A study of the fast wavelet transform method in computational electromagnetics

There exists a great need to develop numerical techniques for efficient solution of multiscale electromagnetic problems, regardless of the computational electromagnetics (CEM) method currently being used to tackle them, be it FEM, FDTD or MoM. Dealing with multiscale objects, which have features that are both small and large compared to the wavelength, is highly challenging and often forces us to compromise the accuracy (relaxing the numerical discretization process when attempting to capture the small-scale features) in order to cope with the limited available resources in terms of CPU memory and time. In this paper, we introduce a new scheme that combines the Recursive Update Frequency Domain (RUFD) method with the Dipole Moment (DM) method, to solve multiscale problems in a numerically efficient manner.

Summary form only given. Numerical analysis of electromagnetic scattering has been widely studied in the past. The method of moments (MoM) is suitable to simulate scattering from an object in an open region. In order to solve problems with complicated geometries and materials, the finite element method (FEM) is preferred. However, when solving unbounded problems with FEM, the use of absorbing boundary conditions or perfectly matched layers will result in truncation errors. To combine the advantages of both MoM and FEM, the finite element-boundary integral (FE-BI) method was developed and widely used (J. M. Jin, The Finite Element Method in Electromagnetics, Second Edition. New York: Wiley, 2002). This algorithm is not only able to deal with highly complicated geometry and material, but also able to eliminate the truncation error. However, since the MoM has O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2)</sup> computational and storage complexities, the computational cost of the BI-related algorithm is still prohibitively high when solving large-scale scattering problems. To alleviate this problem, the multilevel fast multipole algorithm (MLFMA) has been incorporated into the FE-BI method (J. Liu and J. M. Jin, IEEE Trans. Antennas Propag., vol. 51, pp.1157-1167, Jun. 2003). Another important technique to accelerate the computation is to resort to parallel computers. Recently, the graphics processing unit (GPU), as one of many-core computing systems, has received intensive attention from the scientific computing community due to its low price, a massive number of processors, and the resulting high computational efficiency. In this work, we discuss the GPU implementation of the FE-BI algorithm for solving electromagnetic scattering problems. First, the assembly of the BI-related matrix is parallelized on GPU by NVIDIA's compute unified device architecture (CUDA). A massive number of threads are assigned to carry out the assembly, and each thread computes one element in the BI-related matrix. Second, both the direct and iterative solvers are implemented on GPU. For the direct solver, the Intel's math kernel library (MKL) is used for LU decomposition, and the CUSPARSE (a set of basic linear algebra subroutines used for handling sparse matrices) is employed to speedup forward and backward substitutions. For the iterative solver, the high performance CUDA Basic Linear Algebra Subprograms (BLAS) is utilized for the matrix-vector multiplication and the vector-vector multiplication. Extensive numerical experiments are conducted to validate the implementation and demonstrate the achievable computational speedup.

The Origin of Spurious Solutions in Computational Electromagnetics

This study investigates and evaluates applications of the adjoint problem and its solution in frequency-domain computational electromagnetics (CEM). The study establishes and validates adjoint-based applications including higher order parameter sampling, a posteriori error estimate evaluation, and p- and h-refinements. These applications can improve the efficiency, automation, and robustness of CEM methods. We employ a 1-D finite-element-method scattering solver, simplifying the implementation, replicability, clarity, and intuitiveness of analysis results and conclusions, which then extend naturally to higher dimensional solvers and more complicated CEM problems. While demonstrated with a higher order solver, the derived techniques apply to low-order methodology as well. This is the first demonstration of the applicability of adjoint-based a posteriori error estimation techniques to adaptive discretization refinement in frequency-domain CEM with arbitrary-order basis functions. This work introduces the application of dual-weighted residual error estimation and selective adaptivity based on error cancellation. The proposed targeted, adaptive mesh/model p- and/or h-refinement heuristics informed by adjoint element-wise error contribution estimates show near-monotonic reduction of the quantity-of-interest error with increased numbers of refined elements. In general, adjoint techniques are underutilized in CEM, and another goal of this work is to promote their future use for refinement, optimization, and uncertainty quantification.

