

∂t= −∂(ρui)

∂xi, (10)

the mass conservation equation,

∂(ρui)

∂t= −∂(ujρui)

∂xj+

∂τij

∂xj− ∂p

∂xi+ ρgi, (11)

the momentum equation and

∂(ρE)

∂t= − ∂

∂xj(ujρE) +

∂

∂xi

k∂T

∂xi

− ∂

∂xj(ujp) +

∂

∂xj(τijuj) + ρgiui. (12)

the energy equation, where ρ is the specific mass, ui the i− th velocity component, p pressure,τij the deviatoric stress tensor (i, j) component, gi the i − th direction field forces constant, Ethe specific energy, T temperature and k the thermal conductivity.

The Arbitrary Lagrangian-Eulerian (ALE) description is obtained by the introduction ofa reference domain R(t) with arbitrary movement when deriving the governing equations. Itmay be considered as a mapping from initial configuration C(t0) to final configuration C(t),written with respect to the moving reference domain (domain covered by the finite elementmesh) Donea et al. (1982) (see Fig. 1).

Figure 1: Adopted kinematics for ALE description adapted from Sanches and Coda (2014)

The Jacobian J , for the transformation from reference domain R(t) to the material domainC(t0), is given by:

J = det (A) onde Aij =∂ξi

∂ajwith i and j = 1, 2 or 3, (13)

where ξi and aj are the position vectors components regarding respectively to R(t) and C(T0).

Considering a physical property expressed on the reference configuration as f(ξi, t) andequal to F (ai, t) on the initial configuration,considering also that ∂J

∂t = J∇ · w, where w is




the velocity vector of the reference point of coordinates ξ,it is possible to write: Based on thederivative rules applied to ∇(fw) we get:

∂(JF )

∂t= J

∂f

∂t+∇ · (fw)

, (14)

Substituting f = ρ on equation (14) and considering the mass conservation equation ((10)),using index notation, one writes:

∂(ρJ)

∂t= J

∂ (ρ(wi − ui))

∂xj(15)

or, substituting the time derivative of J :

∂ρ

∂t+

∂(ρui)

∂xi= wi

∂ρ

∂xi, (16)

which is the final form for the mass conservation equation on ALE description.

Following the same procedure, now for (10) and (11), the ALE formulations for momentumand energy equation are respectively:

∂(ρui)

∂t+

∂(ujρui)

∂xj− ∂τij

∂xj+

∂p

∂xi− ρgi = wj

∂(ρui)

∂xj(17)

and

∂(ρE)

∂t+

∂(ujρE)

∂xj− ∂

∂xi

k∂T

∂xi

+

∂(ujp)

∂xj− ∂(τijuj)

∂xj− ρgiui = wi

∂(ρE)

∂xi. (18)

It is important to observe that if the velocity w is null, the formulation relies on the Euleriandescription, while if w = u the formulation relies on the Lagrangian description Donea et al.(1982).

3.1 Time integration

If there is no diffusion, the time variation of φ over a characteristic coordinates x is bydefinition null. For the Navier-Stokes equations we can write:

∂φ(x, t)

∂t−Q(x) = 0, (19)

where Q(x) contains all the non convective terms.

We assume the following approximation for Eq. (19) (Zienkiewicz and Taylor, 2000):

φ(y)n+1 − φ(x)n∆t

≈ θ(Q(y)n+1) + (1− θ)(Q(x)n), (20)



where x and y means respectively the characteristic positions at t = n and t = n + 1, θ is aconstant with value 0 for explicit solution and may be chosen larger than zero 0 and smallerthan 1 for semi-implicit or implicit solution.

The product uφ and the term Q(x) may be approximated by Taylor resulting the followingexpressions:

uφ(x)n = uφ(y)n − (y − x)∂(uφ(y))n

∂x+

(y − x)2

2

∂2(uφ(y)n)

∂x2

+O(∆t3),

(21)

Q(x)n = Q(y)n − (y − x)∂Q(y)n∂x

+O(∆t2). (22)

Assuming ∆t = (y − x)/u, from Eqs. (21), (20) and (22), and assuming θ = 0 (explicitform), we have:

φ(y)n+1 = φ(y)n −∆t

∂(uφ(y))n

∂x−Q(y)n

+(∆t)2

2u∂

∂x

∂(uφ(y)n)

∂x−Q(y)n

+O(∆t

2).

(23)

One important point about this procedure is that the high order terms of Eq. (23), ob-tained due to time integration along characteristics, introduce dissipation on stream lines direc-tion, which as shown by (Zienkiewicz and Taylor, 2000) are equivalent to the Petrov-Galerkingschemes when the time interval tends to the critical time interval, and gets smaller effects as thetime interval get larger.

