Compumag 91 - CiteSeerX

Edge-element Computation Of The Force Field In Deformable Bodies IEEE Trans., MAG-28, 2 (1992), pp. 1263-6.

É

A. Bossavit

A b s t r a c t—We present an edge-element method (h-formulation) for the computation of force fields in full generality: deformable bodies, non-linear b--h laws, presence of magnets. It gives not only integrated quantities like resultant or torque, but l o c a l force. The main ingredients are: (1) Lagrangian approach ("co-moving" mesh, and Maxwell's equations expressed in "material" form), (2) Virtual work principle (force as the derivative of coenergy w.r.t. configuration), (3) Edge-based degrees of freedom, which can be interpreted as magnetomotive forces (mmf) along the edges. I. INTRODUCTION

The standard tools for the computation of force fields (Maxwell's tensor, j × b) are not completely satisfactory. The integration of Maxwell's tensor yields global quantities, and its use is hampered by well known difficulties when the surface of integration crosses a small airgap. Problems with the j × b formula are more subtle. Numerical methods generally treat differently the two equations curl h = j and ∂ tb + curl e = 0. When one of them is exactly satisfied, the other one is only enforced in weak form. For this reason, the two fields in the j × b formula are not consistently approximated. To compound these numerical difficulties, there is some hesitation in the eddy-currents community about the very meaning of local force: some argue that only integrated, global quantities make sense. Worse, the wrong belief that force is either q h or (j + j a) × b, where q and ja are equivalent (fictitious) charges or currents, is not eradicated. We propose here a line of approach which overcomes these problems. The discussion is based on the "edge-element h–ϕ formulation" [1], for definiteness, but most ideas transpose easily to methods like T–Ω , or a–ψ, etc. However, the adequation of edge-elements to the present theory of force will be apparent, and may be conceived as another point to their credit. We shall discuss the matter directly at the level of the discretized equations. After a description of how the finite element mesh relates to moving matter, we put forward a system of differential equations, whose solution is the set of "edge magnetomotive forces (mmf)". Coenergy is a quadratic function of the latter and of the mesh-nodes location, hence an easy computation of the force field. Finally, we show that the proposed discrete system does constitute a discretization of eddy-current equations.

Manuscript received July 7, 1991. In red, typos fixed later.

II. SYSTEM DESCRIPTION

Consider a bounded domain of space D, containing flexible bodies, which may be conductive and/or magnetizable, and air. If D is large enough, one may neglect whatever happens outside, which we do. (The size of such a domain can be very much reduced by the use of integral methods which take care of the outer-field reaction, but we ignore this issue.) One (at least) of the conductors is a coil that supports a given time-dependent current distribution. We assume the existence of a reference configuration. A material particle will occupy a definite position in the reference picture, say x, and its changing position in the actual configuration will be given at time t by u(t, x), where u(t, .) (or just u, with time-dependence understood) is a mapping from D into itself. We postulate that u(t, x) = u(x) if x is in the boundary of D, which means that no movement exists outside this domain. We next assume a tetrahedral finite-element mesh of the reference configuration. Let N, E , be the sets of nodes and of edges of this mesh. There are N nodes, E edges, F faces. Call λ n (x) the barycentric coordinate of a material point x with respect to node n: thus λ n = 1 at node n, = 0 at other nodes, and is continuous and piecewise affine. The kinematics of the situation will be described by giving the position of each of the N nodes as a function of time: un (t), where index n spans N. We call u(t), or just u, the (3 × N)-vector {u n : n ∈ N } of all node positions at time t. Now the position at time t of the material point x is given by u(t, x) = ∑n ∈ N un (t) λ n (x). (Note that this applies to nodes in the air as well: these are not necessary to the kinematical description, but will be necessary to the computation of the field. The way they move, however, is left at the discretion of the programmer.) Let us turn to the description of the electromagnetic field. Each edge e of the reference mesh is attributed a scalar quantity he(t), which we shall interpret as the magnetomotive force (mmf) along the material line which supports edge e. (Reasons for such an interpretation are coming.) The actual field h at time t will be given by the following, unavoidably complicated procedure. First define the edge-element basis in configuration u. For this, let u wn be the function of spatial coordinates such that u wn (u(x)) = λ n (x). This is called the "push-forward" of λ n by the mapping u. Then we define u-dependent edgeelement bases. This is done as usual: to each edge e is

