RAPTOR tutorial - Variable timegrid
It is possible to use a time-varying grid, typically to have shorter time steps in the early (transient) part of the shot and fewer time points later.
This is illustrated with a simple ITER-like simulation below
Contents
Set up the simulation
close all hidden; clear; run('../RAPTOR_path.m'); addpath(fullfile(pwd,'..','code'),'-end'); % add RAPTOR path [config] = RAPTOR_config('ITER'); % load default ITER params % set NBI parameters config.nbhcd.params.active = true; config.nbhcd.params.rdep = [0.25]; % config.nbhcd.params.wdep = [0.25]; % broad heating config.nbhcd.params.wdep_out = [0.75]; % broad heating config.nbhcd.params.cd_eff = [0]; % current drive config.nbhcd.params.uindices = [2]; % index in input vector config.echcd = echcd('none'); % switch off alpha power and losses config.palpha.active = false; config.pbrem.active = false; config.prad.active = false; config.pei.active = false; % create model, params, init, geometry g, kinetic profiles v. [model,params,init,g,v,~] = build_RAPTOR_model(config);
Simulation with variable time step
define a monotonic, but not evenly spaced time grid:
tgrid0 = [0:0.1:2.9,3:1:80]; params.tgrid = tgrid0; % inputs rampfun = @(t,tmin,ymin,tmax,ymax) ... max(ymin,min((ymax-ymin)/(tmax-tmin)*(t-tmin),ymax-ymin)+ymin); % anonymous function for ramps % Construct actuator inputs on the time grid tgrid0. U0(1,:) = rampfun(tgrid0,0,4e6,80,12e6); % Ip U0(2,:) = rampfun(tgrid0,30,0,40,40e6); % NBI power
init.te0 = 0.5e3; init.jpow = 0; % Construct kinetic profiles according to the new settings. v = build_kinetic_profiles(model,init); % initial condition init.Ip0 = U0(1,1); x0 = RAPTOR_initial_conditions(model,init,g(:,1),v(:,1)); simres0 = RAPTOR_predictive(x0,g,v,U0,model,params); out0 = RAPTOR_out(simres0,model,params);
it telaps newt res t[ s] dt[ s] Ip[MA] Icd[MA] Ibs[MA] Ioh[MA] qe qmin q0 Vl[V] Te0[keV] Ti0[keV] ne0[e19] f_ss
1 0.12 6 1.5e-12 0 0.1 4 0 0.00219 4 15 7.2 7.2 1.0e+00 0.56 0.56 2.00 3.2e+00
11 0.59 3 1.5e-12 1 0.1 4.1 0 0.0615 4.04 14.6 6.61 6.61 1.0e+00 0.95 0.95 2.00 3.8e+00
21 0.94 2 1.7e-12 2 0.1 4.2 0 0.0695 4.13 14.3 6.09 6.09 9.8e-01 1.05 1.05 2.00 3.8e+00
31 1.3 3 1.3e-12 3 1 4.3 0 0.073 4.23 13.9 5.33 5.33 8.9e-01 1.15 1.15 2.00 3.4e+00
41 1.6 2 1.2e-11 13 1 5.3 0 0.0987 5.2 11.3 3.02 3.02 9.5e-01 1.71 1.71 2.00 3.6e+00
51 1.9 2 2.4e-11 23 1 6.3 0 0.118 6.18 9.51 2.01 2.01 9.5e-01 2.35 2.35 2.00 3.7e+00
61 2.3 5 6.9e-12 33 1 7.3 0 0.227 7.07 8.21 1.56 1.56 6.9e-01 5.46 5.46 2.00 3.6e+00
71 2.8 2 2.6e-09 43 1 8.3 0 0.517 7.78 7.22 1.47 1.47 4.3e-01 11.09 11.09 2.00 2.5e+00
81 3.1 2 1.1e-11 53 1 9.3 0 0.518 8.78 6.44 1.43 1.43 4.4e-01 12.03 12.03 2.00 2.5e+00
91 3.4 2 4.1e-11 63 1 10.3 0 0.518 9.78 5.82 1.39 1.39 4.4e-01 12.97 12.97 2.00 2.6e+00
101 3.8 2 1.9e-11 73 1 11.3 0 0.517 10.8 5.3 1.36 1.36 4.4e-01 13.92 13.92 2.00 2.6e+00
Simulation with constant time step
Repeat the same simulation but with a fixed time step.
