what to use for mass in a 2D FEM simulation

Question

I am trying to find the energy of a wave travelling through a solid material in a 2D Finite Element Method (FEM) - Simulation.

As a general approach I would try to use $E_{kin}=\frac{1}{2}mv^2$ at the highest particle velocity $v$, as this would be the point, where all the energy in the wave is kinetic and potential energy should be zero. So I would know the energy of the wave

But what would I use for mass? Since it is a 2D simulation, I image that, as the third dimension is infinitely small, there is no mass. Now I remember seeing in a paper once (sorry, I don't know the source anymore), that people used density $\rho$ instead of mass in their 2D FEM simulations.

My questions have 2 parts:

• Is using density instead of mass justifiable? If not, what other approach could someone point me to?
• The units would not match up anymore. So if it was ok to use density how would I use it.

Answer 1

Density is mass per unit "something". In a string that means $kgm^{-1}$ on a plane it's $kgm^{-2}$ and in three dimensions it's $kgm^{-3}$. In your simulation you are only concerned with the behaviour in the two dimensions, you are effectively working in 3D with one unit of finite element thick.

Just imagine the density and pressure of the sound wave.

In your model, you are concerned with the energy travelling in your medium. The energy travels like a wave. I would solve the acoustic wave equation, in your 2 dimension plane. You can do this at discrete points, which make up your finite element mesh. The acoustic wave equation actually has density in it.

http://cmst.curtin.edu.au/local/docs/pubs/2000-32.pdf

This looks helpful. I suggest you lean Eulers method. That is a simple way of numerically solving the differential equation. You can do that across your mesh.