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Copy pathFIG11_Boudinage.m
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FIG11_Boudinage.m
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% Iterative finite difference program for 2D viscous deformation, Matlab version
% This code is free software under the Creative Commons CC-BY-NC-ND license
% Published in W.R.Halter, E.Macherel, and S.M.Schmalholz (2022) JSG, https://doi.org/10.1016/j.jsg.2022.104617
clear all, close all, clc, tic, load('colormaps\vik.mat'), colormap(vik) % Colormap from Crameri, F. (2018). Scientific colour maps. Zenodo. http://doi.org/10.5281/zenodo.1243862
% Additional functions perfoming interpolations on the numerical grid
function A1 = n2c(A0) % Interpolation of nodal points to center points
A1 = (A0(2:end,:) + A0(1:end-1,:))/2;
A1 = (A1(:,2:end) + A1(:,1:end-1))/2;
end
function A2 = c2n(A0) % Interpolation of center points to nodal points
A1 = zeros(size(A0,1)+1,size(A0,2));
A1(:,:) = [1.5*A0(1,:)-0.5*A0(2,:); (A0(2:end,:)+A0(1:end-1,:))/2; 1.5*A0(end,:)-0.5*A0(end-1,:)];
A2 = zeros(size(A1,1),size(A1,2)+1);
A2(:,:) = [1.5*A1(:,1)-0.5*A1(:,2), (A1(:,2:end)+A1(:,1:end-1))/2, 1.5*A1(:,end)-0.5*A1(:,end-1)];
end
% Beginning of the actual Script
ps = 1; % ps = 1 models pure shear; ps = 0 models simple shear
etaRatio = 1/50; % Viscosity ratio (matrix/inclusion)
nout = 1e3;
% Numerical parameters
nx = 201; ny = nx; % Numerical resolution
Lx = 1; Ly = 1; % Model dimension
dx = Lx/(nx-1); dy = Ly/(ny-1); % Grid spacing
x = [-Lx/2:dx:Lx/2]; y = [-Ly/2:dy:Ly/2]; % Coordinate vectors
x_vx = [x(1)-dx/2,( x(1:end-1) + x(2:end) )/2,x(end)+dx/2]; % Horizontal vector for Vx which is one more than basic grid
y_vy = [y(1)-dy/2,( y(1:end-1) + y(2:end) )/2,y(end)+dy/2]; % Vertical vector for Vy which is one more than basic grid
% 3 numerical grids due to staggered grid
[X,Y] = ndgrid(x,y); [X_vx,Y_vx] = ndgrid(x_vx,y); [X_vy,Y_vy] = ndgrid(x,y_vy);
% Physical parameters
eta_B = 1; n_exp = 1;
s_ref = 1.5; D_B =-1;
% Initialization
P = zeros(nx, ny); ETA = eta_B*ones(nx, ny);
% Define boudins
inside = find(X<-0.05 & Y>-0.2 & Y<0.2 | X>0.05 & Y>-0.2 & Y<0.2);
ETA(inside) = eta_B/etaRatio; etamin = 0.1*min(ETA(:));
for smo=1:2; Iix = [2:nx-1]; Iiy = [2:ny-1]; % Smoothing of the initial viscosity field
ETA(Iix,:) = ETA(Iix,:) + 0.4*(ETA(Iix+1,:)-2*ETA(Iix,:)+ETA(Iix-1,:));
ETA(:,Iiy) = ETA(:,Iiy) + 0.4*(ETA(:,Iiy+1)-2*ETA(:,Iiy)+ETA(:,Iiy-1));
end
ETA_L = ETA; ETA_XY = n2c(ETA); ETA_PL = ETA;
RES_VX_relaxed = zeros(nx-1,ny-2); RES_VY_relaxed = zeros(nx-2,ny-1);
% Boundary condition
if ps==1, VX = -D_B*X_vx; VY = D_B*Y_vy;
elseif ps==0, VX = D_B*Y_vx; VY = 0*Y_vy; end
% Parameters for pseudo-transient iterations
tol = 1e-6; err_absolute = 1; err_relative = 1;
%%% Numerical parameters for iterative solver %%%
CFLV = 1/5; CFLP = CFLV/nx/20; % Numerical parameter for iterative solver
dpt_P = zeros(nx,ny);
dpt_Vx = CFLV./(max(ETA(1:end-1,2:end-1),ETA(2:end,2:end-1))/dx^2 + max(ETA_XY(:,1:end-1),ETA_XY(:,2:end))/dx/dy);
dpt_Vy = CFLV./(max(ETA(2:end-1,1:end-1),ETA(2:end-1,2:end))/dy^2 + max(ETA_XY(1:end-1,:),ETA_XY(2:end,:))/dx/dy);
dpt_V_dx2 = max(max(dpt_Vx(1:end-1,:),dpt_Vx(2:end,:))/dx^2, max(dpt_Vy(:,1:end-1),dpt_Vy(:,2:end))/dy^2);
dpt_P(2:end-1,2:end-1) = CFLP./dpt_V_dx2;
dpt_P([1 end],:) = dpt_P([2 end-1],:); dpt_P(:,[1 end]) = dpt_P(:,[2 end-1]);
%%% Numerical parameters for iterative solver %%%
iter = 0;
while err_absolute(end)>tol||err_relative(end)>tol; iter = iter+1; % START of iteration loop
DXX = diff(VX,1,1)/dx; % Eq (1)
DYY = diff(VY,1,2)/dy; % Eq (2)
DXY = 1/2*( diff(VX(2:end-1,:),1,2)/dy ...
