Numerical methods for ferromagnetic plates -...

Numerical methods for ferromagnetic plates ∗

M. Fluck, T. Hofer, A. Janka, J. Rappaz.

1 Introduction

We compare some numerical methods for the simulation of ferromagnetic phenomenons in a metal-lic plate, with or without holes.

First we briefly recall the physical model we use for describing the ferromagnetic phenomenon.Next we present the discretization methods we want to compare. We then apply these methods onthe simple test-case of a thin ferromagnetic plate placed in front of a rectilinear electric conductor.We compare the various obtained results: magnetic field on a line perpendicular to the plate andrelative permeability on a given line in the plate; we also compare memory requirements for eachmethod.

2 Modeling of ferromagnetism

Let Λ ⊂ R3 be a domain with boundary ∂Λ occupied by a ferromagnetic material with relativemagnetic permeability µr

(‖ ~H‖

)≥ 1 depending on the euclidean norm of the magnetic field ~H,

denoted by ‖ ~H‖. In the following, we suppose that Λ is a bounded open, possibly non simplyconnected set, surrounded by known stationary electric currents denoted by ~j0. We denote by ~nthe unit normal on ∂Λ, external to Λ. Moreover, we assume that all the external currents are notmodified by the presence of the ferromagnetic material and no electric current flows in the domainΛ. Without the ferromagnetic material, it is possible to explicit the magnetic induction field ~B0

due to ~j0 by using Biot-Savart law:

~B0(~x) = µ0

∫R3

~∇xG(~x, ~y) ∧~j0(~y) dy, ∀~x ∈ R3, (1)

where µ0 is the magnetic permeability of the void, G(~x, ~y) is the Green kernel given by

G(~x, ~y) =14π

1‖~x− ~y‖

with ~x, ~y ∈ R3, ~x 6= ~y, (2)

and ~∇x denotes the gradient with respect to the variable ~x.Let us remark that if ~H0 is the magnetic field corresponding to ~B0, we have in the whole space

R3 without ferromagnetic materials

~B0 = µ0~H0, (3)

div ~B0 = 0, (4)

curl ~H0 = ~j0. (5)

Due to the presence of the ferromagnetic domain Λ, the magnetic field ~H and the induction field~B cannot be explicitly given in function of ~j0, but they are governed by the following relationships,∗This work is supported by Alcan-Pechiney Company.

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true in the whole space R3:

~B = µ0µr ~H, (6)

div ~B = 0, (7)

curl ~H = ~j0. (8)

We note that outside the domain Λ we have µr = 1. Since the magnetization field ~M is definedby ~M = µ0 (µr − 1) ~H, we will be able to compute ~M if we are able to calculate ~H. In thefollowing, we are looking for the field ~H.

2.1 Scalar potential model

By subtracting (5) and (8) we obtain the existence of a continuous function ψ satisfying

~H(~x)− ~H0(~x) = −~∇ψ(~x) ∀~x ∈ R3. (9)

By using equalities (3), (4) and (6), (7) together with (9), we easily verify that

−div(µr ~∇ψ

)= −div (µr − 1) ~H0 in R3. (10)

In order to obtain a finite energy, we assume that

ψ(~x) = O(

1‖~x‖

)when ‖~x‖ tends to infinity . (11)

Let Λ′ be the exterior open domain Λ′ = R3 \ Λ. Since µr = 1 in Λ′, we obtain

∆ψ = 0 in Λ′, (12)

where ∆ is the laplacian operator.In fact, equation (10) is non linear and it is necessary to precise what is µr, which is a discon-

tinuous function since µr = 1 in Λ′ and µr = µr(‖ ~H‖) in Λ. In order to write correctly the model,we define the mapping µ : R3 × R+ → R+ by

µ(~x, s) =

1 if ~x ∈ Λ′, s ∈ R+,

µr(s) if ~x ∈ Λ, s ∈ R+,(13)

where µr(s) is the relative magnetic permeability of the ferromagnetic plates occupying Λ givenin function of s = ‖ ~H‖. Since ~H = ~H0− ~∇ψ, it follows that the model consists to find ψ : R3 → Rsatisfying

− div(µ(·, ‖ ~H0 − ~∇ψ‖)~∇ψ

)= −div

(µ(·, ‖ ~H0 − ~∇ψ‖)− 1

)~H0 in R3, (14)

ψ(~x) = O(

1‖~x‖

)when ‖~x‖ → ∞. (15)

In order to simplify the notation, we will leave out in the following the argument of the mappingµ, knowing that it depends on ~x ∈ R3 and ‖ ~H0 − ~∇ψ‖, in order to write−div

(µ~∇ψ

)= −div (µ− 1) ~H0 in R3,

ψ(~x) = O(

1‖~x‖

)when ‖~x‖ → ∞.

(16)

Remark that ~H0 need not be known in R3 but only on Λ because µ = 1 outside Λ.

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2.2 Vector potential model

From (4) and (7) we obtain div( ~B− ~B0) = 0 in R3. Thus, there exists a vector potential ~A : R3 →R3 satisfying Coulomb’s gauge such that

~B − ~B0 = curl ~A in R3 (17)

with

div ~A = 0 in R3. (18)

By subtracting (5) and (8) we have

curl( ~H − ~H0) = 0 in R3, (19)

and, using (3), (6) and (19), we get

curl(

1µr

curl ~A)

= curl((

1− 1µr

)~B0

)in R3. (20)

Let us remark that outside Λ we have µr = 1 and consequently curl(curl ~A

)= 0. Together

with (18) we obtain

∆ ~A = 0 in Λ′. (21)

To assure the physical meaning of finite magnitude energy, we impose the following condition atinfinity:

‖ ~A(~x)‖ = O(

1‖~x‖

)when ‖~x‖ tends to infinity . (22)

Here, the model consists to find ~A : R3 → R3 satisfying (18),(20) and (22). In (20), µr = 1 outsideΛ and µr is a function of ‖ ~H‖ inside Λ. In fact, Equation (20) is non linear. Let us precise thispoint. Using definition (13), the meaning of µr in Equation (20) is in fact µ(x, ‖ ~H‖) when weadopt this notation. In the applications, the mapping s → µr(s) is smooth, bounded, strictlydecreasing and satisfying

µr(s) + sd

dsµr(s) ≥ 1, ∀s ∈ R+. (23)

It follows that

d

ds(sµ0µr(s)) ≥ µ0 > 0, ∀s ∈ R+ (24)

and we can consider the reciprocal function ν : R+ → R+ of s→ sµ0µr(s) satisfying

ν (sµ0µr(s)) = s, ∀s ∈ R+. (25)

Since s = ‖ ~H‖ in the plate, we have ‖ ~H‖ = ν(µ0µr(‖ ~H‖) ‖ ~H‖

)= ν(‖ ~B‖) and consequently

µr(‖ ~H‖) =1µ0

‖ ~B‖‖ ~H‖

=1µ0

‖ ~B‖ν(‖ ~B‖)

(26)

in the plates. Remark that the function νr defined by

νr(s) = µ0ν(s)s, s ∈ R+ (27)

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satisfies1

µr(‖ ~H‖)= νr(‖ ~B‖). The function νr is called ”relative reluctivity”.

By using (17) we obtain in the ferromagnetic material:

µr(‖ ~H‖) =1

νr(‖ ~B‖)=

1

νr(‖ ~B0 + curl ~A‖). (28)

Now, by defining ν : R3 × R+ → R+ by

ν(~x, s) =

1 if ~x ∈ Λ′, s ∈ R+,

νr(s) if ~x ∈ Λ, s ∈ R+,(29)

we have to find a field ~A : R3 → R3 satisfying the nonlinear problem:

curl(ν(·, ‖ ~B0 + curl ~A‖) curl ~A

)= curl

((1− ν(·, ‖ ~B0 + curl ~A‖)

)~B0

)in R3, (30)

div ~A = 0 in R3, (31)

‖ ~A(~x)‖ = O(1‖~x‖

) when ‖~x‖ → ∞. (32)

In order to simplify the notations, we leave out the argument of the mapping ν knowing it dependson ~x ∈ R3 and ‖ ~B0 + curl ~A‖.

