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Wireless power transfer (WPT) technologies are rapidly attracting attention because of their wide application field. Among them, inductive power transfer (IPT) technology is most promising for daily usage in households. Foreign metal detection (FMD) is indispensable for IPT applications because foreign metal heats up easily in the magnetic field generated by IPT devices and causes a dangerous situation. On the other hand, the representation of the foreign metal as an equivalent circuit model is not readily available. Therefore, we need to use finite element method (FEM)-based electromagnetic field simulators, which require massive computation time and cost. This work proposes a novel simulation method of foreign metal in the magnetic field based on theoretical analysis. We establish an equation-based theory for simulating metal pieces as circuit elements and deliver an equivalent circuit model with a virtual inductance of the metal piece and a mutual inductance between the sense coil and the metal piece. The Z parameter of the extracted circuit model is compared with that of a FEM-based simulator. Evaluation results show that the proposed method returns the result 971 times faster than the FEM-based simulator with the memory usage of 1/3863, while the simulation error is at most 3.88 %.

In tokamak and other fusion devices, magnetic control is the main tool that allows to regulate the plasma current, position and shape; it is in charge of actuating the desired plasma current waveform, steering the plasma position to a given set point and maintain the plasma shape close to a prescribed plasma equilibrium. This work describes the application of several physics concepts and computational tools in order to obtain a novel optimal controller for the plasma centroid position, which has been implemented and tested in the real-time plasma control system at the ISTTOK tokamak. A key point for the development of the new control system was the installation of a recently upgraded hardware, that numerically integrates in real-time the magnetic probes signals.

The paper provides a novel framework to study the accuracy and stability of numerical integration schemes when employed for the time domain simulation of power systems. A matrix pencil-based approach is adopted to evaluate the error between the dynamic modes of the power system and the modes of the approximated discrete-time system arising from the application of the numerical method. The proposed approach can provide meaningful insights on how different methods compare to each other when applied to a power system, while being general enough to be systematically utilized for, in principle, any numerical method. The framework is illustrated for a handful of well-known explicit and implicit methods, while simulation results are presented based on the WSCC 9-bus system, as well as on a 1,479-bus dynamic model of the All-Island Irish Transmission System. [ABSTRACT FROM AUTHOR]

In this paper, we develop a quadrature framework for large-scale kernel machines via a numerical integration representation. Considering that the integration domain and measure of typical kernels, e.g., Gaussian kernels, arc-cosine kernels, are fully symmetric, we leverage a numerical integration technique, deterministic fully symmetric interpolatory rules, to efficiently compute quadrature nodes and associated weights for kernel approximation. Thanks to the full symmetric property, the applied interpolatory rules are able to reduce the number of needed nodes while retaining a high approximation accuracy. Further, we randomize the above deterministic rules by the classical Monte-Carlo sampling and control variates techniques with two merits: 1) The proposed stochastic rules make the dimension of the feature mapping flexibly varying, such that we can control the discrepancy between the original and approximate kernels by tuning the dimnension. 2) Our stochastic rules have nice statistical properties of unbiasedness and variance reduction. In addition, we elucidate the relationship between our deterministic/stochastic interpolatory rules and current typical quadrature based rules for kernel approximation, thereby unifying these methods under our framework. Experimental results on several benchmark datasets show that our methods compare favorably with other representative kernel approximation based methods. [ABSTRACT FROM AUTHOR]

The partial-element equivalent circuit method is a well-known numerical technique that is used to solve Maxwell’s equations in their integral equation form. The application of the partial-element equivalent circuit (PEEC) method to modeling domains with non-orthogonal three-dimensional geometries requires the computation of the interaction integrals to be performed numerically, thus slowing down the overall computation. This work presents a new technique that allows improving the computation of the interaction integrals of the PEEC method for non-orthogonal geometries under the quasi-static hypothesis. To this purpose, a voxelization approach that automatically decomposes non-orthogonal volumes in elementary parallelepipeds is used, allowing the implementation of closed-form formulas for the interaction integrals and completely avoiding numerical integration. The proposed approach is applied to three example problems exhibiting very good accuracy and excellent speed-up compared with the standard one using the numerical integration. [ABSTRACT FROM AUTHOR]

The solenoid has been used extensively in a wide range of electrical applications. In the present work, an analytical model is presented for estimating the 3-D magneto quasi-static (MQS) fields of the cylindrical solenoid with helix-shaped (helical) wire winding. The proposed integral formulation is based on the Schur (Hadamard) vector product of the current vector $\vec {I}$ and the helical tangent vector $\vec {T}$ , to express the magnetic vector potential $\vec {A}$. The model is used to estimate the magnetic flux density vector $\vec {B}$ and the flux linkage $\Psi $ anywhere in 3-D space, excluding the (source) regions of conducting wire. A constant scaling coefficient, based on the measured terminal inductance, is proposed for calibration of the model’s magnetic field. Error analysis is presented on numerical integration, based on truncated series expansion of complete elliptic integrals in terms of Chebychev polynomials. The model may be used to estimate all three cylindrical components of MQS fields of solenoids as a function of core diameter, wire radius, the number of wire turns per winding layer, the number of winding layers, and complex permeability of frequency-dependent linear material. Several numerical examples are provided to validate the helical model against the superposition of circular loops and an idealized circuit model. [ABSTRACT FROM AUTHOR]

