ECE490O: Special Topics in EM-Plasma Simulations JK LEE (Spring, 2006)

1. ECE490O: Special Topics in EM-Plasma SimulationsJK LEE (Spring, 2006)

2. ECE490O: Special Topics in EM-Plasma SimulationsJK LEE (Spring, 2006) • ODE Solvers • PIC-MCC • PDE Solvers (FEM and FDM) • Linear & NL Eq. Solvers

5. Computational Eng./Sci.

7. "대학은 그릇을 키우는 곳 당장 소용되는 것 채워서야 … " 신영복 성공회대 교수가 서울대 신입생에게

8. Gonsalves’ lecture notes (Fall 2005)

9. Gonsalves’ lecture notes (Fall 2005)

10. ECE490O: ODEJK LEE (Spring, 2006)

11. Gonsalves’ lecture notes (Fall 2005)

13. Gonsalves’ lecture notes (Fall 2005)

16. Plasma Application Modeling Group POSTECH Programs of Initial Value Problem & Shooting Method for BVP ODE Sung Jin Kim and Jae Koo Lee

19. Plasma Application Modeling, POSTECH Initial Value Problems of Ordinary Differential Equations Program 9-1 Second-order Runge-kutta Method , y(0)=1, y’(0)=0 Changing 2nd order ODE to 1st order ODE, y(0)=1 (1) z(0)=0 (2)

20. Plasma Application Modeling, POSTECH Second-Order Runge-Kutta Method By 2nd order RK Method

21. Plasma Application Modeling, POSTECH Program 9-1 k1 = h*z; l1 = -h*(bm*z + km*y); k2 = h*(z + l1); l2 = -h*(bm*(z + l1) + km*(y + k1)); y = y + (k1 + k2)/2; z = z + (l1 + l2)/2; } printf( " %12.6f %12.5e %12.5e \n", time, y, z ); } exit(0); } /* CSL/c9-1.c Second Order Runge-Kutta Scheme (Solving the problem II of Example 9.6) */ #include <stdio.h> #include <stdlib.h> #include <math.h> /* time : t y,z: y,y' kount: number of steps between two lines of printing k, m, b: k, M(mass), B(damping coefficient) in Example 9.6 int main() { int kount, n, kstep=0; float bm, k1, k2, km, l1, l2; static float time, k = 100.0, m = 0.5, b = 10.0, z = 0.0; static float y = 1.0, h = 0.001; printf( "CSL/C9-1 Second Order Runge-Kutta Scheme \n" ); printf( " t y z\n" ); printf( " %12.6f %12.5e %12.5e \n", time, y, z ); km = k/m; bm = b/m; for( n = 1; n <= 20; n++ ){ for( kount = 1; kount <= 50; kount++ ){ kstep=kstep+1; time = h*kstep ; 2nd order RK result CSL/C9-1 Second Order Runge-Kutta Scheme t y z 0.000000 1.00000e+00 0.00000e+00 0.100000 5.08312e-01 -6.19085e+00 0.200000 6.67480e-02 -2.46111e+00 0.300000 -4.22529e-02 -1.40603e-01 0.400000 -2.58300e-02 2.77157e-01 0.500000 -4.55050e-03 1.29208e-01 0.600000 1.68646e-03 1.38587e-02 0.700000 1.28624e-03 -1.19758e-02 0.800000 2.83107e-04 -6.63630e-03 0.900000 -6.15151e-05 -1.01755e-03 1.000000 -6.27664e-05 4.93549e-04

22. Plasma Application Modeling, POSTECH Various Numerical Methods h=0.1 Exact Solution: h=0.01 h=0.001

23. Plasma Application Modeling Group POSTECH Error Estimation Error estimation Fourth order Runge-Kutta

24. Plasma Application Modeling, POSTECH Program 9-2 Fourth-order Runge-Kutta Scheme A first order Ordinary differential equation y(0)=1 Fourth-order RK Method

25. Plasma Application Modeling Boundary Value Problems of Ordinary Differential Equations & Scharfetter-Gummel method Sung Soo Yang and Jae Koo Lee Homepage : http://jkl.postech.ac.kr

26. * Three types of boundary conditions 1) : Dirichlet type boundary condition Plasma Application Modeling@ POSTECH 1) : Neumann type boundary condition 1) : Mixed type boundary condition Boundary value problems

27.          Plasma Application Modeling@ POSTECH Program 10-1 Solve difference equation, With the boundary conditions, x = 0 1 2 9 10 i = 0 1 2 9 10 known y(0)=1 Especially for i = 1,

28. Plasma Application Modeling@ POSTECH Program 10-1 For i = 10, Summarizing the difference equations obtained, we write Tridiagonal matrix

29. Plasma Application Modeling@ POSTECH Solution Algorithm for Tridiagonal Equations (1) R2 Based on Gauss elimination R3

30. Plasma Application Modeling@ POSTECH Solution Algorithm for Tridiagonal Equations (2)

31. Plasma Application Modeling@ POSTECH Solution Algorithm for Tridiagonal Equations (3) void trdg(float a[], float b[], float c[], float d[], int n) /* Tridiagonal solution */ { int i; float r; for ( i = 2; i <= n; i++ ) { r = a[i]/b[i - 1]; b[i] = b[i] - r*c[i - 1]; d[i] = d[i] - r*d[i - 1]; } d[n] = d[n]/b[n]; for ( i = n - 1; i >= 1; i-- ) { d[i] = (d[i] - c[i]*d[i + 1])/b[i]; } return; } Recurrently calculate the equations in increasing order of i until i=N is reached Calculate the solution for the last unknown by Calculate the following equation in decreasing order of i

