Compel (Vol. 23, 2004)

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Volume 23 Number 3 2004

ISBN 0-86176-978-3

ISSN 0332-1649

COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Selected papers from the 11th International Symposium on Electromagnetics Fields in Electrical Engineering ISEF 2003 Guest Editor: Professor S. Wiak Co-editors: Professor M. Trlep and Professor A. Krawczyk

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COMPEL

ISSN 0332-1649

The International Journal for Computation and Mathematics in Electrical and Electronic Engineering

Volume 23 Number 3 2004

Selected papers from the 11th International Symposium on Electromagnetics Fields in Electrical Engineering ISEF 2003 Guest Editor Professor S. Wiak Co-editors Professor M. Trlep and Professor A. Krawczyk

Access this journal online __________________________ 596 Editorial advisory board ___________________________ 597 Abstracts and keywords ___________________________ 598 Editorial __________________________________________ 605 Special issue section Application of Haar’s wavelets in the method of moments to solve electrostatic problems Aldo Artur Belardi, Jose´ Roberto Cardoso and Carlos Antonio Franc¸ a Sartori ____________________________________

606

A 3D multimodal FDTD algorithm for electromagnetic and acoustic propagation in curved waveguides and bent ducts of varying cross-section Nikolaos V. Kantartzis, Theodoros K. Katsibas, Christos S. Antonopoulos and Theodoros D. Tsiboukis ______________________________________

613

The highly efficient three-phase small induction motors with stator cores made from amorphous iron M. Dems, K. Kome˛za, S. Wiak and T. Stec __________________________

Access this journal electronically The current and past volumes of this journal are available at:

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625

CONTENTS

CONTENTS continued

Optimal shape design of a high-voltage test arrangement P. Di Barba, R. Galdi, U. Piovan, A. Savini and G. Consogno____________

633

Cogging torque calculation considering magnetic anisotropy for permanent magnet synchronous motors Shinichi Yamaguchi, Akihiro Daikoku and Norio Takahashi_____________

639

Magnetoelastic coupling and Rayleigh damping A. Belahcen ____________________________________________________

647

Modelling of temperature-dependent effective impedance of non-ferromagnetic massive conductor Ivo Dolezˇel, Ladislav Musil and Bohusˇ Ulrych ________________________

655

Field strength computation at edges in nonlinear magnetostatics Friedemann Groh, Wolfgang Hafla, Andre´ Buchau and Wolfgang M. Rucker_____________________________________________

662

Genetic algorithm coupled with FEM-3D for metrological optimal design of combined current-voltage instrument transformer Marija Cundeva, Ljupco Arsov and Goga Cvetkovski___________________

670

Adaptive meshing algorithm for recognition of material cracks Konstanty M. Gawrylczyk and Piotr Putek ___________________________

677

Incorporation of a Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations – application of the Newton-Raphson method J. Gyselinck, P. Dular, N. Sadowski, J. Leite and J.P.A. Bastos____________

685

The modelling of the FDTD method based on graph theory Andrzej Jordan and Carsten Maple _________________________________

694

Inverse problem – determining unknown distribution of charge density using the dual reciprocity method Dean Ogrizek and Mladen Trlep ___________________________________

701

Finite element modelling of stacked thin regions with non-zero global currents P. Dular, J. Gyselinck, T. Zeidan and L. Kra¨henbu¨hl ___________________

707

Reliability-based topology optimization for electromagnetic systems Jenam Kang, Chwail Kim and Semyung Wang _______________________

715

A ‘‘quasi-genetic’’ algorithm for searching the dangerous areas generated by a grounding system Marcello Sylos Labini, Arturo Covitti, Giuseppe Delvecchio and Ferrante Neri __________________________________________________

724

CONTENTS continued

Development of optimizing method using quality engineering and multivariate analysis based on finite element method Yukihiro Okada, Yoshihiro Kawase and Shinya Sano __________________

733

An improved fast method for computing capacitance L. Song and A. Konrad __________________________________________

740

Power losses analysis in the windings of electromagnetic gear Andrzej Patecki, Sławomir Ste˛pien´ and Grzegorz Szyman´ski ____________

748

Finite element analysis of the magnetorheological fluid brake transients Wojciech Szela˛g_________________________________________________

758

Magnetic stimulation of knee – mathematical model Bartosz Sawicki, Jacek Starzyn´ski, Stanisław Wincenciak, Andrzej Krawczyk and Mladen Trlep _______________________________________________ 767

2D harmonic analysis of the cogging torque in synchronous permanent magnet machines M. Łukaniszyn, M. Jagiela, R. Wro´bel and K. Latawiec ________________

774

Determination of a dynamic radial active magnetic bearing model using the finite element method Bosˇtjan Polajzˇer, Gorazd Sˇtumberger, Drago Dolinar and Kay Hameyer __

783

Electromagnetic forming: a coupled numerical electromagnetic-mechanical-electrical approach compared to measurements A. Giannoglou, A. Kladas, J. Tegopoulos, A. Koumoutsos, D. Manolakos and A. Mamalis ________________________________________________

789

Regular section 2D harmonic balance FE modelling of electromagnetic devices coupled to nonlinear circuits J. Gyselinck, P. Dular, C. Geuzaine and W. Legros _____________________

800

Finite element analysis of coupled phenomena in magnetorheological fluid devices Wojciech Szela˛g_________________________________________________

813

Comparison of the Preisach and Jiles-Atherton models to take hysteresis phenomenon into account in finite element analysis Abdelkader Benabou, Ste´phane Cle´net and Francis Piriou _______________

825

Error bounds for the FEM numerical solution of non-linear field problems Ioan R. Ciric, Theodor Maghiar, Florea Hantila and Costin Ifrim ________

