I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM). The equation is;
$c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 p = f$
where the $d$ is the damping parameter, the $i$ is complex number, the f is the forcing, the $\omega$ is the eigenvalue and the $p$ is the eigenvector. In matrix form;
$(\mathbf{A} + \omega \mathbf{B} + \omega^2 \mathbf{C})\mathbf{p} = \mathbf{D}\mathbf{p} $
where $\mathbf{B}$ is the damping matrix and $\mathbf{D}$ is the forcing matrix. For 1D case, the boundary conditions are Neumann boundary at both ends.
I am aiming to damp the system with $\mathbf{B}$ and did try 4 different cases;
- $\mathbf{A} + \omega^2 \mathbf{C} = 0 $ (no damping and no forcing)
- $\mathbf{A} + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0$ (damping and no forcing)
- $(\mathbf{A}-\mathbf{D}) + \omega^2 \mathbf{C} = 0 $ (no damping and forcing)
- $(\mathbf{A}-\mathbf{D}) + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0 $ (damping and forcing)
When I run this cases, I get very weird behaviour. Here is the list of the eigenvalues for the cases;
- $\mathbf{A} + \omega^2 \mathbf{C} = 0 $ gives $\omega=1069.2399-0j$
- $\mathbf{A} + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0$ gives $\omega=1139.6967-0.2463j$
- $(\mathbf{A}-\mathbf{D}) + \omega^2 \mathbf{C} = 0 $ gives $\omega=1074.748+3.2431j$
- $(\mathbf{A}-\mathbf{D}) + \omega \mathbf{B} + \omega^2 \mathbf{C} = 0 $ gives $\omega=1146.0758+3.5129j$
The complex part of the eigenvalue is representing the growth rate of the system. As I impose a damping, the last case's complex part (+3.5129j) should be lower than the 3. case's complex part (3.2431j) according to the law of superposition but it is not.
Would anyone can enlight me about this, I am struggling for more than a week.
Thank you!