To meet the high current measurement requirements of the international thermonuclear experimental reactor polar field (ITER PF) power supply, a noncontact dot matrix Hall high current sensor without a magnetic core is designed. Due to the existence of installation error, the measurement accuracy will be reduced due to conductor eccentricity in practical use. To solve this problem, this article will take the circular dot matrix Hall current sensor composed of eight Hall elements as an example and note the shortcomings of the two algorithms by analyzing the numerical integration algorithm and conductor positioning algorithm. Based on the two algorithms, a digital signal processing (DSP) conditioning algorithm is proposed. The DSP conditioning algorithm detects the output voltage of each Hall element after conductor offset and adjusts the output voltage of each Hall element by self-feedback, which simplifies the complexity of algorithm operation and eliminates conductor eccentricity error at the same time. Through simulation and field measurement verification, DSP self-feedback conditioning of the Hall element output voltage can eliminate the influence of conductor eccentricity error and improve the measurement accuracy of the sensor. [ABSTRACT FROM AUTHOR]

The electromagnetic (EM) coupling effects of microstrip antennas have become increasingly prominent in complicated electric systems. In this article, a hybrid method is proposed based on the combination of cavity eigenmode theory and BLT equation. First, the internal electric field generated by the transmission lines inside the resonant cavity is obtained via cavity eigenmode theory. Then, the coupling mechanism between the electric field and microstrip antenna is established through radiation coupling interference. On this basis, the BLT equation and numerical integration method are applied to determine the induced voltage at the target microstrip antenna. Eventually, the proposed approach is validated by a full-wave simulation method and the impact of the microstrip antenna parameters on the coupling interference is discussed. This article provides an analytical method for calculating the coupling of a microstrip antenna. Compared with numerical simulation, this method has certain potential advantages in solving closed electrically large objects. Furthermore, duplicate modeling can be avoided in the same type of scenario. [ABSTRACT FROM AUTHOR]

We derive closed form bit error rate (BER) expression for orthogonal time frequency space (OTFS) zero-forcing (ZF) receiver with perfect and imperfect receive channel information. Depending on the delay-Doppler locations of the propagation paths of the OTFS channel H, the expression $\mathrm {H}^{H}\mathrm {H}$ is shown to have either distinct or repetitive eigenvalues. When $\mathrm {H}^{H}\mathrm {H}$ has two or less distinct eigenvalues and the remaining ones are repetitive, we derive the probability distribution function (pdf) of the signal-to-noise-plus-interference-ratio (SINR) of the ZF receiver. A closed form BER expression is then derived by averaging the conditional BER over the SINR pdf. When $\mathrm {H}^{H}\mathrm {H}$ has $n$ distinct eigenvalues, we use numerical integration to derive a generalized expression for the SINR pdf. We show that this pdf can be tightly approximated by the Gamma pdf, and then use it to derive the BER expression. The derived OTFS ZF BER expression therefore, unlike the existing ones in the literature, does not require averaging over multiple channel realizations. We show, for different modulation schemes and OTFS system parameters, that the BER calculated using the derived expressions closely matches the one calculated numerically. [ABSTRACT FROM AUTHOR]

In computer graphics, the efficient rendering of a clear sky may greatly enhance the realism of synthesised interactive virtual environments. However, light atmospheric scattering, lying behind a reliable sky synthesis, is a computationally demanding process. Obtaining the highest possible accuracy and quality of the synthesised image in an efficient manner is not trivial as even minor integration errors may result in huge image artefacts. In this paper we propose two new approaches, namely the first based on the splines and the second on the Taylor expansion, in order to obtain the efficient computing of the transmittance (optical depth) component of the physically based single scattering model. We used Bruneton’s framework to compare the proposed solution with state-of-the-art methods in a single, scattering-based clear sky synthesis, in terms of image quality and performance (time). We managed to improve the quality of the single scattering effect and time performance and we removed the necessity of pre-calculating values to look-up tables for the transmittance part of the scattering integral, reducing exploited memory usage.

This contribution reports a comparative analysis of two methodologies for the periodic steady-state solution of linear and nonlinear electric networks in time-domain (TD), based on the state-space model (SSM) and companion-circuit analysis (CCA), respectively. Newton methods based on a conventional Numerical Differentiation (ND) and Enhanced Numerical Differentiation (END) process, respectively, are used. The application of the SSM and CCA methods is done using a brute force (BF) procedure and the application of advanced numerical techniques (ANT) through the forward Euler (FE) and Trapezoidal Rule (TR) numerical integration, respectively. Related analysis show comparisons of each case study in terms of efficiency, speed, computer effort and accuracy between the combination of SSM, CCA and Newton methods, e.g. SSM-ND, SSM-END, CCA-ND and CCA-END, respectively. In particular, the analysis is shown in terms of time-steps per period, CPU processing time and a computer platform used are shown. The results obtained from the reported case studies are validated against the Power Systems Computer Aided Design/Electromagnetic Transients Program including Direct Current (PSCAD/EMTDC®) response, widely accepted by the power industry.

