The accurate interpolation of magnetization tables is of paramount importance in the design of high‐precision magnets used for particle accelerators or for magnetic resonance…
Abstract
The accurate interpolation of magnetization tables is of paramount importance in the design of high‐precision magnets used for particle accelerators or for magnetic resonance imaging of the human body. Cubic spline interpolation is normally used in combination with the fast converging Newton‐Raphson scheme in the two‐dimensional finite element modelling of such magnets. We compare cubic spline interpolation with experiment, using the magnetization tables as a source of carefully measured experimental data. We show that, in all examined cases, cubic spline interpolation introduces errors large enough to invalidate a design. We also propose a simple solution to the problem, thus combining the best of all worlds: the speed and convergence properties of Newton‐Raphson, the accuracy of a good interpolation scheme, and the convenient mathematical properties of cubic splines. We examine both two‐dimensional and three‐dimensional cases.
A simple infinite element for cylindrical problems is considered. The element matches with linear triangular ring elements and bilinear quadrilaterals. It is assumed that the…
Abstract
A simple infinite element for cylindrical problems is considered. The element matches with linear triangular ring elements and bilinear quadrilaterals. It is assumed that the magnetic vector potential decays with p−n towards infinity. For boundaries having a general shape, the element matrix is calculated numerically using m‐point Gaussian quadrature. Analytical expressions are given for elements with edges parallel to one of the coordinate axes.
A magnetization table describing the magnetic properties of the material of interest is the primary input for any computer program expected to calculate magnetic fields or other…
Abstract
A magnetization table describing the magnetic properties of the material of interest is the primary input for any computer program expected to calculate magnetic fields or other magnetic parameters in a nonlinear case. Magnetization tables, however, consist of discrete points, and the program assumes some interpolation rule to calculate values between them. There exists a variety of interpolation schemes, and some of them can produce very large errors and even unphysical results when the intervals are not narrow enough. Unfortunately, it was found that intervals used in practice are seldom narrow enough. The accurate interpolation of magnetization tables thus becomes a central issue in the numerical solution of nonlinear magnetic problems. We discuss several interpolation schemes used in practice. We propose a new one that is guaranteed to give physical results, and we address the question as to how wide the table invervals can be if a desired accuracy is specified. The discussion is illustrated with many examples.
Infinite elements provide one of the most attractive alternatives for dealing with differential equations in unbounded domains. The region where loads, sources, inhomogeneities…
Abstract
Infinite elements provide one of the most attractive alternatives for dealing with differential equations in unbounded domains. The region where loads, sources, inhomogeneities and anisotropics exist is modelled by finite elements and the far, uniform region is represented by infinite elements. We propose a new infinite element which can represent any type of decay towards infinity. The element is so simple that explicit expressions can be obtained for the element matrix in many cases, yet large improvements in the accuracy of the solution are obtained as compared with the truncated mesh. Explicit expressions are in fact given for the Laplace equation and 1/rn decay. The element is conforming with linear triangles and bilinear quadrilaterals in two dimensions. The element can be used with any standard finite‐element program without having to modify the shape function library or the numerical quadrature library of the program. The structure or bandwidth of the stiffness matrix of the finite portion of the mesh is not modified when the infinite elements are used. An example problem is solved and the solution found to be better than several other methods in common usage. The proposed method is thus highly recommended.
Sergio PISSANETZKY and Youqing XIANG
We present a set of new simple closed‐form analytical formulas for the calculation of the magnetic field produced at any point of space by any solid polyhedral conductor with a…
Abstract
We present a set of new simple closed‐form analytical formulas for the calculation of the magnetic field produced at any point of space by any solid polyhedral conductor with a uniform current density j. The formulas have been obtained by analytical integration of Ampère's law under the only assumptions that the conductor is bounded by flat surfaces and that j = constant in the conductor. This includes bars, bricks, tetrahedrons, wedges, prisms, trapezoids, pyramids, and polyhedrons in general. The formulas contain no singularities, and can be used for the numerical calculation of the field at any point, including points inside the conductor, or on its surface, edges or corners. The formulas can easily be extended for conductors of infinite length. Extensive numerical tests of the formulas have been performed.
P. LEE, J.E. PASCIAK and S. PISSANETZKY
In this paper, a parallel preconditioning technique based on the additive variant of overlapping domain decomposition is described and implemented to solve magnetostatic field…
Abstract
In this paper, a parallel preconditioning technique based on the additive variant of overlapping domain decomposition is described and implemented to solve magnetostatic field problems. This technique involves covering the domain with a number of overlapping subdomains. The pre‐conditioner results from adding together approximate inversions on the subdomains, Theoretical estimates for the rate of convergence for the resulting algorithm are available and are based on the properties of underlying differential equations. Numerical experiments are given to demonstrate the effectiveness of this algorithm.
The paper presents infinite elements for axisymmetric electrical field problems with open boundaries. The formulation of the elements is so simple that closed‐form expressions for…
Abstract
The paper presents infinite elements for axisymmetric electrical field problems with open boundaries. The formulation of the elements is so simple that closed‐form expressions for the infinite element matrix are obtained. In order to test the infinite elements, a simple problem, for which an analytical solution exists, is analysed.
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Takayoshi NAKATA and Koji FUJIWARA
Benchmark problem 13 of the TEAM Workshop consists of steel plates around a coil (a nonlinear magnetostatic problem). Seventeen computer codes developed by twelve groups are…
Abstract
Benchmark problem 13 of the TEAM Workshop consists of steel plates around a coil (a nonlinear magnetostatic problem). Seventeen computer codes developed by twelve groups are applied, and twenty‐five solutions are compared with each other and with experimental results. In addition to the numerical calculations, two theoretical presentations are given in order to explain discrepancies between the calculations and the experiment.
Infinite elements are becoming increasingly popular as an efficient and economical means of extending the finite element method to deal with unbounded domains. In this paper an…
Abstract
Infinite elements are becoming increasingly popular as an efficient and economical means of extending the finite element method to deal with unbounded domains. In this paper an infinite element is considered which has three nodes and is compatible with conventional quadratic triangles and quadratic quadrilaterals. The closed‐form expressions for the element matrix are obtained.
Peter Bettess and Jacqueline A. Bettess
This paper is concerned with static problems, i.e. those which do not change with time. Dynamic problems will be considered in a sequel. The historical development of infinite…
Abstract
This paper is concerned with static problems, i.e. those which do not change with time. Dynamic problems will be considered in a sequel. The historical development of infinite elements is described. The two main developments, decay function infinite elements and mapped infinite elements, are described in detail. Results obtained using various infinite elements are given, followed by a discussion of possibilities and likely developments.