Applying the procedure of Eq. (23) to the Navier-Stokes equations, one may write formomentum and energy equations one may write:

∆(ρui)n+1 = ∆t

−∂(ujρui)

∂xj+ wj

∂(ρui)

∂xj+

∂τij

∂xj− ∂p

∂xi+ ρgi

n

+

∆t2

2

uk

∂

∂xk

∂(ujρui)

∂xj− wj

∂(ρui)

∂xj− ∂τij

∂xj+

∂p

∂xi− ρgi

n

, (24)

and

∆(ρE)n+1 = ∆t

−∂(uiρE)

∂xi+ wi

∂(ρE)

∂xi

n

+

∆t

∂

∂xi

k∂T

∂xi

− ∂(uip)

∂xi+

∂(τijuj)

∂xi− ρgiui

+

∆t2

2uk

∂

∂xk

∂(uiρE)

∂xi− wi

∂(ρE)

∂xi

+

∆t2

2uk

∂

∂xk

− ∂

∂xi

k∂T

∂xi

+

∂(uip)

∂xi− ∂(τijuj)

∂xi+ ρgiui

n

.

(25)

where all the right hand side terms are known at the instant t = n.




For mass conservation equation we use the following expression:

∆ρn+1 = −∆t

∂

∂xi(ρui)n + θ

∂ (∆ (ρui))n+1

∂xi+ wi

∂ρ

∂xi

+

∆t2

2

uk

∂

∂xk

−wi

∂ρ

∂xi

n

.(26)

where θ is an arbitrary constant with value between 0.5 and 1.0.

Applying the Galerkin method to Eq. (24), (26) and (25), we obtain the spatial discretiza-tion and solve the resulting system getting the week solution.

We still need to deal with the discontinuities due to the presence of shock waves, as thestandard Galerkin method is unable to deal with strong discontinuities. Therefore, the artificialdiffusion term based on pressure second derivative is added:

fµa = ∆tµa∂

∂xi

∂φ

∂xi

, (27)

where φ is the variable to be smoothed and µa is the artificial viscosity given by:

µa = qdifh3 (|u|+ c)

pav

∂

∂xi

∂p

∂xi

e

, (28)

where |u| is the velocity absolute value, pav is the pressure average over the element, qdif is anuser specified coefficient taken between 0 and 2, c is the sound speed and h is the element size(Zienkiewicz and Taylor, 2000; Nithiarasu et al., 1998).

4 PARTITIONED FLUID-STRUCTURE COUPLING

It is desirable a numerical scheme able to make the exactly prediction of an uniform flow.Lesoinne and Farhat (1996) showed that this condition is satisfied if the scheme under consid-eration satisfies a discrete version of the geometric conservation laws (DGCL), which statesbasically that that the change in volume of each element for a given time interval must be equalto the volume (or area) swept by the element boundary during the time interval.

It occurs if the mesh motion is developed in a way that it is exactly captured by the timeintegrator scheme. In the FSI modeling with different fluid and structure time integrators, en-forcing DGCL implies in different values for mesh and structure velocity for the fluid nodes inthe fluid-structure interface.

Geometric conservation poses a difficulty particularly when the divergence ALE formula-tion is applied, however if a convective ALE formulation is employed, as the one employedhere, constant flow is exactly predicted irrespective of the employed time discretization scheme(Förster, 2007). Boffi and Gastaldi (2004) proved that the GCL is neither necessary nor suffi-cient for stability and, as admitted by Guillard and Farhat (2000), there are recurrent assertionsin the literature stating that enforcing the DGCL may be unnecessary.

Taking this facts into account, although possible to do, we do not care about enforcing theDGCLs in our coupling scheme.



4.1 Forces and velocities transfers

As fluid and shell meshes are different and generated in an independent process, duringthe pre-processing step, for each fluid node i, a closest point Psi at shell mesh domain (Ωs) isidentified and stored (see Fig. 2). Subsequently, the fluid mesh is adapted moving the node i tothe exact position of Psi.

Figure 2: Transfer points

For each shell node k , a closest point Pfk on the fluid boundary related to the structure(Γs) is identified and stored.

The Dirichlet boundary conditions for node i are obtained from the shell point Psi. Forviscous flow uf (i) = us(Psi) while for inviscid flow we adopt:

uf (i) = uf (i) + [(us(Psi)− uf (i)) · n]n. (29)

where uf is the fluid velocity vector, us is the shell velocity vector and n, the unity vector normalto Γs.

The Neumman boundary conditions for node k are obtained from fluid point Pfk by thefollowing expression:

qkj = [τjlnl − pnl]Pfk, (30)

where the indexes j and l represent Cartesian direction and nl is the l component form thenormal vector to Γs.

