Shape and Topological Optimization for Electromagnetism ...The aim of the topological sensitivity analysis is to obtain an asymptotic ex-pansion of shape functional with respect to

Contemporary Engineering Sciences, Vol. 3, 2010, no. 8, 373 - 394

Shape and Topological Optimization

for Electromagnetism Problems

Aminata Diop∗,+

∗Universite Cheikh Anta Diop de DakarDepartement Genie Informatique, ESP

+Laboratoire de Mathematiques de la Decision et d ’ Analyse [email protected]

Ibrahima Faye+

Universite de Bambey BP 30, SenegalUMI UMMISCO 209, [email protected]

Idrissa Ly ∗,+

UMI UMMISCO 209, [email protected]

Diaraf Seck∗,+

UMI UMMISCO 209, [email protected]

AbstractThis paper presents a topological shape optimization problem tech-

nique for electromagnetic problems using the topological sensitivity anal-ysis and topological derivative. The objective function that representsthe design objective is expressed in terms of magnetic field. The ad-joint method is used to optimize the distribution of magnetics fields.Some numerical results that demonstrated the validity of the proposedapproach are presented.

Mathematics Subject classification: 49Q10, 49Q12, 78A25, 78A40,78A45, 78A50, 35J05

Keywords: Topological optimization, shape optimization, topological gra-dient, Helmhotz equation, Numerical simulations

374 A. Diop, I. Faye, I. Ly and D. Seck

1 Introduction

Shape and topological optimization is the most flexible optimization methodthat can simultaneously deal with geometrical and topological distribution.This method involves defining a fixed design domain such that it is larger thanthe resulting domain. In the fixed domain, an arbitrary configuration can beexpress, putting hole, allowing large change in the geometrical and topologicaldesign during the optimization process. Shape and topological optimizationwere originally developed for structures design and recently adapted for manyother areas of design by taking other branches of science into consideration,such as electromagnetic.The aim of the topological sensitivity analysis is to obtain an asymptotic ex-pansion of shape functional with respect to the creation of a small hole insidethe domain. The principe is the following. One consider a cost functionj(Ω) = J(Ω, uΩ) where uΩ is solution to a partial differential equation definedin the domain Ω ⊂ R

d, d = 2 ord = 3, a point x0 ∈ Ω and a fixed domainB ⊂ R

d, containing the origin. One searches for an asymptotic expansion ofj(Ω\(x0 + εB) when ε tends to zeros. In most cases, it reads in the form

j(Ω\(x0 + εB) − j(Ω) = f(ε)g(x0) + o(f(ε)). (1)

Here f(ε) is an explicit positive function going to zeros with ε and functiong, the topological gradient or topological derivative, is in general easy to com-pute. Expression (1) is called the asymptotic analysis. Hence to minimizethe criterion J, one must to create holes at some points x where the topolog-ical gradient is negative. For more details about this approach we refer thereader to S. Garreau, P. Guillaume and Masmoudi [14], M. Masmoudi [21],A.A Novotny and al [24], J. Sokolowski and A.Zochowski [27]. For all thesework perturbing the domain consist to insert hole.B. Samet and al [26] obtain an asymptotic expansion of a functional with re-spect to the creation of a small hole in the domain. Such expansion is obtainedfor Helmholtz equation with Dirichlet condition on the boundary of circularhole. They presented some applications to waveguide problem.

In this paper our study consists to determine the geometrical and topolog-ical distribution of the magnetic fields in a considered space.The electromagnetism takes care with describing phenomena governed by elec-tric and magnetic fields which are connected. The study of phenomena resultsfrom interaction of the electric current and magnetic fields. Useful simulationsare made on precise examples by taking account various pulsations in the caseof the atomic physics.This paper is organized as follows. In section 2 the modeling of electromag-netism phenomena is presented. In the third section a topological sensitivityframework using an adaptation of the adjoint method is introduced and we

Shape and topological optimization 375

compute the expression of the topological sensitivity by using the approachesof Masmoudi [21] and Nazarov and Sokolowski [23]. In the last section, weconclude by some numerical simulations concerning the geometrical and topo-logical distribution of the magnetic fields

−→B =

−→rot

−→A , where

−→A , the potential

vector, is solution to a boundary partial differential equation in the considereddomain.

