How to obtain the Poisson equation as used by FEATool?
I am new to modeling and simulation in general and am trying to find the modes of an acoustic resonator including dissipative effects and thermal conductivity. I am using the Poisson equation as given in FEATool by
dts_poi*p' - d_poi*(px_x + py_y + pz_z) = f_poi
with dts_poi as the time scaling coefficient, d_poi as the diffusion term, and f_poi the source term, set to 0. I have tried at length to figure out how this equation is obtained, or even the Helmholtz equation as provided in the "Resonance Frequencies of a Room" tutorial, given by
u' - (ux_x + uy_y + uz_z) = 1.
Neither one seems like a standard variational construction of the equation. I get the modes predicted but don't understand why. So how can one obtain these equations from the standard (nabla)^2 * f = x?
Re: How to obtain the Poisson equation as used by FEATool?
Thanks for the prompter than expected response! I know that the Helmholtz equation and Poisson's equation are both standard PDEs, but I don't recognize them in the state they are presented. Both equations have a time derivative included, whereas the standard formulations of the equations do not. I don't see in the Poisson's equation section of the documentation where there is any time derivative. or where the FEM formulation of Poisson's equation has to do with what's presented in the documentation. It appears to be more akin to the diffusion equation, which is related to the heat equation, as opposed to the wave equation.
Both me and my research advisor have spent a long while trying to figure it out, to no avail. Any light you can shed would be greatly appreciated.
Since my last response I have learned a lot about FEM formulations of equations. Even so, the time derivative in the Helmholtz equation continues to elude both me and my RA. Please, either reference the derivation you used for the Helmholtz/Poisson equation or explain why the time derivative presents in your usage of the Helmholtz equation. I'm using your (very robust) software for my thesis. There's an excellent chance the modeling I've done in your program will be published, and time is running out very quickly to understand exactly what's being done here to produce such compelling results. Any light you could shed would be much appreciated.