Potential Integral Equation Research Articles

Several Aspects of Human Exposure to Low Frequency Fields; Incident and Internal Field Dosimetry Procedures and Related Legal issues

The paper deals with simple and efficient dosimetry procedures for human exposure to low frequency (LF) electric and magnetic fields at single and multiple frequencies, respectively. The electromagnetic interference (EMI) sources of interest are overhead power lines and transmission substations. Electric and magnetic fields due to EMI sources are either calculated or measured. Theoretical dosimetry is based on scalar potential integral equation (SPIE) for the case of the electric field assessment while the magnetic fields are computed by means of the Biot-Savart law. The internal fields in the human body are determined by using the canonical body models (disk model and cylindrical model). The obtained results are compared to exposure limits proposed by national and international legislation, respectively (National Gazette No 146/2014, 31/2019, 59/2016) and ICNIRP International Commission on Nonionizing Radiation protection). Finally, legal issues pertaining to human exposure to LF fields are addressed.

On the Discontinuous Galerkin Mixed Potential Integral Equation for Half-Space Scattering Analysis

On the Discontinuous Galerkin Mixed Potential Integral Equation for Half-Space Scattering Analysis

Analytic Preconditioners for Decoupled Potential Integral Equations and Wideband Analysis of Scattering From PEC Objects

Many integral equations used to analyze scattering, such as the standard combined field integral equation (CFIE), are not well-conditioned for a wide range of frequencies and multi-scale geometries. There has been significant effort to alleviate this problem. A more recent one is using a set of decoupled potential integral equations (DPIE). These equations have been shown to be robust at low frequencies and immune to topology breakdown. But they mimic the ill-conditioning behavior of CFIE at high frequencies. This paper addresses this deficiency through new Calderón-type identities derived from the Vector Potential Integral Equation (VPIE). We construct novel analytic preconditioners for the vector potential integral equation (VPIE) and scalar potential integral equation (SPIE) constrained to perfect electric conductors (PEC). These new formulations are wide-band well-conditioned and converge rapidly for multi-scale geometries. This is demonstrated though a number of examples that use analytic and piecewise basis sets.

A Generalized Scalar Potential Integral Equation Formulation for the DC Analysis of Conductors

The electrostatic modeling of conductors is a fundamental challenge in various applications, including the prediction of parasitic effects in electrical interconnects, the design of biasing networks, and the modeling of biological, microelectromechanical, and sensing systems. The boundary element method (BEM) can be an effective simulation tool for these problems because it allows modeling three-dimensional objects with only a surface mesh. However, existing BEM formulations can be restrictive because they make assumptions specific to particular applications. For example, capacitance extraction formulations usually assume a constant electric scalar potential on the surface of each conductor and cannot be used to model a flowing current, nor to extract the resistance. When modeling steady currents, many existing techniques do not address mathematical challenges such as the null space associated with the operators representing the internal region of a conductor. We propose a more general BEM framework based on the electric scalar potential for modeling conductive objects in various scenarios in a unified manner. Restrictive application-specific assumptions are not made, and the aforementioned operator null space is handled in an intuitive and rigorous manner. Numerical examples drawn from diverse applications confirm the accuracy and generality of the proposed method.

Analysis of noise attenuation through soft vibrating barriers: an analytical investigation

<abstract><p>In this article, the impact of fluid flow and vibration on the acoustics of a subsonic flow is examined. Specifically, it focuses on the noise generated by a convective gust in uniform flow that is scattered by a vibrating plate of limited size. The study analyzes the interaction between acoustics and structures by considering the scattering of sound waves by a soft finite barrier. To achieve this, the Wiener-Hopf technique is utilized for the analytical treatment of the acoustic model. The approach involves performing temporal and spatial Fourier transforms on the governing convective boundary value problem, then formulating the resulting Wiener-Hopf equations. The product decomposition theorem, an extended version of Liouville's theorem, and analytic continuation are employed to solve these equations. Finally, the scattered potential integral equations are computed asymptotically. This study can be significant for understanding the acoustic properties of structures and how they interact with fluid flow in subsonic environments, which could have applications in fields such as aerospace engineering, noise reduction, and structural acoustics.</p></abstract>