835

EDITORIAL ADVISORY BOARD

Professor O. Biro Graz University of Technology, Graz, Austria Professor J.R. Cardoso University of Sao Paulo, Sao Paulo, Brazil Professor C. Christopoulos University of Nottingham, Nottingham, UK Professor J.-L. Coulomb Laboratoire d’Electrotechnique de Grenoble, Grenoble, France Professor X. Cui North China Electric Power University, Baoding, Hebei, China Professor A. Demenko Poznan´ University of Technology, Poznan´, Poland Professor E. Freeman Imperial College of Science, London, UK Professor Song-yop Hahn Seoul National University, Seoul, Korea Professor Dr.-Ing K. Hameyer Katholieke Universiteit Leuven, Leuven-Heverlee, Belgium Professor N. Ida University of Akron, Akron, USA Professor A. Jack The University, Newcastle Upon Tyne, UK

Professor D. Lowther McGill University, Ville Saint Laurent, Quebec, Canada

Editorial advisory board

Professor O. Mohammed Florida International University, Florida, USA Professor G. Molinari University of Genoa, Genoa, Italy

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Professor A. Razek Laboratorie de Genie Electrique de Paris - CNRS, Gif sur Yvette, France Professor G. Rubinacci Universita di Cassino, Cassino, Italy Professor M. Rudan University of Bologna, Bologna, Italy Professor M. Sever The Hebrew University, Jerusalem, Israel Professor J. Tegopoulos National Tech University of Athens, Athens, Greece Professor W. Trowbridge Vector Fields Ltd, Oxford, UK Professor T. Tsiboukis Aristotle University of Thessaloniki, Thessaloniki, Greece Dr L.R. Turner Argonne National Laboratory, Argonne, USA

Professor A. Kost Technische Universitat Berlin, Berlin, Germany

Professor Dr.-Ing T. Weiland Technische Universitat Darmstadt, Darmstadt, Germany

Professor T.S. Low National University of Singapore, Singapore

Professor K. Zakrzewski Politechnika Lodzka, Lodz, Poland

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 p. 597 # Emerald Group Publishing Limited 0332-1649

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COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 Abstracts and keywords # Emerald Group Publishing Limited 0332-1649

Application of Haar’s wavelets in the method of moments to solve electrostatic problems Aldo Artur Belardi, Jose´ Roberto Cardoso and Carlos Antonio Franc¸ a Sartori Keywords Electrostatics, Density measurement, Optimization techniques Presents the mathematical basis and some results, concerning the application of Haar’s wavelets, as an expansion function, in the method of moments to solve electrostatic problems. Two applications regarding the evaluation of linear and surface charge densities were carried out: the first one on a finite straight wire, and the second one on a thin square plate. Some optimization techniques were used, whose main computational performance aspects are emphasized. Presents comparative results related to the use of Haar’s wavelets and the conventional expansion functions. A 3D multimodal FDTD algorithm for electromagnetic and acoustic propagation in curved waveguides and bent ducts of varying cross-section Nikolaos V. Kantartzis, Theodoros K. Katsibas, Christos S. Antonopoulos and Theodoros D. Tsiboukis Keywords Electromagnetic fields, Acoustic waves, Wave physics, Finite difference time-domain analysis This paper presents a curvilinearlyestablished finite-difference time-domain methodology for the enhanced 3D analysis of electromagnetic and acoustic propagation in generalised electromagnetic compatibility devices, junctions or bent ducts. Based on an exact multimodal decomposition and a higherorder differencing topology, the new technique successfully treats complex systems of varying cross-section and guarantees the consistent evaluation of their scattering parameters or resonance frequencies. To subdue the non-separable modes at the structures’ interfaces, a convergent grid approach is developed, while the tough case of abrupt excitations is also studied. Thus, the proposed algorithm attains significant accuracy and savings, as numerically verified by various practical problems.

The highly efficient three-phase small induction motors with stator cores made from amorphous iron M. Dems, K. Kome˛za, S. Wiak and T. Stec Keywords Inductance, Design, Iron Applies the field/circuit two-dimensional method and improved circuit method to engineering designs of the induction motor with stator cores made of amorphous iron. Exploiting of these methods makes possible computation of many different specific parameters and working curves in steady states for the ‘‘high efficiency’’ three-phase small induction motor. Compares the results of this calculation with the results obtained for the classical induction motor with identical geometric structure. Optimal shape design of a high-voltage test arrangement P. Di Barba, R. Galdi, U. Piovan, A. Savini and G. Consogno Keywords Finite element analysis, Electrostatics, High voltage Discusses the automated shape design of the electrodes supplying an arrangement for highvoltage test. Obtains results that are feasible for industrial applications by means of an optimisation algorithm able to process discrete-valued design variables. Cogging torque calculation considering magnetic anisotropy for permanent magnet synchronous motors Shinichi Yamaguchi, Akihiro Daikoku and Norio Takahashi Keywords Magnetic fields, Torque, Laminates This paper describes the cogging torque of the permanent magnet synchronous (PM) motors due to the magnetic anisotropy of motor core. The cogging torque due to the magnetic anisotropy is calculated by the finite element method using two kinds of modeling methods: one is the 2D magnetization property method, and the other is the conventional method. As a result, the PM motors with parallel laminated core show different cogging torque waveform from the PM motors with the rotational laminated core due to the influence of the magnetic anisotropy. The amplitudes of the cogging torque are different depending on

the modeling methods in the region of high flux density. Magnetoelastic coupling and Rayleigh damping A. Belahcen Keywords Rayleigh-Ritz methods, Finite element analysis, Vibration measurement This paper presents a magnetoelastic dynamic FE model. As first approach, the effect of magnetostriction and strong coupling is not considered. The effect of Rayleigh damping factors on the vibrational behaviour of the stator core of a synchronous generator is studied using the presented model. It shows that the static approach is not accurate enough and the difference between calculations with damped and undamped cases is too important to be ignored. However, the difference between damped cases with reasonable damping is not very important. Modelling of temperature-dependent effective impedance of non-ferromagnetic massive conductor Ivo Dolez˘el, Ladislav Musil and Bohus˘ Ulrych Keywords Modelling, Numerical analysis, Inductance Impedance of long direct massive conductors carrying time-variable currents is a complex function of time. Its evolution is affected not only by the skin effect but also by the temperature rise. This paper presents a numerical method that allows one to compute the resistance and internal inductance of a non-ferromagnetic conductor of any cross-section from values of the total Joule losses and magnetic energy within the conductor, and also illustrates the theoretical analysis based on the field approach on a typical example and discusses the results. Field strength computation at edges in nonlinear magnetostatics Friedemann Groh, Wolfgang Hafla, Andre´ Buchau and Wolfgang M. Rucker Keywords Magnetic fields, Integral equations, Nonlinear control systems, Vectors Magnetostatic problems including iron components can be solved by a nonlinear

indirect volume integral equation. Its unknowns are scalar field sources. They are evaluated iteratively. In doing so the integral representation of fields has to be calculated. At edges singularities occur. Following a method to calculate the field strength on charged surfaces a way out is presented.