tgrid1 = [0:0.1:80]; params.tgrid = tgrid1; % inputs rampfun = @(t,tmin,ymin,tmax,ymax) max(ymin,min((ymax-ymin)/(tmax-tmin)*(t-tmin),ymax-ymin)+ymin); % anonymous function for ramps % Construct actuator inputs on the time grid tgrid1. U1(1,:) = rampfun(tgrid1,0,4e6,80,12e6); % Ip U1(2,:) = rampfun(tgrid1,30,0,40,40e6); % NBI power % initial condition init.Ip0 = U1(1,1); x0 = RAPTOR_initial_conditions(model,init,g(:,1),v(:,1)); simres1 = RAPTOR_predictive(x0,g,v,U1,model,params); out1 = RAPTOR_out(simres1,model,params);
it telaps newt res t[ s] dt[ s] Ip[MA] Icd[MA] Ibs[MA] Ioh[MA] qe qmin q0 Vl[V] Te0[keV] Ti0[keV] ne0[e19] f_ss
1 0.1 6 1.5e-12 0 0.1 4 0 0.00219 4 15 7.2 7.2 1.0e+00 0.56 0.56 2.00 3.2e+00
11 0.6 3 1.5e-12 1 0.1 4.1 0 0.0615 4.04 14.6 6.61 6.61 1.0e+00 0.95 0.95 2.00 3.8e+00
21 0.96 2 1.9e-12 2 0.1 4.2 0 0.0695 4.13 14.3 6.09 6.09 9.8e-01 1.05 1.05 2.00 3.8e+00
31 1.3 2 1.8e-12 3 0.1 4.3 0 0.073 4.23 13.9 5.67 5.67 9.7e-01 1.11 1.11 2.00 3.8e+00
41 1.6 2 1.5e-12 4 0.1 4.4 0 0.0761 4.32 13.6 5.3 5.3 9.7e-01 1.16 1.16 2.00 3.8e+00
51 2 2 1.7e-12 5 0.1 4.5 0 0.079 4.42 13.3 4.97 4.97 9.6e-01 1.21 1.21 2.00 3.7e+00
61 2.4 2 2.1e-12 6 0.1 4.6 0 0.0819 4.52 13 4.67 4.67 9.6e-01 1.27 1.27 2.00 3.7e+00
71 2.8 2 4.4e-12 7 0.1 4.7 0 0.0847 4.62 12.7 4.39 4.39 9.6e-01 1.32 1.32 2.00 3.7e+00
81 3.1 2 5.5e-12 8 0.1 4.8 0 0.0875 4.71 12.5 4.14 4.14 9.6e-01 1.38 1.38 2.00 3.7e+00
91 3.5 2 4.5e-12 9 0.1 4.9 0 0.0901 4.81 12.2 3.9 3.9 9.6e-01 1.44 1.44 2.00 3.7e+00
101 3.8 2 8.3e-12 10 0.1 5 0 0.0927 4.91 12 3.69 3.69 9.6e-01 1.50 1.50 2.00 3.7e+00
111 4.2 2 5.1e-12 11 0.1 5.1 0 0.0952 5 11.7 3.5 3.5 9.6e-01 1.56 1.56 2.00 3.7e+00
121 4.5 2 3.6e-12 12 0.1 5.2 0 0.0976 5.1 11.5 3.32 3.32 9.6e-01 1.62 1.62 2.00 3.7e+00
131 4.9 2 2.8e-12 13 0.1 5.3 0 0.0999 5.2 11.3 3.16 3.16 9.6e-01 1.68 1.68 2.00 3.7e+00
141 5.2 2 8.7e-12 14 0.1 5.4 0 0.102 5.3 11.1 3 3 9.6e-01 1.74 1.74 2.00 3.7e+00
151 5.6 2 5.7e-12 15 0.1 5.5 0 0.104 5.4 10.9 2.87 2.87 9.6e-01 1.80 1.80 2.00 3.7e+00
161 5.9 2 1.4e-11 16 0.1 5.6 0 0.106 5.49 10.7 2.74 2.74 9.6e-01 1.87 1.87 2.00 3.7e+00
171 6.3 2 9.4e-12 17 0.1 5.7 0 0.108 5.59 10.5 2.62 2.62 9.6e-01 1.93 1.93 2.00 3.7e+00
