is there a way to increase the speed of this code

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Volkan Kandemir
Volkan Kandemir on 13 Jan 2014
Answered: Walter Roberson on 14 Jan 2014
The following code is taken from the book "MATLAB Guide to Finite Elements" and it takes too much time to run this code and I am calling this function very often. How can I speed up this code.
function w =
LinearBrickElementStiffness(E,NU,x1,y1,z1,x2,y2,z2,x3,y3,z3,x4,y4,z4,
x5,y5,z5,x6,y6,z6,x7,y7,z7,x8,y8,z8)
% LinearBrickElementStiffness This function returns the element
% stiffness matrix for a linear brick
% element with modulus of elasticity
% E, Poisson’s ratio NU, coordinates of
% node 1 (x1,y1,z1), coordinates
% of node 2 (x2,y2,z2), coordinates of
% node 3 (x3,y3,z3), coordinates of
% node 4 (x4,y4,z4), coordinates of
% node 5 (x5,y5,z5), coordinates of
% node 6 (x6,y6,z6), coordinates of
% node 7 (x7,y7,z7), and coordinates
% of node 8 (x8,y8,z8).
% The size of the element
% stiffness matrix is 24 x 24.
syms s t u;
N1 = (1-s)*(1-t)*(1+u)/8;
N2 = (1-s)*(1-t)*(1-u)/8;
N3 = (1-s)*(1+t)*(1-u)/8;
N4 = (1-s)*(1+t)*(1+u)/8;
N5 = (1+s)*(1-t)*(1+u)/8;
N6 = (1+s)*(1-t)*(1-u)/8;
N7 = (1+s)*(1+t)*(1-u)/8;
N8 = (1+s)*(1+t)*(1+u)/8;
x = N1*x1 + N2*x2 + N3*x3 + N4*x4 + N5*x5 + N6*x6 + N7*x7 + N8*x8;
y = N1*y1 + N2*y2 + N3*y3 + N4*y4 + N5*y5 + N6*y6 + N7*y7 + N8*y8;
z = N1*z1 + N2*z2 + N3*z3 + N4*z4 + N5*z5 + N6*z6 + N7*z7 + N8*z8;
xs = diff(x,s);
xt = diff(x,t);
xu = diff(x,u);
ys = diff(y,s);
yt = diff(y,t);
yu = diff(y,u);
zs = diff(z,s);
zt = diff(z,t);
zu = diff(z,u);
J = xs*(yt*zu - zt*yu) - ys*(xt*zu - zt*xu) + zs*(xt*yu - yt*xu);
N1s = diff(N1,s);
N2s = diff(N2,s);
N3s = diff(N3,s);
N4s = diff(N4,s);
N5s = diff(N5,s);
N6s = diff(N6,s);
N7s = diff(N7,s);
N8s = diff(N8,s);
N1t = diff(N1,t);
N2t = diff(N2,t);
N3t = diff(N3,t);
N4t = diff(N4,t);
N5t = diff(N5,t);
N6t = diff(N6,t);
N7t = diff(N7,t);
N8t = diff(N8,t);
N1u = diff(N1,u);
N2u = diff(N2,u);
N3u = diff(N3,u);
N4u = diff(N4,u);
N5u = diff(N5,u);
N6u = diff(N6,u);
N7u = diff(N7,u);
N8u = diff(N8,u);
N1x = N1s*(yt*zu - zt*yu) - ys*(N1t*zu - zt*N1u) + zs*(N1t*yu - yt*N1u);
N2x = N2s*(yt*zu - zt*yu) - ys*(N2t*zu - zt*N2u) + zs*(N2t*yu - yt*N2u);
N3x = N3s*(yt*zu - zt*yu) - ys*(N3t*zu - zt*N3u) + zs*(N3t*yu - yt*N3u);
N4x = N4s*(yt*zu - zt*yu) - ys*(N4t*zu - zt*N4u) + zs*(N4t*yu - yt*N4u);
N5x = N5s*(yt*zu - zt*yu) - ys*(N5t*zu - zt*N5u) + zs*(N5t*yu - yt*N5u);
N6x = N6s*(yt*zu - zt*yu) - ys*(N6t*zu - zt*N6u) + zs*(N6t*yu - yt*N6u);
N7x = N7s*(yt*zu - zt*yu) - ys*(N7t*zu - zt*N7u) + zs*(N7t*yu - yt*N7u);
N8x = N8s*(yt*zu - zt*yu) - ys*(N8t*zu - zt*N8u) + zs*(N8t*yu - yt*N8u);
N1y = xs*(N1t*zu - zt*N1u) - N1s*(xt*zu - zt*xu) + zs*(xt*N1u - N1t*xu);
N2y = xs*(N2t*zu - zt*N2u) - N2s*(xt*zu - zt*xu) + zs*(xt*N2u - N2t*xu);
N3y = xs*(N3t*zu - zt*N3u) - N3s*(xt*zu - zt*xu) + zs*(xt*N3u - N3t*xu);
