Time-domain Integral Equation Method Research Articles

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.

A New Method to Calculate Induced Current of Thin Wire over Anisotropic Ground under HPM

The time response analysis of wire structures is often carried out in free space or isotropic half-space, but the real ground is usually layered and has anisotropic properties. In this paper, the induced current of a thin wire over layered anisotropic half-space under a high-power microwave (HPM) is calculated by using the time-domain integral equation (TDIE) method. The reflection coefficient of a layered anisotropic medium is obtained by the general transmitting matrix (GTM) method combined with Fourier transform. The variation of the induced current on the thin wire under different incident conditions is analyzed.

Accurate Simulation of Shielding Effectiveness of Metallic Cabins Using an Improved Calderón Preconditioner-Based Time-Domain Integral Equation Method

An efficient simulation of shielding effectiveness of some metallic cabins or enclosures is performed using an improved Calderon preconditioner-based time-domain integral equation method. During its implementation, Rao–Wilton–Glisson and Buffa–Christiansen functions are used as the spatial basis functions for solving the Calderon preconditioned time-domain electric field integral equation, which generate a set of well-conditioned system matrices irrespective of the mesh density. Moreover, a set of approximate prolate spheroidal wave functions is chosen as the temporal basis functions; therefore, both late-time stability and accuracy are improved significantly for simulating electromagnetic pulse responses in some typical cabins. Numerical results are validated in comparison with those obtained by some commercial software. These results are useful for further evaluating electromagnetic environment effects in some complex systems.

Discontinuous Galerkin Method Using Laguerre Polynomials for Solving a Time-Domain Electric Field Integral Equation

A discontinuous Galerkin time-domain electric field integral equation method based on a marching-on-in-degree (MOD) scheme for simulating transient responses of perfect electric conductor objects is proposed in this letter. With the help of half-Rao–Wilton–Glisson spatial basis functions, we can simulate transient scattering responses of three-dimensional structures with either conformal or nonconformal meshes. The weighted Laguerre polynomials are chosen as temporal basis functions and implemented for the MOD scheme. Under such circumstances, the developed algorithm eliminates late-time instabilities and the limitation of Courant–Friedrichs–Lewy criteria. In addition, we adopt a fast-Fourier-transform-based blocking scheme to accelerate the temporal convolution of the MOD procedure. Some numerical examples are given to validate its stability and accuracy.

Electromagnetic Field Analysis of Operation Space of the Switch in the Substation Based on the Time Domain Integration Method

In this paper, the approximate calculation formula is used to calculate the horizontal electric field from lightning which has arbitrary current, with the help of time domain integral equation method solving the lightning induced over voltages on overhead lines. In this paper, the time domain integral equations of horizontal electric field is established. Taking the conductor axis current as the variable, the pulse function is used as the basis function, and the discretization equation is used in space and time. Using time stepping algorithm to solve the resulting linear equations. The calculation process of this method is simple, and it is proved that the method is consistent with the conclusion of the relevant literature.

Nonlinear Analysis of Microwave Limiter Using Field-Circuit Coupling Algorithm Based on Time-Domain Volume-Surface Integral Method

This letter presents the nonlinear analysis of the physical characteristic of a microwave positive intrinsic-negative diode limiter using the field-circuit coupling algorithm based on a time-domain volume-surface integral equation method. In order to accelerate the Newton iteration solution of the nonlinear field-circuit coupling equations, an improved solution scheme of a matrix equation is used here. Moreover, the basic diode charge-control model employed contains both a forward and reverse process as well as emitter recombination. Experimental and numerical results are given to demonstrate the accuracy and the efficiency of the proposed method for the nonlinear analysis of commercial limiter.