On the Low-Frequency Behavior of Vector Potential Integral Equations for Perfect Electrically Conducting Scatterers

Low-frequency behavior of vector potential integral equations (VPIEs) for perfect electrically conducting (PEC) scatterers is investigated. Two equation sets are considered: the first set (VPIE-1) enforces the tangential component of the vector potential on the scatterer surface to be zero and uses the fundamental field relationship on its normal component. The second set (VPIE-2) uses the same condition as VPIE-1 for the tangential component of the vector potential but enforces its divergence to be zero. In both the sets, unknowns are the electric current and the normal component of the vector potential on the scatterer surface and are expanded using the Rao–Wilton–Glisson (RWG) and pulse basis functions, respectively. To achieve a conforming discretization, RWG, scalar Buffa–Christiansen (BC), and pulse testing functions are used. Theoretical and numerical analyses of the resulting matrix systems show that the electric current obtained by solving VPIE-1 has the wrong frequency scaling and is inaccurate at low frequencies.

Modeling and Simulation of Magnetic Balance Current Sensor Based on Magnetic Scalar Potential Volume Integral

In this paper, a field–circuit combined simulation method, based on the magnetic scalar potential volume integral equation (MSP-VIE) and its fast algorithms, are proposed for the transient simulation and nonlinear distortion analysis of the magnetic balance current sensor. The magnetic part of the sensor is modeled and simulated by MSP-VIE with the field matrices extracted by the method of moments. By directly implementing the magnetic field equation in the circuit, these field matrices can be regarded as equivalent circuit parameters of the magnetic part, to construct the corresponding SPICE model. Finally, the field–circuit combined model of the entire sensor is unified in a circuit, so with the SPICE solver, the transient simulation is accomplished in the time domain. Moreover, aiming at the time-consuming problem, this paper presents a corresponding fast method to accelerate the simulation. The comparison of measurement and simulation demonstrates that the proposed method not only realizes the transient simulation of the whole sensor, but also simulates some hidden performance details; thus, it can be applied to practical engineering, to guide and test the early design of the product.

Sensitivity analysis for active electromagnetic field manipulation in free space

ABSTRACT This paper presents a detailed sensitivity analysis of the active manipulation scheme for electromagnetic (EM) fields in free space. The active EM fields control strategy is designed to construct surface current sources (electric and/or magnetic) that can manipulate the EM fields in prescribed exterior regions. The active EM field control is formulated as an inverse source problem. We follow the numerical strategies in our previous works, which employ the Debye potential representation and integral equation representation in the forward modelling. We consider two regularization approaches to the inverse problem to approximate a current source, namely the truncated singular value decomposition (TSVD) and the Tikhonov regularization with the Morozov discrepancy principle. Moreover, we discuss the sensitivity of the active scheme (concerning power budget, control accuracy, and quality factor) as a function of the frequency, the distance between the control region and the source, the mutual distance between the control regions, and the size of the control region. The numerical simulations demonstrate some challenges and limitations of the active EM field control scheme.

On the Poisson equation in exterior domains

We construct a solution of the Poisson equation in exterior domains $\Omega \subset \mathbb R^n,\;n \ge 2,$ in homogeneous Lebesgue spaces $L^{2,q}(\Omega),;1 < q <\infty,$ with methods of potential theory and integral equations. We investigate the corresponding null spaces and prove that its dimensions is equal to $n+1$ independent of $q$.

FPGA Acceleration of Matrix-Assembly Phase of RWG-Based MoM

In this letter, the field-programmable gate array (FPGA) accelerated implementation of matrix-assembly phase of the method of moments (MoM) is presented. The solution is based on a discretization of the frequency-domain mixed potential integral equation using the Rao–Wilton–Glisson basis functions and their extension to wire-to-surface junctions. To take advantage of the given hardware resources (i.e., Xilinx Alveo U200 accelerator card), nine independent processing paths/runtime efficient compute units are developed and synthesized. Numerical results provided for a quadrifilar spiral antenna mounted on a conductive handset box show that the proposed parallelization scheme performs 9.53× faster than a traditional (i.e., serial) central processing unit (CPU) MoM implementation, and about 1.67× faster than a parallel six-core CPU MoM implementation.

