Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
S. Kurz
University of the German Armed Forces,
School of Electrical Engineering
Holstenhofweg 85
22043 Hamburg, GERMANY e-mail: stefan.kurz@unibw-hamburg.de
B. Flemisch, B. Wohlmuth
Universität Stuttgart, Institut für angewandte
Analysis und numerische Simulation
Pfaffenwaldring 57
70569 Stuttgart, GERMANY e-mail: flemisch@ians.uni-stuttgart.de
Abstract
In many engineering applications the interaction between the electromagnetic field and moving bodies is of great interest. E.g., motional induced eddy currents have to be taken into account correctly for the modelling and simulation of high-speed solenoid actuators. In connection with computational electromagnetism, it seems natural to use a Lagrangian (also called material) description. The unknowns are defined on the mesh, which moves and deforms together with the considered objects.
What is the correct form of Maxwell’s and the constitutive equations under such circumstances?
Since the bodies might undergo accelerated motion, this question cannot in general be answered by the application of Lorentz transforms. Consequently, Maxwell’s equations do not necessarily have their usual form in accelerated frames of reference. This was demonstrated in a classical paper by Schiff
[1], where it is shown that a significant difference occurs even at “low” velocities, which are small compared to the velocity of light. In contrast, it is convenient to perform the analysis of rotating induction machines from the rotor’s point of view. Despite the acceleration, starting from the usual form of Maxwell’s equations yields the correct results. How could that be possible?
There are only few publications that address the subject from a general point of view and not only for a restricted class of examples, e.g. [2,3]. The aim of the present paper is to tackle the problem once more by using the language of differential forms (DFs). DFs are especially well suited for such considerations, since they allow to seamlessly migrate from the (3+1)- to the four-dimensional formulation of electrodynamics, which are both briefly reviewed.
Moreover, DFs allow separating the topological from the metric part of the theory. Using a noninertial frame induces a metric that is different from the standard Lorentz metric. This metric enters the formulation only through the coordinate expression for the four-dimensional Hodge operator. A localization transform can be introduced, to revert to a (3+1)-dimensional description. This is connected to the concept of a co-moving observer. The result is a relativistically correct Lagrangian form of Maxwell’s and the constitutive equations. For “small” accelerations, i.e. if the extension of the system is neglectable compared to the radii of curvature, a concise set of transforms for all the relevant field quantities can be derived. These transforms are well suited for the implementation into numerical field computation codes.
1. Introduction: The 3D and 4D formulations
We start from Maxwell's equations for vector fields in the affine Euclidian space E
3
, i.e., curl r
E
= −∂ t r
B , div v
B
=
0 , curl r
H
=
J r
+ ∂ t r
D , div r
D
= ρ
, (1) together with the constitutive relations r
D
= ε r
E , r
B
= µ r
H , valid for simple media at rest. By introducing the corresponding differential forms and the exterior differential operator d
3
, we can equivalently reformulate (1) as d
3
E
= −∂ t
B , d
3
B
=
0 , d
3
H
=
J
+ ∂ t
D , d
3
D
= ρ
, (2) and, denoting by
∗
3
the Hodge operator associated with the Euclidean metric the material laws become
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
D
= ∗
3
ε
E , B
= ∗
3
µ
H .
(3)
In order to obtain a formulation in the Minkowski space tial, and construct the field F
=
B
+
E
∧ d t , the excitation
M
4
, we use d as the fourth basis differen-
G
=
D
−
H
∧ d t , and the 4-current density
J
= ρ −
J
∧ d t . With the differential operator d
= d
3
+ d t
∧ ∂
/
∂ t , we can write (2) as d F
=
0 , d G
= J .
(4)
The material laws (3) can be cast into the covariant form [4, p. 303] i u
( ∗
4
G
+ c
0
ε
F
) =
0 , i u
( ∗
4
F
− c
0
µ
G
) =
0 , (5) where
∗
4
denotes the Hodge operator corresponding to the Minkowski metric, the four velocity u , and c
0
the velocity of light in empty space.
i u
the contraction with
2. The Lagrangian perspektive
Let ( x
µ
) be the laboratory coordinates, where greek indices run from 0 to 3, and x
0 = c be the nor-
0 t malised time. We denote by
µ e
ν
= δ µ
ν
Σ
the natural laboratory frame ( )
=
( d x
µ
) and by ( e
ν
) its dual,
,
⋅ ⋅
being the duality pairing. The laboratory coordinates shall be time-orthogonal w.r.t. the
Minkowski metric the metric coefficients from 1 to 3. If g ( e i
, g e
( k
⋅
,
⋅
)
) , such that g ( e
0
, e
ν
)
= − δ
0
ν
. We denote the Hodge operators associated with g
=
( e
µ
δ ik
, e
ν
) and g ( e i
, e k
) by
∗ Σ
4
and
∗ Σ
3
, respectively, where latin indices run
held, the laboratory coordinates would be Cartesian and we would have
∗ Σ
4
= ∗
4
,
∗
3
Σ = ∗
3
, but this is not necessarily required.
