Time-domain Integral Equation Research Articles

An efficient algorithm for time-domain acoustic scattering in three dimensions by layer potentials

In this paper, we develop a simple and fast algorithm for the time-domain acoustic scattering by a sound-soft or an impedance obstacle in three dimensions. We express the solution to the scattering problem by layer potentials, and then a time-domain boundary integral equation is derived. To numerically solve the resulting boundary integral equation, we propose a full discretization scheme by combining the convolution splines with a Galerkin method. In time, we approximate the density in a backward manner in terms of the convolution splines. In space, we project the density at each time onto the space of spherical harmonics, and then use the spatial discretization of a Nyström type on the surface of an obstacle which is homeomorphic to a sphere. A gallery of numerical examples are presented to show the efficiency of our algorithm. The stability, convergence and accuracy of the algorithm are discussed.

Bragg Resonance of Surface Gravity Waves by Surface-Piercing Vertical Multi-Plate Breakwaters

Abstract The interaction between surface gravity waves and surface-piercing vertical multi-plate breakwaters in finite water depth is investigated under the assumption of potential flow theory. The time-domain desingularized boundary integral equation method (DBIEM) is adopted to solve the boundary value problem. A decomposition method based on the superposition principle is used to obtain reflected waves. The effect of the vertical plates as a floating breakwater by making full use of Bragg resonance is analyzed. Initially, the waveforms calculated without breakwaters are compared with second-order Stokes regular wave to obtain the appropriate desingularization distance. Then, the interaction of surface gravity waves with the breakwaters with the single plate, double, and triple plates are analyzed. The transmission and reflection coefficients of the breakwater with the single plate are verified with the analytical solutions of previous literature. The transmission and reflection coefficients of the breakwaters with double and triple plates are verified with experimental results. Finally, the transmission and reflection coefficients of surface-piercing vertical multi-plate breakwaters with different plate numbers and drafts are also calculated. The occurring condition of Bragg resonance and its values affected by the plate number and the draft are analyzed. The present results are reference values for the optimal design of vertical plates as a floating breakwater for offshore and coastal engineering applications by fully using Bragg resonance.

Analysis of electromagnetic scattering from array of time-modulated graphene ribbons.

An accurate and fast method is presented for the analysis of scattering of electromagnetic waves from an array of time-modulated graphene ribbons. We derive a time-domain integral equation for induced surface currents under subwavelength approximation. Using the method of harmonic balance, this equation is solved for a sinusoidal modulation. The solution of the integral equation is then used to obtain the transmission and reflection coefficients of time-modulated graphene ribbon array. The accuracy of the method was verified through comparison with results of full-wave simulations. In contrast with previously reported analysis techniques, our method is extremely fast and can analyze structures with a much higher modulation frequency. The proposed method also provides interesting physical insights useful for designing novel applications and opens up new vistas in the fast design of time-modulated graphene-based devices.

Port Parameter Extraction-Based Self-Consistent Coupled EM-Circuit FEM Solvers

Self consistent solution to electromagnetic (EM)-circuit systems is of significant interest for a number of applications. This has resulted in exhaustive research on means to couple them. In time domain, this typically involves a tight integration (or coupling) with field and non-linear circuit solvers. This is in stark contrast to coupled analysis of linear/weakly non-linear circuits and EM systems in frequency domain. Here, one typically extracts equivalent port parameters that are then fed into the circuit solver. Such an approach has several advantages; (a) the number of ports is typically smaller than the number of degrees of freedom, resulting in cost savings; (b) is circuit agnostic; (c) can be integrated with a variety of device models. Port extraction is tantamount to obtaining impulse response of the linear EM system. In time domain, the deconvolution required to effect this is unstable. Recently, a novel approach was developed for time domain integral equations to overcome this bottleneck. We extend this approach to time domain finite element method, and demonstrate its utility via a number of examples; significantly, we demonstrate that self consistent solutions obtained using either a fully coupled or port extraction is identical to the desired precision for non-linear circuit systems. This is shown within a nodal network. We also demonstrate integration of port extracted data directly with drift diffusion equation to model device physics.

Moving force identification of simply supported bridges through the integral time domain method

This paper presents a new form of inversion matrix, which aims to obtain more accurate moving force identification results from the responses of a simply supported bridge deck to vehicles passing over it. The load identification process of the modal superposition method discretizes the time-domain integral equation of the load response, obtains the load inversion matrix, and converts the load identification problem into a linear equation solving problem. The inversion matrix formation process of this traditional time domain method (TDM) can introduce the load compression error and the discretization error of the unit impulse response function, which affect the load solution. To reduce the source of error and maintain the stability of the solution, the discretized unit impulse response function is replaced by an integral form, thereby eliminating the discretization error. This method is called the integral time domain method (ITDM). A supported beam subjected to a single force and two forces is illustrated as a numerical example. Through numerical analysis, the effects of the two methods in single time-varying force identification are analyzed. Using two moving forces analysis, when noise is included, the influence of discretization error on the load solution is analyzed. Numerical results show that the discretization error has a greater impact on the load inversion using the acceleration response than the bending moment and displacement response inversion, and the ITDM should be used to avoid the interference of discrete error.

