High order transmission conditions for thin conductive sheets in magneto- quasistatics
Condition de transmission d’ordre ´elev´e pour la mod´elisation de plaques minces en domaine magn´eto- quasistatique INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
High order transmission conditions for thin conductive sheets in magneto-quasistatics
Xxxxxxx Xxxxxxx∗ — Xxxxxxxxx Xxxxxxx†
N° 7254
ISRN INRIA/RR--7254--FR+ENG
ADpormila2in0e101
apport
ISSN 0249-6399
de recherche
High order transmission conditions for thin conductive sheets in magneto-quasistatics
Xxxxxxx Xxxxxxx∗ , S´ebastien Xxxxxxx†
Domaine : Math´ematiques appliqu´ees, calcul et simulation E´quipes-Projets POEMS
Rapport de recherche n 7254 — April 2010 — 28 pages
Abstract: We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are essentially smaller or at the order of the skin depth. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to the small parameter ε and obtain optimal bound for the modelling error outside the sheet of order εN+1 for the condition of order N . Numerical experiments with high order finite elements for sheets with varying curvature verify the theoretical findings.
Key-words: Asymptotic Expansions, Transmission Condition, Thin Conducting Sheets.
∗ Project POEMS, INRIA Paris-Rocquencourt, 78153 Le Chesnay, France, e-mail: xxxxxxx.xxxxxxx@xxxxx.xx
† Institut de Math´ematiques de Toulouse, Universit´e de Toulouse, France, e-mail: xxxxxxxxx.xxxxxxx@xxxx-xxxxxxxx.xx
Centre de recherche INRIA Paris – Rocquencourt
Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex
Téléphone : x00 0 00 00 00 00 — Télécopie : x00 0 00 00 00 00
Conditions de transmission d’ordre ´elev´e pour
les plaques minces conductrices en magn´eto-quasistatique
R´esum´e : En domaine magn´eto-quasistatique, nous nous int´eressons `a la mod´elisation des plaques minces condutrices dont l’´epaisseur ε est de l’ordre de leur ´epaisseur de peau. Nous les mod´elisons par des conditions de transmission d’ordre 1, 2 et 3 pos´ees sur une interface localis´ee en leur centre. La prise en compte dans les codes de calcul de ces mod`eles est ais´ee et ne g´en`ere qu’un couˆt de calcul marginal. Nous d´emontrons le caract`ere bien pos´e des mod`eles approch´es associ´es et obtenons des estimations d’erreurs opitmales d’ordre εN+1 pour le mod`ele d’ordre N . Quelques exp´eriences num´eriques r´ealis´ees avec des ´el´ements finis d’ordre ´elev´e sont en ad´equation avec les r´esultats th´eoriques.
Mots-cl´es : D´eveloppements asymptotiques, Condition de transmission, plaques minces conductrices.
Contents
Introduction 3
1. Problem definition 4
1.1. The geometrical setting 4
1.2. Two-dimensional magneto-quasistatic with xxxx current modelling 5
1.3. Asymptotic expansion with respect to the thickness of the sheet 6
2. Hierarchical asymptotic expansions 7
3. Derivation of the transmission conditions for the exterior fields 8
3.1. The definition of the exterior approximation 8
3.2. The definition of the interior approximation 9
4. Weak formulation, Uniqueness, and Stability 10
4.1. Preliminary material: function spaces and admissible boundary conditions 10
4.3. The models of order 2 and 3 12
4.4. Regularity 17
5. Estimates of the modelling error 18
6. Numerical examples 20
Concluding remarks 25
Appendix A. 26
A.1. The surface operators 26
A.2. The interior operators 27
References 27
In many practical applications, electronic devices are surrounded by casings or other sheets of a highly conductive material to protect them from external electromagnetic fields (e. g., data cables) or to protect the environment from the electromagnetic fiels generated by devices (e. g., transformer or bushings). To minimize the cost, size and weight, these sheets have to be thin. This leads to a non-perfect shielding where the electro- magnetic fields partly penetrate the xxxxxxx and, e. g., external fields have a small but significant effect on the encased electronic devices. The large ratio of characteristic lengths (width of the device against thickness of the sheet) leads to serious numerical problems. Indeed the classical numerical methods such as finite differences or finite elements require a small mesh size and are consequently very costly or simply not able (due to limited memory) to compute a numerical approximation of the solution of such a problem. Another important issue for such problems is related to mesh generation. It is very time consuming to take into account small details in the geometry. Moreover, most of commercial mesh generators generate meshes with poor quality when the geometrical characteristic lengths are too different.Designing methods where the sheet needs not be included in the mesh is really a crucial objective. These two considerations point out the necessity of an appropriate modelling of the shielding behaviour by thin sheets. This is the problem that we address in this paper.
So called impedance boundary conditions (IBC) for thin layers of low order have been proposed by several authors, e. g., [14,16,20], and also for transient analysis [19]. We prefer the notation “transmission condition” to distinguish from IBCs, originally proposed for solid conductors by Xxxxxxxx [28] and Xxxxxxxxxx [17], and derived for higher orders [1, 9, 13, 27], and for perfect conductors with thin coatings [2, 3, 7]. Asymptotic expansion to any order have been derived for the electro-quasistatic equations [22] and time-harmonic Xxxxxxx equations [21] in biological cells with isolating membranes. In a previous article [26] we derived an asymptotic expansion at any order for the eddy current problem in 2D with thin conducting sheets. We have chosen an asymptotical framework of constant shielding when the thickness ε of the sheet tends to zero leading to a non-trivial limit
problem. These derivations lead to the definition of a limit solution and correctors of higher orders, which are computed iteratively.
In this article we introduce approximate transmission conditions for the eddy current problem in 2D with thin conducting sheets which define approximate solutions of order 1, 2 and 3. Like for the IBCs the original problem is equipped with conditions to take into account the shielding behaviour of the thin sheet.
The problem we intestigate is to find the electrical field eε satisfying
−∆eε(x) = f (x), in Ωε
ext
ε
int
(1)
−∆eε(x) + c0 eε(x) = 0, in Ωε ,
where Ω is the whole domain, Ωε
eε = eimp on Γe,
∇eε · n − βeε = ιimp on Γi,
int
ext
is domain of the thin sheet of thickness ε and Ωε
int
:= Ω\Ωε
(see detailed
definition in Sec. 1). The relative conductity is c0, β is some operator related to an impedance boundary condition and the remaining symbols stands for sources and boundary data. We have indexed the function e by ε which shall vary to 0.
In what follows we will design a one step procedure to compute a numerical approximation of eε which does
int
require neither mesh refinement nor meshing of the thin sheet Ωε
. This technique is based on the asymptotic
expansion of eε obtained in [25] and consists in modelling the thin sheet by two approximate transmission conditions which are derived and justified in three steps:
(i) We derive formally an approximate models whose solutions e ε,N are candidates to approximate the
exact solution eε,N . ˜
(ii) We prove that the approximate problems are well posed for small ε and asymptotically stable.
(iii) We prove that e ε,N is an approximation of eε of order N , i. e., e ε,N − eε = o
(εN ).
˜ ˜ ε→0
√
≤
In order to have a presentation as clear as possible, this article will only carry on the cases corresponding to approximation order less than 4, i. e., N 3. These results can be extended to N > 3 even if one has to deal with higher derivative operators on the midline Γ of the sheet which introduces extra difficulties. The role of steps (ii) and (iii) is to give a mathematical background to the formal computations of step (i). In Section 2 the result of [25] will be shortly summarized. The steps (i) will be carried out in Section 3. Section 4 is devoted to step (ii). Here, we observe that with the chosen asymptotical framework of constant shielding the approximate problems are stable for thicknesses not exceeding some multiple of the skin depth dskin := 2/ωµσ. This transfers to the modelling error (step (iii)) with which we will deal in Section 5. Finally, in Section 6 numerical experiments verifying the theoretical results will be shown.
Let us denote by →−x = (x, y, z) a parametrization of R3 and by →−ex, →−ey and →−ez the associated orthogonal unit vectors. To avoid difficulties mostly related to differential geometry, we will be concerned in this article with a
z-invariant configuration. To take care of the two-dimensional phenomenon, we introduce the vector x = (x, y) composed of the two first coordinates of →−x .
int
The computational domain Ω × R is decomposed into a highly conducting sheet Ωε × R and a domain
Ω
ε
ext
× R filled with air which satisfies
(i) Ω is a bounded domain of R2 with Lipschitz boundary.
2
(ii) The conducting sheet
Ω
ε
int
= x ∈ Ω : ∃ y ∈ Γ x − y 2 < ε } (2)
(a) Illustration of the geometry and the local coor-
Ω
Ωε
ext
ε
Ωε
t
0
int
Ωε
ext
s
n
∂Ω
Ω0
ext
Γ
Ω0
ext
n
∂Ω
ext
(b) The limit geometry of Ωε
Ω
is the whole domain
dinate system inside the sheet. without the midline Γ of the sheet.