In many applications, electromagnetic scattering from inhomogeneous objects embedded in multiple layers needs to be simulated numerically. The straightforward solution by the method of moments (MoM) for the volume integral equation method is computationally expensive. Due to the shift-invariance and correlation properties of the layered-medium Green’s functions, the stabilized-biconjugate gradient fast Fourier transform (BCGS-FFT) has been developed to greatly reduce the computational complexity of the MoM, but so far this method is limited to objects located in a homogeneous background or in the same layer of a layered medium background. For those problems with objects located in different layers, FFT cannot be applied directly in the direction normal to the layer interfaces, thus the MoM solution requires huge computer memory and CPU time. To overcome these difficulties, the BCGS-FFT method combined with the domain decomposition method (DDM) is proposed in this article. With the BCGS-FFT-DDM, the objects or different parts of an object are first treated separately in several subdomains, each of which satisfies the 3-D shift-invariance and correlation properties; the couplings among the different objects/parts are then taken into account, where the coupling matrices can be built to satisfy the 2-D shift-invariance property if the objects/subdomains have the same mesh size on the $xy$ plane. Hence, 3-D FFT and 2-D FFT can respectively be applied to accelerate the self- and mutual-coupling matrix-vector multiplications. By doing so, the impedance matrix is explicitly formed as one including both the self- and mutual-coupling parts, and the solver converges well for problems with considerable conductivity contrasts. The computational complexity in memory and CPU time for self-coupling matrix-vector multiplication are $O(N_{{z}}^{q} N_{x} N_{y})$ and $O(N_{{z}} N_{x} N_{y} \log (N_{{z}} N_{x} N_{y}))$ respectively, and for mutual-coupling matrix-vector multiplication are $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y})$ and $O(N_{{z}}^{p} N_{{z}}^{q} N_{x} N_{y} \log (N_{x} N_{y}))$ , respectively, for the proposed method, where $N_{x}$ and $N_{y}$ are the cell numbers of all the subdomains in the $x$ - and $y$ -directions, and $N_{{z}}^{p}$ and $N_{{z}}^{q}$ the cell numbers of different subdomains in the ${z}$ -direction. Several results of different subsurface sensing scenarios are presented to show the capabilities of this method.

A class of challenging problems encountered in geomechanics pertains to the prediction of the transient response of elastic systems of infinite or semi-infinite extent subjected to time dependent loading and boundary conditions. The nature and complexity of such problems restrict the range of available closed form solutions to the ones of linear problems of highly simplified geometries and boundary conditions. Among the numerical techniques, the most popular and well established one is the Finite Element Method (FEM) which can routinely handle complex geometries, medium inhomogeneities, and material non-linearities, and is well suited for both frequency and time domain analysis. However, the FEM by nature encompasses the major disadvantage of modeling infinite, and semi-infinite domains by finite size meshes violating, thus, the radiation damping condition. The problem is confronted, to some extend, by the development of hybrid techniques, and transmitting boundaries, for frequency domain analysis. Yet, time domain FEM solutions involving unbounded 3-D solid media, if available at all, may increase the size of the problem to computationally prohibitive levels. An alternative to the FEM is the Boundary Element Method (BEM), which is based on integral equation formulations, developed in the beginning of the century. The first 3-D time domain BEM formulation, associated with soil-structure interaction problems, has been reported by Karabalis and Beskos [1]. In this formulation the Stokes fundamental solutions are developed on the basis of a step impulse forcing function.

Finite element method (FEM) is widely used in the modeling of electromagnetic systems and as a reliable method to verify analytical modeling of electromagnetic devices in design step. As the method is based on discretization of domains, the electromagnetic continuity conditions at the boundaries of elements should be confirmed. Electromagnetic continuity conditions in FEM modeling were investigated in this article. It was shown that the continuity of tangential component of magnetic field intensity is not guaranteed at the boundaries of elements. The inability of FEM to establish the continuity condition leads to some virtual surface current densities at the boundaries of adjacent elements which does not exist in reality. The existence of virtual surface currents is proved in an example. Then, with the help of a complicated example, it is shown that large number of elements and, consequently, the large number of boundaries between elements may cause great errors in FEM postprocess results, especially in calculating magnetic stored energy and electromagnetic forces acting on moving parts. It has been shown that it is very important to take into account the impact of these virtual current densities in energy conversion analysis by FEM.

A finite element–boundary element method for advection–diffusion problems with variable advective fields and infinite domains

A finite element method (FEM)/boundary element method (BEM)-hybrid approach is presented which allows electromagnetic field calculations in planar layered media including three-dimensional inhomogeneous regions. The fields within the inhomogeneities are modeled by the FEM in terms of the electric field strength Ē with edge elements on tetrahedral meshes. Based on Huygens’ principle the BEM is formulated with equivalent electric and magnetic surface currents on a surface completely enclosing the inhomogeneous regions. Additionally, ideal conducting objects can be included into the BEM-model. The electric field integral equation (EFIE) for the BEM is formulated as a mixed potential integral equation (MPIE). Applications of the method to field calculations in a microstrip resonator with finite metallization thickness and to the coupling of a cylindrical dielectric resonator to a microstrip line are presented.

