Shape function gradients with 3D boudary elements

[jfieldstone]

Hello all,

I am developing a 3D boundary element program where I have only surface elements (4-node quads and 3-node tris), but that are oriented in 3D space. Likewise, I want to compute x,y,z gradients on each of these faces.

I have a nodal solution (phi), and I want to compute dphi/dx, dphi/dy, dphi/dz.

I have successfully done this for quads, but cannot seem to figure it out for triangles.

For the quad-4 with linear shape functions:
N = [1/4*(1+xi)*(1+eta) 1/4(1+xi)*(1-eta) 1/4(1-xi)*(1-eta) 1/4(1-xi)*(1-eta)]
I can compute the differential operators in x,y,z (dNdx, dNdy, and dNdz) by simply computing 2 tangential vectors (dxdxi, dxdeta, dydxi, dydeta, dzdxi, dzdeta):
e1 = dNdxi*[ex,ey,ez]
and
e2 = dNdeta*[ex,ey,ez]

Taking the cross product of these gives me my surface gradient:
e3 = cross(e1,e2)

Then using these 3 vectors as my jacobian:
Jac = [e1; e2; e3];

I can invert my jacobian to compute dxidx,dxidy,dxidz, detadx,detady,detadz, and finally post multiply with dNdxi and dNdeta again to compute dNdx, dNdy, dNdz:

dNdx = Jac(1,1)^-1*dNdxi + Jac(1,2)^-1*dNdeta
dNdy = Jac(2,1)^-1*dNdxi + Jac(2,2)^-1*dNdeta
dNdz = Jac(3,1)^-1*dNdxi + Jac(3,2)^-1*dNdeta

These work well, and I get the correct gradients.

But now for the 3-node triangle, it's not so straight forward. I have formulated the TRI-3 with 3 local coordinates (xi, eta, and zeta), where my shape functions are just:
N = [xi eta zeta],
and so:
dN = I (identity).

How can I compute dNdx, dNdy, and dNdz for the linear 3-node triangle?

Any help is appreciated, and thank you in advance.

 

[aerostress82]

Being not so familiar with Gauss Quadrature in the last 10 years, I found this which also has some directions for a triangle. Not sure if it is explained there but it feels like you are entering a world of pain Good luck!

http://math2.uncc.edu/~shaodeng/TEACHING/math5172/Lectures/Lect_15.PDF

Please let me know if you find the solution later, I would be very interested as well..

Spaceship!!


Aerospace Engineer, M.Sc. / Aircraft Stress Engineer with 7 years of experience
(United States)

 

[jfieldstone]

hi aerostress82, the issue I am talking about is not Guass Quadrature. Although, if I were to integrate my gradient (which I am talking about) over the element, I would use Gauss Quadrature.

The issue I am referring to is computing the discrete exterior gradient on the element face (in 3D, 3 components). Since it's oriented in 3D space, standard 2D formulations do not apply, and since it's a boundary element, standard 3D volume formulations also do not apply.

Sort of tricky, and cannot seem to get the correct derivation, and cannot find anything online about it; though I know somebody has at least attempted it before.

 

[aerostress82]

Thanks for the clarification jfieldstone. I'm not familiar with boundary element method but the quad-4 equation looked very similar to finite element - that's why I sent that link.
Maybe if you happen to have some research assistant friend(s) in university, you could send him/her the sciencedirect links related to your case? (after checking them from sciencedirect)

Other than that, hope someone can help you on this. Good luck with it!

Spaceship!!
Aerospace Engineer, M.Sc. / Aircraft Stress Engineer