diff --git a/CHANGELOG b/CHANGELOG
new file mode 100644
index 0000000000000000000000000000000000000000..a85bb1f41503661e350dc3fdab9b680a4260c153
--- /dev/null
+++ b/CHANGELOG
@@ -0,0 +1,4 @@
+
+
+20210324: Version 1.0, initial release.
+
diff --git a/COPYING b/COPYING
new file mode 100644
index 0000000000000000000000000000000000000000..53d1f3d01864c35841739c55aeba9700657627b7
--- /dev/null
+++ b/COPYING
@@ -0,0 +1,675 @@
+ GNU GENERAL PUBLIC LICENSE
+ Version 3, 29 June 2007
+
+ Copyright (C) 2007 Free Software Foundation, Inc. <https://fsf.org/>
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+DATA OR DATA BEING RENDERED INACCURATE OR LOSSES SUSTAINED BY YOU OR THIRD
+PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS),
+EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF
+SUCH DAMAGES.
+
+ 17. Interpretation of Sections 15 and 16.
+
+ If the disclaimer of warranty and limitation of liability provided
+above cannot be given local legal effect according to their terms,
+reviewing courts shall apply local law that most closely approximates
+an absolute waiver of all civil liability in connection with the
+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+ END OF TERMS AND CONDITIONS
+
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+ <one line to give the program's name and a brief idea of what it does.>
+ Copyright (C) <year> <name of author>
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see <https://www.gnu.org/licenses/>.
+
+Also add information on how to contact you by electronic and paper mail.
+
+ If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+ <program> Copyright (C) <year> <name of author>
+ This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+ You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+<https://www.gnu.org/licenses/>.
+
+ The GNU General Public License does not permit incorporating your program
+into proprietary programs. If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library. If this is what you want to do, use the GNU Lesser General
+Public License instead of this License. But first, please read
+<https://www.gnu.org/licenses/why-not-lgpl.html>.
+
diff --git a/README.md b/README.md
index 697169493243be733820d16d103051cfe504dc27..a61f3ea2f187cf0a0d75965d42d88c679dead180 100644
--- a/README.md
+++ b/README.md
@@ -1,2 +1,27 @@
-# VillaEtAl2021BullMathBiol
+# Public gitlab repository VillaEtAl2021BullMathBiol
+This repository provides Matlab files to perform simulations as described in
+*Mechanical models of pattern and form in biological tissues: the role of stress-strain constitutive equations*
+by *C. Villa, M.A.J. Chaplain, A. Gerisch, T. Lorenzi*
+and to appear in *Bullettin of Mathematical Biology* (2021).
+For details we refer the intereseted reader to this publication and its supplementary material.
+
+If you use this software in your work then please cite the above name paper.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% VillaEtAl2021BullMathBio: simulate mechanical models of pattern formation%
+% Copyright (C) 2021 C. Villa and A. Gerisch %
+% %
+% This program is free software: you can redistribute it and/or modify %
+% it under the terms of the GNU General Public License as published by %
+% the Free Software Foundation, either version 3 of the License, or %
+% (at your option) any later version. %
+% %
+% This program is distributed in the hope that it will be useful, %
+% but WITHOUT ANY WARRANTY; without even the implied warranty of %
+% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %
+% GNU General Public License for more details. %
+% %
+% You should have received a copy of the GNU General Public License %
+% along with this program. If not, see <https://www.gnu.org/licenses/>. %
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
diff --git a/Simulations_1D.m b/Simulations_1D.m
new file mode 100644
index 0000000000000000000000000000000000000000..3efb1da87a6f0dfce55e2bf7ae09a1191bf65b15
--- /dev/null
+++ b/Simulations_1D.m
@@ -0,0 +1,267 @@
+function Simulations_1D(in_K, in_model)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% %%%
+%%% "Mechanical models of pattern and form in biological tissues: %%%
+%%% the role of stress-strain constitutive equations" %%%
+%%% %%%
+%%% C. Villa (*), M.A.J. Chaplain, A. Gerisch (**), T. Lorenzi %%%
+%%% %%%
+%%% Bullettin of Mathematical Biology (2021) %%%
+%%% %%%
+%%% %%%
+%%% (*) cv23[at]st-andrews.ac.uk %%%
+%%% (**) gerisch[at]mathematik.tu-darmstadt.de %%%
+%%% %%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% 1D Simulations (Kelvin-Voigt and Maxwell models) %%%
+%%% For details about the equations and schemes please see the %%%
+%%% manuscript indicated above and the supplementary material %%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% function Simulations_1D(in_K, in_model)
+% Runs the spatially one-dimensional simulation using viscoelastic
+% model in_model (either 'Maxwell' or 'Kelvin-Voigt') on a uniform grid
+% with in_K grid cells.