Figure 1. The two-dimensional geometrical setting for a sheet of thickness ε and the limit geometry for ε → 0.
has constant thickness ε > 0 and is centered around Γ a regular closed curve of Ω with no cross point. To each point of Γ can be associated a curvature κ(t) and a left normal unit vector n(t) (see Fig. 1. Moreover it can be parameterized, for ε small enough, by a local coordinate system as follows. Let Γˆ be a one-dimensional torus of the same length than Γ. Let denote by xm : Γˆ → Ω an injective C∞
int
mapping 1 whose range is Γ and satisfying xm′ = 1. The domain Ωε can then be seen as the range
of the injective mapping
Γˆ×] − ε/2, ε/2[ −→ Ω
(3)
(t, s) −→ x(t, s) = xm(t) + sn(t).
int
This sub-domain is filled with material of constant conductivity σ and permeability µ0.
ext
(iii) The exterior of the sheet Ωε
= Ω\Ωε
has constant permeability µ0 and is not conductive.
1.2. Two-dimensional magneto-quasistatic with xxxx current modelling
When the geometric characteristic length are all much smaller than the wave length, the electromagnetic fields are accurately described by the eddy current model, a quasi-static approximation to the Xxxxxxx equations [5,24],
div(→−E ) = 0,
−r→ot(→−E ) = −∂t→−B ,
−r→ot(→−B ) = µ0→−J ,
(4)
1A C2 mapping would be enough to have a continuous normal vector. We assume C∞ for simplicity.
ext
where the current is impressed, i. e., →−J = −J→0 is known, in a sub-domain of Ωε
(see Fig. 1) and it is by Ohms
int
law proportional to the electric field →−E , i. e., →−J = σ→−E inside Ωε .
We consider a time-harmonic excitation
J−→0(→−x , t) = exp(−iωt) j0(x) →−e z. (5)
Due to z-invariance, the electromagnetic fields has to be sought in frequency domain with the form
→−E (→−x , t) = e(x) exp(−iωt) →−e z and →−B (→−x , t) = bx(x) exp(−iωt) →−e x + by(x) exp(−iωt) →−e y. (6)
int
Inside Ωε
and Ωε
it reads for the out-of-plane electric field [23]
ext
ε
−∆e(x) = −iωµ0j0(x) =: f (x), in Ωext, (7)
ε
ext
−∆e(x) + iωµ0σe(x) = 0, in Ωint. (8) Furthermore, let the electric field satisfies the standard transmission conditions at the two interfaces between
Ω
ε
int
and Ωε
: the function e and its normal are continuous across the interfaces or equivalently
∆
e ∈ H1 (Ω) := nu ∈ H1(Ω) : ∆u ∈ L2(Ω),. (9)
Finally, let equations (7) and (8) be supplemented with suitable boundary conditions on ∂Ω such it provides a unique solution in H1(Ω): Let be given a prescribed electric field eimp on Γe ⊂ ∂Ω (Dirichlet condition) and a general impedance boundary condition on Γi := ∂Ω\Γe, which are given as
e = eimp on Γe,
∇e · n − βe = ιimp on Γi,
with a source term ιimp of the impedance condition and a impedance operator β. The latter could represent for example an integral representation of the electric field in the homogeneous exterior of Ω or a simple Xxxxxxx boundary condition (β = 0).
1.3. Asymptotic expansion with respect to the thickness of the sheet
In this article, we aim in designing a numerical method which does not require any meshing of the sheet. This method will be based on the asymptotic expansions of [25] which were derived for small thickness ε and a
int
conductivity scaled reciprocal to the thickness ε, i. e., inside Ωε
it holds
where c0 := iεωµ0σ is independent of ε.
−∆e(x) +
c0 e(x) = 0, (11)
ε
In real configuration, a thin sheet has a given thickness (ε = 1 mm for example). Consequently looking for an asymptotic expansion with respect to the thickness of the sheet, i. e., varying ε to 0, does not have at first glance a clear meaning. However, this point of view is known for similar problems to be rather efficient [3, 13] to design numerical methods.
→
Vm
| | ∼
| |
Moreover, a real thin sheet has a given conductivity (σ = 5.9 · 107 A for copper). However, scaling ωµ0σ like 1/ε corresponds to a borderline case where the sheet is neither impenetrable (this will happen for ωµ0σ −1 = o(ε)) nor transparent ( ωµ0σ = o(ε−1)) in the limit for ε going to 0. Thus, with the scaling ωµ0σ 1/ε already the limit model for ε 0 is physically relevant and an asymptotic expansion is expected to be accurate already with a few terms.
2. Hierarchical asymptotic expansions
We aim in this section in summarizing the results of [25] obtained for Dirichlet boundary conditions and adapting them to other boundary conditions. More precisely, we derived the complete asymptotic expansion with respect to the width of the sheet ε of the solution of problem (1) for a regular sheet.
We have shown that the use two points of view is necessary in order to describe sharply eε. The first point of view consists in considering the restriction of eε to the exterior of the sheet and in looking for a Xxxxxx expansion of this restriction
N
Ω
ext
ε
ext
ext
ext
ext
ext
eε (x) = eε (x) = eε,N (x) + rε,N (x) with eε,N (x) := Σ εjuj (x), ∀N ∈ N. (12)
j=0
ε
A contrario, the second point of view considers the restriction to the interior of the sheet of eε. We do not look for a Xxxxxx expansion of eε in the original curvilinear coordinates (s, t), but in the normalised curvilinear coordinates (S, t) = ( s , t)
N
Ω
int
ε
int
int
int
int
int(
ε
, t),
eε (x) = eε (x) = eε,N (x) + rε,N (x) with eε,N (s, t) := Σ εjuj s
j=0
∀N ∈ N. (13)
ext
int
The coefficients uj (x) and uj
(S, t) of these Xxxxxx expansions are functions not depending on ε which are
ext
defined on the limit domain Ω0
ε
of Ω
ext
for ε → 0
Ω
0
ext
= Ω \ Γ (14)
2
2
and on the normalised sub-domain of the sheet Γˆ × [− 1 , 1 ], respectively. They are defined hierarchically order
by order by a coupled problem (that will not be detailed here). Moreover, the following two estimates make clear what we mean by Xxxxxx expansion
ε ε,N
ε,N
N+1
ext
eext − eext H1(Ωε
) = rext H1(Ωε
) ≤ CN ε
, ∀N ∈ N, (15)
ε ε,N
ε,N
ext
int
N+ 1
int
eint − eint H1(Ωε
) = rint H1(Ωε
) ≤ CN ε 2 , ∀N ∈ N. (16)
Analysing deeply the coupled system that is solved by the exterior and interior asymptotic expansions, one can remark — this has been done up to order 2 in [25] — that one can define the exterior coefficients with problems involving only the exterior coefficients of lower order and not the interior coefficients. These hierarchical decouple problems take the form
j 0
j
−∆uext(x) = fj(x), in Ωext, (17a)
u
h
i
j
ext
(t) =
ΣA=2
ext
(γAuj−A)(t), on Γ, (17b)
∂nu
j
h
i
j
ext
(t) − c0 n
u
,
j
ext
(t) =
ΣA=1
ext
(ζAuj−A)(t), on Γ, (17c)
with the two transmission operators [·] and {·} defined on the midline Γ by
2
[u] (t) := u(t, 0+) − u(t, 0−), {u} (t) := 1 u(t, 0+) + u(t, 0−) ,
with the differential operators γA and ζA. that are explicitely given for l ≤ 3 in Appendix A.1, and with the source terms fj and boundary conditions which are inherited from the original problem and consequently satisfy (j > 0)
ext
f0(x) = f (x) and fj(x) = 0, in Ω0 ,
u
ext
j
0
ext
= eimp and uj
= 0 on Γe,
(18)
∇u
0
ext
· n − βu0
= ιimp, and ∇uj · n − βu
ext
ext
= 0, on Γi,
ext
Once the exterior coefficients are defined the interior coefficients can be computed in the following way. They are polynomials in the normal coordinate S and result by the exterior fields of the same and the previous orders. More precisely, they can be written
U
j
int
j
Σ −(t, S) = (η u )(t, S) (19)A
j A
ext
A=0
with for l ≤ 3 the ηA given in Appendix A.2.
Remark 2.1. The latter asymptotic expansion can directly be used to obtain a numerical approximation of eε. Indeed one has just to compute eε,N , and eε,N , with N fixed by the desired precision. These computations do
ext int
require neither mesh refinement nor the meshing of the thin sheet. However, this method suffer from a major
drawback: For relatively large ε the model of order 0 does quite possibly not reach the desired precision and one has to compute more further terms of the asymptotic expansion in order to obtain a sharp approximation of eε. The multi-step procedure is not standard and may disencourage to be implemented in a numerical library.
3. Derivation of the transmission conditions for the exterior fields
In this section, we show how one can derive the approximate problems for a regular sheet (Γ is C∞).
3.1. The definition of the exterior approximation
ext
We adopt the point of view of formal series. Due to (12), the formal Xxxxxx series of eε
takes the form
e
ε
ext
(x) ∼
+∞
Σ jjε u
ext
j=0
(x). (20)
ext
where we have adopted the symbol “∼” to mention that this series may diverge or converge but not toward eε .