Biro et al. [4, 5] treated this problem by inserting (31) in (30) in order to obtain an equivalentform:

curl(ν curl ~A

)− ~∇

(ν div ~A

)= curl

((1− ν) ~B0

)in R3,

‖ ~A(~x)‖ = O( 1‖~x‖ ) when ‖~x‖ → ∞.

(33)

Clearly speaking, if ~A is a solution of (33) we obtain by taking the divergence of (33):

∆(ν div ~A

)= 0 in R3. (34)

Since ~A is harmonic outside Λ (see (21)) and if ‖ ~A(~x)‖ tends to zero when ‖~x‖ tends to infinity,we have

div ~A = O(1

‖~x‖2) when ‖~x‖ → ∞. (35)

We conclude with (34) and (35) that div ~A = 0 in R3. Remark that in formulation (33), ~B0 neednot be known in R3 but only in Λ because ν = 1 outside Λ.

3 Formulations of the scalar and vector potential problems

Let us now focus on the weak formulations of the scalar or vector potential models. The maindifficulty with problems (16) resp. (33) , is that we seek a function ψ or ~A defined in the wholespace R3.

We will use two different ways to overcome this problem: the first one uses an integral formu-lation on ∂Λ to replace the so-called “exterior” problem by a relation valid on the boundary of Λ;the numerical approximation leads to a big non sparse matrix to “invert”.

The other way formulates the problem only on a bounded domain, stretching from the ferro-magnetic object Λ to the nearby artificial boundary. The behaviour of the solution in the far-field(“exterior” problem) is simulated through a “harmonic” boundary condition given by the Poissonrepresentation formula.

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3.1 Formulations for the scalar potential model

We have seen that the scalar potential model leads to find a mapping ψ satisfying (16). Sincediv ~H0 = 0 in R3, we can write this problem in the form

div(µ( ~H0 − ~∇ψ)

)= 0 in R3 (36)

with ψ(~x) = O(

1‖~x‖

)when ‖~x‖ → ∞. If W 1(R3) is the Beppo-Levi space given by

W 1(R3) = v : R3 → R :v(~x)

1 + ‖~x‖∈ L2(R3), ~∇v ∈ L2(R3), (37)

it is proven in [12] that there exists a unique ψ ∈W 1(R3) satisfying∫R3µ ( ~H0 − ~∇ψ) · ~∇ϕdx = 0, ∀ϕ ∈W 1(R3). (38)

It follows that Problem (16) possesses a unique weak solution ψ ∈W 1(R3).We now present three different approaches to compute the scalar potential ψ.

3.1.1 Boundary integral formulation of the scalar potential model

It is known [9], that if v is a harmonic function in Λ and in Λ′ satisfying v(~x) = O(‖~x‖−1

)when

‖~x‖ → ∞, and sufficiently regular (say C1 in Λ and in Λ′), then we have for ~x ∈ ∂Λ:

12

(vE(~x) + vI(~x)

)= −

∫∂Λ

⌊∂v

∂~n(~y)

⌋∂Λ

G(~x, ~y) ds(~y) +∫∂Λ

bv(~y)c∂Λ

∂G(~x, ~y)∂~n(~y)

ds(~y), (39)

where vE is the restriction of v to Λ′, vI is the restriction of v to Λ and [v]∂Λ = vE − vI is thejump of v through the boundary ∂Λ of Λ.

If ψ is the solution of (36), let w be a harmonic function in Λ ∪ Λ′ satisfying w = ψ on ∂Λ,w(~x) = O

(‖~x‖−1

)when ‖~x‖ → ∞. Clearly, because ψ is harmonic in Λ′, we have w = ψ in Λ

′.

Moreover, by using relationship (39) with v = w in Λ and v = ψ in Λ′ , we obtain for x ∈ ∂Λ

ψ(~x) = −∫∂Λ

(∂ψE

∂~n(~y)− ∂wI

∂~n(~y)

)G(~x, ~y) ds(~y), (40)

which is equivalent to∫∂Λ

ψη ds = −∫∂Λ

η(~x) ds(~x)∫∂Λ

(∂ψE

∂~n(~y)− ∂wI

∂~n(~y)

)G(~x, ~y) ds(~y), (41)

for all η ∈ H−1/2(∂Λ) 1.By using the fact that div ~H0 = 0 in R3, we have∫

R3

~H0 · ~∇ϕdx = 0, ∀ϕ ∈W 1(R3), (42)

and with the weak formulation (38):∫R3µ~∇ψ · ~∇ϕdx =

∫R3

(µ− 1) ~H0 · ~∇ϕdx, ∀ϕ ∈W 1(R3). (43)

1We note by H1(Λ) the classical Sobolev space of order 1, H1/2(∂Λ) the space of traces on ∂Λ of mappingsbelonging to H1(Λ) and H−1/2(∂Λ) its dual space.

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Since µ = 1 outside Λ we obtain∫Λ

µ~∇ψ · ~∇ϕdx+∫

Λ′

~∇ψ · ~∇ϕdx =∫

Λ

(µ− 1) ~H0 · ~∇ϕdx, ∀ϕ ∈W 1(R3), (44)

and, by integrating by parts (~n is pointing inside Λ′ and ∆ψ = 0 in Λ′):∫Λ

µ~∇ψ · ~∇ϕdx−∫∂Λ′

∂ψE

∂nϕds =

∫Λ

(µ− 1) ~H0 · ~∇ϕdx, ∀ϕ ∈W 1(R3). (45)

Since w is harmonic in Λ, we have w ∈ H1(Λ) satisfying w = ψ on ∂Λ and∫Λ~∇w · ~∇vdx =

0, ∀v ∈ H10 (Λ).

By using the definition (13) of µ, we can replace µ in (45) by µ = µr(‖ ~H0 − ~∇ψ‖) and bysetting λ = ∂ψE

∂n (external Steklov-Poincarre operator) we obtain the non linear problem:find ψ ∈ H1(Λ), w ∈ H1(Λ) and λ ∈ H− 1

2 (∂Λ) satisfying w = ψ on ∂Λ and:∫Λ

µr ~∇ψ ·~∇ϕdx−∫∂Λ

λϕds =∫Λ

(µr − 1) ~H0 ·~∇ϕdx, ∀ϕ ∈ H1(Λ) (46)

∫Λ

~∇w·~∇v dx = 0, ∀v ∈ H10 (Λ) (47)

∫∂Λ

ψη ds = −∫∂Λ

η(~x) ds(~x)∫∂Λ

(λ(~y)− ∂w

∂~n(~y)

)G(~x, ~y) ds(~y), ∀η ∈ H− 1

2 (∂Λ); (48)

here µr = µr(‖ ~H0 − ~∇ψ‖).

3.1.2 Scalar potential problem with Poisson formula boundary condition

Let us now introduce another way to reduce the ”exterior” problem to a problem expressed in abounded domain.

Let BR be the ball of radius R > 0 and Br be the ball of radius r with R > r > 0, bothcentered at the origin, and such that Λ ⊂ Br.

We use the following Poisson formula which says that since ψ is a harmonic function outsidethe ball Br and radially decreasing at infinity (11), then for each point ~x outside the ball Br wehave

ψ(~x) =‖~x‖2 − r2

4πr

∫∂Br

ψ(~y)‖~y − ~x‖3

ds(~y). (49)

By using this formula for ~x ∈ ∂BR we obtain the formulation: find ψ ∈ H1(BR) satisfying forall ϕ ∈ H1

0 (BR), ∫BR

µ~∇ψ ·~∇ϕdx =∫Λ

(µ− 1) ~H0 ·~∇ϕdx, (50)

ψ(~x) =R2 − r2

4πr

∫∂Br

ψ(~y)‖~y − ~x‖3

ds(~y), (51)

for all points ~x ∈ ∂BR; here µ = µ(·, ‖ ~H0 − ~∇ψ‖).