32. Plasma Application Modeling@ POSTECH Program 10-1 /* CSL/c10-1.c Linear Boundary Value Problem */ #include <stdio.h> #include <stdlib.h> #include <math.h> /* a[i], b[i], c[i], d[i] : a(i), b(i), c(i), and d(i) n: number of grid points */ int main() { int i, n; float a[20], b[20], c[20], d[20], x; void trdg(float a[], float b[], float c[], float d[], int n); /* Tridiagonal solution */ printf( "\n\nCSL/C10-1 Linear Boundary Value Problem\n" ); n = 10; /* n: Number of grid points */ for( i = 1; i <= n; i++ ) { x = i; a[i] = -2; b[i] = 5; c[i] = -2; d[i] = exp( -0.2*x ); }

33. Plasma Application Modeling@ POSTECH Program 10-1 d[1] = d[1] + 2; d[n] = d[n]*0.5; b[n] = 4.5; trdg( a, b, c, d, n ); d[0] = 1; /* Setting d[0] for printing purpose */ printf( "\n Grid point no. Solution\n" ); for ( i = 0; i <= n; i++ ) { printf( " %3.1d %12.5e \n", i, d[i] ); } exit(0); } CSL/C10-1 Linear Boundary Value Problem Grid point no. Solution 0 1.00000e+00 1 8.46449e-01 2 7.06756e-01 3 5.85282e-01 4 4.82043e-01 5 3.95162e-01 6 3.21921e-01 7 2.59044e-01 8 2.02390e-01 9 1.45983e-01 10 7.99187e-02

34. a b    i-1 i i+1 Plasma Application Modeling@ POSTECH Program 10-3 An eigenvalue problem of ordinary differential equation We assume

35. ECE490O: PIC & FEMJK LEE (Spring, 2006)

37. PIC Overview • PIC Codes Overview • PIC codes simulate plasma behavior of a large number of charges particles using a few representative “super particles”. • These type of codes solve the Newton-Lorentz equation of motion to move particles in conjunction with Maxwell’s equations (or a subset). • Boundary conditions are applied to the particles and the fields to solve the set of equations. • PIC codes are quite successful in simulating kinetic and nonlinear plasma phenomenon like ECR, stochastic heating, etc.

38. Plasma Application Modeling POSTECH + + + + – – – – – – – – Capacitively Coupled Plasma – 1D PIC-MCC ~ j = 1,  , N + + Sheath + + Bulk Plasma Sheath Substrate + • MCC (Monte-Carlo Collision) Processes • - Electron-Neutral Collisions • (Ionization, Scattering, Excitation) • - Ion-Neutral Collisions • (Charge-exchange, Scattering) ~ • 1D Asymmetric Dual-Freq. Voltage-Driven System

39. PIC-MCC Flow Chart • Particles in continuum space • Fields at discrete mesh locations in space • Coupling between particles and fields I II V IV III IV Fig: Flow chart for an explicit PIC-MCC scheme

40. I. Particle Equations of Motion • Newton-Lorentz equations of motion • In finite difference form, the leapfrog method Fig: Schematic leapfrog integration

41. III. Electrostatic Field Model • Possion’s equation • Finite difference form in 1D planar geometry • Boundary condition : External circuit Fig: Schematic one-dimensional bounded plasma with external circuit

42. Plasma Application Modeling POSTECH Visible Light Sustain Electrode Sustain Electrode Bus Electrode Bus Electrode Front Glass Substrate MgO Discharge Dielectric layer Dielectric Layer Protection Layer Barrier Barrier Rib Address Electrode UV o 90 rotation Phosphor Rear Glass Substrate Phosphor(R,G,B) Address Electrode PDP Structure AC PDP Discharge in PDP

43. Plasma Application Modeling POSTECH 100Torr 200Torr 500Torr Striation Profiles in PDP – 2D PIC/MCC Anode Cathode • Pressure dependence of striations • : Number of peaks depend on the pressure and electrode size.

44. Plasma Application Modeling, POSTECH XOOPIC and MAGIC Codes for Electromagnetic Field S.J. Kim and J.K. Lee Contents • Overview of XOOPIC code • Overview of MAGIC code • Klystron simulation using XOOPIC code

45. Plasma Application Modeling, POSTECH Simulation Domain of Klystron RF output port RF input port 9.55 cm 10.05 cm 13.07 cm E-beam 7.569 cm 6.66 cm Cylindrical Axis 37.2 cm • Simulation condition: • Beam emitter: I= 12 kA, ud =2.48e8 m/s • Input port : Rin=2300 , R=20 , f=7.69 GHz • Output port : R=47.124 

46. Plasma Application Modeling, POSTECH Example of Klystron Simulation Phase space Density uz Kinetic energy

47. Plasma Application Modeling, POSTECH Simulation Results at 0.5 ns

48. Plasma Application Modeling, POSTECH Simulation Results at 10 ns

49. Plasma Application Modeling, POSTECH Simulation Results at 20 ns

50. Plasma Application Modeling, POSTECH KE as a Function of Beam Current