A Fully-Discrete Finite Element Approximation for the Eddy Currents Problem

Un esquema completamente discreto basado en elementos finitos para el problema de corrientes inducidas

Ramiro Acevedo¹ and Gerardo Loaiza²

1 Doctor en Ingeniería Matemática, rmacevedo@unicauca.edu.co, Universidad del Cauca, Popayán, Colombia.

2 Magíster en Ciencias Matemáticas, gloaiza@unicauca.edu.co, Universidad del Cauca, Popayán, Colombia.

Received: 18-04-2012, Acepted: 10-12-2012

Available online: 22-03-2013

MSC:78M10, 65N30

Abstract

The eddy current model is obtained from Maxwell's equations by neglecting the displacement currents in the Ampère-Maxwell's law. The so-called ''A, V - A potential formulation'' is nowadays one of the most accepted formulations to solve the eddy current equations numerically, and Bíró & Valli have recently provided its well-posedness and convergence analysis for the time-harmonic eddy current problem. The aim of this paper is to extend the analysis performed by Bíró & Valli to the general transient eddy current model. We provide a backward-Euler fully-discrete approximation based on nodal finite elements and we show that the resulting discrete variational problem is well posed. Furthermore, error estimates that prove optimal convergence are settled.

Key words: Transient eddy current model, potential formulation, fully-discrete approximation, finite elements, error estimates.

Highlights

• We have proposed a fully-discrete finite element approximation for the eddy current problem. • The eddy current problem is considered in a bounded domain, without topological restrictions on the conductor. • We have shown that at each step the fully-discrete scheme propose is well posed. • We have obtained quasi-optimal error estimates of the typical physical variables of interest of the eddy current problem.

Resumen

El modelo de Eddy Current se obtiene a partir de las ecuaciones de Maxwell, despreciando las corrientes de desplazamiento de la Ley de Ampère-Maxwell. Bíró & Valli realizaron recientemente el análisis de existencia y unicidad de solución y el análisis teórico de convergencia para una de las formulaciones más populares del problema de Eddy Current en regimen armónico, conocida como ''formulación en potenciales A, V - A''. En el presente artículo se extiende el análisis realizado por Bíró & Valli al modelo evolutivo general de Eddy Current. Presentamos un esquema completamente discreto para la formulación, basado en una aproximación temporal usando un método de Euler implícito y una aproximación espacial a través del método de elementos finitos. Además, demostramos que el problema discreto resultante es un problema bien planteado y obtenemos estimaciones del error que muestran convergencia óptima.

Palabras clave: Modelo evolutivo de Eddy Current, formulación en términos de potenciales, esquema completamente discreto, elementos finitos, estimaciones de error.

1 Introduction

In applications related to electrical power engineering (see for instance [1]) the displacement currents existing in a metallic conductor are negligible compared to the conduction current. In such situations the displacement currents can be dropped from Maxwell equations to obtain a magneto-quasistatic submodel usually called eddy current problem ; see for instance [2, Chapter 8]. From the mathematical point of view, this submodel provides a reasonable approximation to the solution of the full Maxwell system in the low frequency range [3].

Among the numerical methods used to approximate eddy current equations, the finite element method (FEM) and methods combining FEM and boundary element method (FEM-BEM) are the most extended, see, for instance, the recent book by Alonso & Valli [4] for a survey on this subject including a large list of references. In the applied mathematical literature, we can find several recent papers devoted to the numerical analysis of the 3D time dependent eddy current model, in bounded domains as well as in unbounded domains by using FEM and FEM-BEM methods: Meddahi & Selgas [5], Ma [6], Acevedo et al. [7], Kang & Kim [8], Prato et al. [9], Acevedo & Meddahi [10], Bermudez et al. [11], Camaño & Rodríguez [12]. The main differences among these works are the selected unknowns to compute the electromagnetic field, the topological assumptions on the conductor domain and the boundary conditions when the problem is solved in a bounded domain.

