Curl-Curl-Eigenvalue Equation - Institut für Allgemeine ...

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Solution of the Eigenvalue Equation (C‘ted) „Ghostmodes“ in inhomogeneous waveguide Ursula van Rienen, Universität Rostock, Institut für Allgemeine Elektrotechnik, AG Computational Electrodynamics Numerical approaches and discretization methods which cannot fulfil the relation div curl=0 on the grid space have no straightforward possibility to distinguish the two solution types. They often apply the so-called penalty method where the grad divterm in the eigenvalue equation is weighted by some factor p (penalty factor). After multiple solution with different values of p the correct (physical) eigensolutions can be distinguished because they do not show any dependence on p. With FIT this effort of multiple solution of the eigenvalue problem is not necessary since the product of the div and curl operator vanishes and thus it is sufficient to apply the dual divergence operator to the electric grid flux to distinguish the two types of solution.
Memory Usage Example: mesh points: N = P 6 10 (double precision: 8 bytes/number) FIT: 39 bands of entries N P (21 have to be stored, symmetry) ⇒ 170 MB (+ some add. vectors) no band structure: fill-in-entries by solver (still symmetric!) ⇒ 0 .6 2 13 .5 (3N P ) = 3 ∗10 Byte Ursula van Rienen, Universität Rostock, Institut für Allgemeine Elektrotechnik, AG Computational Electrodynamics Our eigenvalue problem has the dimension N=3 N P and thus also N eigensolutions. The number N P of grid points may be very large in some applications (up to a few millions) which involves that a complete solution is generally hardly possible. In the following, we will describe an iteration method for the solution of such kind of eigenvalue problems. In the practically interesting case of only a few 10-100 eigenvalues searched for, this method provides good results with acceptable effort. All methods described here exploit the fact that the system matrix is very large but sparse: In the iterative solution of the eigenvalue problem only so-called „matrix-vector-operations“ are used for which the matrix pattern can directly be exploited for an efficient implementation. In general, methods which afford a decomposition of the matrix should not be applied since they destroy the band structure. As example let us assume a computational grid of N P = 10 6 grid points. According to the previously found structure the Curl-Curl-System matrix has 39 bands of length N P where only 21 of those have to be stored because of symmetry reasons. For a double-precision calculation we need 8 Byte per floating point number which yields a storage requirement of about 170 MB. Additionally, we need to store some auxiliary vectors (depends on the solution method). This is easily affordable on modern PC´s or workstations. Yet, if we would loose our band structure and would need to store the complete matrix (because of fill-ins produced by the solution method) we would end up with 0.5* (3N P )² = 4.5 * 10 12 numbers or 3.6 * 10 13 Byte (still for a symmetric matrix!)
Page 1 and 2: j ≡ 0 S Curl-Curl-Eigenvalue Equa
Page 3 and 4: Solution Space of Eigenvalue Proble
Page 5 and 6: Solution Space of Eigenvalue Proble
Page 7 and 8: Grid Dispersion Equation Plane wave
Page 9 and 10: Discrete induction law: Grid Disper
Page 11 and 12: Grid Dispersion Equation Electric m
Page 13 and 14: Grid Dispersion Equation The grid d
Page 15: Solution of the Eigenvalue Equation
Page 19 and 20: Eigenvalue Solver Rayleigh-quotient
Page 21 and 22: Spectral Shift The von Mises-iterat
Page 23 and 24: Sub-Space-Iteration Very large, spa
Page 25 and 26: Sub-Space-Iteration 0 ( r ) ( r ) (

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