Therefore multiplying for all j ∈ N system (17b), (17c) by εj and summing we get with
h Σ
+∞
j=0
j
j j
ε u
ext
i(t) =
+∞
Σ
εj
j=0
ΣA=0
ext
(γAuj−A)(t), on Γ, (21a)
h∂n
+∞
Σ
j=0
j
j j
ε u
ext
i(t) − c0
+∞
n Σ
j=0
j j
ε u
ext
,(t) =
+∞
Σ
εj
j=0
ΣA=0
ext
(ζAuj−A)(t), on Γ, (21b)
ext
with the convention γ0 = γ1 = ζ0 = 0. Interchanging the two sums and identifying eε
, we find
[e
ε
ext
ε ε
ext
] (t) ∼ (γεeε
)(t), on Γ, (22a)
ext
ext
[∂neε
] (t) − c0 {eε
} (t) ∼ (ζ eext
)(t), on Γ, (22b)
with the two formal operator series γε and ζε given by
+∞ +∞
γε = Σ εjγj, and ζε = Σ εjζj. (23)
j=0
j=0
These two transmission conditions appear to be perfect. However, the question of convergence of the series (23) remains (we think they diverge potentially). Moreover, it seems not possible to get a simple formula for the sum if it exists. Consequently, these two perfect transmission conditions could not directly be used for numerical computations. However, truncating these two series at a given order N
N N
γε,N = Σ εjγj, and ζε,N = Σ εjζj, (24)
j=0
j=0
we get well defined transmission conditions to model the highly conductive thin sheet.
ext
ext
This series truncation leads to approximate problems for the approximate solutions e˜ε,N ∈ H1(Ω0 )
ε,N 0
−∆e˜ext (x) = f (x), in Ωext, (25a)
ext
ext
h∂ne˜ε,N i(t) − c0 ne˜ε,N ,(t) − (ζε,N e˜ε,N )(t) = 0, on Γ, (25c)
he˜ε,N i(t) − (γε,N e˜ε,N )(t) = 0, on Γ, (25b)
ext
ext
ext
ext
e˜ε,N = eimp on Γe, (25d)
ε,N ε,N
∇e˜ext · n − βe˜ext = ιimp onΓi. (25e)
The approximate solutions are indexed by N which is related to the order of the approximation. For N > 1, the reader can note that this approximate solution is no more continuous across Γ and therefore does not belong to H1(Ω).
3.2. The definition of the interior approximation
The same strategy can be applied to the derivation of an interior approximation. Due to (13), the Xxxxxx
series of eε reads eε (t, s) ∼ Σ+∞ xxX x (t, s ). Inserting (19) we have
int
int
eε
j=0
Σ
j
(t, s) ∼
(ηAuj−A)(t, s ) =
εAηA
εjuj
(t, s ) ∼ ηεeε
(t, s ).
ext
ε
ext
ε
Σ
+∞
εj
j=0
A=0
int ε
ext
Σ+∞
ε
l=0
Σ+∞
j=0
int
with the formal series operator ηε and it associated partial sums defined by
+∞ N
ηε = Σ εAηA and ηε,N := Σ εAηA. (26)
l=0
A=0
It leads to the introduction of the interior approximation of order N
e˜ε,N (t, s) = (ηε,N e˜ε,N )(t, s ). (27)
int
ext ε
Remark 3.1. In the continuation we will not carry on the justification of these approximations. Note, however,
that it can be proved that e˜ε,N is an approximation of order N of eε
. More precisely, it holds
int
ε
ε,N
ε,N
int
N+ 1
int
int
eint − e˜int H1(Ωε
) = rint H1(Ωε
) ≤ CN ε
2 . (28)
4. Weak formulation, Uniqueness, and Stability
˜
The question of existence and uniqueness of eε,N is not only an interesting mathematical question. Very often problems arising from an asymptotic expansion are not well posed. Consequently, it is crucial to check the existence and uniqueness of solution of problem (25). Moreover, the convergence proof needs a stability result that is closely related to the existence and uniqueness of the solution of the collected models.
Since offset functions allow to exchange Dirichlet data eimp against source terms f , we will only deal with homogeneous Dirichlet boundary conditions i. e., eimp = 0.
ext
As the jump of the external field e˜ε,1 in contrast to the higher orders is vanishing in general due to γε,1 = 0 we will propose for order 1 an own weak formulation in H1(Ω). For the models of order 2 and higher we will impose the jump condition (25b) weakly with an additional equation.
4.1. Preliminary material: function spaces and admissible boundary conditions
In what follows, homogeneous Dirichlet boundary conditions at Γe will be incorporated in the trial and test spaces which are
Γe
for N = 1 H1 (Ω) = nv ∈ H1(Ω) : v = 0 on Γe,, (29a)
Γe
ext
ext
for N = 2 or 3 H1 (Ω0 ) = nv ∈ H1(Ω0 ) : v = 0 on Γe,. (29b)
Moreover, we consider impedance operators that provides coercive variational weak formulations. This leads to the notion of admissible boundary conditions.
Definition 4.1 (Admissible boundary conditions). Let V = H1 (Ω) or H1 (Ω0
) A boundary condition (10)
∫
is V -admissible if for all v ∈ V
Γe Γe ext
Im
Γi
β|v|2 dS ≥ 0, Re ∫ β|v|2 dS ≥ 0,
Γi
and if for any constant δ = |δ|eiφ ∈ C\{0} with φ ∈ [0, π), the bilinear form
∫
bδ(u, v) := ∫
Ω
0
ext
∇u · v dx + ∫
Γi
βu v dS + δ
Γ
{u}{v} + [u][v] dt. (30)
is V -continuous and V -elliptic with an ellipticity constant γ ≥ h(δ) and h a non-negative continuous function defined for δ = |δ|eiφ /= 0 with φ ∈ [0, π).
Γe
Remark 4.2. In H1 (Ω), ones has {u} = u and [u] = 0 on Γ. Consequently, the bilinear form bδ can be
simplified into
bδ(u, v) := ∫
Ω
0
ext
∇u · v dx + ∫
Γi
βu v dS + δ
∫
Γ
uv dt. (31)
these three are H1 (Ω)- and H1 (Ω0
i φ
Remark 4.3. It is easy to show, that either Dirichlet (prescribed electric field) or Xxxxxxx, or general impedance boundary conditions with Im βv, v Γi ≥ 0, Re βv, v Γi ≥ 0, or any boundary condition mixed out of
ext)-admissible. This follows by testing the bilinear form with v = e 2 u, tak-
Γe
Γe
ing the real part and applying the Poincar´e-Friedrich inequality [10]. The general impedance boundary conditions
with purely real β-operator are included.
Γe
ext
Moreover, due to the presence of the differential operator ∂t in (25c) for N = 2 and 3 the following function space will be introduced
Γe
ext
H1,1(Ω0
) := {v ∈ H1 (Ω0
) : {v} ∈ H1(Γ)}
with its associated norm defined by
2 2 2 2 2 2 2
H (Ω
v 1,1 0
ext
) := v
1 0
H (Ω
ext
) + ε
∂t{v}L2(Γ) = v
1 0
H (Ω
ext
) + ε |{v}|H1(Γ), (32)
where the norm has been weigthed by ε in order to simplify the proofs that follow.
Definition 4.4. The topological dual space of a space V is denoted by V ′.
Definition 4.5. Let A be a bounded or unbounded operator of L2 equipped with the inner product ⟨·, ·⟩. We denote by A the adjoint operator of A meaning that ⟨Au, v⟩ = u, Av and with c = Re(c) −i Im(c) the complex conjugate of a complex number c ∈ C.
Note, that due to the definition of the scalar product we have ⟨cu, v⟩ = ⟨u, cv⟩ for c a multiplication operator.
4.2.1. The weak formulation
ext
The first order approximation is given by (25) with N = 1. Since γε,1 = 0, e˜ε,1
belongs to H1 (Ω). The
Γ
e
system of order 1 is given by
e˜ε,1 ∈ H1 (Ω), (33a)
ext
Γe
ε,1 0
n
ext
h∂ e˜ε,1 i(t) − c
−∆e˜ext(x) = f (x), in Ωext, (33b)
0
6
ε,1
ext
1 + εc0 ne˜ε,1 ,(t) = 0, on Γ. (33c)
ε,1
∇e˜ext · n − βe˜ext = ιimp, on Γi, (33d)
ext
ext
where one can replace {e˜ε,1 } by e˜ε,1 if it is necessary. Moreover, this system is equivalent to the variational
ext
Γe
Γe
formulation: Seek e˜ε,1 ∈ H1 (Ω) such that for all e′ ∈ H1 (Ω)
ext
a1(e˜ε,1 , e′) = ⟨l, e′⟩ , (34)
with the bilinear form a1 and the linear form l defined by
a1(e, e′) = ∫
Ω
0
− ∫
ext
∇e · ∇e′ dx + ∫
Γi
βe e′ dS + c0
∫
Γ
1 + εc0 {e}{e′} dt, (35)
⟨l, e′⟩ = ∫
Ω
0
ext
fe′ dx
Γi
6
ιimp e′ dS. (36)
4.2.2. Well-posedness and stability of the order 1 model
Lemma 4.6. Let εm := m
|c0|
for any m > 0, let the boundary conditions be H1 (Ω)-admissible, and l ∈
Γe
(H1 (Ω))′. Then, there exists a unique solution e˜ε,1 ∈ H1 (Ω) of (34) and a constant C > 0 such that
Γe
ext
¨e˜ε,1 ¨
H1(Ω)
ext
≤ Cm¨l¨
Γe
Γe
(H1 (Ω))'
∀ε ∈ (0, εm)
6
Proof. Let δε := c0 1 + ε c0 . Since c0 = i|c0| /= 0 we have Im(δε) = |c0|, Re(δε) ∈ (−εm|c0|2/6, 0) and so
Γe
δε = |δε|eiφm with φm only depending on m. Since the bilinear form a1 is H1 (Ω)-elliptic (see Def. 4.1) we get
the uniform coercivity of a1 (h is continuous)
∃ ε
2 1
RR n 7254
γm = min
∈]0,εm[
h(δε) > 0 : ∀ε ∈]0, εm[ |a1(e, e)| ≥ γm e H1(Ω) ∀e ∈ HΓe (Ω). (37)
γm
Application of the Lax-Milgram lemma [6] completes the proof with Cm = 1 .
