Question

Write a C or Matlab code for E11(1, 6), consider the point G = (2, 7). Compute the multiples of G from 2G through 13G

Answer 1

% function abc = cowin_coefficients_cowins_method(E, A)

%

% Compute Cowin's coefficients a, b, and c as defined in his 1985 paper.

% This method uses a quasi-direct solution procedure proposed by himself.

%

% E is a 6x6 symmetric matrix representing an orthogonal stiffness tensor

% in sqrt2 notation. The coefficients of E should correspond to

% the principal axis system spanned by the eigenvectors of the fabric tensor.

% A is a 3x1 vector containing the eigenvalues of the fabric tensor.

%

% Return: abc is a 9x1 vector containing coefficients [a; b; c].

% This file is part of MMTensor. % %

% %

% Redistribution and use in source and binary forms, with or without %

% modification, are permitted provided that the following conditions are %

% met: %

% * Redistributions of source code must retain the above copyright %

% notice, this list of conditions and the following disclaimer. %

% * Redistributions in binary form must reproduce the above copyright %

% notice, this list of conditions and the following disclaimer in %

% the documentation and/or other materials provided with the %

% distribution. %

% * Neither the name of the Katholieke Universiteit Leuven nor the %

% names of its contributors may be used to endorse or promote %

% products derived from this software without specific prior written %

% permission.

function abc = cowin_coefficients_cowins_method(E, A)

%% Initialize fabric tensor A

% Keep only the diagonal, the other coefficients should be zero.

if isequal(size(A),[3,3])

A = diag(A);

end

% Make sure A is a 3x1 vector

A = reshape(A, 3, 1);

% Precompute the square of A

A2 = A.^2;

%% Initialize stiffness tensor E

E_diag = diag(E(1:3,1:3));

E_off = [E(2,3); E(1,3); E(1,2)]; % Off-diagonal terms

E_shear = 0.5*diag(E(4:6,4:6)); % Division by 2 due to sqrt2 notation.

%% Initialize the coefficient matrices

Z = [ ones(3,1) 2*A 2*A2];

B = repmat(A2, 1, 3) .* [ ones(3,1) 2*A A2];

C = [ 1 A(2)+A(3) A2(2)+A2(3);

1 A(1)+A(3) A2(1)+A2(3);

1 A(1)+A(2) A2(1)+A2(2) ];

  

D = [ A(2)*A(3); A(1)*A(3); A(1)*A(2) ];

D = [ D C(:,2).*D D.*D ];

%% Solve the system according to Cowin's proposal

C_inv = inv(C);

F = B - Z*C_inv*D;

F_inv = inv(F);

c = C_inv * E_shear;

b = F_inv*E_diag - F_inv*Z*C_inv*[E_off+2*E_shear];

a = C_inv * (E_off - D*b);

%% Put everything together

abc = [a;b;c];

end