IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002
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Modeling Magnetization Using Whitney Facet Elements Alexander M. Vishnevsky, Alexander G. Kalimov, and Andrew A. Lapovok
Abstract—An integral method for computation of static magnetic fields exploits Whitney facet elements for approximation of magnetization of arbitrary solids or thin shells. A co-tree gauge of unknown variables allows constructing an exactly divergence-free magnetization inside homogeneous media. The approach leads to positive definite and symmetric matrices of algebraic equations. Comparisons of the method with gradient representation of magnetization, and also with usual piecewise constant approximation show some advantages, which are attributed to proper approximation of the operator null-space. Index Terms—Approximation methods, integral equations, magnetostatics.
I. INTRODUCTION
T
HE USE of two-Whitney (facet) elements for approximation of equivalent electric currents in problems of steady conduction has been proposed recently [1]. This approach seems to be very powerful as it allows incorporating different geometrical structures (thin plates and shells, rods and wires, solid blocks), and also lumped elements of electrical network within a unified integral formulation. The analogy between the equivalent current and magnetization is straightforward; therefore there is a temptation to apply the method to magnetostatic problems, first of all, to problems with open boundaries. The generation of a finite-element mesh in the infinite open region is very objectionable. It requires a complicated preprocessing and seems a waste of computational resources. To avoid this, plenty of different methods [2] have been developed, but neither of them seems to be optimal. Hybrid finite- and boundary-element methods are, obviously, more universal, but as a rule lead to algebraic equations with nonsymmetrical and indefinite system matrices. Considering the pattern of nonzero entities, the proposed integral approach gives a matrix similar to a hybrid one. But, since the quadratic form of this matrix exhibits the magnetostatic energy, the matrix is symmetrical and positive definite. Earlier, facet functions (under the name of RWG-basis) were intensively used for representation of surface electric currents in patch antennas [3]. Since a generalized gauging procedure for plates and shells has been already considered in [1], in this paper, we focus attention on solid structures. Manuscript received July 2, 2001; revised October 25, 2001. This work was supported in part by the Russian Ministry of Education under Grant TOO-2.42230. A. M. Vishnevsky and A. J. Lapovok are with the Krylov Shipbuilding Research Institute, 196158 St. Petersburg, Russia (e-mail:
[email protected]). A. G. Kalimov is with the St. Petersburg State Technical University, 195251 St. Petersburg, Russia (e-mail:
[email protected]). Publisher Item Identifier S 0018-9464(02)01224-4.
The governing integral equation can be traced back to the nineteenth century, to the works of Poisson and Thomson, and since then has been theoretically studied well enough. Nevertheless, there are a lot of numerical methods where magnetization is replaced by nongauged piecewise polynomial approximation or even by point dipoles. Such approximations are partially or completely out of the magnetization space and lead to unreliable results for complicated geometry and coarse meshes. Alternative approaches exploit Maxwell equations and inside magnetic region as calibration conditions for magnetization spaces. One cannot construct a discrete - and -conformal; one of space, which is simultaneously the properties should be sacrificed. So, in the first group of numerical methods, the magnetization is expressed via an exact gradient. This can be implemented using nodal potentials (see, e.g., [4]) or with the help of edge elements. The paper presents another alternative. By using facet elements together with a co-tree gauging, exactly divergence-free magnetization is constructed inside each homogeneous magnetic region. The magnetization field is created by (piecewise constant) surface charges disposed only on the interfaces between regions. However, the gradient property is satisfied in a weak form. II. FORMULATION OF THE METHOD To illustrate the basics of the method, we consider a single region with constant magnetic permeability. Generalization onto piecewise homogeneous media is straightforward; nonlinear case may be considered as iterations of linear problems. having permeability and arbitrary A magnetic region shaped boundary (multiply connected regions and cavities are allowed) is placed into infinite space of permeability and sub. At first we assume that jected to a known external field can be represented as a gradient, i.e., . The Poisson–Thomson equation in this case reads (1) where is the unknown magnetization vector; is the magnetic susceptibility; and is the distance between an observation point and the integration point , both are disposed inside . , Since there are no internal magnetic sources when magnetization is found in the subspace
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where is a unit normal vector outward to ; the second condition establishes square integrable surface magnetic sources and guarantees the finiteness of magnetostatic energy. Given a subdivision of onto a tetrahedral mesh, so as the boundary is approximated by facets of tetrahedra. Then, a discrete approximation of can be constructed using Whitney of each tetrahedron facet functions
Fig. 1.
