Three-dimensional Electromagnetic Problems Research Articles

A Parallel 3-D HIE-FDTD Method using the MPI Library

This paper presents the implementation of the parallel hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method using the Message Passing Interface (MPI) library. The method proves to be very effective in simulating large-scale three-dimensional electromagnetic problems with fine structures in one direction. For the decomposition of the computational volume in the HIE-FDTD method, an MPI Cartesian 2D topology is implemented, allowing arbitrary division of the volume in two directions. Derived data types provided in the MPI library are employed to optimize inter-process communication. High accuracy and efficiency are subsequently demonstrated through a numerical example of a frequency-selected surface (FSS). It shows that the proposed method is very suitable for parallel computing, and the parallel efficiency maintains above 80% for different numbers of processes.

A Symmetric FDTD Subgridding Method With Guaranteed Stability and Arbitrary Grid Ratio

A finite-difference time-domain (FDTD) subgridding method is proposed to efficiently and accurately solve three-dimensional electromagnetic problems. Based on the reciprocal and symmetric interpolation operators between coarse and subgridding meshes, spatial coupling matrices are carefully designed to guarantee long-time stability. To further enhance its capability of handling multiscale structures, arbitrary grid ratios and nested subgridding meshes are extended to be supported in the proposed method. In addition, the rigorous analysis shows that the proposed FDTD subgridding method is theoretically stable. Five numerical examples including a simple cavity with perfect electric conductors, a rectangular TEM waveguide, a dielectric resonator, a single-layer substrate integrated waveguide, and a large airplane platform with a dipole are carried out to validate its effectiveness. Results show that it is stable, accurate, efficient, and easy to model complex structures.

Turbogenerator Electromagnetic Analysis with Changing Reactive Load

Comparative numerical analysis of electromagnetic processes in the end zone of powerful turbogenerators has been carried out at changes in reactive load from rated duty to underexcitation, as well as performance parameters of static overload that determine the stability of synchronous operation of turbogenerators in the power system. It was demonstrated that the representation of magnetic field sources and calculation results in the form of rotating waves allows simplifying and speeding up the solution of threedimensional electromagnetic problems in the turbogenerator end zone. Using a previously developed calculation method, the electromagnetic fields and processes in over- and underexcitation modes, in particular, peculiarities of shielding in turbogenerators are studied. It was established that the added losses and electrodynamic forces in underexcitation modes acting on the outermost core packages, pressure plate, and stator shield increased by 1.5–3.0 times, resulting in accelerated aging of the turbogenerator end zone. An appropriate margin shall be provided in the cooling system and the structural design of the turbogenerator end zone. Static overload capacity in nominal conditions is determined by the ratio of maximum power to rated power. An analytical functional relationship between static overload capacity and load conditions was established. It was demonstrated that when switching to the underexcitation mode while maintaining the active load due to a decrease in the excitation current and maximum power, the static overload factor decreases to unity, that there is virtually no static stability margin of the turbogenerator in this mode. Calculations were made for turbogenerators with a capacity of 200–1111 MVA. It is recommended to limit the operation of turbogenerators in underexcitation modes and to use specialized compensators more extensively.

Investigation of electromagnetic wave propagation in the bicomplex 3D-FEM using a wavenumber Whitney Hodge operator

PurposeThe purpose of this paper is to show the applicability of a discrete Hodge operator in the context of the De Rham cohomology to bicomplex-valued electromagnetic wave propagation problems. It was applied in the finite element method (FEM) to get a higher accuracy through conformal discretization. Therewith, merely the primal mesh is needed to discretize the full system of Maxwell equations.Design/methodology/approachAt the beginning, the theoretical background is presented. The bicomplex number system is used as a geometrical algebra to describe three-dimensional electromagnetic problems. Because we treat rotational field problems, Whitney edge elements are chosen in the FEM to realize a conformal discretization. Next, numerical simulations regarding practical wave propagation problems are performed and compared with the common FEM approach using the Helmholtz equation.FindingsDifferent field problems of three-dimensional electromagnetic wave propagation are treated to present the merits and shortcomings of the method, which calculates the electric and magnetic field at the same spatial location on a primal mesh. A significant improvement in accuracy is achieved, whereas fewer essential boundary conditions are necessary. Furthermore, no numerical dispersion is observed.Originality/valueA novel Hodge operator, which acts on bicomplex-valued cotangential spaces, is constructed and discretized as an edge-based finite element matrix. The interpretation of the proposed geometrical algebra in the language of the De Rham cohomology leads to a more comprehensive viewpoint than the classical treatment in FEM. The presented paper may motivate researchers to interpret the form of number system as a degree of freedom when modeling physical effects. Several relationships between physical quantities might be inherently implemented in such an algebra.

