Accuracy improvement using a modified Gauss-quadrature for integral methods in electromagnetics

Schlemmer, E.; Steffan, J.; Rucker, W.M.; Richter, K.R.

doi:10.1109/20.124044

Cited by 17 publications

(8 citation statements)

References 5 publications

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“…The boundary element formulation used in this study is based on eight-noded quadrilateral isoparametric surface patches [12,14]. Any point r q on the surface of a boundary element (BE) is described by second-order shape functions W fc (£ 77) and nodes r fc s Γ, = 1 k=l (23)…”

Section: Forward Calculations Using the Boundary Element Methodsmentioning

confidence: 99%

“…From Equation (14) and the area element dA defined previously we obtaiñ -sin0d0d0 8 *sin£ (15) Applying the half-angle formula for trigonometric functions the integral equation kernel g(0 s , 0 S , 0, 0) of a spherical medium can be defined by (16) To the term set into parentheses in Eq. (16), the following series expansion of Legendre polynomials can be applied [17] 1 1 l vl -(2cosy-l) ^To * In terms of the associated Legendre functions, the Le-n (cos ) sin(0) cos( ηθ)άθ (19) where, ^v are the normalized Legendre polynomials defined by (20)…”

Section: Computing the Integral Equation Kernel On The Spherementioning

confidence: 99%

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Two-Dimensional Fourier Representation Used in the Bioelectric Forward Problem - Zweidimensionale Fourier-Darstellung, angewandt im bioelektrischen Vorwärtsproblem

Wach

Lafer²,

Tilg³

et al. 1999

Biomedizinische Technik/Biomedical Engineering

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The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.Schlüsselwörter: Bioelektrizität, doppelte Fourierreihen, diskrete Fouriertransformation, boundary element method, bioelektrisches Vorwärts-Problem Ziel dieser Studie war die Anwendung der diskreten zweidimensionalen Fouriertransformation zur Lösung der Integralgleichung im bioelektrischen Vorwärtsproblem. Das Potential, der Quellterm und der Kern der Integralgleichung wurden als gleichmäßig diskret abgetastet angenommen. Somit können die kontinuierlichen Funktionen des Problems durch deren diskrete Fouriertransformierte im räumlichen Frequenzbereich dargestellt werden. Die Methode wurde zur Berechnung des Potentials auf einer homogenen leitfähigen Kugel angewandt. Die Integralgleichung für das Potential wurde im räumlichen Frequenzbereich gelöst. Das Potential in den Abtastpunkten wurde durch Anwendung der inversen Fouriertransformation berechnet. Die Lösung der vorgestellten Methode wurde mit einer analytischen Lösung und mit einer Lösung durch Anwendung der Boundary Element-Methode verglichen. Isoparametrische quadrilaterale Boundary-Elemente wurden zur Modellierung des kugelförmigen Volumenleiters herangezogen. In der vorgestellten Methode wurde die sphärische Oberfläche durch eine parametrische Approximation der Randelemente beschrieben.

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Section: Forward Calculations Using the Boundary Element Methodsmentioning

confidence: 99%

Section: Computing the Integral Equation Kernel On The Spherementioning

confidence: 99%

Two-Dimensional Fourier Representation Used in the Bioelectric Forward Problem - Zweidimensionale Fourier-Darstellung, angewandt im bioelektrischen Vorwärtsproblem

Wach

Lafer²,

Tilg³

et al. 1999

Biomedizinische Technik/Biomedical Engineering

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“…The mass matrices M K,h and M Ex K,h can be computed analytically, whereas numerical integration is used to compile the others. In the realization, we utilize an advanced adaptive integration scheme, which is based on local subdivision of edges and shifting of Gaussian points [21] according to the singularities of the integral kernels. By the use of this quadrature, integrals of the form (6.6) (V K ν, μ) L2(∂K) and (K K ν, μ) L2(∂K) for ν ∈ τ 0 E , τ 1 E j : E ⊂ ∂K, j = 1, .…”

Section: Numerical Realizationmentioning

confidence: 99%

Higher Order BEM-Based FEM on Polygonal Meshes

Rjasanow¹,

Weißer²

2012

SIAM J. Numer. Anal.