181 6.6 2 1.3e-11 18 0.1 5.8 0 0.11 5.69 10.3 2.51 2.51 9.6e-01 1.99 1.99 2.00 3.7e+00
191 7 2 1.2e-11 19 0.1 5.9 0 0.112 5.79 10.2 2.41 2.41 9.6e-01 2.06 2.06 2.00 3.7e+00
it telaps newt res t[ s] dt[ s] Ip[MA] Icd[MA] Ibs[MA] Ioh[MA] qe qmin q0 Vl[V] Te0[keV] Ti0[keV] ne0[e19] f_ss
201 7.3 2 1.0e-11 20 0.1 6 0 0.114 5.89 9.99 2.31 2.31 9.5e-01 2.13 2.13 2.00 3.8e+00
211 7.7 2 1.1e-11 21 0.1 6.1 0 0.116 5.98 9.82 2.23 2.23 9.5e-01 2.19 2.19 2.00 3.8e+00
221 8 2 1.8e-11 22 0.1 6.2 0 0.118 6.08 9.66 2.14 2.14 9.5e-01 2.26 2.26 2.00 3.8e+00
231 8.4 2 1.2e-11 23 0.1 6.3 0 0.119 6.18 9.51 2.07 2.07 9.5e-01 2.33 2.33 2.00 3.8e+00
241 8.7 2 1.1e-11 24 0.1 6.4 0 0.121 6.28 9.36 1.99 1.99 9.5e-01 2.39 2.39 2.00 3.8e+00
251 9.1 2 1.2e-11 25 0.1 6.5 0 0.122 6.38 9.22 1.93 1.93 9.5e-01 2.46 2.46 2.00 3.8e+00
261 9.4 2 1.7e-11 26 0.1 6.6 0 0.124 6.48 9.08 1.86 1.86 9.5e-01 2.53 2.53 2.00 3.8e+00
271 9.8 2 9.8e-12 27 0.1 6.7 0 0.125 6.57 8.94 1.81 1.81 9.4e-01 2.60 2.60 2.00 3.8e+00
281 10 2 2.6e-11 28 0.1 6.8 0 0.127 6.67 8.81 1.75 1.75 9.4e-01 2.67 2.67 2.00 3.8e+00
291 10 2 1.2e-11 29 0.1 6.9 0 0.128 6.77 8.68 1.7 1.7 9.4e-01 2.74 2.74 2.00 3.8e+00
301 11 2 1.1e-11 30 0.1 7 0 0.129 6.87 8.56 1.65 1.65 9.4e-01 2.80 2.80 2.00 3.8e+00
311 11 2 8.7e-09 31 0.1 7.1 0 0.164 6.94 8.44 1.61 1.61 8.8e-01 3.39 3.39 2.00 4.0e+00
321 12 2 4.0e-10 32 0.1 7.2 0 0.22 6.98 8.32 1.59 1.59 7.8e-01 4.41 4.41 2.00 3.9e+00
331 12 2 9.7e-11 33 0.1 7.3 0 0.272 7.03 8.21 1.58 1.58 6.9e-01 5.45 5.45 2.00 3.7e+00
341 12 2 2.0e-11 34 0.1 7.4 0 0.317 7.08 8.1 1.57 1.57 6.3e-01 6.38 6.38 2.00 3.4e+00
351 13 2 3.1e-11 35 0.1 7.5 0 0.356 7.14 7.99 1.55 1.55 5.8e-01 7.22 7.22 2.00 3.2e+00
361 13 2 1.8e-11 36 0.1 7.6 0 0.391 7.21 7.88 1.54 1.54 5.4e-01 7.99 7.99 2.00 3.0e+00
371 13 2 2.6e-11 37 0.1 7.7 0 0.424 7.28 7.78 1.53 1.53 5.1e-01 8.71 8.71 2.00 2.9e+00
381 14 2 1.8e-11 38 0.1 7.8 0 0.454 7.35 7.68 1.52 1.52 4.8e-01 9.39 9.39 2.00 2.8e+00
391 14 2 2.7e-11 39 0.1 7.9 0 0.482 7.42 7.58 1.51 1.51 4.6e-01 10.04 10.04 2.00 2.7e+00
it telaps newt res t[ s] dt[ s] Ip[MA] Icd[MA] Ibs[MA] Ioh[MA] qe qmin q0 Vl[V] Te0[keV] Ti0[keV] ne0[e19] f_ss