In this article, we report on the Fano resonance characteristics tunability in a coupled system of magnonic stripes and resonators by variating the geometric parameters, the magnitude of the coupling between them, and the intensity of spin waves (SWs). Using numerical integration of the coupled-wave equation system, we calculated the transfer characteristics of the considered system and the parameters of Fano resonance considering the cubic nonlinearity of magnetic media. We demonstrate the features of constructive and destructive interference of SWs at Fano resonance. Our results reveal the mechanism of the SW spectra control of the nonlinear Fano resonance (asymmetry coefficient and resonance frequency shifts) via the intensity of SW. [ABSTRACT FROM AUTHOR]

For outdoor millimeter-wave mobile radio systems, the main service area is expected to be in line-of-sight (LOS) and semi-line-of-sight (SLOS) environments. The modeling of propagation in a mixed LOS/SLOS environment is thus important in system design. For modeling propagation in an SLOS environment, the application of the Loo model developed for land mobile satellite service with roadside shadowing is promising because this model reflects the physical mechanism of radio propagation. However, the model is inconvenient in practice because its calculation involves numerical integration over an infinite range. In this article, we approximate the Loo distribution using three basic distributions, namely the Nakagami-Rice distribution (or Rician distribution), Nakagami-m distribution, and lognormal distribution. Based on the evaluation results, we propose the use of the Nakagami-m distribution in place of the Loo distribution. We demonstrate the conversion of Loo distribution parameters to Nakagami-m distribution parameters. Finally, we examine a multistate model that expresses environments composed of LOS, SLOS, and non-LOS conditions. [ABSTRACT FROM AUTHOR]

The large-scale adoption of power electronics systems enables a more sustainable and flexible power grid. The offline simulation tools support the analysis of such systems. However, despite of the advancement in computer hardware and software, the accurate and efficient simulation of large-scale power converters is still a significant challenge; the simulation is usually time-consuming. In this article, we propose a state-variable-interfacd decoupling (SVID) strategy based on the discrete state event-driven (DSED) approach. The SVID strategy partitions the large-scale circuit into smaller subsystems with state variables as interfaces so that the numerical integration can be performed in a decoupled manner. Meanwhile, it ensures that no accuracy is sacrificed after decoupling by representing the interfaced variables with the high-order Taylor expansion. A 10-kV 2-MW solid-state transformer with 578 power switches is studied as a simulated case, and more than 1000-fold acceleration (from 3 h to 12 s) is achieved compared with commercial software, where the SVID strategy contributes to five- to eightfold. The simulated results are compared with both explicit and implicit solvers in commercial software and experimental results. The DSED approach with SVID strategy makes it possible to analyze and design a high-power converter more reliably and perform studies that are very difficult or impossible to do otherwise. [ABSTRACT FROM AUTHOR]

We present a novel numerical integration technique, Neural Network Integration, or NNI, where shallow neural network design is used to approximate an integrand function within a bounded set. This function approximation is such that a closed-form solution exists to its definite integral across any generalized polyhedron within the network's domain. This closed-form solution allows for fast integral evaluation of the function across different bounds, following the initial training of the network. In other words, it becomes possible to “pre-compute” the numerical integration problem, allowing for rapid evaluation later. Experimental tests are performed using the Genz integration test functions. These experiments show NNI to be a viable integration method, working best on predictable integrand functions, but worse results on singular and non-smooth functions. NNI is proposed as a solution to problems where numerical integrations of higher dimension must be performed over different domains frequently or rapidly and with low memory requirements, such as in real-time or embedded engineering applications. The application of this method to the optimization of integral functions is also discussed.

Static potential integrals for constant and linear sources on triangles are derived in a straightforward way. The new representations, as presented, are robust with respect to machine evaluation in important limiting cases. The potential integrals comprise up to six functions, each dependent on the relative position and orientation (with respect to an observation point) of a vertex and edge, respectively, of the source triangle. Gradients of the potentials are derived by differentiation, thus preserving relations between the representations. Each such vertex function reveals any anomalous functional behavior near its associated vertex or edge, which is useful information for devising test integral schemes. Potential plots in the source plane of an equilateral triangle illustrate such behavior, as do similar plots for each vertex function and gradient components near their associated edge and vertex.

We propose novel adaptive quadrature schemes based on an active learning procedure. We consider an interpolative approach for building a surrogate posterior density, combining it with Monte Carlo sampling methods and other quadrature rules. The nodes of the quadrature are sequentially chosen by maximizing a suitable acquisition function, which takes into account the current approximation of the posterior and the positions of the nodes. This maximization does not require additional evaluations of the true posterior. We introduce two specific schemes based on Gaussian and Nearest Neighbors bases. For the Gaussian case, we also provide a novel procedure for fitting the bandwidth parameter, in order to build a suitable emulator of a density function. With both techniques, we always obtain a positive estimation of the marginal likelihood (a.k.a., Bayesian evidence). An equivalent importance sampling interpretation is also described, which allows the design of extended schemes. Several theoretical results are provided and discussed. Numerical results show the advantage of the proposed approach, including a challenging inference problem in an astronomic dynamical model, with the goal of revealing the number of planets orbiting a star.