Abstracts and keywords

599 Genetic algorithm coupled with FEM-3D for metrological optimal design of combined current-voltage instrument transformer Marija Cundeva, Ljupco Arsov and Goga Cvetkovski Keywords Transformers, Genetic algorithms, Magnetic fields The combined current-voltage instrument transformer (CCVIT) is a complex non-linear electromagnetic system with increased voltage, current and phase displacement errors. Genetic algorithm (GA) coupled with finite element method (FEM-3D) is applied for CCVIT optimal design. The optimal design objective function is the metrological parameters minimum. The magnetic field analysis made by FEM-3D enables exact estimation of the four CCVIT windings leakage reactances. The initial CCVIT design is made according to analytical transformer theory. The FEM-3D results are a basis for the further GA optimal design. Compares the initial and GA optimal output CCVIT parameters. The GA coupled with FEM-3D derives metrologically positive design results, which leads to higher CCVIT accuracy class. Adaptive meshing algorithm for recognition of material cracks Konstanty M. Gawrylczyk and Piotr Putek Keywords Sensitivity analysis, Mesh generation, Optimization techniques Describes the algorithm allowing recognition of cracks and flaws placed on the surface of conducting plate. The algorithm is based on sensitivity analysis in finite elements, which determines the influence of geometrical parameters on some local quantities, used as objective function. The methods are similar to that of circuit analysis, based on differentiation of stiffness matrix. The algorithm works iteratively using gradient method. The information on the gradient of the goal

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function provides the sensitivity analysis. The sensitivity algorithm allows us to calculate the sensitivity versus x and y, so the nodes can be properly displaced, modeling complicated shapes of defects. The examples show that sensitivity analysis applied for recognition of cracks and flaws provides very good results, even for complicated shape of the flaw. Incorporation of a Jiles-Atherton vector hysteresis model in 2D FE magnetic field computations: application of the Newton-Raphson method J. Gyselinck, P. Dular, N. Sadowski, J. Leite and J.P.A. Bastos Keywords Finite element analysis, Vector hysteresis, Magnetic fields, Newton-Raphson method This paper deals with the incorporation of a vector hysteresis model in 2D finite-element (FE) magnetic field calculations. A previously proposed vector extension of the well-known scalar Jiles-Atherton model is considered. The vectorised hysteresis model is shown to have the same advantages as the scalar one: a limited number of parameters (which have the same value in both models) and ease of implementation. The classical magnetic vector potential FE formulation is adopted. Particular attention is paid to the resolution of the nonlinear equations by means of the Newton-Raphson method. It is shown that the application of the latter method naturally leads to the use of the differential reluctivity tensor, i.e. the derivative of the magnetic field vector with respect to the magnetic induction vector. This second rank tensor can be straightforwardly calculated for the considered hysteresis model. By way of example, the vector Jiles-Atherton is applied to two simple 2D FE models exhibiting rotational flux. The excellent convergence of the Newton-Raphson method is demonstrated. The modelling of the FDTD method based on graph theory Andrzej Jordan and Carsten Maple Keywords Modelling, Finite difference time-domain analysis, Magnetic fields, Graph theory Discusses a parallel algorithm for the finitedifference time domain method. In particular, investigates electromagnetic field propagation

in two and three dimensions. The computational intensity of such problems necessitates the use of multiple processors to realise solutions to interesting problems in a reasonable time. Presents the parallel algorithm with examples, and uses aspects of graph theory to examine the communication overhead of the algorithm in practice. This is achieved by observing the dynamically changing adjacency matrix of the communications graph. Inverse problem – determining unknown distribution of charge density using the dual reciprocity method Dean Ogrizek and Mladen Trlep Keywords Density measurement, Reciprocating engines, Algorithmic languages Presents the use of the dual reciprocity method (DRM) for solving inverse problems described by Poisson’s equation. DRM provides a technique for taking the domain integrals associated with the inhomogeneous term to the boundary. For that reason, the DRM is supposed to be ideal for solving inverse problems. Solving inverse problems, a linear system is produced which is usually predetermined and ill-posed. To solve that kind of problem, implements the Tikhonov algorithm and compares it with the analytical solution. In the end, tests the whole algorithm on different problems with analytical solutions. Finite element modelling of stacked thin regions with non-zero global currents P. Dular, J. Gyselinck, T. Zeidan and L. Kra¨henbu¨hl Keywords Laminates, Finite element analysis, Eddy currents Develops a method to take the eddy currents in stacked thin regions, in particular lamination stacks, into account with the finite element method using the 3D magnetic vector potential magnetodynamic formulation. It consists in converting the stacked laminations into continuums with which terms are associated for considering the eddy current loops produced by both parallel and perpendicular fluxes. Non-zero global currents can be considered in the