Remark 4.7. For sheet thicknesses up to some multiple m of 1 , e. g., m = 10, uniform stability holds with
|c0|
a stability constant well bounded away from 0. Only if the sheet is much thicker than 1 and so the skin depth
|c0|
dskin the stability constant approaches 0.
4.3. The models of order 2 and 3
In this section, a weak formulation for the models of order 2 and 3 will be derived. Their solutions solve transmission problems (25) that involve the operators γj and ζj given in (69). They are differential operators in t acting on the mean trace and the mean normal trace on Γ. Hence, we can rewrite the operators γε,N and ζε,N as follows
ext
0
ext
1
ext
(γε,N e˜ε,N )(t) := γε,N (t) e˜ε,N }(t) + γε,N (t) ∂ne˜ε,N }(t), (38a)
ext
0
ext
1
ext
(ζε,N e˜ε,N )(t) := ζε,N (t) ne˜ε,N ,(t) + ζε,N (t) ∂ne˜ε,N }(t). (38b)
The multiplication operators γε,N (٨), γε,N (٨٨), ζε,N (٨٨٨٨), and the second order differential operator ζε,N (٨٨٨)
0 1 1 0
can be read from the transmission conditions (25b) and (25c), which are given for N = 2 by
e˜ε,2 (t) − − ε2 c0 κ(t) {e˜ε,2}(t) − − ε2 c0 {∂
e˜ε,2}(t) = 0, (39a)
24
12
ext
e˜ε,2
(t) −
+ εc0 + ε2 c0
6
12
20
2
∂
n
ext
c
0
` (˛٨¸)
0
x }
t
ext
n
7 c2 − ∂2
e˜ε,2
(t) − ε2 c0 κ(t){∂
e˜ε,2}(t) = 0 (39b)
24
` (˛٨¸٨) x
n
and for N = 3
` (٨˛٨¸٨) x
` (٨˛٨¸٨٨) x
e˜ε,3 (t) − − ε2 c0 κ(t) 1 − ε c0 {e˜ε,3}(t) − − ε2 c0 1 − ε c0 {∂
e˜ε,3}(t) = 0, (40a)
12
24
10
ext
e˜ε,3
+ ε c0 + ε2 c0
7 c2 − ∂2
17 c2 − 1 κ2(t) − ∂2 e˜ε,3 }(t) (40b)
x
x
2 `
(˛٨¸) 2
` (˛٨¸٨)
10 n
∂
(t) −
c
n ext
0 6 12
`
20 0
t 40
+ ε3 c0
(٨˛٨¸٨)
84 0
3 t
2 c0 x
ext
c0
ε,3
4.3.1. The weak formulation: a mixed formulation
− ε 24 κ(t) 1 − ε 10
` (٨˛٨¸٨٨)
{∂ne˜
x
}(t) = 0.
The original problem (1) is posed in H1(Ω). Consequently it is natural to intend to find a formulation posed
ext
in H1(Ω0
). However, this function space does not impose a sufficient regularity to deal with derivative of
mean trace and with mean normal traces. As a remedy, the mean normal trace will be considered as a new unknown function of L2(Γ)
ext
ext
λ˜ε,N = {∂ne˜ε,N }, (41)
and the solution e˜ε,N will be searched in the trial space H1,1(Ω0 ). A variational formulation of (25) will be
ext
derived as a problem coupling e˜ε,N and λ˜ε,N .
Γe ext
We will use the notation
u, v
Γ for the integral
Γ uv dS. Integration by parts is applied to the tangential
derivatives such that the highest tangential derivative is one (− ∂2u, v
= ∂tu, ∂tv
ex t ext ∫
t
Γ
Γ
).
To derive the variational formulation we multiply (25a) with a test function e′ ∈ H1,1(Ω0
) and using Green’s
formula we get
Γe ext
ext
∂Ω
∫ ∇e˜ε,N · ∇e′ dx + ∫
Ω
0
ext
(∇e˜ε,N · n) e′ dS + ∫
[∂ne˜ε,N ]{e′} + {∂ne˜ε,N }[e′] dt = ∫
Ω
Γ
f e′ dx.
ext
ext
ext
0
ext
Inserting the boundary conditions (10) we obtain
ext
Γi
∫ ∇e˜ε,N · ∇e′ dx + ∫
Ω
0
ext
βe˜ε,N e′ dS + ∫
[∂ne˜ε,N ]{e′} + {∂ne˜ε,N }[e′] dt = ⟨l, e′⟩ (42)
ext
Γ
ext
ext
with l defined by (36). Inserting the second transmission condition (25c) with the expression (38) into (42) and multiplying the first transmission condition (25b) with a test function λ′ integrating over Γ we obtain the
ext
Γe
ext
ext
variational formulation: Seek e˜ε,N ∈ H1,1(Ω0 ) and λ˜ε,N ∈ L2(Γ) such that
λ
a
ext
,
= ⟨l, e′⟩ , ∀e′ ∈ H1,1(Ω0
), ∀λ′ ∈ L2(Γ) (43)
N
˜ε,N
ext
λ′
Γe
ext
with
a e , e′ = ∫
Ω
∇e · ∇e′ dx + ∫
βe e′ dS + (c
i
+ ζε,N ){e}, {e′}
1
0
1
N
λ
λ′
0
ext
Γ
0
0
Γ
(44)
Γ
Γ
Γ
+ ζε,N λ,{e′} + λ, [e′] + [e], λ′ − γε,N {e}, λ′ − γε,N λ, λ′ .
Γ Γ
Conversely, it is easy to check if e˜ε,N and λ˜ε,N are solutions of (43) that e˜ε,N satisfies (25) and (41).
ext
ext
ext
4.3.2. Well-posedness and stability
In the following lemma we collect some bounds on the operators defined in (38).
≤
Lemma 4.8 (Bounds on the operators of the models of order 2 and 3). Let N = 2, 3 and ε 5 . Then,
|c0|
(i)
¨γε,N ¨
L∞(Γ) =
¨ζε,N ¨
L∞(Γ) ≤ ε
c
| |02
20 κ L∞(Γ),
0
1
1
(ii) ¨γε,N ¨
≤ ε2 |c0| , ¨(γε,N )−1¨
≤ 12 ε−2,
ε,N
L∞(Γ)
10
1
L∞(Γ)
|c0|
(iii) Im γ1 u, u Γ = −Im γ1 u, u Γ = 12 |c0| u L2(Γ) ,
ε,N
ε2 2
0
(iv) ζε,N u, u
≤ 5 ε|c | u 2
3
2
H1(Γ)
u L2(Γ) +
3 |u|H1(Γ)
.
|u| ,
Γ
16
0
L2(Γ) + 2 ε
(v) Im (c0 + ζ0 )u, u Γ = −Im (c0 + ζ0 )u, u Γ ≥
Proof. The proof will be divided into the cases of the lemma.
ε,N
ε,N
|c0| 2
4
ε2 2
10
5
−1c c60 0
|c0|
(i) With the assumption ε ≤ 5 and c0 = i|c0| we have 1 − ε c0 ≤ 6 , and the inequalities follow.
10
5
10
(iii) Since c0 = i|c0| and γε,N are multiplication operators the equality holds.
(ii) The first inequality is a consequence of 1 − ε ≤ and the second of 1 − ε ≤ 1.
1
(iv) With 1
2
ε
− 20
2
κ
L∞(Γ)
< 1 and ε 5 we get
≤
|c0|
ε,N
|c0|2 2
3 |c0|2 2
|c0|2 2
2 |c0| 2
Re ζ0 u, u Γ ≤ ε
6 u L2(Γ) + ε
40 |u|H1(Γ) ≤ ε
6 u L2(Γ) + ε
8 |u|H1(Γ),
0
Γ
240
0
L2(Γ)
12
H1(Γ)
48
0
L2(Γ)
Im ζε,N u, u ≤ 7 ε2|c |3 u 2 + ε2 |c0| |u|2 ≤ 7 ε|c |2 u 2
+ ε |c0| |u|2 ,
12
H1(Γ)
and with the triangle inequality the desired bound results.
≤
(v) With the assumption ε 5 it follows
Im (c0 + ζ0 )u, u Γ = −Im c0 + ζ0 u, u Γ = |c0| 1 − 240 ε |c0|
|c0|
ε,N
ε,N
7 2
2 2
ε2 2
u L2(Γ) + 12 |c0||u|H1(Γ),
≥
u L2(Γ) +
3 |u|H1(Γ)
.
|c0| 2 ε2 2
4
Remark 4.9. The optimal bound for ε0 is of order 1
∼
|c0|
and consequently in the order of the skin depth dskin.
This is the typical setting where we would like to apply the model where ε dskin or ε dskin. For thicker
sheets than the skin depth we refer to Chap. 7 in [23] where the author consider an optimal basis approach.
The next lemma is in terms of mathematics the most technical of the paper but it is also the key argument that makes the second and third order models work.
