Neumann Boundary Conditions for Heat Equation

In summary, the speaker wrote a program using the FEM to approximate a solution to the heat conduction equation, but only allowed for Neumann boundary conditions at first. When testing the program with low values for heat flux, reasonable results were obtained. However, when the heat flux was increased by 1%, the results became chaotic and the maximum temperature reached a constant 2 billion. The speaker then realized that the stiffness matrix had one zero eigenvalue, making it non-positive definite. To solve this issue, Dirichlet boundary conditions need to be implemented. The program uses 3x3 Gaussian quadrature for the values of K and a 2 point Gaussian quadrature for the values of F.
  • #1
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I wrote a program that uses the FEM to approximate a solution to the heat conduction equation. I was lazy and wanted to test it, so I only allowed Neumann boundary conditions (I will program in the Dirichlet conditions and the source terms later).

When I input low values for the heat flux, I seem to get reasonable results, as you can see in this http://i222.photobucket.com/albums/dd24/RabidUndeadMonkeys/Neumann_Low.png" [Broken].

When I increase the flux by just a little bit (about 1%), though, it all goes http://i222.photobucket.com/albums/dd24/RabidUndeadMonkeys/Neumann_Chaos.png" [Broken]

In these pictures, red is hot (obviously!), and maximum red value is normalized so that the hottest temperature is pure red. What you can't see is that the temperatures in the first image were very close to zero. The temperature in the second image is about a constant 2 billion. I checked to make sure that the result I got was indeed a solution to Kx = F, (since I am using the conjugate gradient method, it would fail for a non-positive definite matrix), and it was.

My question is this: Is this result actually supposed to happen? Is it that the problem is ill-posed for certain Neumann conditions? Or if it is not, what is likely the problem (I know it's a long shot, but maybe someone has seen this problem before)?

I am using 3x3 Gaussian quadrature for the values of K, and a 2 point Gaussian Quadrature over each side in the elements for the values of F. The values of K(i,j) and F(j) were calculated using the functions given in "The Finite Element Method" by Thomas J.R. Hughes.

UPDATE::
I looked at my stiffness matrix and realized it has exactly one zero eigenvalue, so it's not positive-definite. That is a problem!
 
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  • #2
I need to implement Dirichlet boundary conditions to make it positive definite. Thanks for everyone's help.
 

What is the Neumann Boundary Condition for Heat Equation?

The Neumann Boundary Condition for Heat Equation is a type of boundary condition used in solving the heat equation, a partial differential equation that describes the flow of heat in a given system. It specifies the rate of heat transfer across the boundary of the system.

Why are Neumann Boundary Conditions important in studying heat transfer?

Neumann Boundary Conditions are important because they allow us to model the flow of heat in a system that is not closed, meaning that heat can enter or leave the system. This is a common scenario in real-world systems, such as in buildings or electronic devices, where heat is constantly being exchanged with the surrounding environment.

What is the difference between Neumann Boundary Conditions and Dirichlet Boundary Conditions?

The main difference between Neumann Boundary Conditions and Dirichlet Boundary Conditions is that Neumann conditions specify the rate of heat transfer at the boundary, while Dirichlet conditions specify the temperature at the boundary. In other words, Neumann conditions deal with heat flux, while Dirichlet conditions deal with temperature.

How do Neumann Boundary Conditions affect the solution to the heat equation?

Neumann Boundary Conditions are essential in determining a unique solution to the heat equation. They provide additional information about the behavior of the system at the boundaries, which is necessary for solving the partial differential equation. Without these conditions, the solution to the heat equation would not be unique.

What are some applications of Neumann Boundary Conditions?

Neumann Boundary Conditions have numerous applications in various fields, including engineering, physics, and chemistry. They are commonly used in modeling heat transfer and thermal insulation in buildings, electronic devices, and industrial processes. They are also important in studying diffusion processes and other physical phenomena involving the flow of heat.

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