The aim of this paper is to analyze a finite element fully-discrete approximation for the time-dependent eddy current problem in a bounded domain, based in a formulation in terms of a vector magnetic potential and an electric scalar potential. Numerical experiments showing the efficiency of this approach, were reported by Bíró & Preis (1989) [13], and nowadays, it is the basis of several commercial codes to compute the solution of the eddy current model. Bíró & Valli (2007) [14] have studied that potential formulation for the time-harmonic eddy current model and have proved its well-posedness and theoretical convergence in a general geometric situation. However, a similar analysis for the transient case has not been realized yet. Although, Kang & Kim (2009) in [8] have recently provided quasi-optimal error estimates, showing the theoretical convergence of the method in the case of a simply-connected conductor and by assuming homogeneous conditions for the electromagnetic fields on the boundary of the conductor, these assumptions are very restrictive for many real applications (e.g., metallurgical electrodes [15] and power transformers [16]). Moreover, the decay conditions on the fields at infinity (see [3]) allow to assume that the electromagnetic field is weak far away from the conductor and not on its boundary, as it was assumed by Kang & Kim when the artificial homogeneous boundary conditions are supposed.

In order to improve the results obtained by Kang & King for more realistic applications, we consider a multiply-connected conductor and we opt for a typical approach: to restrict the eddy current equations to a sufficiently large computational domain containing the region of interest and impose a convenient artificial homogeneous boundary conditions for the electromagnetic fields on its border. We provide a backward-Euler fully-discrete approximation based on nodal finite elements to approximate the solution of the resultant model. This fully-discrete scheme solves in each time an elliptic problem and we use the techniques used in [14] and [17] to prove the ellipticity of its bilinear form. Furthermore, by using this ellipticity we define projection operators to the discrete finite element subspaces and obtain quasi-optimal error estimates, which allows us to approximate the typical physical variables of interest of the eddy current problem: the eddy currents in the conductor and the magnetic induction in the computational domain.

The outline of the paper is as follows: In section 2, we summarize some results concerning tangential traces in H(curl; Ω) and recall some basic results for the study of time-dependent problems. In Section 3, we introduce the eddy current model in a bounded domain and deduce a potential-based formulation (the so-called ''A, V - A potential formulation'') for the time-dependent eddy current problem. In Section 4, we obtain a variational formulation for the problem and its fully-discrete approximation scheme is analyzed in Section 5. Finally, the results presented in Section 6 prove that the resulting fully discrete scheme is convergent in an optimal way. We end this paper by summarizing its main results in Section 7.

2 Preliminaries

We use boldface letters to denote vectors as well as vector-valued functions and the symbol represents the standard Euclidean norm for vectors. In this section Ω is a bounded open set in ³ with a Lipschitz boundary. We denote by Γ its boundary and by n the unit outward normal to Ω. Let

2.1 Normal and tangential traces

We recall that

which is bounded, surjective and possesses a right inverse. Moreover, the following Green's identity holds for any v H(div; Ω) and φ H¹(Ω)

where, as usual, v n denotes γ_n(v). Furthermore, we denote by H₀(div; Ω) the kernel of γ_n in H(div; Ω), i.e.,

We also recall that

endowed with the standard norm in L²(Γ)³.

We define the tangential trace γ_τ:C^∞ L²τ (Γ) and the tangential component trace π_τ:C^∞ as γ_τv:= v_Γxn and π_τv:= n x (v_Γ x n) respectively. The previous traces can be extended by continuity to H¹.

The spaces := γ_τ(H¹) and := π_τ (H¹), are respectively endowed with the Hilbert norms

Let us notice that the density of H^1/2(Γ)³ in L²(Γ)³ ensures that and are dense subspaces of L²τ (Γ). The dual spaces of and with as pivot space, are denoted by and respectively.

it is easy to deduce that γ_τ and π_τ can be extended to define bounded tangential mappings from H(curl; Ω) onto onto respectively. The space H₀(curl; Ω) stands for the kernel of γ_τ in H(curl; Ω), i.e.,

where, as usual, v_Γ x n denotes γ_τv.