Lemma 4.10 (Well-posedness and stability of the models of order 2 and 3). Let N = 2, 3, the boundary
conditions be H1,1(Ω0
)-admissible (see Def. 4.1) f ∈ (H1,1(Ω0
))′, g ∈ L2(Γ) and ε ≤ min( 5 , 2 κ −1 ).
Γe ext
Γe ext
|c0|
L∞(Γ)
Γe
ext
Then, there exists a unique solution (e, λ) ∈ H1,1(Ω0 ) × L2(Γ) of
⟨f, e′⟩ + g, λ′ = a e , e′ , ∀(e′, λ′) ∈ H1,1(Ω0
) × L2(Γ) (45)
and it holds
Γ N λ λ′
Γe ext
¨e¨
H
1,1
Γe
ext)
+ ε¨λ¨
L2(Γ)
≤ C ¨f ¨
(H
1,1
Γe
0
(Ω
ext
))'
+ ε−2¨g¨
L2(Γ)
(Ω
0
with a constant C independent of ε.
· ·
Proof. The continuity of aN ( , ) follows by the Xxxxxx-Xxxxxxx inequality. For convenience we will omit N in the superscripts of the operators. The proof of the stability result (46) falls into four steps.
1
1
(i) Testing (45) with e′ = e and λ′ satisfying
[e] − γελ′ + ζε{e} = 0 (47)
e
we obtain
2
H (Ω
1 0
ext
)
Γi
+ βe, e
+ (c
+ ζε){e}, {e}
+ [e], λ′
1
1
Γ
0
Γ
1
Γ
Γ
.
1
0
0
Γ
Due to (47), ζελ,{e}
+ λ, [e] Γ
− γελ, λ′
= 0. Therefore, we have
H (Ω
2
e 1 0
ext
+ βe, e Γi
+ (c0
Γ
+ λ, [e] Γ
Γ
+ ζελ,{e}
− γε{e}, λ′
− γελ, λ′
= ⟨f, e⟩ + g, λ′
Γ
)
+ ζε){e}, {e} + γελ′, λ′
0
1
1
Γ
1
1
Γ
1
0
Γ
Γ
Γ
= ⟨f, e⟩ + (γε)−1g, [e]
− (γε)−1ζεg, {e}
+ (ζε + γε){e}, λ′ .
Now, taking the imaginary part and bounding with the estimates (i) of Lemma 4.8 we get
Im βe, e Γi
+ Im (c0 + ζε){e}, {e} + Im γελ′, λ′
0
1
1
1
1
Γ
20
L∞(Γ)
Γ
Γ
≤ | ⟨f, e⟩ | + | (γε)−1g, [e]
| + | (γε)−1ζεg, {e}
| + ε2 |c0| κ
¨{e}¨
¨λ′¨2
. (48)
Γ
L2(Γ)
L2(Γ)
Since Im βe, e ≥ 0 (see Definition 4.1) and due to (v) of Lemma 4.8 we obtain for the left hand side
4
{e}L2(Γ) +
3 |{e}|H1(Γ) +
λ
3
|c0| 2
ε2 2
ε2 ′ 2
L2(Γ)
1
Γ
1
1
Γ
≤ | ⟨f, e⟩ | + | (γε)−1g, [e] | + | (γε)−1ζεg, {e}
| + C ε2 ¨{e}¨
¨λ′¨ .
α
L2(Γ)
L2(Γ)
Since for all α > 0 we have 2ab ≤ αa2 + 1 b2 it holds
κ L∞(Γ)
2¨{e}¨
2
L2(Γ)
¨λ′¨
L2(Γ)
≤ 8 κ L∞(Γ)
1
2
′λ
{e}L2(Γ) + 8
L2(Γ) .
2
κ L∞(Γ))
{e}
So, we can incorporate the last term of the right hand side of (48) to its left hand side
(1 − 5 ε
|c0|
2 2 2 2
2
ε
λ
L2(Γ)
≤
1
Γ
1
1
Γ
¨ ′¨2
2
ε
2
2
5
` ≤˛¸1 x
| ⟨f, e⟩ | + | (γε)−1g, [e]
| + | (γε)−1ζεg, {e}
| =: rhs. (49)
L2(Γ) + 24 {e} H1(Γ) + 12
¨ ¨
The right hand side of (49) can then be bounded with the help of (i) and (ii) of Lemma 4.8
rhs ≤ ¨f ¨
(H
1,1
Γe
0
(Ω
ext
))'
¨e¨
H
1,1
Γe
ext)
+ Cε−2¨g¨
L2(Γ)
¨[e]¨
L2(Γ)
+ C¨g¨
L2(Γ)
¨{e}¨
L2(Γ).
(Ω
0
The trace theorems leads to
rhs ≤ C ¨f ¨
(H
1,1
Γe
0
(Ω
ext
))'
+ ε−2¨g¨
L2(Γ)
¨e¨
H
1,1
Γe
0
(Ω
ext
). (50)
Thus, due to (49) and (50), we obtain with an ε independent constant C > 0
2
{e}
L2(Γ) + ε
¨ ¨ 2
¨ ¨ −2
(H1,1(Ω0
λ
L2(Γ) ≤
f
(H1,1(Ω0
))' + ε
g L2(Γ)
e H1,1(Ω0
). (52)
¨ ′¨2
Γe
2
{e} H1(Γ) ≤ C
f
))' + ε
g L2(Γ)
e H1,1(Ω0
), (51)
Γe
C ¨ ¨
ε
2
ext
Γe ext
ext
ext
−2
Γe
Using (47), (51), (52), Xxxxxx inequality and the upper bounds for γε and ζε given in (i) and (ii) of
1 1
Lemma 4.8 we get
2
ε 2
2
¨[e]¨ ≤ 2¨γελ′¨ + 2¨ζ {e}¨
L2(Γ)
1
ε 2
ε 2
2
1
L∞(Γ)
L2(Γ)
1
L∞(Γ)
L2(Γ)
≤ 2¨γ ¨
L2(Γ)
¨λ′¨2
1 L2(Γ)
+ 2¨ζ ¨
¨{e}¨
(Ω
(H
≤ Cε2 ¨f ¨
1,1
Γe
0
ext
))'
+ ε−2¨g¨
H
L2(Γ)
¨e¨
1,1
Γe
ext)
(Ω
0
with an ε independent constant C > 0.
Γe
ext
(ii) Let us bound f˜, e with f˜∈ (H1,1(Ω0 ))′ defined by
1
0
Γ
Γ
f˜, e′ := ⟨f, e′⟩ − ζελ,{e′} − λ, [e′]
Testing (45) with e′ = 0 and λ′ = γε −1 [e] + ζε{e} , we get
Γ
− ζε{e}, {e′} − c0[e], [e′] .
Γ
1 1
ζελ,{e} + λ, [e] = λ, [e] + ζε{e} = (γε)−1([e] − γε{e} − g), [e] + ζε{e} .
1
Γ
Γ
1
Γ
1
0
1
Γ
Using the bounds of Lemma 4.8 and the Xxxxxx-Xxxxxxx inequality we find
1
Γ
| ζελ,{e}
+ λ, [e] Γ
| ≤ Cε−2 ¨[e]¨
4 2
2
L2(Γ)
+ Cε2¨{e}¨
L2(Γ)
+ ¨g¨
L2(Γ)
¨[e]¨
L2(Γ)
+ Cε2¨{e}¨
L2(Γ)
L2(Γ)
≤ Cε−2 ¨[e]¨2
+ ε ¨{e}¨L2(Γ)
+ ¨g¨L2(Γ) .
By (i) of Lemma 4.8, we have
| ζε{e}, {e} + c0[e], [e] | ≤ C ε¨{e}¨
2
0
L2(Γ)
Γ
Γ
+ ε {e} H1(Γ)
+ ¨[e]¨L2(Γ) .
2 2
2
Inserting (51), (52), (53) leads to
L2(Γ)
| f˜, e | ≤ | f, e | + C ε¨{e}¨2
+ ε {e} H1(Γ)
+ ε−2¨[e]¨2
+ ε−2¨g¨2
≤ C ¨f ¨
(H1,1(Ω0
))'
+ ε−2¨g¨
L2(Γ)
L2(Γ)
¨e¨
2 2
L2(Γ)
)
H1,1(Ω0
+ Cε−2¨g¨2 .
Γe ext Γe ext
L2(Γ)
¨ ¨+ α e
Since 2ab ≤ αa2 + b2/α for all α > 0, we have
∀α > 0, ∃Cα
> 0 : | f˜, e | ≤ Cα
¨f ¨
(H1,1(Ω0
))'
+ ε−2¨g¨
L2(Γ)
2 2
H1,1(Ω0 )
(54)
Γe ext Γe ext
(iii) Now, we bound the traces of e on Γ. Testing (45) with λ′ = 0 we obtain a variational formulation
bc0 (e, e′) := ∫
Ω
0
ext
∇e · ∇e′ dx + βe, e′ Γi + c0 {e}, {e′} Γ + [e], [e′] Γ
1
0
Γ
Γ
= ⟨f, e′⟩ − ζελ,{e′} − λ, [e′] − ζε{e}, {e′} − c0[e], [e′]
Γ Γ
= f˜, e′ . (55)
The left hand side of (55) is H1 (Ω0
)-elliptic (see Definition 4.1) due to the assumption on the boundary
Γe ext
conditions, and it exists a constant γ > 0 such that
2
∀α > 0, ∃Cα > 0 : γ¨e¨H1 (Ω0
) ≤ | f˜, e |
¨ ¨+ α e
Γe ext
≤ Cα
¨f ¨
(H1,1(Ω0
))'
+ ε−2¨g¨
L2(Γ)
2 2
H1,1(Ω0 ).