+% The results are saved in file saved_y1D_[in_model]_[in_K].mat and
+% a video in saved_y1D_[in_model]_[in_K].avi .
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Simulations_1D.m: simulates a 1D mechanical model of pattern formation%
+%%% Copyright (C) 2021 C. Villa and A. Gerisch %
+%%% %
+%%% This program is free software: you can redistribute it and/or modify %
+%%% it under the terms of the GNU General Public License as published by %
+%%% the Free Software Foundation, either version 3 of the License, or %
+%%% (at your option) any later version. %
+%%% %
+%%% This program is distributed in the hope that it will be useful, %
+%%% but WITHOUT ANY WARRANTY; without even the implied warranty of %
+%%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %
+%%% GNU General Public License for more details. %
+%%% %
+%%% You should have received a copy of the GNU General Public License %
+%%% along with this program. If not, see <https://www.gnu.org/licenses/>.%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+
+clc
+close all
+
+set(0,'DefaultAxesFontName', 'Times New Roman')
+set(0,'DefaultAxesFontSize', 14)
+set(0,'defaultaxeslinewidth',1)
+set(0,'defaultpatchlinewidth',1)
+set(0,'defaultlinelinewidth',2)
+set(0,'defaultTextInterpreter','latex')
+
+
+%% Numerical set up
+par.seed = 44; % Seed for random number generator
+par.model = in_model; % type of viscoelastic model
+%par.model = 'Maxwell';
+%par.model = 'Kelvin-Voigt';
+par.K = in_K; % Number of spatial grid cells
+par.L = 1; % Domain length
+x = linspace(0,par.L,par.K+1); % Discretise spatial domain
+par.dx = x(2)-x(1); % Cell size
+t0 = 0; % Initial time
+tf = 10000; % Final time
+tspan = linspace(t0,tf,201); % Time span
+
+%% Initial conditions - eq.(28)
+steadystate = [ones(2*par.K,1); zeros(par.K,1)];
+rng(par.seed); % initialize random number generator
+randP = [randn(par.K,1); zeros( 2*par.K,1)]; % random perturbation
+factP = 1e-2; % factor for random perturbation
+y0 = steadystate+factP*randP; % perturbed IC [n0, p0, u0] as long column
+
+%% Solve with ODE15i
+%%% Compute consistent yp0
+%res = @(y,yp)(norm(mechanochemical(y,yp,par),'inf'));
+res = @(y,yp)(norm(mechanochemical(y,yp,par)));
+disp(['residuum of steady state = ' ...
+ num2str(res(steadystate,0*steadystate), '%15.10e')]);
+[y0,yp0,resnorm] = decic(@(t,y,yp)(mechanochemical(y,yp,par)), t0, ...
+ y0, ones(3*par.K,1), zeros(3*par.K,1), zeros(3*par.K,1));
+disp(['residuum (from decic) of IC = ' num2str(resnorm, '%15.10e')]);
+disp(['residuum (from res()) of IC = ' num2str(res(y0,yp0), '%15.10e')]);
+%%% Solve
+tic
+[t,y] = ode15i(@(t,y,yp)(mechanochemical(y,yp,par)),tspan,y0,yp0);
+toc
+%%% Save computed solution to file
+filename = ['saved_y1D_' par.model '_' num2str(par.K)];
+save(filename, 't', 'y', 'par', 'x');
+
+%% Plot
+video_on = true; % Record video: YES (true), NO (false)
+video_filename = [filename '.avi'];
+plot_solution(x,y,t,par,video_on,video_filename);
+
+end
+
+%% Main function implementing the model
+function f = mechanochemical(y,yp,par)
+
+ %%% Parameter values
+ eta = 1; % viscosity
+ E = 1; % elasticity
+ D = 0.01; % diffusion
+ alpha = 0.05; % haptotaxis
+ r = 1; % proliferation
+ s = 10; % substrate elasticity
+ beta = 0.005; % long range traction
+ lambda = 0.5; % (cell traction) saturation coefficient
+ tau = 0.2; % cell traction
+
+ %%% Choose constitutive model (see Table 1)
+ switch par.model
+ case 'Kelvin-Voigt' % Kelvin Voigt - eq.(3)
+ [a0,a1,b0,b1] = deal(1,0,E,eta);
+ case 'Maxwell' % Maxwell - eq.(4)
+ [a0,a1,b0,b1] = deal(1/eta,1/E,0,1);
+ otherwise
+ error('Unknown constitutive model')
+ end
+
+ %%% Reshape input vectors
+ [n,p,u] = deal(y(1:par.K),y(par.K+1:2*par.K),y(2*par.K+1:3*par.K));
+ [np,pp,up] = deal(yp(1:par.K),yp(par.K+1:2*par.K),...