where vector is directed to a point within the tetrahedron is the tetrafrom the vertex opposite to the th facet, and is hedron volume. The normal component of each function constant over the th facet (the total flux across this facet is equal to unity) and vanishes over others. Usual approximation by facet functions ensures the absence of magnetic sources on the interfaces between tetrahedra. But each function has nonzero (constant) divergence, and we need an additional gauging procedure to exclude internal sources. This procedure is very similar to one described in [5]. We form a graph whose (oriented) branches correspond to individual facets of tetrahedra, whereas the nodes are associated with tetrahedron centers and interconnections (Fig. 1). All nodes except for lying on the boundary are “nondivergent” and depicted by crosses. The “divergent” boundary nodes are shown by circles. All circles are actually considered as the same graph node while constructing a spanning tree (a subgraph spanning all nodes and having no loops). Each degree of freedom in the representation of magnetizaacross the facet associated with the th tion is the flux branch of the co-tree. By definition of the tree, there is a unique set of the tree branches, which complete, one after another, the co-tree branch to a closed loop. The whole set of facets corresponding to this loop (including the co-tree branch itself) is and referred as a fundamental cycle. At the first denoted by time Kirchhoff introduced this notion for independent contour currents in electrical circuit. The fundamental cycles provide all the information for constructing the discrete space
(2)
(3)
where is the total number of fundamental cycles in the graph; the multiplier ( 1) depends on the orientation of the branch within a cycle. It is plain from Fig. 1 that there are two types of cycles. Some cycles start and finish at the boundary . We enumerate these cycles from 1 to and denote the starting and finishing boundary and correspondingly. elements of the th cycle by The remaining cycles are closed inside .
Gauging procedure (2D example).
Fig. 2. Weight coefficients a for tetrahedron edges.
By projecting (1) onto all cycles (more precisely, onto corresponding basis functions ) and by taking integration by parts, one can obtain the resulting system of algebraic equations -------
----
(4)
where (5)
(6) (7) stand for the areas of boundary elements . is a dense symmetrical nonnegative definite The matrix boundary-element matrix associated with facets lying only on . Formulae for computation of coefficients of this matrix are is associated with all facets, symgiven in [1]. The matrix , but sparse. The more metrical and positive definite when detailed volume mesh, the more sparse the entire matrix of the system (4). we need inner products of two facet For computation of of the same tetrahedron. Such products are functions and calculated as
where is the length of the th tetrahedral edge; the values of are provided in Fig. 2. weight coefficients There is one undetermined step in the algorithm: the choice of a tree. Although the final result does not depend on this choice, the convergence of iterative solvers does.
VISHNEVSKY et al.: MODELING MAGNETIZATION USING WHITNEY FACET ELEMENTS
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III. COILS
Fig. 3. Error in computation of an average demagnetizing factor. Tetrahedral meshes are constructed from one layer of quadrangular or triangular prisms.
