ammari eddy

A JUSTIFICATION OF EDDY CURRENTS MODEL FOR THE MAXWELL EQUATIONS∗ H. AMMARI†, A. BUFFA‡,ANDJ.-C. N ´ED´ELEC†SIAM J. A PPL.MATH. c/circlecopyrt2000 Society for Industrial and Applied Mathematics Vol. 60, No. 5, pp. 1805–1823 Abstract. This paper is concerned with the approximation of the Maxwell equations by the eddy currents model, which appears as a correction of the quasi-static model. The eddy currentsmodel is obtained by neglecting the displacement currents in the Maxwell equations and exhibits anelliptic character in the time-harmonic formulation. Our main concern in this paper is to show thatthe eddy currents model approximates the full Maxwell system up to the second order with respect to the frequency if and only if an additional condition on the current source is fulfilled. Otherwise, itis afirst-order approximation to the Maxwell equations. We also study the well-posedness of the eddy currents model and investigate the time-dependent case. All our results strongly depend onthetopology properties of the domains under consideration. This dependence which is specific to Maxwell’s equations does not appear for the two- or the three-dimensional Helmholtz operator. Key words. eddy currentproblems, Maxwell’s equat ions, low-frequency analysis, validit y of approximation AMS subject classifications. 35C20, 35B40, 35Q60 PII.S0036139998348979 1. Introduction. In many problems encountered in electromagnetism, the nu- mericalresolutionofthefullsystemofMaxwell’sequationscanbeextremelyexpensivein terms of computer time. In some cases, however, it is possible to use a simplifiedmodelwhichapproximatesMaxwell’sequationsinsomesenseandwhichcanbesolvedin a more economic way. Such a situation happens, for instance, in problems in con-nection with machines working at power frequencies. In that particular case, onecan use the so-called eddy currents model which differs from Maxwell’s equations byneglecting the displacement currents. Significant mathematical and numerical results on eddy current problems have been obtained by Hariharan [11], Hariharan and MacCamy [12], and MacCamy andStephan [16] for a two-dimensional problem. The boundary integral equations meth-ods are successfully applied in these papers to solve two- and three-dimensional eddycurrent problems. It is now allowed that in many situations the topology of a conductor strongly influences the surface current density in the presence of an exciting electromagneticfield. This is not the case for a dielectric material. The topology of the surface ofa dielectric material does not have any significant effect. Our main concern in thispaperistoshowthatthetopologyoftheconductoraffectstheuseoftheeddycurrentsmodel for approximating Maxwell’s equations. We will prove that the eddy currentsmodel approximates the full Maxwell system up to the secondorder with respect to the frequencyifandonlyif the limit of the source current as the frequency goes to zero is divergence free in the conductor and is orthogonal to all the Neumann fields in ∗Received by the editors November 25, 1998; accepted for publication (in revised form) November 22, 1999; published electronically May 26, 2000. This research was partially supported by IAN-CNR(Italy) and CNRS (France). http://www.siam.org/journals/siap/60-5/34897.html †Centre de Math´ ematiques Appliqu´ ees, CNRS UMR 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France ([email protected], [email protected]). ‡Dipartimento di Matematica, Universit´ a di Pavia, Via Abbiategrasso 209, 27100 Pavia, Italy ([email protected]). 1805 1806 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC the conductor. Otherwise, it is a first-order approximation to the Maxwell equations. This influence of the topology is very specific to full Maxwell’s equations and doesnot appear in the models studied by MacCamy and Stephan [16], [17]. Note that ifthe conductor is connected, and simply connected, then the only necessary conditionis that the limit of the source current as the frequency goes to zero is divergence free. In this paper we also prove the well-posedness of the eddy currents model and investigate the time-dependent case. These results also depend on the topologyprop- ertiesof the domain under consideration. Note that this dependance is specific to Maxwell’s equations. It does not appear for the Helmholtz equation or for the modelsinvestigated in [16] and [17]. To the best of our knowledge, the results established inthis paper are new and the topological considerations discussed here are significant insome applications related to complex electronic systems and the design of subshieldsand grounding networks as shown in the paper by Baum [6]. The paper is organized as follows. Section 2 is concerned with the formulation of the model problem and some auxiliary results. The time-harmonic Maxwell systemand the eddy currents model are presented. In section 3, we study the well-posednessof the eddy currents model. The variational formulation of the eddy currents modelrelies on a classical