Solution of the magnetic field integral equation, which is obtained by the classical marching-on-in-time (MOT) scheme, becomes inaccurate when the time-step is large, i.e., under low-frequency excitation. It is shown here that the inaccuracy stems from the classical MOT scheme's failure to predict the correct scaling of the current's Helmholtz components for large time-steps. A recently proposed mixed discretization strategy is used to alleviate the inaccuracy problem by restoring the correct scaling of the current's Helmholtz components under low-frequency excitation.

Several preconditioning techniques have been developed to control the condition number of the linear system matrices arising from the majority of the integral formulations for electromagnetics problems. The performance of these techniques is usually assessed in two regimes: (i) the case where the number of unknowns is kept constant and the frequency decreases (low-frequency regime) and (ii) the case where the frequency is kept constant and the number of unknowns increases (dense discretization regime). A third regime, however, can be identified: (iii) the case where both frequency and the number of unknowns grow since the discretization size in wavelengths is kept constant and the frequency increases (high frequency regime). This contribution will focus on formulations that are free from high-frequency resonances (combined-type equations) and are well-conditioned in both regimes (i) and (ii) and it will study and regularize their behavior in regime (iii).

Algorithmic improvements to the parallel, distributed-memory Multilevel Fast Multipole Algorithm (MLFMA) have resulted in implementations with favorable weak scaling properties. This allows for the simulation of increasingly larger electromagnetic problems, provided that sufficient computational resources are available (B. Michiels et al., “Weak Scalability Analysis of the Distributed-Memory Parallel MLFMA”, IEEE Trans. Antennas Propag., 61(11, 2013). Recently, we were able to benchmark our implementation on the Flemish Supercomputing Centre's (VSC) Tier 1 supercomputer. This cluster consists of 512 nodes interconnected by an FDR Infiniband network. Each node contains two 8-core Intel Xeon E5-2670 processors and 64 GByte of RAM. The complete system hence provides 8192 CPU cores and 32 TByte of RAM in total.

Algorithmic improvements to the parallel, distributed-memory multilevel fast multipole algorithm (MLFMA) have resulted in implementations with favorable weak scaling properties. This allows for the simulation of increasingly larger electromagnetic problems, provided that sufficient computational resources are available. This is demonstrated by presenting the full-wave simulations of extremely large perfectly electrically conducting (PEC) sphere and Thunderbird geometries. Both problems are formulated using the combined field integral equation (CFIE) and discretized in over respectively 3 and 2.5 billion unknowns. They are solved using 4096 CPU cores and 25 TByte of memory. To the best of our knowledge, this is the largest number of unknowns and the highest amount of parallel processes reported to date, for this type of simulation. Additionally, it is demonstrated that the implementation attains a high parallel speedup and efficiency.

The standard and mixed discretizations for the magnetic field integral equation (MFIE) and the Muller integral equation (MUIE) are investigated in the context of low-frequency (LF) scattering problems involving simply connected scatterers. It is proved that, at low frequencies, the frequency scaling of the nonsolenoidal part of the solution current can be incorrect for the standard discretization. In addition, it is proved that the frequency scaling obtained with the mixed discretization is correct. The reason for this problem in the standard discretization scheme is the absence of exact solenoidal currents in the rotated RWG finite element space. The adoption of the mixed discretization scheme eliminates this problem and leads to a well-conditioned system of linear equations that remains accurate at low frequencies. Numerical results confirm these theoretical predictions and also show that, when the frequency is lowered, a finer and finer mesh is required to keep the accuracy constant with the standard discretization.

Gauss-Legendre quadrature rules are of considerable theoretical and practical interest because of their role in numerical integration and interpolation. In this paper, a series expansion for the zeros of the Legendre polynomials is constructed. In addition, a series expansion useful for the computation of the Gauss-Legendre weights is derived. Together, these two expansions provide a practical and fast iteration-free method to compute individual Gauss-Legendre node-weight pairs in O(1) complexity and with double precision accuracy. An expansion for the barycentric interpolation weights for the Gauss-Legendre nodes is also derived. A C++ implementation is available online.

We introduce a new magnetic field integral equation that does not suffer from low-frequency numerical cancellations even for extremely low frequencies (10 -40 Hz). The new equation is obtained by symmetrizing the standard MFIE with its dual equation and by using appropriately chosen quasi-Helmholtz projectors. When compared to mixed discretized MFIEs, the new equation maintains the favorable properties of mixed discretizations even without using high precision integration rules. Numerical results confirm the theoretical developments and show the effectiveness of the new scheme.

Recently, a novel high-order finite-element space for wires, quadrilaterals, and hexahedrons was presented [M. Kostic and B. Kolundzija, "Maximally Orthogonalized Higher Order Bases Over Generalized Wires, Quadrilaterals, and Hexahedra," IEEE Trans. Antennas Propag., vol. 61, no. 6, pp. 3135-3148, 2013]. Numerical results have shown a very favorable behavior of the condition number of the Gram matrix of this finite-element space as a function of the polynomial degree. In this paper, this high-order finite-element space is recognized to be expressible in terms of Jacobi polynomials, which can be easily computed using a three-term recurrence. In addition, the condition number of the Gram matrix of the one-dimensional finite-element space is rigorously analyzed for the general case of a piecewise smooth (possibly curved) geometry. An explicit upper bound for the condition number in terms of the mesh quality is proved. This bound implies that the one-dimensional finite-element space is stable for arbitrarily high polynomial degree. Numerical results corroborate the theoretical results and show that the basis can be used to perform hp-refinement, leading to an accurate handling of both large smooth regions and corners.