2 Modeling

The model of electromagnetism is well known in the literature. But in orderto facilitate the justification of work in other sections we present the Maxwellequations.Let us begin with the acknowledgement that the expression of the Lorentzforce

�f(P ) = q{−→E (P ) + �v ∧−→B (P )} (2)

not only gives meaning to the fields solutions of Maxwell equations when ap-plied to point charges , but yields new predictions. The velocity appearing inthe expression of Lorentz force is the velocity of the charge.−→B defines an axis of rotation, f(P ) : Strength exercised on the particle P,−→E (P ) : electric field (volts per meter),

−→B (P ) magnetic induction (areal den-

sity of magnetic flux q is related to P : electric charge of the particle.In the vacuum in the sense we Condensed Matter

1. Electric field E0 creates a point M by a stationary charge q, located atP.

−→E 0(M) =

q

4πε0

−−→PM

|PM |3 (3)

ε0 : dielectric constant of vacuum; ε0 = 10−9

36πFarad/metter

2. Electric field E�v creates at M by a charge q located at P and with avelocity v straight �v = v�k

E�v(M) = (1 − v2

c2)1/2(E0.�k)�k + (1 − v2

c2)−1/2(E0 − (E0.�k)�k) (4)

c speed of light in vacuum ∼ 3.108m/s

3. Magnetic induction B�v located at M by a charge q located at P and witha velocity v straight �v = v�k

−→B (M) =

�v(P ) ∧ −→E �v(M)

c2(5)


4. Electric field−→E created at M by a volume charge densityρ(x)

−→E (x) =

1

4πε0

∫Ω

ρ(ξ)(ξi − xi)�ki

|ξ − x|3 dξ (6)

All that precedes is valid provided reflect all charges (the charges of polariza-tion and those of magnetization)

1. Approximation of−→E v and

−→B v:

In a conductor, �v ∼ cm/s =⇒ v2

c2is important in front of 1 then

−→E �v �=−→

E 0

−→B �v(M) =

μ0

4πq�v(P ) ∧ −−→

PM

|PM |3 (7)

μ0 =1

ε0c2(8)

μ0 : Magnetic permeability of the space ∼ 4π10−7H/m, H: Henry

2. Strength of current I in a conductorThe material contains positive and negative ions. The strength of cur-rent in a driver is the quantity of loads(responsibilities) which crossesthe straight(right) section of a driver by unit of time. The intensity isexpressed in ampere.

Remark 1 In electromagnetism the canonical greatness is L : length,T : time, M : the mass I : the intensity.

3. Current Density

J =I

S(9)

The current vector is going to take into account the direction of themovement of loads.

�J(x) = ρ(x)�v (10)

The variation of intensity is

dI = �J.�ndS = J. �dS (11)


2.1 General laws of the electromagnetism

The law of preservation of the electricity is given by the relation between �Jand ρ In the Maxwell equations the unknowns are

−→E and

−→B

The law of conservation of the electricity is given by the following relation

Q =

∫D

ρdV, φ =

∫∂D

�J.�ndS (12)

where D is an arbitrary domain in R3, ρ the volume density of charges.

All charges through ∂D is exactly offset by the change in the total load.

d

dtQ + φ = 0

d

dt

∫D

ρ(x, t)dv +

∫∂D

�J(x, t).�ndS = 0 ∀ D ⊂ R3

∂ρ

∂t+ div �J = 0 (13)

There may be a source term, but it is not natural.

