Issue 
Eur. Phys. J. Appl. Phys.
Volume 85, Number 1, January 2019



Article Number  10901  
Number of page(s)  7  
Section  Physics of Energy Transfer, Conversion and Storage  
DOI  https://doi.org/10.1051/epjap/2018180161  
Published online  14 January 2019 
https://doi.org/10.1051/epjap/2018180161
Regular Article
Modeling stationary moving medium by static magnetoelectric material
Budapest University of Technology and Economics, Egry J. 18, 1111 Budapest, Hungary
^{*} email: gyimothy@evt.bme.hu
Received:
25
May
2018
Received in final form:
14
September
2018
Accepted:
12
November
2018
Published online: 14 January 2019
The electromagnetic equivalence of a stationary moving medium to magnetoelectric materials is studied. The equivalent material characteristics of the medium at rest is obtained in terms of the diffusion tensor coefficient of the governing partial differential equation. Special attention is paid to the transition condition of field quantities on the boundary of the moving medium; it is found that the nonmoving magnetoelectric equivalent model must be supplied with surface sources. The method is demonstrated through examples and verified by finite element simulation.
© EDP Sciences, 2019
1 Introduction
Electromagnetic phenomena involving moving medium are frequently analyzed in practice. In a subclass of such problems, the moving domain can be taken unbounded in the direction of motion and its cross section invariant to the movement (e.g., translation or rotation), that is, the geometry is stationary. A nonexhaustive list of topical applications includes magnetic levitation and braking [1], electromagnetic launchers [2], and Lorentzforce velocimetry [3]. In turn, some wellknown academic examples are the homopolar generator (Faraday disc), the magnet falling in a tube [4], and the − rather exotic − Wilson & Wilson experiment [5].
In the physical description of a general movingbody problem, the effect of motion appears in a timedependent geometry, as well as in the motional terms of constitutive relations and interface conditions. In the specific problem type considered, however, the first one is obviously not the case. Moreover, the interface conditions for fields reduce to their original form at rest [6]. Hence, the main place where motion appears is the set of constitutive relations, which are normally related to material characteristics.
Moving media usually generate a kind of magnetoelectric coupling. It is known that similar coupling is realized by specific composite metamaterials at rest. This similarity has already been exploited in transformation optics (TO) among others to raise the illusion of motion [7]. In principle, any linear transformation of the electromagnetic fields (such as the Lorentztransform) can be imitated by specific − often fictitious − bianisotropic “field transforming” materials [8].
The goal of this work is to study the electromagnetic equivalence of moving media and field transforming materials, with its practical utilization in mind. A method is introduced later by which the equivalent material characteristics of the medium at rest can be obtained in terms of the diffusion tensor coefficient of the governing partial differential equation (PDE); the method is verified by numerical simulation. On the other hand, the realization of such materials are not dealt with; this may be part of future work.
The main idea behind the present work has been already outlined in a conference digest [9] aiming at developing a specific numerical technique for the finite element (FE) method. The fact is that a standard FE software does not facilitate the simulation of moving medium, at least not at the toplevel user interface. In more advanced programs (like, e.g., [10]), a “motional term” for conductors is provided, but not for moving magnets or dielectrics. At the same time, most FE programs are capable of treating inhomogeneous and anisotropic media outofthebox. Therefore, the original idea was to convert the motional term into a diffusive material characteristics, so that the user who is committed to a particular software can define such problems without having to implement the motional term at a lower access level of the software.
Another goal envisaged in reference [9] was to improve the convergence of FE computations. It is common to problems involving moving medium that the governing secondorder partial differential equation, as derived from Maxwell's equations, contains a first order convection term representing the motion effect. If this convection term dominates, the FE formulation suffers from numerical instability as spurious oscillating components occur in the solution, see e.g., [1,2,11]. However, if one could imitate the motion with materials, this convection term would be missing from the PDE, so we expect that we could get rid of the instability too (eventually it proved to be wrong).
It was soon realized, however, that the developed conversion method goes beyond being a mere numerical trick. In fact, the FE implementation of the model with nonmoving medium must be “tweaked” by a specific interface condition in order to obtain perfect equivalence. Technically it derives from the weak form of the PDE, which happens to involve an additional surface source term. Nevertheless, it is believed that such surface sources must be provided in a real device too, which is used for imitating motion by static media.
