A Novel Efficient Technique for the Calculation of the Green's Functions in Rectangular Waveguides Based on Accelerated Series Decomposition

Soler, Francisco Javier Pérez; Pereira, Fernando Daniel Quesada; Rebenaque, David Cañete; Melcón, Alejandro Álvarez; Mosig, J. R.

doi:10.1109/tap.2008.929438

Cited by 13 publications

(4 citation statements)

References 15 publications

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“…The Ewald summation technique has been applied to the vector potential dyadic Green's function of the rectangular cavity [5,6,9,10]. The derivation of the Ewald summation for the rectangular cavity here closely follows that of [10], and center of coordinate system is shifted such that 0 ≤ x j ≤ L j .…”

Section: Ewald Summation Techniquementioning

confidence: 99%

“…On the other side, spectral expansion does not converge in the proximity of the source as a consequence of the singular behavior of the Green's function. The famous Ewald's technique is about to obtain a hybrid spectral-spatial summation that has an exponential convergence rate [9][10][11][12] which is a successful technique of taking advantage of both spectral and spatial expansions. Another method based on the Chebyshev polynomial approximation is reported [13] that provides an efficient way of evaluation of the Green's function in the rectangular cavity.…”

Section: Introductionmentioning

confidence: 99%

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Fast and Broad Band Calculation of the Dyadic Green's Function in the Rectangular Cavity; An Imaginary Wave Number Extraction Technique

Sanamzadeh¹,

Tsang²

2019

PIER C

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An analytical approach for the calculation of the dyadic Green's functions inside the rectangular cavity over a broad range of frequency is presented. Both vector potential and electric field dyadic Green's functions are considered. The method is based on the extraction of the Green's function at an imaginary wavenumber from itself to obtain a rapidly convergent eigenfunction expansion of the dyadic Green's function. The extracted term encompasses the singularity of the Green's function and is computed using spatial expansions. Results are illustrated for a rectangular cavity up to 5 wavelengths in size with thousands of cavity modes obtained by the 6th order convergent expansion. It is shown that for an accurate and broadband simulation, the proposed method is many times faster than the Ewald method.

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Section: Ewald Summation Techniquementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fast and Broad Band Calculation of the Dyadic Green's Function in the Rectangular Cavity; An Imaginary Wave Number Extraction Technique

Sanamzadeh¹,

Tsang²

2019

PIER C

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“…The discrete complex image method (DCIM) is widely used, either to obtain approximate closed-form spatial Green's functions [40], or the asymptotic expressions [41]. However, this procedure may necessitate formidable task to take care of the spatial and lateral waves for layered medium [65], and the computation of complementary error function (Erfc) with complex arguments is also of large computational costs [66]. In [42], the spectral Green's function is approximated with real exponentials for certain ranges of transverse wave number.…”

Section: Discussionmentioning

confidence: 99%

Boundary integral equation method for electromagnetic and elastic waves

Chen

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“…Unfortunately, the terms in the images part inside Ewald's method requires the evaluation of the complementary error function (erfc), which is computationally expensive, mainly for the complex arguments needed in frequency dependent problems. To mitigate this problem, the GF could be split into a static and dynamic part using Kummer's transform [45]. But still, the static part contains the evaluation of the erfc with real-valued arguments.…”

Section: Green's Functionsmentioning

confidence: 99%

Multilevel adaptive cross approximation and direct evaluation method for fast and accurate discretization of electromagnetic integral equations