-2u

u

u

u

u

assigned the vector field we = wn grad wm − wm grad wn , where n and m are the nodes connected by edge e in the mesh. Let us stress that the u wes depend on time via u. Now, the actual magnetic field at time t will be h(t) = ∑e ∈ E he(t) u(t)we. Note that its circulation along the image of edge e by u, which is the mmf between the points un (t) and u m(t) along this line, is equal to he(t), hence the interpretation of these scalar quantities as mmf's. It is absolutely crucial not to confuse the actual magnetic field h(t) and the vector of edge-values h(t), or "mmf-vector": the former is configuration-dependent, the latter is not. (Mathematically speaking, they are two objects of different kinds: h(t) is a bona fide vector field, whereas h(t) is what is called a "cochain" in differential geometry.) III. DISCRETE EDDY-CURRENT EQUATIONS, WITH MOVEMENT

What follows will be easier to understand if one keeps in mind some facts about the standard h-formulation of eddycurrent equations with non-moving bodies. In strong form, ∂t (µh) + rot(σ−1 rot h) = 0 where σ > 0,

(1)

rot h = js(t) where σ = 0,

(2)

s

where j is the given source-current density. The corresponding weak form is: find h(t) ∈ Hs(t) such that ∫ ∂t (µh) · h' + ∫ σ−1 rot h · rot h' = 0 0

(3)

s

for all test-fields h' ∈ H , where H (t) is the class of fields satisfying condition (2) at time t, and H 0 those for which rot h = 0 where σ = 0. By substituting in this an expansion of h as a linear combination of edge-fields, and successively substituting each of the E edge-fields for h', one obtains a system of algebraic-differential equations for the E degrees of freedom (DoF). This system has the form: ∂t (A h) + B h = Ct a,

(4)

C h = js(t).

(5)

Here h is the vector of edge-DoF, js(t) the vector of current intensities through "impervious" faces of the mesh (i.e., those in the air, or in media where σ = 0, or bounding such media) and a a Lagrange multiplier corresponding to the constraint (5). The dimension of a is the number of impervious faces, say F0 . The symmetric matrices A and B are obtained by the standard assembly process. The rectangular (F 0 × E)-matrix C, whose non-zero entries are 1 or − 1, is the discrete analog of curl in the non conducting region: (C h)f is the flux of rot h through face f. Users of the edge-element method won't stop at (4) (5), but will either introduce nodal (scalar) DoF ϕn in the region where σ = 0 (hence the "h–ϕ" formulation, with edge variables in conductors, node-variables elsewhere), or look for a spanning-tree in the mesh graph, in order to get rid of a and obtain a genuine differential system. We ignore these issues here.

The procedure we now propose, and which is valid for moving bodies, is formally the same. The only difference is that A and B will depend on the (changing) configuration, and thus will have to be reassembled at each time-step. We denote by µu and σu the push-forwards of µ and σ: µu (u(x)) = µ(x), and the same for σ. We'll set A = A(u) and B = B(u) (not A(u) and B(u): there is a slight conceptual difference, worth keeping in mind.) The entries of these matrices, indexed by pairs of edges e and ε, are Ae ε(u) = ∫ µu u we · u wε, Be ε(u) = ∫ σu

−1

rot u we · rot u wε.

Beware the integrals are over physical space, not over the reference configuration. This is well seen by considering the assembly process. Assembly consists in obtaining Aeε(u) and B eε(u) as sums of the following contributions of each individual tetrahedron: ATe ε(u) = ∫u(T) µu u we · u wε,

(6)

BTe ε(u) = ∫u(T) σu

(7)

−1

rot u we · rot u wε.

