A MOVING MESH ALE APPROACH TO SOLVE 2D INCOMPRESSIBLEMULTI-FLUID FLOWS

Fabricio Simeoni de Sousafsimeoni@icmc.usp.brDepartamento de Matematica Aplicada e EstatısticaInstituto de Ciencias Matematicas e de ComputacaoUniversidade de Sao Paulo (USP) - Campus de Sao CarlosCaixa Postal 668, 13560-970, Sao Carlos, SP, Brasil.

Norberto Mangiavacchinorberto@uerj.brDepartamento de Engenharia MecanicaFaculdade de EngenhariaUniversidade Estadual do Rio de Janeiro (UERJ)Rua Sao Francisco Xavier, 524, 20550-900, Rio de Janeiro, RJ, Brasil

Abstract. An Arbitrary Lagrangian-Eulerian (ALE) moving-mesh method to solve incompress-ible two-dimensional multi-fluid flows is presented. The Navier-Stokes equations are discretizedby a Galerkin Finite Element method. A projection method based on approximated LU decom-position is employed to decouple the system of non-linear equations. The interface betweenfluids is sharply represented by marker points and edges of the computational mesh plus anadditional discrete Heaviside function to correctly distribute the continuum surface forces. Ourmethod employs a technique which moves the nodes of the Finite Element mesh with arbitraryvelocity. The quality of the mesh is controlled by a remeshing procedure, avoiding bad trianglesby flipping edges, inserting or removing vertices from the triangulation. The relative velocity inthe ALE approach is designed to allow for a continuous improvement of the mesh, thus reducingthe amount of remeshing required to control the quality of the mesh. Results of numerical simu-lations are presented, illustrating the improvements in computational cost, mass conservation,and accuracy of this new methodology.

Keywords: Two-fluid flows, Incompressible flows, Arbitrary Lagrangian-Eulerian, Level-set,Moving mesh

1. Introduction

Simulations of multi-fluid flows are known to be difficult to perform due to discontinu-ities at the fronts separating the different fluids. A numberof methods have been developedto approximate the fronts, and they can be classified in two main groups: Front-tracking andFront-capturing methods (Unverdi & Tryggvason 1992). In Front-tracking methods, the frontsare represented by computational elements like marker particles, that move through the domainwith the fluid velocity field. A number of papers dealing with Front-tracking methods can befound in the literature (Esmaeeli & Tryggvason 1998, Esmaeeli & Tryggvason 1999, Tryg-gvasonet al. 2001). The Front-tracking methodology is more accurate than Front-capturing,introducing very small mass variation of the fluids involvedin the simulation. However, itsimplementation is more difficult, in particular when the flows undergo topological changes,like either coalescence or splitting of the interfaces. On the other hand, Front-capturing meth-ods represent the interfaces by a region of high gradient variation, where the fronts are recon-structed at each time-step. Among these, the level-set method, introduced by Osher and Sethian(Osher & Sethian 1988), has acquired popularity because of its algorithmic simplicity. In thismethod, the fronts are represented by the zero level set of a function, that is advected by solvingφt +u ·∇φ = 0 whereu is the velocity field. Most numerical procedures designed tosolve thisequation will introduce artificial diffusion leading to pronounced mass conservation errors.

The above described methods can be applied to both structured and unstructured grids.Methods employing unstructured grids can produce more accurate and efficient methods byselectively refining regions of the domain where the interface and other important small scalefeatures of the flow occur. For instance, Chen, Minev and Nandakumar (Chenet al. 2004)present a finite element method for incompressible multiphase flows with capillary interfaceson a fixed Eulerian grid, in which the fluid phases are identified and advected using a levelset function, and the grid is temporarily adapted around theinterfaces. The resulting tech-nique can be considered as a compromise between the arbitrary-Lagrangean-Eulerian (ALE)approach and the fixed grid, Eulerian approach. Perot and Nallapati (Perot & Nallapati 2003)developed a front tracking method for free-surface flows that employs an unstructured meshthat dynamically adjusts to the free surface. The points in the interior of the domain do notmove in a Lagrangian fashion to avoid strong distortion of the mesh. Instead, each edge of themesh is treated as a linear spring, requiring the solution ofan equilibrium equation at each step.The flow field is then updated using an Arbitrary Lagrangian Eulerian (ALE) formulation. Theproposed method, however, does not deal with splitting or reconnections of the free surface.

