Matlab-Programs/symMinimization.m

+
+function [G, angle, Type, kappa] = symMinimization(print_Input,print_statPoint,print_Output,make_FunctionPlot, InputPath) %(Q_hom,B_eff)
+
+
+syms v1 v2 q1 q2 q3 q12 b1 b2 b3
+
+
+% -------- Options ----------
+% % print_Input = false;  %effective quantities
+% print_Input = true;
+% % print_statPoint = false;
+% print_statPoint = true;
+% print_Minimizer = true;
+% % make_FunctionPlot = false;
+% make_FunctionPlot = true;
+
+print_Uniqueness = false;         
+check_StationaryPoints = false;   % could be added to Input-Parameters..
+compare_with_Classification = false;  %maybe added later --- 
+
+
+%fprintf('Functions to be minimized:')
+f_plus(v1,v2,q1,q2,q3,q12,b1,b2,b3) = q1*v1^4 + q2*v2^4+2*q3*v1^2*v2^2-2*(q1*b1*v1^2+ q2*b2*v2^2+sqrt(2)*q3*b3*v1*v2)...
+                                        + q12*(v1^2*v2^2-b2*v1^2-b1*v2^2+b1*b2);
+f_minus(v1,v2,q1,q2,q3,q12,b1,b2,b3) = q1*v1^4 + q2*v2^4+2*q3*v1^2*v2^2+2*(q1*b1*v1^2+ q2*b2*v2^2+sqrt(2)*q3*b3*v1*v2)...
+                                        + q12*(v1^2*v2^2+b2*v1^2+b1*v2^2+b1*b2);
+
+
+% ---- Fix parameters
+
+
+% Epsilon used:
+epsilon = 1e-8;
+
+
+if ~exist('InputPath','var')
+     % third parameter does not exist, so default it to something
+      absPath = "/home/klaus/Desktop/DUNE/dune-microstructure/outputs";
+ end
+
+
+% 1. Import effective quantities from CellSolver-Code:
+%read as sparse Matrix...
+try %absolutePath
+    Qmat = spconvert(load(absPath + '' + "/QMatrix.txt"));
+    Bmat = spconvert(load(absPath + '' + "/BMatrix.txt"));
+%     fprintf('Use absolute Path')
+catch ME  % use relativePath
+    Qmat = spconvert(load('../outputs/QMatrix.txt'));
+    Bmat = spconvert(load('../outputs/BMatrix.txt'));
+%     fprintf('Use relative Path')
+end
+%convert to full matrix... 
+Qmat = full(Qmat);
+Bmat = full(Bmat);
+
+
+
+
+% --- TODO CHECK: assert if Q is orthotropic ???  check ifq13=q31=q23=q32= 0 ?
+
+
+if print_Input
+    fprintf('effective quadratic form:')
+    Qmat
+    fprintf('effective prestrain')
+    Bmat
+    
+    % check if Q is (close to..) symmetric
+    % könnte Anti-symmetric part berechnen und schauen dass dieser klein?
+    % Test: issymmetric(Qmat) does not work for float matrices?
+    % symmetric part 0.5*(Qmat+Qmat')
+    % anti-symmetric part 0.5*(Qmat-Qmat')
+    if norm(0.5*(Qmat-Qmat'),'fro') < 1e-8
+        fprintf('Qmat (close to) symmetric \n')
+        norm(0.5*(Qmat-Qmat'),'fro') % TEST
+    else
+        fprintf('Qmat not symmetric \n')
+    end
+    
+    
+    % Check if B_eff is diagonal this is equivalent to b3 == 0
+    if abs(Bmat(3)) < 1e-8
+        fprintf('B_eff is diagonal (b3 == 0) \n')
+    else
+        fprintf('B_eff is  NOT diagonal (b3 != 0)  \n')
+    end
+end
+
+
+
+
+% CAST VALUES TO SYM FIRST? This is done anyway..
+% % Substitute effective quantitites      
+f_plus = subs(f_plus,{q1, q2, q3, q12, b1, b2, b3}, {Qmat(1,1), Qmat(2,2), Qmat(3,3), Qmat(1,2), ...
+                                                     Bmat(1), Bmat(2), Bmat(3)});
+
+f_minus = subs(f_minus,{q1, q2, q3, q12, b1, b2, b3}, {Qmat(1,1), Qmat(2,2), Qmat(3,3), Qmat(1,2), ...
+                                                     Bmat(1), Bmat(2), Bmat(3)});
+
+
+% Compute the Gradients 
+df_plusx = diff(f_plus,v1);
+df_plusy = diff(f_plus,v2);
+df_minusx = diff(f_minus,v1);
+df_minusy = diff(f_minus,v2);
+
+% Setup Equations Grad(f) = 0 
+eq1 = df_plusx == 0;
+eq2 = df_plusy == 0;
+eqns_plus = [eq1, eq2];
+eq3 = df_minusx == 0;
+eq4 = df_minusy == 0;
+eqns_minus = [eq3, eq4];
+
+% -------  Symbolically Solve Equations: 
+% More robust (works even for values b_3 ~ 1e-08 ): 
+S_plus = solve(eqns_plus,v1,v2,'MaxDegree' , 5);  
+S_minus = solve(eqns_minus,v1,v2,'MaxDegree' , 5);
+
+A_plus = S_plus.v1;
+B_plus = S_plus.v2;
+A_minus = S_minus.v1;
+B_minus = S_minus.v2;
+
+if check_StationaryPoints
+    %---------- TEST if Grad(f) = 0 ---------------------
+    fprintf('Testing equation grad(f) = 0  with stationary points')
+    for i = 1:size(A_plus,1)
+        fprintf('Testing %d.point (f_plus): ',i )
+        [ double(subs(subs(df_plusx,v1,A_plus(i)),v2,B_plus(i))), double(subs(subs(df_plusy,v1,A_plus(i)),v2,B_plus(i))) ]
+    end
+    for i = 1:size(A_minus,1)
+        fprintf('Testing %d.point (f_minus): ',i )
+        [double(subs(subs(df_minusx,v1,A_minus(i)),v2,B_minus(i))), double(subs(subs(df_minusy,v1,A_minus(i)),v2,B_minus(i)))]
+    end
+    % ------------------------------------
+end
+
+% --- Extract only Real-Solutions
+% fprintf('real stationary points of f_plus:')
+tmp1 = A_plus(imag(double(A_plus))==0 & imag(double(B_plus)) == 0);
+tmp2 = B_plus(imag(double(A_plus))==0 & imag(double(B_plus)) == 0);
+A_plus = tmp1;
+B_plus = tmp2;
+SP_plus = [A_plus,B_plus];
+
+% fprintf('real stationary points of f_minus:')
+tmp1 = A_minus(imag(double(A_minus))==0 & imag(double(B_minus)) == 0);
+tmp2 = B_minus(imag(double(A_minus))==0 & imag(double(B_minus)) == 0);
+A_minus = tmp1;
+B_minus = tmp2;
+SP_minus = [A_minus,B_minus];
+
+% TODO one should use f_plus.subs(A_plus..) to compute function value symbolically?
+% in the end only the stationaryPoints are used.. should be ok to compare function values numerically
+
+% Determine global Minimizer from stationary points:
+% fprintf('function values at stationary points (f_plus):')
+T_plus = arrayfun(@(v1,v2) double(f_plus(v1,v2,q1,q2,q3,q12,b1,b2,b3)),A_plus,B_plus,'UniformOutput', false);
+T_plus = cell2mat(T_plus);
+
+%Test: use Substitution
+% subs(f_plus,{v1, v2}, {A_plus,B_plus})
+
+% fprintf('function values at stationary points (f_minus):')
+T_minus = arrayfun(@(v1,v2) double(f_minus(v1,v2,q1,q2,q3,q12,b1,b2,b3)),A_minus,B_minus,'UniformOutput', false);
+T_minus = cell2mat(T_minus);
+
+%Test: use Substitution
+% T_minus = subs(f_minus,{v1, v2}, {A_minus,B_minus})
+% double(T_minus)
+
+if print_statPoint
+    fprintf('real stationary points of f_plus: ')
+%     SP_Plus     %alternative: output as symbolic (can be unwieldy)
+    double(SP_plus)
+    fprintf('real stationary points of f_minus:')
+%     SP_Minus     %alternative: output as symbolic (can be unwieldy)
+    double(SP_minus)
+    fprintf('function values at stationary points (f_plus):')
+    T_plus
+    fprintf('function values at stationary points (f_minus):')
+    T_minus
+end
+
+% --- Find Stationary Point(s) with minimum Value 
+[Min_plus,MinIdx_plus] = min(T_plus, [], 'all', 'linear');    %find one min...