Electromagnetic Modeling of Lossy Materials With a Potential-Based Boundary Element Method

The boundary element method (BEM) enables solving three-dimensional electromagnetic problems using a two-dimensional surface mesh, making it appealing for applications ranging from electrical interconnect analysis to the design of metasurfaces. The BEM typically involves the electric and magnetic fields as unknown quantities. Formulations based on electromagnetic potentials rather than fields have garnered interest recently, for two main reasons: (a) they are inherently stable at low frequencies, unlike many field-based approaches, and (b) potentials provide a more direct interface to quantum physical phenomena. Existing potential-based formulations for electromagnetic scattering have been proposed primarily for perfect conductors. We develop a potential-based BEM formulation which can capture both dielectric and conductive losses, and accurately models the skin effect over broad ranges of frequency. The accuracy of the proposed formulation is validated through canonical and realistic numerical examples.

Well Posedness and Finite Element Approximability of Three-Dimensional Time-Harmonic Electromagnetic Problems Involving Rotating Axisymmetric Objects

A set of sufficient conditions for the well posedness and the convergence of the finite element approximation of three-dimensional time-harmonic electromagnetic boundary value problems involving non-conducting rotating objects with stationary boundaries or bianisotropic media is provided for the first time to the best of authors’ knowledge. It is shown that it is not difficult to check the validity of these conditions and that they hold true for broad classes of practically important problems which involve rotating or bianisotropic materials. All details of the applications of the theory are provided for electromagnetic problems involving rotating axisymmetric objects.

Unconditionally stable meshless method for solving 3D transient electromagnetic problems based on LOD scheme

One of the main problems of the conventional meshless methods in the time domain is their conditional stability. Up to now, some algorithms have been introduced to remove the stability condition, like alternating-direction-implicit (ADI) and locally one-dimensional (LOD) schemes. ADI meshless method has been investigated completely, but there is no report to exploit the full efficiency of the LOD meshless method. So, in this study, LOD scheme is employed to a meshless method for solving three-dimensional transient electromagnetic problems. The initial form of the proposed meshless method solves time-domain Maxwell's equations in three sub-steps, so it is called LOD3. LOD3 meshless method has first-order temporal accuracy. Moreover, LOD5 meshless method is introduced to upgrade temporal accuracy. Also, the results show that LOD5 meshless method is more accurate than ADI and LOD3 meshless methods. The accuracy of the proposed method is due to two main factors; Crank–Nicolson scheme and solving the equation in five sub-steps instead of one sub-step. Stability and accuracy of the proposed method are assessed through numerical experiment. Also, the unconditional stability of the proposed method is proved, analytically.

Automated netlist generation for 3D electrothermal and electromagnetic field problems

We present a method for the automatic generation of netlists describing general three-dimensional electrothermal and electromagnetic field problems. Using a pair of structured orthogonal grids as spatial discretisation, a one-to-one correspondence between grid objects and circuit elements is obtained by employing the finite integration technique. The resulting circuit can then be solved with any standard available circuit simulator, circumventing the need for the implementation of a custom time integrator and dealing with non-linearities. Additionally, the approach straightforwardly allows for field-circuit coupling simulations by appropriately stamping the circuit description of lumped devices. As the computational domain in wave propagation problems must be finite, stamps representing absorbing boundary conditions are developed as well. Representative numerical examples are used to verify the approach. The results obtained by circuit simulation on the generated netlists are compared with appropriate reference solutions.

Solution to a three-dimensional axisymmetric inverse electromagnetic casting problem

ABSTRACTA new method based on topology optimization is proposed for solving a three-dimensional axisymmetric inverse problem regarding the configuration of electric currents used in the electromagnetic casting technique of the metallurgical industry. These electric currents must generate an electromagnetic field such that when certain mass of liquid metal is placed in the field it levitates acquiring a predefined axisymmetric shape. The method proposed is able to obtain a suitable configuration of electric currents, and is very attractive from the computational point of view, since it is based on a sparse convex quadratic programming formulation for which there are very efficient interior-point solution algorithms.