In order to minimize mass conservation errors and improve accuracy of the level-set tech-nique for incompressible two-phase flows, a new approach wasproposed by Sousa and Mangia-vacchi (Sousa & Mangiavacchi 2005) which can be classified asa Lagrangian level-set method.It uses the same pseudo-concentration functionφ to represent the fronts, but its advection is per-formed by moving the nodes of the mesh, where the values ofφ are stored. The method thus hasfeatures of both front-tracking and front-capturing methods, and can take advantage of them. Inthis work, the Lagrangian method employed by Sousa and Mangiavacchi is extended to allowfor the relative movement between fluid particles and mesh points, in an ALE approach. Therelative velocity is designed to improve the quality of the mesh, thus reducing the amount ofinsertions and deletions of nodes required to mantain its quality. The interface representation ismodified in order to get rid of the pseudo-concentration function. The method employed heresharply represents the interface by segments belonging to the triangulation. In the followingsections, the formulation, discretization and underlyingnumerical method employed in an un-structured mesh ALE approach for the simulation of multiphase flows will be shortly described.Some implementation issues regarding the interface discretization will be discussed.

2. Formulation

The conservation equations modeling incompressible multi-fluid flows are the equation ofmotion in a moving computational mesh

∂(ρu)

∂t+ ((u − u) · ∇)ρu = −∇p +

1

Re∇ ·

[

µ(

∇u + ∇uT)]

+1

Fr2ρg +

1

Wef , (1)

and the continuity equation∇·u = 0, whereu is the velocity field,p is the pressure field,µ andρ are the discontinuous viscosity and density,g represents the gravitational acceleration fieldandf is a source term representing the surface tension. In this equation,Re, Fr andWe arethe non-dimensional Reynolds, Froude and Weber numbers. Using the CSF model (Brackbillet al. 1992), the source term can be written asf = σκ∇H, whereσ is the surface tensioncoefficient,κ is the curvature andH is a Heaviside function which is 1 inside one fluid and 0outside. Here,u is the mesh velocity which is computed as a combination between fluid velocityand an elastic velocity, more specifically

u = β1u + β2ue (2)

whereue is an elastic velocity computed based on a Laplacian filter applied to the node po-sitions, in order to improve the quality of the elements. In eq. (2), parametersβ1 andβ2 arechosen between 0 and 1 to control the final velocity. Notice that β1 = 1 is set for the interfacenodes, where they move in a Lagrangian fashion.

3. Discretization

The domain is discretized by an unstructured triangular mesh which is initially a Delaunaytriangulation. The element used in this approximation is the mini-element (P1+ − P1), with 4velocity nodes (one in each vertex plus one in the centroid ofthe triangle) and 3 pressure nodes(one in each vertex). The shape functions interpolating thediscrete approximations are assumedto be linear plus a bubble function in the centroid for the velocity, and linear for pressure, bothcontinous. ConsideringV = H1(Ω)m = v = (v1, . . . , vm) : vi ∈ H1(Ω), ∀ i = 1, . . . , m,whereH1(Ω) is a Sobolev space, and the sub-spacesVuΓ

= v ∈ V : v = uΓ emΓ1, PpΓ=

q ∈ L2(Ω) : q = pΓ emΓ2, the weak formulation of the problem can be written as: findu(x, t) ∈ VuΓ

andp(x, t) ∈ PpΓsuch that

m(∂(ρu)

∂t, w) + a(u − u, ρu, w) − g(p, w) +

1

Rek(µ, u, w)

−1

Fr2m(ρg, w) −

1

Wem(f, w) = 0 (3)

d(q, u) = 0, (4)

for all w ∈ V0 andq ∈ P0, where the functionals in (3)-(4) are given by

m(v, w) =

∫

Ω

v · w dΩ (5)

a(u, v, w) =

∫

Ω

(u · ∇)v · w dΩ (6)

k(µ, v, w) =

∫

Ω

µ[

(∇v + ∇vT ) : ∇wT]

dΩ (7)

g(p, w) =

∫

Ω

∇p · w dΩ (8)

d(p, w) =

∫

Ω

(∇ · w)p dΩ (9)

The discretization of (3)-(4) using the shape functions forthe mini-element and Galerkinweighting, results in the following ODE system

