sliding interfaces for eddy current simulationshiptmair/studentprojects/casagrande/thesis.pdf ·...

Sliding Interfaces for EddyCurrent Simulations

Raffael CasagrandeMaster Thesis

Supervisor:Prof. Dr. Ralf Hiptmair

Zürich, April 2013

Contents

Contents i

1. Introduction 1

2. Eddy Current Problem 3

2.1. Coulomb gauged formulation . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2. Temporal gauged formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3. H-formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4. The eddy current problem in a moving, solid body . . . . . . . . . . . . . 9

3. Discontinuous Galerkin formulation 11

3.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.2. Broken spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1.3. Discrete Friedrichs inequality . . . . . . . . . . . . . . . . . . . . . 17

3.1.4. Trace inequalities and polynomial approximation . . . . . . . . . . 17

3.2. DG formulation of Eddy Current problem . . . . . . . . . . . . . . . . . . 18

3.2.1. Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4. Aspects of the Implementation 32

4.1. Reduction to 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CONTENTS

4.2. Details of the imlementation. . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2.1. The mesh data structure . . . . . . . . . . . . . . . . . . . . . . . . 36

4.3. Variational formulation for moving systems . . . . . . . . . . . . . . . . . 40

5. Results 45

5.1. Convergence to analytical solution . . . . . . . . . . . . . . . . . . . . . . 45

5.2. Translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3. Rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6. Conclusion and Outlook 54

A. Maxwell’s equations in a moving frame of reference 55

B. Documentation of the code 60

C. Acknowledgements 61

Bibliography 62

ii

1. Introduction

We present a discontinuous Galerkin (DG) approach for solving the transient eddy cur-rent problem in the presence of moving, rigid bodies which share common interfaces.Such problems are important in the modelling of electromechanical devices. In partic-ular the induced currents can have positive or negative effects and are often essentialfor the functioning of electric engines. Consider for example the setting depicted in Fig.1.1. Here multiple bodies are moving with respect to each other and sliding interfacesexist. We attach to each body a separate coordinate frame and mesh so we can solvethe equations in Lagrangian variables.

Figure 1.1.: An example configuration with 3 solid bodies. Separate coordinate systemsare attached to each one.

In order to deal with the non-matching grids we use symmetric-interior penalty tech-niques at the interfaces. We use conforming finite element spaces in the interior of eachbody. Together with the appropriate transformation formulas we then link the differentdescriptions into one consistent framework. This approach doesn’t need any remeshingand allows us to model the problem from a Lagrangian perspective. In particular noconvective terms appear in the formulation and the method is therefore expected to bestable for bodies moving at high speed.

In time, we use an implicit Euler discretization which results in a sparse, symmetric-positive-definite linear system that has to be solved in every iteration. The proposeddiscretization is analyzed in a general 3 dimensional DG framework but only 2 dimen-sional model problems have been implemented and tested. In space, the problem isdiscretized by means of first order Lagrangian elements as well as first order Nedelec

1. Introduction

elements of the first kind.

Another commonly used approach to solve the same physical problem is based on mortar-methods (see for instance [14],[15], [4],[8]). These methods have typically symmetric-positive definite matrices as well and need no remeshing. However the implementationis harder because additional degrees of freedom are introduced along the interface. Thisis especially true for translational settings where the interface itself is changing overtime. There exist many other approaches which rely on conforming meshes, but theyalmost always need a partial or full remeshing in every time step which increases thecomputational cost heavily (cf. [14, Introduction]). More recently Ferrer [7],[1] hasapplied the DG theory to the Navier-Stokes equations for treating sliding interfaces.

The material is presented in five chapters: We start off in chapter 2 by giving a briefintroduction to the eddy current model and its possible formulations. The chapter endswith a discussion of the Galilean transformation properties of the previously presentededdy current formulations. Chapter 3 starts by introducing mathematical tools from theDG world and continues with a presentation of the DG variational formulation. Thelatter is then analyzed and a proof of convergence is given. Following this rather theo-retical material we present some implementation details in chapter 4. Finally, chapter 5presents and discusses all the numerical results. The thesis ends with a short summaryand some concluding remarks (chapter 6).

2

2. Eddy Current Problem

In this chapter we will derive three different formulations of the transient eddy currentproblem by neglecting the displacement current in Maxwell’s equations. By requiring aunique solution we will derive appropriate initial and boundary conditions for the givenproblems. In the last part of this chapter we will show that the three formulations areinvariant under rotation and translation if the quantities are transformed correctly.

It should be mentioned that there are many other ways of treating the eddy currentproblem. In this short presentation, we focus our attention on the Coulomb gaugedpotential formulation (unknowns: A, ϕ), the temporal gauged potential formulation(unknown: A) and the H formulation (unknown: the magnetic field H). We refer thereader to Albanese and Rubinacci [2] for a survey of different formulations.

Maxwell’s equations The eddy current problem is a simplified version of Maxwell’sequations,

Magnetic Gauss’s law: div B = 0 (2.1a)

Faraday’s law of induction: curl E + ∂B∂t

= 0 (2.1b)

Electric Gauss’s law: div D = ρ (2.1c)

Ampère’s law: curlH− ∂D∂t

= jf + ji. (2.1d)

Here B denotes the magnetic induction ([ Vs/m2]), E the electric field ([ V/m]), Dthe electric induction ([ N/Vm]), ρ the free charge density ([ C/m3]), H the magneticfield ([ A/m]), jf the free current density ([ A/m2]) and ji the impressed current density([ A/m2]). The latter is a prescribed current density which can be used to model coilsin the computational domain.

We consider only linear, isotropic materials with immediate response in this thesis andtherefore the constitutive relations

D = εE B = µH jf = σE


apply. ε, µ and σ are all material dependent properties and are termed permittivity,permeability and conductivity, respectively. They are simple scalars which are only afunction of the material.

In the following we are going to derive the eddy current problem. For this we need tomake the centralAssumption 2.1. The displacement current, ∂D

∂t in Ampères law (2.1d) is zero. Thisis justified if the electric field is varying only slowly (Quasistatic simplification).

This assumption is typically justified for very high conductivities respectively if theelectric field E is varying only slowly.

2.1. Coulomb gauged formulation

Remark. In this chapter we assume that A ∈ [C∞(Ω)]3 and ϕ ∈ C∞(Ω) to simplify thederivations and manipulations. This assumption is relaxed later when we consider thevariational formulation in chapter 3. In addition we assume that the domain Ω is simplyconnected.

By virtue of the Helmholtz decomposition (cf. [18]) we can express B and E throughthe vector potential A∗ and the scalar potential ϕ∗:

B = curl A∗, (2.2a)

E = − gradϕ∗ − ∂A∗∂t

. (2.2b)

This way the first two Maxwell equations (2.1a),(2.1b) are fulfilled automatically andonly Ampères law remains. Expressed in the variables A∗ and ϕ∗ it reads as

σ∂A∗∂t

+ curl 1µ

curl A∗ + σ gradϕ∗ = ji. (2.3)

It is well known that this equation doesn’t possess a unique solution. Indeed, we canuse for example the gauge transformation

A′∗ = A∗ + gradψ, ϕ′∗ = ϕ∗ − ∂ψ

∂t,

to transform A and ϕ without affecting the observable quantities B and E (ψ(x, t) is anarbitrary function). In practice the Coulomb gauge condition,

div A∗ = 0 (2.4)

is often introduced to fix this ambiguity. Together with appropriate boundary conditionsthis fixes the value of A and ϕ and leads to a well posed problem. We will refer to thesystem of equations represented by Eq. 2.3 together with Eq. 2.4 as the Coulomb gaugededdy current problem.

4

2.2. Temporal gauged formulation

Remark. If σ = 0 the Eqs. (2.3) and (2.4) become the elliptic boundary value problem

curl 1µ

curl A∗ = ji, div A∗ = 0.

This shows that the eddy current model is principally not capable of modelling electro-magnetic waves.


Another possibility to guarantee that Eq. (2.3) has a unique solution consists in usingthe temporal gauge transformation

A := A∗ +∫ t

0gradϕ∗(t)dt, ϕ := 0. (2.5)

The electromagnetic fields can then be calculated through the relations

B = curl A and E = −∂A∂t

.

The gauge transformation 2.5 effectively corresponds to a gauge condition in whichϕ ≡ 0. If we apply the temporal gauge transform to the Coulomb gauged eddy currentproblem we arrive at the temporally gauged eddy current problem,

curl 1µ

curl A + σ∂A∂t

= ji (2.6)

Boundary and Initial Conditions So far we have omitted the boundary and initial con-ditions for the eddy current problem tactically. In this section we will derive suchconditions by requiring Eq. (2.6) to have a unique solution.

Notice that if A1 and A2 are both solutions of Eq. (2.6) then A = A1 −A2 must be asolution of the homogeneous equation,

curl 1µ

curl A + σ∂A∂t

= 0.

Multiplying the equation by A, integrating over Ω and time and using Stokes theorem(cf. [20, Thm. 3.6]) we get

∫Ω

∫ T

0σ∂A∂t· A +

∫ T

0

∫Ω

curl A · 1µ

curl A =∫ T

0

∫∂Ω

1µ

(curl A× n

)· A.

5


The integral over time of the first term can be evaluated exactly if σ is independent oftime:∫

Ω

12σ(A(T )

)2− 1

2σ(A(0)

)2+∫ T

0

∫Ω

curl A · 1µ

curl A =∫ T

0

∫∂Ω

1µ

(curl A× n

)· A.

(2.7)

From the above equation it is clear that the cases σ = 0 and σ > 0 must be handledseparately. For this we split the domain Ω into two distinct subdomains Ω = Ω1 ∪ Ω2such that σ|Ω1

> 0 and σ|Ω2≡ 0. In the following we will therefore discuss the solution

to (2.6) for each subdomain separately and consider the coupling of the two solutionslater.

In Ω1 we prescribe the following initial and boundary conditions:

A(x, y, z, t = 0) = A(0) on Ω1, (2.8a)curl A× n = g1 on ∂Ω1, (2.8b)

This causes the second term on the left-hand side and the boundary term on the right-hand side of Eq. (2.7) to drop out:∫

Ω1

12σ(A(T )

)2+∫ T

0

∫Ω1

curl A · 1µ

curl A = 0

Because there are only quadratic terms left, A(T ) ≡ 0 and curl A ≡ 0 in Ω1, i.e.A1 = A2. Therefore Eq. (2.6) together with the boundary/initial conditions (2.8) hasa unique solution.

In Ω2 things are more involved; Here we only require the boundary condition,

curl A× n = g2 on ∂Ω2, (2.9)

so that Eq. (2.7) reduces to∫ T0∫

Ω2curl A · 1

µ curl A = 0. Once again the terms on

the left hand side are quadratic, so that curl A !≡ 0 in Ω2. This implies that A isirrotational and therefore there exists a scalar function ψ such that A = gradψ. Torestore uniqueness one usually requires additionally the Coulomb gauge condition

div A = 0 on Ω2 with b.c. (2.10)A · n = h on ∂Ω2. (2.11)

This fixes ψ to zero or a constant function so that A = 0 and therefore A1 = A2.

It remains to connect the subproblems on Ω1 and Ω2 by suitable transmission conditions.For this we define Γ12 := Ω1 ∩Ω2. Let p be a planar plane with boundary ∂p and define

6


l = Γ12 ∩ p to be the intersection of Γ12 and p. Now integrating Eq. (2.6) over p,assuming that no surface currents exist and using Stokes theorem we get:

0 =∫p(·)−

∫p∩Ω1

(·)−∫p∩Ω2

(·)

= −∫l

1µ|Ω1

curl A|Ω1· dτ1 −

∫l

1µ|Ω2

curl A|Ω2· dτ2

= −∫l

(1µ|Ω1

curl A|Ω1− 1µ|Ω2

curl A|Ω2

)· dτ1

where τ1 = −τ2 denotes the line elements on either side of Γ12. Because p is arbitrarythis results in the jump condition[ 1

µcurl A

]× n12 = 0. (2.12)

where n12 points from Ω1 to Ω2. I.e. the tangential component of 1µ curl A is continuous

across Γ12 (cf. Lemma 3.8 in chapter 3 for a more elegant and general proof of thisjump/interface condition). From this and the boundary conditions (2.8b) and (2.9) wesee that g1(x, y, z, t)|Γ12

= g2(x, y, z, t)|Γ12. We are now in the position to state the eddy

current problem in temporal gauge with the appropriate boundary and initial conditions:

curl 1µ

curl A + σ∂A∂t

= ji on Ω (2.13a)

div A = 0 on Ω2 (2.13b)A(x, y, z, t = 0) = A(0) on Ω (2.13c)

1µ

curl A× n = g on ∂Ω (2.13d)

A · n = h on ∂Ω2 (2.13e)[ 1µ

curl A]× n12 = 0 on Γ12 = Ω1 ∩ Ω2 (2.13f)

Remarks.• By taking the divergence of Eq. (2.13a) we see that div ji must fulfil the com-

patibility conditions div ji = 0 in Ω2. In addition h must fulfil the Neumanncompatibility condition

∫∂Ω2

h ds = 0.• For proving the uniqueness of the IBVP (2.13) we assumed that σ is independent

of time. This assumption is probably not needed to guarantee uniqueness butsimplifies the proof.

• This is the first possible interpretation of the eddy current equation. In contrastto the interpretation presented next, this formulation works for σ = 0 as well asfor σ > 0.

