1. Ghost-Cell method
NFRI
16/07/2014

2. Ghost-Cell method
If the shape of a boundary (Plasma/Wall interface) is complex => 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]

3. Bibliography Paper Dim Reynolds spatial Time Extrapolation Boundary
condition
other
Tseng[1]
2/3
Low/high
FVM
Linear/quadratic
Piecewise linear segment
Neuman/Dirichlet
Mirror
Mittal ARFM
[2]
FDM
Rigid/moving
Ortho. Projection or middle segment
Majunmbar [3] 2
~100 - ~1000
implicit
rigid
Gibou [4] 3
explicit
Linear
moving D
Mark [5]
~1 - ~1000
trilinear
D/N
Mittal JCP [6]
~10 - ~1000
Van-Kan
linear
Berthelsen [7]
~10 - ~100
WENO RK
Cubic / 1D in the same direction
Rigid
highly irregular
Pan [8]
~100
backward
Gao [9]
Adams–
Bashforth
Linear or Taylor series
kim_choi [10]
RK+CN
Shinn [11]
time-marching
fractional-step
mirror

4. Fluid-Surface Interaction
Solid = inside, fluid = outside
Solid
Fluid GC
From [1]

5. (Simple plasma shape assumed)
Tokamak
Solid = outside, fluid = inside
(Simple plasma shape assumed) GC
Plasma
(Fluid)
Tokamak
Wall
(Solid)

6. 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

7. Algorithm
Initialization step: 1- Identify all the GC in the grid 2- 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)

8. 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)=100
For all solid point, test if neighbor = plasma => F(A)=1
GC definition depends on WENO method: (md parameter in the code)

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

10. Algo: Interpolation coefficient
From [1]
GC (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 =>

11. Interpolation coefficient - Result

12. Mirror
Problem if the interpolated point too close of one of the neighbors => high negative weight => can lead to numerical instability Use image of ghost node Interpolate I Then get G:
From [1]

13. Algo: update GC values Use this formulation:
For each GC, at the initial step, need to save: w1, x1,y1,w2, x2,y2, w0, + …,

14. 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)=5
ystart(4)=4 yend(4)=6 xstart(5)=3 xend(5)=6
ystart(5)=4 yend(5)=6 xstart(6)=3 xend(6)=6
ystart(6)=5 yend(6)=6
No periodicity in X & Y direction (mesh big enough to contain plasma+GC)

15. Start/end - Result

16. Bibliography
[1]Tseng et al.: A ghost-cell immersed boundary method for flow in complex geometry, Journal of Computational Physics 192 (2003) 593–623 [2] R. Mittal and G. Iaccarino. Immersed boundary methods. Annu. Rev. Fluid Mech., 37: , doi: [3] S. Majumdar, G. Iaccarino, and P. Durbin. Rans solvers with adaptive structured boundary nonconforming grids. Center for Turbulence Research Annual Research Briefs, pages , [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): , doi: [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): , doi: [6] R. Mittal, H. Dong, M. Bozkurttas, F.M. Najjar, A. Vargas, and A. von Loebbecke. A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. Journal of Computational Physics, 227(10): , doi: [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, doi: [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): , doi: [9] T. Gao, Y.H. Tseng, and X.Y Lu. An improved hybrid Cartesian/immersed boundary method for fluid-solid flows. Int. J. Numer. Meth. Fluids, 55(12): , doi: [10] D. Kim and H. Choi. Immersed boundary method for flow around an arbitrarily moving body. Journal of Computational Physics, 212(2): , doi: [11] A.F. Shinn, M.A. Goodwin, and S.P. Vanka. Immersed boundary computations of shear- and buoyancydriven flows in complex enclosures. International Journal of Heat and Mass Transfer, 52(17-18): , doi: