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

README

+Preparing the Sources
+=========================
+
+Additional to the software mentioned in README you'll need the
+following programs installed on your system:
+
+  cmake >= 3.1
+
+Getting started
+---------------
+
+If these preliminaries are met, you should run
+
+  dunecontrol all
+
+which will find all installed dune modules as well as all dune modules
+(not installed) which sources reside in a subdirectory of the current
+directory. Note that if dune is not installed properly you will either
+have to add the directory where the dunecontrol script resides (probably
+./dune-common/bin) to your path or specify the relative path of the script.
+
+Most probably you'll have to provide additional information to dunecontrol
+(e. g. compilers, configure options) and/or make options.
+
+The most convenient way is to use options files in this case. The files
+define four variables:
+
+CMAKE_FLAGS      flags passed to cmake (during configure)
+
+An example options file might look like this:
+
+#use this options to configure and make if no other options are given
+CMAKE_FLAGS=" \
+-DCMAKE_CXX_COMPILER=g++-5 \
+-DCMAKE_CXX_FLAGS='-Wall -pedantic' \
+-DCMAKE_INSTALL_PREFIX=/install/path" #Force g++-5 and set compiler flags
+
+If you save this information into example.opts you can pass the opts file to
+dunecontrol via the --opts option, e. g.
+
+  dunecontrol --opts=example.opts all
+
+More info
+---------
+
+See
+
+     dunecontrol --help
+
+for further options.
+
+
+The full build system is described in the dune-common/doc/buildsystem (Git version) or under share/doc/dune-common/buildsystem if you installed DUNE!

cmake/modules/CMakeLists.txt

+set(modules "DuneMicrostructureMacros.cmake")
+
+install(FILES ${modules} DESTINATION ${DUNE_INSTALL_MODULEDIR})

cmake/modules/DuneMicrostructureMacros.cmake

+# File for module specific CMake tests.

config.h.cmake

+/* begin dune-microstructure
+   put the definitions for config.h specific to
+   your project here. Everything above will be
+   overwritten
+*/
+
+/* begin private */
+/* Name of package */
+#define PACKAGE "@DUNE_MOD_NAME@"
+
+/* Define to the address where bug reports for this package should be sent. */
+#define PACKAGE_BUGREPORT "@DUNE_MAINTAINER@"
+
+/* Define to the full name of this package. */
+#define PACKAGE_NAME "@DUNE_MOD_NAME@"
+
+/* Define to the full name and version of this package. */
+#define PACKAGE_STRING "@DUNE_MOD_NAME@ @DUNE_MOD_VERSION@"
+
+/* Define to the one symbol short name of this package. */
+#define PACKAGE_TARNAME "@DUNE_MOD_NAME@"
+
+/* Define to the home page for this package. */
+#define PACKAGE_URL "@DUNE_MOD_URL@"
+
+/* Define to the version of this package. */
+#define PACKAGE_VERSION "@DUNE_MOD_VERSION@"
+
+/* end private */
+
+/* Define to the version of dune-microstructure */
+#define DUNE_MICROSTRUCTURE_VERSION "@DUNE_MICROSTRUCTURE_VERSION@"
+
+/* Define to the major version of dune-microstructure */
+#define DUNE_MICROSTRUCTURE_VERSION_MAJOR @DUNE_MICROSTRUCTURE_VERSION_MAJOR@
+
+/* Define to the minor version of dune-microstructure */
+#define DUNE_MICROSTRUCTURE_VERSION_MINOR @DUNE_MICROSTRUCTURE_VERSION_MINOR@
+
+/* Define to the revision of dune-microstructure */
+#define DUNE_MICROSTRUCTURE_VERSION_REVISION @DUNE_MICROSTRUCTURE_VERSION_REVISION@
+
+/* end dune-microstructure
+   Everything below here will be overwritten
+*/

doc/CMakeLists.txt

+add_subdirectory("doxygen")

doc/doxygen/CMakeLists.txt

+# shortcut for creating the Doxyfile.in and Doxyfile
+add_doxygen_target()

doc/doxygen/Doxylocal

+# This file contains local changes to the doxygen configuration
+# please use '+=' to add files/directories to the lists
+
+# The INPUT tag can be used to specify the files and/or directories that contain
+# documented source files. You may enter file names like "myfile.cpp" or
+# directories like "/usr/src/myproject". Separate the files or directories
+# with spaces.
+
+INPUT                 += @top_srcdir@/dune/
+# see e.g. dune-grid for the examples of mainpage and modules
+# INPUT                 += @srcdir@/mainpage \
+#                          @srcdir@/modules
+
+# The EXCLUDE tag can be used to specify files and/or directories that should
+# be excluded from the INPUT source files. This way you can easily exclude a
+# subdirectory from a directory tree whose root is specified with the INPUT tag.
+
+# EXCLUDE               += @top_srcdir@/dune/microstructure/test
+
+# The EXAMPLE_PATH tag can be used to specify one or more files or
+# directories that contain example code fragments that are included (see
+# the \include command).
+
+# EXAMPLE_PATH          += @top_srcdir@/src
+
+# The IMAGE_PATH tag can be used to specify one or more files or
+# directories that contain image that are included in the documentation (see
+# the \image command).
+
+# IMAGE_PATH            += @top_srcdir@/dune/microstructure/pics

dune-microstructure.pc.in

+prefix=@prefix@
+exec_prefix=@exec_prefix@
+libdir=@libdir@
+includedir=@includedir@
+CXX=@CXX@
+CC=@CC@
+DEPENDENCIES=@REQUIRES@
+
+Name: @PACKAGE_NAME@
+Version: @VERSION@
+Description: dune-microstructure module
+URL: http://dune-project.org/
+Requires: dune-common dune-istl dune-geometry dune-localfunctions dune-uggrid dune-grid dune-typetree dune-functions dune-solvers
+Libs: -L${libdir}
+Cflags: -I${includedir}

dune.module

+################################
+# Dune module information file #
+################################
+
+# Name of the module
+Module: dune-microstructure
+Version: 1.0
+Maintainer: klaus.boehnlein@tu-dresden.de
+# Required build dependencies
+Depends: dune-common dune-istl dune-geometry dune-localfunctions dune-uggrid dune-grid dune-typetree dune-functions 
+# Optional build dependencies
+#Suggests:

dune/CMakeLists.txt

+add_subdirectory(microstructure)

dune/microstructure/CMakeLists.txt

+#install headers
+install(FILES microstructure.hh DESTINATION ${CMAKE_INSTALL_INCLUDEDIR}/dune/microstructure)