401 14 2 2.1e-11 40 0.1 8 0 0.509 7.49 7.49 1.5 1.5 4.4e-01 10.66 10.66 2.00 2.6e+00
411 15 2 2.1e-11 41 0.1 8.1 0 0.516 7.58 7.4 1.5 1.5 4.3e-01 10.89 10.89 2.00 2.5e+00
421 15 2 2.8e-11 42 0.1 8.2 0 0.517 7.68 7.31 1.49 1.49 4.3e-01 10.99 10.99 2.00 2.5e+00
431 15 2 1.9e-11 43 0.1 8.3 0 0.517 7.78 7.22 1.48 1.48 4.3e-01 11.08 11.08 2.00 2.5e+00
441 16 2 1.9e-11 44 0.1 8.4 0 0.517 7.88 7.13 1.48 1.48 4.3e-01 11.18 11.18 2.00 2.5e+00
451 16 2 4.2e-11 45 0.1 8.5 0 0.517 7.98 7.05 1.47 1.47 4.3e-01 11.27 11.27 2.00 2.5e+00
461 16 2 1.9e-11 46 0.1 8.6 0 0.517 8.08 6.97 1.47 1.47 4.3e-01 11.36 11.36 2.00 2.5e+00
471 17 2 4.2e-11 47 0.1 8.7 0 0.517 8.18 6.89 1.46 1.46 4.3e-01 11.46 11.46 2.00 2.5e+00
481 17 2 2.8e-11 48 0.1 8.8 0 0.517 8.28 6.81 1.46 1.46 4.3e-01 11.55 11.55 2.00 2.5e+00
491 17 2 4.1e-11 49 0.1 8.9 0 0.518 8.38 6.73 1.46 1.46 4.3e-01 11.64 11.64 2.00 2.5e+00
501 18 2 2.1e-11 50 0.1 9 0 0.518 8.48 6.66 1.45 1.45 4.3e-01 11.74 11.74 2.00 2.5e+00
511 18 2 2.9e-11 51 0.1 9.1 0 0.518 8.58 6.58 1.45 1.45 4.3e-01 11.83 11.83 2.00 2.5e+00
521 18 1 9.8e-09 52 0.1 9.2 0 0.518 8.68 6.51 1.44 1.44 4.4e-01 11.93 11.93 2.00 2.5e+00
531 19 1 8.4e-09 53 0.1 9.3 0 0.518 8.78 6.44 1.44 1.44 4.4e-01 12.02 12.02 2.00 2.5e+00
541 19 1 7.1e-09 54 0.1 9.4 0 0.518 8.88 6.37 1.43 1.43 4.4e-01 12.11 12.11 2.00 2.5e+00
551 19 1 6.0e-09 55 0.1 9.5 0 0.518 8.98 6.31 1.43 1.43 4.4e-01 12.21 12.21 2.00 2.6e+00
561 19 1 4.9e-09 56 0.1 9.6 0 0.518 9.08 6.24 1.43 1.43 4.4e-01 12.30 12.30 2.00 2.6e+00
571 19 1 4.0e-09 57 0.1 9.7 0 0.518 9.18 6.18 1.42 1.42 4.4e-01 12.40 12.40 2.00 2.6e+00
581 20 1 3.3e-09 58 0.1 9.8 0 0.518 9.28 6.11 1.42 1.42 4.4e-01 12.49 12.49 2.00 2.6e+00
591 20 1 2.8e-09 59 0.1 9.9 0 0.518 9.38 6.05 1.41 1.41 4.4e-01 12.59 12.59 2.00 2.6e+00
it telaps newt res t[ s] dt[ s] Ip[MA] Icd[MA] Ibs[MA] Ioh[MA] qe qmin q0 Vl[V] Te0[keV] Ti0[keV] ne0[e19] f_ss
601 20 1 2.6e-09 60 0.1 10 0 0.518 9.48 5.99 1.41 1.41 4.4e-01 12.68 12.68 2.00 2.6e+00
611 20 1 2.6e-09 61 0.1 10.1 0 0.518 9.58 5.93 1.41 1.41 4.4e-01 12.78 12.78 2.00 2.6e+00
621 20 1 2.9e-09 62 0.1 10.2 0 0.518 9.68 5.87 1.4 1.4 4.4e-01 12.87 12.87 2.00 2.6e+00
