Ignace Bogaert

A Nondirective Plane Wave MLFMA Stable at Low Frequencies

A novel method, called the nondirective stable plane wave multilevel fast multipole algorithm (NSPWMLFMA), is presented to evaluate the low-frequency (LF) interactions that cannot be handled by the multilevel fast multipole algorithm (MLFMA). It is well known that the MLFMA cannot be used for LF interactions, since it suffers from numerical instability. Contrary to current techniques, the proposed technique is not based on the spectral representation of the Green function. Instead the addition theorem of the MLFMA is manipulated into a form that allows numerically stable translations along the z axis. The translation operator for these translations is derived in closed form. A QR-based method is devised to allow stable translations in all the other directions. Interpolations and anterpolations are also provided, allowing a full multilevel algorithm. Since the NSPWMLFMA is based on the same mathematical foundations as the MLFMA, it requires limited adaptations to existing MLFMA codes. The fact that a QR is needed limits this algorithm to LF interactions. However, a coupling with the MLFMA is straightforward, allowing the easy construction of a broadband algorithm. The DC limit of the algorithm is also presented and it is shown that the algorithm remains valid for static problems. Finally, it is shown that the error introduced in the different steps of the algorithm is controllable, and a single-level vectorial version of the algorithm is applied to a generic scattering application to demonstrate its validity.

On a Well-Conditioned Electric Field Integral Operator for Multiply Connected Geometries

All known integral equation techniques for simulating scattering and radiation from arbitrarily shaped, perfect electrically conducting objects suffer from one or more of the following shortcomings: (i) they give rise to ill-conditioned systems when the frequency is low (ii) and/or when the discretization density is high, (iii) their applicability is limited to the quasi-static regime, (iv) they require a search for global topological loops, (v) they suffer from numerical cancellations in the solution when the frequency is very low. This work presents an equation that does not suffer from any of the above drawbacks when applied to smooth and closed objects. The new formulation is obtained starting from a Helmholtz decomposition of two discretizations of the electric field integral operator obtained by using RWGs and dual bases respectively. The new decomposition does not leverage Loop and Star/Tree basis functions, but projectors that derive from them. Following the decomposition, the two discretizations are combined in a Calderon-like fashion resulting in a new overall equation that is shown to exhibit self-regularizing properties without suffering from the limitations of existing formulations. Numerical results show the usefulness of the proposed method both for closed and open structures.

O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing

A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument

An efficient hybrid MLFMA-FFT solver for the volume integral equation in case of sparse 3D inhomogeneous dielectric scatterers

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A low frequency stable plane wave addition theorem

. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is

Iteration-Free Computation of Gauss--Legendre Quadrature Nodes and Weights

\mathcal{O}(1)

Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns

. The proposed algorithm also immediately yields an

Embedding Calderón Multiplicative Preconditioners in Multilevel Fast Multipole Algorithms

\mathcal{O}(1)

Simulation and Experimental Verification of W-Band Finite Frequency Selective Surfaces on Infinite Background with 3D Full Wave Solver Nspwmlfma

algorithm for computing an arbitrary Gauss--Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss--Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the

Weak Scalability Analysis of the Distributed-Memory Parallel MLFMA

\mathcal{O}(1)