7


2.3. H-formulation

One can give a simpler but more restrictive meaning to the eddy current equation (2.13a)by dividing it on both sides by σ, applying the curl operator and makingAssumption 2.2. We assume σ > 0 in Ω. This implies that Ω2 = ∅ and Ω1 = Ω.

curl 1σ

curl 1µ

curl A + ∂ curlA∂t

= curl 1µ

ji, ⇔

curl 1σ

curlH+ ∂µH∂t

= curl 1σ

ji on Ω (2.14)

Of course we have to give a new meaning to the gauge condition (2.13b) and rephrase itin terms of H. By taking a glimpse at the magnetic Gauss law we easily state the newgauge condition:

divµH = 0 on Ω. (2.15)

By using a similar reasoning as before for the temporally gauged formulation it is possibleto show that the following initial-boundary value problem (IBVP) has a unique solution:

curl 1σ

curlH+ ∂µH∂t

= curl 1σ

ji on Ω (2.16a)

divµH = 0 on Ω (2.16b)H(x, y, z, t = 0) = H(0) on Ω (2.16c)

H× n = g on ∂Ω (2.16d)

Remark. The gauge condition (2.16b) in this formulation is necessarily required for thewhole computational domain Ω by virtue of the magnetic gauss law. However by takingthe divergence of Eq. (2.16a) we see that ∂ divµH

∂t = 0. So if the gauge condition isfulfilled for time t = 0 it is automatically fulfilled for all t > 0. Usually one still includesthe gauge condition in practice for the sake of numerical stability.

It remains to show that the solution to the IBVP (2.16) fulfils Maxwell’s equationsunder assumptions 2.1 and 2.2. First we notice that Ampères law (2.1d) reduces tocurlH = jf + ji = σE + ji. Solving this relation for E and inserting it into Faraday’slaw of induction (2.1b) we get the eddy current equation (2.16a). I.e. Faraday’s law isfulfilled and

jf = curlH− ji E = jfσ.

Remark. If we compare the continuity equation for electric charges, div (jf + ji) = ∂ρ∂t ,

with the currents computed by the eddy current model we see ∂ρ∂t = div curlH = 0. I.e.

the eddy current model preserves space charges.

8

2.4. The eddy current problem in a moving, solid body

2.4. The eddy current problem in a moving, solid body

Later on we are interested in settings where one solid body is moving inside the labo-ratory frame. It is not straightforward to calculate the electromagnetic fields in such asetting. In particular Lorentz forces appear as source terms in the eddy current equa-tions. For example the Coulomb gauged eddy current equation (2.13a) in the laboratoryframe of reference reads as (cf. appendix A, Eq. (A.7))

σ∂A∗∂t

+ curl 1µ

curl A∗ + σ gradϕ∗ = ji + σV× curl A∗. (2.17)

In appendix A we show that this equation is invariant under translation and rotationand takes exactly the same form in the moving frame of reference,

σ∂A∗∂t

+ ˜curl 1µ

˜curlA∗ + σ ˜grad ϕ∗ = ji + σV× ˜curlA∗, (2.18)

where we used a tilde to denote field quantities which are defined in the moving frameof reference. The field quantities of the laboratory frame are related to the fields in thebody’s coordinate frame by the relations

TE = E + V ×B TB = BTji = ji TH = HTjf = jf TA∗ = A∗

ϕ = ϕ− V ·A∗

(2.19)

Here T(t) stands for the rotation matrix of the moving frame (the principal coordinatesin the moving system are the columns of T) and V stands for the velocity of of themoving frame (both are defined in the laboratory frame).

Usually we will pick the moving frame of reference such that it coincides the body’sprincipal coordinate axes. In that case V ≡ 0, i.e. the Lorentz forces disappear in themoving frame of reference. In other words the Lorentz force term is due to a change incoordinates.

The H-formulation is of course also invariant under rotation and translation and thequantities transform according to the transformation rules (2.19). It remains to derive atransformation formula for the temporal gauged formulation of the eddy current problem.

The idea is to start from the Coulomb gauged formulation and use the temporal gaugetransformation (2.5) to express A in the moving frame of reference. In other wordswe are imposing the gauge condition ϕ ≡ 0 in the moving frame. We can then usethe transformation rules (2.19) to expand the expression in terms of the laboratory

9


coordinates:

A := A∗ +∫ t

0˜grad ˜ϕ(t)dt

= TTA∗ +∫ t

0TT gradϕ−

∫ t

0TT grad (V ·A∗)

= TTA−∫ t

0TT grad (V ·A∗) (2.20)

Where we have used the transformation rule for the del-operator,∇ = TT ∇. We sum-marize the transformation rule in the

Lemma 2.3 (Transformation rule for A). The temporally gauged eddy current problemwith σ > 0,

curl 1µ

curl A + σ∂A∂t

= ji + σV× curl A (2.21a)

A(x, y, z, t = 0) = A(0) (2.21b)curl A× n = g (2.21c)

is invariant under translation and rotation if ji transforms according to the rules (2.19)and if A and g transform as

TA = A−T∫ t

0TT grad (V ·A). (2.22)

Tg = g (2.23)

Proof. We can interpret A as a solution of (2.17) with A∗ = A, ϕ∗ = 0. Owing torelation (2.20) transformation rule (2.22) follows immediately and it is clear that A is asolution to (2.21). The transformation rule for g follows from the relation T ˜curl( TTg) =curl g (see Appendix A, Eq. (A.4)).

Remark. If we were to use the simple transformation rule TA = A instead of Eq. (2.22)the system would actually change gauge when switching from the laboratory to themoving frame of reference. In the rest frame the system is gauged such that ϕ ≡ 0 butin the moving frame ϕ = −V · A. I.e. an additional term would appear if the eddycurrent equation was expressed in the principal coordinates of the moving frame. Toovercome this problem we applied the temporal gauge transformation to A and ϕ.

We have thus provided transformation formulas for all quantities appearing in theCoulomb gauged, the temporal gauged and the H-formulation of the eddy current equa-tion. In chapter 4 we will use these relations to derive the appropriate transmissionconditions at sliding interfaces.

10

3. Discontinuous Galerkin formulation

In the previous chapter we have seen that the solution of the eddy current problem isnot necessarily continuous across interfaces. For instance in the Eddy current problemonly the tangential components of the magnetic field H are continuous across interfaces.The normal component of the field can jump. Thus edge/Nédelec elements (see [3], [13,Sect. 5.5]) are often used in the context of finite element methods. They are tangentiallycontinuous by construction and are thus conforming in the space H(curl,Ω) (cf. Sect.3.1.2).

In the framework of discontinuous Galerkin methods one allows the numerical solutionto be discontinuous across element faces. In particular we will allow the numericalsolution of the eddy current problem to contain tangential discontinuities. Consequentlythis solution will not lie in the space H(curl,Ω) any more and thus the method isnon-conforming.

Because the numerical solution isn’t required to be continuous, additional conditionsare required to assert uniqueness and convergence. There are various ways to achievethis; We will focus on the symmetric-weighted-interior-penalty (SWIP) method whichessentially punishes tangential jumps of the numerical approximation. Using certainassumptions on the analytical solution and on the mesh we will be able to prove theconvergence of the approximative solution to the analytical solution in a well definedsense (Sec. 3.2.1).

The relaxed constraint on the continuity of the tangential components has the big advan-tage that non-matching meshes can easily be included in the discussion. This is becausethe resulting formulation is local to each element and the elements are interconnectedonly by their fluxes. As we will see later, the discontinuous Galerkin approach is indeedvery well suited for non-conforming meshes resulting from sliding interface problems.

3.1. Preliminaries

The discontinuous Galerkin approach relies on an extensive machinery of mathematicaltools. We will present many of them without proof in this section and refer the readerto the excellent book of Di Pietro and Ern [6] for further information.


3.1.1. Notation

Later on we will need the following spaces:

Lp(Ω) :=f : Ω 7→ R : f is Lebesque measurable,

∫Ω|f(x)|p dµ(x) <∞

,

Wm,p(Ω) := f ∈ Lp(Ω)| ∂αf ∈ Lp(Ω) for all |α|`1 ≤ m ,Hm(Ω) := Wm,2(Ω),

H(curl,Ω) : =

f ∈[L2(Ω)

]3∣∣∣∣ ‖curl f‖L2(Ω) <∞

where ∂αf and curl refer to the weak/distributional derivative of f and α is a multi-index. The spaces are equipped with the norms and seminorms

‖f‖Lp(Ω) :=(∫

Ω|f |p

)1/p

‖f‖Wm,p(Ω) :=

∑|α|`1≤m

‖∂αf‖pLp(Ω)

1/p

|f |Wm,p(Ω) :=

∑|α|`1=m

‖∂αf‖pLp(Ω)

1/p

‖f‖Hm(Ω) := ‖f‖Wm,2(Ω) |f |Hm(Ω) = |f |Wm,2(Ω)

‖f‖H(curl,Ω) :=(‖f‖2L2(Ω) + |f |2H(curl,Ω)

)1/2|f |H(curl,Ω) := ‖curl f‖L2(Ω) .

We used the convention that the L2 norm and the Hm norms of vector valued function‖f‖ =

√∑di=1 ‖fi‖

2. We also define the L2 inner product by (f, g) :=∫

Ω fg, and equiv-alently for vector fields (f ,g) :=

∫Ω f · g. For time dependent problems we consider the

space

C l(V ) := C l([0, tF ];V )

of V -valued functions which are l-times continuously differentiable in the interval [0, tF ].I.e. if ψ ∈ C l(V ), then

ψ : (0, tF ) 3 t 7→ ψ(t) ≡ ψ(·, t) ∈ V.

Meshes We will assume that there exists a mesh of the domain Ω and define it’s facesas (cf. Di Pietro and Ern [6, Definition 1.16])

Definition 3.1 (Mesh faces). Let Th be a mesh of the domain Ω. We say that a (closed)subset F in Ω is a mesh face if F has positive (d−1)-dimensional Hausdorff measure (indimension 1, this means that F is nonempty) and if either one of the following conditionsis satisfied:

1. There are distinct mesh elements T1 and T2 such that F = ∂T1 ∩ ∂T2; in such acase, F is called an interface

12

3.1. Preliminaries

2. there is T ∈ Th such that F = ∂T ∩∂Ω; in such a case, F is called a boundary face.

Based on this we introduce the following sets of faces:

F ih := set of interfaces in Th,Fbh := set of all boundary faces in Th,Fh := F ih ∪ Fbh,FT := F ∈ Fh|F ⊂ ∂T .

For a given element/face T ∈ Th or F ∈ Fh we denote by hT resp. hF the diameter ofthe T resp. F . Further we define

h := maxT∈Th

ht.

Later on we will state polynomial approximation properties and trace inequalities whichrely on a sequence of meshes TH := (Th)h∈H where the sequence H = (hn)n∈N is strictlydecreasing and satisfies hn > 0 ∀n and hn → 0 as n → ∞. Further we assume that themesh sequence TH is admissible (see [6, Definition 1.57]) and weakly quasi-uniform:

Definition 3.2 (Weakly quasi-uniform mesh sequence (cf. [12])). A mesh sequence His weakly quasi-uniform if there is a constant µ∗ with 0 < µ∗ < 1 such that

hh−µ∗

min → 0 as h→ 0 where hmin = minT∈Th

hT

One can obtain an admissible and weakly quasi-uniform sequence by starting out witha coarse, tetrahedral nonconforming mesh and then subdividing every tetrahedron uni-formly in 8 smaller tetrahedrons. In 2D the same procedure works with triangles.

On each mesh element we can further define the polynomial space Pkd(T ),

Pkd(T ) :=

p : Rd ⊃ T 3 x 7→ R∣∣∣ p(x) =

∑|α|`1≤k

γαxα

where we used the multi-index α and T ∈ Th is a mesh element.

Following the idea of Di Pietro and Ern [6] we define the broken polynomial space by

Pkd(Th) :=v ∈ L2(Ω)

∣∣∣ ∀T ∈ Th, v|t ∈ Pkd(T ).

Later on in this chapter we will look for discrete solutions Ah to the eddy current problem(2.13) in broken polynomial space

[Pk3(Th)

]3. This means that Ah can be discontinuous

across faces in tangential and normal direction. This motivates the introduction of a

13


tangential jump resp. weighted average of a function Ah ∈[Pk3(Th)

]3across a face

F ∈ Fh:

for F ∈ F ih, F = ∂Ti ∩ ∂Tj :for F ∈ Fbh, F = ∂Ti ∩ ∂Ω :

[Ah]T = nF ×(

Ah|Ti − Ah|Tj)

[Ah]T = nF × Ah|Ti(jump)


Ahω = ω1 Ah|Ti + ω2 Ah|TjAhω = Ah|Ti .

(average)

where nF always points from Ti to Tj and ωi ∈ [0, 1] are weighting factors which fulfil theconstraint ω1 +ω2 = 1. Notice that the tangential jump as well as the weighted averageboth act on vectors and produce vectors. In a similar fashion we define the usual jumpsand averages for scalar valued functions ϕ ∈ Pk3(Th),


[ϕ] = ϕ|Ti − ϕ|Tj[ϕ] = ϕ|Ti

(jump)


ϕω = ω1 ϕ|Ti + ω2 ϕ|Tjϕω = ϕ|Ti .

(average)

Remark. In the above definitions the corresponding traces of Ah, ϕ always exist becauseAh resp. ϕ lies in the broken polynomial space. Later on we will use the same definitionof the tangential jump and weighted average also for functions belonging to more generalfunction spaces. In order for this to work the traces of the function must exist. E.g. itdoesn’t make sense to define a the tangential jump of a L2 function over an interfacebecause there is no suitable definition for the trace of the function. We will fix thisambiguity by placing additional assumptions on the analytical solution and defer thedetails to Sec. 3.2.1.Remark. For the scalar valued jumps it is necessary to fix the orientation of nF uniquelyfor every edge.

3.1.2. Broken spaces

In this section we extend the concepts of Sobolev spaces and the H(curl,Ω) space byallowing discontinuities across element faces (assuming there exists a suitable mesh).Following Di Pietro and Ern [6, Section 1.2.5] we define the broken Sobolev spaces as

Wm,p(Th) := v ∈ Lp(Ω) | ∀T ∈ Th, v|T ∈Wm,p(T ) ,

Hm(Th) :=v ∈ L2(Ω)

∣∣∣ ∀T ∈ Th, v|T ∈ Hm(T ).

Analogous we define the broken version of the H(curl,Ω) space by

H(curl, Th) :=

v ∈[L2(Ω)

]3 ∣∣∣∣ ∀T ∈ Th, v|T ∈H(curl, T ).

14

3.1. Preliminaries

In order to simplify notation we introduce the broken gradient and broken curl operator.

