livrable-maillage-nsmb - ovetset grid theory

8/12/2019 Livrable-Maillage-nsmb - Ovetset grid theory

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3rd step : Calculation of the interpolations parameters. In this step, the

donor cells and the associated weights that compose the interpolation areevaluated for each valid overlapped cell.

4th step : Calculation of the interpolation values.

grid 3 (flat)

grid 1 (airfoil)

grid 2 (flap)

a) b)

A B C

WNS

n

WHC

nWNS

n

WHC

n

interpolation

grid 2(flap)

grid 1(airfoil)

d)

D

grid 2

(flap)

grid 1(airfoil)

overlapped cellsc)

A B C D E

e)

g(a

Figure 1: Main steps of the chimera method.

In the present work, a three-dimensional structured multi-block chimeramethod is implemented in order to simulate the free falling sphere falling ina tube (Fig? 2). A basic chimera method is used and the main works focus onthe increasing of the speed process with a study on each step of chimera algo-rithm. This work allowed to find out a new criteria for a quickly and efficientlyhole cutting process. The developped chimera method is fully automatic with

few, if not none, of user input and pre-processing.

Y

cylindri

spherical grid

XY-plane

YZ-plane

Figure 2: Representation of the cylindrical grid that discretizes the pipe spaceand the spherical grid representing the sphere.

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The paper is organized as follows: the section 2, the details of each step of

the implemented chimera method are discussed. In this section is described thestudy of the bucket method for detection of overlapped cells, the definition of anew criteria for hole cutting algorithm and the justification of the interpolationscheme. In Section 3 we report the results of test cases to validate the accuracyand the efficiency. We find here the results of the flow around a sphere, thenthe flow around the sphere close to a plane wall and, to finish, the results of freefalling sphere in a tube.

2 The flow solver

We implemented the chimera method in the flow solver called Navier-StokesMulti-Block (NSMB) [15, 16]. The NSMB code solves the compressible Navier-

Stokes equations using a finite volume formulation on Multi-Block structuredgrids and parallelized using the Message Passing Interface.

Among the many discretization schemes present in NSMB, in this study weuse a central scheme with 4th order artificial dissipation [17] for spatial discreti-sation and an implicit dual time stepping solved by Lower-Upper SymmetricGauss-Seidel method (LS-SGS) for time integration. The artificial compress-ibility method [18] is employed to fulfil the incompressible flow.

The governing equations are the preconditioned unsteady Navier-Stokes equa-tion written in conservative form with dual-time stepping. Using the Cartesiancoordinates (x,y,z), these equations can be expressed in a conservative form asfollows :

P1

(W) + I t

(W) + x

(f fv) + y

(g gv) + z

(h hv) =0 (1)

The state vector Wand the the inviscid fluxes f, g andh are given by :

W = (p, u, v, w)T

f=

u, u2 +p, uv, uwT

g =

v,vu,v2 +p, vwT

h=

w,wu,wv,w2 +pT

where is the density, u v and w are the Cartesian components of velocity, pis the pressure, dt is the physical time step and d the pseudo-time step. Theviscous fluxes fv, gv and hv are given by :

fv = (0, xx, xy , xz)T

gv = (0, yx, yy , yz )T

hv = (0, zx, zy , zz )T

with the shear stress tensor given by

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xx = 23

2 ux

vy

wz

xy =yx =

vx

+ uy

yy =2

3

u

x+ 2

v

y

w

z

xz =zx =

w

x +

u

z

zz =2

3

u

x

v

y+ 2

w

z

yz =zy =

v

z+

w

y

where is the dynamic viscosity (Stokes hypothesis).The modified unit matrix I and the pre-conditioning matrix P for the pseudo-

time equation system are given by

I=

0 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

and P =

2 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

The coefficientcan be seen as a relaxation parameter for the pseudo-time

solution and is defined as

2 =max( 2minU 2

, CU

2)

with min = 5 and C = 0.05.

3 The chimera method

Details and improvements to meet our constraints (free fall body in a confinedspace) are described following the same steps of the description (detection ofoverlapped cells, determination of the type of overlapped cells, ...)