2.2 Non stationary Maxwell’s Equations

The unknowns are−→E the electric field

−→B the magnetic field and �J the electric

current density with Jtotal being the total current including the displacementcurrent. The generalized Ampere theorem is given by

∂ρ

∂t+ div �J = 0 (14)

We find �α such that

−→rot

−→B = μ0( �J + �α) (15)

The complete set of Maxwell’s equations are⎧⎪⎪⎪⎨⎪⎪⎪⎩

div−→B = 0

div−→E = ρtot

ε0−→rot

−→B = μ0( �J + ε0

∂−→E

∂t)−→

rot−→E = −∂

−→B

∂t

(16)

According to Maxwell’s equations, we have −→α = ε0∂−→E

∂t. By Helmholtz’s theo-

rem−→B can be written in terms of vector field

−→A , called the magnetic potential

−→B =

−→rot

−→A . (17)


Plugging this relation into Faraday’s law, we get

−→rot(

−→E +

∂−→A

∂t) = 0. (18)

By Helmhotz’s theorem the quantity in parenthesis can be written in terms ofscalar function V called the electric potential

−→E +

∂−→A

∂t= −−−→

grad V, that is−→E = −−−→

grad V − ∂−→A

∂t(19)

We also have

−→rot(

−→B ) =

−→rot

−→rot( �A) = μ0(J + ε0

∂−→E

∂t) =

−−→grad div �A − Δ �A = ∇(∇ · −→A ) −∇2−→A .

(20)

The equation (20) implies

∇(∇ · −→A ) −∇2−→A = μ0J + μ0ε0∂∇V

∂t− μ0ε0

∂2−→A∂t2

. (21)

The equality (21) implies

μ0ε0∂2−→A∂t2

−∇2−→A = μ0J −∇(∇ · −→A + μ0ε0∂−→V

∂t). (22)

A more sensible choice is the so called Lorentz gauge

div−→A = ∇ · −→A = −ε0μ0

∂−→V

∂t. (23)

If we adopt the Lorentz gauge the last term on the right hand side of (22)becomes zeros. Substituting the Lorentz gauge condition into the expression

−∇2V − ∂

∂tdiv

−→A − ρ

ε0

(24)

we obtain

ε0μ0∂2V

∂t2−∇2V =

ρ

ε0. (25)

Thus we find that Maxwell’s equations reduce to the following

ε0μ0∂2V

∂t2−∇2V =

ρ

ε0

ε0μ0∂2A∂t2

−∇2A = μ0J. (26)

The Maxwell’s equations can be expressed by

V =[ 1

c2

∂2

∂t2− Δ

]V =

ρtot

ε0

−→A =

[1c2

∂2

∂t2− Δ

]−→A = μ0

�Jtot (27)

where c = 1√μ0ε0

is the vaccum velocity of the light and is the d’Alembertianoperator.


2.2.1 General laws and special laws of behavior

1. Electric polarization The electric polarization or polarization densityor simply polarization is the vector field that expressed the density ofpermanent or induced electric dipole moment per unit volume.The behavior of electric fields (E and D), magnetic field B, chargedensityρ and current density J by Maxwell’s equations.The polarization density P defines the electric displacement field D

D = ε0E + P (28)

where ε0 is the electric permeability.Electric polarization corresponds to rearrangement of the bounds elec-trons in the material, which creates on additional charge, known as thebound charge density ρb : ρb = −∇ · −→P = −div

−→P so that the total

charge density that enters Maxwell’s equations is given by

ρtot = ρext + ρb (29)

where ρext is the free charge density describing charges brought fromoutside. We obtain the continuity equation. That is the charge beingconserved, the net flow of chosen volume must equal to the net chargein charge held inside the volume∫

S

J.dA − d

dt

∫V

ρtotdV −∫

V

ρtot

∂tdV (30)

where ρtot is the charge density per unit volume and dA is surface elementof the surface S enclosing the volume V. From the divergence theorem,we have ∫

