FEM: periodic boundary conditions (1D)

In summary, the conversation discussed setting up a mass matrix for a 1D system using finite elements. The mass matrix was defined as M = \int{NN^T}dL, where N is the finite element linear basis functions. The issue was how to construct the mass matrix for the 10th node, taking into account periodic boundary conditions and the values for M_{10,10} and M_{10,1}. It was clarified that x_11 and x_1 are not the same in reality, but they have the same role in the matrix. The main focus was to maintain the periodic nature of x and ensure that the values at the edge were consistent with the rest of the matrix.
  • #1
Niles
1,866
0
I am trying to set up the mass matrix for a 1D system which I want to solve using finite elements. So the mass matrix is defined as

[tex]
M = \int{NN^T}dL,
[/tex]
where N is the finite element linear basis functions. I use hat functions.

Say I have 10 elements, corresponding to 11 nodes running from -5 to 5 so the spacing is 1. Node 1 is equal to node 11 since I want to employ periodic boundary conditions.

My issue is that I am not sure how to construct the mass matrix for the 10th node. As shown here, the elements for the 10th node will be (I use periodic boundary conditions, so [itex]x_{N+1}=x_1[/itex])

[tex]
M_{10,10} = \frac{x_{1}-x_{10}}{3} = -10/3\\
M_{10,1} = \frac{x_{1}-x_{10}}{6} = -10/6
[/tex]
All other elements have positive values given by 1/3 and 1/6, respectively.

Are my values for [itex]M_{10,10}[/itex] and [itex]M_{10,1}[/itex] correct? I find it odd that their values are so much different than the values in the "bulk".
 
Technology news on Phys.org
  • #2
This is 1D, so is it a line?
Is x_11 actually x_1 or is it x_1 + P where P is your period in X?
Thus f(x_11) = f(x_1) but x_11 is not actually x_1.
If this is the case, use x_11 for x_11 and in the matrix assign it the position of x_1 since it will be multiplied by that node.
 
  • #3
Thanks, that is also what I thought. But then my matrix has dimensions 10x10, but my field will have 11 values since x_11 and x_1 are not the same. But that won't work when I multiply them together (?)

Am I missing something?
 
  • #4
x_11 and x_1 are not the same in reality, but in the matrix they have the same role.
The main thing is that you are keeping with the periodic nature in your x, if all other entries are 1/3 and 1/6, then so should the ones at the edge.
You aren't actually putting the 11th value in, but when you plot it, you can manually put the 11th point in.
 

FAQ: FEM: periodic boundary conditions (1D)

What are periodic boundary conditions in 1D Finite Element Method (FEM)?

Periodic boundary conditions are a type of boundary condition used in 1D FEM simulations to model a repeating structure. They define the behavior of the solution at the edges of the domain by imposing a periodicity, meaning that the solution at one edge is identical to the solution at the opposite edge.

Why are periodic boundary conditions important in 1D FEM simulations?

Periodic boundary conditions are important because they allow for the simulation of a larger system using a smaller computational domain. This is useful in situations where the structure being studied is repetitive or infinite in nature, such as in the case of a periodic lattice or a wave propagating through a medium.

How are periodic boundary conditions implemented in 1D FEM simulations?

Periodic boundary conditions can be implemented in 1D FEM simulations by modifying the stiffness matrix and load vector to account for the periodicity. This can be done by adding additional equations to the system or by using special basis functions that satisfy the periodicity condition.

What are the advantages of using periodic boundary conditions in 1D FEM simulations?

There are several advantages to using periodic boundary conditions in 1D FEM simulations. Firstly, they allow for the simulation of larger systems with a smaller computational effort. They also preserve the symmetry of the system, which can result in more accurate and efficient solutions. Additionally, they can be used to study the behavior of a system over a longer time or distance than would be possible with a finite computational domain.

Are there any limitations to using periodic boundary conditions in 1D FEM simulations?

While periodic boundary conditions are useful in many cases, they may not be appropriate for all situations. For example, they may not accurately represent the behavior of a non-repetitive structure or a system with discontinuities at the edges. Additionally, the implementation of periodic boundary conditions may be more complex and require more computational resources compared to traditional boundary conditions.

Back
Top