+[Min_minus,MinIdx_minus] = min(T_minus, [], 'all', 'linear');
+% [Min_minus,MinIdx_minus] = min(T_minus)                     % works with symbolic too
+
+% Compare Minimizers of f_plus & f_minus
+[globalMinimizerValue,GlobalIdx] = min([Min_plus,Min_minus]); 
+
+if GlobalIdx == 1     %Min_plus   % i.e. Global Minimizer is given by f_plus
+    GlobalMinimizer = SP_plus(MinIdx_plus,:);
+    sign = 1.0;
+elseif GlobalIdx == 2  %Min_minus % i.e. Global Minimizer is given by f_minus
+    GlobalMinimizer = SP_minus(MinIdx_minus,:);
+    sign = -1.0; 
+end
+
+% ------ Check if there are more SP with the same value...
+ MinIndices_minus = find(T_minus(:) == globalMinimizerValue);    % Find indices of All Minima
+ MinIndices_plus  = find(T_plus(:) == globalMinimizerValue);    % Find indices of All Minima                   
+ numMinSP_minus = size(MinIndices_minus,1);   % One of these is always >= 2 due to the structure of the roots..
+ numMinSP_plus  = size(MinIndices_plus,1);
+    
+%     AllMinSP_minus = SP_minus(MinIndices_minus,:)
+%     AllMinSP_minus = double(SP_minus(MinIndices_minus,:))
+%     AllMin = T_minus(MinIndices)  %bereits klar dass diese selben funktionswert haben..
+      
+    Minimizer = sign*(GlobalMinimizer'*GlobalMinimizer);  % global minimizing Matrix G*
+    MinimizerCount = 1;
+    % different Stationary Points might correspond to the same minimizing
+    % Matrix G*... check this: 
+    
+    % Compare only with other StationaryPoints/Minimizers
+    % remove Index of Minimizer
+    if GlobalIdx == 1     
+        MinIndices_plus = MinIndices_plus(MinIndices_plus~=MinIdx_plus);
+    elseif GlobalIdx == 2  
+        MinIndices_minus = MinIndices_minus(MinIndices_minus~=MinIdx_minus);
+    end
+    
+    MinIndices = cat(1,MinIndices_plus,MinIndices_minus);  %[Minimizers-Indices f_plus, Minimizer-Indices f_minus]
+    
+    for i = 1:(numMinSP_minus+numMinSP_plus-1)  % -1: dont count Minimizer itself.. 
+        idx = MinIndices(i);   
+        
+        if i > numMinSP_plus
+            SP = SP_minus(idx,:);
+        else
+            SP = SP_plus(idx,:); 
+        end
+        
+%         SP_value = T_minus(idx) % not needed? 
+        Matrix = sign*(SP'*SP);
+
+        if norm(double(Matrix-Minimizer),'fro') < 1e-8   %check is this sufficient here?
+%             fprintf('both StationaryPoints correspond to the same(Matrix-)Minimizer')
+        else
+%             fprintf('StationaryPoint corresponds to a different (Matrix-)Minimizer')
+            MinimizerCount = MinimizerCount + 1;
+        end
+    end
+% ----------------------------------------------------------------------------------------------------------------
+
+
+
+% Output Uniqueness of Minimizers:
+if print_Uniqueness
+    if MinimizerCount == 1
+        fprintf('Unique Minimzier')
+    elseif MinimizerCount == 2
+        fprintf('Two Minimziers')
+    else
+        fprintf('1-Parameter family of Minimziers')
+    end
+end
+
+
+
+
+
+% --- determine the angle of the Minimizer
+% a1 = Minimizer(1,1)
+% a2 = Minimizer(2,2)
+a1 = double(Minimizer(1,1));
+a2 = double(Minimizer(2,2));
+
+% compute the angle <(e,e_1) where Minimizer = kappa* (e (x) e)
+e = [sqrt((a1/(a1+a2))), sqrt((a2/(a1+a2)))];    % always positive under sqrt here .. basically takes absolute value here
+angle = atan2(e(2), e(1)); 
+
+% compute curvature kappa
+kappa = (a1 + a2);
+
+% % CHeck off diagonal entries:
+% sqrt(a1*a2);
+% double(Minimizer);
+
+G = double(Minimizer);
+
+
+
+% --- "Classification" / Determine the TYPE of Minimizer by using 
+% the number of solutions (Uniqueness?)
+% the angle (axial- or non-axial Minimizer)
+
+% (Alternative compute det[GlobalMinimizer' e1']  where e1 = [1 0] ?)