Mρu + Au + 1

ReKu − Gp − 1

Fr2 Mρg − 1

WeMf = 0

Du = 0 , (10)

In (10), the quantitiesu, v, p, g, andf are vectors of dimensions, wheres is the total numberof nodes in the discretized domain. The ODE’s are discretized in time using a semi-implicittime integration technique, where the non-linear terms arekept explicit. Thus, the matrices tobe solved are symmetric positive definite and fast linear system solvers can be applied, e.g. theconjugate gradient method. Using a semi-implicit time integration, eq. (10) becomes

Mρ

(

un+1−un

∆t

)

+ Aun + 1

ReKun+1 − Gpn+1 − 1

Fr2 Mρg − 1

WeMf = 0

Dun+1 = 0 . (11)

4. Numerical method

The numerical procedure employed to solve the Navier-Stokes equations is a projectionmethod based in approximated blockLU decomposition (Perot 1993, Changet al.2002, Leeetal. 2001). The system of discrete equations (11) can be rewritenas

[

B −∆tGD 0

] [

un+1

pn+1

]

=

[

rn

0

]

+

[

bc1

bc2

]

(12)

whereun+1 = [un+11 , . . . , un+1

Nu , vn+11 , . . . , vn+1

Nv ]T , pn+1 = [pn+11 , . . . , pn+1

Np ]T , with Nu, Nv andNp being the number of free-nodes for the velocity (x andy direction) and pressure. The matrixB is given by

B = Mρ +∆t

ReK (13)

and the right hand side are the known values at timen,

rn = −∆t

(

Aun −1

Fr2Mρg −

1

WeMf

)

+ Mρun , (14)

plus the contribution of the known values from the boundary conditions. Applying a canonicalblockLU decomposition (Leeet al.2001) to the matrix of the system (12) results

[

B 0D ∆tDB−1G

] [

I −∆tB−1G0 I

] [

un+1

pn+1

]

=

[

rn

0

]

+

[

bc1

bc2

]

(15)

According to Chang et al. (Changet al. 2002), the method originating from eq. (15)is named as Uzawa method. However this method is computationally expensive due to theinversion of matrixB. To avoid this operation, we substituteB−1 by an approximationM−1

L,ρ,which is a diagonal matrix, computed from the lumping of the mass matrix apearing in eq. (13),and can be easily inverted. This results the following procedure

1. Solveu from Bu = rn + bc1 ;

2. Solvepn+1 from ∆tDM−1

L,ρGpn+1 = −Du + bc2 ;

3. Compute the final velocity fromun+1 = u + ∆tM−1

L,ρGpn+1 .

As we used the same approximation forB−1 in steps 2 and 3, the computed final velocity isdivergence-free at discrete level.

Fluid 1

Fluid 2 Interface

Figure 1: Representation of the interface using vertices and edges from the triangulation.

5. Interface representation

The interface between fluids is represented by vertices and edges belonging to the trian-gulation (see fig. 1), and is moved by the fluid velocity. Additionally, a discrete Heavisidefunction is also used to represent the interface, and its gradient is computed to distribute theinterface tension force to the free-nodes where velocity isevaluated, as it was done in a workdue to Sousa et al. (Sousaet al.2004) for structured grids.

The discrete Heaviside function is defined as

Hλ,j =

1, if vertex j belongs to phaseλ1

2, if vertex j belongs to the interface of phaseλ

0, if vertex j do not belongs to phaseλ(16)

such that the interface is identified as the level-set1

2of Hλ,j. Thus, the gradient of this function

is used to distribute the interfacial force. In practice, itis not necessary to compute∇H inwhole domain, but only in the neighborhood of the interface.

Discretizingf in a variational fashion, and applying the Galerking weighting, results

MfD = ΣGhλ , (17)

whereΣ is a diagonal matrix with elements given byσκ1, . . . , σκNV , that are approxima-tions for the intensity of the capilary pressure in the velocity nodes. Additionally,hλ =[Hλ,1, . . . , Hλ,NP ]T is the discreteHeavisidefunction computed in the pressure nodes. No-tice that formally,κ is not defined away from the interface. We defineκ in the velocity nodesadjacent to the interface as the weighted average from the known values in the interface nodes.Finally, eq. (17) becomes

fD = M−1(ΣGhλ) (18)

that can replacef in eq. (14).

