I FEM Method for the Wave Equation

bob012345
Gold Member
Messages
2,283
Reaction score
1,010
TL;DR Summary
How to set up the Finite Element Method of a 1D Wave Equation.
I am trying to understand how to apply the finite element method for a simple 1D wave equation with four elements for learning purposes.

$$\frac{d^2 T(x)}{dx^2} = -f(x)$$

I am stuck because the structure of the equations set up in Numerical Methods for Engineers by Chapra and Canale as shown here seems to assume one knows what the function ##f(x)## is. What if ##f(x)## is ##T(x)## as in the wave equation? Is that case doable? Thanks.

0.jpg
 
Physics news on Phys.org
I assume N_i are test functions with T approximated by T_1N_1(x) + T_2N_2(x). If so, then \frac{d^2T}{dx^2} + kT(x) = -f(x) where f is a known source term gives <br /> \begin{multline*}<br /> \int_{x_1}^{x_2} N_i(x)\frac{d^2T}{dx^2} + kN_i(x)T(x) + N_i(x)f(x)\,dx \\=<br /> \left[ N_i(x)\frac{dT}{dx}\right]_{x_1}^{x_2} <br /> +\int_{x_1}^{x_2} -\frac{dN_i}{dx}\frac{dT}{dx} + kN_i(x)T(x) + N_i(x)f(x)\,dx<br /> \end{multline*} which results in <br /> \begin{multline*}<br /> \begin{pmatrix}<br /> \int_{x_1}^{x_2} \left(\frac{dN_1}{dx}\right)^2\,dx &amp; \int_{x_1}^{x_2} \frac{dN_1}{dx}\frac{dN_2}{dx} \,dx \\<br /> \int_{x_1}^{x_2} \frac{dN_1}{dx}\frac{dN_2}{dx} \,dx &amp; \int_{x_1}^{x_2} \left(\frac{dN_2}{dx}\right)^2\,dx<br /> \end{pmatrix}<br /> \begin{pmatrix} T_1 \\ T_2 \end{pmatrix} \\<br /> - k\begin{pmatrix} \int_{x_1}^{x_2} N_1^2(x)\,dx &amp; \int_{x_1}^{x_2} N_1(x)N_2(x)\,dx \\<br /> \int_{x_1}^{x_2} N_1(x)N_2(x)\,dx &amp; \int_{x_1}^{x_2} N_2^2(x)\,dx \end{pmatrix}<br /> \begin{pmatrix} T_1 \\ T_2 \end{pmatrix} \\<br /> = <br /> \begin{pmatrix}<br /> N_1(x_2) \frac{dT}{dx}(x_2) - N_1(x_1) \frac{dT}{dx}(x_1) \\<br /> N_2(x_2) \frac{dT}{dx}(x_2) - N_2(x_1) \frac{dT}{dx}(x_1) \end{pmatrix} +<br /> \begin{pmatrix}<br /> \int_{x_1}^{x_2} N_1(x)f(x)\,dx \\ <br /> \int_{x_1}^{x_2} N_2(x)f(x)\,dx<br /> \end{pmatrix}.\end{multline*} I'm sure the authors will give an example of this at some point.

The wave equation is actually a PDE, so T_1 and T_2 are not constants but functions of time. You therefore end up with the system of ODEs <br /> A\begin{pmatrix} \frac{d^2T_1}{dt^2} \\ \frac{d^2T_2}{dt^2} \end{pmatrix} = -c^2 B \begin{pmatrix} T_1 \\ T_2 \end{pmatrix} + (\mbox{boundary terms}) where B is the stiffness matrix and <br /> A = \begin{pmatrix} \int_{x_1}^{x_2} N_1^2(x)\,dx &amp; \int_{x_1}^{x_2} N_1(x)N_2(x)\,dx \\<br /> \int_{x_1}^{x_2} N_1(x)N_2(x)\,dx &amp; \int_{x_1}^{x_2} N_2^2(x)\,dx \end{pmatrix}.
 
  • Like
Likes Delta2 and bob012345
I worked it out and it works great! Thanks. Here is a plot of the points calculated by the Finite Element Method as compared to the exact analytical solution ##\psi(x)= \sqrt{\frac{2}{10}}sin(\frac{\pi x}{10})## of an infinite potential well of width 10 units.;

desmos-graph (7).png
 
Last edited:
What does the input function ##f(x)## represent in the case of a wave equation? I know from experience it is a constant and it is necessary to solve the linear equations. I also noticed it can be scaled to get the amplitude I want but I am having trouble justifying it and what value to give it.
 
Thread 'Direction Fields and Isoclines'
I sketched the isoclines for $$ m=-1,0,1,2 $$. Since both $$ \frac{dy}{dx} $$ and $$ D_{y} \frac{dy}{dx} $$ are continuous on the square region R defined by $$ -4\leq x \leq 4, -4 \leq y \leq 4 $$ the existence and uniqueness theorem guarantees that if we pick a point in the interior that lies on an isocline there will be a unique differentiable function (solution) passing through that point. I understand that a solution exists but I unsure how to actually sketch it. For example, consider a...
Back
Top