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3.1.3 Mixed formulation of the scalar potential problem with Poisson formula bound-ary condition

We finally present a variant formulation of the previous one, inspired by the fact that we seeka good (numerical) approximation of ~H, not of ψ which is just an auxiliary potential. To thisaim, we remark that equation (10) just expresses that the divergence of ~p vanishes, where ~p =µr ~∇ψ − (µr − 1) ~H0; moreover we have µ0~p = ~B0 − ~B which corresponds to the physical field weare interested in.

At this point we remark by using (17) that ~p = − 1µ0

curl ~A and by using (28),(29) we have

µ(·, ‖ ~H0 − ~∇ψ‖)−1 = ν(·, ‖ ~B0 − µ0~p ‖). (52)

So, for the system:

1µ~p = ~∇ψ − (1− 1

µ) ~H0, in BR, (53)

div ~p = 0, in BR, (54)ψ = ψ0, on ∂BR, (55)

when ψ0 is given on the boundary of BR (say ψ0 ∈ H1/2(∂BR)), we obtain with (52) the naturalformulation:find ψ ∈ L2(BR), ~p ∈ H(div,BR) such that∫

BR

ν ~p·~q dx+∫BR

ψ div ~q dx =∫∂BR

ψ0 ~q ·~n ds−∫Λ

(1− ν) ~H0 ·~q dx, (56)

∫BR

ϕ div ~p dx = 0, (57)

for all ~q ∈ H(div,BR) and for all ϕ ∈ L2(BR). Here, H(div,BR) = ~v ∈ L2(BR)3; div~v ∈ L2(BR)and ν = ν(·, ‖ ~B0 − µ0~p ‖).

Now we would like to replace in (56) ψ0 on the boundary ∂BR by the Poisson formula (49) ,i.e.:

ψ0(~x) =R2 − r2

4πr

∫∂Br

ψ(~y)‖~y − ~x‖3

ds(~y), ∀~x ∈ ∂BR.

Strictly taken, this formulation leads to a difficulty since ψ is a priori found in L2(BR) and thetrace of ψ on ∂BR does not exist a priori. For this reason, we are obliged to introduce a regularizingoperator Rε defined by

Rε(f)(~x) =∫BR

Jε(~x− ~y)f(~y)dy, ∀f ∈ L2(BR), ∀~x ∈ BR, (58)

where Jε is a mollifier, and we add to (56),(57) :

ψ0(~x) =R2 − r2

4πr

∫∂Br

Rεψ(~y)‖~y − ~x‖3

ds(~y), ∀~x ∈ ∂BR. (59)

Clearly speaking, a solution (~p, ψ) of Problem (56),(57),(59) depends on ε.

3.2 Formulations for the vector potential model

Let us come now to the vector potential model. In this case we will use only the second way totreat the exterior problem, i.e., boundary condition on ∂BR obtained by Poisson’s formula.

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In order to establish a weak formulation of Problem (33) we suppose the geometric settingΛ ⊂ Br ⊂ BR of Section 3.1.2 and we formally multiply the first equation of (33) by ~w, with ~w = 0on ∂BR, for obtaining, after integrating by parts:∫

BR

ν curl ~A · curl ~w dx+∫BR

ν div ~A div ~w dx =∫Λ

(1− ν) ~B0 · curl ~w dx. (60)

As ~A has to be harmonic in B′r ≡ R3\Br ⊂ Λ′, we can write the Poisson representation formulafor the exterior problem in B′r,

~A(~x) =‖~x‖2 − r2

4πr

∫∂Br

~A(~y)‖~y − ~x‖3

ds(~y), for ~x ∈ B′r. (61)

The weak formulation of (33) restricted only on BR ⊂ R3, with (61) in the role of boundarycondition for all ~x ∈ ∂BR, gives the following problem: find ~A ∈ H(curl,BR) ∩ H(div,BR) suchthat for all ~w ∈ H(curl,BR) ∩H(div,BR), ~w = 0 on ∂BR, we have∫

BR

ν curl ~A · curl ~w dx+∫BR

ν div ~A div ~w dx =∫Λ

(1− ν) ~B0 · curl ~w dx, (62)

~A(~x) =R2 − r2

4πr

∫∂Br

~A(~y)‖~y − ~x‖3

ds(~y), (63)

for all boundary points ~x ∈ ∂BR. Here, H(curl,BR) = ~w ∈ L2(BR)3; curl ~w ∈ L2(BR)3 andH(div,BR) = ~w ∈ L2(BR)3; div ~w ∈ L2(BR) and ν = ν(·, ‖ ~B0 + curl ~A ‖).

4 Discretization and numerical methods

Before we describe the methods we derived from the above formulations, let us introduce somenotations of discretization spaces, common for all of them.

Let us assume that the ferromagnetic domain Λ is a polyhedron. Let us also consider polyhedraBRh and Brh approximating the big and the small balls BR, resp. Br.

Definition 1 (interior and boundary mesh). Let Ω be a polyhedral domain.

1. Let us denote τh(Ω) the tetrahedral mesh of Ω with conforming tetrahedra, in the finite-element sense. The tetrahedron K ∈ τh(Ω) is understood as a closed tetrahedron.

2. The set of all internal and external faces of the tetrahedral mesh τh(Ω) is denoted Fh(Ω).

3. Let us denote τh(∂Ω) the trace of the mesh τh(Ω) on the boundary ∂Ω. The boundary meshτh(∂Ω) is composed of all triangular faces F ∈ Fh(Ω)) such that F ⊂ ∂Ω.

Definition 2 (finite element spaces). Let Ω be a polyhedral domain, τh(Ω) its tetrahedral meshand τh(∂Ω) the corresponding boundary mesh.

1. On the tetrahedral mesh τh(Ω), we define:

(a) The finite element space P1h(Ω) of continuous functions which are piecewise polynomialof degree 1 on K ∈ τh(Ω),P1h(Ω) = w ∈ C0(Ω),∀K ∈ τh(Ω) ∃a ∈ R,~b ∈ R3 s.t. w(~x) = a+~b · ~x,∀~x ∈ K.

(b) The finite element space P1h0(Ω) of functions w ∈ P1h(Ω) such that w = 0 on ∂Ω.

(c) The finite element space P0h(Ω) of piecewise constant functions on K ∈ τh(Ω),P0h(Ω) = w ∈ L2(Ω),∀K ∈ τh(Ω) ∃a ∈ R s.t. w(~x) = a,∀~x ∈ K.

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(d) The finite element space RT 0h(Ω), the lowest-order Raviart-Thomas space of piecewiselinear vector-functions on K ∈ τh(Ω), with normal component continuous through eachinternal face F belonging to Fh(Ω),RT 0h(Ω) = ~w ∈ L2(Ω)3,∀K ∈ τh(Ω) ∃~a ∈ R3, b ∈ R s.t. ~w(~x) = ~a+ b~x,∀~x ∈ K

and b~w · ~ncF = 0,∀F ∈ Fh(Ω), F 6⊂ ∂Ω.

2. On the boundary mesh τh(∂Ω) we define

(a) The finite element space P0h(∂Ω) of piecewise constant functions on the faces F ∈ τh(∂Ω),P0h(∂Ω) = w ∈ L2(∂Ω),∀F ∈ τh(∂Ω) ∃a ∈ R s.t. w(~x) = a,∀~x ∈ F.

(b) The finite element space P1h(∂Ω) of continuous functions which are piecewise polyno-mial of degree 1 on each face F ∈ τh(∂Ω),P1h(∂Ω) = w ∈ C0(∂Ω),∀F ∈ τh(∂Ω) ∃a ∈ R,~b ∈ R3 s.t. w(~x) = a+~b · ~x,∀~x ∈ F.

