Gauging in Whitney spaces. Lauri Kettunen, Kimmo Forsman. Tampere University of Technology, Lab. of El&romagnetics, P.O. Box 692, FIN-33101 Tampere, ...
IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, MAY 1999
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Gauging in Whitney spaces Lauri Kettunen, Kimmo Forsman Tampere University of Technology, Lab. of El&romagnetics, P.O.Box 692, FIN-33101 Tampere, Finland Alain Bossavit hxtricit6 de France, 1 AV.du Gal de Gaulle, 92141 Clamart, France Abstract-In this paper gauging is approached as a problem of selecting a representative in classes of equivalent representations. In this light we interpret how different gauging techniques are related to each other, and examine how they can be imposed on the discrete level using Whitney elements. Indeo terms-Gauge condition, Finite element method, Whitney elements
I. INTRODUCTION When operating with potentials instead of fields, one typically encounters the problem of imposing a gaugecondition to gain a one-to-one correspondence between the potentials and the fields. Due to this fact one may be biased towards interpreting gauging simply as a mechanism to gain this 1-to-1 relation. However, in a broader sense all potentials yielding the same field can be regarded as one class and then, gauging corresponds with selecting a representative of each class. In this paper we shall approach gauging from this angle of view. At first, we study gauging without assuming any metric structures and seek for a single geometrical interpretation which applies for all operators grad, curl and div. In the second part the paper is extended to some metric-dependent gauging techniques. In this case the overall idea is t o restrict the kernel of the operator, be it grad, curl or div, to a single element. Again, only one “recipe” is needed which applies to all cases. The motivation of working with all operators at once is the idea that on the practical level the same should still hold: only one subroutine is needed for gauging which does the job for all cases. However, vector algebra is not powerful enough t o make this point clear, which is the motivation for us to work in terms of differential geometry.
11. EQUIVALENCE CLASSES
AND GAUGING
We assume domain R is a bounded open set of three dimensional Euclidean space E , and I?, the boundary of R, is smooth. We also assume that the boundary of each connected component of R consists of c+ 1 closed surfaces rj, j = 0,. . . ,c, and R is locally on one side of I?. For example, R can be like the bretzel-shaped domain shown in Fig. 1. Suppose there is a field b such that b = da, where d is the exterior operator [l].Although the meaning of symbols b and a is arbitrary, the coincidence with magnetic flux density and magnetic vector potential, respectively, Manuscript received June 3, 1998. Lauri Kettunen, e-mail Lauri.KettunenQcc.tut.fi
Fig. 1. A bretzel-sha ed domain. The exterior boundary is denoted by I?’ and r ?is the boundary of a cavity. For a later use we have also drawn the “C-cuts” and a “C-link” connecting the boundary of onto I’O. is deliberate. One may well first interpret d as the curl operator and understand b = da to mean b = curl a in the sense of vector calculus. As is well known, a is not unique since one may add, for instance, df to a without affecting b, i.e. b = da = d(a df). But now, instead of trying to select a unique a representing b, let us consider all the a’s yielding the same Meld as a single object. For this reason, we introduce an equivalence relation R: Definition 1. Let the domain of operator d be denoted by dom(d). We say that a1,a2 E dom(d) are equivalent if dal = da2, and we write al R a2 to say a1 and a2 are thus related. The equivalence relation R creates a partition of dom(d) into equivalence classes, the set of which (the so called “quotient” of dom(d) by R)is denoted by dom(d)/R. The idea here is: as each class consists of elements which yield the same b field, there is evidently a one-to-one correspondence between the classes and the b’s. Moreover, the important point is that each class - taken as a whole - is the same object as the field [2]. Gauging is now a step forward and it corresponds with selecting a unique representative in each class of equivalent representations of the field, as displayed graphically in Fig. 2.
+
I
I Fig. 2. Equivalence classes iLpierced”by a gauge condition. The intersection with the gauge condition is the unique representative of each class.
In practice the use of potentials is typically motivated by the fact that, if b = da then we have automatically 0018-9464/99$10.00 0 1999 IEEE
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db = 0, since for any f E dom(d), theproperty d(df) = 0 holds [I]. For instance, if the problem to be solved is, find b and h in appropriate spaces such that db = 0, dh = j, h = *ub, and if we may set b = da, then the single equation dwda = j
DISCRETE BASIS
aim is to construct a discrete basis for the equivalence classes. We start by assuming field b of degree (p+l) from which y e construct an array of degrees of freedom (DoF-array) b such that
bs = / b ,
(2)
yields the same solution for b as problem (1) There are now two different strategies by which to solve (2). We may (i) operate directly with the equivalence classes, or (ii) set a gauge, and look for a unique representative from each class. Remark 1. Assuming exact arithmetic, the choice between using fields, potentials or equivalence classes should not affect the field solution.
111. CHARACTERIZATION OF EQUIVALENCE CLASSES Let us now assume that R is divided into a simplicial finite element mesh. The sets of psimplices are denoted by S P , p = 0 , . .,3. For instance, So is the set of nodes, S1is the set of edges and so on. Our first problem is to characterize the equivalence classes. For this we need some tools: Definition 2. A p-chain is an assembly of oriented p simplices, each with an integer weight (usually 0, +1 or -1). We say a chain “lies in” a subset S of S P if all psimplices outside S bear the weight 0. A path of edges, for instance, is a 1-chain with weights +1 or -1, depending on whether the edge is oriented along the path or the other way. Chains can be added, by summing up weights for each simplex. The boundary of a psimplex is a chain, containing all its (p - 1)-faces with weights 1or -1, depending whether the orientation match or not. (For example, the boundary of a facet is an assembly of three edges with weigths 1 or -1.) The boundary of a chain is then, by linearity, a chain, obtained by taking the sum of the boundaries of its simplices. Definition 3. A cycle is a chain with null boundary. Definition 4. A bounding cycle is a pcycle which is the boundary of a (p 1)-chain. The boundary of a node is, by convention, null. Any pair of nodes in the same connected component of Cl is an example of a bounding cycle. Definition 5. A subset S of S P is cycle-free if no pchain lying in S can be a pcycle. Definition 6. A subset of SP is boundary-freeif no pchain lying in S is a bounding cycle. If p = 1 the m&mal cyclefree subsets, i.e. subsets to which no psimplex can be added without having a p cycle lying in it, are commonly called 6ctrees,,.The