dune/microstructure/matrix_operations.hh

+#ifndef DUNE_MICROSTRUCTURE_MATRIXOPERATIONS_HH
+#define DUNE_MICROSTRUCTURE_MATRIXOPERATIONS_HH
+
+namespace MatrixOperations {
+
+	using namespace Dune;
+
+	using MatrixRT = FieldMatrix< double, 3, 3>;
+	using VectorRT = FieldVector< double, 3>;	
+
+	using std::sin;
+	using std::cos;
+
+	static MatrixRT Id (){ 
+		MatrixRT Id(0);
+	    for (int i=0;i<3;i++)
+	        Id[i][i]=1.0;
+	    return Id;
+	}
+	
+	
+    static double norm(VectorRT v){
+		return sqrt(pow(v[0],2) + pow(v[1],2) + pow(v[2],2));
+	}
+
+	static double norm(MatrixRT M){
+		return sqrt(
+			  pow(M[0][0],2) + pow(M[0][1],2) + pow(M[0][2],2)
+			+ pow(M[1][0],2) + pow(M[1][1],2) + pow(M[1][2],2)
+			+ pow(M[2][0],2) + pow(M[2][1],2) + pow(M[2][2],2));
+	}
+	
+
+	static MatrixRT sym (MatrixRT M) { // 1/2 (M^T + M)
+    	MatrixRT ret(0);
+    	for (int i = 0; i< 3; i++)
+    		for (int j = 0; j< 3; j++)
+    			ret[i][j] = 0.5 * (M[i][j] + M[j][i]);
+    	return ret;
+	}
+
+	static VectorRT crossProduct (VectorRT v, VectorRT w) { // v otimes w
+    	return {v[1]*w[2] - v[2]*w[1], -1*(v[0]*w[2] - v[2]*w[0]), v[0]*w[1] - v[1]*w[0]};
+	}
+
+	static MatrixRT rankoneTensorproduct (VectorRT v, VectorRT w) { // v otimes w
+    	return 
+    	{{v[0]*w[0], v[1]*w[0], v[2]*w[0]}, 
+	     {v[0]*w[1], v[1]*w[1], v[2]*w[1]}, 
+	     {v[0]*w[2], v[1]*w[2], v[2]*w[2]}};
+	}
+
+	static MatrixRT nematicLiquidCrystal (double p, VectorRT n){ //B = 1/6*p*Id + 1/2*p*(n otimes n)
+		MatrixRT B(0);
+	    for (int i=0;i<3;i++)
+	        B[i][i]=p/6.0;
+	    MatrixRT n_ot_n = rankoneTensorproduct(n,n);
+	    n_ot_n*=p/2.0;
+	    B += n_ot_n;
+	    return B;
+	}
+
+	static MatrixRT biotStrainApprox (VectorRT U, VectorRT k, VectorRT e_cs){ //E_h = (U +  k x e_cs, 0, 0)
+	    VectorRT k_x_ecs = crossProduct(k, e_cs);
+	    VectorRT U_plus_k_x_ecs = U + k_x_ecs;
+	    VectorRT e_1 = {1, 0, 0};
+	    return rankoneTensorproduct(U_plus_k_x_ecs, e_1);
+	}
+
+	static MatrixRT crossSectionDirectionScaling(double w, MatrixRT M){
+		return {M[0], w*M[1], w*M[2]};
+	}
+
+	static double trace (MatrixRT M){ 
+	    return M[0][0]+ M[1][1] + M[2][2];
+	}
+
+	static double scalarProduct (MatrixRT M1, MatrixRT M2){ 
+		double sum = 0;
+		for (int i=0; i<3; i++)
+		    for (int j=0; j<3; j++)
+		    	sum += M1[i][j] * M2[i][j];
+	    return sum;
+	}
+
+// 	static double linearizedStVenantKirchhoffDensity(double mu, double lambda, MatrixRT E1, MatrixRT E2){    // ?? Check with Robert
+// 		E1= sym(E1);
+// 		E2 = sym(E2);
+// 		double t1 = scalarProduct(E1,E2);
+// 		t1 *= 2* mu;
+// 		double t2 = trace(E1)*trace(E2); 
+// 		t2 *= lambda;
+// 		return t1 + t2;
+// 	}
+	
+    static double linearizedStVenantKirchhoffDensity(double mu, double lambda, MatrixRT E1, MatrixRT E2)  // CHANGED
+    {  
+        auto t1 = 2.0 * mu * sym(E1) + lambda * trace(sym(E1)) * Id();
+        return scalarProduct(t1,E2);
+
+	}
+
+    // --- Generalization: Define Quadratic QuadraticForm
+    
+    static double QuadraticForm(const double mu, const double lambda, const MatrixRT M){
+        
+        auto tmp1 = sym(M);
+        double tmp2 = norm(tmp1);
+        return lambda*std::pow(trace(M),2) + 2*mu*pow( tmp2 ,2);
+//         return lambda*std::pow(trace(M),2) + 2*mu*pow( norm( sym(M) ),2);
+    }
+
+    
+	static double generalizedDensity(const double mu, const double lambda, MatrixRT F, MatrixRT G){
+     /// Write this whole File as a Class that uses lambda,mu as members ? 
+        
+     // Define L via Polarization-Identity from QuadratifForm
+     // <LF,G> := (1/2)*(Q(F+G) - Q(F) - Q(G) ) 
+        return (1.0/2.0)*(QuadraticForm(mu,lambda,F+G) - QuadraticForm(mu,lambda,F) - QuadraticForm(mu,lambda,G) );
+    }
+
+	static MatrixRT matrixSqrt(MatrixRT M){
+		std::cout << "matrixSqrt not implemented!!!" << std::endl;//implement this
+		return M;
+	}
+