laminations, in particular for studying the effect of imperfect insulation between their ends. The method is based on an analytical expression of eddy currents and is adapted to a wide frequency range. Reliability-based topology optimization for electromagnetic systems Jenam Kang, Chwail Kim and Semyung Wang Keywords Design, Optimization techniques, Topology, Sensitivity analysis This paper presents a probabilistic optimal design for electromagnetic systems. A 2D magnetostatic finite element model is constructed for a reliability-based topology optimization (RBTO). Permeability, coercive force, and applied current density are considered as uncertain variables. The uncertain variable means that the variable has a variance on a certain design point. In order to compute reliability constraints, a performance measure approach is widely used. To find reliability index easily, the limit-state function is linearly approximated at each iteration. This approximation method is called the first-order reliability method, which is widely used in reliability-based design optimizations. To show the effectiveness of the proposed method, RBTO for the electromagnetic systems is applied to magnetostatic problems. A ‘‘quasi-genetic’’ algorithm for searching the dangerous areas generated by a grounding system Marcello Sylos Labini, Arturo Covitti, Giuseppe Delvecchio and Ferrante Neri Keywords Programming, Algorithmic languages, Soil testing Sets out a method for determining the dangerous areas on the soil surface. The touch voltages are calculated by a Maxwell’s subareas program. The search for the areas in which the touch voltages are dangerous is performed by a suitably modified genetic algorithm. The fitness is redefined so that the genetic algorithm does not lead directly to the only optimum solution, but to a certain number of solutions having pre-arranged ‘‘goodness’’ characteristics. The algorithm has been called ‘‘quasi-genetic’’ algorithm

and has been successfully applied to various grounding systems. Development of optimizing method using quality engineering and multivariate analysis based on finite element method Yukihiro Okada, Yoshihiro Kawase and Shinya Sano Keywords Multivariate analysis, Finite element analysis, Torque, Optimization techniques Describes the method of optimization based on the finite element method. The quality engineering and the multivariable analysis are used as the optimization technique. In addition, this method is applied to a design of IPM motor to reduce the torque ripple. An improved fast method for computing capacitance L. Song and A. Konrad Keywords Capacitance, Computer applications, Production cycle In the design of chip carriers, appropriate analysis tools can shorten the overall production cycle and reduce costs. Among the functions to be performed by such computer-aided engineering software tools are self and mutual capacitance calculations. Since the method of moments is slow when applied to large multi-conductors systems, a fast approximate method, the average potential method (APM), can be employed for capacitance calculations. This paper describes the improved average potential method, which can further reduce the computational complexity and achieve more accuracy than the APM. Power losses analysis in the windings of electromagnetic gear Andrzej Patecki, Sławomir Ste˛pien´ and Grzegorz Szyman´ski Keywords Power measurement, Electromagnetic fields, Eddy currents Presents 3D method for the computation of the winding current distribution and power losses of the electromagnetic gear. For a prescribed current obtained from measurement, the transient eddy current

Abstracts and keywords

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field is defined in terms of a magnetic vector potential and an electric scalar potential. From numerically obtained potentials the power losses are determined. The winding power losses calculation of an electromagnetic gear shows that a given course of the current generates skin effect and significantly changes the windings resistances. Also presents the designing method for reducing power losses.

! potential T and magnetic scalar potential V . Since the problem is of low frequency and the electric conductivity of biological tissues is very small, consideration of electric vector potential only is quite satisfactory.

Finite element analysis of the magnetorheological fluid brake transients Wojciech Szela˛g Keywords Newton-Raphson method, Fluid dynamics, Finite element anaylsis Deals with coupled electromagnetic, hydrodynamic, thermodynamic and mechanical motion phenomena in magnetorheological fluid brake. Presents the governing equations of these phenomena. The numerical implementation of the mathematical model is based on the finite element method and a step-by-step algorithm. In order to include non-linearity, the NewtonRaphson process has been adopted. The method has been successfully adapted to the analysis of the coupled phenomena in the magnetorheological fluid brake. Present the results of the analysis and measurements.

2D harmonic analysis of the cogging torque in synchronous permanent magnet machines M. Łukaniszyn, M. Jagiela, R. Wro´bel and K. Latawiec Keywords Magnetic devices, Torque, Flux density, Fourier transforms Presents an approach to determine sources of cogging torque harmonics in permanent magnet electrical machines on the basis of variations of air-gap magnetic flux density with time and space. The magnetic flux density is determined from the twodimensional (2D) finite element model and decomposed into the double Fourier series through the 2D fast Fourier transform (FFT). The real trigonometric form of the Fourier series is used for the purpose to identify those space and time harmonics of magnetic flux density whose involvement in the cogging torque is the greatest relative contribution. Carries out calculations for a symmetric permanent magnet brushless machine for several rotor eccentricities and imbalances.

Magnetic stimulation of knee – mathematical model Bartosz Sawicki, Jacek Starzyn´ski, Stanisław Wincenciak, Andrzej Krawczyk and Mladen Trlep Keywords Finite element analysis, Bones, Vectors, Mathematical modelling Arthritis, the illness of the bones, is one of the diseases which especially attack the knee joint. Magnetic stimulation is a very promising treatment, although not very clear as to its physical background. Deals with the mathematical simulation of the therapeutical technique, i.e. the magnetic stimulation method. Considers the low-frequency magnetic field. To consider eddy currents one uses the pair of potentials: electric vector

Determination of a dynamic radial active magnetic bearing model using the finite element method ˘ tumberger, Bos˘tjan Polajz˘er, Gorazd S Drago Dolinar and Kay Hameyer Keywords Magnetic fields, Modelling, Nonlinear control systems The dynamic model of radial active magnetic bearings, which is based on the current and position dependent partial derivatives of flux linkages and radial force characteristics, is determined using the finite element method. In this way, magnetic nonlinearities and cross-coupling effects are considered more completely than in similar dynamic models. The presented results show that magnetic nonlinearities and cross-coupling effects can

change the electromotive forces considerably. These disturbing effects have been determined and can be incorporated into the real-time realization of nonlinear control in order to achieve cross-coupling compensations.

Electromagnetic forming: a coupled numerical electromagnetic-mechanicalelectrical approach compared to measurements A. Giannoglou, A. Kladas, J. Tegopoulos, A. Koumoutsos, D. Manolakos and A. Mamalis Keywords Electromagnetic fields, Finite element analysis, Manufacturing systems, Numerical analysis Undertakes an analysis of electromagnetic forming process. Despite the fact that it is an old process, it is able to treat current problems of advanced manufacturing technology. Primary emphasis is placed on presentation of the physical phenomena, which govern the process, as well as their numerical representation by means of simplified electrical equivalent circuits and fully coupled fields approach of the electromagneticmechanical-electric phenomena involved. Compares the numerical results with measurements. Finally, draws conclusions and perspectives for future work.