+
+% Check Uniqueness -- Options: unique/twoMinimizers/1-ParameterFamily
+if MinimizerCount == 1
+%     fprintf('Unique Minimzier')
+    % Check if Minimizer is axial or non-axial:
+    if (abs(angle-pi/2) < 1e-8 || abs(angle) < 1e-8) % axial Minimizer
+     Type = 3;
+    else % non-axial Minimizer   
+     Type = 1;
+    end
+elseif MinimizerCount == 2
+%     fprintf('Two Minimziers')
+    % Check if Minimizer is axial or non-axial:
+    if (abs(angle-pi/2) < 1e-8 || abs(angle) < 1e-8) % axial Minimizer
+        Type = 3;
+    else % non-axial Minimizer   
+        fprintf('ERROR: Two non-axial Minimizers cannot happen!')
+    end
+else
+%     fprintf('1-Parameter family of Minimziers')
+    % Check if Minimizer is axial or non-axial:
+    if (abs(angle-pi/2) < 1e-8 || abs(angle) < 1e-8) % axial Minimizer
+%         fprintf('ERROR: axial Minimizers cannot happen for 1-Parameter Family!')
+    else % non-axial Minimizer   
+        Type = 2;
+    end
+end
+% ------------------------------------------------------------------------------------------------------
+
+
+
+
+if print_Output
+    fprintf(' --------- Output symMinimization --------')
+    fprintf('Global Minimizer v: (%d,%d) \n', GlobalMinimizer(1),GlobalMinimizer(2) )
+    fprintf('Global Minimizer Value f(v): %d \n', sym(globalMinimizerValue) ) %cast to symbolic
+    %     fprintf('Global Minimizer Value : %d', globalMinimizerValue )
+    
+    fprintf('Global Minimizer G: \n' )
+    G 
+    fprintf("Angle = %d \n", angle)
+    fprintf("Curvature = %d \n", kappa)
+    fprintf("Type = %i \n", Type)
+    fprintf(' --------- -------------------- --------')
+end
+
+
+
+if make_FunctionPlot  
+    fsurf(@(x,y) f_plus(x,y,q1,q2,q3,q12,b1,b2,b3)) % Plot functions
+    hold on
+    plot3(double(A_plus),double(B_plus),T_plus,'g*')
+    %Plot GlobalMinimizer:
+    hold on
+    plot3(double(GlobalMinimizer(1)),double(GlobalMinimizer(2)),globalMinimizerValue, 'o', 'Color','c')
+    % view(90,0)
+    % view(2)
+
+    figure
+    fsurf(@(x,y) f_minus(x,y,q1,q2,q3,q12,b1,b2,b3))
+    hold on
+    plot3(double(A_minus),double(B_minus),T_minus,'g*')
+    hold on
+    plot3(double(GlobalMinimizer(1)), double(GlobalMinimizer(2)),globalMinimizerValue, 'o', 'Color','c')
+end
+
+
+return 
+
+
+% Write symbolic solution to txt-File in Latex format
+% fileID = fopen('txt.txt','w');
+% fprintf(fileID,'%s' , latex(S_plus.v1));
+% fclose(fileID);
+
+
+
+
+
+
+

Matlab-Programs/symSolveSystem.m

+
+
+syms q1 q2 q3 q12 b1 b2  a1 a2 
+
+% eqn1 = q1 * a1 + (q3 + (q12/2)) *a2 == -2*q1*b1 - q12*b2
+% 
+% eqn2 = (q3 + (q12/2)) * a1 + q2 *a2 == -2*q2*b2 - q12*b1
+
+
+eqn1 = q1 * a1 + (q3 + (q12/2)) *a2 == q1*b1 + (q12*b2/2)
+
+eqn2 = (q3 + (q12/2)) * a1 + q2 *a2 == q2*b2 + (q12*b1/2)
+
+
+
+[A,B] = equationsToMatrix([eqn1, eqn2], [a1,a2])
+
+X = linsolve(A,B)
+
+
+% check special case
+fprintf('special case q12 = 0 ') 
+subs(X,q12,0)
\ No newline at end of file

Paraview-Scripts/ParaViewSettings.txt

+
+Colormap: Cool-to-Warm
+
+Representation: Point-Gaussian
+Opacity: 0.01
+PointSize: 2
+Gaussian-radius: 0.0075
+Shader-Preset: Plain-Circle