Hodge decomposition lemma (Lemma 2.1) and Proposition 3.1,which asserts that the electric and magnetic fields, solutions of the eddy currentsmodel, are square integrable on all R 3. In section 4, we focus our attention on the Neumann fields in the conductor. Section 5 is devoted to the asymptotic analysis ofthe eddy currents model at low frequencies. The analysis is based on an asymptoticexpansion of the solution in powers of the frequency. Recalling the low-frequencyanalysis of Maxwell’s equations in section 6, we prove in section 7 the main result ofthis paper which is concerned with the approximation of the Maxwell equations bythe eddy currents model. Finally in section 8 we deal with the time-dependent case.We prove that the same results are valid under suitable hypothesis on the spectrumof the source current. This justifies the use of the eddy currents model as a second-order approximation if and only if the additional condition on the source current is fulfilled. Otherwise,the eddy currents model gives a first-order approximation to the Maxwell equations. 2. Notationsandproblemformulation. Inthissectionwegiveaprecisedefi- nitionofthetypeofgeometryinwhichweareworkingandwemakesomeassumptionson the material functions: the electric permittivity, the magnetic permeability, andthe conductivity. Next, we present the dissipative Maxwell equations and the eddycurrents model and discuss compatibility conditions on the source current. LettheconductorΩ Cbeaboundedopensubsetof R3withaLipschitz-continuous boundaryΓ. WedenotebyΓ iwithi=1,...,ptheconnectedcomponentsofΓandby ntheouternormaltoΓ. AssumethattheconductorΩ Cisnotsimplyconnected,there exists Σ j⊂ΩC, withj=1,...,q,qregular disjoint manifolds such that Ω C\∪q j=1Σj is simply connected. We set Ω e=R3\ΩC. Let the material functions ε,µ,andσbe the electric permittivity, the magnetic permability, and the conductivity, respectively.We make the following general assumptions: •ε(x),µ(x), andσ(x) are all real valued bounded functions. Furthermore, ε(x)≥ε 0>0,µ(x)≥µ0>0, andσ(x)≥σ0>0i nΩ C, whereε0,µ0, and σ0are positive constants. •There existsR>0 such that outside the sphere SR={|x|=R}the material parametersεandµare constants. We denote by BR={|x|<R}the corresponding ball and by Ω R=Ω e∩BR. EDDY CURRENTS MODEL 1807 •σ(x)≡0i nΩ e. •The source current Jis such that its support is included in the ball BR. Furthermore, it is square integrable on BRand it may depend on the angular frequency. In order to shorten our notations we set ε(x)≡µ(x)≡1 forx∈R3\BR. LetL2(Ωe) (resp., L2(ΩC)) be the space of square integrable complex functions on Ω e(resp., Ω C) and H−1/2(Γ) the classical Sobolev space on Γ of order −1/2. We also introduce the classical Sobolev spaces H1(Ωe)={ϕ∈L2(Ωe);gradϕ∈L2(Ωe)3}, H1(ΩC)={ϕ∈L2(ΩC);gradϕ∈L2(ΩC)3}, provided with the norm ||ϕ||H1(Ωe)=(||ϕ||2 L2(Ωe)+||gradϕ||2 L2(Ωe)3)1/2, ||ϕ||H1(ΩC)=(||ϕ||2 L2(ΩC)+||gradϕ||2 L2(ΩC)3)1/2, respectively, and the following functional spaces H(curl,Ωe): ={u∈L2(Ωe)3;curlu∈L2(Ωe)}, H(curl,ΩC): ={u∈L2(ΩC)3;curlu∈L2(ΩC)3}, H(div0 ε,Ωe): ={u∈L2(Ωe)3;div(εu)=0i nΩ e}, H(div0 σ,ΩC): ={u∈L2(ΩC)3;div(σu)=0i nΩ C} normed by ||u||H(curl ,Ωe)=(||u||2 L2(Ωe)3+||curlu||2 L2(Ωe)3)1/2, ||u||H(curl ,ΩC)=(||u||2 L2(ΩC)3+||curlu||2 L2(ΩC)3)1/2, ||u||H(div0 ε,Ωe)=(||u||2 L2(Ωe)3+||εu.n||2 H−1/2(Γ))1/2, ||u||H(div0 σ,ΩC)=(||u||2 L2(ΩC)3+||σu.n||2 H−1/2(Γ))1/2. The system of time-harmonic dissipative Maxwell equations is curlHm=+iεωEm+σEm+JinR3, (2.1) curlEm=−iµωHminR3, (2.2) lim/parenleftbigg Hm×x |x|−Em/parenrightbigg =0, uniformly for |x|→∞, (2.3) whereωis the angular frequency and the sup-index mstands for Maxwell. The dissipation in the system of equations (2.1)–(2.3) is due to the conductivity functionσ(x), which satisfies σ(x)≥σ 0>0 in the conductor Ω C. Note that the system of equations (2.1)–(2.2) is written in the sense of distributions. Since the problem 1808 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC will be to find the fields EmandHmlocally in L2(R3), then curl Emand curlHm will be locally in L2(R3), so there is no jump of the tangential components of the fieldsEmandHmacross any interface [14]. However, the normal components are not continuous. In fact, there is no jump of ( iεωEm+σEm+J).nandµHm.n, wherenis a unit normal to the interface. By this skew an interface separating two materials with different parameters plays a key role in the formulation of the systemof Maxwell’s equations. Dealingwithproblemsinconnectionwithmachinesworkingatpowerfrequencies, the full system of the Maxwell equations (2.1)–(2.3) is usually simplified. Since theangular frequency ωis very small, displacement currents are usually neglected, which means thatωεis set to zero in Ω e. Moreover, the Silver–M¨ uller radiation condition (2.3) is replaced by (2.7)–(2.8) below, which holds uniformly for |x|→+∞. All these modifications together yield the eddy currents model, which appears as a correctionof the quasi-static model. We refer the reader to Bossavit [8] and Kettunen, Fors-man, and Bossavit [15] for a derivation of the eddy currents model and its numericalimplementation. Note that the eddy currents model has the advantage to exhibit an elliptic char- acter. In the time-dependent case, the transient eddy currents model yields parabolic-elliptic interface problems; mainly, the Laplace equation in one domain coupled withthe heat equation in the other domain. On these problems, significant mathematicalandnumericalresultshavebeenobtainedbyMacCamyandSuri[18], Costabel, Ervin,and Stephan [10], and Lee, Hariharan, and Ida [13]. Assume that div( J)=0i nΩ e. The mathematical model of the eddy currents has then the following form: curlH=σE+J inR3, (2.4) curlE=−iµωH inR3, (2.5) div(εE)=0 i nΩ e, (2.6) H(x)=O/parenleftBigg 1 |x|/parenrightBigg ,uniformly for |x|→∞, (2.7) E(x)=O/parenleftBigg 1 |x|/parenrightBigg ,uniformly for |x|→∞. (2.8) From the above discussion on interface conditions, standard jump conditions may be deduced from the above system. In fact, the tangential parts of the electric andmagnetic fields are continuous across the interface Γ. Then we have [σE.n] Γ=σEint.n=−[J.n]Γ, where [f]Γdenotes the jump of the function facross the interface Γ and Eint≡E|ΩC. Due to (2.4) the current source Jhas to satisfy some compatibility conditions. In particular, outsidetheconductorΩ C, thetermσEisidenticallyzeroandthefunction Jis then the curl of the magnetic field H. In order to characterize this constraint on J, we introduce three other functional spaces: gradH1 0(Ωe): ={gradϕ;ϕ∈H1(Ωe),ϕ|∂ΩC=0}, Dεext:={d∈L2(Ωe)3;curld= 0 in Ω e; EDDY CURRENTS MODEL 1809 div(εd)=0i nΩ e;d×n|Γ=0}, curlH(curl,Ωe): ={curlu;u∈H(curl,Ωe)}. Note that the space Dεextis formed only by the Dirichletfields ; it is well known that it is finite-dimensional and its dimension is equal the number of connected com-ponents of the boundary Γ. A complete characterization of this space is given in [20],[21], and [23], and, in particular, D ε ext≡span{gradϕj;ϕj∈H1(Ωe),div(εgradϕj)=0i nΩ e, ϕj|Γi=δi,j; fori,j=1,...,p }, whereδi,jdenotes the Kronecker index. The following lemma is a simple consequence of the Hodge decomposition pre- sented in [20] and of some results on the Dirichlet fields obtained in [23]. Lemma 2.1.ThefollowingHodgedecompositionholds: L2(Ωe)3= curl H(curl,Ωe)⊥⊕gradH1 0(Ωe)⊥⊕Dε ext, (2.9) where⊥⊕denotes the direct sum of spaces which are orthogonal with respect to the L2-scalarproduct. From the previous Lemma and (2.4), it follows that J⊥gradH1 0(Ωe)⊥⊕Dεext⇐⇒div(J)|Ωe=0 and/integraldisplay ΓjJ·nd Γ=0 ∀j=1,...,p.(2.10) From now on, we assume that Jsatisfies the right-hand side (or the left one) of (2.10). Ourconcerniswiththelow-frequencysolutionsoftheMaxwellsystem(2.1)–(2.3) and the eddy currents model (2.4)–(2.8). Our aim is to show that ( E,H) is a second- order approximation of ( Em,Hm) when the frequency goes to zero if and only if the following additional condition on the source current Jis fullfilled: the zero-order term in the asymptotic expansion of the source current with respect to the frequency ωis divergence free in the conductor Ω Cand it is orthogonal to all the Neumann fields in ΩC. 3. Existence and uniqueness for the eddy currents model. Inthissection, we investigate questions on existence and uniqueness for the eddy currents model(2.4)–(2.8). Our main result is that problem (2.4)–(2.8), together with the additionalcondition on the electric field /integraldisplay ΓjεE.n= 0 forj=1,...,p, (3.11) is well posed. Let us state a useful result on the behavior of the electric and magnetic fields as |x|goes to + ∞. Proposition 3.1. LetHandEbetwovectorfieldssatisfyingthesystem (2.4)– (2.8). Thentheyhavethefollowingbehaviorattheinfinity: H(x)=O/parenleftBigg 1 |x|2/parenrightBigg ,E(x)=O/parenleftBigg 1 |x|2/parenrightBigg ,uniformly for |x|→∞. (3.12) 1810 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC Proof. It suffices to prove the result for the electric field E. From (2.4)–(2.8) we have, by making use of the identity ∆ = grad div −curlcurl and recalling that divεE= 0 in Ω e, ∆E=0 i n R3\BR, divE=0 i n R3\BR, E(x)=O/parenleftbigg1 |x|/parenrightbigg ,uniformly for |x|→∞.(3.13) Let{Ym l}−l≤m≤lbe an orthonormal sequence of spherical harmonics of order lon the unit sphereS1, normalized such that /integraldisplay S1Ym l¯Ym/prime l/prime=δl,l/primeδm,m/prime. According to [9] it follows from (3.13) that the vector field Ehas the following expan- sion: E=E(r,θ,ϕ)=∞/summationdisplay l=0l/summationdisplay m=−l1 rl+1Em lYm l(θ,ϕ), (3.14) wherer>R,θ∈[0,2π) andϕ∈[−π/2,π/2) are the spherical coordinates in R3and {Em l}l,mare constant vectors in R3. By imposing (3.13), we obtain divE(r,θ,ϕ)=∞/summationdisplay l=0l/summationdisplay m=−lgrad/parenleftBigg 1 rl+1Ym l(θ,ϕ)/parenrightBigg ·Em l=0 ∀r>R,θ,ϕ. (3.15) Recalling that grad Ym l=1 rgrad S1Ym land grad S1Y0 0= 0, we arrive at 0 = divE=1 r2E0 0·nr+O/parenleftbigg1 r3/parenrightbigg , (3.16) wherenr=nr(θ,ϕ) denotes the unit radial vector outer to the ball BR. Since the first term on the right-hand side of (3.16) needs