# Finite Element Homework Solution

Let us find the finite element system of equations for one element. In this problem, we discretize the time variable into elements. Let the element extend from time t 1 \displaystyle t_1 to time t 2 \displaystyle t_2 .

## Finite Element Homework Solution

We have seen that instead of thinking of the weighting function as a series of the same form as θ h ( t ) \displaystyle \theta _h(t) , we can just consider these functions to have the form N i \displaystyle N_i - where the index i \displaystyle i stands for a row of the finite element system of equations. Therefore, we write

Use ANSYS or some other tool of your choice (including your own code) to solve the linear equation of motion via finite elements. Compare with the exact solution. You can use any reasonable values for time ( t \displaystyle t ), mass ( m \displaystyle m ), and length ( l \displaystyle l ).

The example code implements the finite element method as described above through 3header/source file pairs, a main source file, and a couple of supporting Python scripts.The header/source file pairs are:

where the two command line arguments are the number of points in the radial and azimuthaldirections respectively. The total number of unknowns that the finite element code has tosolve for will be the product of these numbers (well, slightly less). I can report that evenup to 200 radial points and 400 azimuthal points will give a solution fairly quickly, thoughthe python plotting routines will start to struggle.

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You may want to use LaTeX to typeset your homework solutions. LaTeX is the standard document preparation system used in the mathematical sciences. Using LaTeX makes it easier for you to revise and edit your solutions and for us to read them, so you will never lose points for illegibility.

Numerical Approximation of Partial Differential Equations, A. Quarteroni and A. ValliDescription and GoalsDescription. (Credit hours: 3), Introduction to partial differential equations and fundamental concepts of modeling elliptic, parabolic, and hyperbolic equations. Emphasis will be placed on finite element simulations. Students will have the opportunity to explore select topics in greater depth.

I take academic integrity very seriously. If a homework solution or take-home portion of an exam you turn in is not your own work as outlined above, you will receive a zero on that assignment. If multiple assignments are found not to be your own work, you will receive an F in the class.