In the "Spectral Domain Integration" section of the dialog, you can set a value to '''Max Spectral Radius in k0''', which has a default value of 30. This means that the infinite spectral-domain integrals in the spectral variable k<sub>ρ</sub> are pre-calculated and tabulated up to a limit of 30k<sub>0</sub>, where k<sub>0</sub> is the free space propagation constant. These integrals may converge much faster based on the specified Convergence Rate for Integration described earlier. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k<sub>0</sub> might be needed to warrant adequate convergence. For spectral-domain integration along the real k<sub>ρ</sub> axis, the interval [0, Nk<sub>0</sub>] is subdivided into a large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, '''No. Radial Integration Divisions per k<sub>0</sub>''', determines how small these intervals should be. By default, 2 divisions are used for the interval [0, k<sub>0</sub>]. In other words, the length of each integration sub-interval is k<sub>0</sub>/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a polar integration is performed. You can set the '''No. of Angular Integration Points''', which has a default value of 100.
[[File:PMOM79.png]] Figure 1: The Planar MoM Engine Settings dialog. === Planar Module's Linear System Solvers === After the MoM impedance matrix '''[Z]''' (not to be confused with the impedance [[parameters]]) and excitation vector '''[V]''' have been computed through the matrix fill process, the planar MoM simulation engine is ready to solve the system of linear equations: :<math> \mathbf{[Z]}_{N\times N} \cdot \mathbf{[I]}_{N\times 1} = \mathbf{[V]}_{N\times 1} </math><!--[[File:PMOM81.png]]--> where '''[I]''' is the solution vector, which contains the unknown amplitudes of all the basis functions that represent the unknown electric and magnetic currents of finite extents in your planar structure. In the above equation, N is the dimension of the linear system and equal to the total number of basis functions in the planar mesh. [[EM.Cube]]'s linear solvers compute the solution vector'''[I]''' of the above system. You can instruct [[EM.Cube]] to write the MoM matrix and excitation and solution vectors into output data files for your examination. To do so, check the box labeled "'''Output MoM Matrix and Vectors'''" in the Matrix Fill section of the Planar MoM Engine Settings dialog. These are written into three files called mom.dat1, exc.dat1 and soln.dat1, respectively. There are Picasso provides a large number of numerical methods for solving systems selection of linear equations. These methods are generally divided into two groups: system solvers including both direct solvers and iterative solversmethods. Iterative solvers are usually based on matrix-vector multiplications. Direct solvers typically work faster for matrices of smal to medium size (N<3,000). [[EM.Cube]]'s [[Planar Module]] offers five linear solvers: # LU Decomposition Method# Biconjugate Gradient Method (BiCG)# Preconditioned Stabilized Biconjugate Gradient Method (BCG-STAB)# Generalized Minimal Residual Method (GMRES)# Transpose-Free Quasi-Minimum Residual Method (TFQMR) Of the above list, LU is a direct solver, while the rest are iterative solvers. BiCG is a relatively fast iterative solver, but it works only for symmetric matrices. You cannot use BiCG for periodic structures or planar structures that contain both metal and slot traces at different planes, as their MoM matrices are not symmetric. The three solvers BCG-STAB, GMRES and TtFQMR work well for both symmetric and asymmetric matrices and they also belong to a class of solvers called '''Krylov Sub-space Methods'''. In particular, the GMRES method always provides guaranteed unconditional convergence. [[EM.Cube]]'s [[Planar Module]]Picasso, by default, provides a "'''Automatic'''" solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. If the size of the linear system exceeds N = 15,000, the sparse version of the iterative solvers is used, utilizing a row-indexed sparse storage scheme. You can override the automatic solver option and manually set you own solver type. This is done using the '''Solver Type''' dropdown list in the "'''Linear System Solver'''" section of the Planar MoM Engine Settings dialog. There are also a number of other instruct [[parametersEM.Cube]] related to write the solvers. The default value of '''Tolerance of Iterative Solver''' is 1E-3, which can be increased MoM matrix and excitation and solution vectors into output data files for more ill-conditioned systemsyour examination. The maximum number of iterations is usually expressed as a multiple of the systems size. The default value of '''Max No. of Solver Iterations / System Size''' is 3. For extremely large systems, sparse versions of iterative solvers are used. In this caseTo do so, check the elements of the matrix are thresholded with respect to the larges element. The default value of '''Threshold for Sparse Solver''' is 1E-6, meaning that all the matrix elements whose magnitude is less than 1E-6 times the large matrix elements are set equal to zero. There are two more [[parameters]] that are related to the Automatic Solver option. These are box labeled "''' User Iterative Solver When System Size >'''" with a default value of 3,000 Output MoM Matrix and "''' Use SParse Storage When System Size >Vectors''' " with a default value of 15,000. In other words, you control in the automatic solver when to switch between direct and iterative solvers and when to switch to the sparse version Matrix Fill section of iterative solvers. If your computer has an Intel CPU, then [[EM.Cube]] offers special versions of all the above linear solvers that have been optimized for Intel CPU platformsPlanar MoM Engine Settings dialog. These optimal solvers usually work 2-3 time faster than their generic counterpartsare written into three files called mom. When you install [[EM.Cube]]dat1, the option to use Intel-optimized solvers is already enabledexc. However, you can disable this option (edat1 and soln.g. if your computer has a non-Intel CPU). To do thatdat1, open the [[EM.Cube]]'s Preferences Dialog from '''Menu > Edit > Preferences''' or using the keyboard shortcut '''Ctrl+H'''. Select the Advanced tab of the dialog and uncheck the box labeled "''' Use Optimized Solvers for Intel CPU'''"respectively.
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<td> [[Image:PMOM79.png|thumb|550px|EM.Picasso's Planar MoM Engine Settings dialog.]] </td>
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=== Running Frequency Sweep Simulations in EM.Picasso ===