OD2-4
Source Field Modeling by Mesh Incidence Matrices Zsolt Badics and Zoltan J. Cendes Ansoft Corporation, 225 W. Station Square Dr., Pittsburgh, PA 15219, U.S.A.
[email protected] and
[email protected] Abstract – The discrete forms of the topological laws for source magnetic fields can be formulated by mesh incidence matrices using cohomology concepts. In this paper, an algorithm utilizing these mesh incidence matrices is developed for the fast generation of source magnetic fields. The algorithm is based on an integer Gaussian elimination procedure.
One efficient way to solve magnetostatic problems is to represent the magnetic field H in terms of a magnetic scalar potential Φ and a current vector potential T as H = T + ∇Φ . The division between the source field T and ∇Φ is arbitrary provided that T is represented by edge elements and satisfies
∇×T = J ,
∇⋅J = 0
(1)
where J is a prescribed solenoidal current density. Several excellent approaches to generating this non-unique T exist. (See for example [1].) Here, we describe an alternative approach based on the incidence matrices of the mesh. I. COHOMOLOGY CONCEPTS
Consider a triangulation Ωh of the computational domain Ω. The mesh is a collection of p-facets Sp that are cells S3, faces S2, edges S1, and vertices S0. Note that (1) implies that T and J must only satisfy topological laws. Thus, we can state these laws in the calculus of cochains [2-4]. A p-cochain on Ωh is a mapping from S p to the real numbers ℝ : S p ֏ ℝ . 1-
solution because then the rhs is in the range space of D1 . It turns out that we can solve (2a) very fast and efficiently by the following algorithm: (i) sort the rows of D1 by the occurrence of the first nonzero elements, (ii) perform Gaussian elimination on the system until the matrix takes the form
U D 0 tr 0 0 t = b 0
(3)
where U is an upper triangular square matrix with dimensions equal to the rank of D1 . This Gaussian elimination is very fast because the entries of the matrix remain in the set {-1, 0, 1}. Since (3) is overdetermined and rank deficient, we can set t0 to zero. Then, we compute tr by back-substitution. We have developed a similar algorithm for enforcing (2b) exactly based on the same principles described above. The differences between the two algorithms come from the fact that (2b) is underdetermined and rank-deficient. Details will be provided in the full paper.
cochain t and 2-cochain j correspond to the source magnetic field T and the current density J, respectively. We denote the vector space of p-cochains by C p and we can identify it with ℝ N where Np is the cardinality of S p . p
Similarly to differential forms, we can define a linear operator as the exterior derivative d h : C p ֏ C p +1 . Therefore,
Fig. 1. Curl of the magnetic field in the conduction path
we can associate incidence matrices D p ∈ ℝ N , N of p-facets and (p+1)-facets. The incidence matrices are sparse with entries {-1, 0, 1} determined by adjacency relations and relative orientations [5]. The topological laws (1) then become
Figure 1 shows a multiply connected conductor driven by a prescribed current. T is determined by the algorithm (3) and then H is calculated. Figure 1 depicts the distribution of curl H and validates the correctness of the algorithm.
p +1
D1 t = j ,
D2 j = 0
p
(2)
The Whitney map [4], which makes discrete differential (Whitney) p-forms from simplicial p-cochains, states that the entries of t are the coefficients of the edge element approximation of T. The same is true for j and the Whitney 2-form approximation of J.
III. REFERENCES [1]
[2]
[3] II. ALGORITHM [4]
To utilize (2a) to compute t we note that the equation system (2a) is very sparse, overdetermined and rank-deficient. If we ensure that j satisfies (2b) exactly we can obtain a
1-4244-0320-0/06/$20.00 ©2006 IEEE
343
[5]
Z. Ren, P. Zhou, and Z.J. Cendes, “Computation of current vector potential due to excitation currents in multiply connected conductors,” in ICEF 2000, Tianjin, China, September 2000. T. Tarhasaari and L. Kettunen, “Topological approach to computational electromagnetism,” Progress In Electromagnetics Research, PIER 32, pp. 189–206, 2001 R. Hiptmair, “Finite elements in computational electromagnetism,” Acta Numerica, 11, pp. 237-339, 2002. P.W. Gross, and P.R. Kotiuga, “Data structures for geometric and topological aspects of finite element algorithms,” Progress in Electromagnetics Research, PIER 32, pp. 151-169, 2001. A. Bossavit, Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elements, Vol. 2 of Electromagnetism Series, Academic Press, San Diego, CA, 1998.