If electromagnetic field problems including moving or deforming bodies have to be solved, it is useful to proceed to a Lagrangian description. To this end, we introduce material coordinates ( x 0
,
µ
) , where x 0
, 0 = x
0
. The ( x 0
, i
)
∈
M describe the considered body in a certain reference state, say at t
=
0 , and
M is called material manifold. The actual state of the body is given by a smooth placement mapping
U t
: M
×
R
→
E
3
: ( x 0
, i
, t ) a ( x j
) .
(6)
We denote by
Σ
0 the natural material frame ( 0
,
)
=
( d x 0
,
µ
) and by ( e 0
ν
) its dual. Note that general not time-orthogonal. By pullback, (4) and (5) could be reformulated w.r.t.
Σ
0 is in
Σ
0 . While the
Hodge operators in (5) would have to be adapted to the new metric coefficients g ( e 0
µ
, e 0
ν
) yielding
∗ Σ
4
0
, all other coordinate expressions remain invariant, thanks to the exterior calculus.
We introduce still another frame
Σ′
, the co-moving inertial frame bases are defined as
(
′ ,
µ
) and its dual ( e
′
ν
) . These e
′ , i
=
= e 0
0
0
, i
= u
,
/ c
0
,
′ , 0 = e k
′ = e 0 k
− g ( e 0 k
, e
′
0
) e
′
0
,
0
, 0
/
γ + g ( e 0 k
, e
′
0
)
′ , k
,
(7a)
(7b) where
γ =
( g ( e 0
0
, e 0
0
)
) −
1 / 2
is the usual Lorentz factor. Eqs. (7) exhibit the first step of a Gram-Schmidt orthonormalisation procedure. Note that c
0 e
′
0
is the four-velocity and d t
′ = ′ , 0
/ c
0
the proper time
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31 differential of the considered material element.
Σ′
is no natural frame, thus anholonomic [5, p. 70], since d
′ , 0 ≠
0 in general.
Each of the considered frames goes along with its specific (1+3)-decomposition of the fields. According to the hypothesis of locality [6], the frame
Σ′
defines the measurable fields. By definition, a considered material element is momentarily at rest in this frame, so the material laws have their simple form (3). However, the Hodge operator
∗ Σ′
3
has to be understood w.r.t. the spatial metric coefficients
γ ik
= g ( e
′ i
, e
′ k
) . If we formulated (1+3)-Maxwell's equations in this frame, they would have a rather complex form, due to its anholonomity. This road has been pursued in [3].
In contrast, (1+3)-Maxwell's equations have their simple form (2) in the natural frame
Σ
0 . If we wrote the material laws in this frame, we would end up with involved expressions. Even in vacuum a coupling between the electric and magnetic fields would occur [2, p. 272], due to the lack of time-orthogonality. The key point of this paper is to use both frames simultaneously, in order to keep (1+3)-
Maxwell's equations as well as (1+3)-material laws as simple as possible. The equations have to be complemented by transformation laws, which relate both (1+3)-decompositons to each other.
3. (1+3)-decomposition and transformation laws
In order to keep as much flexibility as possible, we employ the techniques presented in [5, p. 117]. The topological structure of the Minkowski space M
4
guarantees the existence of a family of nonintersecting 3-dimensional hypersurfaces choose a fixed vector field n satisfying i n h
σ
parametrised by the "would-be time" variable
σ
. We d
σ =
1 . Any given p -form can be decomposed w.r.t.
n into a part
||
parallel to h
σ
and a remaining part
⊥
such that
= || + ⊥
,
|| = i n
( d
σ ∧
)
,
⊥ ∧
⊥
, where
⊥
= i n
.