S

JdS =

∫V

divJdV (31)

Hence ∫V

divJdV = −∫

V

∂ρtot

∂tdV (32)

This relation is valid for any volume, we obtain the continuity’s equation

div �J = −∂ρtot

∂t= −∂ρext

∂t− ∂ρb

∂t= −(∂ρext

∂t− div ∂P

∂t). (33)

By including

−→rot

−→B = μ0[ �J + ε0

∂−→E

∂t]


we have

div �J = div(−→rot

−→B

μ0

− ε0∂−→E

∂t) = −∂ρext

∂t+ div

∂ �P

∂t(34)

Let us introduce �Jext defined by

div �Jext = −∂ρext

∂t(35)

conduction current density.

div[−→rot

−→B

μ0− ε0

∂−→E

∂t− �Jext − ∂ �P

∂t] = 0 (36)

There exists−→M such that

−→rot

−→B

μ0− ε0

∂−→E

∂t− �Jext − ∂ �P

∂t=

−→rot �M (37)

Let

Jp =∂ �P

∂t(38)

This is the current density due to variation of system.

�Ja = �Jext + �JP (39)

2. Linear laws:Di = εijEj εij = ε0δij + Pij

Bi = μijHj μij = μ0δij + Mij

Ji = σijEj σij is the conduction tensor.

3. Isotropic law Pij = πδij

Mij = μδij

σij = σδij

3 Topological optimization problem

The goal of the topological optimization problem is to find an optimal de-sign with an a priori poor information on the optimal shape of the struc-ture. The shape optimization problem consists in minimizing a functionalj(Ω) = J(Ω; uΩ) where the function uΩ is defined, for example, on an open


and bounded subset of RN . For ε > 0; let Ωε = Ω\ωε be the set obtained by

removing a small part, where Ω and ω are two regular subset containing theorigin. Then, using general adjoint method, an asymptotic expansion of thefunction will be obtained in the following form

j(uΩε) − j(Ω) = f(ε)g(x0) + o(f(ε), limε→0f(ε). = 0 (40)

The topological sensitivity g(x0) provides information when the creating asmall hole located at x0 : Hence, the function g will be used as descent directionin the optimization process.

The model of the electromagnetism gives

Δ−→A − 1

c2

∂2−→A∂t2

= −μ0Jtot (41)

where

−→A = (A1, A2, A3) (42)

is the potential vector. Let us take for j = 1, 2, 3

Aj = Φj(x)eiwjt (43)

then we obtain the following equation

ΔΦj + w2jΦj =

μ0

Jtot, j = 1, 2, 3. (44)

In this equation we will add a Dirichlet boundary condition Φj = gi on ∂Ω or

a Neumann condition∂Φj

∂n= 0 sur ∂Ω Then we obtain the following equation

{ΔΦ + w2Φ = μ0

Re(eiwt)Jtot in Ω

Φ = g or ∂Φ∂n

= g on ∂Ω(45)

where g ∈ (H− 12 )3 and ΔΦ = (ΔΦ1, ΔΦ2, ΔΦ3) is the vectorial Laplacian and

w2 = (w21, w

22, w

23)

3.1 Position of the problem

Consider a regular and bounded domain Ω ⊂ RN(N = 2 or 3) for example of

class C2, Φ is solution to (45). Let ω ⊂ Ω a bounded open set of RN containing


the origin and x0 ∈ Ω. For all ε > 0 let us defined ωε = x0 + εω and Ωε = Ω\ωε.Let Φε be the solution in Ωε of:⎧⎨

⎩ΔΦε + w2Φε = μ0

Re(eiwt)Jtot in Ω

∂Φε

∂n= 0 on ∂Ω

Φε = 0 or ∂Φε

∂n= 0 on ∂ωε

(46)

