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Advect_and_Calculate_3DSteady.m
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Advect_and_Calculate_3DSteady.m
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% Input arguments:
% tspan : Advection timesteps (e.g. tspan=linspace(t_initial,t_final,100)
% xx,yy,zz : Initial positions for advection
% U_Interp,V_Interp,W_Interp : Velocity field interpolants with inputs (tspan,xx,yy,zz)
% NCores : Number of Cores for parpool
% Output arguments:
% xt,yt,zt : xx-component, yy-component of trajectory final position
% time_note : tspan time if a trajectory left interpolant domain
% TSE_Bar,TSE,TRA_Bar,TRA : Single trajectory metrics from
% section II and III in [1]
%--------------------------------------------------------------------------
% Author: Nikolas Aksamit naksamit@ethz.ch
%--------------------------------------------------------------------------
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% References:
% [1] Haller, G., Aksamit, N. O., & Bartos, A. P. E. (2021). Quasi-Objective Coherent Structure Diagnostics from Single Trajectories.
% Chaos, 31, 043131-1–17. https://doi.org/10.1063/5.0044151
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Calculate TRA and TSE diagnostics for steady 3D flows.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [xt,yt,zt,time_note,TSE_Bar,TSE,TRA_Bar,TRA] = Advect_and_Calculate_3DSteady(tspan,xx,yy,zz,U_Interp,V_Interp,W_Interp,NCores)
%%%%% Start parallel pool with NCores. Make sure the size of pool
%%%%% >= number requested by spmd
if isempty(gcp)
parpool(NCores);
end
Np=numel(xx);
cpu_num = min(NCores,Np);
id = ceil( linspace(0,Np,cpu_num+1) );
% Shared variables
xt = zeros(2,length(xx));
yt = zeros(2,length(yy));
zt = zeros(2,length(zz));
xt(1,:) = xx;
yt(1,:) = yy;
zt(1,:) = zz;
spmd(cpu_num)
tic
%%%% Split up initial conditions for separate cores
Range = id(labindex)+1:id(labindex+1);
x_spmd=xt(1,Range);
y_spmd=yt(1,Range);
z_spmd=zt(1,Range);
time_note=zeros(size(x_spmd));
TSE_spmd=zeros(size(x_spmd));
TRA_spmd=zeros(size(x_spmd));
TSE_spmd_NM=zeros(size(x_spmd));
for s=1:numel(tspan)-1
%%%% This only records the last point, not entire path of particles
ds = tspan(s+1) - tspan(s);
[UK1,VK1,WK1] = r_prime(x_spmd(1,:),y_spmd(1,:),z_spmd(1,:),U_Interp,V_Interp,W_Interp);
xx = x_spmd(1,:) + 0.5 * ds * UK1;
yy = y_spmd(1,:) + 0.5 * ds * VK1;
zz = z_spmd(1,:) + 0.5 * ds * WK1;
[UK2,VK2,WK2] = r_prime(xx,yy,zz,U_Interp,V_Interp,W_Interp);
xx = x_spmd(1,:) + 0.5 * ds * UK2;
yy = y_spmd(1,:) + 0.5 * ds * VK2;
zz = z_spmd(1,:) + 0.5 * ds * WK2;
[UK3,VK3,WK3] = r_prime(xx,yy,zz,U_Interp,V_Interp,W_Interp);
xx = x_spmd(1,:) + ds * UK3;
yy = y_spmd(1,:) + ds * VK3;
zz = z_spmd(1,:) + ds * WK3;
[UK4,VK4,WK4] = r_prime(xx,yy,zz,U_Interp,V_Interp,W_Interp);
%increment in trajectories (RK4 displacement)
deltax(2,:) = ds / 6 * (UK1 + 2 * UK2 + 2 * UK3 + UK4);
deltay(2,:) = ds / 6 * (VK1 + 2 * VK2 + 2 * VK3 + VK4);
deltaz(2,:) = ds / 6 * (WK1 + 2 * WK2 + 2 * WK3 + WK4);
%update particle positions
x_spmd(2,:) = x_spmd(1,:) + deltax(2,:);
y_spmd(2,:) = y_spmd(1,:) + deltay(2,:);
z_spmd(2,:) = z_spmd(1,:) + deltaz(2,:);
%%%% If particle leaves domain, keep record of last position.