6. Mesh control

As the mesh is moved, elements can be distorted, resulting inbad elements to the finiteelement approximation. To avoid bad elements to appear in the mesh, a mesh control procedureis employed. Insertions and deletions of nodes are periodically made in the mesh, in order toremove elements considered bad from the triangulation.

A bad triangle can be classified in two types: a cap triangle, possesses a large circum-scribing circle radius comparing to its edges. It can be removed by flipping the largest edge,inserting a new point in the middle of the largest edge or deleting the vertex which possessesthe largest angle; a thin triangle, possesses a very small edge comparing to the other edges and

Grid spacing Ne ∆p E max|u|, |v|

h = 0.2 248 1.97847 1.088% 1.638 × 10−4

h = 0.1 946 1.99485 0.258% 2.015 × 10−5

h = 0.05 3782 1.99872 0.064% 4.535 × 10−6

h = 0.025 14972 1.99969 0.015% 4.252 × 10−6

Table 1: Comparison between pressure jumps and relative errors for several grid spacings. In this table,Ne is the number of elements,∆p is the pressure jump at the interface andE is the percentage of theapproximation error.

to the circumscribing circle radius. It can be removed from the triangulation by deleting theshortest edge or inserting a point in the largest edge.

To garantee the quality of the resulting mesh, the vertices and edges are inserted into andremoved from the mesh such that the resulting triangulationis Delaunay in the vicinity of thechanged region. Although the entire mesh is not Delaunay, this ensures that the local triangula-tion is optimal.

Additionally, the elastic velocityue computed by a Laplacian filter tends to centralize theposition of the vertices in relation to its neighbors. This movement minimizes the amount ofinsertions and deletions depending on the parametersβ1 andβ2 from eq. (2).

7. Numerical results

In this section, results for the static bubble and oscillating drop computations are presentedin order to validate the proposed method. Additionally, results for rising bubble and bubblecoalescence are presented, and they are compared with othertechniques, demonstrating theadvantages of this method.

7.1 Static bubble

A static bubble immersed into another fluid is simulated to verify the surface tension calcu-lation and measure the influence of parasitic currents in theflow. This problem was simulatedin a2× 2 domain discretized by four different unstructured meshes with constant grid spacing,h = 0.2, h = 0.1, h = 0.05 andh = 0.025. The validation is performed comparing the pressurejump at the interace with known analytical value, given by the Laplace formula (Brackbilletal. 1992) as

∆p = pb − pf = σκ =σ

R(19)

wherepb is the internal bubble pressure,pf is the external pressure andR is the radius of thebubble. Figure 2 shows the pressure profiles in a horizontal line in the middle of the domain.The profiles obtained are flat, which illustrate the precision and the sharpness of the interfacerepresentation. Table 1 shows the number of elements, pressure jump, relative error and themagnitude of the parasitic currents for the several grid spacings. Note that in this table, forh = 0.05 andh = 0.025, the magnitude of the parasitic currents have reached the numericalprecision used for these simulations.

The results are in good agreement with analytical values. From the results reported in table1, it can be concluded second order convergence for the interface tension computation.

0 0.5 1 1.5 2

0

0.5

1

1.5

2

x

p

h = 0.2

h = 0.1

h = 0.05

h = 0.025

Figure 2: Pressure profiles in the static bubble simulation: flat profiles were obtained using the proposedforce distribution.

0 0.5 1 1.5 20.4

0.405

0.41

0.415

0.42

t

Dro

pd

iam

eter

Figure 3: Oscillation of the bubble diameter, showing the decay due toviscosity. The error found infrequency of oscillation is about9%.

7.2 Oscillating drop

To verify that the transfer between the surface energy and kinetic energy is correctly ac-counted for, which is important when the flow is dominated by surface tension effects, wevalidated the method on an eliptic oscillating drop, for which an analytical solution for the os-cillation frequency exists. This problem consists of simulating an elliptic drop immersed in acontinuous lighter and less viscous phase without a gravityfield. The drop has a small initialperturbation with respect to its equilibrium circular formand, driven by the interfacial forces,it tends to oscillate. The non-dimensional parameters chosen for the simulation are diameterD = 0.4, densityρd = 1 e viscosityµd = 0.01, with the initial horizontal diameter5% greaterthan the vertical diameter. The density ratio between drop and external fluid isρd/ρf = 20 andviscosity ratioµd/µf = 10. This validation was simulated in an unitary domain discretized bya non-uniform mesh withhmin = 0.02 on the interface andhmax = 0.05 on the boundaries.

