XFEM: crack tip function derivatives

Hi,

I am using a X-FEM Matlab code found here http://people.civil.gla.ac.uk/~bordas/xfemMatlab.html"]http://people.civil.gla.ac.uk/~bordas/xfemMatlab.html

In branch.m/branch_node.m (both functions of radius r and angle theta) crack tip fields are defined...and questions arise.

% Functions



f(1) = r2 * st2 ;

f(2) = r2 * ct2;

f(3) = r2 * st2 * ct;

f(4) = r2 * ct2 * ct;

-these functions are not the same defined e.g. in (Fleming,1997) or (Sukumar,2000) etc...why?

 their derivatives are then considered:

(here for function f(3))

% third function

dPhidr  = fac * st2 * (2*st*st-ct) ;

dPhidt  = fac * ct* cos(3*theta/2.);

dfdx(3) = dPhidr * drdx + dPhidt * dtdx  ;

dfdy(3) = dPhidr * drdy + dPhidt * dtdy  ;

-these derivatives..do not seem correct...(e.g. dPhidt should simply give fac * st2 * ct). But if I change these values, crack intensity factors are not correctly computed in the Matlab code. Do you have any idea? (I miss something for sure, but...)

In the following I give some notations: 

r2 = sqrt(r);

fac  = 0.5/r2 ;

st2  = sin(theta/2.);

ct2  = cos(theta/2.);

st   = sin(theta);

ct   = cos(theta);



drdx = cos(alpha);

drdy = sin(alpha);

dtdx = -sin(alpha);

dtdy = cos(alpha);

thanks for helping!