Fast Analysis of Transient Electromagnetic Scattering Using the Taylor Series Expansion-Enhanced Time-Domain Integral Equation Solver

This paper concentrates on an efficient fast algorithm for accelerating the marching-on-in-time-based time-domain integral equation method for the analysis of the transient electromagnetic scattering from perfect electrically conducting bodies. Taking advantage of the approximate evaluation of the vector and scalar potential terms and the Taylor series expansion of far field, the computational complexity and memory requirement of the proposed algorithm scale as ${\mathrm{ O}}(N_{t} N_{s} \log N_{s} )$ and ${\mathrm{ O}}(N_{s}^{1.5} )$ , respectively, where $N_{t} $ and $N_{s}$ denote the number of simulation time steps and spatial basis functions, respectively. Several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed algorithm.

A Discontinuous Galerkin Time-Domain Integral Equation Method for Electromagnetic Scattering From PEC Objects

A discontinuous Galerkin time-domain integral equation (DG-TDIE) method based on the marching-on-in-time (MOT) scheme is presented to analyze the transient scattering from arbitrarily shaped perfect electrically conducting (PEC) objects. In this paper, the mono-polar Rao–Wilton–Glisson (RWG) functions are chosen as the trial and the test spatial functions, while the triangular basis functions are for the temporal expansion. Therefore, it is flexible and convenient to analyze objects with nonconformal discretization or targets containing more than three surfaces in contact with each other. Numerical results are presented to demonstrate the accuracy and the efficiency of the DG-TDIE method.

Stability analysis of the marching-on-in-time boundary element method for electromagnetics

The Time Domain Integral Equation method for electromagnetics is an appealing computational method for many applications in industry. However, its applicability has long been suffering from instabilities. A rigorous analysis of the variational formulation is imperative to the successful design of stable and robust numerical schemes. In this paper, an established functional framework and stability theorem will be extended to the differentiated version of the electric field integral equations, which can be discretized more efficient and is more often used in engineering literature. The extended stability theorem, combined with efficiency requirements, will give guidelines on the choice of test and basis functions of the space–time Petrov–Galerkin scheme. A discrete equivalence with the collocation method results in the recommendation to choose the quadratic spline basis function in the standard Marching-on-in-Time scheme. Computational experiments confirm that the quadratic spline basis functions have superior stability characteristics compared to the conventional quadratic Lagrange basis functions in time.

Preface to BIT 55:1

This special issue of BIT contains articles related to the Workshop on Computational Electromagnetics andAcoustics, held at theMathematisches Forschungsinstitut Oberwolfach, Germany, in January 2013. This workshop was organized by R. Hiptmair (ETH Zurich, Switzerland), R. H.W. Hoppe (Univ. Augsburg, Germany and Houston, Texas), P. Joly (INRIA, Paris, France), and U. Langer (Univ. Linz, Austria). It was attended by 52 invited researchers in the field and had its focus on recent developments of mathematical theory and algorithms. A total of 26 presentations were given at the workshop, their topics covering major current developments in the field of Computational Electromagnetics and Acoustics. Extended abstracts of all contributions have been collected in the Oberwolfach Report 03/2013 (http://www.ems-ph.org/journals/ show_issue.php?issn=1660-8933&vol=10&iss=1, doi:10.4171/OWR/2013/03). Now, several participants of theworkshop, all leaders in their fields, have heeded the organizers’ call for papers for this special issue of the journal BIT. Their manuscripts provide a glimpse of current progress in the mathematical and numerical analysis of problems arising in acoustics or electromagnetics. The contribution P. Monk and J. Chen is in the very topical area of time-domain boundary integral equation methods, and contains deep numerical analysis of a convolution quadrature based scheme for transient electromagnetics. The article by X. Claeys, E. Spindler and myself proposes and analyses a novel boundary integral formulation for scattering transmission problems, whose Galerkin boundary discretization will always yield well conditioned linear systems In their paper M. Dauge, R. A. Norton, and R. Scheichl study the regularity of Bloch modes in photonic crystals, which arise as solutions of eigenvalue problems. They obtain new results in elliptic regularity theory, which are important for designing good approximation spaces for numerical methods.