Let us recapitulate the novelty of this work comparing with others also dealing with the electromagnetic equivalence between static bianisotropic and moving media, like references [7,8]: (i) a direct approach is proposed based on rewriting the convection term of an advectiondiffusion equation into an anisotropic diffusion term, without referring to material parameters, (ii) particular attention is paid to the continuity and jump conditions on the interface between moving and nonmoving media, (iii) issues related to FE implementation are dealt with as well.
The remainder of the paper is organized as follows. In the theoretical section, the relevant interface conditions and constitutive relations are reviewed first. Then the new technique is introduced by which the motional term of the PDE is converted and incorporated into the diffusion term of that PDE. After that the interface conditions are revisited by means of the weak formulation. For simplicity, only 2D problems with inplane velocity fields are investigated, but the main findings and statements may be generalized to 3D problems as well. This part is followed by a practical section, in which the FE solution of two test problems − involving rigid translation and rotation, respectively − are presented.
2 Theory
2.1 Movingbody interface conditions
Let V _{1} represent the domain of a rigid body whose motion is characterized by a velocity field v . The domain of the surrounding rest medium is V _{2}. On the boundary of the moving body, S _{12}, the following electromagnetic relationships (interface, continuity, or jump conditions) ensue from the integral form of Maxwell's equations [6]:(1) (2) (3) (4)where n is the surface normal vector of S _{12} and [ ⋅ ] denotes the jump of a given field quantity in the direction of the normal vector upon that surface. The vector fields E , B , H , D stand for the electric field, magnetic flux density, magnetic field, and electric displacement, respectively, measured in the rest frame (or laboratory frame). Finally, J _{s} and ρ _{ s } are the surface current density and charge density, respectively.
Let us now consider stationary movement − as it was defined above, and cf. Figure 1 – in the absence of surface sources. Since v and n are orthogonal at S _{12}, equations (1)–(4) simplify to(5) (6) (7) (8)
These are completed with the interface condition for the electric current density J , which follows from charge conservation and equation (8):(9)
Fig. 1 Interface between stationary moving and nonmoving domains. 
2.2 Constitutive relations in moving medium
In a sourcefree linear isotropic medium, the following relations apply:(10)where the primed vector fields are all measured in the comoving frame of the medium. Symbols ϵ, μ, and σ stand for the permittivity, permeability, and specific conductivity of the medium, respectively.
By taking the Lorentztransform of the fields, one can easily obtain (see, e.g., the derivation in [12]) the fully relativistic form of the constitutive relations as seen in the laboratory frame:(11) (12) (13)where v is the velocity of the medium, c _{0} the speed of light in vacuum, and the Lorentz factor. When deriving equation (13) we assumed that the volume density of free charges in the comoving frame vanishes, i.e., ρ ^{′} = 0.
In the lowvelocity limit, v = c _{0}, one can get a quasirelativistic firstorder approximation of the constitutive relations by setting ,(14) (15) (16)
Equations (14) and (15) represent one particular form of the wellknown Minkowski relations [6]. Lately it was pointed out that, under certain circumstances, two distinct lowvelocity limiting cases of equations (11) and (12) can be identified [12]; a socalled magnetic limit:(17) (18)and an electric limit:(19) (20)
Many practical moving body problems involve motioninduced eddy currents in conductors, where only equation (16) is considered. But there are some particular problems, such as the abovementioned Wilson & Wilson experiment or the plain wave propagating in a drifting magnetoionic plasma [13], in which the other constitutive equations play a role too. The decision, whether to use equations (14) and (15), equations (17) and (18), or equations (19) and (20), respectively, must be based on the dimensional analysis of the problem at hand [14].
Strictly speaking, the above relationships hold in the context of two inertial frames only. However, they can be extended to accelerating (e.g., rotating) frames or even to more general velocity fields based on the concept of “instantaneous inertial frame,” which is valid up to very high velocities [6]. This poses some limitation − albeit rather loose − on the velocity fields that can be treated by the method.