2D harmonic balance FE modelling of electromagnetic devices coupled to nonlinear circuits J. Gyselinck, P. Dular, C. Geuzaine and W. Legros Keywords Finite element analysis, Nonlinear control systems, Harmonics, Frequency multipliers This paper deals with the two-dimensional finite element analysis in the frequency domain of saturated electromagnetic devices coupled to electrical circuits comprising nonlinear resistive and inductive components. The resulting system of nonlinear algebraic equations is solved straightforwardly by means of the NewtonRaphson method. As an application example we consider a three-phase transformer feeding

a nonlinear RL load through a six-pulse diode rectifier. The harmonic balance results are compared to those obtained with timestepping and the computational cost is briefly discussed.

Finite element analysis of coupled phenomena in magnetorheological fluid devices Wojciech Szela˛g Keywords Couplers, Electromagnetism, Fluids, Finite element analysis This paper deals with coupled electromagnetic, hydrodynamic and mechanical motion phenomena in magnetorheological fluid devices. The governing equations of these phenomena are presented. The numerical implementation of the mathematical model is based on the finite element method and a step-by-step algorithm. In order to include non-linearity, the NewtonRaphson process has been adopted. A prototype of an electromagnetic brake has been built at the Poznan´ University of Technology. The method has been successfully adapted to the analysis of this brake. The results of the analysis are presented.

Comparison of the Preisach and Jiles-Atherton models to take hysteresis phenomenon into account in finite element analysis Abdelkader Benabou, Ste´phane Cle´net and Francis Piriou Keywords Finite element analysis, Electromagnetism, Energy In this communication, the Preisach and JilesAtherton models are studied to take hysteresis phenomenon into account in finite element analysis. First, the models and their identification procedure are briefly developed. Then, their implementation in the finite element code is presented. Finally, their performances are compared with an electromagnetic system made of soft magnetic composite. Current and iron losses are calculated and compared with the experimental results.

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Error bounds for the FEM numerical solution of non-linear field problems Ioan R. Ciric, Theodor Maghiar, Florea Hantila and Costin Ifrim Keywords Error analysis, Magnetic fields, Field testing A bound for a norm of the difference between the computed and exact solution vectors for static, stationary or quasistationary

non-linear magnetic fields is derived by employing the polarization fixed point iterative method. At each iteration step, the linearized field is computed by using the finite element method. The error introduced in the iterative procedure is controlled by the number of iterations, while the error due to the chosen discretization mesh is evaluated on the basis of the hypercircle principle.

Editorial

Editorial This special issue is devoted to the papers that were presented at the International Symposium on Electromagnetic Fields in Electrical Engineering ISEF’03. The symposium was held in Maribor, Slovenia on 18-20 September 2003. The city of Maribor is known for its beauty, charm and academic flavour as well. Therefore, the participants of ISEF’03 found there very good atmosphere to present their papers and debate on them. After the selection process, 159 papers have been accepted for the presentation at the symposium and almost 90 per cent were presented at the conference both orally and in the poster sessions. The papers have been divided into the following groups: . Computational Electromagnetics; . Electromagnetic Engineering; . Coupled Field and Special Applications; . Bioelectromagnetics and Electromagnetic Hazards; . Magnetic Material Modelling.

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It is the tradition of the ISEF meetings that they comprise quite a vast area of computational and applied electromagnetics. Moreover, the ISEF symposia aim at joining theory and practice, thus the majority of papers are deeply rooted in engineering problems, being simultaneously of high theoretical level. Bearing this tradition, we hope to touch the heart of the matter in electromagnetism. The present issue of COMPEL contains 27 papers which have been selected by the editors on the basis of the reviewing process done by the chairmen of the sessions. This selection, however, gave the number of papers much bigger than the number imposed by the COMPEL. Thus, going to the number required we also considered differentia specifica of COMPEL – the selected papers are of more computational aspect than the remaining part of high-qualified papers. The latter ones are expected to be published elsewhere. We, the Editors of the special issue, would like to express our thanks to COMPEL for giving us the opportunity to present, at least, the flavour of the ISEF meeting. We also thank our colleagues for their help in reviewing the papers. Finally, we would like to wish the prospective readers of the issue to find within many subjects of interest. Mladen Trlep Chairman of the Organising Committee Andrzej Krawczyk Scientific Secretary Sławomir Wiak Chairman of the ISEF symposium

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 p. 605 q Emerald Group Publishing Limited 0332-1649

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The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

Application of Haar’s wavelets in the method of moments to solve electrostatic problems

606

Aldo Artur Belardi Centro Universita´rio de FEI, Sa˜o Paulo, Brazil

Jose´ Roberto Cardoso and Carlos Antonio Franc¸a Sartori Escola Polite´cnica, Universidade de Sa˜o Paulo, Sa˜o Paulo, Brazil Keywords Electrostatics, Density measurement, Optimization techniques Abstract Presents the mathematical basis and some results, concerning the application of Haar’s wavelets, as an expansion function, in the method of moments to solve electrostatic problems. Two applications regarding the evaluation of linear and surface charge densities were carried out: the first one on a finite straight wire, and the second one on a thin square plate. Some optimization techniques were used, whose main computational performance aspects are emphasized. Presents comparative results related to the use of Haar’s wavelets and the conventional expansion functions.

