Differentiating Matrices

I have a 2 by 2 matrix and i need to differentiate each term in this matrix by each term in another 2 by 2 matrix so that i end up with a 4 by 4 result. Eventually i will be doing the same thing with 20 by 5 matrices. Is there any way of doing this? Can anybody help please?

[bym]

what have you got so far?

[Haast]

sorry what do you mean? You want me to paste the code?

[Walter Roberson]

Pasting the code would be a good start. Commenting the code would help after that.

[Haast]

clc

clear

%% Defining variables

%symbolic variables

syms c11 c12 c21 c22 x y

%array

c_ij = [c11 c12; c21 c22];

%dimensions

L = 20;

W = 20;

T = 0.125;

%unit loads

Nx = 1;

Ny = 1;

Nxy = 1;

%material properties

E = 72.4*10^9;

nu = 0.3;

D = (E*T^3)/(12*(1-nu.^2));

G = E/(2*(1+nu));

%% Defining and differentiating Omega

%Omega

w = c11*sin((pi*x)/L)*sin((pi*y)/W) + c12*sin((pi*x)/L)*sin((2*pi*y)/W) + c21*sin((2*pi*x)/L)*sin((pi*y)/W) + c22*sin((2*pi*x)/L)*sin((2*pi*y)/W);

%derivatives

dw_x = diff(w, x);

dw_y = diff(w, y);

dw_xy = diff(w, x, y);

%second derivatives

dw2_x = diff(dw_x, x);

dw2_y = diff(dw_y, y);

dw2_xy = diff(dw_xy, x, y);

%% FINDING KMN and PMN

F = ((dw2_x + dw2_y)^2)-2*(1-nu)*((dw2_x*dw2_y)-(dw2_xy)^2);

KMN = int(F,x,0,L);

KMN2 = D*int(KMN,y,0,W);

S = (Nx*dw_x^2+Ny*dw_y^2+2*Nxy*dw_xy^2);

PMN = int(S,0,L);

PMN2 = int(PMN,0,W);

%% Differentiation wrt cij

KMNF = diff(KMN2, c_ij);

this works up until i try and do the KMNF :(

Answer 1

[Walter Roberson]

Is this symbolic or numeric differentiation ? If it is symbolic, is the second matrix containing just one symbol per entry or does it contain expressions? Differentiating with respect to an expression is not easy.

Answer 2

[Haast]

it is symbolic. Actually i realised there was an error in my code and i am trying to differentiate one expression with respect to a matrix. I haven't typed it up here because it is a very long expression but it is essentially an expression in terms of c11 c12 c21 and c22 and i want to differentiate it with respect to each of these and because i am going to eventually be doing the same with many c values i thought the neatest way to set it out would be as a matrix!

[Walter Roberson]

At the MuPad level (not in MATLAB directly):

exprvars := indets(YourExpression) minus Type::ConstantIdents:

partialdiffs := map((thisvar,expression) -> diff(expression,thisvar), exprvars, YourExpression);

Then partialdiffs will be the partial differentials and exprvars will be the variables used for the partials.

You should be able to turn this into something you can feval(symengine,...) without great difficulty.

[Haast]

I am sorry I am very much a beginner with MATLAB and have never worked with anything to do with MuPad before. Am I just able to create a new notebook file with Mupad and enter these expressions into it??

[Walter Roberson]

You could do that, but there are alternatives. For example, at the MATLAB level, you might be able to use

diffs_list = simple(subs('map(proc(thisvar,expression) diff(expression,thisvar) end_proc, indets(YourExpression) minus Type::ConstantIdents, YourExpression)', 'YourExpression', your_symbolic_expression));

Unfortunately I do not have the symbolic toolbox myself, so I am not certain it is possible to activate the internal toolkit map() operation in this manner.