Figure 3 displays the oscillation of the bubble diameter in time. The results obtained usinga Lagrangian fashion to move the grid points showed an error about9% in the frequency of os-

µb, ρb

µf , ρf

R

4R

6R

Figure 4: Domain of simulation (left) and initial meshes used for the convergence analysis, with uni-formely distributed points, forh = 0.2, h = 0.1 andh = 0.05 respectively.

t = 0.15 t = 0.3

Method‖uh1

−uh2‖1

‖uh2−uh3

‖1n

‖uh1−uh2

‖1

‖uh2−uh3

‖1n

Lagrangian 3.83 1.74 3.98 1.88ALE 3.34 1.60 3.77 1.79Segment projection 3.78 1.92 – –Level-set 3.20 1.68 – –Front-tracking 3.48 1.80 – –

Table 2: Convergence analysis fort = 0.15 andt = 0.3, compared to the methodsSegment projection,Level-setandFront-trackingwith static meshes. The data for these methods were taken from Tornberg,2000.

cillation, which in comparable to other results in literature for equivalent resolution and domain(Esmaeeli & Tryggvason 1998).

7.3 Convergence analysis

To assess the convergence rate for a single problem, we simulated a rising bubble similar tothe one simulated by Tornberg (Tornberg 2000). The domain used for this case can be found onfigure 4. The nondimensional parameters, taken from (Tornberg 2000), areM = 0.1, Eo = 10,ρf/ρb = 100, µf/µb = 2 andR = 0.5. We consider for this analysis meshes of uniformelydistributed points, with displacements ofh = 0.2, h = 0.1 andh = 0.05. The initial meshescan be seen in figure 4.

This problem was simulated with the Lagrangian and ALE schemes, usingβ1 = 1 andβ2 = 0 for the Lagrangian, andβ1 = 1 andβ2 = 1 for the ALE scheme.

Results for the convergence analysis are reported in table 2. TheL1 norm appearing in thistable is given by

‖u‖1 =1

N

N∑

i=1

(u2i + v2

i )1

2 , (20)

whereN is the number of points used for the discretization ofu. The values obtained with theLagrangian and ALE schemes are slightly bellow, but still comparable, to the values reportedin (Tornberg 2000) fort = 0.15. We also computed the convergence rate fort = 0.3, whichunfortunatelly are not reported in the same work. All these methos are compatible with their

(a) (b) (c) (d) (e)

Figure 5: Initial mesh (a) and comparison between the meshes att = 12.5 for the parameters: (b)β2 = 0, (c) β2 = 0.1, (d) β2 = 0.5, (e)β2 = 1.0.

formal convergence rate, which is second order.

7.4 Rising bubble

In order to compare the influence of the elastic velocity on the results, a rising bubble testcase was simulated for several parameterβ2, which determines the amount of elastic velocitypresent in the mesh velocity. The nondimensional parameters, taken from (Tornberg 2000), areM = 0.1, Eo = 10, ρf/ρb = 100, µf/µb = 2 andR = 0.5, whereρb, µb are the density andviscosity for the bubble, andρf , µf are density and viscoty for the fluid. The mesh is initiallydiscretized by 1306 elements withhmin = 0.08, in a 2 × 6 domain. The parameters for thecomputation of the mesh velocity areβ1 = 1 for all cases, and each case simulated with adifferentβ2: β2 = 0, β2 = 0.1, β2 = 0.5 andβ2 = 1. Figure 5 displays the meshes obtained ineach simulation.

Figure 6 shows a comparison of the Reynolds number and mass conservation between thesimulated cases. In these figures we can observe an oscillation in the rising velocity of thebubble when increasing the parameterβ2, while the rising velocity forβ2 = 0 is smooth. Thisoscilation can be explained by the amount of mesh changes done at once in these simulations.The higher the parameterβ2, the harder the mesh is to move, and as the front is moving freelyacross the domain, the mesh has to adapt killing and creatingmany cells at once, which gener-ates the rising velocity oscillations.

