How to make contour plots for 2D Finite Element problems

[maverick280857]

To the moderator: Please shift this to the appropriate forum if this isn't the place for it. EDIT -- It seems this (https://www.physicsforums.com/forumdisplay.php?f=189) is the right place for it? Sorry I overlooked this forum.

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Hi everyone,

I am solving some 2D boundary value problems in electromagnetics, using the Finite Element Method to familiarize myself with the method before I can apply it to my senior project. I have written a program in MATLAB which to solve the Laplace Equation using the FEM, on a rectangular solution domain defined by the coordinates (0,0), (1,0), (1,1), (0,1) [a square in this case]. The boundary conditions are

V(x = 0) = V(x = 1) = 0
V(y = 0) = 0
V(y = 1) = V_0 (some fixed constant value)

I have been able to write a program that generates a mesh for this region (please refer to attached diagram), and computes the value of the potential at every non-boundary node of the region (points of the triangular elements). Now, I need to use the element shape functions to interpolate the potential at points interior to an element using

(a) the values of the potential at its 3 nodes
(b) Lagrange polynomials (element shape functions)

I want to obtain a contour plot for the solution in the entire solution domain. What is the best and easiest way to go about this?

I thought of specifying several points within each triangular region and use the interpolation equation to calculate the values of the potential at those points. Then I would have a set of points (x, y, V(x,y)) which I could use to make a surface plot. I got a very poor plot when I tried this, so I am guessing there is something wrong in the way I've plotted this.

 

[BigJDubs]

I would really appreciate it if someone could help me with this and provide some guidance on how to proceed.Thanks in advance!

 

[charng]

Any suggestions on how to proceed would be greatly appreciated!

Hello,

Thank you for your question. Contour plots are a useful tool for visualizing 2D Finite Element problems. They provide a clear representation of the solution over the entire solution domain, making it easier to identify patterns and trends in the data.

To make a contour plot for a 2D Finite Element problem, you will need to follow these steps:

1. Generate a mesh for the solution domain: This can be done using the coordinates of the solution domain and dividing it into smaller triangular elements.

2. Solve the problem using the Finite Element Method: Use the mesh generated in the previous step to solve the problem and obtain the values of the solution at each node of the mesh.

3. Interpolate the solution at points within each element: Use the element shape functions and the values of the solution at the nodes to interpolate the solution at points within each element.

4. Create a grid of points over the solution domain: Define a grid of points over the solution domain where you want to plot the contour plot. These points should be evenly spaced and cover the entire solution domain.

5. Use the interpolated values to plot the contour lines: Use the interpolated values of the solution at the grid points to plot the contour lines. You can use a function like contourf in MATLAB to plot the contour lines.

6. Add labels and legends to the plot: Finally, add appropriate labels and legends to your plot to make it easier to understand.

I hope this helps. Good luck with your senior project!