The gauging procedure requires no modifications for multiply -conconnected regions, since the considered space is not satisfies the Ampere formal and the total internal field law only in a weak form. But the representation of the external field in case of multiply connected regions magnetized by coils has to be changed. ensures the The potentiality condition orthogonality of the right-hand side of (4) to “spurious” circulating magnetization. External fields due to geomagnetism and permanent magnets automatically satisfy this condition, but coils involve some difficulty. The problem can be resolved using a combination of the – method together with the “thick cut” idea [6]. This approach, besides the necessity in the of a (finite-element) modeling of vector potential coil region, requires an additional procedure for definition of fundamental cycles, which intersect the cutting surfaces. Once such a cycle is found, the magnetomotive force (MMF) of the coil should be added to the right-hand side of the corresponding formula in (4). But, in fact, the potentiality and cuts are not obligatory for facet element methods. All we need is an exact keeping the Ampere law inside the magnetic region. So, as a first approximation, we replace the coil with separate closed wires. Each wire, in turn, is divided into line-segments with uniform currents. The magnetic field, produced by line-segments, is integrated analytically along fundamental cycles. IV. TEST EXAMPLES
Fig. 4.
Example of nonuniform magnetization.
Fig. 5.
Magnetic field along the line x
= y = 0 inside sphere.
Providing the magnetization in the space, the original operator of the Poisson–Thomson equation is positive definite if , and negative definite if [7]. The suggested approximation of the magnetization retains the definiteness in , i.e., for all paramagnetic mathe discrete space for terials. In this case the equation (4) is proven to be the Euler equation for a functional associated with magnetostatic energy of the magnetized body. Following the theory, the worst situation for facet-element ) methods is represented by specific case of insulator ( with cavity. The original operator has a gradient null-space [8], and one should expect a loss of accuracy “near” this case, since there is no place for gradients in our approximation. As illustrated in Fig. 3, the presence of cavity results in a significant error for negative . This effect can be attributed to a whole bunch of nonzero, but very small eigenvalues in the discrete approximation of the operator (the same idea has been suggested for another methods, see, e.g., [8]). In the next example, we consider a nonuniform magnetization. The solid sphere is magnetized by the ring-shaped coil with 10 mm 10-mm rectangular cross section as shown in Fig. 4. Other dimensions (in millimeters) are presented in the figure. For discretization, we use a four-layer mesh, which results in 1280 tetrahedra for one-fourth part of the sphere. Typical distrifor is bution of internal magnetic field shown in Fig. 5. This distribution is compared with results obtained by a potential method [4], and by ordinary -formulation. The latter
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Fig. 6.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 38, NO. 2, MARCH 2002
Accuracy of computation of dipole moment of the sphere.
In the whole, numerical results obtained with facet-elements for homogeneous bodies are quite satisfactory and agree well with theoretical assumptions. However, for nonlinear media, direct application of the method is limited by the density of the system matrix, since the boundary-element matrix in this case expands interior to magnetic regions. An obvious profit in the nonlinear case can be achieved if a main part of ferromagnetic structure allows approximation with thin rods and thin layers (shells), because different types of facet-elements can be easily combined [1]. Advantages of the shell elements can be illustrated by solution of the TEAM Workshop Problem 13 [9]. The results given in Fig. 7 are obtained using only 727 triangular elements (for one-half of the problem geometry) and 1028 unknowns for approximation of magnetization of steel plates. An agreement with measurements presented in [9] is good enough for both MMFvalues of the exciting coil. is much greater than the Since the plate thickness air-gap width , an additional specific contact resistance is introduced for approximation of air-gap regions. It should be noted that in this case the problem becomes multiply connected.
V. CONCLUSION Test results presented in the paper confirm the effectiveness of Whitney facet elements for approximation of magnetization for homogeneous solids, and for shells with nonlinear magnetic properties. Further development of the method concerns rational in generic implementation of the gauge condition nonlinear case.
REFERENCES
Fig. 7. Problem 13: magnetic flux in steel plates.
utilizes a collocation method with nongauged piecewise constant approximation of magnetization. Unlike -formulation, both the facet-element and the potential methods give physically reliable solution. Calculations of dipole moments (Fig. 6) show that the potential method loses some accuracy comparing to facet elements for large values of . Similar to the cavity problem (see Fig. 3), the reasoning of such behavior is an ability of null-space approximation. But now . it is another, pure rotational space for
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