4.1 Discretization of scalar potential formulations

4.1.1 Boundary integral method for the scalar potential model

Let us approximate the spaces H1(Λ), resp. H− 12 (∂Λ) by the space P1h(Λ) of piecewise linear

functions, resp. by the space P0h(∂Λ) of piecewise constant functions on the boundary mesh. Wecan write the discrete formulation corresponding to the problem (46)-(48):find (ψh, λh, wh) ∈ P1h(Λ)× P0h(∂Λ)× P1h(Λ), wh = ψh on ∂Λ such that∫

Λ

µ~∇ψh ·~∇ϕh dx−∫∂Λ

λhϕhds =∫Λ

(µ− 1) ~H0 · ~∇ϕh dx, (64)

∫∂Λ

ψh ·ηh ds+∫∂Λ

ηh(~x) ds(~x)∫∂Λ

(λh(~y)−

∂wh∂~n

(~y))G(~x, ~y) ds(~y) = 0, (65)

∫Λ

~∇wh · ~∇vh dx = 0, (66)

for all (ϕh, ηh, vh) ∈ P1h(Λ)×P0h(∂Λ)×P1h0(Λ). The function µ ∈ P0h(Λ) is the approximationof µr in Λ defined by

µ = µr

(‖Qh ~H0 − ~∇ψh‖

), (67)

where Qh : L2(Λ)3 → P0h(Λ)3 is the L2-orthogonal projection.Note that all integrals are done exactly except for the one involving the Green’s kernel G which

must be numerically approximated.The nonlinear problem (64)-(66) is solved using a standard fixed point method, cf. Algorithm 1.

The convergence of this fixed point method is proven in [12].

Algorithm 1 (Boundary integral method for scalar potential, fixed point).Let us set ψ0

h ∈ P1h(Λ), ψ0h = 0.

For k = 1, . . . , J do

1.) Evaluate µk = µr

(‖Qh ~H0 − ~∇ψk−1

h ‖).

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2.) Find (ψkh, λkh, w

kh) ∈ P1h(Λ)× P0h(∂Λ)× P1h(Λ), wkh = ψkh on ∂Λ such that∫

Λ

µk ~∇ψkh ·~∇ϕh dx−∫∂Λ

λkhϕhds =∫Λ

(µk−1) ~H0 ·~∇ϕh dx,

∫∂Λ

ψkh ·ηh ds+∫∂Λ

ηh(~x) ds(~x)∫∂Λ

(λkh(~y)−

∂wkh∂~n

(~y))G(~x, ~y) ds(~y) = 0,

∫Λ

~∇wkh · ~∇vh dx = 0,

for all (ϕh, ηh, vh) ∈ P1h(Λ)× P0h(∂Λ)× P1h0(Λ).

until estimated convergence.

In the applications, the ferromagnetic structures are often thin and their discretizations containa lot of triangles on their surfaces. The main drawback arizing from formulation (64)-(66) is thatthe second term of equation (65) leads in this case to a full matrix with a big order. Consequently,formulation (64)-(66) imposes an important restriction on the mesh, which is not operational in alot of applications on the presented form.

4.1.2 Scalar potential problem with Poisson formula boundary condition

Let us now discretize the formulation (50)-(51). We are approximating the functional spaceH1(BR) by the space P1h(BRh) by assuming that ∂Λ and ∂Brh are made of faces of Fh(BRh).

The discrete formulation for (50)-(51) reads: find ψh ∈ P1h(BRh) such that for all ϕh ∈P1h0(BRh) we have ∫

BRh

µ~∇ψh · ~∇ϕh dx =∫Λ

(µ− 1) ~H0 · ~∇ϕh dx, (68)

ψh(~xi) =R2 − r2

4πr

∫∂Brh

ψh(~y)‖~y − ~xi‖3

ds(~y), (69)

pointwise for all vertices ~xi of the mesh τh(∂BRh). Here, µ ∈ P0h(BRh) is the piecewise-constantapproximation of µ defined by

µ =

µr

(‖Qh ~H0 − ~∇ψh‖

)in Λ,

1 otherwise,(70)

where Qh : L2(Λ)3 → P0h(Λ)3 is the L2-orthogonal projection.Note that all integrals are done exactly except for the one on ∂Brh. In that case, first recall

that the sphere is approximated by a triangular mesh, second on each of these triangles we use asimple Gauss integration scheme.

The problem (68)-(70) is nonlinear. Moreover, the coupling of (68) and (69) is non-local, thusfilling the matrix of the underlying linear system with full blocks. This is why we propose inAlgorithm 2 a fixed point iteration, combined with a multiplicative Dirichlet-Dirichlet domaindecomposition between the meshed interior and the exterior, represented by the Poisson represen-tation formula (69) (see [12] for the convergence).

Algorithm 2 (Domain decomposition for scalar potential, fixed point).Let us set ψ0

h ∈ P1h(BRh), ψ0h = 0.

For k = 1, . . . , J do

10

1.) Define ψk−12

h ∈ P1h(∂BRh) such that

ψk− 1

2h (~xi) =

R2 − r2

4πr

∫∂Brh

ψk−1h (~y)

‖~y − ~xi‖3ds(~y), (71)

pointwise on each vertex ~xi of the mesh τh(∂BRh).

2.) Evaluate µk ∈ P0h(BRh) by

µk =

µr

(‖Qh ~H0 − ~∇ψk−1

h ‖)

in Λ,1 otherwise.

3.) Find ψkh ∈ P1h(BRh), ψkh = ψk− 1

2h on ∂BRh, such that for all ϕh ∈ P1h0(BRh) there is∫

BRh

µk ~∇ψkh · ~∇ϕh dx =∫Λ

(µk − 1) ~H0 · ~∇ϕh dx (72)


To solve (72) approximatively, we use several iterations of an algebraic multigrid AMG [11].

4.1.3 Mixed P1h-P1h method with Poisson boundary conditions

Consider the problem (56),(57),(59) for finding (~p, ψ) ∈ H(div,BR)×L2(BR), with the regularizingoperator Rε of (58).

If we approximate the involved functional spacesH(div,BR), resp. L2(BR) with spaces P1h(BRh)3,resp. P1h(BRh) of piecewise-linear functions, we have to add a stabilisation term to the weak for-mulation.

We choose the GLS stabilisation (see [8]) with the stabilization parameter α = O(1) andwrite the discretized problem: find (~ph, ψh) ∈ P1h(BRh)3 × P1h(BRh) such that for all(~qh, ϕh) ∈ P1h(BRh)3 × P1h(BRh) there is∫

BRh

ν ~ph · ~qh dx+∫

BRh

(ψh div ~qh + ϕh div ~ph) dx+ α

∫BRh

(ν ~ph − ~∇ψh

) (ν ~qh − ~∇ϕh

)dx

=∫

∂BRh

ψh~qh · ~n ds− (ν α+ 1)∫Λ

(1− ν) ~H0 · ~qh dx+ α

∫Λ

(1− ν) ~H0 · ~∇ϕh dx,(73)

ψh(~xi) =R2 − r2

4πr

∫∂Brh

ψh(~y)‖~y − ~xi‖3

ds(~y), (74)

pointwise for all vertices ~xi in the boundary mesh τh(∂BRh). Due to continuity of ψh, we do notneed the regularization operator Rε here, it is replaced by identity.

The function ν is the piecewise-constant approximation of ν, defined here through the relativereluctivity ν (see 29). Recall that ν is a function of ‖ ~B‖, while µ is a function of ‖ ~H‖. We set

ν =

νr

(‖Qh( ~B0 − µ0~ph)‖

)in Λ,

1 otherwise.(75)

Here, as before, Qh : L2(BRh)3 → P0h(BRh)3 is the L2-orthogonal projection.As before, all integrals are done exactly except for the one on ∂Brh.Again, the problem (73)-(74) is non-linear and the coupling of (73) and (74) is non-local. That

is why, we employ an iterative fixed-point scheme to solve it, cf. Algorithm 3.

11

Algorithm 3 (Domain decomposition for mixed P1h-P1h formulation, fixed point).Let us set ψ 0

h ∈ P1h(BRh), ψ0h = 0, and calculate ~p 0

h ∈ P1h(BRh)3, ~p 0h =

(1− µ(·, ‖Qh ~H0‖)

)Qh ~H0.

