The stability of time-domain (TD) integral equation (TDIE) approaches to the computation of electromagnetic scattering is profoundly affected by the accuracy of the underlying numerical integration methods used for computation of the kernel elements. In most publications, the accuracy of the quadratures used to compute kernel elements is never measured, and higher order computation is assumed to deliver high accuracy. In this communication, we examine the complicated relationship between the actual accuracy of kernel element computation and the resulting stability of integral equations. The numerical results show that stability may be improved for higher integral accuracy. Although integral accuracy is not always improved by higher order integration rules, more careful integration (as delivered by adaptive integration methods) is often helpful. The numerical results for a range of problems demonstrate these contentions.

Stability of time domain integral equation approaches to the computation of electromagnetic scattering is profoundly affected by the accuracy of the underlying numerical integration methods used for computation of the kernel elements. In most publications, the accuracy of the quadratures used to compute kernel elements is never measured, and higher order computation is assumed to deliver high accuracy. In this paper, we examine the complicated relationship between the actual accuracy of kernel element computation and the resulting stability of integral equations. Numerical results show that stability may be improved for a higher integral accuracy. Moreover, while integral accuracy is not always improved by higher order integration rules, more careful integration (as delivered by adaptive integration methods) is often helpful. Numerical results for a range of problems demonstrate these contentions.

A balanced (Weston-type) absorber, characterized by both electric and magnetic loss mechanisms, can be placed within the null field inside the equivalent surface currents replacing a perfectly conducting or homogeneous scatterer. The absorber, referred to as the filler, substantially reduces interactions between pairs of opposing basis/testing functions. The resultant moment matrix, formulated with the filler Green's function, is thinned accordingly. Moreover, most annulled elements need not be computed at all, thereby reducing substantially the matrix fill time. The lossy nature of the Green's function also serves to eliminate spurious internal resonances and thus makes the electric or magnetic field integral equation matrix well conditioned without resorting to a combined field integral equation.

The form of Maxwell's equations when expressed in material coordinates has been debated for decades. The most commonly accepted formulation is known to be inconsistent, even to first order approximation. It has been shown that self-consistent formulations are possible when convected coordinates are used. In this presentation, we review these formulations. We then demonstrate their use in solving for the coupled EM/Structural response of a simple parallel plate capacitor bent into an arc.

A balanced (Weston-type) absorber, characterized by both electric and magnetic loss mechanisms, can be placed within the null field inside the equivalent surface currents replacing a perfectly conducting or homogeneous scatterer. The absorber, referred to as the filler, substantially reduces the interactions between pairs of opposing basis/testing functions. The resultant moment matrix, formulated with the filler Green's function (GF), is thinned accordingly. Moreover, most annulled elements need not be computed at all, thereby reducing substantially the matrix fill time. The lossy nature of GF also serves to eliminate spurious internal resonances and thus makes the electric or magnetic field integral equation matrix to be well conditioned without resorting to a combined field integral equation.

Because the analysis of the time domain integral equations of electromagnetics is complicated and cannot easily take into account all sources of error in any given implementation, judgements of stability tend to rest on experience. While such an approach eliminates the worst methods immediately, subtle implementation issues may affect the stability of more promising approaches. In this work, we examine the effects of integration rules on stability. Numerical results demonstrate that increasing accuracy does not always increase stability.

The integration rules and orders for the computation of kernel elements directly effect the stability of the discretization of time domain integral equations for electromagnetic analysis. Our previous works examined this effect for closed objects using an analysis based on the combined field integral equation. In this work, we use adaptive quadrature to show the impact of integration accuracy in kernel computation on stability for open structures analyzed with the electric field integral equation. Numerical results show a high integral order does not necessarily bring a high accuracy, and that a higher accuracy in the near field integration can be helpful to achieve stability.

This work proposes a new approach to determine the spatial location and orientation of an object using measurements performed on the object itself. The on-board triangulation algorithm we outline could be implemented in lieu of, or in addition to, well-known alternatives such as Global Positioning System (GPS) or standard triangulation, since both of these correspond to significantly different geometric pictures and necessitate different hardware and algorithms. We motivate the theory by describing situations in which on-board triangulation would be useful and even preferable to standard methods. The on-board triangulation algorithm we outline involves utilizing dumb beacons which broadcast omnidirectional single frequency radio waves, and smart antenna arrays on the object itself to infer the direction of the beacon signals, which may be used for onboard calculation of the position and orientation of the object. Numerical examples demonstrate the utility of the method and its noise tolerance.