Consider the coast functional defined by

J(Φε) =

∫Ωε

|−→rot(Φε) − Bd|2 (47)

where Bd, a reference magnetic field, can be expressed in the form Bd =−→rotAd with Ad a reference potential vector.Our aim is to evaluate the difference J(Φε)− J(Φ0) if ε tends to zero by usingthe generalized adjoint method.Multiplying (46) by a test function v and integrating we have∫

Ωε

∇Φε : ∇v − w2Φεv +

∫∂Ω

∂Φε

∂n.v +

∫∂ωε

∂Φε

∂n.v = −

∫ωε

μ0

cos wtJtotv

LetVε = {v ∈ H1(Ωε), v = 0 on ∂ωε},

aε(Φε, v) =

∫Ωε

∇Φε : ∇v − w2Φεv

and

Lε(v) = −∫

Ωε

μ0

coswtJtotv

Following the same idea, in Ω, let

V = {v ∈ H1(Ω), v = 0 ∂Ω},

a0(Φ, v) =

∫Ω

∇Φ : ∇v − w2Φv,

aε(Φε, v)−a0(Φ, v) =

∫Ωε

∇(Φε−Φ)∇V −w2(Φε−Φ)v+

∫ωε

∇Φ∇v−w2

∫ωε

Φv

3.2 A generalized adjoint method

The mathematical framework for domain parametrization introduced by theMurat-Simon work [13] cannot be used here. Alternatively, it is possible how-ever to invoke the adjoint method, as described in [21], in application to topo-logical optimization. A basic feature of the adjoint method is yield of anasymptotic expansion of a functional J(Φε) which depends of a parameter Φ,


using a adjoint state V which does not depend on the parameter. This impliesthe need to solve a certain system of equations in order to obtain an approx-imation of the topological gradient g(x); accordingly, let V be a fixed Hilbertspace and L(V) (respL2(V)) denotes the spaces of linear (resp bilinear) formson V. We are able then to state the following hypothesis:

1. H − 1 There exists a real function f defined in R+ , a bilinear and

continuous form a0 defined in L2(V) and a linear form δa such that:

limε→0

f(ε) = 0 (48)

‖aε − a0 − f(ε)δa‖L2(V) = 0, (49)

‖lε − l0 − f(ε)δJ‖L(V) = 0, (50)

2. H − 2 The bilinear form a0 is coercive: There exists a constant α > 0such that

a0(u, u) ≥ α‖u‖2, ∀u ∈ V (51)

According to (51), the bilinear form aε depends continuously on ε; hencethere exists ε0 and β > 0 such that ∀ ε ∈ [0; ε0] the following uniformcoercivity condition holds.

aε(u, u) ≥ α‖u‖2, ∀u ∈ V (52)

Using inequality (52) for Φ1 = Φε − Φ0, for all ε ≥ 0, the function Φε issolution to (46), the inequality (48) and (49) and the continuity of δa,we obtain the following lemma

Lemma 1 We have

‖Φε − Φ‖ = O(f(ε)) (53)

3. H − 3 Consider a cost function j(ε) = J(Φε); where the functional Jis differentiable. For u ∈ V there exists a linear and continuous formDJ(u) ⊂ L(V) and δJ such that:

Jε(v) − J0(u) = DJ0(u)(v − u) + f(ε)δJ + o(‖v − u‖ + f(ε) (54)

The Lagrangian is defined by

Lε(u, v) = Jε(u) − aε(u, v) − lε(v) ∀u, v ∈ V (55)


Theorem 3.1 If hypothesis H− 1, H − 2 and H − 3 are satisfied and let Φε

the solution to (46). Then the functionals Jε admits the following asymptoticexpansion

Jε(Φε) − J0(Φ) = f(ε)δL(Φ, U) + o(f(ε))

where δL(u, v) = δJ (u) + δa(u, v) − δl and U is the solution of the adjointproblem: to look for U ∈ V such that

a0(W, U) = −DJ0(Φ)W, ∀W ∈ V

In order to get the asymptotic expansion of the cost functional, we will use thefact that variation of the Lagrangian is equal to the one of the cost functional.Then