+ yp(2*par.K+1:3*par.K));
+
+ %%% Equation for n
+ % Advection velocity at grid cell interfaces - eq.(S.6)
+ up_Avx1 = Avx1(up);
+ vx1 = alpha*Mx1_face(p,par) + up_Avx1;
+ % fn(n,n',p,u') = 0 - eq.(S.5)
+ fn = np - D*Mxx1(n,par) + MA1(n, vx1, par) - r*n.*(1-n);
+
+ %%% Equation for p
+ % fp(p,p',u') = 0 - eq.(S.10)
+ fp = pp + MA1(p, up_Avx1, par);
+
+ %%% Equation for u
+ % Traction term - eq.(S.12)-(S.14)
+ pexp = 2;
+ fn1 = n./(1+lambda*n.^pexp); % Lambda_1
+ fn2 = ((1-(pexp-1)*lambda*n.^pexp)./((1+lambda*n.^pexp).^2)); %Lambda_2
+ fp1 = p + beta*Mxx1(p, par); % M_T1 P
+ fp2 = pp + beta*Mxx1(pp, par); % M_T1 P'
+ Tr = tau*(a0*fn1.*fp1 + a1*(fn2.*np.*fp1+fn1.*fp2));
+ % fu(n,n',p,p',u,u') = 0 - eq.(S.11)
+ fu = b1*Mxx1(up, par) + b0*Mxx1(u, par) ...
+ + Mx1_center(Tr, par) - a1*s*(p.*up + pp.*u) - a0*s*p.*u;
+
+ %%% Full system - eq.(S.1)
+ f = [fn; fp; fu];
+end
+
+
+%% Annexed functions
+
+%%% Compute variable at grid cell interfaces - def.(S.3)
+function avx1 = Avx1(y)
+ avx1 = 0.5*(y +y([2:end,1]));
+end
+
+%%% Compute first order derivative on grid cell interfaces
+%%% (second order approximation, central) - def.(S.4)
+function dx1 = Mx1_face(y, par)
+ persistent Mdx;
+ if size(Mdx,2) ~= size(y,1)
+ disp('Mx1_face : resetting persistent matrix.');
+ c = (1/par.dx)*sparse([-1; zeros(par.K-2,1); 1]); % First column
+ Mdx = toeplitz(c,[c(1), c(end:-1:2)']);
+ end
+ dx1 = Mdx*y;
+end
+
+%%% Compute first order derivative on grid cell centers
+%%% (second order approximation, central) - def.(S.2)
+function dx1 = Mx1_center(y, par)
+ persistent Mdx;
+ if size(Mdx,2) ~= size(y,1)
+ disp('Mx1_center: resetting persistent matrix.');
+ c = (1/(2*par.dx))*sparse([0;-1; zeros(par.K-3,1); 1]);
+ Mdx = toeplitz(c,[c(1), c(end:-1:2)']);
+ end
+ dx1 = Mdx*y;
+end
+
+%%% Compute second order derivative on grid cell centers
+%%% (second order approximation, central) - def.(S.2)
+function dxx1 = Mxx1(y, par)
+ persistent Mdxx;
+ if size(Mdxx,2) ~= size(y,1)
+ disp('Mxx1 : resetting persistent matrix.');
+ c = (1/(par.dx^2))*sparse([-2; 1; zeros(par.K-3,1); 1]);
+ Mdxx = toeplitz(c,[c(1), c(end:-1:2)']);
+ end
+ dxx1 = Mdxx*y;
+end
+
+%%% Compute advection at grid cell centers using first order upwinding
+%%% with advective velocity given at grid cell interfaces - def.(S.7)
+function fluxdiffx1 = MA1(y, vel, par)
+ % find pos/neg velocity entries
+ velPos_ind = (vel>0);
+ velNeg_ind = (~velPos_ind);
+ % compute flux accross cell interfaces using first order upwinding
+ % def.(S.8)-(S.9)
+ flux = NaN(size(y));
+ flux(velPos_ind) = y(velPos_ind).*vel(velPos_ind);
+ yshift = y([2:end 1]);