Because the mathematical analysis of time domain integral equations is fraught with difficulty, researchers are often forced to use experimental means to characterize the stability of their approaches. The difficulty inherent in this approach is that the variables affecting the success or failure of an experiment necessarily depend on a computer implementation of potentially hundreds of thousands of lines of instructions hiding unknown assumptions and even errors. In this work, the importance of different integration orders for near and far basis functions is investigated, and demonstrated to have an enormous effect on the stability of the implementation. Methods previously thought unstable for difficult problems are shown reliable with tiny changes over broad choices of parameter values.

A balanced (Weston-type) absorber with both electric and magnetic conductances can be placed within the null field inside a scatterer, helping reduce interactions between pairs of opposing basis functions. The resultant MoM matrix is substantially thinned. Moreover, the annulled elements need not be computed, thereby reducing the matrix fill-up time.

A balanced absorber with both electric and magnetic conductances can be placed within the null field inside a scatterer, helping reduce interactions between pairs of opposing basis functions. The resultant MoM matrix is severely thinned with all but a few percent of the original elements being annulled.

Time domain integral equations have become a major tool in the computational analysis of electromagnetic scattering problems. Classically it was difficult to ensure a stable numerical solution of standard boundary integral formulations of the problem. In this chapter we shall discuss both theoretical and numerical aspects of one approach that solves the stability problem: convolution quadrature. We start with scattering from a perfectly conducting object and develop the electric field integral equation, as well as an error analysis of the fully discrete problem using finite elements in space. After presenting a brief discussion of some special numerical features of this problem, and some numerical results, we move on to scattering by a penetrable object. We end with a general discussion of computational electromagnetism illustrating the role that time domain integral equations can play.

Time-domain integral-equation methods for the simulation of electromagnetic wave scattering have historically been subject to two sources of instability. The first source of instability causes the solution yielded by the process to oscillate wildly, and is caused by spatial integrations that are unable to capture the rapid change in the integrand at shadow boundaries. The second source of instability results in a slow growth of the current on the structure, and is due to the ill-conditioned nature of the electric-field integral equation at low frequencies. In this work, the shadow region instability is eliminated using a convolution quadrature method, and the low-frequency difficulties are improved using a loop-tree decomposition. Numerical results will demonstrate the efficacy of the combination.

Convolution quadtrature (CQ) is a method for discretizing continuous convolution integrals by substituting a discrete Ƶ domain approximation for the Laplace domain frequency parameters. The model CQ provides is inherently dispersive, and so gives rise to a discrete Green's function with expanding temporal support. This work investigates two approaches to alleviating this problem: dispersion halting and fast Fourier transform methods. Numerical results will be used to compare the methods with each other in both dispersive and nondispersive media.

The convolution quadrature (CQ) method of temporal discretization has several properties that make it an ideal method for the construction of marching-on-in-time schemes (Q. Chen, P. Monk, X. Wang, and D. S. Weile, Commun. Comput. Phys, 11, 383–399, 2012). In particular, the temporal discretization they yield is always renders accurate spatial integrations without any sharp shadow transition, and they can represent the Green's functions in complicated, dispersive media with no special difficulty (X. Wang and D. S. Weile, IEEE Trans. Antenn. Propag., 59, 4651–4663, 2011). Finally, because of their close relationship with finite difference methods, they are ideal for the construction of boundary operators for FDTD (Y. Q. Lin and D. S. Weile, IEEE Trans. Antenn. Propag., 61, 2655–2663, 2014; S. Malevsky, E. Heyman, and R. Kastner, IEEE Trans. Antenn. Propag., 58, 3602-3609, 2010.)

Despite the importance of electromagnetomechanical physics to processes ranging from piezoelectricity to the dynamics of electron beams, confusion abounds in the continuum mechanics literature as to how Maxwell’s equations of electrodynamics should be formulated in the material frame of continuum mechanics. Current formulations in the literature conflict as to the manner in which the authors define fields, derive constitutive relations, and interpret contradictory formulations. The difficulties persist even when the phenomena described are electrostatic. This paper will demonstrate that the perplexity arises from two sources: a misunderstanding of the limitations of material frame descriptions, and the failure to appreciate the centrality of relativity theory to the formulation of electrodynamic equations in the vicinity of mechanical motion. Two new formulations of Maxwell’s equations are provided that avoid the paradoxes of earlier formulations and thus describe the physics clearly and without self-contradiction.

Convolution quadrature (CQ) methods are one of a host of new techniques for creating stable implementations of time domain integral equations. Like all such methods, CQ methods provide an accurate approach for spatial integration in view of the natural shadow region created by the Galerkin testing process. The success of CQ in this regard, however, does nothing to eliminate the slowly growing instability connected with the separation of electromagnetic problems into electrostatic and magnetostatic problems at low frequency. In this work, therefore, we add the loop-tree low frequency stabilization technique to the CQ method. Numerical results demonstrate the stability and accuracy of the technique.