A Full-Wave Generalized PEEC Model for Periodic Metallic Structure With Arbitrary Shape

In this work, a full-wave partial element equivalent circuit (PEEC) model for 2-D periodic metallic structures is proposed. The mixed potential integral equations (MPIEs) with periodic Green’s function (GF) are interpreted as loop equations following Kirchhoff’s voltage law (KVL) in a circuit domain. The impact of gauge freedom in the MPIE on the corresponding PEEC model is discussed. A tailored gauge function is introduced to form a generalized PEEC model for periodic structures, which demonstrates superior passivity and frequency stability. With the proposed PEEC model, lumped and active elements can be easily integrated with conducting bodies for field-circuit co-simulation. Through the distributed circuit including coupling elements, the PEEC model provides a clear physical insight into the electromagnetic behavior of the physical models for design and optimization. Three numerical examples, including frequency-selective surface (FSS), array antenna, and pixelwise meta-atom, are given to illustrate the accuracy and efficiency of the proposed PEEC model.

Decoupled Potential Integral Equation for Electromagnetic Scattering From Arbitrarily Shaped Dielectric Objects

Recently, integral equation formulations that use potentials as opposed to fields as unknown quantities have been developed for scattering from dielectric objects. It has been shown that these formulations can be construed so that they are well-conditioned across a broad frequency spectrum, a result that has been theoretically proven for spherical systems. Unfortunately, to date, this formulation has not been implemented on practical discretizations of objects. This is the goal of this article. Specifically, we present a well-conditioned and well-tested decoupled potential integral equation (DPIE) formulation and all the necessary implementation details for electromagnetic scattering from homogeneous, dielectric, arbitrarily shaped objects. The resulting decoupled systems do not suffer from low-frequency breakdown. Results that demonstrate these properties are presented for a number of different dielectric targets. Furthermore, in order to fully validate each of the two integral equations for the potentials, we develop analytical solutions for spherical systems.

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.

Broadband Analysis of Multiscale Electromagnetic Problems: Novel Incomplete-Leaf MLFMA for Potential Integral Equations

Recently introduced incomplete tree structures for the magnetic-field integral equation are modified and used in conjunction with the mixed-form multilevel fast multipole algorithm (MLFMA) to employ a novel broadband incomplete-leaf MLFMA (IL-MLFMA) to the solution of potential integral equations (PIEs) for scattering/radiation from multiscale open and closed surfaces. This population-based algorithm deploys a nonuniform clustering that enables to use deep levels safely and, when necessary, without compromising the accuracy resulting in an improved efficiency and a significant reduction for the memory requirements (order of magnitudes), while the error is controllable. The superiority of the algorithm is demonstrated in several canonical and real-life multiscale geometries.

Convergence of the MPIE Galerkin MoM Thin Wire Formulation

Convergence studies of the Method of Moments solution of thin wires have long been handicapped by the issue of conditional convergence: when the segment length approaches the wire radius, the approximations typically made result in exponentially diverging solutions. In this communication, it is shown that it is the approximation of the singularity which is the main cause of this problem. Using the mixed potential integral equation, paralleling the classic RWG formulation for surfaces, the formulation in this communication approximates the surface (as opposed to equivalent filamentary) current, and applies Galerkin testing on the wire surface. The use of contemporary singularity-canceling integral transforms for highly efficient and accurate quadrature is demonstrated. Results are presented for typical thin-wire dipoles for both radiation and scattering problems. It is shown that with a rigorous treatment of the singularity, a far more comprehensive convergence study may be undertaken than previously demonstrated. Furthermore, the results shown justify the many years of successful use in practice of simple source models with modest mesh requirements.