Implicit/Explicit, BEM/FEM Coupled Scheme for Acoustic Waves with the Wave Equation in the Second Order Formulation

Abstract Acoustic scattering of waves by bounded inhomogeneities in an unbounded homogeneous domain is considered. A symmetric coupled system of time-domain boundary integral equations and the second order formulation of the wave equation is described. A fully discrete system consists of spatial discretization by boundary and finite element methods (BEM/FEM), leapfrog time-stepping in the interior, and convolution quadrature for the boundary integral equations. Convolution quadrature is based on BDF2, trapezoidal rule, or a newly introduced truncated trapezoidal rule that has some favourable properties for both the implementation and quality of approximate solution. We give a stability and convergence analysis under a CFL condition of the fully discrete system. The theoretical results are illustrated by numerical experiments in two dimensions.

On the spurious resonance modes of time domain integral equations for analyzing acoustic scattering from penetrable objects.

The interior resonance problem of time domain integral equations (TDIEs) formulated to analyze acoustic field interactions on penetrable objects is investigated. Two types of TDIEs are considered: The first equation, which is termed the time domain potential integral equation (TDPIE), suffers from the interior resonance problem, i.e., its solution is replete with spurious modes that are excited at the resonance frequencies of the acoustic cavity in the shape of the scatterer. Numerical experiments demonstrate that, unlike the frequency-domain integral equations, the amplitude of these modes in the time domain could be suppressed to a level that does not significantly affect the solution. This is achieved by increasing the numerical solution accuracy through the use of a higher-order discretization in space and the band limited approximate prolate spheroidal wave function with high interpolation accuracy as basis function in time. The second equation is obtained by linearly combining TDPIE with its normal derivative. The solution of this equation, which is termed the time domain combined potential integral equation (TDCPIE), does not involve any spurious interior resonance modes but it is not as accurate as the TDPIE solution at non-resonance frequencies. In addition, TDCPIE's discretization calls for treatment of hypersingular integrals.

Induction mechanism and propagation characteristics of current in line conductors due to an electron avalanche

Transfer of bulk electric power required in the modern world can be realized only through Extra High Voltage and Ultra High Voltage transmission lines. The scenario is dominated by overhead lines in which electromagnetic noise generated by corona is an important concern. Corona-induced currents propagate along line conductors producing electromagnetic noise, which is essential to be quantified. Towards this end, the present work aims to investigate basic mechanism of current induction using an isolated avalanche developing under space charge modulated background field. Beginning with the complete electric field produced by an avalanche, the corresponding induced currents are quantified, and the structure of electric and magnetic fields is extracted. The well-known time-domain electric field integral equation is employed for the purpose. The basic issue with long transmission line modelling is amply demonstrated. Since single avalanche is the basic mechanism behind corona, general inferences drawn are applicable for corona on transmission lines.

Modeling of Resonant Tunneling Diode Oscillators Based on the Time-Domain Boundary Element Method

We demonstrate how the coupling of a full-wave time-domain boundary element method (BEM) solver with a circuit solver can be used to model 1) the generation of high frequency oscillations in resonant tunneling diode (RTD) oscillators, and 2) the mutual coupling and synchronization of non-identical RTDs with significant differences in frequencies to achieve coherent power combination. Numerical simulations show a combined output power of up to 3.7 times a single oscillator in synchronized devices. The non-differential conductance of the RTD is modeled as a lumped component with a non-linear current-voltage relationship. The lumped element is coupled to the radiating structure using a finite-gap model in a consistent and discretisation independent manner. The resulting circuit equations are solved simultaneously and consistently with time-domain electric field integral equations that model the transient scattering of electromagnetic (EM) fields from conducting surfaces that make up the device. This paper introduces three novel elements: (i) the application of a mesh independent feed line to the modelling of feed lines of RTD devices, (ii) the coupling of the radiating system to a strongly non-linear component with negative differential resistance, and (iii) the verification of this model with circuit models where applicable and against the experimental observation of synchronisation when two RTDs are placed in close proximity. These three elements provide a methodology that create the capacity to model RTD sources and related technology.