Definition 3.3 (Broken gradient, cf. [6]). The broken gradient grad h : W 1,p(Th) →[Lp(Ω)]d is defined such that, for all v ∈W 1,p(Th),

∀T ∈ Th, ( grad hv)|T := grad (v|T ) .

Definition 3.4 (Broken curl). The broken curl operator curlh : H(curl, Th) 7→ [L2(Ω)]3is defined such that, for all v ∈H(curl, Th),

∀T ∈ Th, (curlh v)|T := curl (v|T ) .

Intuitively it is clear that the Sobolev spaces Wm,p(Ω) and Hm(Ω) are subspaces ofthe more general broken Sobolev spaces Wm,p(Th) and Hm(Th). The same holds forthe H(curl,Ω) spaces. The following statements are more precise and allow us tocharacterize the involved spaces.

Lemma 3.5 (Broken gradient on usual Sobolev spaces). Let m ≥ 0 and let 1 ≤ p ≤ ∞.There holds Wm,p(Ω) ⊂ Wm,p(Th). Moreover, for all v ∈ W 1,p(Ω), grad hv = grad vin [Lp(Ω)]d.

Lemma 3.6 (Characterization of W 1,p(Ω)). Let 1 ≤ p ≤ ∞. A function v ∈ W 1,p(Th)belongs to W 1,p(Ω) if and only if

[v] := 0 ∀F ∈ F ih.

Lemma 3.7 (Broken curl operator on usual H(curl,Ω) space). There holds H(curl,Ω) ⊂H(curl, Th). Moreover for all v ∈H(curl,Ω), curlh v = curl v in [L2(Ω)]3.

Proof. Let v ∈ H(curl,Ω) and Φ ∈ [C∞0 (T )]3 with T ∈ Th. Using the definition of thebroken curl operator and the notion of a weak derivative we can write:∫

Tcurlh v ·Φ =

∫T

curl (v|T ) ·Φ =∫T

v · curl Φ

Let now EΦ be the extension of Φ by zero to Ω.∫T

curlh v ·Φ =∫

Ωv · curlEΦ =

∫Ω

curl v · EΦ =∫T

(curl v)|T · Φ

I.e. curlh (v|T ) = (curl v)|T since Φ is arbitrary. Because T ∈ Th is arbitrary as well,curlh v = curl v.

Lemma 3.8 (Characterization of H(curl,Ω)). A function v ∈H(curl, Th)∩[W 1,1(Th)]3belongs to H(curl,Ω) if and only if

[v]T = 0 ∀F ∈ F ih

15


Proof. Let v ∈H(curl, Th) ∩ [W 1,1(Th)]3 and let Φ ∈ [C∞0 (Ω)]3.∫

Ωv · curl Φ =

∑T∈Th

∫T

v · curl Φ

=∑T∈Th

−∫∂T

(nT × v) · Φ +∫T

(curl v) · Φ

=∫

Ωcurlh v · Φ−

∑F∈Fi

h

∫F

[v]T · Φ (3.1)

where we have used that v ∈ [W 1,1(Th)]3 so that v|T ∈ [L1(T )]3 which allows us to splitthe boundary integral into the integrals over the faces. Now assume that [v]T = 0∀F ∈F ih, then ∫

Ωv · curl Φ =

∫Ω

curlh v · Φ.

If we compare this to the definition of the weak derivative,∫Ω

v · curlφ =∫

Ωcurl v · φ ∀φ ∈ [C∞0 (Ω)]3,

we see that curl v = curlh v because Φ is arbitrary. This implies v ∈ H(curl,Ω).Conversely assume v ∈H(curl,Ω), then relation (3.1) becomes

∑F∈Fi

h

∫F

[v]T · Φ =∫

Ωcurlh v · Φ−

∫Ω

v · curl Φ

=∫

Ω(curlh v− curl v) · Φ

Lemma 3.7= 0

Because Φ is arbitrary this implies [v]T = 0 on all F ∈ F ih.

Lemmas 3.5 and 3.6 are proven in a similar fashion as Lemmas 3.7 and 3.8. We referthe reader to the book of Di Pietro and Ern [6, Sect. 1.2.5] for the actual proofs.

Remark. Lemma 3.8 gives another explanation for the continuity of the tangential com-ponent of the magnetic field H previously discussed in Sect. 2.1: In order for H tofulfil the strong form of Ampères law (2.1d) its curl must exist, hence H ∈ H(curl,Ω)(assuming that no surface currents exist).

16

3.1. Preliminaries

3.1.3. Discrete Friedrichs inequality

In order to prove convergence of the DG formulation later on we need another theoremwhich is based on the spaces:

Sh :=ph ∈ H1

0 (Ω) : ph|T ∈ Pk+13 (T ), ∀T ∈ Th

Xh :=

vh ∈ [Pk3(Th)]3 : (vh, grad ph) = 0,∀ph ∈ Sh

X :=

v ∈

[W 1,1(Th)

]3 ∣∣∣∣ div v = 0 in Ω

where the space Xh contains only ”discrete divergence free” functions. Note that ifv ∈ X then (v, grad ph) = 0 for all ph ∈ Sh. Indeed, if v ∈ X then [v · nF ] = 0 over allfaces F ∈ F ih (see [6, Lemma 1.24]) and we can write

(v, grad ph) =∑T∈Th

v · grad ph =∑T∈Th

∑F∈FT

∫F

(v · nT )ph =∑F∈Fi

h

∫F

[phv] · nF = 0

where we have used that v is divergence free and that neither ph nor v jumps in normaldirection.

The following theorem is due to Creusé and Nicaise [5, Theorem 4.5]:

Theorem 3.9. Let TH be an admissible, weakly quasi-uniform sequence of conformingmeshes which consist only of tetrahedrons or hexahedrons. Then there exists a positiveconstant Cf > 0 independent of h such that for all h ∈ H, one has

‖Ah‖L2(Ω) ≤ Cf

‖curlh Ah‖L2(Ω) +∑F∈Fh

∫Fh−1F |[Ah]T |

2

, ∀Ah ∈ Xh (3.2)

Remark. This is the discrete version of the first Friedrich inequality (see e.g. [13, Corol-lary 3.51]). It is based on a discrete compactness property which has been proven forconforming/matching meshes only (see [12]).

3.1.4. Trace inequalities and polynomial approximation

Lemma 3.10 (Discrete trace inequality). Let Th be an admissible mesh sequence. Then,for all h ∈ H, all vh ∈ Pkd(Th), all T ∈ Th,

h1/2T ‖vh‖L2(∂T ) ≤ CtrN

1/2∂ ‖vh‖L2(T ) (3.3)

where Ctr and N∂ are both independent of h and T .

Proof. See [6, Lemma 1.46 and Eq. (1.40)]

17


Lemma 3.11 (Polynomial approximation of curl operator). Let Th be an admissiblemesh sequence and πh be the L2-orthogonal projection onto [Pk3(Th)]3. Then for allA ∈ [Hk+1(Th)]3, there holds

‖curlh (A− πhA)‖L2(Ω) ≤ Chk|A|Hk+1(Ω)

where C is independent of h.

Proof. Let u = (ux, uy, uz)T ∈ [H1(T )]3 then we have:

‖curl u‖2L2(T ) = ‖∂yuz − ∂zuy‖2L2(T ) + ‖∂zux − ∂xuz‖2L2(T ) + ‖∂xuy − ∂yux‖2L2(T )

≤ 2|u|2H1(T )

Applying [6, Lemma 1.58],

‖curl (A− πhA)‖L2(T ) ≤ C′|A− πhA|H1(T ) ≤ Ch

kT |A|Hk+1(T ).

Summing over all T ∈ Th and using hT ≤ h for all T ∈ Th yields the assertion.

3.2. DG formulation of Eddy Current problem

In this section the symmetric-weighted-interior-penalty discretization of the eddy currentIBVP (2.13) is presented and convergence to the analytical solution is proven. Manyof the lemma’s and proofs in this section are inspired by lemmas/proofs in the book ofDi Pietro and Ern [6]. In order to simplify the problem we make the additional

Assumptions 3.12.1. The computational domain Ω is a simply connected polyhedron with Lipschitz

boundary and can therefore be tiled by a mesh Th consisting of polyhedral ele-ments T ∈ Th.

2. There exists a partition PΩ = Ωi1≤i≤Np of Ω such that

a) Each Ωi is a polyhedron.

b) The permeability µ and the conductivity σ are constant on each Ωi and inde-pendent of time. Moreover there exist constants µmin, µmax, σmax such that0 < µmin ≤ µ ≤ µmax and 0 < σ ≤ σmax.

3. The mesh sequence TH is compatible with the partition PΩ: For each Th ∈ TH,each element T ∈ Th belongs to exactly one Ωi ∈ PΩ.

4. The initial condition A(t = 0) and the impressed current ji(t) are both divergencefree and the boundary traces exist for all t ∈ [0, tF ], i.e. A(t = 0), ji(t) ∈ X.

18


5. For the Neumann boundary condition there holds (g × n) ∈ C1(∂Ω)2, i.e. thetangential components of g are continuously differentiable.

Assumption 2b implies that Ω2 in (2.13) is empty (only conductors are present) andtherefore it suffices to solve the following IBVP:

curl 1µ

curl A + σ∂A∂t

= ji on Ω (3.4a)

A(t = 0) = A(0) on Ω (3.4b)1µ

curl A× n = g× n on ∂Ω. (3.4c)

Using the finite element space Vh := [Pk3(Th)]3, the SWIP space semi-discretized varia-tional form of the strong form is

Find Ah ∈ C0(Vh) ∩ C1([L2(Ω)]3) such that for all t ∈ [0, tF ], all A′h ∈ Vh,(σ∂Ah

∂t,A′h

)+ aSWIP

h (Ah,A′h) =(ji,A′h

)+∫∂Ω

(g× n) ·A′h (3.5a)

Ah(t = 0) = πh(A(0)) (3.5b)

Where g ∈ L2(∂Ω) and the SWIP bilinear form is

aSWIPh (Ah,A′h) =

∫Ω

1µ

curlh Ah · curlh A′h −∑F∈Fi

h

∫F

1µ

curlh Ah

ω

·[A′h]T −

∑F∈Fi

h

∫F

1µ

curlh A′hω

· [Ah]T +∑F∈Fi

h

ηγµ,FhF

∫F

[Ah]T ·[A′h]T

.

(3.6)

The last three summands of aSWIPh are usually referred to as consistency, symmetry and

penalty terms, respectively. The penalty parameter η must be choosen big enough forconvergence (see lemma 3.16). The factor γµ,F and the average weights are given by

γµ,F := 2µ1 + µ2

, ω1 := µ1µ1 + µ2

, ω2 := µ2µ1 + µ2

. (3.7)

where µi denotes the (constant) value of µ on either side of a face F .Remark. If we choose the conforming finite element space V ′h such that V ′h ⊂H(curl,Ω),the consistency, symmetry and penalty terms in aSWIP

h would disappear and we wouldend up with the usual finite element formulation of the eddy current problem. Alsonotice that aSWIP

h is a symmetric bilinear form and thus the resulting stiffness matrix issymmetric as well.

19


Remark. The bilinear form aSWIPh is equivalent to the one stated by Creusé and Nicaise

[5] if we set µ ≡ 1 everywhere. The main difference is, that the present form also supportsdiscontinuous µ in a stable manner.Remark. Because we have Neumann boundary conditions there is no need to integrateover the boundary faces. If we had Dirichlet boundary conditions aSWIP

h would look thesame but the face integrals would include the boundary faces.

The problem (3.5) is a parabolic problem with Neumann boundary conditions. We willtherefore use implicit Euler timestepping to avoid stability problems. For this we dividethe time interval [0, tF ] into N equal intervals, δt = tF/N. We use the superscripts (·)(i)

to denote the value of a quantity at time t(i) := i · δt so that the fully discretized versionof (3.5) reads as

Find A(i)h ∈ Vh, i = 1, . . . , N such that for all A′h ∈ Vh, we have

(σ

A(i+1)h −A(i)

h

δt,A′h

)+ aSWIP

h (A(i+1)h ,A′h) =

(ji,(i+1),A′h

)+∫∂Ω

(n×A′h) · g(i+1),

(3.8a)

A(0)h = πh(A(0)) (3.8b)

3.2.1. Convergence

In this section we will prove that the approximate solution Ah obtained by Eq. (3.8)converges to the analytical solution A and give an upper bound for the error. Of courseA must exist and we additionally require

Assumption 3.13. We assume that A ∈ C0(V∗)∩C2(L2(Ω)3) where V∗ := H(curl,Ω)∩H2(PΩ) and that A is a solution of the strong form (3.4).

The first assumption is needed to assure that the traces of A on the element facesexist which in turn allows us to extend the meaning of aSWIP

h to the space V ∗: aSWIPh :

V ∗h × Vh 7→ R where V ∗h := Vh ∪ V ∗. (The usual trace theorem for H(curl, T ) gives atrace in H−1/2

t (∂T ) which doesn’t suffice: We need the trace to be in at least L2(∂T ) sothat the surface integral can be split into face integrals, cf. [13, Remark 3.32] and [6,Remark 1.25]) Together with the second assumption this allows us to prove consistency:

Lemma 3.14 (Consistency). The bilinearform aSWIPh is consistent: If A ∈ C0(V ∗h ) ∩

C1(L2(Ω)) is the strong solution of the IBVP (3.4) then

aSWIPh (A(t),A′h) =

∫Ω

curl 1µ

curl A ·A′h +∫∂Ω

(g× n) ·A′h. (3.9)

20


Moreover,

(σ∂A(t)∂t

,A′h)

+ aSWIPh (A(t),A′h) =

(ji,A′h

)+∫∂Ω

(g× n) ·A′h (3.10)

for all A′h ∈ Vh, all t ∈ [0, tF ], all h ∈ H.

