{toc:outline=true|indent=0px|minLevel=2} h2. Introduction Omega3P is a parallel finite-element electrogmagnetic code for high-fidelity modeling of cavities. It calculates the resonant frequencies and other rf parameters of a cavity, as well as damping effects due to external couplings. It uses Nedelec-type hierarchical vector basis up to 6th order with quadratic (10-points) tetrahedral elements for improved solution accuracy. h2. Mathematical Modeling h3. Lossless cavities Maxwell's equations in the frequency domain for a perfectly conducting cavity can be written as the following PDE, {latex} \begin{eqnarray*} \nabla \times \left(\frac{1}{\mu} \nabla \times \vec{\mathbf E}\right) - k^2 \epsilon \vec{\mathbf E} & = 0 & on \quad \Omega \\ \vec{\mathbf n} \times \vec{\mathbf E} & = 0 & on \quad \Gamma_{E} \\ \vec{\mathbf n} \times \left( \frac{1}{\mu} \nabla \times \vec{\mathbf E} \right) & = 0 & on \quad \Gamma_{M} \\ \end{eqnarray*} {latex} We use Neelec-type vector basis functions to discretize the electric field: {latex}\[\vec{\mathbf E}=\sum_i x_i \vec{\mathbf N}_i\]{latex} We obtain the following generalized eigenvalue problem: {latex} \begin{eqnarray*} {\mathbf Kx} = k^2 {\mathbf Mx} \quad where & \\ {\mathbf K}_{ij} = \int_{\Omega} \left( \nabla \times \vec{\mathbf N}_i \right) \cdot \frac{1}{\mu} \left( \nabla \times \vec{\mathbf N}_j \right) d\Omega & \\ {\mathbf M}_{ij} = \int_{\Omega} \vec{\mathbf N}_i \cdot \epsilon \vec{\mathbf N}_j d\Omega & \\ \end{eqnarray*} {latex} The matrix *K* is real symmetric while *M* is real symmetric positive definite. If there are lossy materials in the cavity, the matrix *M* and/or *K* will be complex. h3. Waveguide loaded cavities For modeling waveguide-loaded cavities, we use either the absorbing boundary condition (ABC) or the waveguide boundary condition (WBC). The following picture illustrates a cavity with 3 waveguides, and each of them is modeled with ABC. !Absorbing.png|width=500,align=center! The discretized system is a complex nonlinear eigenvalue problem, {latex} \[ {\mathbf Kx} + i \sum_j \sqrt{k^2-k^2_{cj}} {\mathbf W}_j {\mathbf x} = k^2 {\mathbf Mx} \] {latex} where the damping matrix *W* is {latex} \[ ({\mathbf W}_j)_{mn} = \int_{\Gamma} \left( \vec{\mathbf n} \times \vec{\mathbf N}_m \right) \cdot \left( \vec{\mathbf n} \times \vec{\mathbf N}_n \right) d\Gamma \] {latex} The following picture illustrates a cavity with waveguides, which are modeled with WBC. !Waveguide.png|align=center,width=500! The corresponding discretized system becomes a complex nonlienar eigenvalue problem, {latex} \[ {\mathbf Kx} + i \sum_{m,n} \sqrt{k^2-(k^{TE}_{mn})^2} {\mathbf W}^{TE}_{mn} {\mathbf x} + i \sum_{m,n} \frac{k^2}{\sqrt{k^2-(k^{TM}_{mn})^2}} {\mathbf W}^{TM}_{mn} {\mathbf x} = k^2 {\mathbf Mx} \] {latex} where the waveguide matrices are {latex} \[ ({\mathbf W}^{TE}_{mn})_{ij} = \int_{\Gamma} \vec{\mathbf e}^{TE}_{mn} \cdot \vec{\mathbf N}_i d\Gamma \int_{\Gamma} \vec{\mathbf e}^{TE}_{mn} \cdot \vec{\mathbf N}_j d\Gamma \] \\ \[ ({\mathbf W}^{TM}_{mn})_{ij} = \int_{\Gamma} \vec{\mathbf e}^{TM}_{tmn} \cdot \vec{\mathbf N}_i d\Gamma \int_{\Gamma} \vec{\mathbf e}^{TM}_{tmn} \cdot \vec{\mathbf N}_j d\Gamma \] {latex} h2. Numerical Methods The following is a graphical description of the four different types of physics problems that Omega3P can solve and the solver options that can be used. |