On perfectly matched layers for discontinuous Petrov–Galerkin methods

In this article, several discontinuous Petrov–Galerkin (DPG) methods with perfectly matched layers (PMLs) are derived along with their quasi-optimal graph test norms. Ultimately, two different complex coordinate stretching strategies are considered in these derivations. Unlike with classical formulations used by Bubnov–Galerkin methods, with so-called ultraweak variational formulations, these two strategies in fact deliver different formulations in the PML region. One of the strategies, which is argued to be more physically natural, is employed for numerically solving two- and three-dimensional time-harmonic acoustic, elastic, and electromagnetic wave propagation problems, defined in unbounded domains. Through these numerical experiments, efficacy of the new DPG methods with PMLs is verified.

Application of the generalized finite difference method to three-dimensional transient electromagnetic problems

We apply the generalized finite difference method (GFDM), a relatively new domain-type meshless method, for the numerical solution of three-dimensional (3D) transient electromagnetic problems. The method combines Taylor series expansions and the weighted moving least-squares method. The main idea here is to inherit the high-accuracy advantage of the former and the stability and meshless attributes of the latter. This makes the method particularly attractive for problems defined in 3D complex geometries. Three benchmark 3D problems governed by the Maxwell's equations with both smooth and piecewise smooth geometries have been analyzed. The convergence, accuracy and stability of the method with respect to increasing the number of scattered nodes inside the domain are studied.

Non‐conforming finite element tearing and interconnecting method with one Lagrange multiplier for solving large‐scale electromagnetic problems

A non-conforming finite element tearing and interconnecting method is proposed for the numerical analysis of large-scale three-dimensional electromagnetic problems. The whole computational domain is partitioned into many smaller subdomains, and a Robin-type transmission condition is introduced at the shared interfaces to exchange data among subdomains. A set of orthogonal polynomials is introduced to approximate auxiliary unknowns between adjacent subdomains. Then, a one-Lagrange multiplier scheme is applied to deal with non-conforming meshes at the interfaces. With the help of the Schur complement approach, the method formulates a reduced-order interface problem, which can be solved using an iterative algorithm. Once the resulting interface problem is solved, the unknown electric field in each subdomain can be calculated in parallel. Numerical examples are given to demonstrate the efficiency of the method.

A preconditioned dual–primal finite element tearing and interconnecting method for solving three-dimensional time-harmonic Maxwell's equations

A new preconditioned dual–primal nonoverlapping domain decomposition method is proposed for the finite element solution of three-dimensional large-scale electromagnetic problems. With the aid of two Lagrange multipliers, the new method converts the original volumetric problem to a surface problem by using a higher-order transmission condition at the subdomain interfaces to significantly improve the convergence of the iterative solution of the global interface equation. Similar to the previous version, a global coarse problem related to the degrees of freedom at the subdomain corner edges is formulated to propagate the residual error to the whole computational domain at each iteration, which further increases the rate of convergence. In addition, a fully algebraic preconditioner based on matrix splitting is constructed to make the proposed domain decomposition method even more robust and scalable. Perfectly matched layers (PMLs) are considered for the boundary truncation when solving open-region problems. The influence of the PML truncation on the convergence performance is investigated by examining the convergence of the transmission condition for an interface inside the PML. Numerical examples including wave propagation and antenna radiation problems truncated with PMLs are presented to demonstrate the validity and the capability of this method.

Fast and accurate finite element analysis of large-scale three-dimensional photonic devices with a robust domain decomposition method

A fast and accurate full-wave technique based on the dual-primal finite element tearing and interconnecting method and the second-order transmission condition is presented for large-scale three-dimensional photonic device simulations. The technique decomposes a general three-dimensional electromagnetic problem into smaller subdomain problems so that parallel computing can be performed on distributed-memory computer clusters to reduce the simulation time significantly. With the electric fields computed everywhere, photonic device parameters such as transmission and reflection coefficients are extracted. Several photonic devices, with simulation volumes up to 1.9×10(4) (λ/n(avg))3 and modeled with over one hundred million unknowns, are simulated to demonstrate the application, efficiency, and capability of this technique. The simulations show good agreement with experimental results and in a special case with a simplified two-dimensional simulation.

Low-Frequency Fast Multipole Method Based on Multiple-Precision Arithmetic

We present a low-frequency fast multipole method for the solution of three-dimensional electromagnetic problems involving small objects and their dense discretizations with respect to wavelength. The diagonalization of the Green's function is stabilized using a multiple-precision arithmetic (MPA) for accurate and efficient computations of subwavelength interactions. MPA provides a direct remedy for the low-frequency breakdown of the standard diagonalization based on plane waves, and it enables straightforward implementations for low-frequency problems.