For k = 1, . . . , J do

1.) Define ψk−12


ψk− 1

2h (~xi) =

R2 − r2

4πr

∫∂Brh

ψk−1h (~y)

‖~y − ~xi‖3ds(~y),

pointwise on each vertex ~xi of the mesh τh(∂BRh).


νk =

νr

(‖Qh( ~B0 − µ0~p

k−1h )‖

), in Λ,

1 otherwise.

3.) Find (~p kh , ψkh) ∈ P1h(BRh)3 × P1h(BRh), such that∫

BRh

ν ~p kh · ~qh dx+∫

BRh

(ψ kh div ~qh + ϕh div ~ph) dx+ α

∫BRh

(ν ~p kh − ~∇ψh

) (ν ~qh − ~∇ϕh

)dx

=∫

∂BRh

ψk− 1

2h ~qh · ~n ds− (ν α+ 1)

∫Λ

(1− ν) ~H0 · ~qh dx+ α

∫Λ

(1− ν) ~H0 · ~∇ϕh dx,

(76)

for all (~qh, ϕh) ∈ P1h(BRh)3 × P1h(BRh).


4.1.4 Mixed RT 0h method with Poisson boundary conditions

Let us consider again the problem (56),(57),(59). We are approximating now the functional spacesL2(BR), resp. H(div,BR) by the space P0h(BRh) of piecewise-constant functions, resp. by thespace RT 0h(BRh), the lowest-order Raviart-Thomas space. Note that the dimension of RT 0h isthe number of all faces F ∈ Fh(BRh).

The discrete formulation of the problem (56),(57),(59) reads in this case: find (~ph, ψh) ∈RT 0h(BRh)× P0h(BRh) such that for all (~qh, ϕh) ∈ RT 0h(BRh)× P0h(BRh) there is∫

BRh

ν ~ph · ~qh dx+∫

BRh

ψh div ~qh dx =∫

∂BRh

ψh~qh · ~nds−∫Λ

(1− ν) ~H0 · ~qh dx, (77)

∫BRh

ϕh div ~ph dx = 0, (78)

ψh(~xi) =R2 − r2

4πr

∫∂Brh

Rhψh(~y)‖~y − ~xi‖3

ds(~y), (79)

pointwise for all barycenters ~xi of boundary faces F ∈ τh(∂BRh). The regularizing operator Rh

needs be only given for functions with trace on ∂Brh. Let us define Rh : P0h(BRh) → P0h(∂Brh)in the following way: for each face F ∈ ∂Brh and for each function ηh ∈ P0h(BRh), the value ofRh ηh on this face is the arithmetic mean of ηh on the two tetrahedra in τh(BRh) sharing the faceF .

12

Again, the function ν ∈ P0h(BRh) is the piecewise-constant approximation of ν which satisfies

ν =

νr

(‖Qh( ~B0 − µ0~ph)‖

)in Λ,

1 otherwise.(80)

where Qh : L2(BRh)3 → P0h(BRh)3 is the L2 orthogonal projection.As before, all integrals are done exactly except for the one on ∂Brh.As before, the problem (77)-(80) is non-linear, and the coupling of (77),(78) with (79) is non-

local. Therefore, we solve the non-linear problem by using a fixed point iteration. For one fixed ν,we solve the interior-exterior subproblem (77)-(79) by a multiplicative Dirichlet-Dirichlet domaindecomposition.

Algorithm 4 (Domain decomposition for mixed RT 0h formulation).Let us set ψ0

h ∈ P0h(BRh), ψ0h = 0, and let ~p 0

h ∈ RT 0h(BRh) be an approximation of(1− µ(·, ‖ ~H0‖)

)~H0.

For k = 1, . . . , J do


νk =

νr

(‖Qh( ~B0 − µ0~p

k−1h )‖

)in Λ,

1 otherwise.

2.) Define ψk−12


ψk− 1

2h (~xi) =

R2 − r2

4πr

∫∂Brh

Rhψk−1h (~y)

‖~y − ~xi‖3ds(~y),

for all barycenters ~xi for faces F ∈ Fh(∂BRh).

3.) Find (~p kh , ψkh) ∈ RT 0h(BRh)× P0h(BRh) such that∫

BRh

νk ~p kh · ~qh dx+∫

BRh

ψkh div ~qh dx =∫

∂BRh

ψk− 1

2h ~qh ·~n ds−

∫Λ

(1− νk) ~H0 · ~qh dx,

∫BRh

ϕh div ~p kh dx = 0,

for all (~qh, ϕh) ∈ RT 0h(BRh)× P0h(BRh).


4.2 Discretization of vector potential formulation

Let us discretize the problem (62)-(63). We approximate the space H(curl,BR) ∩H(div,BR) bythe space P1h(BRh)3. The discrete formulation then reads: find ~Ah ∈ P1h(BRh)3 such that forall ~wh ∈ P1h0(BRh)3 there is∫

BRh

[ν curl ~Ah · curl ~wh + ν div ~Ah div ~wh

]dx =

∫Λ

(1− ν) ~B0 · curl ~wh dx, (81)

~Ah(~xi) =R2 − r2

4πr

∫∂Brh

~Ah(~y)‖~y − ~xi‖3

ds(~y), (82)

13

for all nodes ~xi of the boundary mesh τh(∂BRh). The function ν ∈ P0h(BRh) is the piecewise-constant approximation of ν defined here through the relative reluctivity ν = 1

µ ,

ν =

νr

(‖Qh ~B0 + curl ~Ah‖

), in Λ,

1 otherwise,

where Qh : L2(Λ)3 → P0h(Λ)3 is the L2-orthogonal projection.As before, all integrals are done exactly except for the one on ∂Brh.The discrete problem (81)-(82) is non-linear and the coupling of (81) with (82) is non-local. This

is why we solve this problem by a fixed-point iteration combined with a multiplicative Dirichlet-Dirichlet interior-exterior domain decomposition, cf. Algorithm 5.

Algorithm 5 (Domain decomposition for gauged vector-potential, fixed point).Let us set ~A0

h ∈ P1h(BRh)3, ~A0h = 0.

For k = 1, . . . , J do

1.) Define ~Ak− 1

2h ∈ P1h(∂BRh)3 such that

~Ak− 1

2h (~xi) =

R2 − r2

4πr

∫∂Brh

~Ak−1h (~y)

‖~y − ~xi‖3ds(~y),

pointwise for each vertex ~xi of the boundary mesh τh(∂BRh).

2.) Evaluate νk ∈ P0h(BRh) by

νk =

νr

(‖Qh ~B0 + curl ~Ak−1

h ‖), in Λ,

1 otherwise.

3.) Find ~Akh ∈ P1h(BRh)3, ~Akh = ~Ak− 1

2h on ∂BRh, such that∫

BRh

[νk curl ~Akh · curl ~wh + νk div ~Akh · div ~wh

]dx =

∫BRh

(1− νk

)~B0 · curl ~wh dx,

for all ~wh ∈ P1h0(BRh)3.


5 Numerical experiments

Let us proceed with numerical experiments, in order to compare the results and the computationalcomplexity of the above-mentioned algorithms. As it was said at the beginning, special focus isbrought to ferromagnetic plates and their screening effects.

5.1 Definition of the test-case.

We consider a ferromagnetic rectangular plate, 5 meters wide, 4 meters high and 2 centimetersthick which is placed in front of an idealized infinitely long wire with zero section (see Figure 1,where the plate is represented in dark red). A given constant electric current runs in the wire.Omitting the plate, this current would produce an induction field ~B0 (suggested on Figure 1) givenby Biot-Savart law. In the presence of the plate, the induction field ~B will be modified, due toferromagnetic response of the steel plate.

14

Figure 1: Geometry of the test-case: the rectangular plate and current support.

We want to simulate:

1o) The screen effect of the ferromagnetic plate, i.e., the attenuation of the induction field”behind” the plate; we will compare ~B and ~B0.

2o) The magnetization in the plate, i.e., the physical phenomenon which is responsible for thescreen effet; we will compare the relative permeability µr(‖ ~H‖) in the plate.

