I just need to know what does i, j represent in K(i,j) other than representing the row and column of the Sparse Matrix K?

for elx = 1:nelx
    for ely = 1:nely
        n1 = (nely+1)*(elx-1)+ely; 
        n2 = (nely+1)* elx   +ely;
        edof = [2*n1-1; 2*n1; 2*n2-1; 2*n2; 2*n2+1; 2*n2+2; 2*n1+1; 2*n1+2];
        K(edof,edof) = K(edof,edof) + x(ely,elx)^penal*KE;
    end
end

As we are feeding the K here the whole dof matrix containing the degree of freedom of all the node of an element at once. How is it creating? Like for 1st element, the node numbers will be 1, 5, 6 and 2 in a cloclwise way from top left to right, where n1 and n2 represt 1 and 5, and 2 and 6 respectively?

[Voss]

Construct a similar/simpler case on the command line or in another script, and see what happens

K = magic(6)

K = 6×6

35 1 6 26 19 24 3 32 7 21 23 25 31 9 2 22 27 20 8 28 33 17 10 15 30 5 34 12 14 16 4 36 29 13 18 11

n1 = 1;
n2 = 2;
edof = [2*n1-1; 2*n1; 2*n2-1; 2*n2; 2*n2+1; 2*n2+2; 2*n1+1; 2*n1+2]

edof = 8×1

1 2 3 4 5 6 3 4

K(edof,edof)

ans = 8×8

35 1 6 26 19 24 6 26 3 32 7 21 23 25 7 21 31 9 2 22 27 20 2 22 8 28 33 17 10 15 33 17 30 5 34 12 14 16 34 12 4 36 29 13 18 11 29 13 31 9 2 22 27 20 2 22 8 28 33 17 10 15 33 17

[Rohan Nathiya]

I get this, but I am more concerned about what

i, j in K(i,j)

actually represent in this particular 99 line code of Topology Optimization, like F(m,n) in this very code gives force at a particular node and m is the dof and n represents the node number.

Thank You

Answer 1

[Shivam]

Hi,

From the code snippet provided, I understand that you want to know the meaning of indices 'i' and 'j' while indexing into the global stiffness sparse matrix, K.

In the Finite Element Method context, the global stiffness matrix K is a sparse matrix that stores the relationship between the applied forces and resulting nodal displacement in the system. The entry K(i, j) represents the force at the degree of freedom 'i' due to displacement at the degree of freedom 'j,' while all other displacements are zero.

I hope it helps.