An Efficient Solution for the Transient Electromagnetic Scattering From Discrete Body of Revolution

This letter concentrates on an efficient time-domain integral equation (TDIE) method based on the marching-on-in-time (MOT) scheme for the analysis of the transient electromagnetic scattering from discrete body of revolution (DBOR). Taking advantage of the discrete circular symmetry, the storage requirements can be reduced significantly. Further computational efficiency can be improved by using the fast Fourier transform (FFT) technique to accelerate the matrix vector product. Numerical examples are given to validate the accuracy and efficiency of the proposed method.

Stable TDIE-MOT Solver for Transient Scattering by Two-Dimensional Conducting Structures

A new numerical approach for transient scattering by two-dimensional structures using the time domain integral equation (TDIE) method in conjunction with the marching-on-in-time (MOT) scheme is developed. Through proper choice of temporal basis functions and precise evaluations of matrix elements, the approach is unconditionally stable in terms that no extra stabilized measure is necessary. Quadratic B-spline function is employed as the temporal basis function, and closed-form integrals with respect to time are derived for calculating the matrix elements. Three examples with closed, open and coupling structures are examined under illuminations by both TM and TE waves. Numerical validations are illustrated by comparisons with analytical or numerical results in frequency domain via discrete Fourier transforms.

The Influence of the Exact Evaluation of Radiation Fields in Finite Precision Arithmetic on the Stability of the Time Domain Integral Equation Method

Transient electromagnetic scattering phenomena can effectively be simulated with time domain integral equation methods. The stability and accuracy of the marching on in time scheme is usually established with exact evaluation of the radiation fields. Due to singularities in their analytical expressions, straightforward evaluation in finite precision arithmetic can jeopardize the accuracy of the radiation fields. Computational experiments confirm this and show that it can even lead to late-time instability of the numerical model. Hence, a reformulation is necessary to remove the singular behavior and obtain the robustness required for industrial application. To this end, highly accurate and well-behaved expressions for the quasi-exact integration method in finite precision arithmetic will be derived in this paper. Numerical experiments confirm the robustness and stability of the improved Marching on in Time scheme.

Computational Electromagnetism and Acoustics

Computational electromagnetics and acoustics revolve around a few key challenges, among which are the non-local nature of the underlying phenomena and resonance effects. The bulk of the contributions to the workshop addressed mathematical and numerical approaches meant to grapple with these two difficulties. Frequency domain integral equation methods continue to receive much attention, with a particular focus on (i) frequency robust matrix compression algorithms through so-called directional schemes or “butterfly algorithms”, and (ii) domain decomposition approaches. Time domain integral equation methods still enjoy rapid development and much progress was made in their numerical analysis. Of course, efficient and accurate absorbing boundary conditions remain a persistent topic and were covered in a few contributions. Resonance induced phenomena in a broad sense affect the analytical and numerical model for meta-materials, periodic structures, and micro-structured media. There is a lot left to be explored in this field in terms of analysis and algorithm development and a few presentations were devoted to such issues.

A hybrid DGTD-TDIE method for solving complex electromagnetic problems

This paper presents a novel hybrid method that effectively combines two capable numerical methods – the discontinuous Galerkin time domain (DGTD) method and the time domain integral equation (TDIE) method. The hybrid DGTD-TDIE method is highly applicable to the interaction problem between arbitrarily shaped thin wires and dielectric structures with complex media compositions. The Huygens equivalence principle is employed to divide the original problem into two subproblems: (1) the region containing the thin wires, which is calculated by using the TDIE method, and (2) the dielectric region modeled by unstructured grids and analyzed by the DGTD method. A time-stepping scheme is used to match the different time steps between the DGTD and TDIE methods. To validate the hybrid method, several numerical results have been presented, which prove it a promising scheme to solve the complex coupling problems.