A Novel Framework of Singularity Cancellation Transformations for Strongly Near-Singular Integrals

In the solution of frequency- and time-domain integral equations, the singularity cancellation transformations are well-known for the treatments of the singular integrals involving Green’s functions and their gradients. The cancellation transformations for strongly near-singular integrals become inaccurate and inefficient for an extremely deformed triangular patch, and this phenomenon is referred to as shape-dependent problem in this article. As the degree of the singularity increases, the shape-dependent problem of strong near-singularity is more severe than that of the weakly near-singular integrals. Furthermore, if the source triangle is deformed, the accuracy of the cancellation methods decreases when no near singularity exists. In this work, we first investigate the reason for these problems via the theoretical analysis with numerical verification. Second, an updated framework of singularity cancellation methods for strongly near-singular integrals is proposed, which has a fast and consistent convergence rate for both regular and irregular triangles. Third, some numerical experiments are presented to illustrate the effectiveness of the theoretical framework and the proposed transformations.

Pulsed Electromagnetic Scattering by Metasurfaces—A Numerical Solution Based on the Cagniard–DeHoop Method of Moments

A novel time-domain (TD) integral-equation (IE) technique for analyzing the pulsed electromagnetic (EM) scattering by a 2-D metasurface, represented by a thin highly contrasting screen with combined dielectric and magnetic properties, is presented. The problem under consideration is formulated with the aid of the EM reciprocity theorem of the time-convolution time. Subsequently, upon discretizing the space–time solution domain, the reciprocity relation is cast into two independent matrix forms whose elements are derived analytically using the classic Cagniard–DeHoop (CdH) technique. The thus obtained convolution-type systems of equations are solved for induced electric and magnetic currents through a stable step-by-step updating procedure. The presented numerical method is finally validated using alternative TD solutions.

Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements

Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided.

Target Identification with Improved 2D-VMD for Carrier-Free UWB Radar.

In recent years, the interest in radar automatic target recognition (RATR) based on the carrier-free ultra-wideband (UWB) radar has been increasing. Compared with narrow-band and other bandwidth radars, the echo signal of the carrier-free UWB radar includes more comprehensive and detailed information with respect to the targeted object. In this paper, we first utilized 3ds Max to acquire accurate geometric models and applied a time-domain integral equation (TDIE) for echo signal acquisition under the condition that the transmitted signals had an extremely short duration period. By comparing the simulated waveform with the actual one, the accuracy of the electromagnetic modeling is verified. Furthermore, given that the actual environment is full of noise and clutter, we propose an improved two-dimensional variational mode decomposition (2D-IVMD), and an algorithm is proposed to eliminate noise and extract edge features preliminarily, which lays a foundation for further in-depth feature extraction. Then, the deep conventional neural network (DCNN) is introduced for the final recognition. The results show that the proposed methods achieve promising classification performance under the condition of low signal-to-noise ratio (SNR) values.

Potential-Based Time Domain Integral Equations Free From Interior Resonances

Potential-based integral equations are being explored to develop numerical methods that avoid low-frequency breakdown issues so that they can analyze multiscale structures that are becoming increasingly important in electromagnetic engineering. This work continues the development of potential-based time domain integral equations by presenting three new formulations in the Lorenz gauge that do not support interior resonances. These modifications allow the methods to be applied to deeply multiscale structures of practical importance that contain significantly subwavelength- and wavelength-sized geometric features. Appropriate marching-on-in-time discretization schemes are developed for each formulation that fully conform to the temporal Sobolev space properties of the integral equations. It is shown that following this approach leads to a discrete system with improved stability properties. Further, it is demonstrated that this new system can produce accurate results over broad bandwidths and analyze deeply multiscale systems that previous potential-based time domain integral equations could not.

Lorenz Gauge Potential-Based Time Domain Integral Equations for Analyzing Subwavelength Penetrable Regions

Potential-based integral equations are being explored to develop numerical methods that avoid low frequency breakdown issues so that they can analyze multiscale and subwavelength structures that are becoming increasingly important in electromagnetic engineering. This work continues the development of potential-based time domain integral equations by presenting a new formulation in the Lorenz gauge that is applicable to penetrable regions. An appropriate marching-on-in-time discretization scheme is developed that fully conforms to the spatial and temporal Sobolev space properties of the integral equations. It is shown that following this approach leads to a discrete system with improved stability properties. Further, it is demonstrated that this new system can produce accurate results for objects that are significantly smaller than a wavelength where previous methods have failed.