+ flux(velNeg_ind) = yshift(velNeg_ind).*vel(velNeg_ind);
+ % add entries for flux along x1=0 by periodicity
+ flux = flux([end,1:end]);
+ % compute flux difference per grid cell - def.(S.7) and (S.4)
+ fluxdiffx1 = (1/par.dx)*(flux(2:end)-flux(1:(end-1)));
+end
+
+%%% Plot solution
+function plot_solution(x,y,t,par,video_on,video_filename)
+ if video_on % Initialise video
+ vid = VideoWriter(video_filename);
+ open(vid);
+ figure('Units','normalized','Position',[0 0 0.5 0.45])
+ end
+ maxu = 10^(-6);
+ for i=1:length(t)
+ clf
+ n = [y(i,1*par.K),y(i,1:par.K)];
+ p = [y(i,2*par.K),y(i,par.K+1:2*par.K)];
+ u = [y(i,3*par.K),y(i,2*par.K+1:3*par.K)];
+ if max(abs(u))>maxu
+ maxu = max(abs(u))+0.1*max(abs(u));
+ end
+ subplot(1,3,1)
+ plot(x,n)
+ title('$n(t,x)$')
+ ylim([0,2])
+ axis square
+ subplot(1,3,2)
+ plot(x,p)
+ title('$\rho(t,x)$')
+ ylim([0,max(2,max(p))])
+ axis square
+ subplot(1,3,3)
+ plot(x,u)
+ title('$u(t,x)$')
+ axis square
+ ylim([-maxu,maxu])
+ a = axes;
+ t1 = title([par.model ' (t=',num2str(t(i)),')']);
+ a.Visible = 'off';
+ t1.Visible = 'on';
+ drawnow
+ if video_on % Record video
+ frame = getframe(gcf);
+ size(frame.cdata);
+ writeVideo(vid,frame);
+ pause(0.1)
+ end
+ end
+ if video_on % Close video
+ close(vid)
+ end
+end
diff --git a/Simulations_2D.m b/Simulations_2D.m
new file mode 100644
index 0000000000000000000000000000000000000000..aab36d88f6f0aff7a2d90b03aca2111401816277
--- /dev/null
+++ b/Simulations_2D.m
@@ -0,0 +1,453 @@
+function Simulations_2D(in_K, in_model)
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% %%%
+%%% "Mechanical models of pattern and form in biological tissues: %%%
+%%% the role of stress-strain constitutive equations" %%%
+%%% %%%
+%%% C. Villa (*), M.A.J. Chaplain, A. Gerisch (**), T. Lorenzi %%%
+%%% %%%
+%%% Bullettin of Mathematical Biology (2021) %%%
+%%% %%%
+%%% %%%
+%%% (*) cv23[at]st-andrews.ac.uk %%%
+%%% (**) gerisch[at]mathematik.tu-darmstadt.de %%%
+%%% %%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% 2D Simulations (Kelvin-Voigt and Maxwell models) %%%
+%%% For details about the equations and schemes please see the %%%
+%%% manuscript indicated above and the supplementary material %%%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%
+% function Simulations_2D(in_K, in_model)
+% Runs the spatially two-dimensional simulation using viscoelastic
+% model in_model (either 'Maxwell' or 'Kelvin-Voigt') on a uniform
+% square grid with in_K grid cells in each dimension.
+% The results are saved in file saved_y2D_[in_model]_[in_K].mat and
+% a video in saved_y2D_[in_model]_[in_K].avi .