Design of Temporal Basis Functions for Time Domain Integral Equation Methods With Predefined Accuracy and Smoothness

A key parameter in the design of integral equation methods for transient electromagnetic scattering is the definition of temporal basis functions. The choice of temporal basis functions has a profound impact on the efficiency and accuracy of the numerical scheme. This paper presents a framework for the design of temporal basis functions with predefined accuracy and varying smoothness properties. The well-known shifted Lagrange basis functions naturally fit in this framework. New spline basis functions will be derived that have the same interpolation accuracy as shifted Lagrange basis functions and with the added advantage of being smooth. Numerical experiments show the positive influence of smoothness on the quadrature error in the numerical integration procedure. The global accuracy in time of the numerical scheme based on shifted Lagrange and spline basis functions has been experimentally analyzed. For a given interpolation error the experiments confirm the expected accuracy for the shifted Lagrange basis functions, but remarkably show a higher order of accuracy for the spline basis functions.

Size reduction technique for the marching-on-in-order time-domain integral equation method in analysis of transient electromagnetic scattering

The aggregative basis functions (ABFs) are introduced to construct a size-reduced system for the marching-on-in-order (MOO) time-domain integral equation (TDIE) method to analyse transient electromagnetic scattering from conducting objects. Based on the previously developed characteristic basis function method (CBFM), a set of orthogonal vectors that expand the original unknown current coefficients are obtained via the singular value decomposition (SVD). The ABF method can be considered as an application and counterpart of CBFM in TDIE with some differences. The ABFs are aggregations of the weighted Laguerre polynomials and RWG basis functions, which are the elemental temporal and spatial basis functions, respectively. The ABFs are defined over the entire geometry and effective in each order of the MOO scheme. The proposed method gives significant reduction to matrix size and also the storage by several orders of magnitude. This is achieved because of the much less number of ABFs or the orthogonal vectors than the inner edges of the geometry. Several numerical results are presented to illustrate the validity of the proposed method.

Numerical Approach for Analysis of Transient Scattering by an Accelerated Body

A numerical approach for analysis of transient scattering by an accelerated body based on the frame hopping method (FHM) is developed. First, transform formulae for coordinates and fields between an inertial laboratory frame and a non-inertial co-moving frame are derived. The solution of retarded potential equations in the accelerated frame is given. Then, the scattering analysis is performed in the co-moving accelerated frame using the time domain integral equation (TDIE) method. Last, the scattering fields found in the co-moving frame are ransformed back to the laboratory frame. A moving conducting sphere is used to validate the procedure. The proposed approach is unified in terms of suiting stationary, uniform and accelerated objects.

Marching-on-in-order time-domain volume-surface integral equations in analysis of transient electromagnetic scattering from composite conducting-dielectric objects

In this letter, the matching-on-in-order time-domain volume-surface integral equation method is used to analyze transient electromagnetic scattering from objects comprising both conductor and dielectric material. SWG and RWG basis functions are used as the spatial basis functions in dielectric region and on the conducting surface, respectively. Different orders of weighted Laguerre polynomials are used as temporal basis functions, and the relative expansion coefficients are solved recursively order by order. The spatial and temporal testing procedures are separate, and both of them are carried on with the Galerkin's method. Stable solutions are ensured, and several numerical results are given to verify the proposed method. © 2011 Wiley Periodicals, Inc. Microwave Opt Technol Lett 53:1315–1319, 2011; View this article online at wileyonlinelibrary.com. DOI 10.1002/mop.25987

Computational Electromagnetism and Acoustics

The challenge inherent in the accurate and efficient numerical modeling of wave propagation phenomena is the common grand theme in both computational electromagnetics and acoustics. Many excellent contributions at this Oberwolfach workshop were devoted to this theme and a wide range of numerical techniques and algorithms were mustered to tackle this challenge. Among these methods boundary integral equation methods received particular attention, both in frequency and time domain. Combined with phase modulation techniques they pave the way for the construction of frequency robust schemes for wave scattering. This is an exciting recent development, which is complemented by the invention of high-order spectral integral equation methods. In parallel, we also witnessed soaring interest in time-domain integral equation methods. The alternative volume based schemes for wave propagation problems can also boast innovations, most notably in the context of plane wave methods. Though wave propagation is paramount, several other important aspects were addressed during the workshop ranging, among others, from the design of novel Galerkin schemes, asymptotic methods for geometrically singularly perturbed problems, analysis of spectral edge elements, to the design of preconditioners for quasi-static electromagnetics.