An electromagnetism module is under development in the commercial software LS-DYNA in order to perform coupled mechanical/thermal/electromagnetic simulations. The main part of the module is a so-called eddy current solver. This module allows us to introduce source electrical currents into solid conductors, and to compute the associated magnetic fields, electric fields, induced currents, Lorentz forces, Joule heating and so forth. The Maxwell equations are solved using a Finite Element Method (FEM) for the solid conductors coupled with a Boundary Element Method (BEM) for the surrounding air (or insulators). Both the FEM and the BEM are based on discrete differential forms (Nedelec-like elements). The use of the BEM for the air allows to handle complex 3D geometries with multiply connected conductors, very small air gaps, and motion of the conductors, due, in particular, to the electromagnetic forces. The variable of the BEM is a so-called surface current allowing the connection of the model to external current sources by simple dirichlet constraints, as well as direct computations of the self and mutual inductances of the system. In order to handle the large and dense BEM matrices, a domain decomposition is performed and low rank approximations are done on the off-diagonal blocks of the resulting block matrices. The diagonal blocks are used as an efficient preconditioner when solving the BEM part of the system. The singularities arising in the computation of the self and neighbor matrix elements of the Galerkin BEM are taken into account using a method based on Duffy transforms. The BEM method will be presented as well as benchmarks and real life application examples.

SummaryThis paper presents a numerical approach for the design sensitivity analysis of modal frequencies of elastic structures submerged in an unbounded heavy fluid domain. Because the feedback of sound pressure onto the fluid‐loaded structures has to be taken into account in this case, a fully coupled scheme which combines the structural finite element method (FEM) and the acoustic boundary element method (BEM) is employed in the numerical simulation. The resulting nonlinear eigenvalue problem created by the coupled finite element and boundary element (FE‐BE) method is converted into a generalized eigenvalue problem (GEP) of reduced dimension by using a contour integral method. The design sensitivity formulation for non‐repeated eigenvalues is derived based on an adjoint method which uses both the left and right eigenvectors, and a small GEP is formed to calculate the gradients of repeated eigenvalues. The Burton‐Miller formulation is employed to shift the fictitious eigenfrequencies and their derivatives, and the parameter setting which is beneficial to the filtering of fictitious eigenfrequencies is investigated. Numerical examples are given to verify the accuracy and effectiveness of the developed method.

Capacitance parameter extraction of converter system is crucial to analyze the voltage distribution of valve tower under different overvoltages (such as lightening, partial discharge, etc) in HVDC converter station. According to the actual dimension of valves, the Method of Moments(MOM) based on the point matching method is applied to extract the capacitances, including the self-capacitances and the mutual capacitances between the valve modules in the valve hall. Firstly the surfaces of conductors are meshed into small area elements, and the surface charge density of each element is considered as a constant. Then the potential coefficient matrix corresponding to elements can be achieved by MOM. The capacitance matrix can be acquired through matrix transformation. The proposed method is flexible and fast compared to the Finite Element Method (FEM) and Boundary Element Method (BEM), since it is unnecessary to solve the conductor surface charge and the electric field. The calculation results agree very well with the capacitance testing experiments.

For antennas that are mounted on the same object, coupling through surface waves typically is very significant. The accurate characterization of this coupling is instrumental to the design of antennas with required isolation between its multiple functions. When the antennas consist of only conducting elements, they can be efficiently modeled using the method of moments (MoM). However, most MoM programs developed so far are not suitable to the accurate calculation of mutual coupling. For example, MoM programs based on wire-grid models do not accurately model surface-wave coupling. Furthermore, the MoM yields a full matrix equation, the solution of which is very memory intensive and extremely time consuming for a large scale modeling problem when the number of unknown is large. This is because both the memory requirements and the CPU time per matrix vector multiply are proportional to N/sup 2/. To overcome these problems, we considered the application of the multilevel fast multipole algorithm (MLFMA) to reduce the memory requirements and computational complexity per matrix-vector multiply to O(NlogN) (Chew et al. 1997, and Chao 1998). Our preliminary investigation shows that the MLFMA can predict accurately the S-parameters that characterize the mutual coupling/isolation between antennas. It is a useful tool for the design and performance evaluation of multi-antenna systems. We consider two antenna configurations to demonstrate the accuracy of our algorithms. The first configuration consists of two inverted-L antennas on a finite ground plane and the second composes two loop antennas, also on a finite ground plane.

Analysis and application of an overlapped FEM-BEM for wave propagation in unbounded and heterogeneous media

A Monte Carlo solution method for linear elasticity