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Electric and Magnetic Fields, From Numerical Models to Industrial Applications (Proc. Louvain, May 1994, A. Nicolet & R. Beulmans, eds.), Plenum Press (New York, London), 1995, pp. 237-40.

ON "HYBRID" ELECTRIC-MAGNETIC METHODS A. Bossavit Électricité de France, 1 Av. du Gal de Gaulle, 92141 Clamart, France

INTRODUCTION "Hybrid", "mixed", etc., are dangerous words: you never know what people mean. Here, we deal with hybrid methods in the following sense. Due to a well-known duality inherent in Maxwell equations, and to the impossibility to enforce all continuity conditions with finiteelements, basic methods can be classified as "e-oriented" or "h-oriented" (or as "electric" versus "magnetic") according to whether they enforce the tangential continuity of e or of h (hence the normal continuity of b or of j, respectively) at element interfaces. Now, in "hybrid" approaches, one makes a patchwork of methods, using h-oriented ones in some regions of space and e-oriented ones elesewhere. Hybrids we specifically study here are h-oriented in the air (cf. Fig. 1) and e-oriented in the conductor. B S jg C n

n

D

Figure 1. Situation and notation. The computational domain, D, contains a coil with given current density jg (time-dependent), a conductor C with non-zero conductivity σ, and possibly some "magnetic" regions, where σ = 0 and µ ≠ µ 0. The boundary B is supposed to be an "electric" one, on which n × e = 0 (and hence n · b = 0). The air-conductor interface is S. We assume topological triviality (no current loops, no inside holes). Note the orientation of the unit normal field (outward with respect to D – C).

We use L2 (R) to denote the space of square-integrable functions over some region R, (and IL2 (R) for fields). Then we set L 2 grad(R) = {ϕ ∈ L2 (R) : grad ϕ ∈ IL2 (R)}, and IL2 rot(R) = {u ∈ IL2 (R) : rot u ∈ IL2 (R)}. All derivatives are denoted with a ∂. Constructs like ϕS, hS, grad Sϕ, etc., should respectively be understood as the restriction of ϕ to S, the tangential part of h on S, the surface gradient of ϕ on S, etc. FORMULATION We denote IL2 rot(C) by U, and L2 grad(D – C) by Φ. Let us introduce the field a g (t, x) = (4π)−1 ∫ |x − y| −1 j g (t, y) dy, where dy is the volume element, and define the "source-field" hg as h g = rot ag . There exists, in the case of Fig. 1, a region containing C in which h g = grad ϕg , for some suitable potential ϕg (not defined, of course, over all D). This way, we may look for the magnetic field h in the air (the region D – C) in the form h = hg + grad ϕ, where the magnetic potential ϕ lies in Φ. Elements of U will be "modified vector potentials" of sorts: more precisely, calling e(t) the electric field at time t, and assuming that all fields, including the given current t density j g , were null up to time t = 0, we set u(t) = − ∫ 0 e(s) ds. This way, e = − ∂tu, and h = µ−1 rot u.

Note that knowing u in the conductor and ϕ in the air is enough to fully know the electromagnetic situation1 , provided the required continuity conditions at the interface S are enforced. These so-called "transmission conditions" are n · rot u = n · [µ(hg + grad ϕ)], n × (hg + grad ϕ) = n × [µ−1 rot u].

(1)

Unfortunately, this doesn't make much mathematical sense if taken a priori 2 , so we cannot simply say "search for a pair {u, ϕ} that satisfy (1) and ...", where the dots would stand for some weak formulation that should also be enforced. This difficulty is typical of hybrid methods.3 There is a way around it. Let's provisionally pretend that both hS and n · b (that is, µ n · h) are known on S. Then, starting from Ohm's law in C, σ ∂tu + rot h = 0, taking the scalar product with a test-field u', and integrating by parts, we get ∫C σ ∂tu · u' + ∫C µ−1 rot u · rot u' = ∫S n × h · u'

(2)

as a necessary condition to be satisfied by u, for all u' in U. On the other hand, and because div b has to be zero in D – C, and n · b ≡ µ n · h has to vanish on B, ϕ should satisfy ∫D – C µ (hg + grad ϕ) · grad ϕ' = ∫S µ n · h ϕ'

(3)

for all ϕ' in Φ. Since eventually n · b has to equal n · rot u, and n × h has to equal n × (hg + grad ϕ), we see that the pair {u, ϕ} must satisfy the coupled equations: ∫C σ ∂tu · u' + ∫C µ−1 rot u · rot u' − ∫S n × grad ϕ 〈 u' = ∫S n × hg · u' ∀ u' ∈ U, ∫D – C µ (hg + grad ϕ) · grad ϕ' − ∫S n · rot u ϕ' = 0 ∀ ϕ' ∈ Φ, where now all integrals make perfect sense (and have, as we see in a moment, unambiguously defined discrete counterparts). Thanks to the identity divS(n × u) = − (n · rot u)S, we may give a more symmetrical appearance to the previous equations, and finally specify the problem as find {u, ϕ} in U × Φ such that ∫C σ ∂tu · u' + ∫C µ−1 rot u · rot u' + ∫S ϕ n · rot u' = ∫S n × hg · u',

(4)

∫S n · rot u ϕ' − ∫D – C µ grad ϕ · grad ϕ' = ∫D – C µ hg · grad ϕ'

(5)

for all test-pairs {u', ϕ'} in U × Φ. DISCRETIZATION This was a one-way derivation, so we still have to verify that problem (4)(5) is well-posed, that is, has a unique solution that continuously depends on the data h g . (If so, it will yield 1

2

The outside e is rarely needed, and if so, may be recovered by solving a standard electrostatic problem in the air, once h is known.