+%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Simulations_2D.m: simulates a 2D mechanical model of pattern formation%
+%%% Copyright (C) 2021 C. Villa and A. Gerisch %
+%%% %
+%%% This program is free software: you can redistribute it and/or modify %
+%%% it under the terms of the GNU General Public License as published by %
+%%% the Free Software Foundation, either version 3 of the License, or %
+%%% (at your option) any later version. %
+%%% %
+%%% This program is distributed in the hope that it will be useful, %
+%%% but WITHOUT ANY WARRANTY; without even the implied warranty of %
+%%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the %
+%%% GNU General Public License for more details. %
+%%% %
+%%% You should have received a copy of the GNU General Public License %
+%%% along with this program. If not, see <https://www.gnu.org/licenses/>.%
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+clc
+close all
+
+set(0,'DefaultAxesFontName', 'Times New Roman')
+set(0,'DefaultAxesFontSize', 14)
+set(0,'defaultaxeslinewidth',1)
+set(0,'defaultpatchlinewidth',1)
+set(0,'defaultlinelinewidth',2)
+set(0,'defaultTextInterpreter','latex')
+
+
+%% Numerical set up
+par.seed = 44; % Seed for random number generator
+par.model = in_model; % type of viscoelastic model
+%par.model = 'Maxwell';
+%par.model = 'Kelvin-Voigt';
+par.K = in_K; % Number of spatial grid cells
+par.L = 1; % Domain length
+x1 = linspace(0,par.L,par.K+1); % Discretise spatial domain
+par.K2 = par.K^2; % Total number of grid cells
+par.dx = x1(2)-x1(1); % Cell size
+t0 = 0; % Initial time
+tf = 10000; % Final time
+tspan = linspace(t0,tf,251); % Time span
+
+%% Initial conditions
+steadystate = [ones(2*par.K2,1); zeros(2*par.K2,1)];
+rng(par.seed); % initialize random number generator % use seed 44 in random number generator
+randP = [randn(par.K2,1); zeros(3*par.K2,1)]; % random perturbation
+factP = 1e-2; % factor for random perturbation
+y0 = steadystate+factP*randP; % perturbed IC [n0, p0, u10, u20] as long column
+
+%% Solve with ODE15i
+solve_on = true; % run simulation and save result: YES (true), NO (false)
+if solve_on
+ tic
+ %%% Create Jacobian pattern
+ [SPDY,SPDYP] = Jpattern(par.K);
+ options = odeset('Jpattern',{SPDY,SPDYP});
+ %options = [];
+ %%% Compute consistent yp0
+ %res = @(y,yp)(norm(mechanochemical(y,yp,par),'inf'));
+ res = @(y,yp)(norm(mechanochemical(y,yp,par)));
+ disp(['residuum of steady state = ' ...
+ num2str(res(steadystate,0*steadystate), '%15.10e')]);
+ [y0,yp0, resnorm] = decic(@(t,y,yp)(mechanochemical(y,yp,par)), ...
+ t0, y0, ones(4*par.K2,1), ...
+ zeros(4*par.K2,1), zeros(4*par.K2,1), options);
+ disp(['residuum (from decic) of IC = ' num2str(resnorm, '%15.10e')]);
+ disp(['residuum (from res()) of IC = ' num2str(res(y0,yp0), '%15.10e')]);
+ toc
+ %%% Solve
+ tic
+ [t,y] = ode15i(@(t,y,yp)(mechanochemical(y,yp,par)),tspan,...
+ y0,yp0,options);
+ toc
+ %%% Save computed solution to file
+ filename = ['saved_y2D_' par.model '_' num2str(par.K)];
+ save(filename, 't', 'y', 'par', 'x1');
+end
+
+
+%%% Plot
+if isempty(filename)
+ filename = 'saved_y2D_Maxwell_50'; % make sure it exists...
+end
+load(filename, 't', 'y', 'par', 'x1');
+video_on = true; % Record video: YES (true), NO (false)
+video_filename = [filename '.avi'];
+plot_solution(x1,y,t,par,video_on,video_filename);
+
+end
+
+%% Main function implementing the model
+function f = mechanochemical(y,yp,par)
+
+ %%% Parameter values
+ eta = 1; % viscosity
+ E = 1; % elasticity
+ D = 0.01; % diffusion
+ alpha = 0.05; % haptotaxis
+ r = 1; % proliferation
+ s = 10; % substrate elasticity
+ beta = 0.005; % long range traction
+ lambda = 0.5; % (cell traction) saturation coefficient
+ tau = 0.2; % cell traction
+ nu = 0.25; % Poisson ratio
+ Ep = E/(1+nu); % eq.(33)
+ nup = nu/(1-2*nu); % eq.(33)
+ % mu = nup*eta: % Simplifying assumption (A.4)
+
+ %%% Choose constitutive model - see eq.(31) and Table 3
+ switch par.model
+ case 'Kelvin-Voigt' % Kelvin Voigt - (A.2)
+ [a0,a1,b0,b1,c0,c1] = deal(1/eta,0,Ep/eta,1,Ep*nup/eta,nup);
+ case 'Maxwell' % Maxwell - (A.3)
+ [a0,a1,b0,b1,c0,c1] = deal(1/eta,1/Ep,0,1,0,nup);
+ otherwise
+ error('Unknown constitutive model')
+ end
+ %%% Reshape input vectors
+ [n,p,u1,u2] = vec2mats(y,par.K);
+ [np,pp,u1p,u2p] = vec2mats(yp,par.K);
+
+ %%% Equation for n
+ % Advection velocity at grid cell interfaces - eq.(S.22)
+ u1p_Avx1 = Avx1(u1p);
+ vx1 = alpha*Mx1_face(p,par)+u1p_Avx1;
+ u2p_Avx2 = Avx1(u2p')';
+ vx2 = alpha*Mx1_face(p',par)'+u2p_Avx2;
+ % fn(t,n,n') = 0 - eq.(S.21)
+ fn = np ...