On the other hand, the mass conservation reported on figure 6 clearly shows that the higherβ2, the better the mass conservation. While in a ALE fashion, higher values ofβ2 results ininsertions and deletions of many vertices at once, the number of mesh changes is far less thanin a Lagrangian fashion. In simulations withβ2 = 0, the mesh has to be adapted more often,mainly over the interface, which causes the higher loss of mass.

The time evolution of horizontal and vertical diameters canbe seen in figure 7. In thisfigure, it is clear that the loss of mass for the Lagrangian simulation is worse on the sides of thebubble, which is explained by the strong circulation imposed by the flow in this region. Becauseof that, migration of the points over the interface from top to the sides is intense, requiring largeamount of point insertion on the top and point deletion on thesides of the bubble. Since point

0 5 10 150

0.5

1

1.5

2

2.5

3R

e

t

β2 = 0β2 = 0.1β2 = 0.5β2 = 1.0

0 5 10 150.74

0.75

0.76

0.77

0.78

0.79

t

Bu

bb

lear

ea

β2 = 0β2 = 0.1β2 = 0.5β2 = 1.0

Figure 6: Bubble Reynolds number and mass conservation for the simulated cases.

0 5 10 15

1

1.05

1.1

1.15

1.2

1.25

Dx

t

β2 = 0β2 = 0.1β2 = 0.5β2 = 1.0

0 5 10 15

0.8

0.85

0.9

0.95

1

1.05

Dy

t

β2 = 0β2 = 0.1β2 = 0.5β2 = 1.0

Figure 7: Evolution in time of the horizontal (Dx) and vertical (Dy) bubble diameters.

insertion is performed on the middle of the mesh segments, this operation is mass conservative.However, deletion does not conserve mass, which causes losson the sides of the bubble.

7.5 Bubble coalescence

To show that the method can easily deal with topological changes at the interface, wesimulated the coalescence of two rising bubbles. The parameters for this problem are the sameas the previous simulation, with two bubbles of the same fluidand same sizeR = 0.5 beingreleased at a distance of0.1D. As two interfaces get closer to each other, at a distance ofone element, coalescence takes place. We simply check for the existence of elements in thesurrounding fluid that have three interface vertices, and change the material of this element ifthis will not result in a singularity. Removal of vertices that no longer belongs to the interfaceis also performed to avoid singularities. Figure 8 illustrates the coalescence of two bubbles.

This result shows good qualitative agreement with other results reported in the literature(Sousaet al.2004, Sousa & Mangiavacchi 2005).

8. Conclusions

This paper presents a method to simulate incompressible multi-fluid flows in which themesh moves in an ALE fashion. The interface between fluids wasrepresented by vertices andedges of the triangulation plus a level-set of a Heaviside function, which is also used to computethe interfacial force distribution. The discussion focuses the computation of the mesh velocity,

(a) t = 0.0 (b) t = 0.6 (c) t = 1.2 (d) t = 1.9 (e) t = 2.5 (f) t = 3.1

Figure 8: Colaescence of two bubbles of the same fluid rising in an quiescent fluid.

as well as implementations on moving finite element unstructured meshes, using remeshing pro-cedures to avoid bad elements. The conservation equations are solved by a projection methodbased on approximated blockLU decomposition of the system of equations, to decouple theacceleration and pressure.

Validations were performed for the approximation of the interfacial force, by the simulationof static bubble and oscillating drop. The static bubble simulation showed that the parasitic cur-rents are very small, as expected, and a convergence study reveals second order convergence forthe curvature calculation. The oscillating drop simulation also showed good agreement for theoscillation frequency, with error comparable to other works in literature. A convergence studyof the rising bubble problem reveals the convergence rate isslightly lower, but still comparableto other methods found in literature.

Results for the rising bubble are compared for several mesh velocities, showing the bettermass conservation properties obtained with the introduction of an elastic velocity, computed bya Laplacian filter. On the other hand, if this smoothing is toostrong, oscillations in the bubblevelocity can appear, effect that can be easily controlled bythe parameterβ2. Additionally,a bubble coalescence was performed in order to demonstrate the capabilities of the proposedmethod to deal with topologic changes at the interface.

Acknowledgements

The authors thanks brazillian agencies FAPESP (Fundacaode Amparo a Pesquisa do Es-tado de Sao Paulo, grant number 01/12623-5) and CAPES (Coordenacao de Aperfeicoamentode Pessoal de Nıvel Superior, grant number 0582/03-4) for the financial support.

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