IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007
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Reading Whitney and Finite Elements With Hindsight Jari Kangas, Timo Tarhasaari, and Lauri Kettunen Institute of Electromagnetics, Tampere University of Technology, Tampere FIN-33101, Finland
Maxwell laws and constitutive laws are different in nature and, consequently, they impose different requirements on the basis functions used in numerical techniques. We employ Whitney’s framework to address these issues, and introduce a connection to the practice of finite elements exploiting Whitney forms, Mur’s elements, and nodal elements as examples. Index Terms—Finite elements, quasistatics, Whitney forms.
I. INTRODUCTION
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LTHOUGH Whitney elements, and especially the so-called “edge elements” are commonly employed, only a few papers—such as [1]–[3]—have been written about the principles on which the finite-element practice of edge elements is based. In this paper, we approach finite elements relying on Whitney’s own framework of Geometric integration theory [4]. The very idea is to reverse the common reasoning behind computational electromagnetism fitting Whitney elements into the framework of the finite-element method. Instead, we will read Whitney’s theory with hindsight and examine what it suggests on solving electromagnetic field problems in finite dimensional spaces, and examine how finite elements and some other numerical techniques follow from such an approach. The theme of Whitney’s theory is integration over an -dimensional object in -dimensional space, such as integration over a line or surface in ordinary three-dimensional space. The theory is built to tell how the integral over depends on the position of . In short, the theory is established by introducing as a linear combination of ( -dimensional) oriented -cells with real numbers as coefficients. This is to say, object is a so-called polyhedral -chain. Next, by giving some norm appropriate to complete the linear space of polyhedral -chains, then curved lines and surfaces can be meaningfully understood as elements of the Banach space [5] obtained by completion (Fig. 1). Whitney’s theory provides us with a tool to represent electromagnetic fields in finite dimensional spaces by introducing equivalence relations which tell in which way fields are considered equivalent. This equivalence relation partitions the underlying space of fields into equivalence classes, and then one seeks for the numerical solution from the spaces consisting of such classes. Now, if one reads the finite-element technique in the same manner, the emphasis is not on choosing some basis functions needed for interpolation, but instead, the basis functions are a tool to find a unique representative for each equivalence class.
Digital Object Identifier 10.1109/TMAG.2007.892276
Fig. 1. Dashed and the thin lines are two polyhedral 1-chains. To say how close the 1-chains are to each other, a norm in the space of polyhedral 1-chains is needed. Once such a norm is introduced, the underlying space can be completed as a Banach space. Thereafter, “curved lines,” such as the thick one, become meaningful as elements of this completed space.
II. PRELIMINARIES Let us first give enough status to the basic ideas of Whitney’s integration theory. An -chain is a formal sum , where the ’s are -cells, and the ’s are real numbers. The linear function which maps to such that holds, is called an -cochain. Next, can be considered as a definite “integrand” giving a number when applied to a permissible domain. Obviously, to model electromagnetism, the permissible domains must include curved objects. To find such objects, they are constructed as limits of polyhedral -chains. To specify such a limit of a sequence of polyhedral chains, some norm in the space of polyhedral chains is needed. The two norms Whitney introduced are the so-called flat and sharp norm. In this context, it is not necessary to introduce the definitions of these norms. (For the definitions, see [4]). The norm introduced to chain spaces induces also a norm for the dual space of chains, whose elements are called cochains. For our purposes, it is enough to to say the flat cochains given by the flat norm fulfill the following two continuity conditions. 1) Given the -cochain , there exists such that for all oriented -cells, and where is the -dimensional volume of . 2) Given the -cochain , there exists such that for all oriented -cells, where is the boundary of . Sharp cochains satisfy an additional condition. To describe it, we first name the translation (“rigid motion”) of cell by vector . And now, in addition to the first two conditions, sharp cochains fulfill also as follows.
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Fig. 2. Left: Whitney “edge element” (six degrees of freedom, one on each edge). Center: consistently linear edge elements by Mur (12 degrees of freedom, one on each end of each edge). Right: nodal elements (12 degrees of freedom, three on each node).