+ - D*(Mxx1(n,par) + Mxx1(n',par)') ...
+ + MA1(n, vx1, par) + MA1(n', vx2', par)' ...
+ - r * (n.*(1-n));
+
+ %%% Equation for p
+ % fp(t,p,p') = 0 - eq.(S.23)
+ fp = pp ...
+ + MA1(p, u1p_Avx1, par) + MA1(p', u2p_Avx2', par)';
+
+ %%% Equation for u1 and u2
+ % Traction term - eq.(S.29) with (S.13) and (S.30)
+ pexp = 2;
+ fn1 = n./(1+lambda*n.^pexp); % Lambda_1
+ fn2 = ((1-(pexp-1)*lambda*n.^pexp) ./ ((1+lambda*n.^pexp).^2)); % Lambda_2
+ fp1 = p + beta*(Mxx1(p, par)+ Mxx1(p', par)'); % M_T2 P
+ fp2 = pp + beta*(Mxx1(pp, par)+ Mxx1(pp', par)'); % M_T2 P'
+ Tr = tau*(a0*fn1.*fp1 + a1*(fn2.*np.*fp1+fn1.*fp2));
+ % fu1(n,n',p,p',u1,u1',u2,u2') = 0 - eq.(S.27)
+ fu1 = (b1+c1)*Mxx1(u1p, par) + 0.5*b1*(Mxx1(u1p', par))' ...
+ +(0.5*b1+c1)* Mx1x2(u2p, par) ...
+ +(b0+c0)*Mxx1(u1, par) + 0.5*b0*(Mxx1(u1', par))' ...
+ +(0.5*b0+c0)* Mx1x2(u2, par) ...
+ + Mx1_center(Tr, par) ...
+ -a1*s*(p.*u1p + pp.*u1) - a0*s*p.*u1;
+ % fu2(n,n',p,p',u1,u1',u2,u2') = 0 - eq.(S.28)
+ fu2 = (b1+c1)*Mxx1(u2p', par)' + 0.5*b1*Mxx1(u2p, par) ...
+ +(0.5*b1+c1)* Mx1x2(u1p, par) ...
+ +(b0+c0)*Mxx1(u2', par)' + 0.5*b0*Mxx1(u2, par) ...
+ +(0.5*b0+c0)* Mx1x2(u1, par) ...
+ +Mx1_center(Tr', par)' ...
+ -a1*s*(p.*u2p + pp.*u2) - a0*s*p.*u2;
+
+ %%% Full system - eq.(S.15)
+ f = mats2vec(fn, fp, fu1, fu2);
+
+end
+
+%% Annexed functions
+
+%%% Vector to matrices
+function [n,p,u1,u2] = vec2mats(y,K)
+ K2 = K^2;
+ n = reshape(y(0*K2+1:1*K2), K, K);
+ p = reshape(y(1*K2+1:2*K2), K, K);
+ u1 = reshape(y(2*K2+1:3*K2), K, K);
+ u2 = reshape(y(3*K2+1:4*K2), K, K);
+end
+
+%%% Matrices to vector
+function y = mats2vec(n, p, u1, u2)
+ y = [n(:); p(:); u1(:); u2(:)];
+end
+
+%%% Compute advection at grid cell centers using first order upwinding
+%%% in the x1 direction with advective velocity given at grid cell
+%%% interfaces - def.(S.7) and (S.20)
+function fluxdiffx1 = MA1(y, vel, par)
+ % find pos/neg velocity entries
+ velPos_ind = (vel>0);
+ velNeg_ind = (~velPos_ind);
+ % compute flux accross cell interfaces using first order upwinding
+ % def.(S.8)-(S.9)
+ flux = NaN(size(y));
+ flux(velPos_ind) = y(velPos_ind).*vel(velPos_ind);
+ yshift = y([2:end 1],:);
+ flux(velNeg_ind) = yshift(velNeg_ind).*vel(velNeg_ind);
+ % add entries for flux along x1=0 by periodicity
+ flux = flux([end,1:end],:);
+ % compute flux difference per grid cell in x1 direction - def.(S.7) and (S.4)
+ fluxdiffx1 = (1/par.dx)*(flux(2:end,:)-flux(1:(end-1),:));
+end
+
+%%% Compute second order mixed derivative in grid cell centres
+%%% (second order approximation, central) - def.(S.18)
+function dx1x2 = Mx1x2(y, par)
+ persistent Mdx;
+ if size(Mdx,2) ~= size(y,1)
+ disp('Mx1x2 : resetting persistent matrix.');