3.1 Detection of overlapped cells

The search of overlapped cells is based on a test of inclusion on the coordinates.In order to accelerate the search, the mapping of coordinates in a virtual uniformCartesian grid is performed. This method is well known as thebucket method[19, 20, 21]. An inverse mapping algorithm creates an index array that linkvirtual grid to the real coordinates. The search begins on the virtual grid andit continues on the physical associated grid points. The creation of link array istime consuming but the gain in the search time is significant (at least a factor20).

This algorithm is well known but the number of the virtual cells ( Nv) is aparameter not clearly defined. A simple test is performed in order to determi-nate the optimum value ofNv in function of the number of real coordinates N.We consider two uniform three-dimensional Cartesian grids (Nx =Ny =Nz =N = 128) overlapping entire the cube 1x1x1 domain. Identical results wereobtained on a coarser and a finer grid. The cpu-time is determined for range1 Nv 256. In addition to the compromise between cpu-time decrease forvirtual grid creation and cpu-time increase for searching the overset, the sizerequired by the link array must be taken into account. The results, represented

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on figure 3, show that the optimum value for the virtual cell number Nv is the

half of the number of physical coordinates N.The same test is performed for agrid with tangent hyperbolic distribution and, in this case, the optimum valuecorrespond to N.

N

cpu-time

0 50 100 150 200 250

100

101

102

10

10

10cpu-time of creation of link arraycpu-time for searchsum of cpu-time (creation+search)array size of virtual grid link

v

64=N/2 128=N

Figure 3: Representation of the cpu-time used for the indexation of real coor-dinates in virtual grid, cpu-time for searching overset and the size of the indexbetween both grids versus the number of virtual box Nv in one direction.

Another issue concerns the points that are inside a solid body. In this case,the cells are not detected as overlapped cells. To remove these cells, two meth-ods are implemented. For simple cases (cylinder, sphere) we use an analytical

function and for general cases. We compute the dot product of the vector fromthe nearest wall cell center to cell center and the vector of the associated wallnormal vector. If the dot product is negative, the cell is in the solid region,otherwise the cell is out of the solid region. This second method is 30 timesmore expensive than the first one, for the configuration of a sphere in a tube,with 6 millions of cells and 65 000 wall cells.

3.2 Determination of the type of overlapped cells

The second step of the method is to define the type of the overlapped cells(calculated, interpolated, hole). In this step, among several overlapped cell,the best candidate to predict the flow state (calculated cell) and to give theflow solution to the other cells (interpolated cells) is chosen. The criterionthat justifies the choice is based on the best resolution of the physical problem.We distinguish four techniques based on the criteria associated to the followquantities : user-defined grid ranking, user-defined cell quality, cell size, cellsize in the wall-normal direction.

For simple overset, the definition on an overlapped hierarchy of each blockis enough for the cell type selection. All the cells of the highest ranking ofthe grid are defined as calculated cells. This solution is simple and fast butnot suitable for complex overlapped layers where the definition of chimera gridranking by the user is then needed. Based on the fact that the cell size isa criterion indicating the best discretization, Siikonen et al. [21] or Liao et al.[22] justify this choice by assigning thecalculated cellstype to the smallest ones.

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This criterion is locally adapted, automatic and requires no user input. A third

criterion is added to the previous one and based on a user-defined cell quality.This way the user can force a specific kind of cell (calculated or interpolated) byassociating to this cell a high quality value. This technique can force cell type byprotecting or immunizing cells but the user input is binding and does not followthe grid movement. The cell size in the wall-normal direction was introducedby Landmann [14]. The selection use the intersection segment between normalnearest wall and cell boundary. This technique is fully automatic and it is basedon an important physical quality : the boundary layer. This test requires tosearch for the nearest wall and to calculate the intersection for each cell. Inorder to find a lighter, accurate and automatic criterion, we developed a similarcriterion based on the distance of a cell to the nearest local wall. The nearestlocal wall is the wall in the same block where the cell is defined (figure 4). Withthis criterion, the overlapped cell with the smallest local near wall distance

is calculated and the others are interpolated. This approach guarantees theresolution of Navier-Stokes equations in the nearest wall region and consequentlyeach boundary layer is accurately computed. Moreover, this technique is basedon the local wall distance and this quantity does not change with moving grid(an update is needed for deformed meshes) whereas the criterion based on thewall normal intersection need an update after each grid movement.