3) Given the -cochain , there exists such that for all oriented -cell and vector . The first two conditions give status to the Stokes theorem. The intuitive idea of the third one is whatever translation is made for cell , the change in integral is not abrupt. Notice also that all sharp cochains are flat (for all sharp cochains fulfill the first two conditions), but the converse is not true (for flat cochains need not to satisfy the third condition). Example: Whitney elements and the so-called “Consistently linear edge elements” by Mur [6], [7], [8] are flat functions, whereas the nodal elements approximating vector fields are sharp (Fig. 2). In engineering parlance, Whitney “edge elements” and Mur’s elements impose “tangential continuity” on the interelement interfaces fulfilling the first two continuity conditions. Thus, they are flat functions. Nodal elements force also “normal continuity” which coincides with the third continuity condition. Consequently, nodal elements are sharp functions. III. WHITNEY ELEMENTS To support intuition, let us say one has measured some phenomenon along the edges or faces of a simplicial cell complex. For instance, say we have measured the circulations or fluxes along the edges or on the faces, respectively, of a complex. This corresponds to having a so-called algebraic cochain. Now, to construct a field—such as the electric or magnetic field—knowing circulations along the edges or fluxes on the faces is not enough. Instead, the electric field is formally about knowing all voltages and magnetic flux about knowing fluxes on all 2-chains. Consequently, to build a theory, one needs a transition from the edge circulations or face fluxes to all circulations and fluxes, respectively. In formal words, this is about introducing a map from algebraic cochains to some space of cochains which yield numbers on “all” chains. Now, Whitney elements is a particular example of such kind of map. Whitney introduced a linear map from algebraic cochains to so-called elementary flat cochains satisfying certain properties: Let and denote the space of -chains and -cochains in simplicial complex , respectively. Furthermore, let be the space of flat cochains in the set of -simplices in and domain the star of simplex . Now, map fulfills properties (1)
(2) (3) (4)
The term “elementary” emphasizes the idea that the question is of very primitive kind of flat cochains. To make a distinction between cochains and differential forms, one may well call the elementary flat cochains by name “Whitney cochains.” Consequently, “Whitney forms” [9] are the elementary flat forms one-to-one with the elementary flat cochains. Remark: Although Mur’s linearly consistent edge elements are flat, they are not elementary ones. Whitney edge elements is a subspace of Mur’s elements. The additional component in Mur’s elements consists of higher order gradients. IV. ELECTROMAGNETIC FIELDS Without losing generality, let us next choose the static magnetic field as a standard example in explaining the main ideas. The static magnetic field is formally the pair containing cochains and yielding all magnetomotive forces and fluxes, respectively. Assume now we have a need to solve numerically some magnetostatic boundary value problem. However, at this point, let us not choose any specific approach (such as the finite-element method), but instead, find out what Whitney’s theory suggests to do when we need to find numerical solutions. On a very general level, creating a numerical solution is about finding a finite dimensional approximation of the “continuous level” pair .1 Mathematically, the transition from “continuous level” to a finite dimensional space has to do with equivalence relations and equivalence classes. That is, introducing an appropriate equivalence relation which tells how fields are considered the same, then one ends up with finite dimensional spaces of equivalence classes. Then the original boundary value problem can be solved in terms of the equivalence classes thus obtained. This kind of reasoning is a reverted process compared to the one needed in building Whitney’s theory (in which the idea was to generate -cochains, such as the elementary flat cochains, out of algebraic -cochains). So, by reversing Whitney’s approach 1The very name of various approaches finite elements, finite-difference method, finite-integration technique, etc., suggest the main idea is to find a “finite” solution.
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we end up with algebraic cochains and denoting the equivalence classes of magnetomotive forces and magnetic fluxes obtained by introducing equivalence relations. are equivalent, if they yield the same 1) Cochains and magnetomotive forces on the edges of a chosen set. 2) Cochains and are equivalent, if they yield the same magnetic fluxes on the faces of a chosen set. Obviously, each choice of the sets of edges and faces yields a of equivalence classes. different pair
a flat form is an element of . Furthermore, as the space of flat forms is a Banach space but not a Hilbert one—for there is no inner product—the Riesz theorem [5] cannot be employed to give status to the constitutive law between spaces of flat forms. Summing up, to solve the magnetostatic boundary value problem in finite dimensional spaces, some further structure is needed.