to be zero for any value of θandϕ, we necessarily have E0 0= 0 and this, together with (3.14), yields (3.12). One important consequence of Proposition 3.1 is that the vector fields EandH are in L2(R3). This is a key property while writing the variational formulation of the eddy currents problem. Theorem 3.2. Letthefunctionalspace Vbedefinedby H(curl,R3)∩H(div0 ε,Ωe)=V⊕Dε ext. (3.17) Thevariationalformulation: find u∈Vsuchthat /integraldisplay R3µ−1curlu·curl¯ut+iω/integraldisplay ΩCσu·¯ut=−iω/integraldisplay ΩRJ·¯ut∀ut∈V (3.18) hasauniquesolutionin V. Moreover,thevectorfields E≡uandH≡−i(ωµ)−1curluinR3 EDDY CURRENTS MODEL 1811 aretheuniquesolutionsof (2.4)–(2.8)togetherwiththeadditionalcondition (3.11). Proof. First, we note that the bilinear form/integraltext R3µ−1curlu·curl¯utis not coercive onH(curl,R3). Moreover, any field in Dε extyields a solution of the problem /integraldisplay R3µ−1curlu·curl¯ut+iω/integraldisplay ΩCσu·¯ut=0 ∀ut∈H(curl,R3). To avoid these drawbacks, we introduce the identities (2.6) and (3.11) in the varia- tional space Vand consider the variational problem (3.18) on V×V. We set a(u,ut)=/integraldisplay R3µ−1curlu·curl¯ut+iω/integraldisplay ΩCσu·¯ut. By taking ut=uin (3.18) it is easy to see that a(u,u)≥C||curlu||2 L2(R3)3+C/prime||u||2 L2(ΩC)3, (3.19) whereCandC/primeare two positive constants. Let |u|⋆=(||curlu||2 L2(R3)3+||u||2 L2(ΩC)3)1/2. It is obvious that the functional |·|⋆is a seminorm on V. Let us first prove that |v|⋆=0 = ⇒v≡0. Letv∈Vbe such that |v|⋆= 0, then vis a solution of   div(εv)=0 i nΩ e, curlv= 0 in Ω e, v×n|Γ= 0 on Γ .(3.20) Therefore, a function vsatisfying (3.20) is in Dε ext; sincevis also in Vwe may conclude that v= 0. This proves uniqueness of solutions to (3.18). Further, for any sequence (vn)n∈N∈Vsuch that |vn|∗is bounded it is also easy to show that we can extract a subsequence which converges to an element v∈V, so we have existence of a solution to (3.18). Thus, we obtain the existence and uniqueness of solution of the variational for- mulation (3.18) in V. Let nowEbe a solution of the problem (2.4)–(2.8) together with the additional condition (3.11). We shall prove that Eis the solution of the variational formulation (3.18). It is obvious that E∈V. By eliminating the magnetic field Hin the system of equations (2.4)–(2.8), we arrive at curlµ−1curlE=−iωσE−iωJinD/prime(R3), i.e., in the sense of distribution in R3. Note that since curl u∈L2(R3), then auto- matically there is no jump of the tangential component of u, i.e.,E, across interfaces. Now, if we multiply the above equation by a test function ut∈Vand use Propo- sition 3.1, which asserts that the integration by parts over R3is allowed, we get that E∈Vis the solution of the variational formulation (3.18). This ends the proof of Theorem 3.2. 1812 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC 4. Neumann fields. Let (E,H) be the unique solution of the eddy current problem (2.4)–(2.8) together with the additional condition (3.11). We shall analyzein this section some of the properties of this solution. In particular, we focus ourattention on the presence of Neumann fields in the conductor Ω C. Let us start by recalling the definition of Neumannfields . According to [7], the space of Neumann fields in the conductor Ω Cis defined by Nσ int:={u∈H(curl,ΩC)∩H(div0 σ,ΩC);curlu= 0 in Ω C;u·n|Γ=0}. (4.21) Following[5],[20],and[7],weknowthatthisspaceisfinite-dimensionalanditsdimen- sion is equal to q, the number of regular cuts contained in Ω Cwhich are necessary to reduce Ω Cto a simply connected region (see section 2). A basis for this space is char- acterized as the set {gradqi}i=1,...,q,where {qi}i=1,...,qare solutions of the following problem: div(σgradqi)= 0 i n Ω C\q/uniondisplay j=1Σj ∀j=1,...,q [qi]Σj=δi,j across Σ j ∀j=1,...,q [gradqi·n]Σj= 0 across Σ j, ∂nqi = 0 on Γ ,(4.22) where[f]Σjdenotesthejumpofthefunction facrossthemanifoldΣ j. Itisimportant to underline that qi∈H1(ΩC\∪q j=1Σj) but in general qi/negationslash∈ H1(ΩC). However, gradqi∈L2(ΩC\∪q j=1Σj)3and therefore can be extended to L2(ΩC)3. From now on, by gradq,q∈H1(ΩC\∪q j=1Σj), we mean the extension of grad qtoL2(ΩC)3. The Neumann fields coefficients of the electric field E, solution of (2.4)–(2.8), and (3.11) are the integrals /integraldisplay ΩCσE·gradqifori=1,...,q. We shall evaluate them as a function of the source density current J. Using (2.4), we obtain, integrating by parts, /integraldisplay ΩCσE·gradqi=/integraldisplay ΩC(curlH−J)·gradqi =/integraldisplay ΩCH·curl(gradqi)−/integraldisplay ΓH×n·gradqi−/integraldisplay ΩCJ·gradqi. From the jump condition [ H×n]Γ= 0 it follows that div Γ(H×n)=Jext·n, where we setJext:=J|Ωe. This yields /integraldisplay ΓH×n·gradqi=−/integraldisplay Γ∆−1 Γ(Jext·n)∆Γqi. Therefore /integraldisplay ΩCσE·gradqi=αi(J), where αi(J)=/integraldisplay Γ∆−1 Γ(Jext·n)∆Γqi−/integraldisplay ΩCJ·gradqi. (4.23) EDDY CURRENTS MODEL 1813 Note that the first term in the expression of αi(J) takes into account the exterior of the conductor Ω Cwhile the second one