N4y = xs*(N4t*zu - zt*N4u) - N4s*(xt*zu - zt*xu) + zs*(xt*N4u - N4t*xu);
N5y = xs*(N5t*zu - zt*N5u) - N5s*(xt*zu - zt*xu) + zs*(xt*N5u - N5t*xu);
N6y = xs*(N6t*zu - zt*N6u) - N6s*(xt*zu - zt*xu) + zs*(xt*N6u - N6t*xu);
N7y = xs*(N7t*zu - zt*N7u) - N7s*(xt*zu - zt*xu) + zs*(xt*N7u - N7t*xu);
N8y = xs*(N8t*zu - zt*N8u) - N8s*(xt*zu - zt*xu) + zs*(xt*N8u - N8t*xu);
N1z = xs*(yt*N1u - N1t*yu) - ys*(xt*N1u - N1t*xu) + N1s*(xt*yu - yt*xu);
N2z = xs*(yt*N2u - N2t*yu) - ys*(xt*N2u - N2t*xu) + N2s*(xt*yu - yt*xu);
N3z = xs*(yt*N3u - N3t*yu) - ys*(xt*N3u - N3t*xu) + N3s*(xt*yu - yt*xu);
N4z = xs*(yt*N4u - N4t*yu) - ys*(xt*N4u - N4t*xu) + N4s*(xt*yu - yt*xu);
N5z = xs*(yt*N5u - N5t*yu) - ys*(xt*N5u - N5t*xu) + N5s*(xt*yu - yt*xu);
N6z = xs*(yt*N6u - N6t*yu) - ys*(xt*N6u - N6t*xu) + N6s*(xt*yu - yt*xu);
N7z = xs*(yt*N7u - N7t*yu) - ys*(xt*N7u - N7t*xu) + N7s*(xt*yu - yt*xu);
N8z = xs*(yt*N8u - N8t*yu) - ys*(xt*N8u - N8t*xu) + N8s*(xt*yu - yt*xu);
% Next, the B matrix is calculated explicitly as follows:
B = [N1x N2x N3x N4x N5x N6x N7x N8x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 N1y N2y N3y N4y N5y N6y N7y N8y 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 N1z N2z N3z N4z N5z N6z N7z N8z ;
N1y N2y N3y N4y N5y N6y N7y N8y N1x N2x N3x N4x N5x N6x N7x N8x 0 0 0 0 0 0 0 0 ;
0 0 0 0 0 0 0 0 N1z N2z N3z N4z N5z N6z N7z N8z N1y N2y N3y N4y N5y N6y N7y N8y ;
N1z N2z N3z N4z N5z N6z N7z N8z 0 0 0 0 0 0 0 0 N1x N2x N3x N4x N5x N6x N7x N8x];
Bnew = simplify(B);
Jnew = simplify(J);
D = (E/((1+NU)*(1-2*NU)))*[1-NU NU NU 0 0 0 ;
NU 1-NU NU 0 0 0 ;
NU NU 1- NU 0 0 0 ;
0 0 0 (1-2*NU)/2 0 0 ;
0 0 0 0 (1- 2*NU)/2 0 ;
0 0 0 0 0 (1- 2*NU)/2];
BD = transpose(Bnew)*D*Bnew/Jnew;
r = int(int(int(BD, u, -1, 1), t, -1, 1), s, -1, 1);
w = double(r);

Answers (1)

Walter Roberson
Walter Roberson on 14 Jan 2014
Change the final line to
w = r;
Then once in your routine that loops call
syms E NU x1y1 z1 x2 y2 z2 x3 y3 z3 x4 y4 z4 x5 y5 z5 x6 y6 z6 x7 y7 z7 x8 y8 z8
wsym = LinearBrickElementStiffness(E, NU, x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4, x5, y5, z5, x6, y6, z6, x7, y7, z7, x8, y8, z8);
wfun = matlabFunction(wsym, 'vars', [E, NU, x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4, x5, y5, z5, x6, y6, z6, x7, y7, z7, x8, y8, z8] );
Now wfun is the handle of a function to which you can pass all of those variables in numeric form, and you will get back a numeric answer, without it doing all of the symbolic work each time. It would do the symbolic work when you called the function passing in symbolic variables, differentiation and all, and it would do the integration in symbolic form at that time, giving a specific formula as the result.
This approach would not work as well if any of the variables appeared as a exponent, but everything you show is linear or polynomial.

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