Modal network theory has proven to be a useful tool in analyzing waveguide discontinuities. Ports of a modal network correspond to waveguide eigenmodes on either side of the discontinuity. In this work, modal network theory is extended beyond conventional/geometrical waveguide discontinuities. We use the modal network formulation to analyze discontinuities resulting from metasurfaces. Specifically, we analyze azimuthally invariant cascaded sheet metasurfaces placed in a cylindrical waveguide excited by TM0n modes. The metasurface comprises multiple radially varying electric admittance sheets separated by dielectric spacers. We derive the modal representation of a single inhomogeneous, electric sheet admittance using two approaches: a conventional numerical integration approach and a discrete Hankel transform (DHT) approach. We show that the DHT approach is more efficient. Subsequently, the modal matrices of cascaded sheet metasurfaces are derived using the DHT. Finally, we propose an optimization-based procedure to synthesize transparent metasurface-based mode converters. The mode converters transform a set of incident modes on one side to another set of desired modes on the opposite side of the metasurface. Two designs are synthesized at 10 GHz: a single-mode converter, and a mode splitter. The designs are verified using the commercial finite element electromagnetic solvers. [ABSTRACT FROM AUTHOR]

The partial element equivalent circuit (PEEC) method provides an electromagnetic model of interconnections and packaging structures in terms of standard circuit elements. The surface-based PEEC (S-PEEC) formulation can reduce the number of unknowns compared to the standard volume-based PEEC (V-PEEC) method. This reduction is of particular use in the case of high-speed circuits and high-switching power electronics, where the bandwidth extends from low frequencies to the GHz range. In this article, the S-PEEC formulation is revised and cast in a matrix form. The main novelty is that the interaction integrals involving the curl of the magnetic and electric vector potentials are computed through the Taylor series expansion of the full-wave Green's function, leading to analytical forms that are rigorously derived. Therefore, the numerical integration is avoided, with a consequent reduction of the computation time. The proposed formulas are studied in terms of the frequency, size of the mesh, and distance between the basis function domains. Three examples are presented, confirming the accuracy of the proposed method compared to the V-PEEC method and surface-based numerical methods from literature. [ABSTRACT FROM AUTHOR]

We present two computing projects, peridynamics simulation and numerical integration on implicit domains, for which we realized high performance implementations using Python with appropriate packages. The problems are sufficiently compute intensive that a straightforward serial implementation is prohibitively slow. While conventional wisdom suggests moving such problems onto a computing cluster, we very directly produced high-performance parallel implementations that effectively perform the computing tasks on a single GPU. For the peridynamics application, the only package needed in addition to Numpy is Numba whose just-in-time compiler allows us to write kernel functions in Python and compile them to run in parallel on a CUDA-enabled GPU. Our approach to numerical integration on implicit domains invokes two additional packages to support interval arithmetic and dynamic parallelism to enable tree-structured recursive refinement. Use of Python (with only kernels requiring dynamic parallelism written in C) enabled rapid development of concise code that successfully achieves significant performance enhancement. [ABSTRACT FROM AUTHOR]

Based on existing fuzzy simulation algorithms, this article presents two innovative techniques for approximating the expected values of fuzzy numbers’ monotone functions, which is of utmost importance in fuzzy optimization literature. In this regard, the stochastic discretization algorithm of Liu and Liu (2002) is enhanced by updating the discretization procedure for the simulation of the membership function and the calculation formula for the expected values. This is achieved through initiating a novel uniform sampling process and employing a formula for discrete fuzzy numbers, respectively, as the generated membership function in the stochastic discretization algorithm would adversely affect its accuracy to some extent. What is more, considering that the bisection procedure involved in the numerical integration algorithm of Li (2015) is time-consuming and also, not necessary for the specified types of fuzzy numbers, a special numerical integration algorithm is proposed, which can simplify the simulation procedure by adopting the analytical expressions of αoptimistic values. Subsequently, concerning the extensive applications of regular fuzzy intervals, several theorems are introduced and proved as an extended effort to apply the improved stochastic discretization algorithm and the special numerical integration algorithm to the issues of fuzzy intervals. Throughout this article, a series of numerical experiments are conducted from which the superiority of both the two novel techniques over others are conspicuously displayed in aspects of accuracy, stability, and efficiency. [ABSTRACT FROM AUTHOR]

To settle the problem of distributed nonlinear state estimation in sensor networks with naive nodes, a novel distributed high-degree cubature information filter with embedded hybrid consensus (DHCIF) is proposed. The multi-dimensional Gaussian weighted integrals involved in the filtering process are approximated by the fifth-degree cubature rule. A novel scheme, with consideration for the predicted measurement errors, is applied to compute the information contribution. With parallel consensus performed on both prior and measurement information, the proposed DHCIF is derived. The stability analysis with regard to consistency and boundedness of estimation errors for the proposed DHCIF is also developed. Finally, the effectiveness and advantage of the proposed DHCIF is validated by a typical maneuvering target tracing scenario. The experimental results indicate that the proposed DHCIF outperforms the existing algorithms in the aspects of estimation accuracy, consistency and consensus at the cost of a little more extra computation burden.

