Surface flux through a single finite element

[PSantiago]

Hi,

I'm facing a real-life problem and I don't what specific mathematics topic it's related to.

Homework Statement

I know the value of the components of a vector field in three points of space and I have to find the flux of this vector field through the surface defined by those points.

Lets say that the vector field is the following

Q = u(x,y,z)i + v(x,y,z)j

The vertical component of the vector field is omitted because I know that the surface is vertical.

I know the value of the vector's components (u₁,v₁), (u₂,v₂) and (u₃,v₃) in the coordinates (x₁,y₁,z₁), (x₂,y₂,z₂) and (x₃,y₃,z₃), which may be seen as the vertices of a triangular surface.

Linear variations may be assumed for the vector component functions.

Homework Equations

That's what I'm looking for.

The Attempt at a Solution

I've searched a lot for finite element representation of surface fluxes, but I've found nothing what clearly looked similar to my problem.

 

[rude man]

Obviously you first need to find Q(x,y,z). Since you know Q at 3 points only, and you say you can "assume linearity", then you could write ∂Q/∂x = ax and ∂Q/∂y = by. ∂Q/∂z = 0. But now you have two equations with 3 boundary conditions for x, and same for y. So you'd have to assume a second-order expression for those two derivatives, I guess.

Once you knew Q, the flux of course is just a problem in analytic geometry.

 

[LawrenceC]

Here is what I have done in the past with linear triangular elements. You have the value of the field at the nodes of the element. Represent the value of the field over the element as a simple set of 3 equations over the local coordinate system of the element.

The field is represented as

F1 = a0 + a1*x1 + a2*y1

F2 = a0 + a1*x2 + a2*y2

F3 = a0 + a1*x3 + a2*y3


You have 3 equations and 3 unknowns. Solve for a0, a1, a2. Evaluate the flux at the centroid and apply it as a flux over the entire element uniformly. F1, F2, F3 are the nodal values of the field. the x's and y's are the local coordinates of the corresponding nodes.