Γe ext Γe ext
))' + ε
g L2(Γ)
Due to (51), for all α > 0 there exists Cα > 0 such that
2
ε γ {e} H1(Γ) ≤ Cα
f
(H1,1(Ω0
2 ¨ ¨
−2 2 2
),
Γe
We sum the two last equations and get
ext Γe
ext
2
∀α > 0, ∃Cα > 0 : γ¨e¨H1,1(Ω0
) ≤ Cα
¨f ¨
(H1,1(Ω0
))'
+ ε−2¨g¨
L2(Γ)
+ α e H1,1(Ω0
2 2
¨ ¨+ 2α e
H1,1(Ω0 ).
Γe ext
Γe ext
Γe ext
4
Picking α = γ it follows
¨e¨
H
1,1
Γe
ext)
≤ C ¨f ¨
(H
1,1
Γe
0
(Ω
ext
))'
+ ε−2¨g¨
L2(Γ)
(Ω
0
Inserting this bound into (51) and (53) yields
¨{e}¨
L2(Γ)
+ ε {e}
H1(Γ)
+ ε−1¨[e]¨
L2(Γ)
≤ C ¨f ¨
(H
1,1
Γe
0
(Ω
ext
))'
+ ε−2¨g¨
L2(Γ)
1
λ we obtain
(iv) Now, testing (45) with e′ = 0 and λ′ = γε −1
Γ
1
0
Γ
λ, λ = (γε)−1([e] − γε{e} − g), λ .
¨ ¨
Thus, with the Xxxxxx-Xxxxxxx inequality, cancelling a term λ L2(Γ), squaring and using the Young inequality we have
¨λ¨
L2(Γ)
≤ ¨(γε)−1¨
L∞(Γ)
¨[e]¨
L2(Γ)
+ ¨γε¨
L∞(Γ)
¨{e}¨
L2(Γ)
+ ¨g¨
L2(Γ) .
1
0
It follows with the bounds on (γε)−1 and γε given in (i) and (ii) of Lemma 4.8
¨λ¨
L2(Γ)
1
≤ Cε−2 ¨[e]¨
L2(Γ)
0
+ ε2¨{e}¨
L2(Γ)
+ ¨g¨
L2(Γ) .
Finally, estimate (57) allows then to get
¨λ¨
L2(Γ)
≤ C ε−1 ¨f ¨
(H
1,1
Γe
0
(Ω
ext
))'
+ ε−2¨g¨
L2(Γ)
with an ε independent constant C > 0. Summing the bounds for ¨e¨H1,1(Ω0 ) in (56) and for ¨λ¨L2(Γ)
in (58) we have the desired bound and the solution is unique. To show surjectivity we can prove injectivity of the adjoint formulation
Γe ext
Ω
⟨f, e′⟩ + g, e′ Γ = ∫ ∇e · ∇e′ dx + ∫
0
0
ext
βe e′ dS + (c0 + ζε){e}, {e′}
(59)
− γελ,{e′}
+ λ, [e′] Γ + [e], λ′ Γ + ζ {e}, λ − γ λ, λ
0
Γi
Γ
Γ
ε ′ ε ′
1
Γ
1
Γ
similarly to the proof of the original formulation following the items (i-iii), where the assumption of β and the bounds on the respective adjoint operators in Lemma 4.8 are to be used. However, we will not give the proof in detail.
ext
Γe
ext
ext
ext
The variational formulation (43) provides the unique solution e˜ε,N ∈ H1,1(Ω0 ) and λ˜ε,N = {∂ne˜ε,N } ∈ L2(Γ)
ext
∈
of (25) with the concrete transmission conditions in (39) and (40). This means by elliptic regularity theory [18] that e˜ε,N is actually in Hk(ΩΓ) for any k N0 in a neighbourhood ΩΓ of the midline. We will not give a proof of higher regularity, as we do not need it in the following of this article, but refer to [23, Lemma 6.14].
5. Estimates of the modelling error
The derivation of the problems defining e˜ε,N has been formally done. In this section we will prove that e˜ε,N
ext
is indeed an approximation of order N of the solution eε
of the original problem, i. e., e˜ε,N − eε
ext
= o (εN ).
ext
ext
ext
ε→0
ext
This will be done into two steps. First, we will derive the asymptotic expansion of e˜ε,N . Then, we will remark
that the asymptotic expansions e˜ε,N and eε,N coincide up to order N and obtain the final result with triangular
inequalities.
ext
ext
Lemma 5.1 (Asymptotic expansion of the approximate models). Let N = 1, 2, 3 and ΩΓ a neighbourhood of
the midline of the sheet. There exist families of functions of vN,j ∈ H1 (Ω0 )∩H2(ΩΓ) ∞ with vN,j = uj
for j ≤ N satisfying for all m ∈ N
m
ext
Γe ext
j=0
ext
ext
m+1
H (Ω
ext
ext
1 0
ext
)
¨e˜ε,N − Σ εj vN,j¨
j=0
≤ Cm ε , (60a)
m+1
and for N = 2, 3
m
ext
ext
L2(Γ)
¨λ˜ε,N − Σ εj{∂nvN,j}¨
j=0
≤ Cm ε , (60b)
with constants Cm independent of ε.
Proof. Let vN,j = 0 for j < 0 and vN,j for j ∈ N defined by the following system
ext
ext
ext
Γe
ext
v1,j ∈ H1 (Ω0
ext
Γe
ext
) and vN,j ∈ H1,1(Ω0
) for N = 2, 3
N,j 0
N
−∆vext (x) = fj(x), in Ωext,
v
h
i
N,j
ext
(t) =
ΣA=1
ext
(γAvN,j−A)(t)=: γN,j(t), on Γ,
∂nv
N
h
i
N,j
ext
(t) − c0 n
v
,
N,j
ext
(t) =
ΣA=1
ext
(xXxX,j−A)(t)=: δN,j(t), on Γ,
completed with the source term and boundary conditions
ext
f0(x) = f (x) and fj(x) = 0, in Ω0 ,
v
N,0
ext
= 0, and vN,j
= 0, on Γe,
(61b)
N,0
N,0
N,j
ext
N,j
∇vext · n − βvext = ιimp, and ∇vext · n − βvext = 0, on Γi.
ext
Γe
ext
This system uniquely defines the functions vN,0 ∈ H1 (Ω0 ) [26, Lemma 4.2] which do not depend on ε,
ext
and similiarly to [23, Proposition 2.8] we conclude higher regularity and vN,0 ∈ Hk(ΩΓ) for any k ∈ N.
ext
Hence, the right hand sides of the transmission conditions (61a) for N = 1 are given in terms of vN,0 and so γN,1(t) ∈ Hk−3/2(ΩΓ) and δN,1(t) ∈ Hk−5/2(ΩΓ). Repeating these steps we find the well-posedness of (61) for
ext
ext
Γe
ext
vN,j and the regularity vN,j ∈ H1 (Ω0 ) ∩ Hk(ΩΓ) for any k ∈ N and any j ∈ N. Comparing the above system
with (17) leads to the agreement with the terms of the asymptotics of eε,N , i. e., vN,j = uj for j = 0, . . . , N .
ext
ext
ext
Now, let us bound the residual
m m
ext
ext
ext
ext
ext
ext
rε,N,m := u˜ε,N − Σ εjvN,j, λε,N,m := λ˜ε,N − Σ εj{∂nvN,j}. (62)
j=0
j=0
ext
Γe
(i) Let N = 1. The residual rε,1,m ∈ H1 (Ω) solves
ε,1,m 0
−∆rext (x) = 0, in Ωext, (63a)
∂nrext
(t) − c0
1 + ε
rext
(t) = −ε
0
6
vext
(t), on Γ, (63b)
ε,1,m c0 ε,1,m} m+1 c2 1,m}
6
ε,1,m
ε,1,m
∇rext · n − βrext = 0, on Γi. (63c)
Equivalently we can write
Γ
ε,1,m ′ m ′
m+1 c2 ∫
n 1,m, ′ ′ 1
ext
a1(rext , e ) = ⟨lε , e ⟩ := −ε
0
6
vext
(t)e dS, ∀e
∈ HΓe (Ω) (64)
Since v1,m ∈ H1(Ω) does not depend on ε we get
ext
ext
ext
∫ {v1,m}e′ dS ≤ {v1,m}L2(Γ) e′ L2(Γ) ≤ C {v1,m}H1(Ω) e′ H1(Ω) ≤ Cm e′ H1(Ω).
Γ
Γe
m
m+1
Therefore, we get lε (H1 (Ω))' ≤ Cε , and the ε-independent stability of (63) by Lemma 4.6 leads
to the desired result.
(ii) For N = 2, 3, the residual rε,N,m ∈ H1,1(Ω0 ) satisfies
ext
h
Γe ext
ε,N,m 0
N
N
−∆r = 0 in Ω ,
r
ε,N,m
ext
i(t) −
ΣΑ=1
ext
ext
(εΑγΑrε,N,m)(t) = gε,N,m(t) =
(j,Α)Σ∈Jm,N
ext
ext
(εj+ΑγΑvN,j)(t), on Γ
h∂nrε,N,mi (t) − c0 nrε,N,m, (t) − Σ(εΑζΑrε,N,m)(t) = −f ε,N,m(t) = Σ
(εj+ΑζΑvN,j)(t), on Γ
ext
ext
Α=1
ext
(j,Α)∈Jm,N
ext
ext
rε,N,m = 0, on Γe,
ε,N,m ε,N,m
ext
ext
∇r · n − βr = 0, on Γi.
with
Jm,N = {(j, l) ∈ N2 : j ≤ m, l ≤ N and j + l > m}.