V. NUMERICAL APPROACHES Once we make the move to operate with algebraic cochains, imposing Maxwell’s equations becomes immediate as soon as we make sure the idea of a boundary of a simplex is meaningful. For this, it is enough to pick of some complexes and , and then we may impose (the acting on cochains is the coboundary operator [4]) (5) (6) This is to say, the magnetomotive force around any bounding 1-cycle should coincide with the current across the corresponding 2-chain, and the magnetic flux on any bounding 2-cycle should be null.2 To impose magnetostatic boundary value problems, we need also the magnetic constitutive law, which is a relation between differential forms rather than between cochains. Thus, for differential forms and we write or
(7)
where is the Hodge operator [10], and is permeability and reluctivity. Now, the magnetostatic problem (5), (6), and (7) makes sense only if and are elements of some well-defined spaces. It is known that the flat cochains are one-to-one with flat forms [4]. To explain the remaining difficulty, let us denote the largest space of differential forms of degree by . This is to say has no smoothness conditions. The spaces of flat -forms are denoted by , and the spaces of sharp forms by . The relationships between these spaces can be summarized into the following diagram:
A. Galerkin Method Standard finite elements relies on the Galerkin method and circumvents the problems explained by introducing an inner product3 (8) Locally within each tetrahedron, Whitney elements are flat but also sharp. Thus, it is possible to impose elementwise (5) or (6), and (7). Let us say one has decided to impose (5) and (7). Then, by employing the inner product (8), the Gauss law is solved in the “weak sense.” Such a move shifts the emphasis from differentiation to integration, and it is sufficient that locally elementdenotes wise the integral between basis functions and a tetrahedron
is meaningful. And now, if the basis functions are, for instance, Whitney elements, ( is either or ) (9) is integrable, for smoothness is lost only in a set having zeromeasure. B. Finite-Integration Technique In the Finite Integration Technique [11] or the Cell Method [12], one imposes both (5) and (6), but not on the same complex, but instead, on two complexes that are “dual” to each other. Then, one employs pointwise the Hodge operator on the “geometric side” mapping -multivectors to -multivectors [13] and writes for the real numbers that covectors yield on multivectors (10)
This is to say, the Hodge of a sharp form is (at first, pointwise an element of , but then) found to be sharp. However, we are not able to say more than that the exterior derivative of a sharp form is flat. Correspondingly, the of a flat form is also flat. However, in the local sense, we are able to say only that the Hodge of
C
D
2As is well known, (5) and (6) translates to = j and b = 0, respech j ; and b are arrays of degrees of freedom (DoF) and and tively, where h; are the so-called incidence matrices representing the incidence relations of the simplices of the chosen complex.
C
D
Then, these covectors are approximated by cochains making the magnetostatic boundary value problem solvable. VI. PRACTICAL IMPLICATIONS Let us next examine what the aforesaid implies on finite-element practices. 3The inner product can also be understood as a tool to form equivalence classes: Two fields, f and f , are equivalent if (f; f ) = (f; f ) holds for all f . As a result, one gets different equivalence classes than those from Whitney’s algebraic cochains.
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A. Implication of the Chosen Element Type When solving the magnetostatic boundary value with Whitney elements, consistently linear edge elements, or nodal elements, imposing the constitutive law becomes a difficulty. For Whitney elements and consistently linear edge elements are flat and nodal elements are sharp, and the exterior derivative of all these functions is flat. So, say we express the vector potential a in one of these element spaces. The exterior derivative yields a flat function for . Then, the Hodge star of is just an element of , and thus, does not have enough meaning to impose Ampère’s law. The strength of Whitney elements is that the underlying elements spaces and the exterior derivative form an exact sequence [14]. In this sense, Whitney elements lend themselves to Maxwell’s equations. The “built-in” difficulty is that the Hodge of a Whitney element is not a Whitney form. Like Whitney elements, Mur’s consistently linear edge elements are also flat, but they are not elementary. Mur’s elements contains Whitney edge elements as a subspace, and the additional component is about higher order gradients. Thus, Mur’s elements yield no further information in specifying the curl of the basis functions. (This is observed experimentally in [15].) Mur’s elements decompose Whitney edge elements into two components. In the same manner, the Whitney facet elements could easily be split into three components yielding also