On the space-time discretization of variational retarded potential boundary integral equations

This paper discusses the practical development of space-time boundary element methods for the wave equation in three spatial dimensions. The employed trial spaces stem from simplex meshes of the lateral boundary of the space-time cylinder. This approach conforms genuinely to the distinguished structure of the solution operators of the wave equation, so-called retarded potentials. Since the numerical evaluation of the arising integrals is intricate, the bulk of this work is constituted by ideas about quadrature techniques for retarded layer potentials and associated energetic bilinear forms. Finally, we glimpse at algorithmic aspects regarding the efficient implementation of retarded potentials in the space-time setting. The proposed methods are verified by means of numerical experiments, which illustrate their capacity.

Accurate and Efficient Solutions of Densely Discretized Closed Conductors Using a Combined Potential-Field Formulation

We present an accurate, efficient, and stable formulation for rigorous analyses of electromagnetic problems involving closed conductors. The formulation, namely the combined potential-field formulation (CPFF), is constructed from the conventional potential integral equations and the magnetic-field integral equation, together with an additional integral equation using the boundary condition for the normal component of the magnetic vector potential. Being both low-frequency-stable and resonance-free, CPFF is a broadband formulation, which enables accurate and efficient solutions of objects with diverse dimensions and discretization sizes.

A Time-Domain Numerical Method for Multi-Conductor Coaxial Lines Using the Exact Retarded Potential Integral Equations

We present a time-domain numerical method for calculating the current propagating through coaxial lines including the radiation effect. In previous studies of transmission lines using the retarded potential integral equations (RPIE), the so-called thin-wire approximation to the RPIE has been mainly used. However, the thin-wire approximation has the problem that the unphysical oscillation occurs when the spatial mesh size is smaller than the radius, even for one-conductor line. In addition, the application of the thin-wire approximation in multi-conductor coaxial lines causes a problem in the formulation. We solve these problems by developing a numerical method using the exact RPIE. We perform numerical calculations in the time-domain for two-conductor coaxial lines and obtain numerical solutions even for the very small common-mode current that causes radiation emission.

Calculation of Potential Green’s Functions for a Planar Metasurface Excited by an Electric Dipole Using Homogenization Method

One of the rapid and efficient methods to calculate the interaction between electromagnetic fields and periodic surfaces with periodicity much less than the wavelength in the surrounding medium is the homogenization method. In this paper, the Hertzian potential Green’s functions corresponding to electric dipoles which excite a homogenized metasurface are presented. The metasurface includes a periodic array of planar scatterers and is replaced by a thin sheet with equivalent electric and magnetic surface polarization density dyads. The magnetic vector potentials are calculated using the generalized sheet transition conditions on the plane of the metasurface. Also the scalar electric potential corresponding to the structure is calculated with the aid of the Lorentz gauge. These potentials together are necessary to use in the mixed potential integral equation formulation of the method of moment in an electromagnetic problem. Two metasurfaces are investigated to calculate the electric fields; a perfect electric conductor square patch array whose surface polarization densities are calculated analytically, and a Jerusalem cross array whose surface polarization dyads are retrieved from reflection and transmission coefficients.

Method of Moments Based on Equivalent Periodic Problem and FFT with NURBS Surfaces for Analysis of Multilayer Periodic Structures

In this paper, an efficient technique of computation of method of moments (MM) matrix entries for multilayer periodic structures with NURBS surface and Bézier patches modelling is proposed. An approximation in terms of constant pulses of generalized rooftop basis functions (BFs) defined on Bézier patches is proposed. This approximation leads discrete convolutions instead of usual continuous convolution between Green’s functions and BFs obtained by the direct mixed potential integral equation (MPIE) approach. An equivalent periodic problem (EPP) which contains the original problem is proposed to transform the discrete convolutions in discrete cyclic convolutions. The resultant discrete cyclic convolutions are computed by efficiently using the Fast Fourier Transform (FFT) procedure. The performance of the proposed method and direct computation of the MM entries are compared for phases of reflection coefficient. The proposed method is between 9 and 50 times faster than the direct computation for phase errors less than 1 deg. The proposed method exhibits a behaviour of CPU time consumption of O(NbLog10Nb) as the number Nb of BFs increases. This behaviour provides significant CPU time savings with respect to the expected behaviour of O(Nb2) provided by the direct computation of the MM matrix entries.