+	static double lameMu(double E, double nu){
+		return 0.5 * 1.0/(1.0 + nu) * E;
+	}
+
+	static double lameLambda(double E, double nu){
+		return nu/(1.0-2.0*nu) * 1.0/(1.0+nu) * E;
+	}
+
+	static bool isInRotatedPlane(double phi, double x1, double x2){
+		return cos(phi)*x1 + sin(phi)*x2 > 0;
+	}
+
+
+
+	/*
+	template<double phi>
+	static bool isInRotatedPlane(double x1, double x2){
+		return cos(phi)*x1 + sin(phi)*x2 > 0;
+	}*/
+
+}
+
+
+
+// OPTIONAL H1-Seminorm ... 
+
+/*
+template<class VectorCT>
+double semi_H1_vectorScalarProduct(VectorCT& coeffVector1, VectorCT& coeffVector2)
+{
+
+double ret(0);
+auto localView = basis_.localView();
+
+for (const auto& e : elements(basis_.gridView()))
+  	{
+	    localView.bind(e);
+
+	    auto geometry = e.geometry();
+
+	    const auto& localFiniteElement = localView.tree().child(0).finiteElement();
+	    const auto nSf = localFiniteElement.localBasis().size();
+
+	    int order = 2*(dim*localFiniteElement.localBasis().order()-1);
+	    const QuadratureRule<double, dim>& quad = QuadratureRules<double, dim>::rule(e.type(), order);
+
+	   	for (const auto& quadPoint : quad)
+    	{
+    		double elementScp(0);
+
+    		auto pos = quadPoint.position();
+    		const auto jacobian = geometry.jacobianInverseTransposed(pos);
+    		const double integrationElement = geometry.integrationElement(pos);
+
+    		std::vector<FieldMatrix<double,1,dim> > referenceGradients;
+    		localFiniteElement.localBasis().evaluateJacobian(pos, referenceGradients);
+
+    		std::vector< VectorRT > gradients(referenceGradients.size());
+    		for (size_t i=0; i< gradients.size(); i++)
+      		jacobian.mv(referenceGradients[i][0], gradients[i]);
+
+    		for (size_t i=0; i < nSf; i++)
+    			for (size_t j=0; j < nSf; j++)
+    			for (size_t k=0; k < dimworld; k++)
+    				for (size_t l=0; l < dimworld; l++)
+          		{
+          			MatrixRT defGradient1(0);
+          			defGradient1[k] = gradients[i];
+
+          			MatrixRT defGradient2(0);
+          			defGradient2[l] = gradients[j];
+
+		          	size_t localIdx1 = localView.tree().child(k).localIndex(i);
+		          	size_t globalIdx1 = localView.index(localIdx1);
+
+		          	size_t localIdx2 = localView.tree().child(l).localIndex(j);
+		          	size_t globalIdx2 = localView.index(localIdx2);
+
+        			double scp(0);
+        			for (int ii=0; ii<dimworld; ii++)
+            		for (int jj=0; jj<dimworld; jj++)
+        						scp += coeffVector1[globalIdx1] * defGradient1[ii][jj] * coeffVector2[globalIdx2] * defGradient2[ii][jj];
+
+        				elementScp += scp;
+        			}
+
+            ret += elementScp * quadPoint.weight() * integrationElement;
+      } //qp
+  	} //e
+	  return ret;
+}*/
+
+
+
+
+
+
+/*
+	class BiotStrainApprox
+	{
+
+		// E_h = h^{-1} (R^T F_h - Id)
+		//
+		//E_h = (U +  K e_cs, 0, 0)
+		//
+
+		Func2Tensor E_a = [] (const Domain& z) //extra Klasse
+	  	{
+			return biotStrainApprox({1, 0, 0}, {0, 0, 0}, {0, z[1], z[2]}); //MatrixRT{{1.0, 0.0, 0.0}, {0.0, 0.0, 0.0}, {0.0, 0.0, 0.0}}; 
+	  	};
+
+	  	// twist in x2-x3 plane E_K1 = sym (e1 x (0,x2,x3)^T ot e1)
+	    Func2Tensor E_K1 = [] (const Domain& z) 
+	    {
+	    	return biotStrainApprox({0, 0, 0}, {1, 0, 0}, {0, z[1], z[2]}); //MatrixRT{{0.0,-0.5*z[2], 0.5*z[1]}, {-0.5*z[2], 0.0 , 0.0 }, {0.5*z[1],0.0,0.0}}; 
+	  	};
+
+	    //  bend strain in x1-x2 plane	E_K2 = sym (e2 x (0,x2,x3)^T ot e1)
+	  	Func2Tensor E_K2 = [] (const Domain& z) 
+	  	{
+	  		return biotStrainApprox({0, 0, 0}, {0, 1, 0}, {0, z[1], z[2]}); //MatrixRT{{z[2], 0.0, 0.0}, {0.0, 0.0, 0.0}, {0.0, 0.0, 0}}; 
+	  	};
+
+	  	// bend strain in x1-x3 plane	E_K3 = sym (e3 x (0,x2,x3)^T ot e1)
+	  	Func2Tensor E_K3 = [] (const Domain& z) 
+	  	{
+	  		return biotStrainApprox({0, 0, 0}, {0, 0, 1}, {0, z[1], z[2]}); //MatrixRT{{-z[1], 0.0, 0.0}, {0.0, 0.0, 0.0}, {0.0, 0.0, 0.0}}; 
+	  	};
+	};
+*/
+
+#endif