2.3 Transformation of the PDE
Hereinafter, we confine ourselves to 2D planar problems with inplane velocity field. More precisely, we consider a configuration exhibiting translational symmetry w.r.t. the z direction, and a related electromagnetic problem, the computational domain of which, , can be reduced to the transversal x − y plane. Provisionally, we set aside from moving and nonmoving parts, and extend the velocity field v (with v _{ z } = 0) to the whole of V. The PDE covering most of such problems has the general form(21)in which u = u(x, y) ∈ C ^{2}(V) is the unknown scalar field (can be a potential or selected component of a physical field); symbols a, bv , and c are traditionally called the absorption, convection, and diffusion coefficient − the terms they affect have similar names − and f is the source term. The convection term is linear in the velocity vector v . Note that ∇u = (∂ _{ x } u, ∂ _{ y } u) and v = (v _{ x }, v _{ y }) represent 2D vectors in this context, and that c ∈ C ^{1}(V) is required. For brevity, we call equation (21) hereinafter the PDE of the “moving model.”
The objective is to find another PDE without the convection term b v ⋅ ∇ u, so that it is formally equivalent to equation (21). One could certainly deduce the desired bianisotropic material parameters first (as in reference [8]), put them into the Maxwell's equations, and finally derive the appropriate PDE. We choose a shorter way instead: taking immediately equation (21), the convection term is managed to assimilate into the secondorder term; due to linearity we must get the same (or an equivalent) PDE by this approach.
Let us introduce matrix called the “diffusion tensor” so that equation (21) is replaced by(22)
We will refer to equation (22) as PDE of the “equivalent static model.” It was found − based on preliminary works and intuition − that the desired form of is(23)where is the unit matrix, is the 90 counterclockwise rotation matrix, and g = g(x, y) ∈ C ^{1}(V) is an appropriately chosen scalar function. Indeed, if we substitute into the secondorder term of equation (22), we get(24)where we invoked the (scalar valued) 2D cross product, which is defined for any two vectors as p × q = p _{ x } q _{ y } − p _{ y } q _{ x }. Comparing the firstorder derivative terms in equations (21) and (24) we obtain the criteria of equivalence:(25)
These still leave us some freedom in the selection of g, but apparently, its variation (inhomogeneity) within the moving domain is essential. We will return to this point by examining two specific cases in Section 2.5.
In the “traditional” PDE of a physical problem, the diffusion tensor coefficient usually contains material parameters in some expression. Actually, this is the reason why we are allowed to talk about the imitation of movement by specific material, without referring to its concrete parameters though. Certainly, from one may derive appropriate (not necessarily unique) electric and magnetic material properties (which are usually found as inhomogeneous and bianisotropic) but this is not our concern.
2.4 Continuity and weak formulation
We have seen that equations (21) and (22) can be brought in formal correspondence as long as coefficients a, bv , and c in equation (21) are sufficiently smooth all over the domain. However, this is typically not the case on material interface where they change abruptly and, as we will see, this fact introduces some difference between the two models. Therefore, let us consider again the configuration in Figure 1 with , and .
The discontinuity of material properties and the velocity field can be tackled in a way that the validity of the PDE is restricted to the interior of each material domain and, at the same time, jump conditions like equations (5)–(9) are prescribed along their interfaces. In the studied 2D problems governed by scalar PDE, usually only two out of equations (5)–(9) are relevant, which can be written as(26) (27)(note the scalar form of equation (26) in 2D context). The correspondence of equations (26) and (27) to the set of five equations (5)–(9) depends largely on the choice (i.e., physical meaning) of u, but an important point in the latter is that u remains C ^{0} continuous even on the material interfaces; thus, in this case equation (26) satisfies automatically.
Continuity issues can be better accounted for in the socalled weak formulation, where equation (27) appears as a “natural condition” [15]. The weak form of equation (21) is(28)for u = u(x, y) ∈ H ^{1}( V), with w being any test function on V. Without loss of generality, assume that we have a Dirichlet problem, so that we can better focus on the interface condition on S _{12}; in this case, .
Let us break down equation (28) for V _{1} and V _{2}, respectively, integrate by parts, apply the divergence theorem for each, and merge (these steps can be found in many textbooks, e.g., [15]). We arrive at(29)where we find that the surface integral vanishes at the external boundary of V due to w_{∂V } = 0. Note that the jump of n ⋅ (−c ∇ u) appears in the surface integral over S _{12} and, obviously, omitting this term in equation (29) implies equation (27).