The ranges of γ_τ and π_τ are characterized in the following result. We refer to [19, 21] for the definition of the differential operators div_Γ and curl_Γ on piecewise smooth Lipschitz boundaries.

Theorem 2.1. Let

and

Then

are surjective and possess a continuous right inverse.

Proof. See Theorem 4.1 and Lemma 5.6 of [21].

Since in the eddy current problem the physical domain requires to be split into two non-overlapping domains: the conductor and the insulator domain (see Section 3.1 below), we end this subsection by recalling the following classical result that will be used often in the rest of the paper.

Lemma 1. Assume Ω is split into two parts Ω = _{1 2}, where Ω₁ and Ω₂ are two non-overlapping Lipschitz domains. Let Σ := Ω₁ Ω₂ be the interface between the two subdomains and let ν be a unit normal to Σ.

a) A function v belongs to H(div; Ω) if and only if its restrictions v_Ω1 and v_Ω2 belong to H(div; Ω₁) and H(div; Ω₂), respectively, and

b) A function v belongs to H(curl; Ω) if and only if its restrictions v_Ω1 and v_Ω2 belong to H(curl; Ω₁) and H(curl; Ω₂), respectively and

Proof. See, for instance, [18, Lemma 5.3].

2.2 Basic spaces for time dependent problems

Since we will deal with a time-domain problem, besides the Sobolev spaces defined above, we need to introduce spaces of functions defined on a bounded time interval (0, T) and with values in a separable Hilbert space V, whose norm is denoted here by _V. We use the notation C⁰([0, T]; V) for the Banach space consisting of all continuous functions f : [0, T] V. More generally, for any k , C^k([0, T]; V) denotes the subspace of C⁰([0, T]; V) of all functions f with (strong) derivatives of order at most k in C⁰([0, T]; V), i.e.,

We also consider the space L²(0, T; V ) of classes of functions f : (0, T) V that are Böchner-measurable and such that

Furthermore, we will use the space

3 The model problem

3.1 Eddy current problem

We consider a standard eddy current problem: to determine the electromagnetic fields induced in a three-dimensional conducting domain by a given source time-dependent compactly-supported current density. The eddy current problem is in principle posed in the whole space. However, we restrict it to a bounded computational domain containing both, the conductor and the support of the source current, such that adequate boundary conditions can be imposed on its boundary. To this aim, we choose the geometry of the computational domain as simple as possible (e.g., simply connected with a connected boundary).

Let Ω_c ³ be the conducting domain and let us assume that it is an open and bounded set with boundary Γ_c. Let Ω ³ be a simply connected bounded domain with a connected boundary Γ, such that _c Ω. We suppose that both, Ω and Ω_c are Lipschitz domains and we denote by n and n_c the outward unit normal vectors to Ω and Ω_c, respectively. Let Ω_d:= Ω \ _c be the subdomain of Ω occupied by dielectric material, which includes the support of the source current J_d (see Figure 1).

The electric and magnetic fields E : Ω_c x [0, T] ³ and H : Ω x [0, T] ³ are solutions of a submodel of Maxwell's equations obtained by neglecting the displacement currents (see, for instance, [13]):

Let us remark that because of H(, t) must belong to H(curl; Ω) a.e. in [0, T], Lemma 1 implies the magnetic field has to satisfy the coupling conditions, Equation (8).

The magnetic permeability µ and the conductivity σ are bounded functions satisfying:

The data of the problem are the source current density J^d L²(0, T; L²), for which we assume for a.e. t (0, T)

and the initial magnetic field H₀ H(curl; Ω).

3.2 The A, V - A potential formulation

We now recall a classical formulation of the eddy current problem in terms of two potentials: a magnetic vector potential A and an electric scalar potential V. We refer to Bíró & Preis [13] for a detailed discussion, which also includes numerical tests showing the efficiency of this approach.