takes into account the internal part. We also note that for a function ϕ∈H1(ΩC), we have /integraldisplay Γ∆−1 Γ(Jext·n)∆Γϕ=/integraldisplay ΓJext·nϕ. The following holds. Lemma 4.1.Let(E,H)bethesolutionoftheeddycurrentsmodel (2.4)–(2.8) togetherwiththeadditionalcondition (3.11). Then /integraldisplay ΩCσE·gradqi=αi(J)fori=1,...,q, where {αi(J)}i=1,...,qaredefinedby (4.23). Now, if we assume that Jis orthogonal to Nσ intand there is no source current that goes from the exterior to the interior of the conductor Ω C, then the vector field Eis orthogonal to the space of Neumann fields Nσ int. Lemma 4.2.Thefollowingholds: J⊥Nσ intandJext|Γ·n=0⇒/integraldisplay ΩCσE·gradqi=0∀i=1,...,q. (4.24) 5. Low-frequency analysis of the eddy currents model. From now on, we assume that the density current Jadmits the following series expansion: J=/summationdisplay j≥0ωjJj;∀j≥0,div(Jj)|Ωe=0,/integraldisplay ΓiJj·n=0∀i=1,...,p, (5.25) where the last two equalities come from the fact that Jverifies one side of (2.10). As standardinasymptoticanalysisweproceedformallyandwesupposethatthesolutionof (2.4)–(2.8) can also be expanded in power series, namely, E=∞/summationdisplay j=0ωjEj;H=∞/summationdisplay j=0ωjHj. (5.26) In the following, we show first that the terms ( Ej,Hj) can be uniquely determined and, afterward, we study the convergence of the series in (5.26). Using together (2.4)–(2.8) and (3.18), we write Eas the unique solution of the following problem: E∈V, (5.27)/integraldisplay R3µ−1curlE·curl¯Et+iω/integraldisplay ΩCσE·¯Et=−iω/integraldisplay ΩRJ·¯Et∀Et∈V, (5.28) div(σE+J)=0 i n R3, (5.29)/integraldisplay ΩCσE·gradqi=αi(J) fori=1,...,q. (5.30) This system is not minimal due to (5.29)–(5.30), but it turns out to be useful when writing the system of equations satisfied by the asymptotics {Ej}j≥0. 1814 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC Using then (5.26) in (5.27)–(5.30) and writing separately the contribution coming from different powers of the frequency ωin (5.27)–(5.30), we obtain the following systems of equations for the coefficients. For the first one we have E0∈V, (5.31)/integraldisplay R3µ−1curlE0·curl¯Et=0 ∀Et∈V, (5.32) div(σE0+J0)=0 i n R3, (5.33)/integraldisplay ΩCσE0·gradqi=αi(J0) fori=1,...,q, (5.34) while for the other ones, Ej∈V, (5.35)/integraldisplay R3µ−1curlEj·curl¯Et=−i/integraldisplay ΩCσEj−1·¯Et−i/integraldisplay ΩRJj−1·¯Et∀Et∈V, (5.36) div(σEj+Jj)=0 i n R3, (5.37)/integraldisplay ΩCσEj·gradqi=αi(Jj) fori=1,...,q. (5.38) The following theorem holds. Theorem 5.1. Theproblems (5.31)–(5.34)and(5.35)–(5.38)arewellposed. Proof. We focus our attention only on the well-posedness of the system of equa- tions(5.31)–(5.34),sincethewell-posednessofthesystem(5.35)–(5.38)maybeprovedby the same arguments. It is quite natural to split the problem (5.31)–(5.34) into an interior problem defined on the conductor Ω Cand anexteriorproblem defined on Ω e. We then split E0into (EC 0,ENC 0), whereEC 0=E|ΩCandENC 0=E|Ωe. From (5.31)–(5.34) and (5.25) we obtain the following two problems: (5.39) Interior problem Exterior problem   curlEC 0= 0 in Ω C, div ( σEC 0+J0)=0 i nΩ C, σEC 0·n=[J0·n]|Γ on Γ , /integraldisplay ΩCσEC 0·grad qi=αi(J0) for i=1,...,q ;  curlENC 0= 0 in Ω C, div ( εENC 0)=0 i nΩ C, ENC 0 |Γ×n=EC 0|Γ×n on Γ , ENC 0⊥Dε ext. First note that we have to solve the interior problem and then the exterior one. The exterior problem is coupled only to the interior one by the boundary condition onΓ. The study of the interior problem does not require any data from the exteriorproblem. Once the interior problem is solved the problem becomes classical. Weclaim that the interior problem is well posed. The existence of at least one solutionto the interior problem (separately, the exterior problem) has been proved by Picard[20] and Alonso and Valli [1]. Here, we prove only the uniqueness. If we set J 0= 0, then we obtain a homogeneous system since div( J0)=0 , [J0·n]|Γ=0 ,αi(J0) = 0. Thus, from the first three equations of the interior problem EDDY CURRENTS MODEL 1815 in (5.39), we have that EC 0∈Nσ int, while the last one imposes that it is orthogonal to the space of Neumann fields Nσ int. The well-posedness of the exterior problem in (5.39) is classical. For what concerns the terms {Hj}j≥0in the asymptotics expansion of the mag- netic field H, we simply have to use (2.6) and (5.26) in order to derive the following identity: Hj=iµ−1curlEj+1 ∀j≥0. (5.40) The complete variational formulation (3.18) could also be decomposed into an interior and an exterior problem. Also in this case, the interior field can be foundbefore and does not depend directly on the exterior one. For the exterior problem thesolution depends continuously on the boundary data. Proposition 5.2. LetE∈Vbetheuniquesolutionoftheeddycurrentsmodel (2.4)–(2.8)togetherwiththeadditionalcondition (3.11). Thereexists ω 0>0such thatforω∈(0,ω0)thefollowinginequalityholds: ||E||L2(R3)3+||curlE||L2(Ωe)3≤C/parenleftBig ω||J||L2(ΩR)3+||E||H(curl,ΩC)/parenrightBig , (5.41) whereCisapositiveconstantindependentof