GHOST-CELL METHODNFRI16/07/2014

Ghost-Cell methodIf the shape of a boundary (Plasma/Wall interface) is com-plex => use of FDM can be delicate• Use of Ghost-Cells (GC): points in the solid • Value at GC extrapolated from neighbor points (in the

Plasma)• GC method adapted from Tseng et al. [1]

BibliographyPaper Dim Reynolds spatial Time Extrapolation Boundary condition otherTseng[1] 2/3 Low/high FVM   Linear/quadratic Piecewise linear

segmentNeuman/Dirichlet Mirror

Mittal ARFM [2]

2/3 Low/high FDM   Linear/quadratic Rigid/moving   Ortho. Projection or middle segment

Majunmbar [3] 2 ~100 - ~1000 FDM implicit Linear/quadratic rigid   Mirror

Gibou [4] 3   FDM explicit Linear moving D  

Mark [5] 3 ~1 - ~1000 FVM   trilinear rigid D/N Mirror

Mittal JCP [6] 3 ~10 - ~1000 FDM Van-Kan linear moving D/N  

Berthelsen [7] 2 ~10 - ~100 FDMWENO

RK Cubic / 1D in the same direction

Rigidhighly irregular

D/N  

Pan [8] 2/3 ~100 FVM backward Linear rigid    

Gao [9] 2 ~100 FDM Adams–Bashforth

Linear or Taylor series

moving D  

kim_choi [10] 3 ~100 FDM RK+CN linear moving D/N  

Shinn [11] 2/3 ~100 - ~1000 FVM time-marchingfrac-tional-step

linear Rigid/moving D/N mirror

Fluid-Surface Interaction• Solid = inside, fluid = outside

Solid

Fluid

GC

From [1]

• Solid = outside, fluid = inside

(Simple plasma shape assumed)

Tokamak

Plasma(Fluid)

TokamakWall

(Solid)

GC

Tokamak• Boundary = segments• Rigid boundary• 2D (does not depend on Z)• Turbulence: Reynolds >> 1

• Boundary conditions:• ρ =0• ρ.v=0 ?• Bn=0 =>

• Bx = -B// sin α• By = B// cos α

• E = 0.5 * ρ.v2 + P + B// 2/mu0

AlgorithmInitialization step:1- Identify all the GC in the grid2- Compute the interpolation coefficient for each GC

Every time step:3- Use coefficients from 2- to update the GC values

3D = same as 2D (plasma shape does not depend on z)

Algo: Identify GC• Use a mask F(i,j): = 0 for plasma = 1 for GC = 100 for others (no computation)Algo:For all points of the grid (A) cross product with all segments (BC) of the plasma boundary- All z(BAxBC) < 0 => inside plasma => F(A)=0- At least one z(BAxBC) > 0 => outside plasma => F(A)=100For all solid point, test if neighbor = plasma => F(A)=1GC definition depends on WENO method: (md parameter in the code)

Identify Ghost Node - Result• Simple case, with md=4

Algo: Interpolation coefficientGC (G) = extrapolation with 1 projected point (O) and - 2 neighbors (linear) - or 5 neighbors (quadratic)Dirichlet or Neumann=> Need projection of G on boundary and its value(which method? Dirichlet, neumann, linear, quadratic)Ex. Linear dirichlet reconstruction

=>

From [1]

Interpolation coefficient - Result

MirrorProblem if the interpolated point too close of one of the neighbors=> high negative weight => can lead to numerical instability

Use image of ghost nodeInterpolate IThen get G:

From [1]

Algo: update GC values • Use this formulation:

For each GC, at the initial step, need to save: w1, x1,y1,w2, x2,y2, w0, + …,

WENO method• The problem will be solved for each grid point where F(i,j)=0• Near the boundary, the WENO method will use the data of the GC

previously updated• No need to solve the problem on the whole grid

do i=1,nx do i=1,nx do j=1,ny do j=ystart(i),yend(i)

do j=1,ny do j=1,ny do i=1,nx do i=xstart(j),xend(j)

ystart(3)=5 yend(3)=6 xstart(4)=4 xend(4)=5ystart(4)=4 yend(4)=6 xstart(5)=3 xend(5)=6ystart(5)=4 yend(5)=6 xstart(6)=3 xend(6)=6ystart(6)=5 yend(6)=6

No periodicity in X & Y direction (mesh big enough to contain plasma+GC)

Start/end - Result

Bibliography[1]Tseng et al.: A ghost-cell immersed boundary method for flow in complex geometry, Journal of Compu-tational Physics 192 (2003) 593–623[2] R. Mittal and G. Iaccarino. Immersed boundary methods. Annu. Rev. Fluid Mech., 37:239-261, 2005. doi:10.1146 [3] S. Majumdar, G. Iaccarino, and P. Durbin. Rans solvers with adaptive structured boundary noncon-forming grids. Center for Turbulence Research Annual Research Briefs, pages 353-366, 2001.[4] F. Gibou, R.P. Fedkiw, L.T. Cheng, and M. Kang. A Second-Order-Accurate Symmetric Discretization of the Poisson Equation on Irregular Domains. Journal of Computational Physics, 176(1):205-227, 2002. doi:10.1006[5] A. Mark and B.G.M van Wachem. Derivation and validation of a novel implicit second-order accurate immersed boundary method. Journal of Computational Physics, 227(13):6660-6680, 2008. doi:10.1016[6] R. Mittal, H. Dong, M. Bozkurttas, F.M. Najjar, A. Vargas, and A. von Loebbecke. A versatile sharp inter-face immersed boundary method for incompressible flows with complex boundaries. Journal of Compu-tational Physics, 227(10):4825-4852, 2008. doi:10.1016[7] P.A. Berthelsen and O.M. Faltinsen. A local directional ghost cell approach for incompressible viscous flow problems with irregular boundaries. Journal of Computational Physics, 227(9):4354{4397, 2008. doi:10.1016[8] D. Pan and T.T. Shen. Computation of incompressible ows with immersed bodies by a simple ghost cell method. Int. J. Numer. Meth. Fluids, 60(12):1378-1401, 2008. doi:10.1002[9] T. Gao, Y.H. Tseng, and X.Y Lu. An improved hybrid Cartesian/immersed boundary method forfluid-solid flows. Int. J. Numer. Meth. Fluids, 55(12):1189-1211, 2007. doi:10.1002[10] D. Kim and H. Choi. Immersed boundary method for flow around an arbitrarily moving body. Journal of Computational Physics, 212(2):662-680, 2006. doi:10.1016[11] A.F. Shinn, M.A. Goodwin, and S.P. Vanka. Immersed boundary computations of shear- and buoyan-cydriven flows in complex enclosures. International Journal of Heat and Mass Transfer, 52(17-18):4082-4089, 2009. doi:10.1016