In order to introduce the magnetic vector potential, we notice that, by using [23, Theorem I.3.5.], Equation (6) implies that there exists a unique A L²(0, T; H(curl; Ω)) such that

We notice that from (8) and (10) it follows that

Moreover, since A L²(0, T ; H(curl; Ω)), from Lemma 1 and Equation (11) we have

Next, according to Bíró & Preis [13] (see also Bíró & Valli [14]) we introduce an electric scalar potential V : Ω_c x [0, T] , such that

If we define

we obtain

Hence, from (3) there follows that

Moreover, since H(, t) H(curl; Ω) a.e. t in [0, T], we have curl H(, t) H(div; Ω) a.e. t in [0, T], and consequently, from Lemma 1 we have

Then, from (3), (5) and recalling that suppJ^d(, t) Ω_d a.e. t in [0, T], we obtain

Therefore, the Equation (17) implies that

On the other hand, using the Equation (3) together with the Equations (10) and (17), there follows that

Moreover, from the Equations (5) and (10), we have

The Equations (11)-(15) together with the Equations (18)-(21) will be collected in the potential formulation. Hence, we are led to the following formulation of problem (3)-(9) in terms of the potentials A : Ω x [0, T] ³ and v : Ω_c x [0, T] :

and satisfying

Remark 3.1. Equation (33) is obtained from the Equations (7), (9) and (10), and the Equation (32) is a consequence of (16). Furthermore, the initial condition A₀ in (31) must satisfy

and since Ω is a simply connected set, it can be characterized as the unique solution of the boundary value problem (see [23, Theorem I.3.5.]):

Moreover, A₀ can be computed by using a finite element approximation of the following mixed problem [24, Section 4]: find (A₀, λ) H(curl; Ω) x M such that

where

In order to prove the well-posedness of this last problem, we can use the well-known Babuska-Brezzi theory for mixed problems (see for instance [25] and the references given there). Furthermore, it follows easily that the Lagrange multiplier λ vanishes identically.

Remark 3.2. The differential constraint (25) is called the Coulomb gauge condition and it is necessary to assure the uniqueness of potential A [13]. The Coulomb gauge condition is not easy to treat at the discrete level, because it is not simple to construct a suitable space of finite elements which are divergence-free. Here we will follow the ideas from [14], by using the addition of a div - div term to our variational formulation (see Section 4 bellow), which is equivalent to add a penalization term in the Ampère law (see, for instance, [13, 14]).

Another possible alternative can be to introduce a Lagrange multiplier (as λ in the Equation (34)) or use an additional scalar magnetic potential. These approaches have been used to impose some differential constraints, which are necessaries for other eddy current formulations, see for instance [4, Chapters 4-5], [5], [7], [10], [15], [26].

4 Variational formulation

The aim of this section is to give a variational formulation of Problem, Equations(22)-(33). To do this, we introduce the following functional spaces:

where are the connected components of Ω_c. The spaces X and ; (Ω_c) are endowed respectively with the norms

and

Let Z X. By multiplying the Equations (22) and (24) by Z, integrating in Ω_c and Ω_d respectively and summing the resulting equations, we deduce that

We notice now that from (22), (24), (28) and Lemma 1 it follows that

Then, by integrating by parts (see (2)) and using (33), we obtain

Hence, from (37) we have

for all Z X.

On the other hand, by integrating by parts (1) and using (23) and (27)

Summing the last two equations, we obtain the following variational formulation of Problem (22)-(33):

Find (A, v) L²(0, T; X x ; (Ω_c)) H¹(0, T; L²(Ω_c)³ x ; (Ω_c)) such that

and satisfying the initial condition

where

and

Remark 4.1. To our knowledge, the well-posedness of Problem (38)- (39) has not been proved yet. We have tried to apply the theoretical framework for parabolic problems arising from electromagnetism (see Zlámal [27, 28]) and for classical degenerate parabolic problems (see Showalter [29, Chapter III]), without satisfactory results in both cases. However, by adapting the techniques from [27, 28], it is easily seen that Problem (38)-(39) has at most one solution, but this is not the case for the existence result. The main difficulty is that the bilinear form

is not an inner product in L²(Ω_c) x ; (Ω_c).