1. Formulation In order to illustrate the proposed methodology, the main theoretical aspects of the method of moments and of the Haar’s wavelets, concerning one- and two-dimensional configurations, are presented in this paper. 1.1 Method of moments Although the method of moments is a well known numerical technique, and the complete description and details of this method have already been presented in many papers, in order to guide the reader through the overall method explanation, a brief summary of this method is given. In a simplified way, it can be mentioned that the method of moments basis is the application of approximation functions, such as the one represented by the following expression (Harrington, 1968): X X an Lg n ¼ an kLg n ; W m l ¼ k f ; W m l for m ¼ 1; 2. . .; N ð1Þ f ðxÞ ¼ n

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 606–612 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540511

n

In the aforementioned expression, an represents the unknown coefficients; gn is the expansion function, e.g. the pulse or the Haar’s wavelets, “L” a mathematical operator, and “Wm” is a weighting function. Expression (1) can also be represented in a matrix form by ½A
*½a
¼ ½B
; where [a ] is the unknown coefficients column matrix, and the matrixes [A] and [B]: 2 3 2 3 kLg 1 ; W 1 l . . . kLgn ; W 1 l k f ; W 1l 6 7 6 7 ½A
¼ 4 kLg 1 ; W 2 l . . . kLgn ; W 2 l 5 and ½B
¼ 4 k f ; W 2 l 5 ð2Þ kLg 1 ; W n l . . . kLgn ; W n l k f ; W nl As a first application, the potential distribution on a finite and straight wire that can be calculated using the next equation is taken into consideration (Balanis, 1990):

Z

rðr 0 Þ dl 0 Rðx; x 0 Þ

ð3Þ

Application of Haar’s wavelets

Thus, making use of the method of moments, knowing the approximated solution function f(x), the expansion function g(x) and the weighting function W(x), the potential on a finite straight wire can be estimated by the inner product of these functions: Z a 1 gðxÞW ðxÞf ðxÞ dx ð4Þ V ðxÞ ¼ k g; W ; f l ¼ R RðxÞ 2a

607

1 V ðx; y ¼ 0; z ¼ 0Þ ¼ 4p1

Consequently, the surface density r(r 0 ) can be approximated by the N term expansion. If the wire is divided into uniform segments D ¼ L=N ; after applying the weight delta function of Dirac W m ¼ dðxm 2 x 0 Þ ¼ 1; the inner product will become: V ðxÞ ¼ kW m ; f ; Lgl ¼ dðx 2 xm Þ £

Z L N 1 X g n ðx 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 0 an 4p1 n¼1 0 ðxm 2 x 0 Þ2 þ a 2

ð5Þ

Assuming the charges placed in the center of each subdivision in relation to the axis, substituting the values of x by the distance of the charge position to the point P(xm), we will have an integral that is the only function of the x 0 . For a fixed potential V, the equation can be represented, using the matrix notation, ½V m
¼ ½Z mn
½a n
; in which Zmn is defined by: Z mn ¼

Z 0

L

g n ðx 0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 0 ðxm 2 x 0 Þ2 þ a 2

ð6Þ

The same approach can be used, if a two-dimensional application is considered. 1.2 The Haar’s wavelets Different types of functions can be used as expansion functions. Among them are the pulse function, the truncated cosine as well as the wavelets. In this paper, the Haar’s wavelets are used as the expansion function. Thus, considering a two-dimensional application, after applying the method of moments and considering the Haar’s wavelets, a function f(x, y) can be approximated by (Aboufadel and Schlicker, 1999): f ðx; yÞ ¼

1 X k¼21

ck fðx; yÞ þ

1 1 X X

dj;k f P ðx; yÞcj;k ðx; yÞ

ð7Þ

j¼21 k¼21

where “j”, and “k” are, respectively, the resolution and the translation levels. Moreover, once the Haar’s wavelets, and the so-called mother function (8) and the scale or father function (9) are applied, the formulation will result in a product combination of (10) and (11) given by (12): Þ j=2 j cðH j;k ðxÞ ¼ 2 cð2 x 2 kÞ j; k [ Z

ð8Þ

(

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f ðH Þ ðxÞ ¼

608

n

1

0 # x , 1; and

0

for other intervals

ð9Þ

Þ j cðH j;k ðxÞ ¼ ½fðxÞ cðxÞcð2xÞcð2x 2 1Þ. . .cð2 x 2 kÞ


ð10Þ

Þ j cðH j;k ð yÞ ¼ bfðyÞcðyÞcð2yÞcð2y 2 1Þ. . .cð2 y 2 kÞc

ð11Þ

o Þ ðH Þ cðH j;k ðxÞ; cj;k ðyÞ ¼ fðxÞfð yÞ; fðxÞcð yÞ; . . .; cð2x 2 1Þcð2y 2 1Þ

ð12Þ

As an illustration, Figure 1 shows the Haar’s function regarding two dimensions and one level of resolution, for a point P(xm, ym). On the other hand, if the potential in a finite and very thin plane plate is considered as an application, it can be evaluated by (Newland, 1993):

Figure 1. Representation of the Haar’s function for two-dimensional and one level of resolution

V ðx; yÞ4p1 ¼ aj bj

Z

a

2a

þ

Z

b

2b

Application of Haar’s wavelets

fðx; yÞ dx dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxm 2 x 0 Þ2 þ ð ym 2 y 0 Þ2

1 1 X X

aj;k bj;k

j¼21 k¼21

Z

a

2a

Z

b

2b

Þ cðH j;k ðx; yÞ dx dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðxm 2 x 0 Þ2 þ ð ym 2 y 0 Þ2

ð13Þ

609

In a very similar matrix notation that was used for the one-dimensional application, the previous equation can be described as ½V m
¼ ½Z mn
½an
in which Zmn is defined by: Z b Z a gn ðx 0 ; y 0 Þ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy 0 dx ð14Þ Z mn ¼ 2 2 2a 2b 4p1 ðxm 2 x 0 Þ þ ð ym 2 y 0 Þ It should be observed that the previous formulation is indexed by two parameters, “j” and “k”, allowing us to vary the precision of the results through these levels. Concerning the characteristic of the method and the application of the Haar’s wavelets, the main aspects are related to the resulting scattered matrices and null coefficients, an interesting property to be considered regarding the computational aspects. If one remembers that the equation to determine the coefficients of the approximation function can be written as in equation (15), those aspects can be realized based on the following approach: ½Z mn
½r
¼ ½V


ð15Þ

where Zmn is a square matrix that is not necessarily a sparse one, since it depends on the expansion function that was chosen. Moreover, taking advantages of the fact that the Haar’s matrix [ H ] is a sparse matrix, applying the matrix algebra, it will result (Wagner and Chew, 1995): ð16Þ ½Z 0mn
½r 0
¼ ½V 0
else, ½Z 0mn
¼ ½H
½Z mn
½H T
;

½r 0
¼ ½H T
21 ½r
; and ½V 0
¼ ½H
½V


ð17Þ

Consequently, we will obtain: ½H
½Z mn
½H T
½H T
21 ½r
¼ ½H
½V


ð18Þ

Thus, after applying such an approach, we obtained a symmetric matrix and, due to the properties of Haar’s function, a number of “near” zero matrix elements. Additionally, the assumption of threshold levels, a percentage of the difference between the maximum positive value and the minimum negative one, will help us to obtain an additional computing time reduction. It is based on the fact that, once it is adopted, the matrix elements that are smaller than this number will be assumed to be zero. 2. Applications and discussion Applying the aforementioned formulation, we obtained some results related to two applications: the first one related to a finite and straight wire, and another one regarding a thin plane plate. In the two applications it is assumed that a constant potential distribution is equal to 1 V.