+ diff(VY(:,2:end-1),1,1)/dx ); % Eq (3)
TXX = 2.*ETA.*DXX; % Eq (4)
TYY = 2.*ETA.*DYY; % Eq (5)
TXY = 2.*n2c(ETA).*DXY; % Eq (6)
SXX = -P(:,2:end-1)+TXX(:,2:end-1); % Eq (7)
SYY = -P(2:end-1,:)+TYY(2:end-1,:); % Eq (8)
RES_P = -( diff(VX,1,1)/dx + diff(VY,1,2)/dy ); % Eq (16)
RES_VX = diff(SXX,1,1)/dx + diff(TXY,1,2)/dy; % Eq (17)
RES_VY = diff(SYY,1,2)/dy + diff(TXY,1,1)/dx; % Eq (18)
RES_VX_relaxed = RES_VX_relaxed*(1-5/nx) + RES_VX.*dpt_Vx; % Relaxation on residual
RES_VY_relaxed = RES_VY_relaxed*(1-5/ny) + RES_VY.*dpt_Vy; % Relaxation on residual
P = P + dpt_P.*RES_P; % Eq (22)
VX(2:end-1,2:end-1) = VX(2:end-1,2:end-1) + RES_VX_relaxed; % Eq (23)
VY(2:end-1,2:end-1) = VY(2:end-1,2:end-1) + RES_VY_relaxed; % Eq (24)
TII = sqrt(0.5*(TXX.^2 + TYY.^2 + 2*c2n(TXY).^2)); % Eq (13)
if n_exp>1 % Power-law viscosity
ETA_PL_it = ETA_PL; % Viscosity of previous iteration step
ETA_PL = ETA_L.*(TII/s_ref).^(1-n_exp); % Eq (12)
ETA_PL = exp(log(ETA_PL)*0.5+log(ETA_PL_it)*0.5);
ETA = 1./( 1./ETA_L + 1./ETA_PL ); % Eq (14)
ETA(inside) = ETA_L(inside); % Power-law viscosity only applied to matrix
ETA(ETA<etamin) = etamin; % Minimum viscosity
end
err_absolute(iter) = max([max(abs(RES_VX(:))), max(abs(RES_VY(:))), max(abs(RES_P(:)))]);
err_relative(iter) = max([max(abs(RES_VX_relaxed)), max(abs(RES_VY_relaxed)), max(abs(dpt_P.*RES_P))]);
if mod(iter,nout)==0 % Visualization during calculation
subplot(221),pcolor(X,Y,(P-mean(P(:)))/(2*eta_B*abs(D_B))),shading interp,axis equal,axis tight,
caxis([min((P(:)-mean(P(:)))/(2*eta_B*abs(D_B))) max((P(:)-mean(P(:)))/(2*eta_B*abs(D_B)))]/1.5), colorbar
title({'A) Pressure, P / (2\eta_BD_B)'}),ylabel('Height [ ]')
subplot(222),pcolor(X,Y,log10(ETA/eta_B)), shading interp, hold on
title(['B) Log_{10} of Viscosity, \eta / \eta_B, and velocity arrows'])
st = 15; VXC = (VX(1:end-1,:)+VX(2:end,:))/2; VYC = (VY(:,1:end-1)+VY(:,2:end))/2;
quiver(X(1:st:end,1:st:end),Y(1:st:end,1:st:end),VXC(1:st:end,1:st:end),VYC(1:st:end,1:st:end),'w');
axis equal,axis tight,colorbar,axis([-Lx/2,Lx/2,-Ly/2,Ly/2]),hold off
subplot(223),pcolor(X,Y,TII/(2*eta_B*abs(D_B))),shading interp,axis equal,axis tight,colorbar
title(['C) 2nd stress invariant, T_{II} / (2\eta_BD_B)']), xlabel('Width [ ]'), ylabel('Height [ ]')
subplot(224),loglog(1:length(err_absolute),err_absolute,1:length(err_relative),err_relative),title('D) Error (residual) convergence'),xlabel('Iterations'),ylabel('Residual')
axis([1 iter+1e4 tol/10 1e6]), hold on,plot([1 iter+1e4],[tol tol],'--k'),text(1e3,tol*2,'Tolerance'), legend('absolute error','relative error',''), hold off
set(gcf,'position',[281 47.4000 975.2000 724]), drawnow
end
end, toc % END of iteration loop