I am aproximating a radius function with quadratic isoparametric line elements, like this:
d1 = x1*0.5*x*(x-1)+x2*(x+1)*(x-1)+x3*0.5*x*(x+1)
d2 = y1*0.5*x*(x-1)+y2*(x+1)*(x-1)+y3*0.5*x*(x+1)
where x1,y1... -> are nodes from a three noded quadratic element.
By locating a source point I can create a radius function, like this:
r(x) = sqrt( (d1-xsp)^2 + (d2-ysp)^2)
where xsp,ysp... -> are nodes from a located source point.
Obviously I need also the jacobian of this transformation:
d1j = x1*(x-0.5) + x2*(-2*x) + x3*(x+0.5)
d2j = y1*(x-0.5) + y2*(-2*x) + y3*(x+0.5)
JAC = sqrt(d1j^2 + d2j^2)
The question:
What would be the integral or what numerical method should one use to solve the following integral If the source point is located within the quadractic isoparametric element:
Int N2*JAC/(r(x)*e^r(x)) from -1 to 1
with:
N2 = (x+1)*(x-1)
xsp = x2
ysp = y2
The integral becomes singular since the radius goes to zero. Also I think this might be related to the exponential integral but i cant apply it since the radius is a polynomial function. I have tried many things but the standard numeric integration does not work and also mathematica is failing.
Any guess of what kind of integral this is ? or what one should try to solve it?
thanks in advance and sorry for the long question.