As a versatile template characterizing the center of mass movements in legged locomotion, the sagittal spring-loaded inverted pendulum (SLIP) model has been extensively explored in both biomechanics and robotics. Despite concise in mathematical formulation, the accurate analytical representation of the SLIP model is unaccessible due to its intrinsic nonlinearity. This article extends the traditional SLIP model from sagittal hopping into spatially running. A novel perturbation-based approach is proposed to obtain an analytical approximate solution for the 3-D-SLIP model, resulting in a straight-forward closed-form formulation wherein the numerical integration or iteration is avoided. The derived solution does not rely on the negligible gravity assumption as conventional simplification reported in the existing literature and offers satisfactory prediction performance in a wide range of model parameter combinations. The merits of the acquired approximate solution in both critical state of the apex return map and trajectory prediction with high accuracy have been demonstrated via performance evaluation, endowing this approximation the potential in motion planning and gait control for legged robots. [ABSTRACT FROM AUTHOR]

In this article, the low-frequency magnetic shielding effectiveness of a planar screen with magnetic and conductive properties in a coplanar-loops configuration (as required in the IEEE Standard 299) is evaluated through a new closed-form formula under the thin-sheet and small-loop approximations. The formulation clearly shows the separate contribution of the permeability and the conductivity of the screen to the shielding performance. The results obtained with the proposed closed-form formulas are compared with the exact ones derived through numerical integrations to assess the relevant accuracy and limits of validity. [ABSTRACT FROM AUTHOR]

This article provides the implementation procedure, the validation, and some considerations on the computational efforts of the analytical expressions developed for the lightning electromagnetic fields presented in the companion paper. The validation is presented with different configurations in terms of channel-base current, ground conductivity, and distance to the lightning channel, comparing the obtained results with the numerical integration of the classical formulas. The comparative analysis shows a perfect agreement between the proposed analytical approach and reference numerical simulations. Moreover, the computational effort of the proposed method is discussed, focusing the attention on the choice of the points in which the channel has to be divided in order to maximize the CPU time savings without losing accuracy. [ABSTRACT FROM AUTHOR]

This paper proposes a systematic approach to simulating stochastic electromagnetic transients in emtp, through the reliable numerical solution of stochastic differential equations. Actually, numerical integration schemes for stochastic differential equations (sdes) are not available in emtp. It is shown how to set up the Semi-Implicit Backward Euler scheme specific for stochastic differential equations. The implemented interaction strategy between the engine for the solution of electrical circuits and the engine for the Transient Analysis of Control Systems of emtp is exploited. This is a bolt from the blue feature of this simulator that makes it the only one on the shelf able to handle stochastic differential equations. Exploiting this unveiled capability, some basic and tutorial case studies are presented. They pave the way for the simulation of more complex circuits and systems that go beyond the scope of this work. Simulation files are provided as supplementary material and they are available on GitHub. [ABSTRACT FROM AUTHOR]

One of the most accurate approaches to the lightning-related transient analysis of grounding systems in layered soil is based on the method of moments solution of the integral form of the Maxwell equations. The practical application of this rigorous model to large systems is hindered by a great amount of computer time required for the numerical integration of oscillatory and slowly convergent Sommerfeld integrals (SI). As a result, approximate methods that avoid the solution of SI, e.g., image theory-based methods, are often used for computation of the response of large grounding systems. In this article, we extend the application of the rigorous model for large grounding systems in multilayered soil by applying a procedure that minimizes the number of direct computations of the SI using interpolation over a grid of a small number of sample points. We implement a three-dimensional interpolation scheme that adapts to the problem to minimize the interpolation grid, maintaining the accuracy of the rigorous approach. The developed model is applied for the analysis of transient voltages that might couple to secondary cables and disrupt the operation of connected equipment in case of lightning. The analysis shows that there are cases when the image theory-based models might underestimate the possibly harmful transient voltages. [ABSTRACT FROM AUTHOR]

Spherical Fibonacci grids (SFG) yield extremely uniform point set distributions on the sphere. This feature makes SFGs particularly well-suited to a wide range of computer graphics applications, from numerical integration, to vector quantization, among others. However, the application of SFGs to problems in which further refinement of an initial point set is required is currently not possible. This is because there is currently no solution to the problem of adding new points to an existing SFG while maintaining the point set properties. In this work, we fill this gap by proposing the extensible spherical Fibonacci grids (E-SFG). We start by carrying out a formal analysis of SFGs to identify the properties which make these point sets exhibit a nearly-optimal uniform spherical distribution. Then, we propose an algorithm (E-SFG) to extend the original point set while preserving these properties. Finally, we compare the E-SFG with a other extensible spherical point sets. Our results show that the E-SFG outperforms spherical point sets based on a low discrepancy sequence both in terms of spherical cap discrepancy and in terms of root mean squared error for evaluating the rendering integral. [ABSTRACT FROM AUTHOR]

In this communication, a closed-form formulation for calculating the radiated power at sidebands of four-dimensional (4-D) antenna arrays using a generalized element pattern of type $(\sin \theta)^{u}|\cos \theta |^{v}$ is derived. All the expressions exist when the exponents $u, v$ lie in the region: $u>-1$ and $v>-1/2$. Compared with numerical integration, the proposed expression uses less computational resources and time. Examples show the efficiency and accuracy of proposed expression. [ABSTRACT FROM AUTHOR]