These integrals are over the image of tetrahedron T by u, i.e., over the portion of space occupied in configuration u by the matter—the same piece of matter, all the time— contained in T. For convenience, one may compute these terms over T, the reference tetrahedron (and in that case the Jacobian of u will intrude), but (6) (7) well show how A and B explicitly depend on u, hence on time. This time-dependence is the reason why no term looking like v × b appears in the following, which we claim is the right discretized form of the eddy-currents equation, in presence of motion, as described by a "trajectory" t → u(t): ∂t(A(u(t)) h) + B(u(t)) h = Ct a, s

C h = j (t).

(8) (9)

s

(Beware that j f(t) is the intensity through the material face f, and note that C is the same as before.) Indeed the velocity makes itself being felt in (8), because timedifferentiation will apply not only to h but also to A, which depends on u. Eq. (8), or rather, what it becomes after applying the h–ϕ or the spanning-tree technique, will be solved by CrankNicolson, or if needed by a θ-scheme with θ > 1/2. This may be necessary in case of fast variations of u(t) (high velocity), if the alternative of reducing the time step is not acceptable. (There is no reason to expect to avoid, by the present Lagrangian approach, the well-known numerical difficulties experienced in Eulerian computations when the velocity is high, to which "upwinding" is the standard remedy. They just will appear in a different form.) IV. FORCE COMPUTATION

We assume that the dynamics of the flexible bodies is given, and that some structural dynamics code is available which

-3-

can simulate this dynamics: at each time step, given the present-step u and electrodynamic forces as input, it yields the next-step value of u as output. Thus the key ingredient in coupling (8) (9) with the dynamic equations is the computation of forces at each time step. We now show how to do this. Let us call magnetic coenergy of the mmf-vector h in configuration u the quantity Φ(u, h) = 1 /2 ∫ µu | h|2 , where h = ∑ e ∈ E he u we. By the very definition of A above, this is equal to (A(u) h, h)/2, where ( , ) denotes the scalar product in IR E. Note that the variables on which Φ depend are the vector of nodal positions u and the vector h, not the components of the vector field h. Let δu be a virtual displacement. The virtual work done by electromagnetic forces in such a displacement is a linear function of δu, so it can be written as < F, δu> = ∑n ∈ N F n · δun . (In this expression, δun and Fn are ordinary vectors of IR 3 , and "·" is the dot-product.) We shall come back in a moment to the physical meaning of the Fn s. We are entitled to call F "force", since force in mechanics is always that: the mapping which gives virtual work from virtual displacement. It can be proved (see [2]) that < F, δu> = ∂u Φ(u, h). In words: force is the derivative of coenergy, when expressed as a function of nodal positions and edge-mmf's, with respect to the former. Compare this with the standard rule, which says that "force is the derivative of coenergy, while keeping currents constant, with respect to configuration". We have no need for the proviso about currents being "kept constant". This is implied, because 1°- to differentiate w.r.t. to one variable (here u) does mean estimating the variation of the function while keeping other variables (here h) constant, 2°- keeping the mmf's constant along material edges means that their sum over a closed edge-path (taken with signs, + or −, depending on the orientation of each edge) is kept constant, 3°- this sum is nothing else than the intensity through the material surface bounded by this path, by Ampère's theorem. So maintaining unchanged the mmf's amounts to freezing intensities through all conceivable material surfaces at their present values (and this is the only sensible meaning that can be ascribed to the expression "currents kept constant"). So our rule extends the old one and gives it mathematical standing. Since coenergy is (A(u) h, h)/2, the needed differentiation w.r.t. u amounts to taking the derivative of A, and thus of the terms A Teε(u) of (6), w.r.t. u. This must be done at each time step, but the amount of arithmetics necessary can be kept very low (comparable to what is necessary to assemble A itself, because computing ∂ u A is another assembly process, as already remarked in [3]). This delicate exercise in programming has been performed by Dr. Ren [4]. Remark. Matrix A is also a function of µ. If µ depends on stress, thus on u, its derivative w.r.t. u will add to the force considered up to now, which is due to geometrical factors (the distortion of matter as times goes on), a kind of