Equivalently, the residual solves the variational problem
a
ext
,
=
f ε,N,m, {e′}
+
gε,N,m, λ′
, ∀e′ ∈ H1,1(Ω0
), ∀λ′ ∈ L2(Γ). (65)
rε,N,m e′
N
λ
ε,N,m
ext
λ′
Γ
Γ
Γe
ext
The two linear forms of the right hand side can then be bounded as follows. First we remark that the
ext
Γe
ext
vN,j ∈ H1,1(Ω0
) does not depend on ε. Consequently, we get by integration by part due to the weight
in the definition of the H1,1(Ω0
)-norm in (32)
Γe ext
ζAvext , {e }
≤
D N,j ′ E
Γ
CA,j ′
) and γAvext L2(Γ) ≤ CA,j
N,j
H (Ω
1,1 0
ext
e
ε
ext
Γe
ext
ext
where we have used the Xxxxxx-Xxxxxxxx inequality with vN,j ∈ H1,1(Ω0 ) and {∂nvN,j} ∈ L2(Γ).
ε,N,m m+1
Multiplying by εj+A and summing all these expressions over Jm,N we get
Γ
f ε,N,m, {e′}
≤ Cmεm e′
H1,1
0
ext
and g L2(Γ) ≤ Cmε .
(Ω )
ext
Now, Lemma 4.10 leads to
r
ε,N,m
ext
1 0
H (Ω
ext
) ≤ Cmεm−1, λε,N,m
L2(Γ) ≤ Cmεm−2. (66)
This estimate can be made optimal by considering rε,N,m+2 = rε,N,m − εm+1vN,m+1 − εm+2vN,m+2 and
using the triangular inequality
ext
ext
ext
ext
xxxx X (Ω
ε,N,m
1 0
ext
ε,N,m+2
≤ r) H (Ωext
1 0
ext
) + ε
m+1
N,m+1
vext H (Ω
1 0
ext
) + ε
m+2
N,m+2
vext H (Ω
1 0
ext
) ≤ Cmε
m+1,
ext
since each of these terms is bounded by Cmεm+1. Repeating the same arguments with xx,N,m+3 =
ext
ext
ext
ext
ext
λε,N,m−εm+1vN,m+1−εm+2vN,m+2−εm+3vN,m+3 we get the desired estimate for λ˜ε,N , which completes
the proof.
The stability bound for the Lagrange multiplier of the variational formulation in Lemma 4.10 is not uniform
ext
w.r.t. to ε. With the boundness of {∂nvN,0} and Lemma 5.1 we now reveal the uniform stability of λ˜ε,N .
Corollary 5.2 (Uniform stability of the models of order 2 and 3). For N = 2, 3 for the solutions (e˜ε,N , λ˜ε,N )
of (43) it holds with an ε independent constant C > 0
λ˜ε,N L2(Γ) ≤ C.
Theorem 5.3 (Modelling error). Let N = 1, 2, 3. Then, there exists a constant C independent of ε such that
N+1
H (Ω
ext
ext
1 ε
ext
)
¨e˜ε,N − eε ¨ ≤ Cε . (67)
Proof. Inserting eε,N = ΣN
εjuj
, we get by the triangle inequality since Ωε
⊂ Ω0
ext
j=0
ext
ext
¨e˜ε,N − eε
ext
H (Ω
1 ε
ext
¨
≤ ¨e˜ε,N − eε,N ¨
ext
H (Ω
1 0
ext
)
ext
ext
+ ¨eε,N − eε ¨
ext
H (Ω
1 ε
ext
)
.
)
ext
ext
ext
j=0
ext
Due to Lemma 5.1, we have eε,N = ΣN
xxxX,j and the final estimate follows by (60) and (15).
6. Numerical examples
This section is devoted to the numerical validation of the approximate models of order 1, 2 and 3. The experiments will be performed with the numerical C++ library Concepts [8, 11] using exactly curved elements of high order which easily permits discretisation error lying below the modelling error.
The geometrical setting of the experiments is an ellipsoidal thin sheet, a sheet with varying curvature, with two live circular conductors in the middle (with opposite direction of the currents). The problem is completed by perfect magnetic conductor (PMC) boundary condition on the circular outer boundary which turns out to
ε
b
a
f = −iωµ0j0
int
Ωε
Ωε
ext
(a)
(b)
Figure 2. (a) Geometrical setting w√ith elliptic mid-line (dashed line) with the semi-major
axis a = 1.2 and semi-minor axis b = 0.6. The boundary is a circle of radius R = 2. The xxxx
xxxxx are circles of radius 0.25 and midpoints (±0.5, 0). (b) The magnitude and the flux lines of the in-plane magnetic field for ε = 1/16, c0 = 10 and f = 1 in the left wire and f = −1 in the right one – corresponding to an alternating currents j0 with opposite direction. The flux lines of the magnetic field compass the wires and are almost trapped in the interior area enclosed by the thin sheet.
Γ
(a)
Ωε
int
(b)
M
M
Figure 3. (a) Mesh 0 for the finite element solution of the asymptotic expansion models. The mid-line Γ is labelled. (b) Associate mesh ε for the finite element solution of the exact model with the cells in the sheet, here of thickness ε = 1/16.
1
0.5
x2
0
−0.5
−1
1
0.5
x2
0
−0.5
−1
−1 −0.5 0 0.5 1
x1
(a) No shielding sheet.
−1 −0.5 0 0.5 1
x1
(b) Elliptic sheet with ε = 1/16, c0 = 10.
−1 −0.5 0 0.5 1
x1
(c) Same sheet with c0 = 100.
be a homogeneous Xxxxxxx boundary condition. See Fig. 2(a) for a sketch of the geometry and Fig. 2(b) for the flux lines and the absolute value of the magnetic field induced by the two wires and shielded by the thin sheet (computed with the exact model).
In Fig. 4 we illustrate the shielding behaviour more explicitely. In the first row the magnetic field is shown for different (relative) conductivites c0 and the same thickness of the sheet, where in the first figure no sheet is present. The geometrical setting is like in Fig. 2, where just a part is shown. In the second row the corresponding electric field is plotted. To compare the results the same color scaling is used. In the case of no shielding sheet the fields decays slowly away from the two wires, in or inbetween, respectively, they are mainly concentrated. In the presence of the thin sheets the fields are to some degree trapped in the enclosed area, especially pronounced for most right pictures. The skin depth is dskin = 0.079 = 1.26 ε for c0 = 10 and dskin = 0.025 = 0.4 ε for c0 = 100.
The above computations have been done on meshes Mε resolving the sheet (see Fig. 3(b)) using curved cells with polynomial degree p = 10. For simulation with our transmission conditions we use a limit mesh M0 in which the thin sheet is represented by the midline Γ. The purpose of the two layers of cells around Γ in M0 and
L2(Ωε
ext)
10−5
ε,N
e − e˜
ext
10−6
ε
exterior L2 error
10−7
10−8
10−9
10−5
3.0
2.0
3.8
2.0
2.9
2.5
3.5
4.3
1.5
3.3
2.5
order 1
order 2
order 3
ε,N
ε
e − e˜
L2(Ωε
int)
int
10−7
interior L2 error
10−9
10−11
ext
10−13
ε,N
exterior H error |e − e˜ | 1
H (Ωε )
ext
10−5
ε
10−6
1
10−7
10−3
ε,N
interior H error |e − e˜ | 1
H (Ωε )
int
int
10−5
ε
10−7
1
10−9
10−11
10−4 | 10−3 | 10−2 | 10−1 | 10−4 | 10−3 | 10−2 | 10−1 |
thickness ε | thickness ε |
Figure 5. Convergence of the error of the transmission conditions of order 1, 2 and 3 for the geometry shown in Fig. 2 with c0 = 1 and varying thickness ε. The solution with the transmission condition is computed with p = 18 and subtracted from a numerical approximation
ext
ext
(p = 20) to the exact solution to get the error. The error is measured in the L2(Ωε
)-norm
int
(top left), in the L2(Ωε
)-norm (top right), in the H1(Ωε
)-seminorm (bottom left), and in
int
the H1(Ωε
)-seminorm (bottom right). The numerically observed convergence rates verify the
estimates in Theorem 5.3 and Remark 3.1. Note, that the H1-norm of the exact solution in
Ω
ε
ext
int
around Ωε
is of order 1, so the given absolute errors corresponds nearly to the relative errors.
in Mε are practicable representation of the solution on respective other mesh, which we need here
for the computation of norms of the error, but which is no needed for simulation only using the transmission conditions.