These relations link "Dirichlet-like" data (on the left-hand side) to "Neumann-like" ones (on the right), and whereas the former (called "traces") depend on u or ϕ in a continuous way, the latter don't: fields that merely belong to IL2, like rot u or grad ϕ, have no trace, be it tangential or normal. No mathematical pedantry here, but a very fundamental difficulty: a priori constraints on the solution that, like (1) above, involve other than continuous mappings, simply cannot be enforced (as simple algebraic constraints) at the discretized level of finite elements. 3 Contrast the situation with that of a magnetic-magnetic method (h inside, ϕ outside): same space Φ, but h roams in H ≡ IL2rot(C). The transmission condition would then be hS = gradS(ϕ + ϕ g), which defines a closed affine subspace in the Cartesian product H × Φ, because both traces h → hS and ϕ → ϕS are continuous in a suitable sense.

the right electromagnetic field, since conditions (1) result, a posteriori, from (4) and (5), as one may check by doing the integrations by parts.) But first, let us make sure that it discretizes in a straightforward way. Consider a mesh of D. Call G, R, D, the incidence matrices of this mesh (e.g., Gen = 1 if edge e abuts on node n, etc.; cf. [Bo]). They are the discrete counterparts to grad, rot, div. Let λ n be the barycentric function for node n, and we = λ n grad λ m − λ m grad λ n , the edge-element for edge e = {n, m}. For a region R and a scalar function α, define MR(α) as the matrix of entries ∫R α we · w e' (where e and e' are edges). Finally, for a node m and an edge e both in S, set Nm, e = ∫S n · rot we λ m, and let N be the matrix obtained by filling-in with zeros for other node- and edge-indices. Now, by the standard Galerkin treatment, (4)(5) becomes a system of ODE's: find timedependent vectors of degrees of freedom u and ϕ such that ∂t [MC(σ) u] + Rt MC(µ−1 ) R u

+ Nt ϕ = − Nt ϕg ,

Nu − Gt MD − C(µ) G ϕ = Gt MD − C(µ) hg

(6) (7)

(where t stands for "transpose"). Data ϕg (node-values of ϕg ) and hg (edge-circulations of h g ) are easily computed from j g (using now standard "spanning tree" techniques if needed). Note the remarkable structure of the overall matrix of this system (globally symmetric, but not positive definite). It is characteristic of what numerical analysts call "mixed", or "two-field" approaches. Remark also that if µ = µ0 outside C, one may replace the term ∫D – C µ hg · grad ϕ' ≡ ∫S µ n · hg ϕ' by µ0 ∫S n · rot ag ϕ', and thus replace the right-hand side in (7) by µ0 Nag . WELL-POSEDNESS Is (4)(5) well-posed? To study this, let us introduce the "Neumann-to-Dirichlet map" Q, as follows. Let v (a scalar function) be given on S. There is a unique ϕ ∈ Φ such that ∫D – C µ grad ϕ · grad ϕ' = ∫S v ϕ' = 0 ∀ ϕ' ∈ Φ. Then ϕS = Qv defines Q. Let us also define φg as the unique solution to ∫D – C µ grad φ · grad ϕ' = ∫D – C µ hg · grad ϕ' ∀ ϕ' ∈ Φ, that is, eq. (5) with u = 0 and a change of sign. (Note that φg S is not equal to ϕg S.) Now, eq. (5) means that ϕS = Q (n · rot u) − φg S. Substituting in (4), we get ∫C σ ∂tu · u' + ∫C µ−1 rot u · rot u' + ∫S Q(n · rot u) n · rot u' = ∫S (φg − ϕg ) n · rot u' ∀ u' ∈ U,