(8)
As a first step, we establish the transformation between the frames
Σ
and
Σ′
. In terms of the decomposition (8), we have d
σ = 0
, n
= e
0
for the laboratory frame
Σ
, and d
σ ′ = ′ , 0
, n
′ = e
′
0
for the co-moving frame
Σ′
. The p -form and
Σ′
, respectively. We denote by
is decomposed into v
= c
0
( n
′
/
γ − n )
|| ∧
⊥
and
||
′ d
′ ∧
⊥′
w.r.t.
the 3-velocity vector with its Riesz-dual
Σ corresponding to the Minkowski metric, and derive
||
′ = || +
1 c
0
∧
⊥
,
⊥′
= γ
⊥
−
γ c
0
2
∧ i v
⊥
+
γ c
0 i v
||
.
(9)
In terms of Section 1, the p -form is given as and a ( p
−
1 ) -form
= −
(
−
1 ) p c
0
⊥
= + ∧ d t
= ′ + ′ ∧ d t
′
with a
. In this case, the transformation law (9) amounts to p -form
= ||
′ = +
1 c
0
2
∧
,
′ = γ −
γ c
0
2
∧ i v
−
(
−
1 ) p γ i v
.
(10)
When translating (10) into the language of vector fields, it is necessary to establish metric induced translation isomorphisms between differential forms and vector fields [7, p. 242]. For the laboratory frame
E
= 1
( r
E )
Σ
, the isomorphisms corresponding to the standard Euclidean 3-metric can be chosen, e.g.,
and B
= 2
( r
B ) , where p
(
⋅
) denotes the standard isomorphisms ( p =1,2). For the co-moving frame
E
′ = 1
( r
E
′
||
Σ′
/
γ +
E
′
⊥
) and B
′ = 2
( r
B
||
′ + r
B
′
⊥
/
γ
) . In this context,
||
and
⊥
denote the components of a vector parallel and perpendicular to v transformation laws [2, p. 73], e.g., r
B
′ = r
B
||
+ γ
( r
B
⊥
− v
×
E / c
0
2
) , r
E
′ = r
E
||
+ γ
( r
E
⊥
+ v
× r
B ) .
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
4. Consequences and conclusion
(1+3)-Maxwell's equations (2) w.r.t.
between the frames
Σ
0 and
Σ′
Σ
0 , (1+3)-material laws (3) w.r.t.
Σ′
, and a transformation law
, derived in the same way as (10), give a consistent electromagnetic field theory in the material coordinates ( x 0
,
µ
) . How does this description relate to computational electromagnetics?
In technical electromechanical systems the electromagnetic processes are usually low frequent, so that wave propagation can be neglected. The low-frequency limit of the theory can be introduced formally by c
0
→ ∞
. (For a criticism of this approach see [8, p. 369].) Under these circumstances the transformations between the frames
Σ
0 and
Σ′
reduce to identity transformations. Moreover, the transformations (10) reduce to
′ =
,
′ = −
(
−
1 ) p i v
, (11) which is compatible to the classical Galilean transformation. Note that v is not constant in general, but a time- and space-dependent material velocity field. For a translation of (11) to the language of vector fields see [9, p. 260].
We strongly recommend to employ the differential form based framework as a starting point for numerical formulations in the material manifold M . (1+3)-Maxwell's equations (2) keep their usual form, while the (1+3)-material laws (3) “feel” the deformation of the medium through the Hodge operator
∗ Σ
3
0
. This holds as well for Ohm's law commute with the time derivative
J
= ∗ Σ
3
0 κ
E . Note that
∗ Σ
3
0
∂ t
, since the spatial metric coefficients
does in general not
γ ik
might be timedependent. For a good account on an eddy current formulation based on this approach see [10].
In many applications, it is sufficient to restrict the kinematics to rigid body motion. In this case, the placement function induces an orthogonal transformation between
Σ
and
Σ
0 . Since orthogonal transformations preserve the spatial metric, even the Hodge operator keeps its laboratory form
∗ Σ
3
0 = ∗ Σ
3
. This has an important consequence: For the description of low frequency electromagnetics, all rigid frames are equivalent . This goes beyond the standard principle of
Galilean relativity, where only inertial frames are regarded as equivalent. The laws of low frequency electromagnetics are subject to extended Galilean relativity, so to say. This justifies the usual analysis of induction machines from the rotor's point of view. We emphasise that this property is restricted to the low frequency limit of Maxwell's equations. The full equations cannot be expected to keep their form in rotating frames. They have to be treated properly according to the previous sections. This also resolves the classical paradox by Schiff [1].
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