Our aim is here to compare the results obtained using each of the above described formulationsfor ~B inside and outside the plate as well as µr(‖ ~H‖) in the plate. In particular, we will focuson the approximation of ~B inside the plate which for some methods present strong ”oscillations”.Moreover, we want to check the h-convergence for each formulation; to this aim, we will use fourdifferent meshes from coarse to fine.

Let us now introduce the following system of coordinates: the origin O is placed at the centerof the plate; the Ox axis is in the direction of the thickness of the plate; the Oy axis is in thedirection of the length of the plate; the Oz axis is in the direction of the height of the plate.With this coordinate system, the electric current of 143’000A, which is parallel to Oy, goes fromy = −20 meters to y = +20 meters (which simulates correctly an infinite wire) and passes throughthe point (1.01, 0, 0) (see Figure 1 where the current in the wire is suggested by the generated field~B0 in a plane perpendicular to the wire).

The nonlinear material behaviour is characterized by the B − H diagram, or by the relativemagnetic permeability or reluctivity functions µr(‖H‖), resp. νr(‖B‖) given in Fig. 2.

Fig. 3 gives a sketch of some lines we have choosen to compare the results obtained by each ofthe methods. The lines 1, 2 and 3 were chosen to observe the screening effect when we go awayperpendicularly from the plate, while the line 4 is meant to check the behaviour of the field ~B verynear (10 centimeters) the plate. Because it goes through a corner of the plate, line 2 will also beused to check the singular behaviour of the field ~B when approaching this corner. Lines 5 and 6,which are located in the middle of the plate thickness, will be used to measure the magnetizationstate in the plate.

Our aim is now to apply each of the above presented algorithms. They will be referred to nowas given in Table 1.

15

0 2000 4000 6000 8000 10000 12000 140000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

|H|

|B|

H−B diagram of ferro−material without hysteresis

0 2 4 6 8 10x 10

4

0

500

1000

1500

2000

2500

3000

3500

4000

|H|

µ

R(|

H|)

Relative permeability as a function of |H|

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

|B|

ν r(|B

|)

Relative reluctivity νr = 1/µ

r as a function of |B|

Figure 2: Material properties of the ferro-material: the H-B diagram without hysteresis (left),relative permeability µr(‖ ~H‖) as a function of ‖H‖ (middle) and relative reluctivity νr(‖ ~B‖) as afunction of ‖B‖ (right).

Figure 3: Geometry of the test-case with observation lines 1-3, 4 and 5-6, a 3D view).

label short description defined inbem-fem scalar potential formulation with boundary-elements coupled to

finite elementsAlgorithm 1

scal non-mixed scalar potential formulation on P1h nodal finite ele-ment spaces

Algorithm 2

mix mixed scalar potential formulation on P1h-P1h nodal finite ele-ment spaces with GLS stabilisation

Algorithm 3

rt0 scalar potential mixed formulation with RT 0h finite elements Algorithm 4vec vector potential formulation with Coulomb’s gauge on P1h3 nodal

finite element spacesAlgorithm 5

Table 1: Labels for different algorithms to be used for the presentations of results

16

5.2 Mesh for the plate and balls.

Let us now define the balls and meshes used in our algorithms. The small ball Br is the ball ofradius r = 3.5 meters centered at the origin while BR has radius r = 4.4 meters (see Fig. 3).

Finite element mesh of BR: To compare also h-convergence of the finite element discretizationsof the test-case, we are using four levels of refinement of the same coarse mesh. The coarsest quasi-uniform isotropic mesh, called meshH0.25, with a representative mesh-size h = 1/4 has been threetimes refined to produce meshes meshH0.12, meshH0.06 and meshH0.03. This has been doneby successive halving the edges, dividing each triangular face between two tetrahedra into foursmaller similar triangles and dividing each coarse tetrahedron to eight finer tetrahedra. By thisdivision, the quality of tetrahedra might be deteriorated only in the first refinement, all subsequentrefinements do not deteriorate the mesh quality, cf. [1]. Hence the resulting finer finite elementnodal spaces are nested to the coarse ones, except near the small sphere boundary ∂Br, wherefiner meshes have been adapted to capture better the curved surface. Table 2 gives the numberof nodes, edges, faces and tetrahedrons for each mesh while Fig. 4 gives a simple representationof these meshes.

mesh name no. nodes no. edges no. faces no. tetrasmeshH0.25 16.967 114.382 194.657 97.241meshH0.12 131.349 909.976 1.556.556 777.928meshH0.06 1.041.325 7.267.548 12.449.648 6.223.424meshH0.03 8.308.873 58.107.464 99.585.984 49.787.392

Table 2: Statistics of the used finite element mesh with four levels of refinement forh = 1/4, 1/8, 1/16, 1/32.

Figure 4: Mesh of the plate and of the balls BBR and BBr.

Algorithmic complexity of all algorithms: One of the important criteria for measuring theefficiency of the method might be the size of the underlying linear system of equations which is tobe solved in each iteration of the non-linear (domain-decomposition) algorithm.

Table 3 gives the order of the systems of equations as well as the nomber of non-zero terms tostore for each algorithm and for each mesh.

17

mesh algo dofs nonzerosscal 16.967 131.349vec 50.901 1.182.141

meshH0.25 mix 67.868 3.931.696bem-fem 10.396 9.511.092rt0 291.888 2.139.477scal 131.349 909.976vec 394.047 8.189.784

meshH0.12 mix 525.396 31.220.816bem-fem 55.082 151.061.943rt0 2.334.464 17.115.116scal 1.041.325 8.308.873vec 3.123.975 74.779.857

meshH0.06 mix 4.165.300 249.222.736bem-fem 332.490 2.410.017.735rt0 18.673.072 136.918.128

Table 3: Size of the assembled linear system for each algorithm and mesh; column dofs gives thenumber of unknowns and nonzeros gives the number of non zeros terms in the resulting matrix.

5.3 Magnetic induction ~B outside the plate

Let us compare all presented algorithms on the described testcase on the coarsest and the finestmesh. As a criterion of quality, we observe the behaviour of the magnetic induction ~B on each ofthe lines 1-4, defined above.

All plots given in Fig. 5-9 are presented in the same way:

· Column 1 gives plots of componentsBx, By andBz of induction on the coarsest mesh (h = 14 );

· Column 2 gives plots of components Bx, By and Bz of induction on the finest mesh (h = 116 );

note that some algorithms do not appear for the plots in this column, since the mesh is toofine;

· The dotted line refers to induction ~B0 computed with no ferromagnetic plate using Biot-Savart’s formula.

· On the horizontal axis we give the distance to the plate, the plate being at the right of theplot.

We can now discuss the results obtained on observation lines 1-4.

Screen effect along line 1-3 (see Fig. 5-7): All algorithms produce roughly the same results on lines1-3. Even for coarse mesh, the approximation is already acceptable. The biggest differences canbe observed, as expected with finite element approximation, on small components of ~B. Algorithmmix seems to have more difficulties near the plate. We assume it is due to the rather non naturalapproximation for the space H(div,Λ).

Behaviour near the plate - line 4 - (see Fig. 8): Even quite near the plate, approximation seemscorrect for all methods. Once more, algorithm mix presents some difficulties.

Behaviour near singularity - line 2 - (see Fig. 9): It is well known that in the vicinity of edgesand corners of the plate, solution ~B will present a singularity. In order to measure the singularbehaviour on numerically, we superimpose on log-log plots reference lines with prescribed slopes.Let us note that here the mesh of the plate, and as a consequence also the mesh of Br near theplate has been intentionnally refined. Please note that here, for logistic reasons, the plate is placedto the right of the plot.