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Table I presents the results regarding the charge surface density on a 1.0 m straight wire, when it is divided into 16 equal segments, as a function of the resolution ( j) and the translation (k) levels. Those results can be considered as the ones suitable to validate this approach. Figure 2 shows the surface charge density on a 1.0 m straight wire, and diameter equal to 0.0001 m, when the level 4 of resolution is applied for the wavelets, and 32 subdivisions are used. The wire is at a potential of 1.0 V.

Expansion function Point

Table I. Charge surface density ( pC/m) on a straight finite wire as a function of the resolution levels

Figure 2. The surface charge (pC/m) on a 1.0 m straight wire for 32 subdivisions

1 2 3 4 5 ... 12 13 14 15 16

Haar wavelet (level) 2 8.835 8.835 8.835 8.835 7.970 ... 7.970 8.835 8.835 8.835 8.835

3 9.376 9.376 8.274 8.274 8.059 ... 8.059 8.274 8.274 9.376 9.376

Pulse 4 9.957 8.764 8.411 8.219 8.102 ... 8.102 8.219 8.411 8.764 9.957

9.957 8.764 8.411 8.219 8.102 ... 8.102 8.219 8.411 8.764 9.957

Figure 3 shows the surface charge density on a square plate ð1:0 m £ 1:0 mÞ; when 16 subdivisions are considered, and the level 5 of resolution is applied for the wavelets. The plate is at a potential equal to 1.0 V. Table II shows the results obtained after comparing the values of the computing time (Patterson and Hennessy, 2001), function of the number of divisions in each axe of the plate, with and without applying the null value detection (NVD) approach. Moreover, we can compare the results obtained by the proposed methodology considering the expansion function as being the pulse with the other ones using the wavelets. The validation of the aforementioned approach was carried out based on the application of the statistical indexes regarding the paired data and the corresponding average correlation (Papoulis, 1991). For the straight wire, and 32 subdivisions, when the results related to the use of wavelets are compared with the pulse function ones, statistically, we could see no statistical difference between the two approaches (tcalc . 95 per cent). Moreover, it was verified that average comparative values related to the charge density value is less than 0.025 per cent, for the straight finite wire, and for square plane plate applications.

Application of Haar’s wavelets

611

Figure 3. The surface charge ( pC/m) on a 1.0 m £ 1.0 m plate, for 16 subdivisions

Divisions 4£4 8£8 16 £ 16 32 £ 32

Computing time (s) Without NVD With NVD 0.321 7.931 451.960 27,273.738

0.25 5.488 222.60 11,994.487

Difference ( per cent) 22.12 30.80 50.75 56.02

Table II. Computing time as a function of the number of the subdivisions and of the use of the NVD approach

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3. Conclusion This paper presented the main features of wavelets as an expansion function in the method of the moments. Although the proposed methodology can be applied to more complex problems, some applications in electrostatics were demonstrated. Based on the theoretical features and on the statistical indexes applied to comparative results regarding the Haar’s wavelets and the pulse functions, the proposed methodology was validated. The main advantages concerning the matrix arrangements and its numerical treatment, as well as the related computing time were discussed in the paper. References Aboufadel, E. and Schlicker, S. (1999), Discovering Wavelets, Wiley, New York, NY, pp. 1-42. Balanis, C. (1990), Advanced Engineering Electromagnetics, Wiley, New York, NY, pp. 670-95. Harrington, R.F. (1968), “Field computation by moment methods”, Electrical Science, pp. 1-40. Newland, D.E. (1993), Random Vibrations Spectral and Wavelet Analysis, Addison Wesley, Reading, MA, pp. 315-33. Papoulis, A. (1991), Probability Random Variables and Stochastic Processes, McGraw-Hill, New York, pp. 265-78. Patterson, D.A. and Hennessy, J.L. (2001), Computer Organization and Design the Hardware/Software Interface, 1st ed., Morgan Kaufmann, Los Altos, CA, pp. 26-51. Wagner, R.L. and Chew, W.C. (1995), “A study of wavelets for the solution of electromagnetic integral equations”, IEEE Transactions on Antennas and Propagation, Vol. 43 No. 8, pp. 802-10.

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The current issue and full text archive of this journal is available at www.emeraldinsight.com/0332-1649.htm

3D multimodal A 3D multimodal FDTD algorithm AFDTD algorithm for electromagnetic and acoustic propagation in curved waveguides 613 and bent ducts of varying cross-section Nikolaos V. Kantartzis, Theodoros K. Katsibas, Christos S. Antonopoulos and Theodoros D. Tsiboukis Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece Keywords Electromagnetic fields, Acoustic waves, Wave physics, Finite difference time-domain analysis Abstract This paper presents a curvilinearly-established finite-difference time-domain methodology for the enhanced 3D analysis of electromagnetic and acoustic propagation in generalised electromagnetic compatibility devices, junctions or bent ducts. Based on an exact multimodal decomposition and a higher-order differencing topology, the new technique successfully treats complex systems of varying cross-section and guarantees the consistent evaluation of their scattering parameters or resonance frequencies. To subdue the non-separable modes at the structures’ interfaces, a convergent grid approach is developed, while the tough case of abrupt excitations is also studied. Thus, the proposed algorithm attains significant accuracy and savings, as numerically verified by various practical problems.