5 A fully discrete scheme

In what follows we assume that Ω and Ω_c are Lipschitz polyhedra (we recall that Ω is simply-connected). Let {T_h}_h be a regular family of tetrahedral meshes of Ω such that each element K T his contained either in _c or in _d. As usual, h stands for the largest diameter of tetrahedra K in T_h.

Consider the following finite element spaces:

and

We consider a uniform partition {t_n:= nΔt: n = 0, . . . , N} of [0, T] with a step size Δt := . For any finite sequence {θⁿ: n = 0, , N}, let

The fully-discrete version of Problem (38)-(39) reads as follows:

Find such that:

for any (Z, u) X_h x M_h and satisfying

where A_0,h X_h is a suitable approximation of A₀ to obtain optimal error estimates (see Corollary 6.1 below).

In order to prove that Problem (40)-(41) has a unique solution, we first notice that at each iteration step we need to find such that

where

Hence, the existence and uniqueness of solution of Problem (40)-(41) follows by combining the Lax-Milgram's Lemma and the following ellipticity result.

Lemma 2. There exists a constant C > 0 such that

for any Z X and u ; (Ω_c).

Proof. First of all, we notice that for Z X, u ; (Ω_c) we have

Now, since Ω is a Lipschitz and simply-connected set, there exists a constant C₁ > 0 such that (see, for instance, [23, Lemma I.3.6] or [24, Corollary 3.16])

Consequently, from (43), we obtain

Then by noticing that

and using the inequality

it follows that

Consequently, by taking 1/2 < ε < 1, we obtain (42).

6 Error estimates

In this section we will prove error estimates for our fully-discrete scheme. To this end, we have to assume that Problem (38)-(39) has a unique solution and consider the elliptic projection operators P_h: X X_h and Q_h : ; (Ω_c) M_h defined respectively by

and

It is a simple matter to see that Lax-Milgram's Lemma and Lemma 2 imply that P_h and Q_h are well defined. Moreover, the well-known Cea's Lemma yields that there exist positive constants C₁ and C₂, independent of h, such that

and

From now on, let us introduce the following notations:

and

Furthermore, we denote

Lemma 3. There exists a positive constant C, independent of h and Δt, such that

for n = 1, . . . , N.

Proof. Let 1 ≤ n ≤ N and 1 ≤ k ≤ n. It is straightforward to show that

for any (Z, u) X_h x M_h.

and

we obtain

Then, by using the Cauchy-Schwartz inequality in the right hand side, it follows that

for k = 1, . . . , n. In particular,

for k = 1, . . . , n. Then, summing over k, we get

Hence,

Therefore, the discrete Gronwall's Lemma (see, for instance, [31, Lemma 1.4.2]) leads to

for n = 1, . . . , N. Inserting the last inequality in (48), it follows that

for k = 1, . . . , n. Now, summing over k and recalling that = 0, we get the estimates

and therefore

Consequently, using (44), we have

for n = 1, . . . , N.

Then,

and consequently, using the Cauchy-Schwarz inequality on the right hand side, we deduce that

Hence, using the inequality

on the left hand side, leads to

for k = 1, . . . , n. Summing over k, we have

Therefore, by noticing

for k = 0, 1, . . . , n, it follows that

Then, we obtain

for n = 1, . . . , N.

Combining this last estimate with (49), using Lemma 2 and noticing that

we conclude the desired estimate.

Theorem 6.1. Let . If

there exists a constant C > 0, independent of h and Δt, such that

Proof. Since , from the previous lemma we obtain

Hence

The regularity assumptions on A and v allow us commute the time derivative with P_h and Q_h, i.e,

Moreover, if we define := A(t) - P_hA(t) and then from (45) it follows that

and, analogously if := v(t) - Q_hv(t), (46) implies

On the other hand, the Cauchy-Schwarz inequality shows that

thus, from Taylor's Theorem we have

Analogously, we deduce

Finally, the result follows from using (52)-(55) and (45)-(46) in (51).

