Store tensor history into multidimensional array

Dear all,

I read the documentatuin on Multidimensional Arrays and I haven't been able to find a solution to my problem.

I have a tensor history like this:

s11 = [xx1, xx2, xx3];
s22 = [yy1, yy2, yy3];
s33 = [zz1, zz2, zz3];
s12 = [xy1, xy2, xy3];
s13 = [xz1, xz2, xz3];
s23 = [yz1, yz2, yz3];

I want to store these tensor component histories in a single multidimensional array TENSOR, such that each slice of the array is a complete stress tensor at a certain point in time, e.g.;

>>TENSOR(:, :, 1.0)

xx1 xy1 xz1
xy1 yy1 yz1
xz1 yz1 zz1

>>TENSOR(:, :, 2.0)

xx2 xy2 xz2
xy2 yy2 yz2
xz2 yz2 zz2

(and so on)

I know how to create a multidimensional array in individual slices. For example, I could simply define the first slice then use

TENSOR(:, :, n) = [s11(n), s12(n), s13(n); s12(n), s22(n), s23(n); s13(n), s23(n), s33(n)];

To add the next slide. However, I would like to define the array in a single assignment, since the whole point of the exercise is to avoid inefficient loops.

I would greatly appreciate any advice!

Best regards,

Louis

[fsgeek]

Andrei's solution below works, although I had to make a couple of tweaks to get the behaviour just right so that it works in general for any 3x3xL multidimensional array:

%{
    Construct a 3x3xL multidimensional array of the
    stress tensor history, where L is the history
    length
%}
% Direct stress components
normals = [Sxx; Syy; Szz]';

% Shear stress components
shears = [Txy; Txz; Tyz]';

% Assign normal stress component histories to diagonal terms
diagonals = repmat(eye([3.0, 3.0]), 1.0, 1.0, L);
diagonals(diagonals > 0.0) = normals';

% Assign shear stress component histories to non-diagonal terms
nonDiagonals = repmat(tril(ones(3.0), -1.0), 1.0, 1.0, L);
nonDiagonals(nonDiagonals > 0.0) = shears';

% Combine diagonal and non-diagonal terms
multidimensionalStressTensor = diagonals + nonDiagonals + 
permute(nonDiagonals, [2.0, 1.0, 3.0]);

%{
    Get the Eigenvalues for each frame of the tensor data. This code 
    was written by Bruno Luong and can be found on the exchange (File 
    ID: #27680)
%}
eigenvalues = eig3(multidimensionalStressTensor);

%{
    The principal stresses are the maximum, median and
    minimum values of the Eigenvector at each tensor
    frame
%}
s1 = max(eigenvalues);
s2 = median(eigenvalues);
s3 = min(eigenvalues);

Initial tests indicate that this method is approximately 4 times faster than a FOR loop with 2D matrices.

Answer 1

[Andrei Bobrov]

sxx = reshape(1:9,3,[]);    % sxx = [s11;s22;s33];
sxyz = reshape(11:19,3,[]); % sxyz = [s12;s13;s23]

s = size(sxx);
D = repmat(eye(s),1,1,s(2));
D(D>0) = sxx;

ND = repmat(tril(ones(s),-1),1,1,s(2));
ND(ND>0) = sxyz;

tenzor1 = D + ND + permute(ND,[2,1,3]);

[fsgeek]

Thanks Andrei. This works as long as I make the following change to your code:

D(D>0) = sxx';
ND(ND>0) = sxyz';

Otherwise you get tensor histories being assigned to incorrect tensor locations. With the transpose modification it achieves exactly what I wanted. Thanks!

L

Answer 2

[Daniel kiracofe]

As far as I am aware, it is not possible to do what you want in matlab. There is no such thing as a 3-dimensional array literal, which is what you want. It is possible in some programming languages, such as C++, but not matlab. The only way I can think of do it without loops is to create a 1D array and then reshape it. e.g:

X = [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27];
y = reshape(X, [3 3 3])

which would give you this output:

y =

ans(:,:,1) =

             1           4           7
             2           5           8
             3           6           9

ans(:,:,2) =

            10          13          16
            11          14          17
            12          15          18

ans(:,:,3) =

            19          22          25
            20          23          26
            21          24          27

Depending on what you are trying to do, you may find this to be useful. I've not used it, and the documentation on their website is sparse, but it is a toolbox for tensor manipulation in matlab which may helpful

http://www.sandia.gov/~tgkolda/TensorToolbox/index-2.6.html

[fsgeek]

I Daniel,

I think it's possible with the solution from Andrei. I'll check out the TensorToolbox though, because it sounds like something which would be very useful to me.

Thanks for the help!

L