Preconditioned Symmetric Linear BCG Method for Solving FEM Electromagnetism Problems

A new preconditioning technique for solving large linear systems arising from edge-based finite element method (FEM) analysis of three-dimensional (3-D) electromagnetic problems is presented. This method is achieved by applying a shifted-Laplace operator scheme and sparse approximate inverse to symmetric linear BCG (LBCG). The main purpose is to generate a more robust and efficient preconditioner. Numerical results on several electromagnetic problems show that, by comparing with other conventional preconditioning techniques, this technique is more efficient and robust, and can greatly reduce the simulation time.

Nonconformal FETI-DP Methods for Large-Scale Electromagnetic Simulation

Two nonconformal dual-primal finite element tearing and interconnecting (FETI-DP) methods are formulated for the finite element simulation of large-scale three-dimensional (3-D) electromagnetic problems through domain decomposition method (DDM). Both methods implement the Robin-type transmission condition at the subdomain interfaces to preserve the fast convergence of the iterative solution of the global interface problem in the high-frequency region. The first nonconformal FETI-DP method extends the conformal FETI-DP algorithm, which is based on two Lagrange multipliers, to deal with nonconformal interface and corner meshes, and the second method employs cement elements on the interface and combines the global primal unknowns with the global dual unknowns. Similar to the conformal FETI-DP method, the two methods formulate a global coarse problem related to the degrees of freedom at the subdomain corner edges to propagate the residual error to the whole computational domain in the iterative solution of the global interface equation. Numerical results for the simulation of finite antenna arrays, photonic crystal cavities, and dielectric-rod-array metamaterial slabs are presented to demonstrate the application, accuracy, efficiency, and capability of the two proposed nonconformal FETI-DP methods.

Electromagnetic Field Analysis and Modeling of a Relative Position Detection Sensor for High Speed Maglev Trains

The long stator track for high speed maglev trains has a tooth-slot structure. The sensor obtains precise relative position information for the traction system by detecting the long stator tooth-slot structure based on nondestructive detection technology. The magnetic field modeling of the sensor is a typical three-dimensional (3-D) electromagnetic problem with complex boundary conditions, and is studied semi-analytically in this paper. A second-order vector potential (SOVP) is introduced to simplify the vector field problem to a scalar field one, the solution of which can be expressed in terms of series expansions according to Multipole Theory (MT) and the New Equivalent Source (NES) method. The coefficients of the expansions are determined by the least squares method based on the boundary conditions. Then, the solution is compared to the simulation result through Finite Element Analysis (FEA). The comparison results show that the semi-analytical solution agrees approximately with the numerical solution. Finally, based on electromagnetic modeling, a difference coil structure is designed to improve the sensitivity and accuracy of the sensor.

Application of two radial basis function-pseudospectral meshfree methods to three-dimensional electromagnetic problems

In this study the authors present an application of the radial basis function-pseudospectral (RBF-PS) meshfree method as well as a least squares variant thereof to a three-dimensional (3D) benchmark engineering problem defined by the Laplace equation. To their knowledge this is the first such study. The RBF-PS method is a version of the radial basis function (RBF) collocation method formulated in the vein of traditional pseudospectral methods. The least squares RBF-PS method introduced here as a modification of the RBF-PS method allows the authors to work with fewer RBFs while maintaining a high number of collocation points. In addition, the authors use a leave-one-out cross validation algorithm to choose an ‘optimal’ shape parameter for their basis functions. In order to evaluate the accuracy, effectiveness and applicability of their new approach, the authors apply it to a 3D benchmark electromagnetic problem. Their numerical results demonstrate that the proposed methods compare favourably to the finite difference and finite element methods.

A domain decomposition method for the parallelization of a three-dimensional Maxwell solver based on a constrained formulation

Abstract The numerical solution of very large three-dimensional electromagnetic field problems are challenging for various applications in the industry. In this paper, we propose a nonoverlapping domain decomposition approach for solving the three-dimensional Maxwell equations on MIMD computers, based on a mixed variational formulation. It is especially well adapted for the solution of the Vlasov–Maxwell equations, widely used to simulate complex devices like particle injectors or accelerators. This approach has the important property that it leads to reuse without modification most of an existing sequential code. The continuity at the interfaces is imposed by duality using Lagrange multipliers. Hence, the resulting parallel algorithm requires only to add an external preconditioned Uzawa solver. We present the results of some numerical experiments on a parallel distributed memory machine.