Finite Element boundary normal vector

Question

I have a finite element and the meshs coordinates can be described using isoparametric shape functions as:

$x(\xi, \eta) = \sum _i N_i(\xi, \eta)x_i$, $y(\xi, \eta) = \sum_i N_i(\xi, \eta)y_i$

I need to evaluate the outward boundary normal vectors (not the surface normal, but the edges normals, i.e., the vector is contained in the elements plane) of a given element using the isoparametric shape functions $N_i(\xi, \eta)$.

Any idea how can I perform this for any given shape function?

Answer 1

Take $\eta = 0$ for instance. A tangent vector is $$ \left[ \begin{matrix} \frac{\partial x}{\partial \xi}(\xi, 0) \\ \frac{\partial y}{\partial \xi}(\xi, 0) \end{matrix} \right] $$ By rotation of -90 degrees, an outward normal vector is $$ \left[ \begin{matrix} - \frac{\partial y}{\partial \xi}(\xi, 0) \\ \frac{\partial x}{\partial \xi}(\xi, 0) \end{matrix} \right] $$ You can proceed similarly for the other edges.