631 21 1 3.4e-09 63 0.1 10.3 0 0.518 9.78 5.82 1.4 1.4 4.4e-01 12.97 12.97 2.00 2.6e+00
641 21 1 3.9e-09 64 0.1 10.4 0 0.518 9.88 5.76 1.39 1.39 4.4e-01 13.06 13.06 2.00 2.6e+00
651 21 1 4.3e-09 65 0.1 10.5 0 0.518 9.98 5.71 1.39 1.39 4.4e-01 13.15 13.15 2.00 2.6e+00
661 21 1 4.8e-09 66 0.1 10.6 0 0.518 10.1 5.65 1.39 1.39 4.4e-01 13.25 13.25 2.00 2.6e+00
671 21 1 5.3e-09 67 0.1 10.7 0 0.518 10.2 5.6 1.38 1.38 4.4e-01 13.34 13.34 2.00 2.6e+00
681 22 1 5.8e-09 68 0.1 10.8 0 0.518 10.3 5.55 1.38 1.38 4.4e-01 13.44 13.44 2.00 2.6e+00
691 22 1 6.2e-09 69 0.1 10.9 0 0.518 10.4 5.5 1.38 1.38 4.4e-01 13.53 13.53 2.00 2.6e+00
701 22 1 6.6e-09 70 0.1 11 0 0.518 10.5 5.45 1.37 1.37 4.4e-01 13.63 13.63 2.00 2.6e+00
711 22 1 7.0e-09 71 0.1 11.1 0 0.518 10.6 5.4 1.37 1.37 4.4e-01 13.72 13.72 2.00 2.6e+00
721 22 1 7.3e-09 72 0.1 11.2 0 0.517 10.7 5.35 1.36 1.36 4.4e-01 13.82 13.82 2.00 2.6e+00
731 23 1 7.7e-09 73 0.1 11.3 0 0.517 10.8 5.3 1.36 1.36 4.4e-01 13.91 13.91 2.00 2.6e+00
741 23 1 8.0e-09 74 0.1 11.4 0 0.517 10.9 5.26 1.36 1.36 4.4e-01 14.00 14.00 2.00 2.6e+00
751 23 1 8.3e-09 75 0.1 11.5 0 0.517 11 5.21 1.35 1.35 4.4e-01 14.10 14.10 2.00 2.6e+00
761 23 1 8.5e-09 76 0.1 11.6 0 0.517 11.1 5.16 1.35 1.35 4.4e-01 14.19 14.19 2.00 2.6e+00
771 23 1 8.8e-09 77 0.1 11.7 0 0.517 11.2 5.12 1.35 1.35 4.4e-01 14.29 14.29 2.00 2.6e+00
781 24 1 9.0e-09 78 0.1 11.8 0 0.517 11.3 5.08 1.34 1.34 4.4e-01 14.38 14.38 2.00 2.6e+00
791 24 1 9.2e-09 79 0.1 11.9 0 0.517 11.4 5.03 1.34 1.34 4.4e-01 14.47 14.47 2.00 2.6e+00
it telaps newt res t[ s] dt[ s] Ip[MA] Icd[MA] Ibs[MA] Ioh[MA] qe qmin q0 Vl[V] Te0[keV] Ti0[keV] ne0[e19] f_ss
801 24 1 9.4e-09 80 0.1 12 0 0.517 11.5 4.99 1.34 1.34 4.4e-01 14.57 14.57 2.00 2.6e+00
Compare
Comparing the two simulations confirms their equivalence
subplot(311) hh=plot(tgrid1,out1.iota(1:5:end,:),'b.-',tgrid0,out0.iota(1:5:end,:),'ro'); ylabel('iota []'); xlabel('t[s]'); legend(hh([1,end]),'fixed dt','variable dt'); subplot(312) plot(tgrid1,out1.te(1:5:end,:)/1e3,'b.-',tgrid0,out0.te(1:5:end,:)/1e3,'ro'); ylabel('T_e [keV]'); xlabel('t[s]'); subplot(313) plot(tgrid1,out1.psi(1:5:end,:),'b.-',tgrid0,out0.psi(1:5:end,:),'ro'); ylabel('\psi [Wb]'); xlabel('t[s]');