ω. Proof. It is easy to see that E∈Vsatisfies curl1 µcurlE=−iωJin Ω e, (5.42) E×n=Eint×non Γ, (5.43) whereEint≡E|ΩC. From the boundary value problem (5.42)–(5.43) using div( εE)= 0, it is classical to deduce that the following estimate holds: ||E||H(curl,Ωe)≤C/parenleftBig ω||J||L2(ΩR)3+||n×(n×E)||TH−1/2(curl,Γ)/parenrightBig , where the constant Cdoes not depend on the frequency ω. Here, TH−1/2(curl,Γ) is the space of tangential traces of H(curl,Ωe). Since [n×(n×E)]Γ= 0, it follows from the classical trace theorem (see Paquet [19] and Alonso and Valli [1]) that there exists a constant C/prime, independent of the frequencyω, such that ||n×(n×E)||TH−1/2(curl,Γ)≤C/prime||E||H(curl,ΩC), and then we have the result. Theorem 5.3. Let(Ej,Hj),forj≥0,betheuniquesolutionsoftheproblems (5.31)–(5.34),(5.35)–(5.38)and(5.40).L e t(E,H)bethesolutionof (2.4)–(2.8)and (3.11). Then,thereexists ω0>0suchthatforM∈Nandω∈(0,ω0),thefollowing estimateshold: /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE− M/summationdisplay j=0ωjEj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(R3)3≤CωM+1, (5.44) /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH− M/summationdisplay j=0ωjHj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(R3)3≤CωM+1, (5.45) 1816 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC wheretheconstant Cdoesnotdependon ωbutonlyontheconductor ΩC. Proof. Thequantity E−/summationtextM j=0ωjEjsatisfiesthevariationalequation(3.18)when the source density current is replaced by J−/summationtextM j=0ωjJj. By standard manipulations, we have /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublecurl E− M/summationdisplay j=0ωjEj /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(R3)3+σω/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE− M/summationdisplay j=0ωjEj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(ΩC)3 ≤Cω/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleJ− M/summationdisplay j=0ωjJj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(ΩR)3/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE− M/summationdisplay j=0ωjEj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(ΩR)3.(5.46) Using then the Proposition 5.2, we obtain /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE− M/summationdisplay j=0ωjEj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(R3)3≤C/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleJ− M/summationdisplay j=0ωjJj/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(ΩR)3≤C/primeωM+1, /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublecurl E− M/summationdisplay j=0ωjEj /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(R3)3≤C/primeωM+2.(5.47) From (5.47) and (5.40) we obtain the estimates (5.44) and (5.45). 6. Low-frequency analysis of the Maxwell system. In this section, we briefly collect some results on the solutions ( Em,Hm) of the Maxwell equations (2.1)– (2.3) which are useful while comparing the solutions of (2.1)–(2.3) and (2.4)–(2.8). We begin with some general considerations.Due to the compatibility condition (2.10) that we have imposed on J, from (2.1) we also have that εE mhas to verify the same orthogonality condition, namely, εEm |Ωe⊥gradH1 0(Ωe)⊥⊕Dε ext (6.48) or equivalently div(εEm)|Ωe= 0 and/integraldisplay ΓiεEm·n=0∀i=1,...,p. (6.49) It is now classical to prove the following results on the existence and uniqueness for the Maxwell system. The material coefficients ε,µ, andσmay only be assumed bounded and measurable, i.e., jumps are allowed [22], [2]. Proposition 6.1. TheMaxwellsystem (2.1)–(2.3)hasauniquesolution. We pass now to a brief presentation of some results concerning the low-frequency asymptotic analysis of the Maxwell system (2.1)–(2.3). According to [2], [3], [4] we know that the same asymptotic analysis we carried out in the previous sections for the eddy currents model is valid and well posed inthe case of the Maxwell model (2.1)–(2.3) also. Moreover for (2.1)–(2.3) the proofs ofthe same results are more technical since both the electric and the magnetic fields donot belong to L 2(R3). The following theorem is just a collection of the results that we need for our purposes and a complete proof of it can be easily deduced from theseveral situations analyzed in [2], [3], [4]. EDDY CURRENTS MODEL 1817 Theorem 6.2. Let(Em,Hm)bethesolutionof (2.1)–(2.3)andletthesource current Jverifytheassumption (5.25). Boththeelectricandthemagneticfieldscan beexpandedinpowerserieswithrespectto ω,thatis,thereexists ω0suchthatfor anyω∈(0,ω0)andM∈Nthefollowingestimateshold: /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleE m−M/summationdisplay j=0ωjEm j/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(BR)3≤CmωM+1, (6.50) /vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoubleH m−M/summationdisplay j=0ωjHm j/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble L2(BR)3≤CmωM+1, (6.51) where the constant Cmdoes not depend on the frequency ω. Moreover, the terms (Em j,Hm j)aredeterminedby   εE m 0|Ωe⊥gradH1 0(Ωe)⊥⊕Dε ext, curlEm 0=0 i n R3, div(σEm 0+J0)=0 i n R3,/integraldisplay ΩCσEm 0·gradqi=αi(J0) fori=1,...,q,(6.52)   εE m 1|Ωe⊥gradH1 0(Ωe)⊥⊕Dε ext, curlµ−1curlEm 1=−iσEm 0−iJ0 inR3, div(iεEm 0+σEm 1+J1)=0 i n R3,/integraldisplay ΩCσEm 1·gradqi=αi(J1)−iαi(εEm 0) fori=1,...,q,(6.53) and more forj≥2   εE m j+1|Ωe⊥gradH1 0(Ωe)⊥⊕Dε ext, curlµ−1curlEm j+1=εEm j−1−iσEm j−iJjinR3, div(iεEm j+σEm j+1+Jj+1)=0 i n R3,/integraldisplay ΩCσEm j+1·gradqi=αi(Jj+1)−iαi(εEm j) fori=1,...,q.