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Electromagnetic Scattering Analysis of Arbitrary Structures by the Natural Element Method Coupled With Absorbing Boundary Condition
Differential quadrature methods are devised to numerically solve ordinary and partial differential equations by approximating the derivatives of the unknown function at points of a cloud defined on the domain of interest as weighted sums of the values of such function at other points of the cloud. Local versions of this class of meshless methods restrict the points used in such expansion, by establishing suitable supporting regions. In this work, we present the local differential quadrature method (LDQM) and we use it to solve a boundary problem in electromagnetism. In order to do this, we evaluate the numerical solutions of the Poisson equation on a two-dimensional domain. Furthermore, we propose an alternative definition of supporting region that has yielded better solutions than the conventional one. Root mean square errors for the approximations with both (alternative and conventional) definitions of local supports are obtained and their dependencies with the density of nodes are studied. We find out that the best accuracy obtained with the alternative definition of the local support is due to the smaller condition numbers of the linear systems yielded.
The nonstandard finite-difference time-domain (NS-FDTD) method exhibits high accuracy at fixed frequencies, and, thus, proves to be a powerful means of the radar cross section analysis of scattering objects with complex shapes and materials. To this aim, significant enhancement can be pursued in its combination with the FDTD total-field/scattered-field (TF/SF) concept. However, the principal implementation characteristics of the latter in the NS-FDTD technique have yet to be elaborately studied, especially for electrically large domains or structures. Hence, in this paper, a new advanced TF/SF scheme for the 3-D NS-FDTD algorithm is developed and fully validated. Numerical results reveal that the proposed method, considering the features of the NS-FDTD operators, is far more efficient and versatile than the original FDTD one.
This paper presents and discusses the application of the element-free Galerkin method to the study of 2-D electromagnetic-wave scattering problems. The proposed formulation is validated by comparison with a standard finite-element approach, focusing attention on three kinds of model problems, which give rise to localized steep gradients.
This communication deals with a comparison between two methods of discretization: the well known finite element method and the natural element method that is a meshless method. An error estimator, based on the nonverification of the constitutive law, is used. This estimation has been applied to two examples: a device with permanent magnets and a variable reluctance machine.
Summary In this paper, a survey of the most relevant advances in natural neighbour Galerkin methods is presented. In these methods
(also known as natural element methods, NEM), the Sibson and the Laplace (non-Sibsonian) interpolation schemes are used as
trial and test functions in a Galerkin procedure. Natural neighbour-based methods have certain unique features among the wide
family of so-called meshless methods: a well-defined and robust approximation with no user-defined parameters on non-uniform
grids, and the ability to exactly impose essential (Dirichlet) boundary conditions are particularly noteworthy.
A comprehensive review of the method is conducted, including a description of the Sibson and the Laplace interpolants in two-
and three-dimensions. Application of the NEM to linear and non-linear problems in solid as well as fluid mechanics is studied.
Other issues that are pertinent to the vast majority of meshless methods, such as numerical quadrature, imposing essential
boundary conditions, and the handling of secondary variables are also addressed. The paper is concluded with some benchmark
computations that demonstrate the accuracy and the key advantages of this numerical method.
Since the publication of the first edition [see the review in Zbl 0823.65124] of this book about eight years ago, much progress has been made in the development of the finite element method for the analysis of electromagnetics problems, especially in five areas. The first is the development of higher-order vector finite elements, which make it possible to obtain highly accurate and efficient solutions of vector wave equations. The second is the development of perfectly matched layers as an absorbing boundary condition. Although the perfectly matched layers were intended primarily for the time-domain finite-difference method, they have also found applications in the finite element simulations. The third is perhaps the development of hybrid techniques that combine the finite element and asymptotic methods for the analysis of large, complex problems that were unsolvable in the past. The fourth is further development of the finite element-boundary integral methods that incorporate fast integral solvers, such as the fast multipole method, to reduce the computational complexity associated with the boundary integral part. The last, but not least, is the development of the finite element method in the time domain for transient analysis. As a result of all these efforts, the finite element method has gained more popularity in the computational electromagnetics community and has become one of the preeminent simulation techniques for electromagnetics problems. In this second edition, we have updated the subject matter and introduced new advances in finite element technology. Three new chapters have been added: Absorbing boundary conditions; Finite element analysis in the time domain: The method of moments and fast solvers.