Now, if we carry out the same procedure with PDE equation (22) of the static equivalent model, we get(30)Consequently, in this case, the continuity of is implied on material interfaces as natural condition, which contradicts equation (27) though. Let us expand the jump of this expression by substituting equation (23) for :(31)
Taking into account equation (27) we get(32)
This means that for restoring equation (27) in the frame of equation (30) we must explicitly prescribe the interface condition of equation (32) on S _{12}. Such a jump can be interpreted as a surface source (edge source in 2D). It should be pointed out, however, that this source is definitely not due to the motional effect. On the contrary, it is the “price” for substituting moving media with static material.
Further remarks on continuity: (i) the weak forms of equations (29) and (30) merely require the PDE coefficients and the velocity field, respectively, to be integrable on V, (ii) in the FE implementation, the continuity of u, which is a sufficient condition of equation (26), is usually enforced exactly by the discretization scheme. On the other hand, omitting the surface term in equation (29) causes equation (27) to satisfy only approximately depending on the quality of FE discretization.
2.5 Translation and rotation
Two simple cases − which have the most practical interest indeed − are rigid translation and rotation, respectively, of homogeneous media. The latter implies that coefficient b in equation (21) is constant.
In the case of translation, i.e., when v is homogeneous in V _{1}, one can easily deduce a suitable form of function g from the criteria of equation (25):(33)where K is an arbitrary constant. After introducing the position vector r = (x, y) and setting K = 0 for simplicity, we obtain the tensor as(34)Notice that our freedom in the choice of the origin is preserved, because any displacement in r cancels once the second derivative is taken in (22). The corresponding interface condition according to (32) is(35)The reason why the “jump” is not indicated on its righthand side is that , which follows from and K = 0.
In the rotating case, v is expressed with the angular velocity Ω and the position vector as v = (− Ωy ; Ωx), provided that (x, y) = (0, 0) lies on the rotation axis. The appropriate function g that can be deduced from equation (25) is(36)where the arbitrary constant term was omitted for simplicity. Accordingly, the diffusion tensor and the necessary interface condition are(37) (38)respectively, where we utilized again that holds.
3 Test problems
The method is demonstrated below on two test problems involving rigid translation and rotation, respectively, the solutions of which is known in reference [6]. We have solved both the moving model and the equivalent static model of each problem by FEM, using Comsol Multiphysics [10]. In the software we chose the “Coefficient form PDE Interface,” through which a superset of the PDEs (21) and (22) can be solved via the weak formulation. Field jumps like equations (35) and (38) can be prescribed directly by “Flux/Source”type interface conditions in Comsol.
Before presenting the results, we emphasize a common feature. It is not only that the solution of the two models (moving and static) matched in each case but already their respective discretized equation systems agreed “literally.” Hence, there is no point in showing the comparison of the results regarding the two models. This highly conforms their equivalence but, at the same time, no advantage of the static equivalent model can be expected from the point of view of the numerical behavior.
3.1 Sliding contacts
Let us consider a cut down problem of sliding contacts (see reference [6], p. 137). A conducting nonmagnetic slab is moving between two electrodes with constant velocity, while a fixed dc current, I, is injected through it (Fig. 2). Since the current distribution in the slab is stationary from the point of view of the electrodes, it is worth to do the modeling in the rest frame of the latter.
Following the reasoning of reference [6] one can see that the inplane current distribution can be expressed with the perpendicular component of the magnetic field as J = (∂ _{ y } H _{ z }, − ∂ _{ x } H _{ z }), and that for H_{z} the following scalar PDE can be written:(39)
The PDE coefficients regarding equation (21) are thus c = 1, b = μ _{0} σ, a = 0, and f = 0. The unknown variable is u ≡ H _{ z }. The injected current is prescribed by appropriate boundary conditions (not detailed here). It can be seen that equation (26) expresses equation (9), and equation (27) corresponds to equation (5) in this case.
In the FE solution of the moving model, we used the discretized form of equation (29) with the surface integral term omitted, while in the equivalent static model we used equation (30) with equations (34) and (35) substituted. In the present example, an aluminum slab of thickness 10 mm is dragged with a velocity of 6m/s. The streamlines of J (contours of H_{z} ) are plotted in Figure 3.
Numerical instability had been reported about for a similar problem in reference [2] if the Péclet number, P _{ e } = μ _{0} σv _{ x } h/2 exceeds the value 1, where h is the length of an element in the direction of velocity. Indeed, the author experienced the same instability with both models at higher velocities (not shown here).
Fig. 2 The studied problem of sliding contacts. 