A Straight-Forward Method of Moments Procedure to Solve the Time Domain Integral Equation Applicable to PEC Bodies via Triangular Patch Modeling

In this work, a simple and straight-forward method of moments solution (MOM) procedure is presented to obtain the induced current distribution on an arbitrarily-shaped conducting body illuminated by a Gaussian plane wave directly in the time domain using a patch modeling approach. The method presented in this work, besides being stable, is also capable of handling multiple excitation pulses of varying frequency content incident from different directions in a trivial manner. The method utilizes standard Rao-Wilton-Glisson (RWG) functions and simple triangular functions for the space and time variables, respectively, for both expansion and testing. The method adopts conventional MOM and requires no further manipulation invariably needed in standard time-marching methods. The moment matrix generated via this scheme is a block-wise Toeplitz matrix and, hence, the solution is extremely efficient. The method is validated by comparing the results with the data obtained from the frequency domain solution. Several simple and complex numerical results are presented to validate the procedure.

Stable and Accurate Marching-on-in-Time Solvers of Time Domain EFIE, MFIE, and CFIE Based on Quasi-Exact Integration Technique

The development of time-domain integral equation solvers with robust late-time stability properties has been a long-standing quest. Among the many methods that have been investigated, those leveraging quasi-exact integration techniques appear to be most successful. This article presents stable and accurate marching-on-in-time (MOT) solvers for time-domain electric, magnetic, and combined field integral equations (EFIE, MFIE, and CFIE) based on quasi-exact integration techniques. The novel MOT solvers exhibit excellent stability while yielding highly accurate results, as demonstrated by various numerical examples. In addition, the solvers' excellent stability and accuracy properties are used to examine spurious modes encountered when time-domain integral equations are applied to closed surfaces. It is demonstrated that MOT solutions to the time domain EFIE and MFIE often are polluted by spurious modes at the cavity's resonant frequencies whereas those of the CFIE solver are devoid of such contamination.

An Explicit Time Marching Scheme for Efficient Solution of the Magnetic Field Integral Equation at Low Frequencies

An explicit marching-on-in-time (MOT) scheme to efficiently solve the time-domain magnetic field integral equation (TD-MFIE) with a large time step size (under a low-frequency excitation) is developed. The proposed scheme spatially expands the current using high-order nodal functions defined on curvilinear triangles discretizing the scatterer surface. Applying Nyström discretization, which uses this expansion, to the TD-MFIE, which is written as an ordinary differential equation (ODE) by separating self-term contribution, yields a system of ODEs in unknown time-dependent expansion coefficients. A predictor-corrector method is used to integrate this system for the samples of these coefficients. Since the Gram matrix arising from the Nyström discretization is a block-diagonal, the resulting MOT scheme replaces the matrix “inversion” required at each time step by a product of the inverse block-diagonal Gram matrix and the right-hand side vector. It is shown that, on the convergence of the corrector updates, this explicit MOT scheme produces the same solution as its implicit counterpart and is faster for large time step sizes.

A comparison of three temporal basis functions for the time-domain method of moments (TD-MoM)

Three techniques are applied for the resolution of the time-domain electric field integral equation (TD-EFIE) for a thin wire antenna based on the well-known time-domain method of moments. Three temporal bases are used to approximate the unknown temporal variation. The first is derived from Dirac functions, the second from B-spline functions, and the third from Laguerre functions. Piecewise triangular functions are used for the spatial expansion and testing function for the wire structure using Galerkin’s method. The aim is to compare these three techniques in terms of their formulation, complexity, and accuracy. Numerical results are presented to identify which time technique is best suited to solve such transient electromagnetic problems.

High-order, Dispersionless “Fast-Hybrid” Wave Equation Solver. Part I: O(1) Sampling Cost via Incident-Field Windowing and Recentering

This paper proposes a frequency/time hybrid integral-equation method for the time dependent wave equation in two and three-dimensional spatial domains. Relying on Fourier Transformation in time, the method utilizes a fixed (time-independent) number of frequency-domain integral-equation solutions to evaluate, with superalgebraically-small errors, time domain solutions for arbitrarily long times. The approach relies on two main elements, namely, 1) A smooth time-windowing methodology that enables accurate band-limited representations for arbitrarily-long time signals, and 2) A novel Fourier transform approach which, in a time-parallel manner and without causing spurious periodicity effects, delivers numerically dispersionless spectrally-accurate solutions. A similar hybrid technique can be obtained on the basis of Laplace transforms instead of Fourier transforms, but we do not consider the Laplace-based method in the present contribution. The algorithm can handle dispersive media, it can tackle complex physical structures, it enables parallelization in time in a straightforward manner, and it allows for time leaping---that is, solution sampling at any given time $T$ at $\mathcal{O}(1)$-bounded sampling cost, for arbitrarily large values of $T$, and without requirement of evaluation of the solution at intermediate times. The proposed frequency-time hybridization strategy, which generalizes to any linear partial differential equation in the time domain for which frequency-domain solutions can be obtained (including e.g. the time-domain Maxwell equations), and which is applicable in a wide range of scientific and engineering contexts, provides significant advantages over other available alternatives such as volumetric discretization, time-domain integral equations, and convolution-quadrature approaches.