The Partial Element Equivalent Circuit (PEEC) method is promising numerical technique for three-dimension electromagnetic modeling across various application fields. In the framework of the PEEC method, the partial elements modeling the magnetic and electric field coupling between elementary volumes and surfaces are computed by double-folded volume and surface integrals. Assuming the quasi-static hypothesis and an orthogonal mesh, the integrals have been computed by the analytical formulas derived in literature, which significantly reduces the computational time in comparison to the numerical integration. However, the existing analytical formulas are affected by significant numerical errors for certain PEEC structural mesh necessary to model the skin and proximity effects with a higher accuracy. To utilize the full potential of the PEEC method, the calculation of partial elements has to be carefully addressed, which has not been investigated in a comprehensive way so far. Accordingly, this paper presents a systematic accuracy analysis of the existing closed-form analytical formulas and methods for calculating the self and mutual inductances between two rectangular conductors. Additionally, a new strategy to select a proper analytical formula depending on the dimensions and positions of two conductors is proposed, which allows the mutual inductance extraction with a relative error of less than 0.1 $\%$. The new method is systematically validated on examples of 3-D dense PEEC systems using the quadruple precision arithmetic as reference. [ABSTRACT FROM AUTHOR]

In this article, a new iterative numerical method for solving bivariate nonlinear fuzzy Fredholm integral equations is proposed. The method combines two well-proven approaches—successive approximations and mixed trapezoidal and midpoint rules for the numerical integration. Both approaches are elaborated for fuzzy-valued functions. The main advantage of the proposed approach is that, by targeting the particular equation, another subordinate problem was solved under the common constraints. By this, we mean the development of a numerical method for fuzzy integrals. As a result, the proposed method is more accurate in comparison with any other mechanical combination of two separate and independent methods. We give conditions for the existence and uniqueness of a solution and estimate the error of the obtained approximation. We prove the stability and the method is performed on test problems to verify our theoretical results; numerical results are compared with those from existing methods in the literature to confirm the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

Measures of vascular tortuosity—how curved and twisted a vessel is—are associated with a variety of vascular diseases. Consequently, measurements of vessel tortuosity that are accurate and comparable across modality, resolution, and size are greatly needed. Yet in practice, precise and consistent measurements are problematic—mismeasurements, inability to calculate, or contradictory and inconsistent measurements occur within and across studies. Here, we present a new method of measuring vessel tortuosity that ensures improved accuracy. Our method relies on numerical integration of the Frenet-Serret equations. By reconstructing the three-dimensional vessel coordinates from tortuosity measurements, we explain how to identify and use a minimally-sufficient sampling rate based on vessel radius while avoiding errors associated with oversampling and overfitting. Our work identifies a key failing in current practices of filtering asymptotic measurements and highlights inconsistencies and redundancies between existing tortuosity metrics. We demonstrate our method by applying it to manually constructed vessel phantoms with known measures of tortuousity, and 9,000 vessels from medical image data spanning human cerebral, coronary, and pulmonary vascular trees, and the carotid, abdominal, renal, and iliac arteries. [ABSTRACT FROM AUTHOR]

Short-circuit current calculation and its characteristic analysis are the foundation of selecting circuit breakers and designing protection systems. First, a companion circuit-based method is presented to fast calculate the short-circuit current of asymmetric bipolar modular multilevel converter-based multiterminal direct current (MMC-MTDC) grids. With the reduction of the whole MTDC grid into an equivalent RLC circuit, the short-circuit current of the grid is efficiently and accurately solved by companion circuit method instead of the numerical integration. In addition to the efficiency, the proposed method can readily adapt to various types of MTDC grids and all kinds of dc faults. Second, unique short-circuit current characteristics of a bipolar MTDC grid following different types of dc faults are theoretically analyzed. A fault type discrimination method is then proposed for bipolar MMC-MTDC grids with dedicated metallic return. Finally, intensive numerical results of two MTDC grids validate the accuracy, adaptability, and efficiency of the presented dc grid short-circuit current calculation method as well as the correctness of the short-circuit current characteristic analyses. [ABSTRACT FROM AUTHOR]

Importance sampling (IS) and numerical integration methods are usually employed for approximating moments of complicated target distributions. In its basic procedure, the IS methodology randomly draws samples from a proposal distribution and weights them accordingly, accounting for the mismatch between the target and proposal. In this work, we present a general framework of numerical integration techniques inspired by the IS methodology. The framework can also be seen as an incorporation of deterministic rules into IS methods, reducing the error of the estimators by several orders of magnitude in several problems of interest. The proposed approach extends the range of applicability of the Gaussian quadrature rules. For instance, the IS perspective allows us to use Gauss-Hermite rules in problems where the integrand is not involving a Gaussian distribution, and even more, when the integrand can only be evaluated up to a normalizing constant, as it is usually the case in Bayesian inference. The novel perspective makes use of recent advances on the multiple IS (MIS) and adaptive (AIS) literatures, and incorporates it to a wider numerical integration framework that combines several numerical integration rules that can be iteratively adapted. We analyze the convergence of the algorithms and provide some representative examples showing the superiority of the proposed approach in terms of performance. [ABSTRACT FROM AUTHOR]

