Programming Finite Element Method

Question

I am trying to teach myself how about Finite Element Methods.

All of my code is adapted from the following link pages 16-20 http://homepages.cae.wisc.edu/~suresh/ME964Website/M964Notes/Notes/introfem.pdf

I am programming along in Matlab to perform a finite element analysis on a single 8 node cube element. I have defined the xi,eta,zeta local axes (we can think about this as x, y, z for now), so I get the following shape functions:

%%shape functions

zeta = 0:.01:1;
eta = 0:.01:1;
xi = 0:.01:1;

N1 = 1/8*(1-xi).*(1-eta).*(1-zeta);
N2 = 1/8*(1+xi).*(1-eta).*(1-zeta);
N3 = 1/8*(1+xi).*(1+eta).*(1-zeta);
N4 = 1/8*(1-xi).*(1+eta).*(1-zeta);
N5 = 1/8*(1-xi).*(1-eta).*(1+zeta);
N6 = 1/8*(1+xi).*(1-eta).*(1+zeta);
N7 = 1/8*(1+xi).*(1+eta).*(1+zeta);
N8 = 1/8*(1-xi).*(1+eta).*(1+zeta);

The [N] Matrix is to be arranged like this according to the text I am reading:

%N Matrix
N= [N1 0 0 N2 0 0 N3 0 0 N4 0 0 N5 0 0 N6 0 0 N7 0 0 N8 0 0;
    0 N1 0 0 N2 0 0 N3 0 0 N4 0 0 N5 0 0 N6 0 0 N7 0 0 N8 0;
    0 0 N1 0 0 N2 0 0 N3 0 0 N4 0 0 N5 0 0 N6 0 0 N7 0 0 N8];

To find the [B] matrix i have to use the following [D] matrix:

%%Del Matrix for node i
%[  d/dx   0     0
%    0    d/dy   0
%    0     0    d/dz        . . .
%   d/dy  d/dx   0
%    0    d/dz  d/dy
%   d/dz   0    d/dx   ]

which is an operator to go on [N]. (B=DN)

Later on, as the text shows, I will be making calculations involving integrals of this [B] matrix over the volume of this element.

So, my question is, how can I store these polynomial shape functions in a matrix, operate on them with differentiation, and then integrate them numerically. I can tell with the way I have this set up right now, that it wont work because I have defined the functions as a vector over an interval [0,1] and then storing these vectors in the [N] matrix. Then using diff() function to differentiate appropriately to find the [B] matrix. But since the matrix elements of [B] are now vectors over an interval [0,1] I think that is going to cause problems. How would you guys go about these calculations described in the textbook I posted above?

Answer 1

after you have the shape functions you need to substitute it in the stiffness matrix, the stiffness matrix should be 24x24 as you have 24 degrees of freedom. to solve you need to build a linear system (Ax=b), the right hand side is based on the PDE you are solving and you have to include neuman boundary conditions in the right hand side plus the source term. In python for 2d element (4 DOF) will be like:

    def shapefxncoef (Valxy):
    #creating a temporary metrix to store zeros and get the size of the shape
    #function matrix.
    n_temp = np.zeros((4,4))
    #filling the values of the matrix with a loop.
    for i in range(4):
        #the values used in the matrix are from the Valxy x and y components.
        xi = Valxy [0, i];
        yi = Valxy [1, i];
        n_temp[i, 0] = 1;
        n_temp[i, 1] = xi;
        n_temp[i, 2] = yi;
        n_temp[i, 3] = xi*yi;
        #this gives an identity matrix and the stiffness matric can be derived 
        #if we take the inverse.
    n = np.linalg.inv(n_temp);
    return n;

def N (Valxy, x, y):
    n = shapefxncoef (Valxy);
    res = n[0, :] + n[1, :]*x + n[2, :]*y + n[3, :]*x*y;
    return res;

def Be (Valxy, x, y):
    res = np.zeros ((2,4));
    res_temp = shapefxncoef (Valxy);
    for i in range (4):
        res_tempi = res_temp[:, i];
        dNix = res_tempi[1] + res_tempi[3]*y;
        dNiy = res_tempi[2] + res_tempi[3]*x;
        res[0, i] = dNix;
        res[1, i] = dNiy;
    return res;

def Ke (Valxy, conduct):
    a = lambda x, y: conduct * np.dot ((Be(Valxy, x, y)).T, Be(Valxy, x, y));
    k = intr.integrateOnQuadrangle (Valxy.T, a, np.zeros((4,4)));
    return k;

Answer 2

You should check page 24 about how to map from a parametric domain ([0,1]^) to the physical domain.

Answer 3

Although I think you can do as you said, using symbolic. I think symbolic calculation in Matlab is very time-consuming.

I would go for derivate N manually and store as dN, and use it when need it.

Regards,

German

Answer 4

Solved my problem using anonymous functions and storing the polynomials in a symbolic matrix. example:

syms xi eta zeta
N1= ... %type out in terms of xi eta and zeta
.
.
.
dN1dXi = diff(N1,xi) %symbolic differentiation with respect to xi

can also perform symbolic integration when needed:

intN1 = int(N1,xi,lowerLimit,upperLimit) %symbolic integration with respect to xi

and when ready to substitute in actual values to evaluate the symbolic functions:

subs(N1,{xi,eta,zeta},{value1,value2,value3})