(6.54) Finally,theterms (Hm j)j≥0intheasymptoticexpansionofthemagneticfield Hm aredeterminedby Hm j=iµ−1curlEm j+1inR3∀j≥0. (6.55) A complete characterization of all these coefficients should involve some radiation condition or exact external operators as detailed in [2]. Further, from (6.52)–(6.53)and by applying Proposition 3.1 the following Lemma holds. Lemma 6.3.Thevectorfields E m 0andEm 1satisfy /braceleftbigg∆Em 0=0inR3\BR, divEm 0=0inR3\BR,/braceleftbigg∆Em 1=0inR3\BR, divEm 1=0inR3\BR,(6.56) andthentheyarein L2(R3)3. 1818 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC 7. Approximation of the Maxwell system by the eddy currents model. In this section, we shall compare the asymptotic expansions of the solutions of (2.1)–(2.3)and(2.4)–(2.8). Themainideaisthat( E m,Hm)canbeapproximatedby( E,H) as the frequency goes to zero if and only if the first few terms in their asymptoticexpansions are equal. We start then comparing the terms {E m j,Hm j}j≥0and{Ej,Hj}j≥0. •j=0. Comparing the system (5.31)–(5.34) and (6.52) and recalling Lemma 6.3 we have Em 0=E0inR3. (7.57) Moreover, if we assume that div(J0)|ΩC= 0 and αi(J0)=0 ∀i=1,...,q, (7.58) we arrive at Em 0=E0≡0. In the following, it will be clear that (7.58) is a key as- sumptioninordertoshowthevalidityof(2.4)–(2.8)asalow-frequencyapproximationof the full Maxwell system (2.1)–(2.3). •j=1.Comparing (5.35)–(5.38), evaluated for j= 1, and (6.53), we find that E m 1−E1satisfies   ε(E m 1−E1)|Ωe⊥gradH1 0(Ωe)⊥⊕Dε ext, curl(Em 1−E1)=0 i n R3, div(σ(Em 1−E1)) =−idiv(εEm 0)i n Ω C,/integraldisplay ΩCσ(Em 1−E1)·gradqi=−iαi(εEm 0) fori=1,...,q.(7.59) Therefore, there exists a function ψ∈H1(R3) andqreal numbers {βi}i=1,...,qsuch that Em 1−E1= gradψ+q/summationdisplay i=1βigradqi. (7.60) In particular, if Em 0= 0 then the right-hand side of (7.60) is equal to zero and we have Assumption (7 .58)⇒Em 0=E0≡0⇒Em 1=E1. (7.61) Moreover, from (7.59) the following holds: Hm 0=iµ−1curlEm 1=iµ−1curlE1=H0. (7.62) •j=2.If the assumption (7.58) does not hold, nothing can be said about the difference Em 2−E2since the function Em 2is not completely determined by (6.54) evaluated forj= 1 (see Lemma 6.3). Otherwise, if (7.58) is fulfilled, the same argument applied to Em 0andEm 1in Lemma 6.3 works for Em 2, which is completely determined by the system (6.54) with j= 1. As in the case j= 1, the difference Em 2−E2solves the system   ε(E m 2−E2)|Ωe⊥gradH1 0(Ωe)⊥⊕Dε ext, curl(Em 2−E2)=0 i n R3, div(σ(Em 2−E2)) =−idiv(εEm 1)i n Ω C,/integraldisplay ΩCσ(Em 2−E2)·gradqi=−iαi(εEm 1)i=1,...,q.(7.63) EDDY CURRENTS MODEL 1819 In particular, Em 2/negationslash=E2, but we have Hm 1=iµ−1curlEm 2=iµ−1curlE2=H1. (7.64) Now summarizing all these results, our concluding remarks are the following: •If the assumption (7.58) is fulfilled, (2.4)–(2.8) is a good approximation of (2.1)–(2.3) at low frequencies; we use (7.57), (7.61), (7.62), (7.64), (5.44)–(5.45), and (6.50)–(6.51) in order to obtain the following inequalities: ||E m−E||L2(ΩR)3≤(C+Cm)ω2, ||Hm−H||L2(ΩR)3≤(C+Cm)ω2.(7.65) Furthermore, roughly speaking, one can also see how the equations (2.4)– (2.8) are deduced from (2.1)–(2.3). Let Em=ωE1+O(ω2) andHm= H0+ωH1+O(ω2); using these expressions in (2.1)–(2.3), we obtain curlEm=iωµ(H0+ωH1)+O(ω2), (7.66) curlHm=−iωε(ωE1)+σ(ωE1)+J+O(ω2) =σ(ωE1)+J+O(ω2). (7.67) From (7.67) one sees that the term coming from the time derivative of the electric field vanishes since it is of higher order. •If the assumption (7.58) is not fulfilled only the two asymptotic expansions are equal only at order 0. This means that the inequalities (7.65) are replacedby ||E m−E||L2(ΩR)3≤(C+Cm)ω, ||Hm−H||L2(ΩR)3≤(C+Cm)ω.(7.68) Moreover, it is important to underline that in this case it turns out to be useless to solve the problem (2.4)–(2.8) in place of the complete static system   E 0⊥gradH1 0(Ωe)⊥⊕Dε ext, curlE0=0 i n R3, curlH0=σE0+J0 inR3, div(µH)=0 i n R3.(7.69) The well-posedness of this problem is straightforward from section 3 and has already been studied by Ramm et al. in [22]. 8. The time-dependent case. This section is concerned with the extension of the same approximation results obtained in the previous sections to the time-dependent version of the equations (2.1)–(2.3) and (2.4)–(2.7). Namely, the systemof time dependent Maxwell equations reads as follows: curlH m=ε∂Em ∂t+σEm+JinR3, (8.70) curlEm=µ∂Hm ∂tinR3, (8.71) with the initial conditions Em|t=0=0,Hm|t=0=0,andJ|t=0=0 . 