Proof. Because A(t) ∈ H(curl,Ω), all tangential jumps in the definition of aSWIPh are

zero and therefore the symmetry and the penalty term drop out immediately and we areleft with:

aSWIPh (A,A′h) =

∫Ω

1µ

curl A · curlh A′h −∑F∈Fi

h

∫F

1µ

curl Aω

·[A′h]T (3.11)

Next we need the identity

[a × b] · n = (a1 × b1 − a2 × b2) · n,= ((ω1a1 + ω2a2)× (b1 − b2) + (a1 − a2)× (ω1b1 + ω2b2)) · n,= − [b]T · aω + [a]T · bω , (3.12)

where bω := (ω2b1 + ω1b2). Note that this inequality is valid for any weights ω1, ω2.Eq. (3.11) can be written as:

aSWIPh (A,A′h) =

∫Ω

1µ

curl A · curlh A′h +∑F∈Fi

h

∫F

[ 1µ

curl A×A′h]· nF−

∑F∈Fi

h

∫F

[ 1µ

curl A]T

·A′hω

The last term in this equation is zero because A is a strong solution of IBVP (3.4) andtherefore 1

µ curl A ∈H(curl,Ω). The integral over all faces can be split into an integralover all elements,

aSWIPh (A,A′h) =

∑T∈Th

∫T

1µ

curl A · curl A′h +∑T∈Th

∫∂T

( 1µ

curl A×A′h)· nT−∫

∂Ω

( 1µ

curl A×A′h)· n

=∑T∈Th

∫T

curl 1µ

curl A ·A′h −∫∂Ω

(n× 1

µcurl A

)·A′h.

Where we have used an integration by parts formula for H(curl,Ω) functions in the laststep (see e.g. [13, Theorem 3.29]). From this Eq. (3.9) follows immediately. Eq. (3.10)is obtained by multiplying Eq. (3.4a) by A′h, integrating over Ω and using Eq. (3.9).

21


In the following we will also prove boundedness and coercivity for the discrete bilinearform aSWIP

h . With the help of the Lax-Milgram lemma this would allow us to prove thewell posedness of the time independent variational problem arising from (3.5) by settingσ ≡ 0. However we want to prove a convergence result for the time dependent problemso further steps are needed. But let us first define the (semi-)norms with respect towhich we will prove coercivity and boundedness:

|A|SWIP :=(∥∥∥µ−1/2 curlh A

∥∥∥2

L2(Ω)+ |A|2J,µ

)1/2

,

|A|J,µ : =

∑F∈Fh

γµ,FhF‖[A]T ‖

2L2(F )

1/2

,

|A|SWIP,* :=

|A|2SWIP +∑T∈Th

hT∥∥∥µ−1/2 curlh A

∣∣∣T

∥∥∥2

L2(∂T )

1/2

.

Remark. The semi-norm |·|SWIP and the jump semi-norm |·|J,µ are well defined for allfunctions ∈ H(curl,Ω). This is not true for the last semi norm, |·|SWIP,*, because itrequires us to give a meaning to trace of curlh A on ∂T . We avoid this problem byassuming A ∈ V ∗ (Assumption 3.13) which allows for a suitable definition of the trace.

Note that the defined semi-norms are not equivalent to L2(Ω) norms if A ∈ V ∗. Howeverbecause the evolution equation (3.8a) only acts on the discrete divergence free compo-nents of Ah we will be able to prove convergence in the L2(Ω) norm. (A mathematicallymore rigorous statement follows later.)

Before we prove coercivity and boundedness we state a useful lemma to bound theconsistency and symmetry terms later on:

Lemma 3.15. For all (Ah,A′h) ∈ V∗h × V∗h there holds:

∣∣∣∣∣∣∣∑F∈F i

h

∫F

µ−1 curlh Ah

ω·[A′h]T

∣∣∣∣∣∣∣ ≤∑T∈Th

hT2∥∥∥µ−1/2 curlh Ah

∣∣∣T

∥∥∥2

L2(∂T )

1/2 ∣∣A′h∣∣J,µ

Proof. By the Cauchy Schwarz inequality,

∣∣∣∣∫F

µ−1 curlh Ah

ω·[A′h]T

∣∣∣∣ ≤(∫

F

∣∣∣∣ω1µ1

curlh Ah,1 + ω2µ2

curlh Ah,2

∣∣∣∣2)1/2

(∥∥[A′h]T ∥∥2L2(F )

)1/2

22


⇒ Owing to the Cauchy Schwarz inequality,∣∣∣∑ aibi

∣∣∣2 ≤∑ a2i

∑b2i∣∣∣∣ω1

µ1curlh Ah,1 + ω2

µ2curlh Ah,2

∣∣∣∣2 ≤(ω2

1µ1

+ ω22µ2

)( 1µ1

(curlh Ah,1)2 + 1µ2

(curlh Ah,2)2)

≤(∥∥∥µ−1/2

1 curlh Ah,1∥∥∥2

L2(F )+∥∥∥µ−1/2


L2(F )

)1/2

((ω2

1µ1

+ ω22µ2

)∥∥[A′h]T ∥∥2L2(F )

)1/2

⇒ Now use that,ω2

1µ1

+ ω22µ2

= µ1

(µ1 + µ2)2 + µ2

(µ1 + µ2)2 = 1µ1 + µ2

= 12γµ,F ,

and sum over all edges

∣∣∣∣∣∣∣∑F∈Fi

h

∫F

µ−1 curlh Ah

ω·[A′h]T

∣∣∣∣∣∣∣ ≤ ∑F∈Fi

h

γµ,FhF

∥∥[A′h]T ∥∥2L2(F )

1/2

∑F∈Fi

h

hF2∥∥∥µ−1/2


L2(F )+ hF

2∥∥∥µ−1/2


L2(F )

1/2

≤

∑T∈Th

∑F∈FT

hF2∥∥∥µ−1/2 curlh Ah

∣∣∣T

∥∥∥2

L2(F )

1/2 ∣∣A′h∣∣J,µ .observing that hF ≤ hT yields the assertion.

Lemma 3.16 (Discrete Coercivity). The bilinear form aSWIPh is coercive w.r.t. to the

|·|SWIP semi-norm: For all η > C2trN∂/

√2, all Ah ∈ Vh there exists a constant Cstab > 0

such that for all h ∈ H

aSWIPh (Ah,Ah) ≥ Cstab |Ah|2SWIP

with Cstab = η−C2trN∂/

√2

1−η . The constant Ctr and N∂ stem from lemma 3.10 and areindependent of h.

Proof. Lemma 3.15 gives us an estimate on the consistency and symmetry term of aSWIPh .

The factor |Ah|J,µ appears in the definition of the |·|SWIP norm hence only an estimate

23


for the other factor on the right-hand side of lemma 3.15 is needed. Owing to the traceinequality 3.10 (Ah ∈ Vh),

∑T∈Th

hT2∥∥∥µ−1/2 curlh Ah

∣∣∣T

∥∥∥2

L2(∂T )≤∑T∈Th

C2trN∂

2∥∥∥µ−1/2 curlh Ah

∥∥∥2

L2(T )

= C2trN∂

2∥∥∥µ−1/2 curlh Ah

∥∥∥2

L2(Ω). (3.13)

Hence,

aSWIPh (Ah,Ah) ≥

∥∥∥µ−1/2 curlh Ah

∥∥∥2

L2(Ω)− Ctr

√2N∂

∥∥∥µ−1/2 curlh Ah

∥∥∥L2(Ω)

|Ah|Jµ

+∑F∈T i

h

ηγµ,FhF

‖[Ah]T ‖2L2(F ) .

Next we consider the inequality x2−2βxy+ηy2 ≥ η−β2

1−η (x2+y2) which holds for arbitraryβ, η, x, y,. (It can be derived from from (βx − ηy)2 + (x − βy)2 ≥ 0.) Applying this tothe above estimate,

aSWIPh (Ah,Ah) ≥ η − C2

trN∂/√

2

1− η

(∥∥∥µ−1/2 curlh Ah

∥∥∥2

L2(Ω)+ |Ah|2J,µ

)= Cstab |Ah|SWIP .

Note that Cstab > 0 only if the penalty parameter η > C2trN∂/

√2 which completes the

proof.

Lemma 3.17 (Boundedness). There exists a constant Cbnd independent of h such thatfor all (Ah,A′h) ∈ V ∗h × Vh, all h ∈ H

aSWIPh (Ah,A′h) ≤ Cbnd |Ah|SWIP,*

∣∣A′h∣∣SWIP

Proof. Recall the definition of the bilinear form,

aSWIPh (Ah,A′h) =

∫Ω

1µ

curlh Ah · curlh A′h −∑F∈Fi

h

∫F

1µ

curlh Ah

ω

·[A′h]T −

∑F∈Fi

h

∫F

1µ

curlh A′hω

· [Ah]T +∑F∈Fi

h

ηγµ,FhF

∫F

[Ah]T ·[A′h]T

= T1 + T2 + T3 + T4

24


Next we need to bound the individual terms by the respective norms:

|T1| ≤∥∥∥µ−1/2 curlh Ah

∥∥∥L2(Ω)

∥∥∥µ−1/2 curlh A′h∥∥∥L2(Ω)

≤ |Ah|SWIP∣∣A′h∣∣SWIP ,

|T2|Lemma 3.15≤ |Ah|SWIP,*

∣∣A′h∣∣SWIP ,

|T3|Lemma 3.15≤

∑T∈Th

hT2∥∥∥µ−1/2 curlh A′h

∣∣∣T

∥∥∥2

L2(∂T )

1/2

|Ah|J,µ ,

Eq. (3.13)≤

√C2

trN∂

2∥∥∥µ−1/2 curlh A′h

∥∥∥L2(Ω)

|Ah|J,µ ≤ C∣∣A′h∣∣SWIP |Ah|SWIP

|T4| ≤∑F∈Fi

h

ηγµ,FhF

‖[Ah]T ‖L2(F )∥∥[A′h]T ∥∥L2(F ) ≤ η |Ah|J,µ

∣∣A′h∣∣J,µ≤ η |Ah|SWIP

∣∣A′h∣∣SWIP ,

where we have used the Cauchy-Schwarz inequality multiple times to bound T1 andT4.

Thus we have proved the coercivity and boundedness of aSWIPh . Consider for a moment

the discrete problem which is obtained from Eq. (3.5a) by setting σ ≡ 0 in Ω. Wecould now use the Lax-Milgram Lemma to prove the well posedness of this problem ifwe add suitable boundary conditions and require Ah to be discrete divergence free. Thelast condition is needed in order to render Ah unique and implies that the |·|SWIP and|·|SWIP,* semi-norms are actually full norms on the space Vh. This is a more rigorousexplanation for the introduction of the gauge condition which was already discussed inChap. 2.

However, our goal is to proof convergence of the discrete evolution equation (3.8a) ash→ 0. Therefore an expression for the evolution of the approximation error A(i)−A(i)

h

is needed. For this we split the total error,

A(i)h −A(i) = A(i)

h − πhA(i)︸︷︷︸

ξ(i)h

− (A(i) − πhA(i))︸︷︷︸ξ

(i)π

.

The projection error ξ(i)π can always be estimated using polynomial approximation re-

sults. Hence only the development of the term ξ(i)h is of importance for the moment.

Lemma 3.18 (Evolution of ξ(i)h ). Assume the exact solution A ∈ C0(V ∗h ) ∩C2(L2(Ω)),

then (σξ

(i+1)h − ξ(i)

h

δt,A′h

)= −aSWIP

h (ξ(i+1)h ,A′h) + α(i+1) ∀A′h ∈ Vh,

25


where α(i+1) := aSWIPh (ξ(i+1)

π ,A′h) −(σπhθ

(i+1),A′h)

and θ(i+1) := δt−1 ∫ t(i+1)

t(i) (t(i) −t)∂

2A(t)∂t2 dt

Proof. Owing to Taylor’s theorem,

A(i) = A(i+1) − δt∂A(i+1)

∂t−∫ t(i+1)

t(i)(t(i) − t)∂

2A(t)∂t2

dt.

Projecting this equation into Vh we get

πhA(i) = πhA(i+1) − δtπh∂A(i+1)

∂t− δtπhθ(i+1),

Now multiply on both sides with σ and test with A′h ∈ Vh,

(σπhA(i),A′h

)=(σπhA(i+1),A′h

)− δt

(σπh

∂A(i+1)

∂t,A′h

)− δt

(σπhθ

(i+1),A′h),

and owing to the consistency of aSWIPh ,

(σπh

A(i+1) −A(i)

δt,A′h

)= −aSWIP

h (A(i+1),A′h) +(ji,(i+1),A′h

)+∫

∂Ω(g(i+1) × n) ·A′h +

(σπhθ

(i+1),A′h)

Finally we subtract the above equation from the discrete evolution equation (3.8a) toget (

σξ

(i+1)h − ξ(i)

h

δt,A′h

)= −aSWIP

h

(A(i+1)h −A(i+1),A′h

)−(σπhθ

(i+1),A′h)

= −aSWIPh

(ξ

(i+1)h ,A′h

)+ α(i+1)

Next we are going to prove that the solution A(i)h stays discrete divergence free if the

initial value A(0) is divergence free(cf. Sec. 3.1.3),

Lemma 3.19. All solutions A(i+1)h of the evolution equation (3.8) are discrete divergence

free, i.e. they belong to the space Xh. Moreover we have for the exact solution A(t) ofthe IBVP (3.4) (A(t), grad p) = 0 for all p ∈ Sh, t ∈ [0, tF ].

26


Proof. Observe that if p ∈ Sh, then grad p ∈ H(curl,Ω). Indeed, p can be viewedas a discrete 0-form and grad p is therefore a discrete one form and thus tangentiallycontinuous. See [3] for a detailed explanation. Hence we have [ grad p]T = 0 over allFaces F ∈ F ih and grad p = grad hp. Therefore the following terms cancel out:

aSWIPh (A(i+1)

h , grad p) = 0 by definition of aSWIPh(

ji,(i+1), grad p)

= 0 by assumption 3.12∫∂Ω

(g× n) · grad p =∫∂Ω

(n× ( grad p− grad p · n)) · g

= −∫∂Ω

g · curlΓp

= −∫∂Ωp curlΓ g = 0

where curlΓ denotes the vectorial surface curl and curlΓ the scalar surface curl (cf. [13,Sect. 3.4]). In the last step we applied a variant of Stokes theorem (see e.g. [13, Corollary3.21]) and used the fact that p = 0 on ∂Ω.

