Low-frequency Multilevel Fast Multipole Algorithm Research Articles

Vectorial Low-Frequency MLFMA for the Combined Field Integral Equation

A vectorial Low-Frequency Multi-Level Fast Multipole Algorithm (LF-MLFMA) is proposed for acceleration of interactions resultant from the method of moments (MoM) discretization of the combined field integral equation (CFIE). The derivatives relating the scalar Green's function to its dyadic counterparts are defined via recursive identities for scalar wave functions. The method evaluates the matrix vector product in MoM by performing three scalar LF-MLFMA passes. It is demonstrated to be stable for scatterers spanning up to 110 wavelengths in size. As the method does not impose any restrictions on the depth of the MLFMA tree, it is suitable for the solution of both broadband and multiscale problems.

Low-Frequency MLFMA on Graphics Processors

A parallelization of the low-frequency multilevel fast multipole algorithm (MLFMA) for graphics processing units (GPUs) is presented. The implementation exhibits speedups between 10 and 30 compared to a serial CPU implementation of the algorithm. The error of the MLFMA on the GPU is controllable down to machine precision. Under the typical method-of-moments (MoM) error requirement of three correct digits, modern GPUs are shown to handle problems with up to 7.5 million degrees of freedom in dense matrix approximation.

EFFICIENT SOLUTIONS OF METAMATERIAL PROBLEMS USING A LOW-FREQUENCY MULTILEVEL FAST MULTIPOLE ALGORITHM

We present fast and accurate solutions of electromagnet- ics problems involving realistic metamaterial structures using a low- frequency multilevel fast multipole algorithm (LF-MLFMA). Acceler- ating iterative solutions using robust preconditioning techniques may not be su-cient to reduce the overall processing time when the or- dinary high-frequency MLFMA is applied to metamaterial problems. The major bottleneck, i.e., the low-frequency breakdown, should be eliminated for e-cient solutions. We show that the combination of an LF-MLFMA implementation based on the multipole expansion with the sparse-approximate-inverse preconditioner enables e-cient and ac- curate analysis of realistic metamaterial structures. Using the robust LF-MLFMA implementation, we demonstrate how the transmission properties of metamaterial walls can be enhanced with randomly- oriented unit cells.

A multilevel fast multipole algorithm for electrically small composite structures

AbstractA multi‐tree scheme to apply the low‐frequency multilevel fast multipole algorithm to composite structures is presented, so that O(N) CPU time and memory usage are obtained. The convenience of “plug and play” in electromagnetic simulation is illustrated in the scenario of contact‐region modeling with an inconsistent mesh. © 2004 Wiley Periodicals, Inc. Microwave Opt Technol Lett 43: 202–207, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.20419

Three-dimensional multilevel fast multipole algorithm from static to electrodynamic

A normalized three-dimensional (3-D) multilevel fast multipole algorithm (MLFMA) with a computational complexity of O(N) for very low-frequency (LF) problems is developed. This 3-D LF-MLFMA can be used not only independently for very low-frequency cases or very small structures compared to the wavelength, but also to solve large-scale structures with rapidly varying areas when merged with a general dynamic algorithm. From the LF–MLFMA, a more explicit and succinct representation of the static MLFMA is also derived. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 26: 43–48, 2000.

Applying matrix rotation to the three-dimensional low-frequency multilevel fast multipole algorithm

The matrix rotation technique is applied to the three-dimensional (3-D) multilevel fast multipole algorithm (MLFMA) at very low frequencies (LF) to save the storage without increasing the order of the floating-point operations. Some symmetrical properties of the translation matrices along the z-direction for the 3-D LF–MLFMA and the static MLFMA are also derived to further reduce the memory requirements. Numerical results shown that the order of errors introduced by the matrix rotation is much smaller than the order of errors from the plain LF–MLFMA or plain static MLFMA. The implementation of the matrix rotation technique does not change the precision order of the LF–MLFMA and the static MLFMA for fixed truncation of multipole expansions. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 26: 105–110, 2000.

Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies

We develop a new method to precondition the matrix equation resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE). This preconditioning method is based on first applying the loop-tree or loop-star decomposition of the currents to arrive at a Helmholtz decomposition of the unknown currents. However, the MoM matrix thus obtained still cannot be solved efficiently by iterative solvers due to the large number of iterations required. We propose a permutation of the loop-tree or loop-star currents by a connection matrix, to arrive at a current basis that yields a MoM matrix that can be solved efficiently by iterative solvers. Consequently, dramatic reduction in iteration count has been observed. The various steps can be regarded as a rearrangement of the basis functions to arrive at the MoM matrix. Therefore, they are related to the original MoM matrix by matrix transformation, where the transformation requires the inverse of the connection matrix. We have also developed a fast method to invert the connection matrix so that the complexity of the preconditioning procedure is of O(N) and, hence, can be used in fast solvers such as the low-frequency multilevel fast multipole algorithm (LP-MLFMA). This procedure also makes viable the use of fast solvers such as MLFMA to seek the iterative solutions of Maxwell's equations from zero frequency to microwave frequencies.