Corollary 6.1. Let and assume that

and

for some 0 < s < 1. If A_0,h =Π_h(A₀), where Π_h : X H^1+s X_h is the Lagrange interpolant, then there exists a positive constant C, independent of h and Δt, such that

Proof. Let Π_h: ; (Ω_c) H^1+s(Ω_c) M_h be the standard scalar finite element Lagrange interpolant. The result follows by using previous theorem, the regularity assumptions on A and v, and the well-known approximation properties of Π_h and Π_h (see, for instance, [32]).

Remark 6.2. Concerning this convergence result, it has to be noted that the spatial regularity of A (and in particular the regularity of A₀) is not ensured if Ω has reentrant corners or edges, namely, if it is a non-convex polyhedron (see [33], [34]). More important, in that case the space H¹ H(div; Ω) turns out to be a proper closed subspace of X and hence the nodal finite element approximate solution cannot approach an exact solution A L²(0, T; X) with A L²(0, T; H¹ H(div; Ω)), and convergence in X is lost.

However, the result we have proved here above ensures that the nodal finite element approximation is convergent either if the solution is regular (and this information could be available even for a non-convex polyhedron Ω) or if Ω is a convex polyhedron. Actually, in [17, Section 5] is underlined that if Ω is a convex polyhedron and µH H^pwith 0 < p ≤ 1, the vector potential A (which is characterized by (10)-(12)) belongs to H^1+sfor some 0 < s ≤ 1. Let us also note the assumption that Ω is convex is not a severe restriction, as in most real-life applications Ω arises from a somehow arbitrary truncation of the whole space and hence, reentrant corners and edges of Ω can be easily avoided.

A cure for the lack of convergence of nodal finite element approximation for Maxwell equations in the presence of reentrant corners and edges has been proposed by Costabel and Dauge in [35], where they introduce a special weight in the penalization term, thus making it possible to use standard nodal finite elements in a numerically efficient way. Another alternative for the eddy current problem has been reported by Bíró [36]: edge elements are employed for the approximation of the potential A, without requiring that the Coulomb gauge condition be satisfied. However, a complete theory assuring the effectiveness of this last approach, even for the harmonic case, is not available.

To end this section, let us notice that we can approximate at each time t_n the physical quantities of interest: the eddy currents σE(t_n) in the conductor Ω_c and the magnetic induction field µH(t_n) in the computational domain Ω. In fact, the identities (17) and (10) suggest the approximations

Moreover, the last corollary allows us to obtain quasi-optimal error estimates for these approximations in some natural discrete L²-norms. In fact, to obtain the error estimate for the first approximation, we notice that

for any n = 1, 2, . . . , N. Therefore, by using (54), (55) and Corollary 6.1, it follows that

Finally, to get the error estimate for the magnetic induction approximation, we notice that

for n = 1, 2, . . . , N and use Corollary 6.1 to obtain

7 Conclusions

We have proposed a fully-discrete approximation for the so-called ''A, V - A potential formulation'' of a transient eddy current problem in a bounded domain, without topological restrictions on the conductor. The fullydiscrete scheme is obtained by a finite element space discretization of standard nodal finite elements and a time discretization based on a backward-Euler implicit scheme.

We have shown that at each step the fully-discrete scheme solves an elliptic problem and consequently the well-posedness of the discrete problem follows by using the well-known Lax-Milgram's Lemma. Furthermore, under usual regularity assumptions on the solution of the continuous problem, we obtain quasi-optimal error estimates, which allows us to approximate the typical physical variables of interest of the eddy current problem: the eddy currents in the conductor and the magnetic induction in the computational domain.

Acknowledgment

The authors would like to thank the Universidad del Cauca for providing time for this work through research project VRI ID 2665 and anonymous referees for constructive suggestions to the first version of this paper, which allowed us to improve the presentation of this article.

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