In the past decade, several data distribution strategies for the parallel, distributed-memory Multilevel Fast Multipole Algorithm (MLFMA) have been introduced. The common goal is to distribute the computations in the MLFMA uniformly over the different parallel processes, while minimizing dependencies and data communication between them. However, as clusters with thousands of CPU cores are becoming increasingly available, the asymptotic behavior of algorithms for a very large number of processes P and problem size N (weak scaling analysis) starts to play a dominant role. Parallel MLFMA implementations based on a hierarchical distribution of boxes and radiation pattern sampling points that exhibit excellent weak scaling behavior appear to be attractive candidates to tackle even larger problems on future hardware clusters.

This paper investigates the parallel, distributed-memory computation of the translation operator with L+1 multipoles in the three-dimensional Multilevel Fast Multipole Algorithm (MLFMA). A baseline, communication-free parallel algorithm can compute such a translation operator in O(L) time, using O(L2) processes. We propose a parallel algorithm that reduces this complexity to O(logL) time. This complexity is theoretically supported and experimentally validated up to 16 384 parallel processes. For realistic cases, the implementation of the proposed algorithm proves to be up to ten times faster than the baseline algorithm. For a large-scale parallel MLFMA simulation with 4096 parallel processes, the runtime for the computation of all translation operators during the setup stage is reduced from roughly one hour to only a few minutes.

A set of algorithms is proposed for the accurate and efficient computation and storage of the bianisotropic scalar Green's function. The computation is based on an expansion of the Green's function into Chebyshev polynomials. The analytical properties of these polynomials are exploited to allow the accurate computation of the derivatives of the Green's function as well as the Green's function itself. For lossy materials, the proposed computation strategy is provably robust. In addition, a multilevel storage scheme with a favorable complexity, based on the Chebyshev polynomial expansion, is proposed for the storage of the expansion coefficients. Numerical results showcase the accuracy and computational complexity of the proposed algorithms.

Razor blade testing schemes have been proposed in the past for both the EFIE and MFIE. The regularity of these testing functions is, strictly speaking, not sufficient for the discretization to be conforming. However, as will be shown in the contribution, it does yield physical solution currents at low frequencies. This is similar to the low-frequency behavior of the mixed discretization of the MFIE. Nevertheless, in this testing scheme, there is no refined mesh, the impedance integrals are triple integrals instead of quadruple integrals and, in addition, the testing functions are constant instead of linear on their support.

The mixed MFIE and Caldeŕon preconditioned EFIE both can be used to accurately model the scattering of time-harmonic electromagnetic waves by two-dimensional perfect electrical conductors. In the case those conductors are bounded by smooth surfaces, the spectra of the linear systems are clustered around a single non-zero finite value. This configuration is optimal for the iterative solution of these systems by iterative algorithms. Regrettably, it has been demonstrated that this optimal configuration is lost when the methods are applied to scattering by non-smooth surfaces. In this case, the spectrum tends to spread out, negatively influencing the number of iterations required for iterative solvers to converge. In this contribution, this spreading out of the spectrum is studied quantitatively. It is shown that even though the spectrum spreads out, it remains bounded away from zero and oriented along the negative real axis. It can be concluded that iterative solution remains an option, even for non-smooth geometries. In the case the geometry is so complicated that the spectrum is bounded away from zero by only a very small distance, further preconditioning may be required. Here, a quasi-block diagonal preconditioner is introduced that will compress the spectrum. It is explained how this preconditioner can be applied efficiently as expansion in a Neumann series.

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.

Distributed-memory parallelization of the multilevel fast multipole algorithm (MLFMA) relies on the partitioning of the internal data structures of the MLFMA among the local memories of networked machines. For three existing data partitioning schemes (spatial, hybrid and hierarchical partitioning), the weak scalability, i.e., the asymptotic behavior for proportionally increasing problem size and number of parallel processes, is analyzed. It is demonstrated that none of these schemes are weakly scalable. A nontrivial change to the hierarchical scheme is proposed, yielding a parallel MLFMA that does exhibit weak scalability. It is shown that, even for modest problem sizes and a modest number of parallel processes, the memory requirements of the proposed scheme are already significantly lower, compared to existing schemes. Additionally, the proposed scheme is used to perform full-wave simulations of a canonical example, where the number of unknowns and CPU cores are proportionally increased up to more than 200 millions of unknowns and 1024 CPU cores. The time per matrix-vector multiplication for an increasing number of unknowns and CPU cores corresponds very well to the theoretical time complexity.

A method is presented for reducing the number of subsequent integrations in impedance integrals containing the dynamic Green's function. The method works whenever a point can be found that is coplanar (or collinear) to both the basis and test support. This is always the case for singular impedance integrals. As an example, explicit formulas are given for the self-patch impedance integrals for the electric field integral equation.