4.2 Fluid mesh dynamic moving

A good mesh moving algorithm for fluid-structure interaction problems will adapt the fluidmesh to the solid movement with minimal of mesh distortion.

The fluid mesh is usually very fine close to the structure in order to capture shock waves,vortex shedding, limit layer or any other flow structure. However, structural movements areoften much more smooth than the mentioned situations, hardly presenting discontinuity.

Based on these facts, we chose not to deal direct with all the degree of freedom of the fluiddynamic mesh. Instead, we insert another much coarser mesh with high order curved tetrahedralelements which fits to the boundaries and solve the reference domain movement based on thismesh. This mesh we call space mesh.




During the preprocessing phase we find to each fluid node P , the space mesh element elsmwhere it is immersed and its local coordinates ξsm1(P ), ξsm2(P ) and ξsm3(P ) and store theseinformations. The space mesh is dynamically deformed and the positions xf i(P )and velocitieswi(P ) for the fluid mesh is interpolate by:

xf i(P ) = Nsmk(ξsm(P ))xsmi(k), (31)

and

wf i(P ) = Nsmk(ξsm(P ))wsmi(k), (32)

where Nsmk is the space mesh shape function associated to the node k, xsmi(k) is the nodali − thcoordinate of node k, and wsmi(k) is the nodal i − th velocity component associated tothe node k.

The Laplace equation would be a good choice for a mesh moving model (Kanchi andMasud, 2007), however it makes necessary to solve a new equation system. Therefore weadopted the same technique employed by Sanches and Coda (2014), which is based in the oneemployed by Teixeira (2001) . This technique consists on distribute the space mesh velocitiesand positions based on the distance to the boundary according to:

Xsmki =

nej=1 akjXs

jine

j=1 akj +nf

l=1 bkl

, (33)

where ne is the shell nodes number, nf is the fixed boundary (Γf ) nodes number, akj is theweight coefficient that considers the shell movement influence, and bkl is the coefficient thattakes in account the fixed boundary influence and Xsm and Xs may be space mesh and shellvelocities or displacements.

The coefficients a and b are given by:

akj =1

de1kj

(34)

and

bkl =1

de2kl

, (35)

where dkj is the distance between the node k and the node j located at the moving boundary,dkl is the distance between the node k and the node l located at fixed fixed boundary, and e1and e2 are numbers chosen by the operator which enable to adjust the boundary influence overthe mesh movement. A good choice for open flow problems is e1 = e2 = 4, however for closedflows with large displacements a smaller number may present better results.

4.3 Coupling dynamic processAs the shell solver is implicit and the fluid solver explicit, a coupling scheme which enable

sub-cycles of time steps is desirable. It is usual that time steps used for shell analysis are largerthan the ones adopted for fluid modeling, therefore we suggest the scheme depicted on Fig. 3.

This scheme may be summarized as:



Figure 3: Coupling scheme adapted from Sanches and Coda (2013)

• At a given instant ts = i, the structure is solved with the loads imposed by the flow (step1), with a time step ∆ts, so that the velocities and positions at instant ts = i + 1 areobtained (step 2).

• Shell positions variations inside the time interval ∆ts are approximated by a cubic poly-nomial, which is obtained based on xs(i), xs(i), xs(i + 1) and xs(i + 1) (steps 3 and 4).From this polynomial we can find the shell velocities and positions for each fluid timestep ∆tf = (∆ts)/nsf . Following, the fluid mesh movement is obtained from equations(33), (31) and (32) (steps 5 and 7).

• Fluid Dirichlet boundary conditions are obtained from the shell movement and the fluidis solved using time steps ∆tf (steps 6 and 8) until reach the instant tf = i ∗ nsf , whenfluid loads are imposed to the structure and the process restart.

Mesh Moving Tests

In order to verify the proposed mesh moving technique, we choose a panel flutter problem.In this example we study the behavior of an initially flat panel subjected to supersonic flow.

The panel starts at x = 0.25m and has: specific mass ρs = 2710 kg/m3, Young’s modulusE = 77.28 GPa and Poisson’s ratio ν = 0.33, length 0.5 m, width 0.025 m and thickness1.35 mm. The boundary conditions at z = 0 and z = 0.025m are those of symmetry regardingplane xy.

We meshed the panel with 40 8 nodes triangular isoparametric shell elements (cubic order)and 244 nodes while fluid domain with 30224 4 nodes linear tetrahedral elements and 8584nodes (see Fig. 4. The space mesh employed for the adopted moving mesh technique has 16820 nodes tetrahedral isoparametric elements (cubic order) and 1066 nodes (see Fig. 5).