constitutive term, which depends on material properties. The clear distinction thus achieved between these two kinds of force is a point in favor of the present method. Note also that taking into account non-linear b–h laws is no extra theoretical trouble in this theory: ∂t(A(u(t))h) should be replaced with ∂tβ(u(t), h), where β is non-linear. (The numerical side of this question is of course less pleasant.) Note at last that other parameters than u (temperature, for instance) could be necessary to a full description of the system, and that one may always incorporate them into u. So one can obtain the Fn at relatively low cost. Now, what about the force field? Let us call this f, so virtual work is ∫ f · δ u. We shall be satisfied if the force field is expressed by linear interpolation between nodal values, i.e., u as f = ∑ n ∈ N fn wn , where the fn s are node-based ordinary vectors. The virtual displacement field is itself such an interpolation: δu = ∑ n ∈ N δun u wn . Since ∫ f · δ u = u ∑n ∈ N F n · δun , we have the identity ∫ (∑n ∈ N fn wn ) · u wm = F m (for all node-indices m ∈ N ), which shows the connection between the vectorial values of the force field at the nodes (the f n s) and what can be called the generalized nodal forces (the F n s). One should very carefully avoid any confusion between these two sets of quantities. Remark. The integral ∫ f · δu is over the whole space. Does it mean that fn could be non-zero at nodes n in the air? The perhaps unpleasant answer is, yes indeed! But such values of fn have a residual character: they are numerical errors, and one may argue that it is a good thing to have them computed, as an indication of how accurate the other fn s (those in matter) are. These (vectorial) values can also be interpreted as indicators about how to improve the mesh. Numerical tests of this method are reported in [4]. They have been performed on TEAM workshop Problem 6 (hollow sphere in a transient field), for which an analytical solution is available. The accuracy, on a given mesh, is significantly better with this method than by using the j × b formula. V. SOME JUSTIFICATIONS

Now is the time to answer some legitimate questions. First: is it true that, as claimed above, (8) (9) do discretize the right equations? Second: is ∂ u (A(u) h, h) the right force field? (a corollary, if the first question is positively answered). The shortest and most natural way to discuss this is via an expression of Maxwell's equations in "material" form, which calls for the use of some differential geometry: differential forms, the Hodge operator, and the notion of Lie derivative. There is no place here for a tutorial, so we shall only give directions, that should allow the informed reader to check the validity of our above assertions. Details are in [2]. (The unavoidability of this heavy mathematical apparatus is the reason why the present section is placed here at the end, in a reversal of the logical course.)

-4-

A notational point: By "f ∈ U → V", we mean that f maps the set U (or a part of it) into the set V.

imation is consistent with ignoring "electric" forces, i.e., those between electric charges, which we do here.)

Let us distinguish the material manifold X (the above reference configuration of D) and Euclidean space E3 . We shall call placement the mapping u ∈ X → E3 . Let u * ∈ TX → TE3 (≡ E3 ) be the induced tangent mapping, or "push-forward", and u* ∈ T*X → T*E3 the pull-back. A placement u induces a metric on X: if ξ and η are two tangent vectors at x, their scalar product is

Next step is to put these equations in weak form, which can be done by the following heuristic device: follow the same steps as when going from (1) (2) to (3), but whenever the scalar product by a test-field h' would be called for, replace it by a wedge-product of forms: ∫X B ∧ H' instead of ∫Ε3 b · h', etc. One thus obtains the Lagrangian counterpart of Pb. (3).

gu (ξ, η) = u* (x)ξ · u* (x)η. The associated Hodge operator on X is denoted by ∗ u . The Hodge operator on E3 is denoted by ∗ .

Now, one substitutes linear combinations of Whitney one-forms we (the same as the above edge-elements, but considered as one-forms living on X) for H and H', and the result is precisely (8) (9).