%%%% Record this time of leaving domain in time_note.
x_spmd(1,~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))=x_spmd(2,~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)));
y_spmd(1,~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))=y_spmd(2,~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)));
z_spmd(1,~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))=z_spmd(2,~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)));
time_note(1,(isnan(x_spmd(2,:)) & time_note(1,:)==0) | (isnan(y_spmd(2,:)) & time_note(1,:)==0) | (isnan(z_spmd(2,:)) & time_note(1,:)==0))=s;
if s==1
smooth_vx=deltax(2,:)'/ds;
smooth_vy=deltay(2,:)'/ds;
smooth_vz=deltaz(2,:)'/ds;
V1=[smooth_vx,smooth_vy,smooth_vz];
Speed_sqrd(1,:)=sum(V1.^2,2);
deltax(1,:) = deltax(2,:);
deltay(1,:) = deltay(2,:);
deltaz(1,:) = deltaz(2,:);
%%% Save this first veloicty measurement for TRA measurement at end
V1_spmd=V1;
end
%%% After two timesteps, now begin calculating TSE and TRA
if s>1
smooth_vx=deltax(2,:)'/ds;
smooth_vy=deltay(2,:)'/ds;
smooth_vz=deltaz(2,:)'/ds;
V2=[smooth_vx,smooth_vy,smooth_vz];
Speed_sqrd(2,:)=sum(V2.^2,2);
%%%% Calculate instantaneous TSE and TRA values. Add to sum if particle did not leave domain.
TSE_Inst=abs(log(sqrt(Speed_sqrd(2,:))./sqrt(Speed_sqrd(1,:))));
TSE_spmd(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))=TSE_spmd(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))+TSE_Inst(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)));
TSE_Inst_NM=log(sqrt(Speed_sqrd(2,:))./sqrt(Speed_sqrd(1,:)));
TSE_spmd_NM(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))=TSE_spmd_NM(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))+TSE_Inst_NM(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)));
TRA_Inst=acos(sum(V2.*V1,2)./(sqrt(sum(V2.^2,2)).*sqrt(sum(V1.^2,2))))';
TRA_spmd(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))=TRA_spmd(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)))+TRA_Inst(~isnan(x_spmd(2,:)) & ~isnan(y_spmd(2,:)));
deltax(1,:) = deltax(2,:);
deltay(1,:) = deltay(2,:);
deltaz(1,:) = deltaz(2,:);
Speed_sqrd(1,:)=Speed_sqrd(2,:);
V1=V2;
V2_spmd=V2;
end
end
TSE_spmd=TSE_spmd(1,:);
TRA_spmd=TRA_spmd(1,:);
TSE_spmd_NM=TSE_spmd_NM(1,:);
toc
end
%
V1 = cat(1,V1_spmd{:});
V2 = cat(1,V2_spmd{:});
V1=V1./(sqrt(sum(V1.*V1,2)));
V2=V2./(sqrt(sum(V2.*V2,2)));
xt = cat(2,x_spmd{:});
yt = cat(2,y_spmd{:});
zt = cat(2,z_spmd{:});
time_note = cat(2,time_note{:});
TSE=cat(2,TSE_spmd_NM{:})/(tspan(end)-tspan(1));
TSE_Bar=cat(2,TSE_spmd{:})/(tspan(end)-tspan(1));
TRA_Bar=cat(2,TRA_spmd{:})/(tspan(end)-tspan(1));
TRA=real(acos(V1(:,1).*V2(:,1)+V1(:,2).*V2(:,2)))/(tspan(end)-tspan(1));
clear x_spmd y_spmd
disp('Advection and Single Trajectory Metric Calculations Complete')
end
function [uk,vk,wk] = r_prime(xx,yy,zz,U_Interp,V_Interp,W_Interp)
Bly = 1+0*xx;
bounds.x=U_Interp.GridVectors{1};
bounds.y=U_Interp.GridVectors{2};
bounds.z=U_Interp.GridVectors{3};
Bly(yy>max(bounds.y)) = 0; %Freeze when outside domain of interpolant
Bly(yy<min(bounds.y)) = 0;
Bly(xx>max(bounds.x)) = 0;
Bly(xx<min(bounds.x)) = 0;
Bly(zz>max(bounds.z)) = 0;
Bly(zz<min(bounds.z)) = 0;
uk=U_Interp(double(xx),double(yy),double(zz));
uk=uk.*Bly;
vk=V_Interp(double(xx),double(yy),double(zz));
vk=vk.*Bly;
wk=W_Interp(double(xx),double(yy),double(zz));
wk=wk.*Bly;
end