Meshless approximations based on moving least-squares, kernels, and partitions of unity are examined. It is shown that the three methods are in most cases identical except for the important fact that partitions of unity enable p-adaptivity to be achieved. Methods for constructing discontinuous approximations and approximations with discontinuous derivatives are also described. Next, several issues in implementation are reviewed: discretization (collocation and Galerkin), quadrature in Galerkin and fast ways of constructing consistent moving least-square approximations. The paper concludes with some sample calculations.
From the Publisher:
"An IEEE reprinting of this classic 1968 edition, FIELD COMPUTATION BY MOMENT METHODS is the first book to explore the computation of electromagnetic fields by the most popular method for the numerical solution to electromagnetic field problems. It presents a unified approach to moment methods by employing the concepts of linear spaces and functional analysis. Written especially for those who have a minimal amount of experience in electromagnetic theory, this book illustrates theoretical and mathematical concepts to prepare all readers with the skills they need to apply the method of moments to new, engineering-related problems.Written especially for those who have a minimal amount of experience in electromagnetic theory, theoretical and mathematical concepts are illustrated by examples that prepare all readers with the skills they need to apply the method of moments to new, engineering-related problems."
The Natural Element Method (NEM) is a recently proposed novel numerical tool for the solution of partial differential equations. In this work, the development and application of the natural element method to two-dimensional elliptic boundary value problems in solid mechanics is presented. We assume the discrete model of a body $\Omega\subset\IR\sp2$ consists of a set of distinct nodes N, and a polygonal description of the boundary $\partial\Omega .$ In the natural element method, the trial and test functions are constructed using natural neighbor interpolants. These interpolants are based on the Voronoi tessellation of the set of nodes N. The NEM interpolant is strictly linear between adjacent nodes on the boundary of the convex hull, which facilitates imposition of essential boundary conditions. A methodology to model material discontinuities and non-complex bodies such as cracks is described. A standard displacement-based Galerkin procedure is used to obtain the discrete system of linear equations. Application of NEM to various problems in two-dimensional elastostatics is presented. The construction and computational implementation of a $C\sp1$ natural neighbor interpolant for fourth-order elliptic PDEs is presented. By embedding natural neighbor interpolants in the surface representation of a Bernstein-Bezier cubic simplex, a $C\sp1$ interpolant is realized (Farin, 1990b). We present the $C\sp1$ formulation and propose a computational methodology for its numerical implementation for the solution of PDEs. Numerical results for the biharmonic equation with Dirichlet boundary conditions are presented. A mixed formulation for the natural element method in linear elastostatics is presented. A displacement-pressure mixed formulation is adopted with displacements interpolated by $C\sp0$ natural neighbor interpolants; $C\sp0$ and $C\sp{-1}$ interpolation schemes are considered for the interpolation of the pressure. The mixed $C\sp0$-$C\sp{-1}$ NEM formulation alleviates locking in the near incompressible limit $(\nu\to 0.5)$ for the elastostatic boundary value problem; moreover, convergence rates in displacement and energy are optimal for all $\nu\in\lbrack 0,\ 0.5).$ Results for benchmark problems in compressible and incompressible elasticity are presented. Source: Dissertation Abstracts International, Volume: 59-05, Section: B, page: 2278. Adviser: Brian Moran. Thesis (Ph.D.)--Northwestern University, 1998.
The natural-element method, which belongs to the family of meshless methods, is applied in the context of two-dimensional magnetostatics with moving parts. The method is reviewed and its interest for handling discontinuities in electromagnet devices with moving parts is illustrated through a numerical example
Meshless methods: An overview and recent developments
- T Belytschko
- Y Krongauz
- D Organ
- M Fleming
- P Krysl
T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl,
" Meshless methods: An overview and recent developments, " Comput. Methods Appl. Mech. Eng., vol. 139, nos. 1–4, pp. 3–47,
Dec. 1996.