Let us now substitute A′h = grad p into Eq. (3.8a),(σA(i+1)

h , grad p)

=(σA(i)

h , grad p).

Summing the above equation for i = 0, . . . N−1 we see that(A(N)h , grad p

)=(A(0)h , grad p

)=

(πhA(t = 0), grad p) = (A(t = 0), grad p) = 0 which proves the first assertion. For thesecond claim we multiply Eq. (3.4a) by grad p (p ∈ Sh) and split the integral over allelements so we can use the partial integration formula (see e.g. [13, Theorem 3.29]),(

curl 1µ

curl A(t), grad p)

+(σ∂A(t)∂t

, grad p)

=(ji, grad p

)∑T∈Th

∫T

curl 1µ

curl A(t) · grad p+ ∂(σA(t), grad p)∂t

= 0

−∑T∈Th

∑F∈FT

∫F

(( 1µ

curl A(t)× nT)· grad p

)∣∣∣∣T

+ ∂(σA(t), grad p)∂t

= 0.

Now applying the relation (3.12) and noting that[µ−1 curl A(t)

]T = 0 and [ grad p]T =

0

∂(σA(t), grad p)∂t

= 0.

Observe that A(0) ∈ X and therefore (A(t), σ grad p) = 0 for all p ∈ Sh which yieldsthe assertion.

27


Lemma 3.19 is important because it implies that ξ(i+1)h ∈ Xh as well. Indeed we have(

ξ(i+1)h , grad p

)=(A(i+1)h − πhA(i+1), grad p

)= 0 +

(A(i+1), grad p

)= 0 for all

p ∈ Sh.

Lemma 3.20 (Energy estimate). Under the assumptions of lemma 3.18 assume inaddition that the mesh sequence TH is admissible, weakly quasi-uniform and matching,then there holds for i = 0, . . . , N − 1,

∥∥∥√σξ(i+1)h

∥∥∥2

L2(Ω)+∥∥∥√σ(ξ(i+1)

h − ξ(i)h )∥∥∥2

L2(Ω)+ Cstabδt

∣∣∣ξ(i+1)h

∣∣∣2SWIP

≤∥∥∥√σξ(i)

h

∥∥∥2

L2(Ω)

+ Cδt

(∣∣∣ξ(i+1)π

∣∣∣2SWIP,*

+ C2Aδt

2),

where CA = maxt∈[t(i),t(i+1)]

∥∥∥∂2A(t)∂t2

∥∥∥L2(Ω)

.

Proof. Choose A′h = ξ(i+1)h in lemma 3.18 and multiply with δt,∥∥∥√σξ(i+1)

h

∥∥∥2

L2(Ω)−(σξ

(i)h , ξ

(i+1)h

)= −δtaSWIP

h (ξ(i+1)h , ξ

(i+1)h ) + δtaSWIP

h (ξ(i+1)π , ξ

(i+1)h

− δt(σπhθ

(i+1), ξ(i+1)h

),

Now apply the relation −2ab = (a− b)2 − a2 − b2 to the term(σξ

(i)h , ξ

(i+1)h

),

∥∥∥√σξ(i+1)h

∥∥∥2

L2(Ω)−∥∥∥√σξ(i)

h

∥∥∥2

L2(Ω)+∥∥∥√σ(ξ(i+1)

h − ξ(i)h )∥∥∥L2(Ω)

= −2δtaSWIPh (ξ(i+1)

h , ξ(i+1)h )+

2δtaSWIPh (ξ(i+1)

π , ξ(i+1)h )− 2δt

(σθ(i+1), ξ

(i+1)h

),

use the coercivity and boundedness of aSWIPh as well as the Cauchy Schwarz inequality,

∥∥∥√σξ(i+1)h

∥∥∥2

L2(Ω)+∥∥∥√σ(ξ(i+1)

h − ξ(i)h )∥∥∥L2(Ω)

+ 2δtCstab∣∣∣ξ(i+1)h

∣∣∣2SWIP

≤∥∥∥√σξ(i)

h

∥∥∥2

L2(Ω)+

2Cbndδt∣∣∣ξ(i+1)π

∣∣∣SWIP,*

∣∣∣ξ(i+1)h

∣∣∣SWIP

+ 2δt∥∥∥σθ(i+1)

∥∥∥L2(Ω)

∥∥∥ξ(i+1)h

∥∥∥L2(Ω)

.

Because ξ(i+1)h ∈ Xh we can use the discrete Friedrich’s inequality 3.2 for the last term,

∥∥∥√σξ(i+1)h

∥∥∥2

L2(Ω)+∥∥∥√σ(ξ(i+1)

h − ξ(i)h )∥∥∥L2(Ω)

+ 2δtCstab∣∣∣ξ(i+1)h

∣∣∣2SWIP

≤∥∥∥√σξ(i)

h

∥∥∥2

L2(Ω)+

2δt∣∣∣ξ(i+1)h

∣∣∣SWIP

·(Cbnd

∣∣∣ξ(i+1)π

∣∣∣SWIP,*

+ C ′F

∥∥∥σθ(i+1)∥∥∥L2(Ω)

)

28


using Youngs inequality, 2ab ≤ a2/ε+ εb2, with ε = C−1stab we finally get

∥∥∥√σξ(i+1)h

∥∥∥2

L2(Ω)+∥∥∥√σ(ξ(i+1)

h − ξ(i)h )∥∥∥L2(Ω)

+ δtCstab∣∣∣ξ(i+1)h

∣∣∣2SWIP

≤∥∥∥√σξ(i)

h

∥∥∥2

L2(Ω)+

δtCstab

(Cbnd

∣∣∣ξ(i+1)π

∣∣∣SWIP,*

+ C ′F

∥∥∥σθ(i+1)∥∥∥L2(Ω)

)2.

The only thing left to complete the proof is to give an upper bound on∥∥∥σθ(i+1)

∥∥∥L2(Ω)

,

∥∥∥σθ(i+1)∥∥∥2

L2(Ω)≤ σ2

maxδt−2∫

Ω

(∫ t(i+1)

t(i)(t(i) − t)∂

2A(t)∂t2

dt)2

dx,

≤ σ2maxδt

−2∫

Ω

(∫ t(i+1)

t(i)(t(i+1) − t)2dt

)∫ t(i+1)

t(i)

∣∣∣∣∣∂2A(t)∂t2

∣∣∣∣∣2

dt

dx,

≤ 13σ

2maxδt

δt maxt∈[t(i),t(i+1)]

∥∥∥∥∥∂2A(t)∂t2

∥∥∥∥∥2

L2(Ω)

≤ C ′Aδt2,where C ′A depends on the analytical solution.

Lemma 3.20 is essential in proving the convergence of the numerical solution Ah to theexact solution as h → 0. We are now finally in the position to state the main result ofthis chapter.

Theorem 3.21 (Convergence). Let TH be an admissible, weakly quasi-uniform, match-ing mesh sequence. Assume assumptions 3.12 and 3.13 hold, A ∈ C0(Hk+1(PΩ)),Ah ∈ [Pk3(TH)]3 then for all h ∈ H , all integer k > 0,

∥∥∥√σ(A(N) −A(N)h )

∥∥∥L2(Ω)

+(Cstabδt

N∑i=1

∣∣∣A(i) −A(i)h

∣∣∣2SWIP

)1/2

≤ Ct1/2F

(C1h

k + C2δt)

where C1 = maxt∈[0,tF ] |A(t)|Hk+1(Ω) and C2 = maxt∈[0,tF ]

∥∥∥∂2A(t)∂t2

∥∥∥L2(Ω)

. The constantsC1,C2 and C are independent of h and δt.

Proof. In the following C will denote an arbitrary, positive constant which is not alwaysthe same. Let us first proof a polynomial approximation result to bound the term∣∣∣ξ(i)π

∣∣∣SWIP,*

in terms of h,

∣∣∣ξ(i)π

∣∣∣2SWIP,*

=∥∥∥µ− 1

2 curlh ξ(i)π

∥∥∥2

L2(Ω)+∑F∈Fh

γµ,FhF

∥∥∥[ξ(i)π

]T

∥∥∥2

L2(F )+∑T∈Th

hT∥∥∥µ− 1

2 curlh ξ(i)π

∣∣∣T

∥∥∥2

L2(∂T )

= T1 + T2 + T3

29


Owing to lemma 3.11 the first term can easily be bounded by T1 ≤ Ch2k∣∣∣A(i)

∣∣∣2Hk+1(Ω)

.For the second term we have

T2 ≤∑T∈Th

∑F∈FT

2γµ,FhF

∥∥∥ξ(i+1)π

∣∣∣T

∥∥∥2

L2(F )≤ Ch2k

T

∣∣∣A(i)∣∣∣2Hk+1(Ω)

where we have used [6, Lemma 1.59] in the last step. Applying the same lemma to thethird term T3,

T3 ≤ C∑T∈Th

∑F∈FT

hT

3∑j=1

∥∥∥ grad (ξ(i)π )j

∣∣∣T

∥∥∥2

L2(Ω)≤ Ch2k

∣∣∣A(i)∣∣∣2Hk+1(Ω)

Hence∣∣∣ξ(i)π

∣∣∣SWIP,*

≤ Capphk∣∣∣A(i)

∣∣∣Hk+1(Ω)

. Summing up the energy estimate from lemma

3.20 over i = 0, . . . , N − 1 and dropping the non-negative term∥∥∥√σ(ξ(i+1)

h − ξ(i)h )∥∥∥2

L2(Ω),

∥∥∥√σξ(N)h

∥∥∥2

L2(Ω)+ Cstabδt

N∑i=1

∣∣∣ξ(i)h

∣∣∣2SWIP

≤ C(C2

1h2k + C2

2δt2).

⇒ Owing to the triangle inequality,∥∥∥√σ(A(N) −A(N)

h )∥∥∥2

L2(Ω)≤ 2

∥∥∥√σξ(N)h

∥∥∥2

L2(Ω)+ 2

∥∥∥√σξ(N)π

∥∥∥2

L2(Ω)N∑i=1

∣∣∣A(i) −A(i)h

∣∣∣2SWIP

≤ 2N∑i=1

∣∣∣ξ(i)h

∣∣∣2SWIP

+ 2N∑i=1

∣∣∣ξ(i)π

∣∣∣2SWIP

12∥∥∥√σ(A(N) −A(N)

h )∥∥∥2

L2(Ω)+ 1

2CstabδtN∑i=1

∣∣∣A(i) −A(i)h

∣∣∣2SWIP

≤ C(C2

1h2k + C2

2δt2)

+∥∥∥√σξ(N)

π

∥∥∥2

L2(Ω)+ δt

N∑i=1

∣∣∣ξ(i)π

∣∣∣2SWIP

.

The two last terms are also bounded in terms of h2k,

δtN∑i=1

∣∣∣ξ(i)π

∣∣∣2SWIP

≤N∑i=1

∣∣∣ξ(i)π

∣∣∣2SWIP,*

≤ C · C21h

2k,

∥∥∥√σξ(N)π

∥∥∥2

L2(Ω)≤√σC ′apph

2k+2∣∣∣A(N)

∣∣∣Hk+1(Ω)

,

where the last inequality is a consequence of [6, Lemma 1.58]. This completes theproof.

30


The convergence theorem 3.21 is optimal w.r.t. the |·|SWIP norm, i.e. the optimalconvergence rate k is achieved. The only flaw is that for k = 1 the H2 semi-normappears on the right-hand side instead of a term of the form

∣∣∣ 1µ curlh A

∣∣∣H1(Ω)

. Howeverthis is not limitation in light of assumption 3.13.

In contrast to the SWIP-norm the bound on the L2 norm is suboptimal by one powerin h. However it should be possible to get a sharper bound by following the idea ofWheeler (cf. [6, Sect. 4.7.5]).

It is possible to derive the same error estimate if A is approximated with edge functionsand not general polynomials. In this case one can use Theorem 5.41 from [13] in theproof of Thm. 3.21.

The biggest flaw, with respect to this thesis, is that Thm. 3.21 is only valid for matchingmeshes, i.e. hanging nodes are not allowed. This stems from the discrete compactnessproperty which is only valid for matching meshes (see [12]) and on which the discreteFriedrich inequality (Thm. 3.9) depends.

A convergence theorem similar to Thm. 3.21 can be derived for dirichlet boundaryconditions by following a similar approach. The only difference in the aSWIP

h bilinearform is that the sums over all faces also include boundary faces. Moreover it should bepossible to relax the condition on the Neumann boundary condition from g×n ∈ C1(∂Ω)to g × n ∈ H1(∂Ω)2 by splitting the surface integral in the proof of lemma 3.15 intointegrals over the faces of each element.

Finally we want to stress that this convergence result is only valid for h-convergence.For p refinement further investigations are needed.

31

4. Aspects of the Implementation

So far we have always looked at the 3D eddy current problem. In chapter 2 we havederived the model from a physical point of view and in chapter 3 we have given a proofof convergence under the assumptions 3.12 and 3.13. We are now going to simplify theproblem from chapter 3 by

1. reducing it to two dimensions (Sec. 4.1), and by

2. assuming σ > 0 everywhere i.e. the whole computational domain is filled by aconductor.

The second assumption is not a real restriction: The electromagnetic fields in free spacecan be calculated after the fields inside the conductors have been calculated (cf. [2]).

On the other hand we will extend the applicability of the model by dropping the as-sumption of Ω being a simply connected polyhedron. When we discuss the numericalresults in chapter 5 we will consider settings where Ω or a subdomain Ωi ∈ PΩ has theshape of a circle and therefore no polyhedral mesh can exist. To handle such geometriesin a consistent way we introduce special elements with circle shaped boundaries (Sec.4.2).

This chapter starts off by presenting four different ways to discretize the 3D Eddy currentproblem from the previous chapter under the assumption that the whole system is atrest. Following this we present some implementation details. Finally we merge the 2Dvariational formulation obtained from Sec. 4.1 with the transformation laws of Sec. 2.4in Sec. 4.3.