The Figures 5 and 6 show the convergence of the error of the transmission conditions of order 1, 2 and 3
w.r.t. the sheet thickness for the geometry 2 and relative conductivities c0 = 1 and c0 = 250. The error is shown in the L2-norm and the H1-seminorm in the exterior and the interior of the sheet. For the reference solution a polynomial degree p = 20 was choosen higher than that for thex approximative models (p = 18) to observe in the convergence plot when the modelling error falls below the discretisation error. The Lagrange multiplicator was modelled was piecewise continuous elements on the edges of the interface Γ. The polynomial degree has to choosen as high as p = 18 to make the modelling error visible. For the computation of the interior solution we
L2(Ωε
ext)
10−2
10−2
ε,N
ε
e − e˜
ext
L2(Ωε
int)
10−4
exterior L2 error
10−6
10−8
10−5
ε,N
ε
e − e˜
interior L2 error
int
10−8
10−11
H (Ωε )
10−10
10−14
exterior H error |e − e˜ | 1
ext
10−2
ε,N
ext
10−3
ε
10−4
10−5
1
10−6
10−7
10−8
10−4 10−3 10−2 10−1
thickness ε
int
10−2
ε,N
ε
interior H error |e − e˜ | 1
H (Ωε )
int
10−4
1
10−6
10−8
10−10
10−4
2.0
2.9
3.9
3.7
2.0
2.8
2.5
3.5
4.5
1.5
2.5
3.5
order 1
order 2
order 3
10−3 10−2 10−1
thickness ε
Figure 6. Convergence of the error as in Fig. 5, where only the relative conductivity is changed to c0 = 250. The numerically observed rates in ε correspond again to theoretically predicted.
c0
At about ε = 4 · 10−2 = 10 the curves of different orders have a crossing point (dskin ≈ 3 ε).
For larger thicknesses ε the model of order 1 achieves the best results.
used the Lagrange multiplicator for the mean value of the normal derivative and computed locally its second derivative which is present in the model of order 3.
c0
With c0 = 1 (Fig. 5) the range of investigated thicknesses is well below 5
= 5. We observe very low error
t
ext
t
ext
levels for all three models and convergence rates which coincide with the theoretically predicted ones. The convergence stops when reaching the discretisation error gets dominant. For the model of order 3 and the error in the interior this point is reached earlier due to inexact evaluation of ∂2λ˜ε,3 = ∂2{∂ne˜ε,3 }.
For the case c0 = 250 (Fig. 5) we see the same convergence rates as for c0 = 1. In this case a part of
the investigated thicknesses is above the skin depth (above ε = 4 · 10−3), and for about ε = 4 · 10−2 = 10
which correspond to about 3 times the skin depth (dskin = 1/√c =
√ε/√
c0 ≈ 1.3 · 10−3
c0
) the convergence curves
intersect. Below this point it is worthwhile to use transmission conditions of higher orders where above the
point the model of order 1 achieves the best results. Although the last thickness ε = 1 = 31 exceeds largely
8 c0
the proven range of stability (Lemma 4.6 and Lemma 4.10) no stability problem was observed in the presented
numerical experiments.
c0
The error distribution of the electric field in the exterior of the sheet is shown in Fig. 7 for three examples and the transmission conditions of order 1, 2 and 3. The error levels differ largely which are indicated by the indivual colorbars. The first and second examples with sheet thickness ε = 1/16 = 0.026 (relative conductivity c0 = 10)
c
c0
and ε = 1/256 = 3.91 (relative conductivity c0 = 1000) are well and tight, respectively, in the range of validity of the models. We observe also a considerable decrease of the error for the first example when increasing the order, whereas the decrease is lesser in the second example. The thickness ε = 1/16 = 62.5 (relative conductivity
0
c0 = 1000) in the third example lies clearly above the range of validity. The error in this example is smallest
for order 1 and increases for higher model orders. In all the examples the error for the order 1 is mainly located in inner area where the electric field itself has highest values. By increasing the model order the error field is more distributed.
In the context of magneto-quasistatic, we derived three approximate models of order 1, 2 and 3 to take into account the far field behaviour of thin and highly conducting sheets. With these models the sheet has not be discretised, but it is represented by an interface in place of the mid-line on which local transmission conditions are added. Therefore the models can easily be implemented in most of the finite element libraries or codes based on a Galerkin approximation. Once the field outside the sheet (far field) is computed the internal field follows as a polynomial in thickness direction. Our few numerical simulations verify the theoretically achieved estimates for the modelling error and confirm also its range of validity which are thickness up to the order of the skin depth. We end this article with some remarks and open problems.
About regularity of the solution. When the midline Γ of the sheet is regular and the boundary conditions do not create singularities (we think for example to an exterior boundary ∂Ω with corners), the coefficients of the Xxxxxx expansions are also regular. Moreover, a stronger modelling error estimate similar to Theorem 5.3 can be obtained.
It takes the form
N+1
Theorem 6.1. Let N = 1, 2, 3. Then, there exist ε0 > 0 and a constant C independent of ε < ε0 such that
H (Ω
ext
ext
p ε
ext
¨e˜ε,N − eε ¨
≤ Cpε , ∀p ∈ N. (68)
)
We refer to [23] for most of the required ingredients.
About regularity of the boundary. A lot of industrial casings are polygonal or polyhedral and so contains edges. Our first order approximate model can be applied to such geometries. Anyway its justifications require more advance arguments based on a multiscale analysis similar to the one of [7]. The same type of problematic appears for open sheets.
About full Xxxxxxx systems in 2D. When the displacement current is not neglected, one has to face a Helmholtz equation inside the exterior domain. This equation has to be supplemented with radiation condition in order to obtain a well posed problem. This leads to a non-coercive variational formulation and the stability result of Section 4 requires an important modification (one has to act by contradiction). However, the extra ingredients are now classical. One can refer to [15] for a similar problem.
About three-dimensional thin sheets. When the sheet is not z-invariant but completely three-dimensional, the relevant problem is vectorial. However, in the context of IBC, many authors, see [4, 12] for example, have proposed approximate models. One can think to adapt their approach to highly conducting sheets.
ε = 1/16, c0 = 10
6×10−5
4
6×10−6
4
6×10−7
4
2 2 2
0 0 0
-2 -2 -2
-4 -4 -4
-6
-6 -6
ε = 1/256, c0 = 1000
2×10−5
1×10−5
6×10−6
4
1 0.5
2
0 0 0
-2
-1 -0.5
-4
-2 -1
-6
ε = 1/16, c0 = 1000
1×10−3
5×10−3
0.01
0.5
0.005
0 0 0
-0.5
-0.005
-1 -5
-0.01
(a) Order 1.
(b) Order 2.
(c) Order 3.
c0
Figure 7. Error of the electric field in the exterior of the thin conducting sheet for the as- ymptotic models of order 1, 2 and 3 (note the different scalings of the color representation). The configuration in the first row is with a sheet of thickness ε = 1/16 and relative conductivity c0 = 10. The exact solution for this configuration is shown in Fig. 4(b). With this setting the error increases by about one order of magnitude when increasing the order of the model by one (ε = 0.625 ). In the second row the error is plotted for ε = 1/256 and c0 = 1000 and decreases by
c
c0
a factor of 2 per order (ε = 3.91 ). In the third row the configuration is ε = 1/16 and c0 = 1000 and the error increases with increasing order (ε = 62.5 ). Here the skin depth fall below the
0
valid value of the proposed models.
˜ext
The approximate solutions eε,N of order N are defined with the help of two differential operators that consists in the truncation of two formal series of operators, see (25),
N N
γε,N := Σ εjγj and ζε,N := Σ εjζj
INRIA
j=2 j=1
where γ0 = γ1 = ζ0 = 0 and the differential operators γA and ζA that are explicitly given for l ≤ 3 by
(γ u)(t) := − c0 κ(t) {u}(t) + 2 {∂ u}(t) ,
2 24 n
c2
(γ3u)(t) :=
0
240
c2
κ(t) {u}(t) + 2 {∂nu}(t) ,
(ζ1u)(t) := 0
6
{u}(t),
2
12
20
0
t
24
n
1
(ζ u)(t) := c0 7 c2 − ∂2 {u}(t) + c0 κ(t) {∂
u}(t)
(ζ3u)(t) :=
c2 17
c2 −
κ2(t) − ∂2
{u}(t) −
c2
0 κ(t) {∂nu}(t).
0
40
84 0 3 t
240
The interior approximation of order N involves the differential operators
Σ
N
ηε,N := εj ηj
j=0
with
(η0u)(t, S) = {u}(t),
(η u)(t, S) = c0 {u}(t) S2 + 1 + {∂
u}(t) S,
1 2 4 n
0 2 3
3 3
(η2u)(t, S) =
{u}(t) S +
+
{∂nu}(t) S − S
−
0 κ(t) {u}(t) S3 + 3 S
24
6
6
c2 2 c0 c
4
4
4
2
t
n
− 1 κ(t) {∂ u}(t) + ∂2 {u}(t) S2,
+
(η3u)(t, S) =
3
c
c
0
720
0
− 60 κ(t) {u}(t)
c2
{u}(t)
4
S2 + 5
+ S
2
S
4
3 15
4
2
+
16
5
S2 + 1
4
5
2
+ 0
120
S − 0 κ(t) {∂nu}(t) S
c
12
{∂nu}(t) S2 − 5
4
3
2
− 4
2S
S
2
12
t
2
4
8
2
48
− c0 ∂2 {u}(t) S2 + 1 S2 + 1 + c0 κ2(t) {u}(t) S4 + 1 S2 − 1
2
t
3
t
3
2
t
n
+ 1 κ(t)∂2 + 1 κ′(t)∂ {u}(t)S3 + 1 κ2(t) − 1 ∂2 {∂ u}(t)S3.
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