flat but not elementary functions. (These functions could be called “consistently linear facet elements.”) However, the underlying spaces of the extension of Mur’s approach and exterior derivative did not form an exact sequence. Notice also that Mur’s elements are not “algebraic” the same sense as Whitney elements. This implies that it is not immediate how a given field is mapped to a DoF-array. That is, it is not immediately clear, how the two edge values on an edge should be formed out of a given field. Such a map is needed, for example, in setting boundary conditions. Nodal elements are sharp functions, which have the same amount of degrees of freedom as Mur’s edge elements. The spaces of nodal elements and the exterior derivative do not form an exact sequence. B. Post Processing Say now we have solved the magnetostatic boundary value problem with an -formulation using finite elements and obtained a DoF-array corresponding with . The remaining question is, how should one thereafter compute an approximate of from . (Recall that the magnetostatic field is formally the pair .) The custom procedure is simply to multiply locally within each element the proxy vector of by permeability . However, then the approximate of has hardly any properties globally. This generates easily difficulties, for instance, in computing forces acting on magnetic bodies with the Maxwell stress tensor, or in specifying trajectories of charged particles in magnetic field. (The so-called “interelement discontinuities” in the normal component of are typically considered as a source of problems.) A common attempt to avoid such problems is to construct a “nodally averaged” . That is, at each node one forms an av-
IEEE TRANSACTIONS ON MAGNETICS, VOL. 43, NO. 4, APRIL 2007
erage of the proxy , and then one interpolates the magnetic flux density componentwise using barycentric functions. Such a process can be interpreted as creating a sharp function out of a flat one, and in this sense the operation shifts emphasis from the Gauss law (solved in the weak sense) to the constitutive law. However, the problem in the background has to do with the fact that it is not possible to impose both Gauss’s law, Ampère’s laws, and the constitutive law simultaneously in finite dimensional spaces. Therefore, in this sense, nodal averaging does not solve any problem. VII. CONCLUSION Recently, generalizing on Whitney, Harrison has developed a so-called natural norm [16] “between” the flat and sharp norm. The natural norm gives status simultaneously to and . If Whitney elements are to be extended to higher order or to other types of cells than simplices, the construction of Harrison’s natural norm should yield further information in building understanding of finite element kind of techniques.
ACKNOWLEDGMENT This work was supported by the Academy of Finland under Project 53972. REFERENCES [1] A. Bossavit, “A rationale for edge-elements in 3-D fields computations,” IEEE Trans. Magn., vol. 24, no. 1, pp. 74–79, Jan. 1988. [2] ——, “A new rationale for edge-elements,” Int. Compumag Soc. Newslett., vol. 1, no. 3, pp. 3–6, 1995. [3] ——, “Generating Whitney forms of polynomial degree one and higher,” IEEE Trans. Magn., vol. 38, no. 2, pp. 341–344, Mar. 2002. [4] H. Whitney, Geometric Integration Theory. Princeton, NJ: Princeton Univ. Press, 1957. [5] K. Yosida, Functional analysis. Berlin, Germany: Springer-Verlag, 1980. [6] G. Mur and A. T. de Hoop, “A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media,” IEEE Trans. Magn., vol. MAG-21, no. 6, pp. 2188–2191, Nov. 1985. [7] ——, “A finite-element method for computing three-dimensional electromagnetic fields in inhomogeneous media,” IEEE Trans. Magn., vol. MAG-21, no. 6, pp. 2188–2191, Nov. 1985. [8] G. Mur, “Edge elements, their advantages and their disadvantages,” IEEE Trans. Magn, vol. 30, no. 5, pp. 3552–3557, Sep. 1994. [9] A. Bossavit, “Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism,” Proc. Inst. Electron. Eng. A, vol. 135, no. 8, pp. 493–500. [10] H. Flanders, Differential Forms with Applications to the Physical Sciences. New York: Dover, 1989. [11] T. Weiland, “Time domain electromagnetic field computation with finite difference methods,” Int. J. Numer. Modell., vol. 9, pp. 295–319, 1996. [12] E. Tonti, “Finite formulation of electromagnetic field,” IEEE Trans. Magn., vol. 38, no. 2, pp. 333–336, Mar. 2002. [13] T. Tarhasaari and L. Kettunen, “Wave propagation and cochain formulations,” IEEE Trans. Magn., vol. 39, no. 3, pp. 1195–1198, May 2003. [14] A. Bossavit and I. Mayergoyz, “Edge-elements for scattering problems,” IEEE Trans. Magn., vol. 25, no. 4, pp. 2816–2821, Jul. 1989. [15] B. Bandelier and F. Rioux-Damidau, “Modelling of magnetic fields using nodal or edge variables,” IEEE Trans. Magn., vol. 26, no. 5, pp. 1644–1646, Sep. 1990. [16] J. Harrison, “Lectures on chainlet geometry—New topological methods in geometric measure theorey,” 2005 [Online]. Available: http://math.berkeley.edu/~harrison/research
Manuscript received April 30, 2006 (e-mail:
[email protected]).