18

−4 −3 −2 −1 0

−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

2x 10−4

Line 1: X−coordinates in [m]

Bx

meshH0.25: Bx for all methods

B scalB vecB mixB bem−femB rt0B0 initial

−4 −3 −2 −1 0

−20

−18

−16

−14

−12

−10

−8

−6

−4

−2

0

2x 10−4

Line 1: X−coordinates in [m]B

x


B scalB vecB mixB0 initial

−4 −3 −2 −1 0−5−4−3−2−1

0123456789

10111213141516x 10

−5


By

meshH0.25: By for all methods


−4 −3 −2 −1 0−5−4−3−2−1

0123456789

10111213141516x 10

−5


By



−4 −3 −2 −1 0

0.006

0.008

0.01

0.012

0.014

0.016

0.018


Bz

meshH0.25: Bz for all methods


−4 −3 −2 −1 0

0.006

0.008

0.01

0.012

0.014

0.016

0.018


Bz



Figure 5: Magnetic induction ~B = (Bx, By, Bz) on the observation line 1, on the screened side ofthe plate. Left column: computations on coarse mesh, right column: computations on fine mesh.

19

−4 −3 −2 −1 0

−4

−2

0

2

4

6

8

10

12x 10−3

Line 2: X−coordinates in [m], X<Xplate

Bx



−4 −3 −2 −1 0

−4

−2

0

2

4

6

8

10

12x 10−3


Bx



−4 −3 −2 −1 00

1

2

3

4

5

6

7

8x 10−3


By



−4 −3 −2 −1 00

1

2

3

4

5

6

7

8x 10−3


By



−4 −3 −2 −1 04

5

6

7

8

9

10

11

12

13

14

15x 10−3


Bz



−4 −3 −2 −1 04

5

6

7

8

9

10

11

12

13

14

15x 10−3


Bz




20

−4 −3 −2 −1 00

0.002

0.004

0.006

0.008

0.01

0.012

0.014


Bx



−4 −3 −2 −1 00

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Line 3: X−coordinates in [m], X<XplateB

x



−4 −3 −2 −1 00

1

2

3

4x 10−4


By



−4 −3 −2 −1 00

1

2

3

4x 10−4


By



−4 −3 −2 −1 0

5

6

7

8

9

10

11

12

13

14

15x 10−3


Bz



−4 −3 −2 −1 0

5

6

7

8

9

10

11

12

13

14

15x 10−3


Bz




21

−4 −3 −2 −1 0 1 2 3 4−0.024−0.022−0.02

−0.018−0.016−0.014−0.012−0.01

−0.008−0.006−0.004−0.002

00.0020.0040.0060.008

0.010.0120.014

Line 4: Y−coordinates in [m], X=Xplate−11cm

Bx



−4 −3 −2 −1 0 1 2 3 4−0.024−0.022−0.02

−0.018−0.016−0.014−0.012−0.01

−0.008−0.006−0.004−0.002

00.0020.0040.0060.008

0.010.0120.014

Line 4: Y−coordinates in [m], X=Xplate−11cmB

x



−4 −3 −2 −1 0 1 2 3 4−8−7−6−5−4−3−2−1

0123456789

101112131415x 10

−3


By



−4 −3 −2 −1 0 1 2 3 4−8−7−6−5−4−3−2−1

0123456789

101112131415x 10

−3


By



−4 −3 −2 −1 0 1 2 3 4

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028


Bz



−4 −3 −2 −1 0 1 2 3 4

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028


Bz



Figure 8: Magnetic induction ~B = (Bx, By, Bz) on the observation line 4, just near the plate. Leftcolumn: computations on coarse mesh, right column: computations on fine mesh.

22

10−1

100

101

10−3

10−2

Line 2, distance ρ from the ferro−plate in [m], X<Xplate

|Bx|

meshH0.25: |Bx| for all methods

B scalB vecB mixB bem−femB rt0B0 initialreference O(ρ−1/2)reference O(ρ−2/3)reference O(ρ−3/4)

10−1

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101

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Line 2, distance ρ from the ferro−plate in [m], X<Xplate|B

x|

meshH0.06: |Bx| for all methods

B scalB vecB mixB0 initialreference O(ρ−1/2)reference O(ρ−2/3)reference O(ρ−3/4)

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|By|

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|Bz|

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|Bz|

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Figure 9: Observation line 2 with log-log scaled axes,. One can observe the asymptotic behaviourof the singularity in the field ~B when approaching the corner of the plate from the screened side ofthe plate. Left column: computations on coarse mesh, right column: computations on fine mesh.

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5.4 Magnetic induction inside the ferro-plate Λ

Oscillation of induction vector inside the plate. One of the reasons for comparing different solutionmethods for ferromagnetism was the reported oscillations problems when using non-mixed scalarpotential formulation of the problem. Indeed, as we show here, numerical spurious oscillationsof ~M , ~B or ~H are present inside the ferromagnetic plate Λ. Even if such oscillations do notappear outside the plate, it is worth studying this problem because, ~H inside the plate is used forcomputing the relative permeability.

scal

vec

Figure 10: meshH0.12: oscillation of the field ~B inside the plate Λ for methods scal and vec.

It can be shown, using an asymptotic analysis of the solution for infinitely thin plates and highrelative permeabilities from [3], that the magnetic induction vector ~B should be a vector parallelto the plane of the plate. Eventhough our testcase plate is not infinitely thin, let us verify towhich extent the ratio |Bx|/‖ ~B‖ is small in the plate (ie. the field ~B is almost tangential to theplane of the plate).

Fig. 10 shows the results obtained using scal and vec algorithms. The first line of plots givesresults obtained using the scal algorithm while second linge of plots shows results obtained usingthe vec algorithm. Every line of plots is arranged (from left to right) as follows:

· Plots 2, 3, 4: show the elements of the plate for which |Bx|/‖ ~B‖ is bigger than 8%, 4%, resp.2%.

· Plot 1: gives an histogram of the percent of tetrahedra in Λ (vertical axis) having a givenratio |Bx|/‖ ~B‖ (horizontal axis).

We clearly see, that the method vec presents much less oscillations than the scal method.Method rt0 gives results quite similar to the results of vec method while all others give resultscomparable to the scal method. For simplicity we give only results for . As a conclusion here wecan say that only vec and rt0 methods give satisfactory results inside the plate.

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Relative permeability plots. In Fig. 11 and 12 we see the plots of µr on the observation lines 5 and6 inside the plate. The first row of plots corresponds to the observation line 5 (horizontal line),the second one corresponds to the observation line 6 (vertical line). Results for all methods aresuperposed in one plot, for the coarsest mesh meshH0.25 (left), up to the finest mesh meshH0.03(right).

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

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Line 5 from (9.01,−2.5,0)T to (9.01,2.5,0)T, Y in [m]

µ r

meshH0.25: µr for all methods

B scalB vecB mixB edgeB bem−femB rt0

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

10

20

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µ r


B scalB vecB mixB edge

−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.50

10

20

30

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50

60

70

80

90

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µ r


B scal

Figure 11: Relative permeability µr on observation line 5 (horizontal line), for the coarsest mesh(left), up to the finest one (right).

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

500

1000

1500

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Line 6 from (9.01,0,−2)T to (9.01,0,2)T, Z in [m]

µ r


B scalB vecB mixB edgeB bem−femB rt0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

500

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µ r


B scalB vecB mixB edge

−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

500

1000

1500

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µ r


B scal

Figure 12: Relative permeability µr on observation line 6 (vertical line), for the coarsest mesh(left), up to the finest one (right).

While almost all methods give the same result on the observation line 5, unphysical peaks of µrappear for the method mix on the edges of the plate. This is probably caused by the nonphysicalregularity imposed by the P1h continuous finite element space for ~B. For the observation line 6(plots below), we see that all methods tend to converge roughly to the same profile of µr, butfor the methods scal and bem-fem we see spurious oscillations appear, especially for finer meshes(bottom right). This behaviour is caused by spurious oscillations of ~H inside the plate.

5.5 Artificial boundary conditions vs. the Poisson formula

In the domain decomposition method of Algorithm 2-4, resp. Algorithm 5, we take the initialguess ψ0 = 0, resp. ~A0 = 0, in particular, the trace of ψ0, resp. of ~A0 on ∂BR vanish. The domaindecomposition algorithm then iteratively updates the boundary values of ψ resp. ~A on ∂BR byusing the Poisson formula.