Mesh 1 Mesh 2

C1

Global nearest wall distanceLocal nearest wall distance

Figure 4: Definition of local nearest wall distance and global nearest wall dis-tance for the cell C1 of the mesh 1 in the configuration of two overlapped polarmeshes.

In the case of the sphere in a circular tube, the criterion based on the cell sizecomplicates the generation of the mesh. The same problem may be encounteredin the configuration of a sphere near a plane wall. The cell aspect ratio ofthe near wall mesh is not the same as that of the spherical wall mesh and thecell volume is very difficult to control. In our case of a sphere in a tube with acylindrical grid overlapped by a spherical grid, the criterion based on the volumedoes not give a proper solution. The new criterion based on the local near walldistance give the best answer. The figure 5 represents the visualization of thecalculated cells in the 3 planes crossing the sphere center for the both cases.

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Figure 5: Visualization of the calculated cells for the chimera cell selectionbefore cell selection (1st row), using cell volume criterion (2nd row) and usingthe nearest local wall distance criterion (3rd row) for the cross section of thetube (1st line), the streamwise plane in the symmetric plane (2nd line) andstreamwise plane perpendicular to the symmetric plane (3rd line)

3.3 Buffer and hole cells

The next step is the addition of a buffer layer. The buffer cells are cell betweeninterpolated and calculated cells. Their role is to move away the interpolationcommunications and thereby to avoid implicit interpolation. An implicit inter-polation is an interpolation that uses an interpolated cell as a donor cell. Themore the interpolation needs a large number of donor cells, the more the bufferlayer needs to be wide. The importance of wide overset is admitted and de-scribed in several papers [23, 24]. The set-up of the buffer layer is implementedwith an user-input. The user give a fixed number of layer and the algorithmchanges the type of the cell from interpolated to calculated over this layer.

The last type of possible cells is the hole cell. This kind of cell is a non-

necessary cell and it concerns interpolated cells. This cell does not step in anydiscretization schemes. In order to save computation time, the interpolationprocess is avoided for this cell. After the buffer layers, a user-input imposes anumber of interpolated cells in the hole region. These new cells are consideredas ghost-cells. The minimum number of ghost-cells is two because the numericalschemes used in NSMB require two neighbor cells (we have used fourth ordercentral scheme with fourth order artificial dissipation).

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3.4 The interpolation

The interpolation is a key element of the chimera method. It allows the pas-sage of flow information between overlapped grids and it creates the link. Theinfluence of interpolation has already been investigated in several studies (Delfs[25], Sherer & Scott [26] and Sengupta et al. [27]). The simplest treatment ofan interface consists in the interpolation of all state variables [1, 28, 6, 29, 30].However, this approach does not guarantee the global mass conservation as ex-plained Wang et al. [31]. Some studies found a solution to ensure the globalconservation for compressible flows [32] or for incompressible flows (Wright andShyy [33] or Tang et al. [34]) with a correction of interpolated values. The ac-curacy of interpolation is important and different schemes of interpolation areused : interpolation based on Lagrange polynomials [35, 36, 24, 27, 26], trilin-ear interpolation [37, 34, 38, 39], quadratic interpolation [40] or tetravolumic

interpolation [41, 21].The interpolation scheme depends on the physical problem and the oversetwidth. A highly-accurate interpolation needs many donor cells and it requiresa wide overset. In our case of the sphere in a circular pipe, the gap between thesphere wall and tube wall does not provide a wide overset and a large stencil ofdonor cells. For this reason, we chose some interpolation that use a low numberof donor cells. The choice focus itself on the trilinear, tetravolumic and weightedinverse distance interpolations.

The inverse distance weighted interpolation is based on the distance betweenthe points. The implementation is simple and its advantage is the flexibility ofthe stencil of points. The formula used is given by the following relation :

fM =Ni=1 fidiN

i=11

di

(2)

where N is the number of the stencil points, fi the value at the point i, isthe weighting exponent (in our case equal to 2) and di is the distance from thestencil pointi to the interpolation point M.