Tamayo Palau

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El Método de los Momentos (MoM) ha sido ampliamente utilizado en las últimas décadas para la discretización y la solución de las formulaciones de ecuación integral que aparecen en muchos problemas electromagnéticos de antenas y dispersión. Las más utilizadas de dichas formulaciones son la Ecuación Integral de Campo Eléctrico (EFIE), la Ecuación Integral de Campo Magnético (MFIE) y la Ecuación Integral de Campo Combinada (CFIE), que no es más que una combinación lineal de las dos anteriores. Las formulaciones MFIE y CFIE son válidas únicamente para objetos cerrados y necesitan tratar la integración de núcleos con singularidades de orden superior al de la EFIE. La falta de técnicas eficientes y precisas para el cálculo de dichas integrales singulares a llevado a imprecisiones en los resultados. Consecuentemente, su uso se ha visto restringido a propósitos puramente académicos, incluso cuando tienen una velocidad de convergencia muy superior cuando son resuelto iterativamente, debido a su excelente número de condicionamiento. En general, la principal desventaja del MoM es el alto coste de su construcción, almacenamiento y solución teniendo en cuenta que es inevitablemente un sistema denso, que crece con el tamaño eléctrico del objeto a analizar. Por tanto, un gran número de métodos han sido desarrollados para su compresión y solución. Sin embargo, muchos de ellos son absolutamente dependientes del núcleo de la ecuación integral, necesitando de una reformulación completa para cada núcleo, en caso de que sea posible. Esta tesis presenta nuevos enfoques o métodos para acelerar y incrementar la precisión de ecuaciones integrales discretizadas con el Método de los Momentos (MoM) en electromagnetismo computacional. En primer lugar, un nuevo método iterativo rápido, el Multilevel Adaptive Cross Approximation (MLACA), ha sido desarrollado para acelerar la solución del sistema lineal del MoM. En la búsqueda por un esquema de propósito general, el MLACA es un método independiente del núcleo de la ecuación integral y es puramente algebraico. Mejora simultáneamente la eficiencia y la compresión con respecto a su versión mono-nivel, el ACA, ya existente. Por tanto, representa una excelente alternativa para la solución del sistema del MoM de problemas electromagnéticos de gran escala. En segundo lugar, el Direct Evaluation Method, que ha provado ser la referencia principal en términos de eficiencia y precisión, es extendido para superar el cálculo del desafío que suponen las integrales hiper-singulares 4-D que aparecen en la formulación de Ecuación Integral de Campo Magnético (MFIE) así como en la de Ecuación Integral de Campo Combinada (CFIE). La máxima precisión asequible -precisión de máquina se obtiene en un tiempo más que razonable, sobrepasando a cualquier otra técnica existente en la bibliografía. En tercer lugar, las integrales hiper-singulares mencionadas anteriormente se convierten en casi-singulares cuando los elementos discretizados están muy próximo pero sin llegar a tocarse. Se muestra como las reglas de integración tradicionales tampoco convergen adecuadamente en este caso y se propone una posible solución, basada en reglas de integración más sofisticadas, como la Double Exponential y la Gauss-Laguerre. Finalmente, un esfuerzo en facilitar el uso de cualquier programa de simulación de antenas basado en el MoM ha llevado al desarrollo de un modelo matemático general de un puerto de excitación en el espacio discretizado. Con este nuevo modelo, ya no es necesaria la adaptación de los lados del mallado al puerto en cuestión. The Method of Moments (MoM) has been widely used during the last decades for the discretization and the solution of integral equation formulations appearing in several electromagnetic antenna and scattering problems. The most utilized of these formulations are the Electric Field Integral Equation (EFIE), the Magnetic Field Integral Equation (MFIE) and the Combined Field Integral Equation (CFIE), which is a linear combination of the other two. The MFIE and CFIE formulations are only valid for closed objects and need to deal with the integration of singular kernels with singularities of higher order than the EFIE. The lack of efficient and accurate techniques for the computation of these singular integrals has led to inaccuracies in the results. Consequently, their use has been mainly restricted to academic purposes, even having a much better convergence rate when solved iteratively, due to their excellent conditioning number. In general, the main drawback of the MoM is the costly construction, storage and solution considering the unavoidable dense linear system, which grows with the electrical size of the object to analyze. Consequently, a wide range of fast methods have been developed for its compression and solution. Most of them, though, are absolutely dependent on the kernel of the integral equation, claiming for a complete re-formulation, if possible, for each new kernel. This thesis dissertation presents new approaches to accelerate or increase the accuracy of integral equations discretized by the Method of Moments (MoM) in computational electromagnetics. Firstly, a novel fast iterative solver, the Multilevel Adaptive Cross Approximation (MLACA), has been developed for accelerating the solution of the MoM linear system. In the quest for a general-purpose scheme, the MLACA is a method independent of the kernel of the integral equation and is purely algebraic. It improves both efficiency and compression rate with respect to the previously existing single-level version, the ACA. Therefore, it represents an excellent alternative for the solution of the MoM system of large-scale electromagnetic problems. Secondly, the direct evaluation method, which has proved to be the main reference in terms of efficiency and accuracy, is extended to overcome the computation of the challenging 4-D hyper-singular integrals arising in the Magnetic Field Integral Equation (MFIE) and Combined Field Integral Equation (CFIE) formulations. The maximum affordable accuracy --machine precision-- is obtained in a more than reasonable computation time, surpassing any other existing technique in the literature. Thirdly, the aforementioned hyper-singular integrals become near-singular when the discretized elements are very closely placed but not touching. It is shown how traditional integration rules fail to converge also in this case, and a possible solution based on more sophisticated integration rules, like the Double Exponential and the Gauss-Laguerre, is proposed. Finally, an effort to facilitate the usability of any antenna simulation software based on the MoM has led to the development of a general mathematical model of an excitation port in the discretized space. With this new model, it is no longer necessary to adapt the mesh edges to the port.

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A Novel Efficient Technique for the Calculation of the Green's Functions in Rectangular Waveguides Based on Accelerated Series Decomposition

Cited by 13 publications

References 15 publications

Fast and Broad Band Calculation of the Dyadic Green's Function in the Rectangular Cavity; An Imaginary Wave Number Extraction Technique

Fast and Broad Band Calculation of the Dyadic Green's Function in the Rectangular Cavity; An Imaginary Wave Number Extraction Technique

Boundary integral equation method for electromagnetic and elastic waves

Multilevel adaptive cross approximation and direct evaluation method for fast and accurate discretization of electromagnetic integral equations

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