Considering that the potentials ψ or ~A are vanishing at infinity with the rate O(1/‖~x‖), onemay expect that already the initial guess ψ = 0, resp. ~A = 0 on ∂BR is quite reasonable, so that we

25

can avoid computing the Poisson formula. It is an interesting question to see to which extent theupdate of the boundary condition through the domain decomposition algorithm is really necessary.

To this end, we consider formulation (50),(51) for the scalar potential ψ. If we replace equation(51) by ψ(~x) = 0, on ∂BR we get a new formulation we could call ”scalar potential problem withhomogeneous Dirichlet boundary conditions”. We insist on the fact that this formulation does notcorrespond anymore to our original ferromagnetic problem, it is only an approximation of thatproblem. We then call Algorithm 2’ the algorithm obtained from Algorithm 2 by just replacing instep 1.) equation (71) by ψ

k− 12

h = 0, one ∂BRh. Algorithm 2’ will converge to an approximatedsolution for our homogeneous Dirichlet boundary conditions formulation. In Algorithm 2’, we donot use the Poisson formula anymore (and then we do not really need the small ball Brh). It isalso clear that iterations are now only used to solve the non-linear problem.

Clearly, the answer to the question at the beginning of this section will depend mainly on theratio R/r. So we present here two simulations of our test-case using Algorithm 2 and Algorithm2’: one with R/r = 1.25 using a new mesh of BR, one with R/r = 1.5 using a new mesh of BR.In order to also compare convergence of the two algorithms, we define the non-linear residue asfollows: if Ak ~ψk = ~bk is the system of equations resulting from (72) after standard finite elementcalculations, we define the non-linear residue at iteration k to be the vector ~rk = ~bk − Ak ~ψk−1.So, when Algorithm 2 (or 2’) converges, the non-linear residue ~rk should tend to zero when kincreases.

Homogeneous Dirichlet vs. Poisson formula: We call ~B1 the induction field computed usingAlgorithm 2’ and ~B2 the induction field computed using Algorithm 2. We evaluate the relativedifference ‖ ~B2 − ~B1‖/‖ ~B2‖ at every point of BR. Here, ‖ · ‖ denotes the euclidean norm in R3.

Figure 13: meshH0.12 comparison of boundary condition by Poisson formula ( ~B2) vs. the homo-geneous Dirichlet condition ( ~B1) for R/r = 1.5 (top line) and R/r = 1.25 (bottom line)

Fig. 13 compares ~B1 and ~B2 for both R/r = 1.25 and R/r = 1.5. The right, resp. middle plot

26

shows all tetrahedra on which the relative difference ‖ ~B2 − ~B1‖/‖ ~B2‖ exceeds 7% resp. 5%. Theleft plot shows a histogram of how many tetrahedra (vertical axis) there are with a given relativedifference ‖ ~B2 − ~B1‖/‖ ~B2‖ (horizontal axis). More precisely, we cluster all tetrahedra accordingto their corresponding relative error ‖ ~B2− ~B1‖/‖ ~B2‖ into 0.1%-large bins (on the horizontal axis)and we plot the number of tetrahedra in each bin on the vertical axis (in dark blue). In light blue,we plot the corresponding volume of the computational domain BR occupied by the tetrahedra ineach bin.

We clearly see, that the quality of the resulting ~B1 for the homogeneous Dirichlet boundarycondition depends on R/r. For R/r = 1.5 we commit at most 6% of relative error, while forR/r = 1.25 it is already 9%.

It seems from Figs. 13 that we may approximate the decay condition at infinity by a homoge-neous Dirichlet condition on ∂BR, provided that BR is large enough, typically R/r > 2. However,we must consider that the number of unknowns of the underlying discretization is proportional to(R/r)3, once Br has been fixed.

0 5 10 15 20 25 30

10−8

10−6

10−4

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100

no. of iterations (nonlinear iteration)

l2−

norm

of r

elat

ive

nonl

inea

r re

sidu

al

meshH0.12: nonlinear convergence history for R/r = 1.25

Poisson formula on ∂BR

Dirichlet ψ=0 on ∂BR

0 5 10 15 20 25 30

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no. of iterations (non−linear iteration)

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norm

of r

elat

ive

nonl

inea

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al

meshH0.12: nonlinear convergence history for R/r = 1.5

Poisson formula on ∂BR

Dirichlet ψ=0 on ∂BR

Figure 14: meshH0.12: nonlinear convergence history for R/r = 1.25 (left) and R/r = 1.5 (right),domain decomposition with Poisson formula (solid line) vs. fixed ψ = 0 on ∂BR (dashed line) inthe scal method.

On the other hand, we can use smaller BR within the domain decomposition framework withPoisson formula. The solved discrete problem has in this case considerably less unknowns thanin the previous case. Moreover, Fig. 14, which shows the convergence of the non-linear residueas a function of iteration number k for both meshes and both algorithms, seems to indicate, thatthe domain decomposition present just a minor comptational overhead, provided that R/r is largeenough, typically 1.5 ≥ R/r > 1.2.

6 Conclusions

We have presented both a scalar and a vector potential model for a static Maxwell problem in-cluding ferromagnetic effects with different formulations leading to finite element approximations:algorithms scal, bemfem, rt0, mix and vec were developed. We have applied all these algorithmsto a simple but representative test-case of a ferromagnetic plate in front of an electric conductorand compared the results for induction inside and outside the plate. It must be noticed here thatwe had no exact solution for our test-case, we just compared the results obtained from the different

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algorithms.

It comes out from our observations that the scal algorithm is by far the most efficient in termsof CPU time (and memory use) for obtaining quite a reasonable precision in its results.

If we aim the best approximation of induction near the plate or even in the plate, algorithmslike rt0 or vec give better results at the price of much more CPU time (and memory use).

The bemfem algorithm, which gives comparable results as the scal algorihm, could be madeefficient by using multipole techniques which we have not tried here. It is important to note thatthis algorithm is the only one presented here which does not need a mesh outside the ferromagneticparts.

References

[1] J. Bey: Tetrahedral grid refinement, Computing 55 (1995), no. 4, 355–378.

[2] R. Dautray and J-L. Lions: Mathematical Analysis and Numerical Methods for Science andTechnology, vol 4, Springer-Verlag, Berlin 1990.

[3] J. Descloux, M. Flueck, M.V. Romerio: A problem of magnetostatics related to thin plates,RAIRO Model. Math. Anal. Numer. 32 (1998), no. 7, 859–876.

[4] O. Bıro and K. Preis: On the use of magnetic vector potential in the finite element analysisof three-dimensional eddy currents, IEEE Transactions on Magnetics 25 (1989), no. 4, 3145–3159.

[5] O. Bıro, K. Preis, K.R. Richter: On the use of the magnetic vector potential in the nodal andedge finite element analysis of 3D magnetostatic problems, IEEE Transactions on Magnetics32 (1996), no. 3, 651–654.

[6] O.D. Kellogg: Foundations of Potential Theory, Courier Dover Publications 1953.

[7] A. Masserey, J. Rappaz, R. Rozsnyo and M. Swierkosz: Numerical integration of the threedimensional Green kernel for an electromagnetic problem, J. Comput. Phys. 205 (2005), no.1, 48–71.

[8] A. Masud and T.J.R. Hughes: A stabilized mixed finite element method for Darcy flow,Comput. Methods. Mech. Engrg. 191 (2002), 4341–4370.

[9] J-C. Nedelec. Acoustic and Electromagnetic Equations, Integral Representations for HarmonicProblems. Applied Mathematical Sciences, vol. 144. Springer.

[10] S. Russenschuck: Electromagnetic Design and Mathematical Optimization Meth-ods in Magnet Technology, eBook ver. 3.3, April 2006, ISBN 92-9083-242-8,http://russ.home.cern.ch/russ.

[11] P. Vanek, M. Brezina and J. Mandel: Convergence of Algebraic Multigrid Based on SmoothedAggregation, Numer. Math. 88 (2001), no. 3., 559-579.

[12] J. Rappaz: About the ferromagnetic effects. Some mathematical results. Scientific report,Institute of Analysis and Scientific Computing (IACS), EPFL, to appear.

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