The dynamic modeling of an underwater vehicle–manipulator system (UVMS) is still a challenging task, mainly because of nonlinear nature of the hydrodynamic forces acting on the UVMS. This article presents a simplified modeling of the drag forces for the underwater manipulator. The drag force of an underwater body is the function of velocity distribution over the surface of the body. The basis of the proposed drag model is velocity distribution of a rigid link with uniform cross section, which can be uniquely expressed in terms of the velocities of its end points. Therefore, the drag coefficient of the link can be expressed as a function of the velocities of the end points and the ratio of length to cross-sectional dimensions. The drag force was modeled in two parts. First, the drag coefficient was identified with respect to the end point velocities of a one-link manipulator. Second, the identified drag coefficient was used to calculate the drag force. The approach was used to simulate a two-link manipulator in two different configurations. The simulation results were found in good agreement with the experimental results. The proposed drag model does not require any numerical integration or discretization of the links, as in the strip theory used repeatedly by many in the literature. The proposed concept was extended to model the drag force for a ten-degree-of-freedom UVMS and a comparative study was done with the strip theory. The efficacy of the proposed drag modeling and the one based on strip theory was evaluated in terms of accuracy of the simulation results and CPU time. [ABSTRACT FROM AUTHOR]

Numerical integration schemes are mandatory to understand complex behaviors of dynamical systems described by ordinary differential equations. Implementation of these numerical methods involve floating-point computations and propagation of round-off errors. This paper presents a new fine-grained analysis of round-off errors in explicit Runge-Kutta integration methods, taking into account exceptional behaviors, such as underflow and overflow. Linear stability properties play a central role in the proposed approach. For a large class of Runge-Kutta methods applied on linear problems, a tight bound of the round-off errors is provided. A simple test is defined and ensures the absence of underflow and a tighter round-off error bound. The absence of overflow is guaranteed as linear stability properties imply that (computed) solutions are non-increasing. [ABSTRACT FROM AUTHOR]

This article presents the design and optimization of the GPU kernels for numerical integration, as it is applied in the standard form in finite-element codes. The optimization process employs autotuning, with the main emphasis on the placement of variables in the shared memory or registers. OpenCL and the first order finite-element method (FEM) approximation are selected for code design, but the techniques are also applicable to the CUDA programming model and other types of finite-element discretizations (including discontinuous Galerkin and isogeometric). The autotuning optimization is performed for four example graphics processors and the obtained results are discussed. [ABSTRACT FROM AUTHOR]

This article presents a time-domain macromodeling algorithm for buried transmission lines subject to coupling of an incident field and terminated with linear and nonlinear loads. Different from overhead lines, the determination of the transient ground admittance is of great importance due to the complexity of the analytical expressions and the necessity of assuring a good approximation over a wide frequency range. The proposed macromodel adopts the analog behavioral modeling (ABM) library of Spice solvers to model frequency-dependent transmission line parameters directly, and calculate convolution in time domain, making it simple and convenient to treat time-domain field-to-line coupling equations without the need of time and spatial discretization and curve fitting used in other time-domain numerical algorithms. Besides, the model complexity is not limited by the line length. Hence, the proposed algorithm can significantly simplify the modeling process and improve simulation efficiency, especially for long multiconductor transmission lines. Furthermore, an effective and simplified expression for self and mutual ground impedance and admittance of underground lines is proposed in this article. With the proposed expression, the time-consuming numerical integration can be avoided, leading to a saving of 98% of the computational time. Finally, several validation examples of field coupling to buried power cables terminated with linear and nonlinear devices are investigated. The comparison of calculated waveshapes and CPU times shows the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

This article presents new analytical formulas for the efficient computation of the full-wave electric field generated by conductive, dielectric, and magnetic media in the framework of the partial element equivalent circuit (PEEC) method. To this aim, the full-wave Green’s function is handled by the Taylor series expansion leading to three types of integrals for which new analytical formulas are provided in order to avoid slower numerical integration. An orthogonal (Manhattan type) tessellation of the geometries is assumed, and the electrical quantities, i.e., currents, charges, and magnetization, are expanded in space through rectangular basis functions. The full-wave electric field radiated by charges, currents, and magnetization is computed analytically in the postprocessing step. The proposed closed-form computation of the electric field is tested using two examples, comparing the results obtained by the derived analytical formulas with the results from a finite element method solver. [ABSTRACT FROM AUTHOR]

Closed-form expressions are derived that eliminate the need to use numerical integration when computing the Lorentz force and torque in the method of moments-based models of low-frequency magnetic components (i.e., electric machinery). A permanent- magnet synchronous machine (PMSM) is used as an example system to illustrate the applicability of closed-form expressions and to validate them. Validation is provided by comparing the results to those obtained using a commercial finite element-based model. [ABSTRACT FROM AUTHOR]