This is example is chosen due to the facts that initially fluid mesh need to precisely cap-ture small amplitude movements in order to produce dynamical instability and it reaches largeamplitudes along time, being a complete test for the proposed coupling scheme.

Initially we perform 3 different tests only to the mesh moving algorithm where thereis no fluid flow and the panel is submitted to: 1) a positive normal uniformly distributedload (non-conservative load initially oriented to up direction) qn = 5000N/m2; 2) a negativeuniformly distributed normal load (non-conservative load initially oriented to down directionqn = −5000N/m2; and 3) a distributed normal uniform load negative in the left half and posi-tive in right half qn = −15000N/m2 if x < 0.5 and qn = 15000N/m2 if x > 0.5.




(a) fluid mesh (xy view) (b) panel mesh (xz view)

Figure 4: Fluid and panel discretization

Figure 5: Space mesh

For all these cases the panel is fully clamped on the left and with only horizontal displace-ments free on the right.

The shell time step is δt = 0.001. In order to test the coupling procedure, the fluid mesh issolved with one time step δtf = 0.0001, i.e., 10 sub-cycles.

All the three cases showed good results as one can see from figures 6, 7 and 8.



(a) t = 0.007 s s

(b) t = 0.011 s s

Figure 6: Deformed fluid mesh for panel with up oriented uniform distributed load

(a) t = 0.007 s s

(b) t = 0.011 s s

Figure 7: Deformed fluid mesh for panel with down oriented uniform distributed load




Figure 8: Deformed fluid mesh for panel with down oriented uniform distributed load at t = 0.007s

Panel flutter

Based on the results of the preliminary tests, the proposed algorithm seems to be ready forthe true fluid-structure interaction problem.

The flow is considered inviscid and the undisturbed flow has specific mass ρ∞ = 0.339kg/m3 and pressure p∞ = 28 KPa, Mach number M = 2.3 and sound speed c = 340 m/s,while the panel has: specific mass ρs = 2710 kg/m3, Young’s modulus E = 77.28 GPa andPoisson’s ratio ν = 0.33, length 0.5 m, width 0.025 m and thickness 1.35 mm. The boundaryconditions at z = 0 and z = 0.025m are those of symmetry regarding plane xy.

At the beginning of the analysis, the pressure on the inferior face of the panel is reduced0.1 %, introducing a disturbance to the panel. This condition is kept until 4 ms, when pressureon the inferior face is set again to p∞.

This problem has critical Mach number M = 2 according to the simulation performed bySanches and Coda (2014), so that it is expected to get dynamical instability.

Two analysis are performed, one with the boundary conditions fully clamped on the leftand with only horizontal displacements free on the right, in order to get larger displacementsand get results closer to the linear solution, and other with the panel fully clamped on both ends.

Figure 9 depicts the displacements vs. time for x = 6.0 (3/4 of the panel). One can seethat the simulation with free horizontal displacements on the right produced the same resultas Teixeira and Awruch (2005) get for the linear problem, however the fully clamped problemreaches a limit cycle, similar to the one obtained for the geometrical nonlinear analysis obtainedby Rifai et al. (1999), due to the membrane effect. This shows our algorithm to be robust andprecise.

Figure 10 shows a snapshot of pressure distribution and shell displacement distribution atinstant t = 0.1195 s in a 3D view.

Finally, one can observe the good quality of the proposed mesh moving scheme in figures11(a) and 11(b), where the deformed fluid mesh (black) and space mesh (red) are depicted forinstants t = 0.119 s and t = 0.1255 s.



-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

y(m

)

t(s)

Non-linear ref.Linear ref.

Fully clamped pres. work.Free x disp. pres. work

Figure 9: Vertical displacements vs. time at x = 0.6m

Figure 10: Fluid pressure and shell displacements distribution for instant t = 0.1195 s (3D view)

(a) t = 0.119 s (b) t = 0.1255 s

Figure 11: Fluid mesh (black) and space mesh (red)




5 CONCLUSION

In this work we briefly addressed all the aspects of a standard fluid-structure interactionnumerical simulation, as well as presented a compressible fluid flow and a geometric nonlin-ear shell formulations, where the shell formulation has no rotation degrees of freedom, and sois more robust and simple. Following we propose a partitioned coupling technique which al-lows different time-steps for fluid and structure. In order to move and deform the fluid meshwe develop a technique where a coarse high order finite element mesh, called space mesh, isimmersed in the fluid mesh and fitted to the boundaries saving computational work and timeduring the dynamic process. This procedure is tested in a panel flutter problem, showing goodresults, attesting the robustness and efficiency of the proposed technique.

ACKNOWLEDGMENTS

The authors would like to thank CNPq, CAPES and Fundação Araucária for financial sup-port, and professor Humberto Breves Coda for helping with the shell dynamics code.

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