(8)

a perfectly well-posed problem in space U, since Q is positive definite. (More precisely, Q is the "Riesz isomorphism" associated with the space of traces of Φ on S, usually denoted as H 1/2(S): Q maps the dual H−1/2(S) of H 1/2(S) onto the latter. It is well known, on the other hand, that the mapping u → n · rot u sends U into H−1/2(S), continuously.) Eq. (8), by the way, forms the basis of another "hybrid" method, that uses edge-elements for u, or equivalently for e, inside the conductor, and a boundary element method to precompute Q [RR]. Such hybrid methods, which shun finite elements in the air, are better adapted to the case of open (infinite) air regions. VARIANTS From this point on, one can make many variations. One, that seems to be popular these days (cf., e.g., [BP]), consists in expressing u as the sum a + grad ψ, where a is the standard t vector potential and ψ(t) = ∫0 v(s) ds, a suitable primitive of the electric potential v. Thus one deals with triples {a, ψ, ϕ}. Fixing the gauge by requiring (for instance) div a = 0 in C and n · a = 0 on S makes the solution unique. But one may as well penalize the

constraint div a = 0, which is simply done by adding in (4) the term ∫C µ−1 div a div a'. This means that a may be looked for in a subspace of U made of more regular fields, namely A = {a ∈ U : div a ∈ L 2 (C)}, equipped with the scalar product (a, a') = ∫C a · a' + ∫C rot a · rot a' + ∫C div a · div a'. Denoting L 2 grad(C) by Ψ, we arrive at the following formulation: find {a, ψ, ϕ} in A × Ψ × Φ such that ∫C σ ∂t(a + grad ψ) · (a' + grad ψ') + ∫C µ−1 rot a · rot a' + ∫C µ−1 div a div a' + ∫S ϕ n · rot a' = ∫S n × hg · a', ∫S n · rot a ϕ' − ∫D – C µ grad ϕ · grad ϕ' = ∫D – C µ hg · grad ϕ'

(9) (5)

for all test-triples {u', ψ', ϕ'} in A × Ψ × Φ, hence σ(∂ta + grad v) + rot(µ−1 rot a) − grad(µ−1 div a) = − js

(10)

inside the conductor. We know there are solutions: starting from the u that solves (4)(5), take ψ such that ∫C (u − grad ψ) · grad ψ' = ∫C w ψ' ∀ ψ' ∈ Ψ, then set a = u − grad ψ. Here, w is an arbitrary element of H−1/2(S), which shows to what extent the solution of (9)(5) is non-unique. Uniqueness can be enforced by narrowing either A or Ψ (or both), which is done by specifying either n · a on S or ψ on S (or both, on complementary parts of S). Quite independently of these "gauge fixing" procedures, however, µ−1 div a = 0 if (9) holds, which shows that the right problem is solved, in spite of the strange look of eq. (10). (To check this assertion, start from f' ∈ L2 (C), take ψ' such that ψ' = 0 on S and −∆ ψ' = f' in C, and set a' = 0. Then (10) reduces to ∫C µ−1 div a f' = 0 ∀ f' ∈ L2 (C), hence µ−1 div a = 0.) Standard nodal elements (vector-valued) can then be used for a, which seems to be perceived as an advantage by some.4 REFERENCES [Bo] A. Bossavit: Électromagnétisme, en vue de la modélisation, Springer-Verlag, Paris (1993). [B&] I. Bardi, O. Biro, K. Preis, G. Vrisk, K.R. Richter: "Nodal and Edge Element Analysis of Inhomogeneously Loaded 3D Cavities", IEEE Trans., MAG-28, 2 (1992), pp. 1142-5. [BP] O. Biro, K. Preis: "On the Use of the Magnetic Vector Potential in the Finite Element Analysis of 3D Eddy Currents", IEEE Trans., MAG-25, 4 (1989), pp. 3145-59. [Bi] O. Biro, K. Preis: "Finite Element Analysis of Three-Dimensional Eddy Currents", IEEE Trans., MAG-26, 2 (1990), pp. 418-23. [BE] C.F. Bryant, C.R.I. Emson, C.W. Trowbridge: "A Comparison of Lorentz Gauge Formulations in Eddy Current Computations", IEEE Trans., MAG-26, 2 (1990), pp. 430-3. [B&] R. Dyczij-Edlinger, I. Bardi, O. Biro, K. Preis, K.R. Richter: "A Deterministic Approach to the Analysis of Three-Dimensional Waveguide Configurations by Finite Elements and Mode Matching", IEEE Trans., MAG-28, 2 (1992), pp. 1235-8. [P&] K. Preis, I. Bardi, O. Biro, C. Magele, G. Vrisk, K.R. Richter: "Different Finite Element Formulations of 3D Magnetostatic Fields", IEEE Trans., MAG-28, 2 (1992), pp. 1056-9. [RR] Z. Ren, C. Li, A. Razek: "Hybrid FEM-BIM Formulation Using Electric and Magnetic Variables", IEEE Trans., MAG-28, 2 (1992), pp. 1647-50. [R&] W. Renhart, C.A. Magele, K.R. Richter, P. Wach: "Application of Eddy Current Formulations to Magnetic Resonance Imaging", IEEE Trans., MAG-28, 2 (1992), pp. 1517-20.

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[Note of 2007.] This raises further issues, of which I was not well aware when this paper was written, later mentioned in an updated remake: A.B.: " 'Hybrid' electric–magnetic methods in eddy-current problems", Comp. Methods in Appl. Mech. and Engng., 1 7 8 (1999), pp. 383-91.