The second interpolation is the tetravolumic interpolation based on the tetra-hedral volumes composed between the four nearest cells. The interpolationweights are given by the opposite tetrahedral volume :

fM =

i=A,B,C

Wifi

with WA =VBC M

VABC , WB =

VACMVABC andW

C=VABMVABC

with W the weight of the interpolation and VACM the volume of the triangle(ABC). In three dimensions, the triangle volume is replaced by the tetrahedralvolume.

The third interpolation is the trilinear one. We use a trilinear interpolationbased on a linear interpolation on each direction. In two-dimensional exampleof the figure ??, the first linear interpolation yields the values fQ and fP thena second linear interpolation provides fM. In three-dimensions, three steps oflinear interpolation are required.

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3.5 Grid-to-grid interpolation test case

The first results concern the interpolation of data generated by analytic func-tions from one grid onto another. These tests allow us to isolate the interpo-lation properties from other aspects of the solver. The analysis validates theinterpolation and quantifies the accuracy of each interpolation scheme.

The grids are composed of two uniform square Cartesian grids, the back-ground grid and the foreground one which is rotated by 45(Fig.6 (a)). Ananalytical function is superimposed on the background grid and then interpo-lated onto the foreground one. The difference between the interpolated valuesand the analytical solution on the foreground mesh gives the interpolation erroron each cell. Five meshes with different refinement are considered. The spacediscretization (X) is characterized by the number of points (Nx) and rangesfromNx = 34 for the coarse grid to Nx = 514 for the fine grid.

Three two-dimensional analytic functions are considered :

F1(x, y) = cos

x

3

sin

y

3

(3)

F2(x, y) = 52x exp

x2 + y2

2

(4)

F3(x, y) = 28x

x2 + 9(y2 + 9)cos

20x

z+ 3

cos

x

3

sin

z

3

(5)

TheL2-norm error is calculated using the following equation :

E2=

|E|2 dxdy1/2

(6)

with the error E= Fanalytic Finterpolated.The figure 6 represents the L2-norm error for the function F3 versus the

spatial discretization step Xin logarithmic scale. The behavior of all functionsis similar and the order-of-accuracy of each interpolation is calculated. For thetetravolumic interpolation the orders are 2.00, 2.04 and 2.04 for the functionF1, F2 and F3, for the inverse distance weighted 1.33, 1.25 and 1.31 and fortrilinear interpolation 2.03, 2.06 and 2.06. The order-of-accuracy of the trilinearinterpolation (equal to 2.04) is similar to that of the tetravolumic interpolation(equal to 2.00). However, the value ofL2-norm error for trilinear interpolationis smaller than that of the tetravolumic interpolation. The inverse distanceweighted interpolation is less accurate with an order equal to 1.31 at the best.

The advantage of the inverse distance weighted interpolation is the simpleand flexible algorithm but the accuracy is poor. The accuracy of the tetravolu-mic interpolation is better but the stencil of points is more rigid. The trilinearinterpolation is a good compromise with a good accuracy and a flexible stencilof points with the possibility of extrapolation but its implementation is heavy.The trilinear interpolation is used for the following simulations.

One test in this way has been performed. The test is the same kind of grid-to-grid interpolation test presented in the paper, section 4.1. The interpolated gridis a cartesian grid with a fixed distribution, 128x128 cells. The cell size of thedonor cells varies. The L2-norm error of the functionF3(5) is represented versus

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x

y

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Block 1

Block 2

(a)

x

L2-norm

error

0 .0 1 0 .0 2 0 .0 3 0 .0 40 .0 50.060.07

10-5

10-4

10-3

10-2

tetravolumic

inverse distance weightedtrilinear

(b)

Figure 6: Coarse meshes used for the grid-to-grid interpolation test case (a) andL2-norm error versus the x for the function F3 (Eq. 5) (b).

the ratio aspect of the interpolated cell over donor cell (figure 7). The resultsshow that the cell aspect ration have an small influence on interpolated values.The main influence of the big aspect ratio is on the diffusion in the Navier-Stokes equations. When the flow crosses the overset border with a dramaticchangement of cell size, the flow prediction is affected. We didnt had this resultto the paper. To complete our answer, we can had that the chimera connectioncan be compared to a block-block connection (except for the interpolation) :

if the cell aspect ration is too big this will induce numercial dissipation andwe have verified this with turbulent chimera simulations. So one of the a jorrule when building chimera grids is that at the interface we have to respect anequivalent aspect ratio.