4.1. Reduction to 2D

Our goal is to reduce the temporal gauged eddy current formulation 2.13 and the H-formulation into a meaningful two-dimensional representation. There are principally twodifferent ways of doing this: The transverse electric (TE) and the transverse magnetic(TM) formulation (cf. [9, Sec. 8.2], [4]). In both cases one assumes that the 2D domainΩ lies in the x − y plane and that it is the cross-section of a 3D unbounded cylinderwhich is aligned with the z-axis. Further the material coefficients µ and σ as well as the

4.1. Reduction to 2D

source current ji are independent of the z-component.

In the TE model it is assumed that the impressed current ji lies completely in the x− yplane, ji = (jix, jiy, 0). In contrast the TM model assumes that ji is aligned with thez-axis, i.e. ji = (0, 0, jiz). In both cases the form of the remaining electromagnetic fieldsis easily derived. We have summarized them in table 4.1.

Field quantity TM-formulation TE-formulation

ji resp. jf (0, 0, jiz), (0, 0, jfz ) (jix, jiy, 0), (jfx , jfy , 0)

E (0, 0, Ez) (Ex, Ey, 0)

B resp. H (Bx, By, 0), (Hx, Hy, 0) (0, 0, Bz), (0, 0, Hz)

A resp. A∗ (0, 0, Az), (0, 0, A∗z) (Ax, Ay, 0), (A∗x, A∗y, 0)

Table 4.1.: The 2D equivalents of most electromagnetic quantities for the TE and TMformulation.

In order to state the 2D variational form of the eddy current models it is useful tointroduce the following 2D scalar and vector curl operators:

curl2D A := ∂Ay∂x− ∂Ax

∂y, (2D Scalar curl)

curl2D Az :=(

∂Az∂y

−∂Az∂x

). (2D Vector curl)

Let us further introduce the 2D transverse respectively longitudinal version of the symmtric-weighted-interior-penalty bilinearform (3.6):

µaSWIPh,T (Ah,A′h) :=

∫Ω

1µ

curl2D,h Ah · curl2D,h A′h −∑F∈Fi

h

∫F

1µ

curl2D,h Ah

ω

·[A′h]T

−∑F∈Fi

h

∫F

1µ

curl2D,h A′hω

· [Ah]T +∑F∈Fi

h

ηγµ,FhF

∫F

[Ah]T ·[A′h]T

µaSWIPh,L (Az, A′z) :=

∫Ω

1µgrad hAh,z · grad hA

′h,z −

∑F∈Fi

h

∫F

1µgrad hAh,z

ω

· nF[A′h,z

]

−∑F∈Fi

h

∫F

1µgrad hA

′h,z

ω

· nF [Ah,z] +∑F∈Fi

h

ηγµ,FhF

∫F

[Ah,z][A′h,z

]

where the tangential jump for the µaSWIPh,T bilinearform is defined by [Ah]T = nF,xAy −

nF,yAx. The broken, scalar curl operator curl2D,h is defined analogous to the brokengradient in Def. 3.3. Let us further define the bilinearforms µaSWIP

h,T,0 respectively µaSWIPh,L,0

33


in the same way but such that the sums of the consistency, symmetry and penalty termsinclude the boundary faces of the mesh Th. We need those extended forms in the nextsection for solving IBVP with homogeneous dirichlet boundary conditions.

Note that the longitudinal bilinearform µaSWIPh,L,0 is exactly the same as the one given by

Di Pietro and Ern [6, Eq. (4.64)] for the Poisson problem with heterogeneous diffusion.This agrees with the fact that the H-model in the TE formulation is equivalent to the2D heat equation.

Tranverse Electric formulation (TE) By having a peek at the variational formulation 3.8it is easy to state the 2D TE version for the temporal gauged potential formulation (2.13):

Find A(j)h ∈ Rk(Th), j = 1, . . . , N , such that for all A′h ∈ Rk, all i = 0, . . . , N − 1,(

σA(i+1)h −A(i)

h

δt,A′h

)+µ aSWIP

h,T (A(i+1)h ,A′h) =

(ji,(i+1),A′h

)+∫∂Ω

(n×A′h) · g(i+1)

(4.1a)

A(0)h = πh(A(t = 0)) (4.1b)

where Rk(Th) is the space of Nedelec elemens of the first kind of order k over the meshTh (see [13, Sec. 5.5]).Remark. Instead of the space Rk we could have chosen [Pkd]2. However in this thesiswe want to use the DG face integrals only along the interface between two bodies.Everywhere else we use H(curl,Ω) conforming basis functions and thus edge elementsare the natural choice.Remark. The variational formulation 4.1 is only correct for systems at rest. We willstate the correct variational formulation for moving systems in Sec. 4.3.

In a similar way we can readily state the variational formulation for the H-model (2.16)with homogeneous dirichlet boundary conditions (g = 0) in TE-form:

Find H(j)h,z ∈ Pk1(Th), j = 1, . . . , N , such that for all H′h,z ∈ Pk1(Th), all i = 0, . . . , N − 1,µH(i+1)h,z −H(i)

h,z

δt,H′h,z

+σ aSWIPh,L,0 (H(i+1)

h,z ,H′h,z) =(

curl2D,h1σ

ji,(i+1),H′h,z), (4.2a)

H(0)h,z = πh(Hz(t = 0)). (4.2b)

Note how the Neumann source term on the right-hand side is missing. Inhomogeneousdirichlet boundary conditions could easily be implemented by adding another term onthe right-hand side. The idea is that the jumps and averages on the boundary faces areredefined using the dirichlet data (use symmetric averages with ω1 = ω2 = 1/2).

34

4.2. Details of the imlementation.

Transverse Magnetic formulation (TM) The variational formulation of the temporal gaugedpotential model as well as the H model are easily derived in the same manner as for theTM-formulation. For the sake of completeness we will state them here.

For the temporal gauged potential model (2.13) we have:

Find A(j)h,z ∈ Pk1(Th), j = 1, . . . , N such that for all A′h,z ∈ Pk1(Th), all i = 0, . . . , N − 1,

σA(i+1)h,z −A(i)

h,z

δt, A′h

+µ aSWIPh,L (A(i+1)

h,z , A′h,z) =(ji,(i+1)z , A′h,z

)+∫∂Ωg(i+1)A′h,z,

(4.3a)

A(0)h,z = πh(Az(t = 0)). (4.3b)

Similarly the TM-version of the H-formulation (2.16) is,

Find H(j)h ∈ Rk(Th), j = 1, . . . , N , such that for all H′h ∈ Rk(Th), all i = 0, . . . , N − 1,

(µH(i+1)h −H(i)

h

δt,H′h

)+σ aSWIP

h,T,0 (H(i+1)h ,H′h) =

(curl2D,h

1σji,(i+1)z ,H′h

), (4.4a)

H(0)h = πh(H(t = 0)). (4.4b)


In the previous section we have stated four 2D variational formulations. In the nextsections we are discussing some details of their implementation on the Computer. Inparticular we will focus on the non-conforming mesh data structure. For a (almostcomplete) documentation of the code we refer the reader to appendix B and the Doxygendocumentation.

General remarks All the code has been written in C++11 and is based on the Fi-nite Element library NGSolve (http://sourceforge.net/projects/ngsolve/). The latterbuilds in turn on the mesh generator Netgen which also provides the graphical interfacefor meshing and displaying the solution. Most of the code which is contained in theNonConforming-library has been tested for correctness using the unit testing frame-work CUTE developed by Sommerlad and Graf [17].

35


One of the big problems which we faced during the development of the code was the lackof support for nonconforming meshes, both by NGSolve as well as Netgen. Althoughit’s possible to store nonconforming meshes in the data structures of Netgen, NGSolvedoesn’t support this in any way (in particular the Discontinuous Galerkin componentsin NGSolve).

For this reason and some usability reasons the Author decided to come up with a com-plete remake of the mesh data structure. This data structure is discussed in detail inthe next subsection.Remark. Because the code builds on NGSolve a lot of it’s functionality was reused. Inparticular all the shape functions and most of the element mappings stem from NGSolve.The matrix/vector assembly had to be reimplemented in order to support the non-conforming mesh data structure (this was done in the classes NC_Bilinearform resp.NCLinearForm).

4.2.1. The mesh data structure

The newly introduced NonConformingMesh class is in a way a different represen-tation of the data which is available through the MeshAccess interface of NGSolve.The idea is that both data structures coexist at the same time and that they are syn-chronized. The only difference between the two is the representation of the data: TheNonConformingMesh class provides the user with additional information about hang-ing nodes, elements which are not matching etc. This information could be computeddirectly from the MeshAccess class but in a rather complicated way. In other words theNonConformingMesh class can be seen as a buffer for storing additional informationabout the nonconforming nature of the mesh.

Internally the NonConformingMesh stores references to a number of other classeswhich represent the actual geometric objects. We will refer to the entirety of all thoseclasses as the mesh data structure from now on. Figure 4.1 shows all the involved classes.The mesh data structure has been constructed such that it can be extended to threedimensions easily. However the price for this is that the model is more complicated thanwhat is needed for a purely 2D setting.

Let us start with the geometric concepts of surfaces, curves and points (we will providea concrete example a bit later. For now it is a good idea to have Fig. 4.1 at hand whilefollowing the description). These are all implemented in the classes Surface<TRIG>,Curve<LINE>, Curve<CIRC_SEGM> and Point. E.g. the class Surface<TRIG>represents a triangle in 3D/2D space. Notice that these geometric classes derive fromthe purely abstract base classes (interfaces) ISurface, ICurve. The interfaces definea common set of methods related to the geometric nature of surfaces or curves. This isessential because the matrix assembling routines presented later make no difference be-tween circle segments or lines, they treat them in exactly the same way (loose coupling).

36


The Point class doesn’t have such an interface because a point in space has always thesame geometric properties.

A second layer of mesh elements has been built on top of these purely geometric classes.At the top level a 2D NonConformingMesh consists of an arbitrary number of IFaceobjects. The present implementation supports triangles through the class Face<TRIG>which implements IFace. Note that Face<TRIG> is essentially a triangle (implemen-tation is inherited from Surface<TRIG>!) with additional properties. Most notably anIFace object contains references to IFaceSegment objects. In 2D each IFace object”contains” exactly one IFaceSegment object and a one-to-one relationship could beestablished between them. In 3D this is not the case as should become clearer later on.

In a similar fashion each ISurface object contains references to IEdge objects. Theunion of all those IEdge objects represents the geometric boundary of the ISurfaceobject. E.g. each triangle references exactly three Edge<LINE> objects. However eachIEdge object belongs to exactly one IFace and/or IFaceSegment object. In fact in2D every IEdge object corresponds to exactly one face and its facesegment. An edgecannot be associated with two different triangles!

In order to connect the different faces of the mesh with one another IEdgeSegmentshave been introduced: In 2D each IEdgeSegment belongs to exactly one IEdge ob-ject if it lies on ∂Ω. Otherwise it belongs to exactly two IEdge objects. Vice versaeach IEdge object consists of an arbitrary number of IEdgeSegment objects. This isrequired for non-conforming meshes if, for instance, the two sides of a triangle do notmatch. If the mesh is matching, each IEdge object contains exactly one IEdgeSegmentobject but each IEdgeSegment can correspond to one (boundary edge) or two (inneredge) IEdge objects.

Finally each ICurve object consists of exactly two Point objects: the start and endpoint of this curve.

In order to clarify the above description consider the nonconforming mesh in Fig. 4.2.Its associated mesh data structure is provided in table 4.2. Notice that the edges ofa face are ordered in clock-wise direction. Similarly for curves, the index of the startpoint is always lower than the index of the endpoint. For fully discontinuous Galerkinsettings the order doesn’t matter. However if the method is conforming in some parts ofthe mesh, the orientation of the edge is important to fix the orientation of the Nedelecelements. Table 4.2 also shows that in 2D every face belongs to exactly one face segmentand that they both share the same edges.

Remark. Usually the IFace, IEdge and the Point objects have a direct analogue inNGSolve’s MeshAccess interface. The corresponding ID can be obtained by callingGetNGID().

Remark. The edgesegments are constructed by callingNonConformingMesh.ConstructEdgeSegments() in O(N log(N)) time where N

37


Figure 4.1.: UML diagram of the mesh data structure.

38


Figure 4.2.: Example of nonconforming mesh. Table 4.2 shows the associated mesh datastructure.

Faces FaceSegment Edgesf0 fs0 (e0, e1, e2)f1 fs1 (e3, e4, e5)f2 fs2 (e6, e7, e8)

Edge ID EdgeSegments Start-End Pointe0 (es0) (p0, p1)e1 (es1) (p0, p3)e2 (es2, es3) (p1, p3)e3 (es4) (p1, p2)e4 (es2) (p1, p4)e5 (es5) (p2, p4)e6 (es5) (p2, p4)e7 (es3) (p3, p4)e8 (es6) (p2, p3)

Table 4.2.: Mesh data structure for sample mesh in Fig. 4.2.

39


is the number of edges. To achieve this the algorithm ”sorts” all curves accordingto their geometric parametrization (we use the term parametrization to refer to someparameters of the curve which describe it uniquely). After they have been sorted, allcurves which could potentially intersect (have same parameterization) are neighbours.The edge segments are then constructed easily.Remark. All the methods in the Surface<TRIG>, Curve<LINE> and Curve<CIRC_SEGMhave been marked as final (introduced in C++0x), i.e. all deriving classes cannot over-ride the implementation. This allows the compiler to resolve the virtual functions ofthese classes at compile time. Therefore efficient algorithms can be written by usingstatic casts for specific Curve/Surface classes. On the other hand all the Curve resp.Surface objects have a common interface and can therefore easily be stored together inone container.Remark. In principle the IFace, IFaceSegment, IEdge and IEdgeSegment classescould also use composition instead of inheritance to provide the geometric methods fromthe Surface and Curve classes. E.g. the IEdge<LINE> class could internally store areference to a Curve<LINE> object and delegate the respective methods to it. Howeverthis poses a rather big problem for the previously mentioned sorting algorithm. Inthis algorithm IEdge<LINE> objects are compared with each other. The comparisonoperator is defined in the base class Curve<LINE> and needs to access the privatemembers of the other object. This is not possible when using composition.