Introduction The systematic modelling of arbitrary electromagnetic compatibility (EMC) applications or bent ducts with irregular cross-sections remains a fairly demanding area of contemporary research, since curvilinear coordinates complicate the separation of the wave equation and therefore, the extraction of a viable analytical solution. Furthermore, the involved fabrication details of such structures in both electromagnetics and acoustics, having a critical influence on the overall frequency response, enforce regular numerical realisations to utilise extremely fine meshes with heavy computational overheads. Soon after the detection of these shortcomings, several effective techniques have been presented for their mitigation (Farina and Sykulski, 2001; Przybyszewski and Mrozowski, 1998; Sikora et al., 2000) or the evolution of flexible discretisation perspectives (Felix and Pagneux, 2002; Rong et al., 2001) and robust lattice ensembles (Bossavit and Kettunen, 2001; Podebrad et al., 2003; Zagorodov et al., 2003). Among them, the finite-difference time-domain (FDTD) renditions (Taflove and Hagness, 2000) in accordance with the highly-absorptive perfectly matched layers (PMLs) (Berenger, 2003) constitute trustworthy simulation tools, especially in Cartesian grids. In this paper, a novel multimodal FDTD formulation, founded on a 3D curvilinear regime, is introduced for the precise analysis of electromagnetic and acoustic waves inside curved arrangements of varying cross-section. Owing to the essential role of

COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering Vol. 23 No. 3, 2004 pp. 613-624 q Emerald Group Publishing Limited 0332-1649 DOI 10.1108/03321640410540520

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614

the excitation scheme, the framework so developed, projects each component on the basis of a transverse mode series to derive an ordinary differential equation which performs reliable vector representations and launches the initial fields in close proximity to the discontinuity. To annihilate the erroneous oscillations near the curvature or bend, an impedance matrix that can be quantitatively integrated up to a sufficient number of modal counterparts is defined. In this context, the new FDTD concepts for electromagnetics are easily extended to acoustics in a completely dual manner, whereas a higher-order differencing tessellation suppresses the dispersion error mechanisms and evaluates the scattering properties irrespective of geometrical peculiarities or frequency spectrums. Conversely, dissimilar interface media distributions that do not follow the grid lines are handled via a convergent transformation. Numerical results, addressing diverse realistic EMC configurations, waveguides, junctions and ducts – terminated by appropriately constructed curvilinear PMLs – demonstrate the considerable accuracy, the stability as well as the drastic computational savings of the proposed approach. The 3D multimodal FDTD method in general coordinate systems Let us consider the general waveguide of Figure 1(a), including a toroidal-like section of length ut with inner and outer mean radii R1(u) and R2(u), respectively. The two straight parts can have arbitrary cross-sections, while their walls may be flexible or rigid. Actually, the most difficult part of this simulation is the bent waveguide sector that generates non-separable modes not easy to determine because of the cumbersome calculations required at every frequency and the incomplete fulfilment of the suitable continuity conditions regarding the straight parts. Comparable observations may be performed from Figure 1(b), where the more complex three-port Y junction is depicted. Herein, the discontinuity and the two ports prohibit any variable separation of Maxwell’s or the linearised Euler’s equations. To circumvent the prior defects, our methodology introduces a multimodal decomposition for the propagating quantities that is applied to prefixed planes in the bend. Thus, each component f can be written in terms of infinite series as X jk ðr; uÞFk ðr; uÞ with jk ðr; uÞ ¼ Bk cos½kpðr 2 R1 ðuÞÞ=RðuÞ
; ð1Þ f ðr; uÞ ¼ k

and Bk ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2 2 dk0 Þ=RðuÞ

for

Z

R1 þsl

jk ðr; uÞjl ðr; uÞ dr ¼ dkl ;

ð2Þ

R1

where RðuÞ ¼ R2 ðuÞ 2 R1 ðuÞ; Fk are scalar coefficients and jk are eigenfunctions complying with the corresponding transverse electromagnetic or acoustic eigenproblem. Two characteristic functions for R1(u) and R2(u), with au ¼ u=ut ; s ¼ sr 2 sl ; sr ¼ 1:25sl and t ¼ v=c0 ; run into several practical applications, are R1 ðuÞ ¼ st 2 ðau 2 1:5Þ þ sl 2 0:5sr ;

ð3Þ

R2 ðuÞ ¼ 2st 2 ðau 2 1:5Þ þ sl þ 0:5sr :

ð4Þ

The key issue in such systems is the initial mode coupling, occurring in two distinct ways: one due to the curvature of the waveguide or duct and the other due to its

A 3D multimodal FDTD algorithm

615

Figure 1. (a) Geometry of an arbitrarily-curved waveguide with a varying cross-section, and (b) transverse cut of a three-port Y junction comprising two ducts of elliptical cross-section and different dimensions

varying cross-section. The former normally contributes to the generation of higher-order modes and the latter induces the symmetric ones. It is stressed that the existing schemes cannot simulate this intricate situation contaminating so, the final outcomes. The proposed algorithm overcomes this artificial hindrance by projecting the appropriate governing laws to jk in order to extract an equivalent set of ordinary differential equations which combine electric and magnetic fields in electromagnetics or velocity and pressure in acoustics. Their solutions lead to very accurate excitation models and the most substantial; they preserve lattice duality, even when the source is placed quite close to the discontinuity. For illustration and without loss of generality, we concentrate on Maxwell’s curl analogues at a certain plane in the interior of the bend. Then, by means of the respective matrix terminology, one obtains the modified forms of Ampere’s and Faraday’s laws, 1›t ½E
¼ t 21 ðW A þ W B W C ÞH 2 W D E

ð5Þ

m›t ½H
¼ 2tW C E þ ðW E 2 W D ÞH;

ð6Þ

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with E and H the electric and magnetic field intensities defined at the general coordinates (u, v, w) of g(u, v, w) metrics. The elements of matrices W i ði ¼ A; . . .; EÞ describe the fundamental details of the curvature and should be carefully calculated. Hence, after enforcing the boundary constraints, we obtain 8 for k ¼ l