A. J. Pray

Stability Properties of the Time Domain Electric Field Integral Equation Using a Separable Approximation for the Convolution With the Retarded Potential

The state of art of time domain integral equation (TDIE) solvers has grown by leaps and bounds over the past decade. During this time, advances have been made in (i) the development of accelerators that can be retrofitted with these solvers and (ii) understanding the stability properties of the electric field integral equation. As is well known, time domain electric field integral equation solvers have been notoriously difficult to stabilize. Research into methods for understanding and prescribing remedies have been on the uptick. The most recent of these efforts are (i) Lubich quadrature and (ii) exact integration. In this paper, we re-examine the solution to this equation using (i) the undifferentiated form of the time domain electric field integral equation (TDEFIE) and (ii) a separable approximation to the spatio-temporal convolution. The proposed scheme can be constructed such that the spatial integrand over the source and observer domains is smooth and integrable. As several numerical results will demonstrate, the proposed scheme yields stable results for long simulation times and a variety of targets, both of which have proven extremely challenging in the past.

A Higher Order Space-Time Galerkin Scheme for Time Domain Integral Equations

Stability of time domain integral equation (TDIE) solvers has remained an elusive goal for many years. Advancement of this research has largely progressed on four fronts: 1) Exact integration, 2) Lubich quadrature, 3) smooth temporal basis functions, and 4) space-time separation of convolutions with the retarded potential. The latter methods efficacy in stabilizing solutions to the time domain electric field integral equation (TD-EFIE) was previously reported for first-order surface descriptions (flat elements) and zeroth-order functions as the temporal basis. In this work, we develop the methodology necessary to extend the scheme to higher order surface descriptions as well as to enable its use with higher order basis functions in both space and time. These basis functions are then used in a space-time Galerkin framework. A number of results are presented that demonstrate convergence in time. The viability of the space-time separation method in producing stable results is demonstrated experimentally for these examples.

A Singularity Cancellation Technique for Weakly Singular Integrals on Higher Order Surface Descriptions

Accurate integration of singular and near-singular functions is critical to the accuracy of the method of moments solution to surface integral equations. While this problem has been widely addressed for flat geometries, its extensions to higher order surface descriptions have been limited. This letter provides a systematic prescription for the application of the rules for weakly singular integrals on higher order surfaces. Here, we present implementation details and several demonstrative results that compare the accuracy and convergence of the integration rules.

Analysis of transient scattering from PEC objects using the Generalized Method of Moments

Surface integral formulations for the anlysis of transient scattering have become very prevalent over the recent years. These formulations are typically discretized by employing basis functions that are products of spatial and temporal functions. While the design and construction of optimal, stable and accurate temporal basis functions have been the object of considerable study, the de-facto standard for spatial discretization is the traditional set of Rao-Wilton-Glission (RWG) functions. These functions have been carried over to time domain integral equations (TDIE) from their frequency domain counterparts and have been shown to be highly effective in a variety of problems. These functions are defined on a tessellation of the geometry and have spawned a set of higher order and singular variants, though these variations have seen limited use in time domain. The close marriage between the basis function construction and the tessellation stems from the need to satisfy continuity constraints across the interior edges of the tessellation. While this has several advantages, this nature of construction is inherently restrictive. Indeed, even the use of mixed orders of polynomial basis functions is significantly involved and usually requires the careful design of patches and “transition” areas. In the frequency domain, a technique called the Generalized Method of Moments (GMM) was recently introduced to resolve some of these problems [1]. The GMM was designed as an umbrella framework to include arbitrary functions in the basis space. In addition, [1] demonstrates that the basis functions proposed result in a matrix system with stable condition numbers to very low frequencies. In this work, we develop the GMM scheme for the discretization of the TDIE. We will describe a scheme for the extension of the GMM to time domain using products of GMM functions (for spatial basis functions) and interpolatory polynomials (for temporal basis functions). First, we will show that these basis functions provide accurate scattering results over a wide variety of geometries. Further, we will demonstrate the ability of the GMM scheme to arbitrarily mix basis functions across the geometry. We will show that using different basis functions in different areas of the tessellation can be trivially achieved by simply changing an input flag. While the results obtained in this paper are using the magnetic field equation, it will also be shown at the conference that this scheme leads to a stable discretization scheme for the electric field integral equation (EFIE) as well.

Time Domain Integral Equation solver for composite scatterers using a separable expansion for convolution with the retarded potential

Time Domain Integral Equations (TDIEs) have been plagued by instabilities since their inception. The computational community has invested significant time and effort in attempts to remedy the issue to varying success. Recent work on exact integration methods has shed further light on the importance of accurate matrix evaluation to TDIE stability, but their extension to higher order geometries is highly nontrivial. This was largely the inspiration for a method which relied on a separable approximation in space and time of the convolution with the retarded potential. This method was shown to produce late time stable results on a variety of perfectly electrically conducting objects. This work presents the extension of this method to composite scatterers.

A stable higher order TDIE solver using a separable approximation for convolution with the retarded potential

Since the initial development of Time-Domain Integral Equations (TDIEs), the issue of instability has limited their use by computational practitioners. Significant progress notwithstanding, there is still a need for theoretical understanding of the causes of instability, their remedies (in the continuous case) and solution schemes applicable to electrically large, complex structures. It was in an attempt to remedy this that a method based on a separable approximation for the convolution with the retarded potential was recently developed. In this work, we seek to reformulate the Time Domain Electric Field Integral Equation (TDEFIE) in a manner than can be shown to produce bounded currents. We also present a number of results using the non-augmented TDEFIE to demonstrate the methods effectiveness on higher order surfaces.

A singularity cancellation technique on arbitrary higher order patch descriptions

Accurate integration of singular and near-singular functions is critical to the accuracy of the method of moments solution to integral equations. While this problem has been widely addressed for flat geometries, their extensions to higher order surface descriptions have been limited. This paper provides a comprehensive prescription for the application of the singularity cancellation schemes to higher order surfaces. We present implementation details and demonstrative results.

A separable approximation for convolution with the retarded Green's function and its application to time domain integral equations

A scheme for the evaluation of retarded potential integrals is presented wherein the convolution of the retarded potential Greens function with the basis functions is approximated using an expansion of separable functions in time and space. The proposed scheme renders the spatial integrands required in this computation smooth over both the source and testing domains, which is typically not true in the traditional TDIE formulation. Numerical experiments have been performed to quantify the error in this approximation and results to a selection of canonical geometries are presented.

A stable, PWTD-accelerated time domain integral equation solver

Time domain analysis of electromagnetic scattering and radiation problems has a wide variety of applications from nonlinear circuits and antenna feeds to electromagnetic pulse studies. The ability to model nonlinearities and to perform broadband analysis makes time domain methods attractive in comparison to their frequency domain counterparts. An integral equation framework for time domain analysis also offers a number of benefits, such as satisfying radiation boundary conditions, being dispersionless, and providing an effective method for truncating the computation domain in differential equation based solvers.

A stable higher order space time Galerkin marching-on-in-time scheme

We present a method for the stable solution of time-domain integral equations. The method uses a technique developed in [1] to accurately evaluate matrix elements. As opposed to existing stabilization schemes, the method presented uses higher order basis functions in time to improve the accuracy of the solver. The method is validated by showing convergence in temporal basis function order, time step size, and geometric discretization order.