At present, the method of obtaining the voltage of transmission lines by using the inverse problem of the electric field is generally faced with the difficulties of solving data equations. Aiming at the above problems, the method that solves transmission wire voltage through spatial electric field integration based on D-dot sensor is proposed. In this paper, with the ground as the reference potential and the vertical line of the transmission line to the ground as the integral path, D-dot electric field sensor is applied to measure the instant electric field values of several nodes on the integral path and the transmission wire voltage is directly solved combined with Gauss–Legendre numerical integration algorithm. The Gauss–Legendre algorithm is verified by the electric field distribution data on the integration path obtained by simulation. Furthermore, the improvement of the Gauss–Legendre algorithm is realized through optimizing the integral interval so that the location of integral nodes is more reasonable. Finally, the measurement system based on LabVIEW is designed and the experiment platform of the analog transmission line is constructed. Experiment results show that the measurement method of transmission wire voltage through the improved Gauss–Legendre algorithm based on D-dot sensor owns high accuracy with an error of less than 1%. [ABSTRACT FROM AUTHOR]

This paper proposes a novel non-iterative method to solve power system differential algebraic equations (DAEs) using the differential transformation, which is a mathematical tool able to obtain power series coefficients by transformation rules instead of calculating high order derivatives and has proved to be effective in solving state variables of nonlinear differential equations in our previous study. This paper further solves non-state variables, e.g., current injections and bus voltages, directly with a realistic DAE model of power grids. These non-state variables, nonlinearly coupled in network equations, are conventionally solved by numerical methods with time-consuming iterations, but their differential transformations are proved to satisfy formally linear equations in this paper. Thus, a non-iterative algorithm is designed to analytically solve all variables of a power system DAE model with ZIP loads. From test results on a Polish 2383-bus system, the proposed method demonstrates fast and reliable time performance compared to traditional numerical approaches including the implicit trapezoidal rule method and a partitioned scheme using the explicit modified Euler method and Newton Raphson method. [ABSTRACT FROM AUTHOR]

We present our computationally efficient approach to modeling tunable graphene-based active metadevices, where the integral multivariate surface conductivity is reformulated in the time and frequency domains with physically interpretable and fast-to-compute integration-free terms. The derivation is built on an expansion to power series of $ {z}= -\text {exp}(-\mu /k_{B}T)$ that converges very fast for non-zero chemical potential values. The model reveals interesting decomposition of graphene response into a universal constant term plus a damped oscillator (digamma functions in the frequency domain) plus non-oscillating correction terms for near-zero potentials. The number of terms in that series is analyzed theoretically for a given accuracy. In practice, only a few series terms are required, making our approach very efficient for simulation of active metasurfaces compared with direct integration of Kubo’s formulas. A simple performance test comparing run times with our code versus the numerical integration of the original Kubo’s formulas demonstrates a speedup exceeding 103. The proposed models can be critical for the initial ellipsometric characterization of graphene and advanced global optimization of graphene-controlled metadevices. [ABSTRACT FROM AUTHOR]

Radio atmospherics (sferics for short) are electromagnetic (EM) pulses discharged by lightning return strokes and can be used for D region remote sensing. A frequency-domain full-wave (FDFW) numerical model is developed to simulate sferics propagation in earth ionosphere waveguide (EIWG). For a certain sampling frequency, EM fields of sferics are decomposed into a series of plane waves in the spectral domain via 2-D spatial Fourier transforms. The magnetized cold plasma medium of ionosphere is divided into several thin layers with the arbitrary anisotropic dielectric constant for each layer. The EM fields at the sferics receiver site are evaluated by recursively computing the reflection and transmission matrices in all layer boundaries for all the plane wave components and performing 2-D numerical integration in the spectral domain. The earth flattening technique is adopted and the horizontal wave vector of each plane wave component is directly modified in the spectral domain to compensate for earth curvature effects. Interactions between plane waves and the magnetized cold plasma of ionosphere are analyzed and discussed. Comparisons between the FDFW computation and finite-difference time-domain simulation as well as field measured sferics are implemented to verify the accuracy and reliability of the proposed new model. [ABSTRACT FROM AUTHOR]

This article deals with the acceleration of the physical optics (PO) approximation for the backscattering computation in the near and far fields, for both single-bounce (SB) and double-bounce (DB) PO contributions. It is based on physical arguments, which allow us to apply either a closed-form expression (rapid calculation) or a numerical double integration to evaluate the scattered field from subsurfaces of a given mesh. The use of the closed-form expression is governed by two criteria, which must be extended to the bistatic case and also to the DB: the conventional Fraunhofer (related to the far-field zone) criterion and the chord error (related to the surface curvature) criterion. The proposed method is tested on a dihedral geometry. [ABSTRACT FROM AUTHOR]

This article presents a theoretical analysis of an axial field machine using high-temperature superconductors (HTSs) wires and bulks. The air-gap magnetic field obtained with the HTS coil and modulated by the HTS bulks is predicted by two 2-D axisymmetric models. Analytical models are based on the solution of Laplace’s equation by the separation of the variable method. The torque is obtained by a quick numerical integration of the Laplace force that acts on the armature winding. The proposed model is compared with 3-D finite-element simulations and good agreement is obtained. This model can be used with an optimization design procedure with a large reduction of the computational time. [ABSTRACT FROM AUTHOR]