/

L2-norm

error

1 2 30

0.0002

0.0004

0.0006

0.0008

0.001

trilinearinverse distance weightedtetravolumic

x x1 2 x / x

||E|

|2,

/

||E|

|2,=1

1 2 3

1.00

1.05

1.10

1.15

1.20

ttravolumique

inverse la distantrilinaire

1 2=

Figure 7: L2-norm error of the function F3 (5) versus the ratio aspect of theinterpolated cell over donor cell.

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4 Validation of the method

The method described in the previous paragraph will now be validated. Threetest cases are : the simulation of flow around a sphere in an infinty domain, aflow around a sphere translating uniformly along a flat wall, and finally a spherefalling freely in a tube filled with a Newtonian fluid at rest. Each test casecomplexity rises, so one part of a single flow, then an interaction between twowalls defined by two independent chimera meshes, then the case of interactionfluid / solid of a free fall body represented by on overset grid. These test casesare also selected for the presence of numerical or/and experimental results thatmaking the validation deeper.

4.1 Flow past a sphere

The flow around a sphere is the topic of large experimental studies or numericalstudies [42, 43, 44, 45, 46, 47, 48]. The flow state is well known and clearlydefined by a single parameter, the Reynolds number (Re = Ud/, with U thefreestream velocity, d the diameter of the sphere and the kinetic viscosity).For low Reynolds number (Re 212 the axisymmetry is broken into a steady non-axisymmetric flowand the Hopf bifurcation appears atRe = 273 where the flow beacom unsteadywith horse-shoe vortex shedding.

This case test is the first step of validation and it can demonstrate theability and the accuracy of the chimera method in a simple three dimensionalcomputation hugly referenced. The simulations are performed by two set ofgrids : an chiemra grid composed by a cartesian background grid overlapped bya spherical grid and a classical body-fitted mesh.

The representation of the isolines of streamwise velocity in the streamwiseplane (figure 8) shows the wake behind the sphere with a recirculation. Theshape of the vortex is a torus like described in the literature (Johnson et al. [46]for example). The dashed line on the figure 8 represents the external border ofthe spherical grid. The flow information crosses correctly this overset border.

0.20.9

0

0.1

0.2

0.3

0.9

00.2

0.7

0.7

0.8

0.4 0.5

0.6 0.7

0.9

0.9

0

0.1

0.2

0.4

0.60.7

0.80.9

(a)

(b)

Figure 8: Representation, on the streamwise plane, the isovalues of the stream-wise velocity (a) and the streamlines (b)

The comparison of the results is based on three physical quantities : thedrag coefficient (CD), the recirculation length (L) and the separation angle ().For this range of Reynolds number, the flow is axisymmetric and the lift forceis zero. The comparison of the results on the drag coefficient of Bouchet et al.[45] and the present study do not exceed 0.52%. The very closer results show

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the good resolution of overlapped grids. The shape descriptions (L and ) are

in good agreement with the previous works reported on figure 9. The results arecompared with datas obtained by experimental studies (Taneda [42], Nakamura[49] and Roos & Willmarth [50]) and by numerical studies (Tomboulides et al.[48], Mittal [51], Magnaudet et al. [52], Bagchi & Balachandar [53] and Bouchetet al. [45]).

Reynolds number

D

ragcoefficient

50 100 150 200

0.8

1

1.2

1.4

1.6 Bouchetet al.Magnaudetet al.Roos & WilmarthPresent studywith chimera meshPresent studywith no-chimera mesh

Reynolds number

Recirc

ulationlength(L/d)

50 100 150 20

0.4

0.6

0.8

1

1.2

1.4

1.6 Bouchetet al.

MittalTanedaTomboulides et al.Present studywith no-chimera meshPresent studywith chimera mesh

Reynolds number

Separation

angle

(degree)

50 100 150 200

115

120

125

130

135

140

Bouchetet al.Magnaudetet al.MittalTanedaPresent studywith no-chimera meshPresent studywith chimera mesh

Figure 9: Drag coefficientCD, recirculation length (L/d) and separated angleversus Reynolds number for the flow past a sphere and 50 Re 200.