Circle Segments In principal the mesh data structure supports triangular IFace ob-jects which have IEdge<CIRC_SEGM> objects as edges. Strictly speaking this doesn’tmake sense because the IFace objects naturally define an element transform that mapsfrom a reference triangle to another triangle. I.e. points which lie on the boundary ofthe reference triangle would not be mapped to points on the circle segment edges.

Nevertheless we have used this approach to properly model domains with circle shapedboundaries. Consequently the surface integrals are not analytically exact and contributean error of O(h2). Note that the flux integrals have a much smaller error: They arecomputed based on the FaceSegment<CIRC_SEGM> objects which in turn provide aelement transformation which maps from a reference segment to the circle segment. Weuse this transformation together with an integration rule of order 10 which leads to analmost negligible error.

4.3. Variational formulation for moving systems

We have already discussed how the SWIP-formulation of the eddy current problem allowsfor an easy handling of non-matching grids. In this section we further explore the physicaleffects which appear if part of the system is in motion with respect to the other one and

40


Figure 4.3.: The integration area over which the surface integrals are evaluated and thepath along which the Flux integrals are calculated.

how we can incorporate them into the variational formulation. We are going to base ourdiscussion on the transformation properties which have been derived in Sec. 2.4.

Let us first observe that the transformation matrix T which appears in relations (2.19)can be neglected for scalar problems. Indeed, a scalar quantity represents a vector withzero x− y coordinates. However the T-matrix acts only on these components.

If we are dealing with a 2D vector problem we have to keep in mind that the implemen-tation uses edge functions, i.e. TA is actually the quantity which is represented by basisfunctions and not A.

Consequently, the magnetic field H doesn’t change in our implementation under trans-lation and rotation, i.e. TH = H. Hence the problems (4.2) and (4.4) need no modifi-cation if part of the mesh is translated or rotated. In other words the convective termswhich result from the Lorentz forces are automatically and correctly accounted for inthe H − formulation.

Consider now the TM-A potential formulation (4.3) with the primary, scalar unknownAh,z. The correct transformation rule is given by lemma 2.3. However notice that theterm grad

(V · (0, 0, Ah,z)T

)≡ 0 . Therefore ˜Ah,z = Ah,z as well and the formulation

(4.3) needs no modification.

Finally we consider the TE-A potential formulation (4.1). Here the transformation lawis given by 2.3 and reads as

TA = A−T∫ t

0TT grad (V ·A).

Using Eq. (A.11) we can simplify the last term as follows:

grad (V ·A) = −ω ×A + (V · ∇)A + V× curl A.

41


Next we assume that A ∈ R1(Th) and make use of

Lemma 4.1. Let u ∈ R1(T ), T ∈ Th and let V ∈ [L2(T )]3. Then there holds

(V · ∇)u = −12V× curl u

Proof. We start from the representation formula u = λi · ∇λj − λj · ∇λi where λi, λj ∈P1

3(T ) (cf. [13, Eq. 5.47]). Straightforward manipulations then show that the assertionholds.

Therefore the transformation formula for A in (4.1) is

TA = A + T∫ t

0TT (ω ×A)−T

∫ t

0TT (V× curl A). (4.5)

Now consider the two general problems presented in Fig. 4.4. Here Ω2 is always at restwhile Ω1 is either rotating or translating. We will assume that continuous basis functionsare used in both subdomains such that the flux integrals vanish inside of Ω1 and Ω2.The flux is only non-zero at the boundaries resp. interfaces Γ1, Γ2 and Γ12.

(a) Translation problem. (b) Rotation problem

Figure 4.4.: Two possible eddy current problems that involve moving bodies.

In order to calculate the fluxes over Γ12 we transform the vector potential A from the restframe Ω2 to the moving frame Ω1. Notice that the tangential component of V× curl Avanishes at the interface Γ12 in both cases.

Variational formulation for translating bodies We start from the variational formulation(4.1). Because ω = 0, we have for the tangential jump nF ×

(TA1 −TA2

)= nF ×(

TA1 −A2). This means that we don’t have to modify any terms on the left-hand side

42


of (4.1). On the right-hand side the source term stays exactly the same (Tji = ji) andfor the Neumann boundary term we have g = g. Indeed, g = 1

µ˜curlA = 1

µ curl A = gbecause curl gradA = 0.

Therefore the variational formulation (4.1) needs no modification as long as ω = 0.

Variational formulation for rotating bodies Here the boundary Γ1 = ∅ and therefore theright-hand side of (4.1) stays exactly the same. However the left-hand side needs somemodification. First observe that the tangential jump across Γ12 is

nF ×(TA(i+1)

1 −TA(i+1)2

)= nF ×

(TA(i+1)

1 −A(i+1)2 −T

∫ (i+1)·δt

0TT (ω ×A2(t))

),

= nF ×(

TA(i+1)1 −A(i+1)

2 − δt

2 ω ×A(i+1)

−δtT(i+1)i∑

j=1TT,(j)ω(j) ×A(j)

2 −δt

2 T(i+1)(ω(0) ×A(0)2 ) +O(δt2)

,where we have used the trapezoidal integration rule to approximate the integral overtime. Substituting this into the variational formulation (4.1) we get,

Find A(j)h ∈ Vh := (Ah ∈H(curl,Ω1)×H(curl,Ω2))∩R1(Th), j = 1, . . . , N , such that

for all A′h ∈ Vh, all i = 0, . . . N − 1,(σ

A(i+1)h −A(i)

h

δt,A′h

)+ aSWIP

h,rot (A(i+1)h ,A′h) =

(ji,(i+1),A′h

)+∫∂Ω

(n×A′h) · g(i+1) + lirot(A′h),

A(0)h = πh(A(t = 0)),

where aSWIPh,rot is a modified version of the bilinear form µaSWIP

h,T :

aSWIPh,rot (Ah,A′h) =

∫Ω

1µ

curl2D,h Ah · curl2D,h A′h

−∑F∈Γ12

∫F

1µ

curl2D,h Ah

ω

·([

A′h]T −

δt

2 nF ×(ω(i+1) ×A′h,2

))

−∑F∈Γ12

∫F

1µ

curl2D,h A′hω

·(

[Ah]T −δt

2 nF ×(ω(i+1) ×Ah,2

))

+∑F∈Γ12

ηγµ,FhF

∫F

([Ah]T − nF ×

(δt

2 ω(i+1) ×Ah,2

))·

([A′h]T −

δt

2 nF ×(ω(i+1) ×A′h,2

)).

(4.6)

43


Note that this bilinear form is also symmetric. The linearform lirot is given by

lirot(A′h) = −∑F∈Γ12

∫F

1µ

curl2D,h A′hω

·

nF ×

δtT(i+1)i∑

j=0εjTT,(j)

(ω(j) ×A(j)

h,2

)+

∑F∈Γ12

ηγµ,FhF

∫F

nF ×

δtT(i+1)i∑

j=0εjTT,(j)

(ω(j) ×A(j)

h,2

) ·([

A′h]T −

δt

2 nF ×(ω(i+1) ×A′h,2

)),

where ε0 = 1/2 and εj>0 = 1.Remark. We have not only transformed the trial functions Ah but also the test functionsA′h! In theory this is not needed but it gives a symmetric matrix which forms, togetherwith the mass matrix, a symmetric-positive-definite matrix. In practice it was alsoobserved that the transmission conditions are much better fulfilled if the test functionsare transformed in the above way.Remark. We have used the trapezoidal rule to approximate the time integral whichappears in the transformation rule (4.5). In the actual implementation the values ofthe integral are interpolated to the midpoints of every edge and are integrated at thislocation.Remark. We have seen that the eddy current problem in the H-formulation and the timegauged potential formulation need no modification for translating bodies (This can easilybe extended to 3D). In other words just moving the mesh already takes into account theconvective terms. A similar statement was proven by Kurz et al. [10, Section II] for theCoulomb gauged eddy current formulation.

44

5. Results

After having discussed the theoretical aspects of the DG formulation and after we havedescribed the implementation on the computer we finally present some numerical results.In particular we will discuss three examples that shed light on different aspects of theformulations.

In Sec. 5.1 we will consider a radial problem for which an analytical solution is known.We can therefore investigate the theoretical convergence bounds which have been provenin chapter 3. Unfortunately this problem doesn’t contain any convective terms by con-struction. Constructing an analytical solution in which the convective terms play a roleis very hard.

To overcome this problem we resort to comparing the TE-time gauged formulation (4.1)with the TE-H formulation. They both describe the same physical problem so similarresults are expected. Section 5.2 studies this for translational motion whereas Sec. 5.3does the same for a rotating body.

5.1. Convergence to analytical solution

We consider the following problem in polar coordinates on the domain Ω = (x, y) ∈ R2∣∣x2+y2 < 1:

∂Hz∂t− div gradHz = J0(λ1r)

(λ2

12 t

2 + t

), (5.1a)

Hz(r, t = 0) = 0, (5.1b)Hz(r = 1, t) = 0, (5.1c)

where J0 is the first Bessel function of the first kind and λ1 is it’s first root (λ1 ≈2.4048255).

One can show that this IBVP posses the unique solution

Hz(r, t) = 12 t

2J0(λ1r).

5. Results

The problem (5.1) can be discretized straightforwardly and gives the TE-H formulation4.2 introduced in chapter 4 (µ = σ = 1). We split the domain Ω into two subdomainsΩ1 = (x, y) ∈ R2∣∣x2 + y2 < 1/4, Ω2 = Ω\Ω1. The inner circle is then additionallybeing rotated (circle segments have been introduced along Ω1 ∩ Ω2) with ω = 20 rad/s.

Figure 5.1a shows the two subdomains whereas Fig. 5.1b presents the calculated solutionof the problem at t = 0.05 s .

(a) The subdomains Ω1 and Ω2 (b) The solution at time t = 0.05 sec(δt = 0.0005,h = 0.0361371)

Figure 5.1.: Discretization of the IBVP 5.1

Convergence to the exact solution From Theorem 3.21 in chapter 3 we expect to getO(h) respectively O(δt) convergence to the analytical solution. In order to prove thiswe have carried out a number of numerical experiments with different mesh sizes andtimesteps. In particular we study the error in the L2 norm and the (modified) H curlnorm

‖A−Ah‖curl :=∥∥∥A(N) −A(N)

h

∥∥∥L2(Ω)

+(Cstabδt

N∑i=1

∣∣∣A(i) −A(i+1)h

∣∣∣2SWIP

)1/2

.

The error in both norms has been approximated by an integration rule of order 5 oneach triangle and of order 10 on each edge. The total simulation time was fixed to 0.05 sand only the timestep δt and the mesh size h were varied.

Figs. 5.2a and 5.2b show that we get convergence in both norms if δt as well as h arescaled properly. This is exactly what we expect from Thm. 3.21. Next take a look atFigs. 5.3a and 5.3b which show the error in both norms for fixed δt respectively fixed h.

We see that the measured rate of convergence in terms of δt is slightly better than thetheoretical prediction whereas the convergence in h is slightly worse for theH curl-norm.

46

5.1. Convergence to analytical solution

(a) L2 error (b) H curl error

Figure 5.2.: Error vs. δt and h.

(a) δt convergence, h = 0.0180874 (b) h convergence, δt = 2.5e− 4

Figure 5.3.: Convergence plots for h and δt

47

5. Results

Remark. Di Pietro and Ern [6, Sec. 4.7.5] show that the theoretical rate of convergenceis O(h2) in the L2 norm. This estimate is optimal but is not achieved in this examplebecause of the approximation error on the circle segments (see 4.2.1).

5.2. Translational motion

The two previous test cases have shown that the TE-H formulation (4.2) produces theexpected results in the presence of sliding interfaces. In this section we will comparethis formulation with the TE-A formulation and measure the (relative) error betweenthe two approximate solutions.

Figure 5.4.: Dimensions of the translational test case.

We consider the setting depicted in figure 5.4 The bottom rectangular body is at restwhile the top one slides along its edge with velocity V = 1m/s. The material coefficientsare all set to unity (µ = σ = 1) and we use homogeneous Neumann boundary conditionsfor the TE-A formulation. Consequently we apply homogeneous Dirichlet boundaryconditions for the H-field. The system is excited by the impressed source current

ji =

(0, 0)T for x ≤ 1 or x ≥ 3(0, x− 1)T for 1 < x < 2(0, 3− x)T for 2 ≤ x < 3

.

The total simulation time is tEnd = 1 s and we start from the initial condition A = Hz =0Am−1. Figure 5.5 shows the solutions shortly after two timesteps and at the at thefinal time. We can clearly see that the motion of the top body influences the shape ofthe H-field and that both formulations yield similar results.

We now turn to a more quantitative analysis and study the error in terms of meshwidthand timestep size. The error ‖Hz − curl2D,h A‖L2(Ω) is measured slightly different forthe two cases:

h-convergence We simulate both systems with the same timestep δt and the same meshsize and measure the L2 error between them for successive refinements of h (bothformulations are refined).

48

5.2. Translational motion

(a) A-solution (curl2D,h A shown in the back-ground)

(b) H-solution

Figure 5.5.: Solutions of the translational test case at t = 0.01 s (top) respectively t = 1 s(bottom). δt = 5 · 10−3 s, h = 0.1346m.

49

5. Results

δt convergence We compute a (reference) solution of the H-formulation with timestepδt = 10−3 s and meshwidth h = 0.0336515m. The temporal gauged potentialformulation is simulated at the same mesh width but we vary the size of thetimestep. The L2 error is then calculated with respect to the H-formulation.

Remark. In the limit of very small h it doesn’t matter how we measure δt convergence:Taking one model as a reference and while refining the other produces the same rate ofconvergence as refining both models together: For both approaches we essentially plotlog(δt + ε(h)) + const = log(δt) + ε(h)/δt + O(ε(h)2) + const where ε(h) → 0 as h → 0.However if we take one solution as a reference ε(h) is typically smaller than if we refineboth systems together.

Figure 5.6 shows the convergence in terms of h and δt. The measured rates are 0.79 forh and 0.922 for δt. This is really close to the values of Thm. 3.21.

