0
$\begingroup$

I am looking at a tutorial using Fenics for solving PDEs using finite element methods. One example that they use is the Poisson equation with Neumann boundary conditions. The equation itself is:

$$ - \nabla^2 u = f $$ $$ \nabla u \cdot n = g $$

I was just trying to understand the physical intuition behind Neumann boundary conditions in the poisson problem. So for the simple poisson equation component of the system above, a force $f$ is deforming a mesh in the interior region of the domain. That much I get. But I was not sure how to interpret or visualize the Neumann boundary condition. Does this mean that the deformation of the mesh in the interior of the domain is "pulling" or deforming some of the mesh on the exterior of the domain. is that the intuition or am I just getting that wrong.

of course the area outside of the mesh or domain is not modeled, so we don't know the amount of deformation.

Hence, I would appreciate it someone could explain the physical interpretation of what the Neumann boundary condition is doing in the Poisson equation. Thanks.

Improve this question
$\endgroup$
6
  • $\begingroup$ It is unclear what your question is about. Are you asking for an intuitive explanation or a physical example of the Neumann boundary condition, or how it is being used in numerical analysis? If the latter than you should ask your question at math SE. $\endgroup$
    – hyportnex
    Commented Sep 4, 2023 at 23:12
  • $\begingroup$ @hyportnex thanks for the response. Yeah, I am asking for an intuitive explanation of the physics. So I can already solve the math, meaning I can setup the weak form and solve the system of equations using Fenics. But I am not clear how to understand the physics forces that are applied here. So in the heat equation, a Neumann boundary condition means that some of the heat escapes through the boundary--so I would be able to feel the heat radiated from an imperfectly insulated rod. In the case of the Poisson equation, what does it mean to radiate deformation stress. $\endgroup$
    – krishnab
    Commented Sep 4, 2023 at 23:21
  • $\begingroup$ I do not know about deformation stress but in an ohmic electric current conducting material the interface between the conductor and an ideal insulator must be such that $J_n=0$, and since $\mathbf J = \sigma \mathbf E = -\sigma \nabla \phi$ the boundary condition for $\phi$ is $\frac{\partial \phi}{\partial n}=0$. The qTEM mode of a microstrip waveguide between two parallel metal plates has the sides that approximately satisfy the Neumann boundary conditions, and in general the so-called high impedance surfaces, that is "open circuit", are of type Neumann boundary. $\endgroup$
    – hyportnex
    Commented Sep 4, 2023 at 23:56
  • $\begingroup$ @hyportnex okay, I see what you mean. Yes, so the idea of $\frac{\partial \phi}{\partial u} = 0$ represents that there is no flow across the boundaries. But then how do you get a unique solution--since just declaring that $\frac{\partial \phi}{\partial u} = 0$ could mean an infinite number of solutions up to some constant, right. Like are there some auxiliary constraints required? $\endgroup$
    – krishnab
    Commented Sep 5, 2023 at 0:29
  • $\begingroup$ The homogeneous Neumann condition arises naturally from the minimization of an integral without prescribing the boundary, here minimize $\mathcal V[\phi] = \int ((\nabla \phi)^2+f\phi)$, for example, see here. But if you want to get details on when and how this exists and is unique you would likely get a better answer at the math SE than here, and especially from me... And regarding the arbitrary constant in $\phi$, you do get such in any potential field, it is an indefinite integral... $\endgroup$
    – hyportnex
    Commented Sep 5, 2023 at 0:46

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.