Applying Neumann Boundary Conditions in 1D

[Carla1985]

Hi,

I've been doing some work on the finite element method. I have been able to calculate the stiffness matrix and load vector and apply both homogeneous and inhomogeneous Dirichlet conditions but am stuck on calculating the Neumann conditions. I have the definition of it as:

$\int_{\partial\Omega_N}\phi_ig_N$

where $\phi_i$ are my basis functions and $g_N$ is the condition. I am not sure how to evaluate this integral though. Surely in 1D the boundary is just a single point so any integral over it is going to be 0? Could someone please explain how this works.

Thanks
Carla

 

[Aaron Bex]

The Neumann condition is used in the finite element method to define the behavior of a function on the boundaries of the domain. This is done by specifying a normal derivative of the function on the boundary. The Neumann condition can be expressed mathematically as an integral over the boundary:
$\int_{\partial\Omega_N}\phi_ig_N$
where $\phi_i$ are basis functions and $g_N$ is the Neumann condition.

In 1D, the boundary is usually just a single point. Therefore, the integral over the boundary simplifies to just evaluating the basis function at the boundary point and multiplying it by the Neumann condition. For example, if you have a linear basis function $\phi(x) = a*x + b$, then evaluating the integral at the boundary point would look like this:
$\int_{\partial\Omega_N}\phi_ig_N = (a*x_b + b)*g_N$
where $x_b$ is the boundary point.

I hope this helps!