1820 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC The eddy currents approximation is obtained by neglecting the term ε∂Em ∂tin (8.70). We first introduce some notation. Let Fbe the Fourier transform operator with respect to the time variable tandˆJ(·,x)=F(J(·,x)). In order to shorten our nota- tion, from now on, we shall write J(·),ˆJ(·) instead of J(·,x) andˆJ(·,x), respectively; the norms and integrals are referred to the time variable or the frequency variable ω, which is the dual variable. Next, we make the following general assumptions:•The source current Jsatisfies (2.10) and (5.25) ∀t≥0. •J∈H 1(R,L2(R3))3. •The Fourier transform ˆJofJis compactly supported. There exists ¯ ωsuch that supp{ˆJ}⊂]−¯ω,¯ω[. •¯ω<ω 0, whereω0is such that the low-frequency estimates (5.44)–(5.45) and (6.50)–(6.51) hold for any 0 <ω<ω 0. •The time-dependent Maxwell system (8.70)–(8.71) and the corresponding time- dependent eddy currents model, with the initial conditions Em|t=0=0,Hm|t=0=0, andJ|t=0= 0, have a unique solution, denoted by ( Em(t),Hm(t)) and (E(t),H(t)), respectively. The main result of this section is the following theorem.Theorem 8.1. Thereexistsaconstant Csuchthatthefollowingestimateholds: ||E m(t)−E(t)||L2(0,T,H(curl,BR))≤C¯ω2, (8.72) ||Hm(t)−H(t)||L2(0,T,H(curl,BR))≤C¯ω2. (8.73) Proof. We only focus our attention on (8.72). The proof of (8.73) follows from the same arguments. We first state a useful estimate. For u=Eandu=Emthe following stability estimate holds: ||u||2 L2(ΩC)+/integraldisplayT 0||u||2 L2(ΩR)+/integraldisplayT 0||curlu||2 L2(BR)≤C/integraldisplayT 0/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂J ∂t/vextenddouble/vextenddouble/vextenddouble/vextenddouble2 L2(BR), (8.74) where the constant Cis independent of T. For the Maxwell system this estimate is standard, while for the eddy currents model it can be obtained from (5.41). Nowthemainideaistoapproximatethenonperiodic(withrespecttothevariable t) density current Jby a sequence Jnof source density currents where each Jnis a superposition of periodic modes ( eiωlt). Therefore, thanks to the linearity of the equations, we know that the solution of the Maxwell and the eddy currents systemare also superpositions of periodic modes and, for these fields, the theory developedin the previous sections can be applied. Our aim is now to approximate ˆJby a sequence {ˆJ n}n∈Nin which every element ˆJnis a sum of Dirac measures. The inverse Fourier transforms JnofˆJnare then superpositions of periodic modes. Let the function ˆJpbe defined by ˆJp(ω): =/integraldisplayω −¯wˆJ(s)ds. (8.75) We decompose the interval [ −¯ω,¯ω]i n t onsubintervals of the same size 2¯ ω/n and we denote by ωi,i=0,...,n, the nodes of this grid and by χithe characteristic EDDY CURRENTS MODEL 1821 functionoftheinterval[ ωi,ωi+1],i=0,...,n −1. Thenwedefineapiecewiseconstant approximation of ˆJpby ˆJp n(ω): =n−1/summationdisplay i=0ˆJp(ωi)χi. (8.76) It is standard to prove that ||ˆJp−ˆJp n||L2(R)≤2C¯ω n||ˆJp||H1(]¯ω,¯ω[)≤2C/prime¯ω n||ˆJ||L2(R), (8.77) where the constants CandC/primedepend neither on nnor on ¯ω. By taking the derivative of (8.76) in the sense of distributions, we construct a sequence ˆJnapproximating ˆJ, ˆJn(ω): =n/summationdisplay i=1(ˆJp(ωi)−ˆJp(ωi−1))δ(ω−ωi)=n/summationdisplay i=1/bracketleftBigg/integraldisplayωi ωi−1ˆJ(s)ds/bracketrightBigg δ(ω−ωi), (8.78) whereδ(ω−ωi) denotes the Dirac measure centered in ωi. Applying F−1to both sides of (8.78), we obtain Jn(t)=n/summationdisplay l=1Jn(l)eiωltwithJn(l): =/integraldisplayωl ωl−1ˆJ(s)ds. (8.79) Using (8.79), we easily deduce that ∂sJ ∂ts−∂sJn ∂ts=/integraldisplay¯ω −¯ωˆJ(ξ)/bracketleftBig (iξ)seiξt−(iχ(ξ))seiχ(ξ)t/bracketrightBig dξfors=0,1, (8.80) whereχ(ξ)=ωiforξ∈]wi−1,ωi[,i=1,...,n. From (8.80), we obtain ||J−Jn||H1(]0,T[)≤C(T,||J||L2(R),¯ω) n, (8.81) where the quantity C(T,||J||L2(R)) depends linearly on ||J||L2(R). NowwesolvethesystemofMaxwell’sequations(8.70)–(8.71)andthecorrespond- ing eddy currents model when the data Jis replaced by Jndefined by (8.79). We denote by ( Em(n),Hm(n)) and (E(n),H(n)) the corresponding solutions. We focus our attention on the electric fields, since the estimates on the magnetic fields can bededuced by using the same arguments. Using the stability estimate (8.74) togetherwith the inequality (8.81), we obtain ||E m−Em(n)||L2(0,T,H(curl,BR)) +||E−E(n)||L2(0,T,H(curl,BR))≤C/prime(T,||J||L2(R,L2(BR)),¯w) n,(8.82) where the constant C/primedepends linearly on ||J||L2(R,L2(BR)). In order to apply the results established in the previous sections we first observe that Em(n)=n/summationdisplay l=1Em l(n)eiωltandE(n)=n/summationdisplay l=1El(n)eiωlt, (8.83) 1822 H. AMMARI, A. BUFFA, AND J.-C. N ´ED´ELEC whereEm l(n) andEl(n) are the solution of the time harmonic systems (2.1)–(2.3) and (2.4)–(2.8), respectively, with ω=ωl. Usingthetheorydevelopedinsections5and6,weknowthenthatthesequantities can be expanded in power series with respect to the frequency ω. Using the results established in section 7, under the suitable hypothesis on the load J(detailed in section 7) we know that for any ¯ ω∈(0,ω0) Em l(n)−El(n)=O(¯ω2)∀l=1,...,n. 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