+ c = (1/(2*par.dx))*sparse([0;-1; zeros(par.K-3,1); 1]);
+ Mdx = toeplitz(c,[c(1), c(end:-1:2)']);
+ end
+ dx1x2 = (Mdx*(Mdx*y)')';
+end
+
+%%% Compute first order derivative in x1 direction on grid cell interfaces
+%%% (second order approximation, central) - def.(S.16)
+function dx1 = Mx1_face(y, par)
+ persistent Mdx;
+ if size(Mdx,2) ~= size(y,1)
+ disp('Mx1_face : resetting persistent matrix.');
+ c = (1/par.dx)*sparse([-1; zeros(par.K-2,1); 1]);
+ Mdx = toeplitz(c,[c(1), c(end:-1:2)']);
+ end
+ dx1 = Mdx*y;
+end
+
+%%% Compute first order derivative in x1 direction on grid cell centres
+%%% (second order approximation, central) - def.(S.16)
+function dx1 = Mx1_center(y, par)
+ persistent Mdx;
+ if size(Mdx,2) ~= size(y,1)
+ disp('Mx1_center: resetting persistent matrix.');
+ c = (1/(2*par.dx))*sparse([0;-1; zeros(par.K-3,1); 1]);
+ Mdx = toeplitz(c,[c(1), c(end:-1:2)']);
+ end
+ dx1 = Mdx*y;
+end
+
+%%% Compute variable at grid cell interfaces in the x1 direction -
+%%% def.(S.3) and (S.19)
+function avx1 = Avx1(y)
+ avx1 = 0.5*(y +y([2:end,1],:));
+end
+
+%%% Compute second order derivative in x1 direction in grid cell centres
+%%% (second order approximation, central) - def.(S.2) and (S.17)
+function dxx1 = Mxx1(y, par)
+ persistent Mdxx;
+ if size(Mdxx,2) ~= size(y,1)
+ disp('Mxx1 : resetting persistent matrix.');
+ c = (1/(par.dx^2))*sparse([-2; 1; zeros(par.K-3,1); 1]);
+ Mdxx = toeplitz(c,[c(1), c(end:-1:2)']);
+ end
+ dxx1 = Mdxx*y;
+end
+
+%%% Plot solution
+function plot_solution(x1,y,t,par,video_on,video_filename)
+ if video_on % Initialise video
+ vid = VideoWriter(video_filename);
+ open(vid);
+ figure('Units','normalized','Position',[0 0 0.5 0.45])
+ end
+ for i=1:length(t)
+ clf
+ [N, P, U1, U2] = vec2mats(y(i,:),par.K);
+ n = [N(par.K,par.K) N(par.K,:)
+ N(:,par.K) N];
+ p = [P(par.K,par.K) P(par.K,:)
+ P(:,par.K) P];
+ u1 = [U1(par.K,par.K) U1(par.K,:)
+ U1(:,par.K) U1];
+ u2 = [U2(par.K,par.K) U2(par.K,:)
+ U2(:,par.K) U2];
+ subplot(2,2,1)
+ surf(x1,x1,n)
+ view(0,90)
+ shading flat
+ axis square
+ xlim([0,par.L])
+ ylim([0,par.L])
+ title('$n(t,x)$')
+ colormap(parula);
+ colorbar;
+ subplot(2,2,2)
+ surf(x1,x1,p)
+ view(0,90)
+ shading flat
+ axis square
+ xlim([0,par.L])
+ ylim([0,par.L])
+ title('$\rho(t,x)$')
+ colormap(parula);
+ colorbar;
+ subplot(2,2,3)
+ surf(x1,x1,u1)
+ view(0,90)
+ shading flat
+ axis square
+ xlim([0,par.L])
+ ylim([0,par.L])
+ title('$u_1(t,x)$')
+ colormap(parula);
+ colorbar;
+ subplot(2,2,4)
+ surf(x1,x1,u2)
+ view(0,90)
+ shading flat
+ axis square
+ xlim([0,par.L])
+ ylim([0,par.L])
+ title('$u_2(t,x)$')
+ colormap(parula);
+ colorbar;
+ a = axes;
+ t1 = title([par.model ' (t=',num2str(t(i)),')'],...