4.2 Flow around a sphere translating along a plane wallTo complete the study of the three-dimensional chimera method, we consider aconfiguration with an interaction between two walls yielding a border of over-lapping (Fig. 10). We simulate the flow past a sphere in a uniform translationparallel to a plane wall which has been previously reported by Zeng et al. [54]and Takemura & Magnaudet [55]. The parameters for this configuration arethe distance between the sphere center and the wall (L/d) and the Reynoldsnumber Re = U.d/, with U the velocity of the sphere translation and thekinematic viscosity of the fluid. The geometry is meshed with a first Cartesiangrid, which is refined along the plane wall, and a second spherical one, whichis refined near the sphere wall. The criterion for chimera cell selection is based

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on the definition of a boundary layer for the background grid where the cells

are always calculated. Elsewhere the cells of spherical mesh are calculated. Themotion of the sphere is implemented with the wall condition of the plate andwith an inlet boundary condition. The mesh for parallel simulations is decom-posed into 32 sub-blocks. The range of the Reynolds number is 10 Re 250and the studied distance is L/d = 1 and L/d = 0.75.

The presence of the plane wall breaks the axisymmetric geometry, and atall Reynolds numbers a lift force exists. Two mechanisms are the source of thisforce (Takemura & Magnaudet [55]). The first is linked to the strong interactionbetween the wall and the wake of the sphere. The figure 11 shows the isovaluesof the streamwise velocity in the wall normal plane for the case Re= 200 andL/d= 1.00. The distribution of the flow looses the up/top asymmetric and itresults into a lift force directed away from the wall. The second mechanism islinked to the high velocity in the gap associated to a low pressure. This behavior

gives a force with opposite direction (attractive force) to the first mechanism.The resultant force tends to push the sphere away from the wall. The streamlinesplotted along the streamwise wall normal plane (figure 12) represents the vortexstructure forRe = 200 andL/d= 0.75. The same case performed by Zeng et al.[54] is plotted sideline our results for comparison (figure 12). The overset borderis not visible and proves one more time that the chimera communication issufficient. Zeng et al. [54] found the same representation of the flow reproducedin figure 12.

Figure 10: The chimera mesh in the plane perpendicular of the plane wall acrossthe sphere center for L/D= 1.

Figure 11: Isovalues of the streamwise velocity in the plane normal to the wall(-0.3 to 1.1 with a increment of 0.2)

The results on the aerodynamic coefficients are in agreement with those ofZeng et al. [54] both for the drag and lift coefficients (figure 12). The differences

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between the results of Zeng et al. [54] and ours remain below 1%.

Reynolds number

Drag

coefficient(C

D)

50 100 150 200 2500

1

2

3

4

5

Present studyZeng et al.

CD

CL

a)

Figure 12: The drag and lift coefficient for L/d = 1.00 and versus Reynoldsnumbers and b,c) streamlines in the plane normal to the wall for the case Re =200, L/d = 0.75 for the present study (left) and extract to Zeng et al. [54](right)

5 Conclusion

We have developed an overlapped grid methodology in NSMB allowing the sim-ulation of steady/unsteady, three-dimensional, incompressible flows in complexand confined geometries. Arbitrary overlapped sub-domain meshes can be re-composed automatically.

We carried out a series of numerical test cases to validate the algorithm andto assess its feasibility in complex geometries. The computed results demon-strated that our method gives good results both for simple two-dimensionalconfiguration and for complex confined three-dimensional ones. For the three-dimensional test cases of the unconfined and confined sphere, the differenceswith the literature are less than 1%. This proves that the communications inoverlapping interfaces do not induce any distortion.

The automatic detection of overlapped cell allows the simulation of moving

bodies and the next step is to simulate particles transported in confined flows.

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livrable-maillage-nsmb - ovetset grid theory

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