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The BEM-based finite element method is reviewed and extended with higher order basis functions on general polygonal meshes. These functions are defined implicitly as local solutions of the underlying homogeneous problem with constant coefficients. They are treated by means of boundary integral formulations and are approximated using the boundary element method in the numerics. To obtain higher order convergence, a new approximation of the material coefficient is proposed since previous strategies are not sufficient. Following recent ideas, error estimates are proved which guarantee quadratic convergence in the H 1 -norm and cubic convergence in the L 2 -norm. The numerical realization is discussed and several experiments confirm the theoretical results. Introduction.In the field of numerical methods for partial differential equations there is an increasing interest in nonsimplicial meshes. Several applications in solid mechanics, biomechanics, and geological science show a need for general elements within a finite element simulation. Discontinuous Galerkin methods [9] and mimetic finite difference methods [3] are able to handle such meshes. Nevertheless, these two strategies yield nonconforming approximations in their original versions which are not in the Sobolev space in which the exact solution lies.In 1975, Wachspress [26] proposed the construction of conforming rational basis functions on convex polygons with any number of sides. In recent years, several improved basis functions on polygonal elements have been introduced and applied in linear elasticity [24] or computer graphics [15,17], for example. There are even the first attempts which seek to introduce quadratic finite elements on polygons [20]. The recent publication [2] extends the nodal mimetic discretization strategy to arbitrary order on polygonal meshes.In [7] and the detailed work [6], a new kind of conforming finite element method on polygonal meshes has been proposed which uses basis functions that fulfill the differential equation locally. In the local problems constant material parameters and vanishing right-hand sides are prescribed. In case of the diffusion equation harmonic basis functions are recovered. This method has been studied in several articles concerning convergence [12,13] and residual error estimates for adaptive mesh refinement [27]. In the following, the theory is extended to a higher order method on polygonal meshes.The outline of this article is as follows. In section 2, some notation as well as a model problem are introduced. We review the first order method proposed in [7] and give a definition for regular meshes. The BEM-based FEM is extended to a higher order scheme in section 3. Afterward, we introduce interpolation operators in section 4 and prove interpolation properties which yield error estimates for the finite element

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“…To solve these problems, an integral algorithm is introduced into the D-dot sensor voltage measurement system. A comparison of the Gaussian integral algorithm, Gaussian Legendre integral algorithm, Newton Coates algorithm and other commonly used numerical integration algorithms, shows that the Gaussian integration algorithm has the advantages of high precision, a fast convergence speed and a simple solution process [ 10 , 11 , 12 ]. Hence, a D-dot sensor transmission line voltage measurement method based on Gaussian integrals is proposed in this paper.…”

Section: Introductionmentioning

confidence: 99%

Research on Transmission Line Voltage Measurement Method of D-Dot Sensor Based on Gaussian Integral

Wang

Zhao

et al. 2018

Sensors

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D-dot sensors meet the development trend towards the downsizing, automation and digitalization of voltage sensors and is one of research hotspots for new voltage sensors at present. The traditional voltage measurement system of D-dot sensors makes possible the reverse solving of wire potentials according to the computational principles of the electric field inverse problem by measuring electric field values beneath the transmission line. Nevertheless, as it is limited by the solving method of the electric field inverse problem, the D-dot sensor voltage measurement system is struggling with solving difficulties and poor accuracy. To solve these problems, this paper suggests introducing a Gaussian integral into the D-dot sensor voltage measurement system to accurately measure the voltage of transmission lines. Based on studies of D-dot sensors, a transmission line voltage measurement method based on Gaussian integrals is proposed and used for the simulation of the electric field of a 220 kV and a 20 kV transmission line. The feasibility of the introduction of the Gaussian integral to solve transmission line voltage was verified by the simulation results. Finally, the performance of the Gaussian integral was verified by an experiment using the transmission line voltage measurement platform. The experimental results demonstrated that the D-dot sensor measurement system based on a Gaussian integral achieves high accuracy and the relative error is lower than 0.5%.

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Accuracy improvement using a modified Gauss-quadrature for integral methods in electromagnetics

Cited by 17 publications

References 5 publications

Two-Dimensional Fourier Representation Used in the Bioelectric Forward Problem - Zweidimensionale Fourier-Darstellung, angewandt im bioelektrischen Vorwärtsproblem

Two-Dimensional Fourier Representation Used in the Bioelectric Forward Problem - Zweidimensionale Fourier-Darstellung, angewandt im bioelektrischen Vorwärtsproblem

Higher Order BEM-Based FEM on Polygonal Meshes

Research on Transmission Line Voltage Measurement Method of D-Dot Sensor Based on Gaussian Integral

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