+ 'Position', [0.5, 1, 2]);
+ a.Visible = 'off';
+ t1.Visible = 'on';
+ drawnow
+ if video_on % Record video
+ frame = getframe(gcf);
+ size(frame.cdata);
+ writeVideo(vid,frame);
+ pause(0.1)
+ end
+ end
+ if video_on % Close video
+ close(vid)
+ end
+end
+
+function [SPDY,SPDYP] = Jpattern(K)
+ % init return matrices to empty sparse matrices
+ SPDY = sparse(4*K^2,4*K^2);
+ SPDYP = sparse(4*K^2,4*K^2);
+
+ % vector of first entry (start index) of n, p, u1, u2 in y and yp
+ si = 1+(0:3)*(K^2);
+ % vector of the "grid cell numbers-1" from 0 to K^2-1; this vector
+ % represents the order of the grid cells
+ self = (0:(K^2-1))';
+ % indices of all entries in y and yp of the four solution components
+ in=si(1)+self;
+ ip=si(2)+self;
+ iu1=si(3)+self;
+ iu2=si(4)+self;
+
+% % Example stencil: second order central difference of 2D Laplace
+% centre = [2,2]; % [row index, column index] of stencil centre
+% % stencil: visual mode, i.e. increasing row index is decreasing
+% % x2-direction and increasing column index is increasing x1-direction
+% stencil = [0 1 0
+% 1 1 1
+% 0 1 0];
+
+ % Here: for now generic full stencil of size 5x5
+ stencilsize = 5;
+ centre = [1, 1]*((stencilsize+1)/2);
+ stencil = ones(stencilsize); % square matrix of all 1
+
+ % rather simple, not optimal but sufficient choice:
+ % assume each equation depends on n, p, u1, u2 and their
+ % derivatives using the same stencil as defined above.
+ mat = SPDY;
+ mat = add(mat, in, in, stencil, centre, K);
+ mat = add(mat, in, ip, stencil, centre, K);
+ mat = add(mat, in, iu1, stencil, centre, K);
+ mat = add(mat, in, iu2, stencil, centre, K);
+ %
+ mat = add(mat, ip, in, stencil, centre, K);
+ mat = add(mat, ip, ip, stencil, centre, K);
+ mat = add(mat, ip, iu1, stencil, centre, K);
+ mat = add(mat, ip, iu2, stencil, centre, K);
+ %
+ mat = add(mat, iu1, in, stencil, centre, K);
+ mat = add(mat, iu1, ip, stencil, centre, K);
+ mat = add(mat, iu1, iu1, stencil, centre, K);
+ mat = add(mat, iu1, iu2, stencil, centre, K);
+ %
+ mat = add(mat, iu2, in, stencil, centre, K);
+ mat = add(mat, iu2, ip, stencil, centre, K);
+ mat = add(mat, iu2, iu1, stencil, centre, K);
+ mat = add(mat, iu2, iu2, stencil, centre, K);
+ SPDY=mat;
+
+ mat = SPDYP;
+ mat = add(mat, in, in, stencil, centre, K);
+ mat = add(mat, in, ip, stencil, centre, K);
+ mat = add(mat, in, iu1, stencil, centre, K);
+ mat = add(mat, in, iu2, stencil, centre, K);
+ %
+ mat = add(mat, ip, in, stencil, centre, K);
+ mat = add(mat, ip, ip, stencil, centre, K);
+ mat = add(mat, ip, iu1, stencil, centre, K);
+ mat = add(mat, ip, iu2, stencil, centre, K);
+ %
+ mat = add(mat, iu1, in, stencil, centre, K);
+ mat = add(mat, iu1, ip, stencil, centre, K);
+ mat = add(mat, iu1, iu1, stencil, centre, K);
+ mat = add(mat, iu1, iu2, stencil, centre, K);
+ %
+ mat = add(mat, iu2, in, stencil, centre, K);
+ mat = add(mat, iu2, ip, stencil, centre, K);
+ mat = add(mat, iu2, iu1, stencil, centre, K);
+ mat = add(mat, iu2, iu2, stencil, centre, K);
+ SPDYP=mat;
+
+ % final tuning
+ SPDYP(SPDYP>1) = 1;
+ SPDY(SPDY>1) = 1;
+
+ function M = add(M, c, r, stencil, centre, K)
+ toR = @(ind,count)(reshape(circshift(reshape(ind,K,K),-count,1),1,K^2));
+ toU = @(ind,count)(reshape(circshift(reshape(ind,K,K),-count,2),1,K^2));
+ [x2dir,x1dir]= find(stencil);
+ x1dir = x1dir-centre(2);
+ x2dir = -(x2dir-centre(1));
+ %[x1dir,x2dir]
+ for ii=1:length(x1dir)
+ rtmp = toR(r , x1dir(ii));
+ rtmp = toU(rtmp, x2dir(ii));
+ M = M + sparse(c,rtmp,ones(size(c)),4*K^2,4*K^2);
+ end
+ end
+end