(a) h convergence, δt = 5 · 10−3. (b) dt convergence, h = 0.03365

Figure 5.6.: Convergence for the translational test case.

5.3. Rotational motion

Finally we want to compare theH-model with the temporal gauged potential formulationfor a rotating body. In contrast to the previous simulation additional transmission termsappear in the variational formulation of A (see Eq. (4.6)).

We consider the setting depicted in Fig. 5.7: The inner circle is rotating at a speed ofω = π rad/s and the total simulation time is 1 s. The material coefficients are µ = σ = 1in both sub-domains.

We consider the TE formulations (4.2) and (4.1) for the H-field respectively the A-field. We use homogeneous Neumann boundary conditions for A and correspondingly

50


Figure 5.7.: The setup of the numerical experiment considered in this section.

homogeneous Dirichlet boundary conditions for H. On the right hand side we imposethe source current ji = (4y,−2x). Both simulations are carried out at the same timeand we start from the initial condition A = Hz = 0.

Figure 5.8 shows the evolution of the A field (first column) and the Hz field (last col-umn) over time. In addition the solution which is obtained for the ”simple” tangentialboundary conditions is also included in the middle column (i.e. we used aSWIP

h,T from Sec.4.1 instead of aSWIP

h,rot ) . At the beginning the system equilibrates (0 ≤ t / 0.08). In thisphase the curlh A respectively Hz of all three solutions look qualitatively similar but astime evolves only the H field and the curlh A (with additional transmission conditions)look alike. The middle column converges clearly to the wrong solution. It’s interestingto see that the curlh A field (with additional transmission condition) is somewhat time-periodic; after one full rotation (t = 1 s) the vector field looks very similar to the vectorfield at t = 0.05 s. Indeed we observe that the A vector field periodically rotates up tothe position at t ≈ 0.95 s and then switches back to the initial position. The period isapproximately 1 second.

Figure 5.9 shows the convergence rates as δt resp. h decrease. We measure the error inthe same way as for the previous test case: For h-convergence we refine both systems atthe same time. In order to measure the δt convergence we use the H-formulation as areference solution (δtH = 10−3) and measure the error to the A-formulation for differentsizes of the timestep δtA. We see clearly that we get convergence in both variables.Although this is only the relative error between theH-formulation and the A-formulationwe can still conclude that they approximate the analytical solution asymptotically at arate of approximately 0.5 resp. 0.7 (assuming that neither of the two converges to thewrong solution).

It remains unclear as to why the observed rates are not of order 1. If the discretecompactness property (which was used in the derivation of Thm. 3.21) can be extendedto the more general jump-seminorm appearing in the aSWIP

h,rot (Eq. (4.6)) it should be

51

5. Results

t = 0.05

t = 0.5

t = 0.95

t = 1.04

(a) A (vectors), curlh A (back-ground) if transmission con-dition (4.5) is used

(b) A (vectors), curlh A (back-ground) if only tangentialcontinuity is enforced.

(c) Hz field (background)

Figure 5.8.: Different solutions of the rotational test case. (h = 0.101761)52


(a) δt convergence for different mesh sizes (b) h convergence for δt = 0.001 sec.

Figure 5.9.: Convergence of the relative error ‖Hz − curl2D,h A‖L2(Ω) at tend = 1 (onefull rotation)

possible to proof a result similar to Thm. 3.21. In fact the author has considered amuch simpler test case with a constant, impressed current ji on the right-hand side.With this setup linear O(δt) convergence was observed.

53

6. Conclusion and Outlook

Maxwell’s equations are invariant under the Lorentz transformation if the moving frameof reference is travelling at uniform speed. This is not true for rotation because alreadysmall accelerations have a significant effect (see [19], [10]). However, in the quasi-staticlimit of slowly varying electric fields we were able to prove the invariance of the EddyCurrent model under solid body motion (x = T(t)x + r(t)).

This invariance justifies a Lagrangian description. In order to deal with the non-matchinginterfaces we have adopted the Discontinuous Galerkin technique and were able to givea proof for the simple case of matching grids and (sufficiently smooth) H(curl,Ω)conforming solutions. This result was verified with the 2D, scalar H-formulation fornon-matching grids which suggests that the convergence theorem also holds on non-conforming meshes. In fact the theorem could be easily extended if the discrete com-pactness property ([12]) could be proven for non-matching meshes. Furthermore it wouldmake sense to test the convergence theorem in a 3D setting.

Because the transformation law TB = B holds for the magnetic induction the varia-tional formulation needs no change when treating problems with moving bodies (if edgefunctions are used!). However a special transmission condition must be used for the Atemporal gauged potential formulation in rotational settings; the transmission condi-tions from the steady case do not suffice! This was verified with numerical experimentsfor the TE-formulations: For the translational test case we have measured the expectedrates of convergence. For the rotational setting we would expect a rate of O(δt) but wehave measured the convergence rate O(δt1/2). It is not clear what causes the differenceand further investigations are needed.

For the numerical experiments we have developed a (rather complex) mesh data structureto represent the non-conforming nature of the mesh. The algorithm to detect edgesegments worked without problems for very fine meshes in O(N log(N)) time. It isunclear if the algorithm’s idea can be extended in an efficient way to 3D, respectively2D Face segments.

A. Maxwell’s equations in a moving frame ofreference

Consider a body which, from the view of the laboratory system, is moving in a directionr(t) rotating around an axis ω. Then we can establish a body-centered coordinate systemx such that the laboratory coordinates x are given by,

x = T(t)x + r(t), (A.1a)t = t (A.1b)

where T is a (orthogonal) rotation matrix only depending on time. The velocity of afixed point x in the moving system is given by (in laboratory coordinates):

V := ∂x∂t

= ∂T∂t

x + ∂r(t)∂t

,

= ∂T∂t

TT (x− r(t)) + ∂r(t)∂t

.

On the other hand we can express the velocity also in terms of the angular velocity ω:

V = ω × (x− r(t)) + ∂r(t)∂t

Comparing the two expressions we see that for any vector a ∈ R3:

∂T∂t

TTa = Wa = ω × a with W =

0 −ω3 ω2ω3 0 −ω1−ω2 ω1 0

. (A.2)

Our goal is to state the reduced Maxwell equations in in terms of x and it’s derivatives.In the laboratory system the equations are,

div B = 0 (A.3a)

curl E + ∂B∂t

= 0 (A.3b)

curlH = ji + jf . (A.3c)

A. Maxwell’s equations in a moving frame of reference

Remark. We have left out the electric Gauss law on purpose because there seems to beno meaningful transformation of the electric induction D under transformation (A.1a).This is not a problem because this equation is completely decoupled from the rest andis not needed to formulate the eddy current problem.

Theorem A.1. Assuming the constitutive relations of linear materials,the system ofequations (A.3) is invariant under the transformation (A.1a) if the field quantities trans-form in the following form (cf. [10]):

TE = E + V ×B TB = BTji = ji TH = HTjf = jf .

Proof. First notice that the del-operator in the body’s coordinate system can be writtenas: ∇ = Dxx∇ = TT∇. Therefore Eq. (A.3a) in the moving frame of reference can berewritten as

∇ · B =(TT∇

)T (TTB

)= ∇TTTTB = ∇ ·B = 0.

This means that divB = 0 holds and therefore the magnetic Gauss law holds also in themoving system. In order to prove the same for Faraday’s law of induction (A.3b) wefirst notice that because (Ma)× (Mb) = (det M)M−T (a × b), we have

∇ ×A = (det TT )T−1 (∇× (TA)) = TT (∇× (TA)) . (A.4)

Similarly we can express ∂B∂t

as

∂B(x(x, t), t(t)

)∂t

= ∂TTB∂t

= ∂TT

∂tB + TT ∂B

∂t(A.2)= (WT)T B + TT ∂B

(x(x, t), t(t)

)∂t

= −TT (ω ×B) + TT(∂B∂t

+(∂x

∂t· ∇)

B).

Notice that ∂x∂t

= V and use the vector calculus identity ∇× (V×B) = V(∇·B)−B(∇·V) + (B · ∇)V− (V · ∇)B. The first term is obviously zero because magnetic charges donot exist, the second term simplifies to −B(∇ · V) = −B

(∇ ·

(ω × (x− r(t)) + ∂r

∂t

))=

−B (∇ · (ω × x)) = −x · (∇× ω) + ω · (∇× x) = 0 and for the third term we get:

(B · ∇)V = (B · ∇)(ω × x) = (B · ∇)(Wx) = WB = ω ×B. (A.5)

56

Therefore,

∂B(x(x, t), t(t)

)∂t

= TT(−ω ×B + ∂B

∂t+∇× (V ×B) + ω ×B

)= TT

(∂B∂t−∇× (V ×B)

). (A.6)

Now substituting Eqs. (A.4), (A.6) into Faraday’s law and using the transformationrules of Thm. A.1 we obtain

˜curlE + ∂B∂t

= TT(

curl E + curl (V ×B) + ∂B∂t− curl (V ×B)

)= TT

(curl E + ∂B

∂t

)= 0.

I.e. Faraday’s law of induction is also invariant under the transformation (A.1a).

For Ampères law (A.3c) we use Eq. (A.4) to get,

˜curlH − ji − jf = TT(curlH− ji − jf

)= 0.

Remark. It is well known that the full Maxwell equations are invariant under the Lorentztransformation (see e.g. [16, chp. 22]). In the quasistatic limit (i.e. c → ∞) thelorentz transformation reduces to a Galilean transform and the fields are transformedas described by Thm. A.1. Thm. A.1 is however more general than this as it considersgeneral solid body motion which is generally not a Galilean transform. (T(t) as well asr(t) can depend on time!)

This is however only possible in the quasistatic limit. For high frequencies where thedisplacement current can generally not be neglected, the acceleration of a rotating frameof reference can have significant effects on the field quantities.

Generalized Ohm’s law It should be stressed that Ohm’s law, jf = σE is only valid ifthe underlying material is at rest. Let us assume that a body is moving according toEq. (A.1a). Then obviously jf = σE. Therefore, by virtue of Thm. A.1, the relation

jf = σ(E + V ×B) (A.7)

holds in the laboratory’s system. The second term on the right-hand side accommodatesfor the Lorentz forces acting on the moving charges. This is consistent with a "physical"derivation of Ohm’s law for a rotating body (cf. [11]).

57

A. Maxwell’s equations in a moving frame of reference

Transformation rules for potential formulation In the potential formulation of the eddycurrent problem one uses a a vector and scalar potential such that,

B = curl A∗ E = − gradϕ∗ − ∂A∗∂t

. (A.8)

The question arises now as to how these potentials transform under Eq. (A.1a):

Corollary A.2. Under the assumptions of Theorem A.1, the equations (A.8) are in-variant under transformation (A.1a) if the field quantities transform as follows:

TA∗ = A∗ ϕ∗ = ϕ∗ − V ·A∗

Together with Thm. A.1 this implies that if A∗ and ϕ∗ are solutions to Eqs. (A.3) thenA∗ and ϕ∗ are the respective solutions in the moving frame.

Proof. By using formula (A.4), it is easily seen that B = ˜curlA∗ = TT curl A∗ = TTB.This is exactly the transformation law of B and therefore the first equation of (A.8) isinvariant if TA∗ = A.

In order to prove that E = − gradϕ∗ − ∂A∗∂t is invariant, we start from the laboratory

frame and transform the equation to express it in terms of the body’s coordinates:

E = − gradϕ∗ − ∂A∗∂t

⇔ TE− V × curl A∗ = − grad ϕ∗ − grad (V ·A∗)− ∂TA∗∂t

(A.9)

Now use that,

grad (V ·A∗) = (A∗ · ∇)V + (V · ∇)A∗ + A∗ × (∇× V) + V × (∇×A∗)(A.5)= ω ×A∗ + (V · ∇)A∗ + A∗ × (∇× (ω × x)) + V × (∇×A∗)

⇒∇× (ω × x) = ω(div x)− x(∇ · ω) + (x · ∇)ω − (ω · ∇)x = 3ω − ω = 2ω= −ω ×A∗ + (V · ∇)A∗ + V × (∇×A∗), (A.10)

and,

∂TA∗∂t

= ∂TA∗∂t

+ (V · ∇)(TA∗)

⇔ ∂T∂t

A∗ + T∂A∗∂t

= ∂TA∗∂t

+ (V · ∇)(TA∗)

⇔ ∂TA∗∂t

= ω ×A∗ + T∂A∗∂t− (V · ∇)A∗. (A.11)

58

Therefore Eq. (A.9) can be rewritten as,

TE− V × curl A∗ = −T ˜grad ϕ∗ + ω ×A∗ + (V · ∇)A∗ − V × curl A∗ − ω ×A∗−

T∂A∗∂t− (V · ∇)A∗

⇔ TE = T(− ˜grad ϕ∗ − ∂A∗

∂t

)

59

B. Documentation of the code

This chapter together with together with the Doxygen documentation provides a detaileddescription of the code which has been written as a part of this thesis.

The source folder of this thesis contains five folders:

NonConforming In this folder lies the NonConforming library. It provides almost allthe reusable functionality that is needed to solve the eddy current problem onnon-conforming meshes.

The Doxygen documentation of this library is containted in the file refman.pdfwhich lies in the NonConforming folder. It gives a very detailed description ofthe actual classes.

NumProcs In this folder all the numerical experiments which have been presented inchapter 5 lie in this folder. It is essentially a collection of custom NumProc classeswhich can be loaded at runtime by NGSolve/Netgen.

NonConformingTest All the numerical tests which have been written with the unit-testing framework CUTE [17] lie in this folder.

Latex contains the sources of this document.

NGSolve Contains the sources of NGSolve 4.9. The author designed the NonConforminglibrary such that NGSolve’s and Netgens code needed as little modification aspossible. However a few minor changes were necessary to make everything workingand they are included in this folder.

Remark. Everything was built on NGSolve 4.9 and Netgen 4.9.14-RC.

C. Acknowledgements

I would like to thank Prof. Hiptmair for the interesting discussions and the many helpfulcomments which lay the foundation for this thesis.

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