Numerical analysis of 3D magnetostatic fields

Electromagnetic devices can be analysed by the coupled BE-FE method, where the conducting and magnetic parts are discretized by finite elements. In contrast, the surrounding space is described with the help of the boundary element method (BEM). This discretization scheme is well suited especially for problems including moving parts (see [12]). The BEM discretization of the boundary integral operators usually leads to dense matrices without any structure. A naive strategy for the solution of the corresponding linear system would need at least OðN 2 Þ operations and memory, where N ist the number of unknowns. Methods such as fast multipole [6] and panel clustering [9] provide an approximation to the matrix in almost linear complexity. These methods are based on explicitly given kernel approximations by degenerate kernels, i.e. a finite sum of separable functions, which may be seen as a blockwise low-rank approximation of the system matrix. The blockwise approximant permits a fast matrix-vector multiplication, which can be exploited in iterative solvers, and can be stored efficiently. In contrast to the methods mentioned we will generate [2] the low-rank approximant from the matrix itself using only few entries and without using any explicit a priori known degenerate-kernel approximation. Special emphasis is put on the handling of symmetry conditions in connection with ACA. The feasibility of the proposed method is demonstrated by means of a numerical example. IntroductionThe application of the boundary element method for the solution of linear electromagnetic problems has many advantages. Only the boundaries of the considered domains need to be discretized, open boundary problems pose no additional difficulties, and problems including motion can be treated elegantly. The BE-FE coupling procedure is elaborated in [12]. However, application of the BEM leads to dense matrices. The storage requirements and computational costs are of OðN 2 Þ, where N is the number of unknowns. One remedy could be the use of an approximation algorithm with almost linear complexity. Section 2 presents the adaptive cross approximation (ACA) algorithm [13] which generates blockwise low rank approximants for the BEM matrices without using an explicit kernel expansion. The exploitation of symmetry is another possibility to reduce computational costs and has been presented in [1, 4, 5] using linear representation theory for finite groups. The aim is a decomposition of function spaces into orthogonal subspaces of symmetric functions, such that each subproblem is defined on a so called symmetry cell. The global solution can then be reconstructed from these components. Sections 3.1-3.2 give an overview of using symmetry in FE and BE methods. Next, in Sect. 3.3 the major contribution of this paper, a symmetry-exploiting ACA algorithm is discussed, which not only reduces the problem size due to symmetry but also yields additional benefit by compression of BEM matrices and possesses an almost linear complexity w.r.t. the number of unknowns. Numerica...

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“…TEAM problem 10 is treated as a magnetostatic problem (for details see [15]). In all computations we set the ACA accuracy e ¼ 10 À4 .…”

Section: Numerical Resultsmentioning

confidence: 99%

Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Kurz

Rain

Rjasanow

2003

Computational Mechanics

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“…The so-called penalty term 27 1=m 7 · A in equation (1) indirectly enforces the Coulomb gauge and improves in a finite element approach the numerical stability of the solution of the coupled system of equations (Preis et al, 1991). Furthermore, the following boundary conditions are set:…”

Section: Hybrid Numerical Formulationmentioning

confidence: 99%

Numerical modeling of a MEMS actuator considering several magnetic force calculation methods

Preisner

Bolzmacher²,

Gerber

et al. 2011

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

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Purpose -The purpose of this paper is to investigate the accuracy of different force calculation methods and their impact on mechanical deformations. For this purpose, a micrometer scaled actuator is considered, which consists of a micro-coil and of a permanent magnet (PM) embedded in a deformable elastomeric layer. Design/methodology/approach -For the magnetic field evaluation a hybrid numerical approach (finite element method/boundary element method (FEM/BEM) coupling and a FEM/BEM/Biot-Savart approach) is used, whereas FEM is implemented for the mechanical deformation analysis. Furthermore, for the magneto-mechanical coupling several force calculation methods, namely the Maxwell stress tensor, the virtual work approach and the equivalent magnetic sources methods, are considered and compared to each other and to laboratory measurements. Findings -The numerically evaluated magnetic forces and the measured ones are in good accordance with each other with respect to the normal force acting on the PM. Nevertheless, depending on the used method the tangential force components differ from each other, which leads to slightly different mechanical deformations. Research limitations/implications -Since the force calculations are compared to measurement data, it is possible to give a suggestion about their applicability. The mechanical behavior of the actuator due to the acting forces is solely calculated and therefore only an assumption concerning the deformation can be given. Originality/value -A new kind of micrometer scaled actuator is numerically investigated by using two different hybrid approaches for the magnetic field evaluation. Based on those, the results of several force calculation methods are compared to measurement data. Furthermore, a subsequent structural analysis is performed, which shows slightly different mechanical deformations depending on the used force calculation method.

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“…For the purpose of ensuring the uniqueness of the magnetic vector potential and making the solution of the coupled system of equations numerically stable (Preis et al, 1991) the Coulomb gauge:…”

Section: Formulationsmentioning

confidence: 99%

“…Furthermore, the accuracy of the resulting field strength near protruding corners increases by using these elements (Mur, 1994). Anyhow, some studies have shown that edge elements are numerically less stable than nodal elements (Preis et al, 1991) and they are known to be less efficient in storage requirement and computation time by having the same accuracy than nodal elements (Mur, 1994;Jiang, B-N., et al, 1996). In addition some of the expressed problems, especially the discontinuities at almost flat material boundaries using nodal elements, can be softened by implementation of double layer nodes at these boundaries.…”

Section: Introductionmentioning

confidence: 99%

Numerical computation of magnetic fields applied to magnetic force microscopy

Preisner

Greiff

Bala

et al. 2009

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

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PurposeThe purpose of this paper is to introduce a method which allows the calculation of the interactions of tip and sample of a magnetic force microscope as a first step to increase the accuracy of this technique.Design/methodology/approachThe emerging magnetic interactions between the cantilever tip and an arbitrary magnetized sample can be evaluated by the use of several numerical methods. For modelling this magnetically and mechanically coupled multiscale problem the finite element method is implemented.FindingsThe evaluated magnetic fields interact in such a manner that a constructive overlap at the tip apex occurs. This leads to attractive forces acting on the cantilever.Research limitations/implicationsIn order to include the magneto‐mechanical coupling, the implementation of a detailed force calculation is necessary. Furthermore, a hysteresis model is not yet considered.Practical implicationsMagnetic force microscopy is a very sensitive technique. For instance, ideally the end of the tip consists of only one atom, but this is not realizable. Measurement errors cannot be avoided. This approach is the first step in developing an opportunity to soften them.Originality/valueOne opportunity to verify real‐time magnetic force microscope measurements is the comparison with theoretical considerations and calculations of the occurring magnetic distribution by using this technique. For this reason this paper deals with a new micromagnetic model to simulate the interactions between tip and sample of a scanning process of a magnetic force microscope.

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Numerical analysis of 3D magnetostatic fields

Cited by 92 publications

References 9 publications

Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Application of the adaptive cross approximation technique for the coupled BE-FE solution of symmetric electromagnetic problems

Numerical modeling of a MEMS actuator considering several magnetic force calculation methods

Numerical computation of magnetic fields applied to magnetic force microscopy

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