Fig. 3 Current distribution near the contacts at v = 6 m/s. 
3.2 Scattering from a rotating dielectric cylinder
As a second example, let us take a rotating dielectric cylinder illuminated by an Spolarized plane wave (see reference [6] p. 298, cf. Fig. 4). We can decompose the zdirected electric field into incident and scattered parts, E = E _{ i } + E _{ s }, where E _{ i } = exp(− jk _{0} x) V/m is the incident wave, k _{0} = ω/c _{0} is the vacuum wave number, ω = 2πν is the angular frequency with the frequency ν, and j is the imaginary unit. Starting from the Maxwell's equations,(40)with the constitutive laws, equations (14) and (15), and applying usual algebra (details are omitted), we arrive at the scalar PDE,(41)Comparing this with the PDE template, equation (21), the unknown variable is u ≡ E _{ s }, and the respective coefficients are(42)
We see that equation (26) corresponds to equation (5), and equation (27) to equation (7), respectively. In the FE solution of the moving (rotating) model, we used the discretized form of equation (29) with the surface integral term omitted, while in the equivalent static model we used equation (30) with equations (37) and (38) substituted.
In this example, r _{0} = 0.3 m, ϵ _{1} = 10ϵ _{0}, Ω = 10^{7} rad/s, and ν = 1 GHz. The peripheral speed of the cylinder is Ωr _{0} = 0.01c _{0}, which is chosen such high because a relativistic effect is studied, but the firstorder approximation used in equations (14) and (15) is still valid. The scattering diagram is plotted in Figure 5 and compared with that of a standing cylinder, i.e., when Ω = 0. The distortion of the pattern toward the direction of rotation is apparent.
Fig. 4 Rotating dielectric cylinder illuminated by an Spolarized plane wave. 
Fig. 5 Farfield scattering pattern of E_{s}  (normalized values) for Ω = 0 rad/s (solid line), and Ω = 10^{7} rad/s (dashed line). 
4 Conclusions
In this article, the hypothesis was verified, that the effect of motion can be converted to rest material characteristics in the case of bulk media with stationary geometry, and its consequences were examined.
Instead of deriving the adequate material parameters, a formal way was found to directly modify the PDE coefficients. Also it was shown that an additional surface source has to be prescribed for the modified PDE at the interface between moving and nonmoving domains in order to ensure the continuity of the physical fields.
The proposed approach provides a common framework, by which various types of problems (e.g., quasistatics, wave propagation) and media (metal, dielectrics, etc.) can be treated in a uniform manner, as it was demonstrated by the test problems. The method may be extended for nonrigid motion (e.g., bounded fluid flow).
The new approach may have several advantages. From a theoretical point of view, it provides an alternative perspective of the moving body problems. In turn, in the practice it can actually facilitate the design of appropriate magnetoelectric materials that raise the illusion of motion, e.g., in the field of optics and radar technique. Finally, as concerning the numerical computations, one is able to solve moving body problems by FE software in which the moving terms are not implemented.
One further (yet unexploited) idea of utilization is the convergence improvement of FE computations. The stability problems mentioned in Section 3.1 are mostly overcome by a technique called “upwinding” [2], but they can be resolved by adaptive anisotropic meshing, too [16]. In the latter case, the diffusion tensor (23) of the modified PDE might give guidelines for the desired mesh anisotropy.
Acknowledgments
The work was created in commission of the National University of Public Service under the priority project KÖFOP2.1.2VEKOP15201600001 titled “Public Service Development Establishing Good Governance” in the Bay Zoltán Ludovika Workshop. The author would like to thank his colleague Sándor Bilicz for his helpful comments.
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Cite this article as: Szabolcs Gyimóthy, Modeling stationary moving medium by static magnetoelectric material, Eur. Phys. J. Appl. Phys. 85, 10901 (2019)
All Figures
Fig. 1 Interface between stationary moving and nonmoving domains. 

In the text 
Fig. 2 The studied problem of sliding contacts. 

In the text 
Fig. 3 Current distribution near the contacts at v = 6 m/s. 

In the text 
Fig. 4 Rotating dielectric cylinder illuminated by an Spolarized plane wave. 

In the text 
Fig. 5 Farfield scattering pattern of E_{s}  (normalized values) for Ω = 0 rad/s (solid line), and Ω = 10^{7} rad/s (dashed line). 

In the text 
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