Multilevel field interpolation algorithm (mlfia) for fast iterative solution of integral equations with method of moments
- Autores: Hugo Gerzain Espinosa Arroyo
- Directores de la Tesis: Juan Manuel Rius Casals (dir. tes.)
- Lectura: En la Universitat Politècnica de Catalunya (UPC) ( España ) en 2008
- Idioma: español
- Tribunal Calificador de la Tesis: Eduard Úbeda Farré (secret.), Sebastián Blanch Boris (voc.), Josep Parrón Granados (voc.)
- Materias:
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- Resumen
The Method of Moments (MoM) solution of the integral equation formulation of Maxwells equations is an efficient tool for solving electromagnetic (EM) boundary value problems. The kernel of the integral equations (Green's function) is a nonlocal operator which results in a dense impedance matrix. To solve electrically large EM problems, conventional iterative solutions require O(N²) memory for storing the matrix elements and O(N²) operations for the matrix-vector multiplication, where N is the number of unknowns describing the current on the scatterer. In previous years, some successful techniques have been developed in order to reduce the memory and the computational cost. The Multilevel Fast Multipole Algorithm (MLFMA) is a good example of these techniques, but it has the disadvantage of being kernel dependent, which means that for every kernel the code must be reformulated.
Another technique is the Multilevel Domain Decomposition Algorithm (MLMDA). This is a kernel independent approach based on the distribution of equivalent sources distributed in the observation cell. In both techniques the memory requirements and the computational complexity is of the order of O(NlogN) and O(Nlog²N), respectively. In this research we have developed another multilevel algorithm with the same requirements as the MLMDA, also independent of the Green's function and with interesting features in terms of its ease of implementation and simplicity of interpretation. Following the conventional acronyms, we have named our algorithm the Multilevel Field Interpolation Algorithm (MLFIA) In contrast to the MLMDA, the proposed technique works on the basis of the interpolation of fields by distributing a smaller number of samples (evaluation points) over the observation cell and interpolating them to new points going from levels 2 to L-1, then to the field basis functions at the finest level L, where L is the total number of levels. Similar to the MLMDA, our algorithm permits a fast matrix- vector multiplication by the application of a "three-stage" method where several blocks of sparse matrices are obtained instead of the traditional full matrix. The multiplication of each sparse block by a trial solution vector is executed by using a multilevel scheme that resembles a Fast Fourier transform (FFT).
Before and after the interpolation, the phase of the Electric Field Integral Equation (EFIE) must be extracted and restored, respectively, in order to eliminate the oscillations produced by the Green's function.
We have developed and implemented the MLFIA technique in one and two dimensions. For the first we have used the 3-D Green's Function into the Electric Field Integral Equation (EFIE), but applied to a straight thin wire; hence the structure was discretized and decomposed into a multilevel binary-tree. For the second case, we developed a multilevel quad-tree decomposition in 2-D, where the source cells and the far-field cells were analysed by three mechanisms of interaction. The first one corresponds to the near-field interaction (touching cells). We used this part of the matrix to implement an incomplete LU preconditioner, in order to accelerate the convergence of the iterative solution in the GMRES solver. In the second and third mechanism we analysed the interactions between source and far-field cells using MLFIA.
The crucial part of the algorithm is concentrated in the field interpolation and that is the reason why we rigorously investigated different interpolation types; going from the simplest one (Nearest-Neighbour), to Chebyshev interpolation.
The results that we have presented show very good agreement with those of the well established ACA method (Adaptive Cross Approximation) in terms of the accuracy in the relative error of the current distribution and in the bistatic radar cross section, by analysing two-dimensional large PEC structures in free space for both TMz and TEz modes.