If t → u(t) is a smooth trajectory (i.e., a one-parameter family of placements), we denote by v = ∂ tu the vector field (in E3 ) of material velocities. Thus, v(u(x), t) is the velocity at time t of the material particle x. Let V ∈ TX be such that u* V = v. We denote differential forms living on X with small capitals ( H, B, J, ...), in contrast with h, b, j, ..., living in E3 . Boldface: h, b, j, etc., is reserved for the corresponding vector fields. Recall the definition of the Lie derivative: if ω is a differential form (DF) on E3 , L v ω = (i v d + di v )ω, where d is the exterior derivative, and i v the inner product, iv ω = {ξ2 , ..., ξ p } → ω(v, ξ2 , ..., ξ p ). One has u*d = du*, LVu*ω = u*Lv ω , and ∗ u u* = u* ∗ . Now, eddy-current equations in material form are, in the linear case, ∂tB + dE = 0, dH = J, B=

µ ∗ u H,

J

= σ ∗ u E + Js(t),

(10) (11)

s

where J (the source-current) is a given time-dependent two-form on X. The link between this Lagrangian expression and the more familiar Eulerian one is via the Lie derivative, because of the following relation: ∂tu* = u*Lv , which implies ∂ tB = ∂t (u*b) = u*∂ tb + u*L v b. One then defines e, not as (u*)−1E, but such that E = u*(e − iv b), hence, with B = u*b, H = u*h, J = u*j, Js = u*j s, the Eulerian equations, still in terms of DFs: ∂t b + de = 0, dh = j, b = µu ∗ h, j = σu ∗ (e − ivb) + js, with µu (such that µu* = u*µu) and σu the same as above. In the language of vector fields, these equations are just ∂tb + rot e = 0, rot h = j, b = µu h, j = σu (e + v × b) + j s, and this validates our claim that (10) (11) are the right equations to consider. (This, of course, if one accepts these eddy-current equations, i.e., without displacement currents, as the right ones. Let us mention incidentally that doing this approx-

During this process, one will remark that the above expression for the coenergy was the discrete form of this: Φ(u, H) = (∫X µ

H

∧ ∗ u H)/2.

Let us perform the differentiation w.r.t. u, in the case in which µ = µ0 all over. The virtual power is then ∂t ∫X µ/2

H

∧ ∗ u H ≡ ∂t ∫E3 µ/2 h ∧ ∗ h

= ∫E3 µ ∗ h ∧ ∂t h = − ∫E3 µ ∗ h ∧ Lv h = − ∫E3 b ∧ iv j ≡ ∫E3 j × b · v, so we do recover the j × b formula. Only the relative novelty of the differential forms setup makes this appear intricate. In fact, computations are easier and cleaner in this language. When µ is allowed to be a function of the material point (but still independent of the magnitude of H), a similar computation gives, at the bottom line, f = j × b − |h|2 /2 grad µ, a reassuring result, since it coincides with what serious books indicate [5]. REFERENCES [1] A. Bossavit and J.C. Vérité: "A mixed FEM-BIEM method to solve 3D eddy-current problems", IEEE Trans. Magn., vol. 18, 2, pp. 431-35, 1982. [2] A. Bossavit: "Eddy-currents and forces in deformable conductors", in Mechanical Modellings of New Electromagnetic Materials (Proc. IUTAM Symp., Stockholm, April 1990), R.K.T. Hsieh, Ed., Amsterdam: Elsevier, 1990, pp. 235-42. [3] J.L. Coulomb: "A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness", IEEE Trans. Magn., vol. 19, 6, pp. 2514-19, 1983. [4] Z. Ren and A. Razek: "Local force computation in deformable bodies using edge elements", IEEE Trans. Magn., vol. 28, 2 (1992), pp. 1212-5. [5] F.N.H. Robinson, Macroscopic Electromagnetism , Oxford: Pergamon Press, 1973.