Solution Of Time-domain Integral Equations Research Articles

Integral Accuracy and the Stability of Two Methods for the Solution of Time-Domain Integral Equations for Scattering From Perfect Conductors

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.

Laguerre–Galerkin methods with reduced sum-products

The always stable solution of time-domain integral equations by plain implementation of the classical marching-on-in-degree (MOD) methods demands O(M2N3) operations, where M and N are, respectively, the spatial and temporal degrees of freedom. Diverse caching and basis reordering approaches are first explored to reduce the number of matrix–vector multiplications necessary to compute the right hand side of the MOD system. The new schemes primarily eliminate the existing O(N) innermost vector summations on all past polynomial orders. This complexity reduction facilitates the usage of FFT approaches to accelerate the calculation of existing temporal convolutions. In an additive secondary stage, the recursive time convolution products of the Toeplitz block aggregates of the retarded interaction matrices are expedited by array multiplications in spectral domain. When the translation invariancy on time-order indices are placed on the outermost possible nested Toeplitz levels due to the space shift invariancy, the overall computational complexities and memory requirements can be further reduced to O(MNlog(MN)) and O(MN), respectively.

A Higher Order Nyström Scheme for a Marching-on-in-Time Solution of Time-Domain Integral Equation

A marching-on-in-time (MOT) solution of the time-domain integral equation (TDIE) using the higher order Nystrom scheme is developed to analyze the transient electromagnetic scattering from arbitrarily shaped perfect electrically conducting (PEC) objects. In this scheme, the curvilinear triangular patches are utilized to discretize the PEC objects for their great ability to model geometric configuration. Moreover, we use the Lagrange interpolation polynomials to stand for the spatial basis functions, and the triangle functions to represent the temporal basis functions. Furthermore, the spatial and temporal point-matching is applied. Numerical examples demonstrate that the proposed scheme is stable, accurate, and efficient.

A Higher Order Hierarchical Solution to the Time-Domain Electric Field Integral Equation for Thin-Wire Antennas

In this paper, a new time-domain integral equation solution method for arbitrarily shaped, perfectly conducting thin-wire antennas is presented. The space-time discretization of the circumferentially invariant electric current along the wire axis is achieved by employing higher order hierarchical polynomials for the spatial basis functions and discretely causal piecewise polynomial interpolation functions for the temporal basis functions. The solution of the transient current along the wire is obtained by marching-on-in-time (MOT). The hierarchical nature of the spatial basis functions enables the efficient assembly of the space-time impedance matrices. Rigorous evaluation of the time-domain wire kernels is presented. Numerical test cases demonstrate the validity, accuracy, and applicability of the new method.

Robust solution of time-domain integral equations using loop-tree decomposition and bandlimited extrapolation

A stabilization method that eradicates low frequency instabilities in the solution of the time-domain integral equations of electromagnetic scattering is presented. The method uses the loop-tree decomposition originally suggested for the solution of low frequency scattering problems via the method of moments. Specifically, a temporally differentiated form of the pertinent integral equation is tested using tree basis functions, and the undifferentiated form is tested using solenoidal basis functions. The underlying solution method uses bandlimited interpolation functions (BLIFs) and the higher-order divergence-conforming vector bases of Graglia et al. for the temporal and spatial discretization of the integral equations, respectively. An extrapolation technique has been implemented to overcome the noncausality introduced into the system by the BLIFs. Numerical results will demonstrate the stability and accuracy of the proposed technique.

An Accurate Scheme for the Solution of the Time-Domain Integral Equations of Electromagnetics Using Higher Order Vector Bases and Bandlimited Extrapolation

Despite the numerous advances made in increasing the computational efficiency of time-domain integral equation (TDIE)-based solvers, the stability and accuracy of TDIE solvers remain problematic. This paper introduces a new numerical method for the accurate solution of TDIEs for scattering from arbitrary perfectly conducting surfaces. The work described in this paper uses the higher order divergence-conforming basis functions of Graglia et al. for spatial discretization and bandlimited interpolation functions for the temporal discretization of the relevant integral equations. Since the basis functions used for the temporal representation are noncausal, an extrapolation scheme is employed to recover the ability to solve the problem by marching on in time. Numerical results demonstrate that the proposed method is stable and that it exhibits superlinear convergence with regard to the spatial discretization and exponential convergence with respect to the temporal discretization.

Implicit solution of time-domain integral equations for arbitrarily shaped dielectric bodies

Implicit solution of time-domain integral equations for arbitrarily shaped dielectric bodies

Finite-difference approach to the solution of time-domain integral equations for layered structures

A numerical algorithm for the analysis of transient electromagnetic fields in planar structures is proposed based on the time-domain magnetic-field integral equation (MFIE), electric-field integral equation (EFIE), and the marching-on-in-time approach. The field vectors are represented in terms of vector potential functions which are calculated either by integration or by the three-dimensional (3-D) wave equation according to the geometry of the structure. Thus, the algorithm combines the advantages of integral equation techniques and finite-difference schemes. While this approach is applicable to any geometries, it is especially suitable for multilayered planar structures and is competitive to the finite-difference time-domain (FDTD) method in the case of open and radiating problems. Theoretical results are verified by the analysis of a pulse propagation in a homogeneous open-end microstrip line.

A least-square iterative technique for solving time-domain scattering problems

An alternative method for the numerical solution of time-domain integral equations pertaining to three-dimensional, transient scattering problems is developed and applied to two representative acoustic scattering configurations. It is based on the iterative minimization of an integrated square error. For the cases considered, the method gives more accurate results than the frequently used ‘‘marching-on-in-time’’ method (within comparable computation times).

Time domain integral equation solution for acoustic scattering from fluid targets

The responses of fluid targets are computed using space–time integral equations formulated in the time domain. The incident pressure wave is a ’’smoothed-impulse’’ with a Gaussian shaped time dependence whose width is of the order of a target dimension. A space–time integral equation for the pressure field on the outside of the target surface and a space–time integral equation for the pressure field on the inside of the target surface are solved simultaneously for the pressure and the pressure gradient by stepping on in time and making use of the boundary conditions (continuity of pressure and displacement). The farfield is then computed from these source fields. The technique is applicable to targets of arbitrary contour and is demonstrated for a sphere and right circular cylinder at various angles of incidence. Fluid targets support interior compression waves; sound-hard and sound-soft targets are treated as limiting cases of this formulation. The technique has been verified for the case of a sphere by comparison with the response computed by classical expansions.