J(Φε) − J(Φ) = Lε(Φε, v) − L0(Φ, v), ∀ v ∈ V (56)

3.3 Variation of the cost functional

J(−→A ) =

∫Ω

‖−→rot(−→A ) − Bd‖2 =

∫Ω

3∑i=1

((−→rot

−→A )i − Bdi

)2

with

−→rot(

−→A ) =

⎛⎝ ∂2A3 − ∂3A2

∂3A1 − ∂1A3

∂1A2 − ∂2A1

⎞⎠ (57)

Let−→h = (h1, h2, h3) in H1

0 (Ω, R3) then we have

J(−→A +

−→h ) =

∫Ω

‖−→rot(−→A )+−→rot(

−→h )−Bb‖2dx =

∫Ω

3∑i=1

((−→rot

−→A )i−Bdi

+(−→rot(

−→h ))i)

2

=

∫Ω

3∑i=1

{(−→rot−→A )i − Bdi)2 + (

−→rot(

−→h ))i)

2 + 2(−→rot

−→A )i − Bdi

)(rot(−→h ))i)

J(−→A +

−→h )−J(

−→A ) = 2

∫Ω

3∑i=1

(−→rot

−→A )i −Bdi

)(−→rot(h))i)+

∫Ω

3∑i=1

(−→rot(

−→h ))i)

2dx

= 2

∫Ω

−→rot(

−→h )(

−→rot(

−→A ) − Bd)dx +

∫Ω

‖−→rot(−→h )‖2dx

But we get∫Ω

−→rot(

−→h )(

−→rot(

−→A )−Bd)dx =

∫Ω

(∂2h3−∂3h2)K1+(∂3h1−∂1h3)K3+(∂1h2−∂2h1)K3


where Ki =−→rot(

−→A ))i − Bdi

This gives

∫Ω

−→rot(h)(

−→rot(

−→A )−Bd)dx =

∫Ω

−h3∂2K1+h2∂3K1−h1∂3K2+h3∂1K2−h2∂1K3+h1∂2K3

=

∫Ω

h1(∂2K3 − ∂3K2) + h2(∂3K1 − ∂1K3) + h3(∂1K2 − ∂2K1)

then

J(−→A +

−→h ) − J(

−→A ) = 2

∫Ω

−→rot(

−→rot(

−→A ) − Bd)

−→h

+

∫Ω

[(∂2h3 − ∂3h2)

2 + (∂3h1 − ∂1h3)2 + (∂1h2 − ∂2h1)

2

︸︷︷︸o(‖h‖2

W )

where W = H10 (Ω, R3) is a functional space; then DJ(

−→A ) = 2

−→rot(

−→rot(

−→A )−Bd

and the adjoint problem takes the form

{ −ΔU = 2(−→rot

−→rot(

−→A ) − Bd) in Ω

U = 0 or ∂U∂n

= 0 on ∂Ω(58)

3.4 Variation of the bilinear form

Let wε = Φε − Φ, then wε is solution to⎧⎨⎩

Δwε + w2wε = 0 in Ωε∂wε

∂n= 0 on ∂Ω

∂wε

∂n= −∂Φ

∂non ∂ωε

(59)

Let us approximate the solution wε by hε solution to⎧⎨⎩

Δhε + w2hε = 0 in R2\ωε

∂hε

∂n= −∂Φ

∂nin ∂ωε

hε = 0 at ∞(60)

Let hε = εHε(xε) then we get

⎧⎨⎩

ΔH = 0 in R2\ω

∂H∂n

= −∂Φ∂n

on ∂ωH = 0 at ∞

(61)

Using potential theory the solution H of problem (61) is obtained by

H(x) =

∫∂ω

λ(y)E(x − y)ds(y) ∀x ∈ R2\ω (62)


where λ ∈ H−1/20 (∂ω) is the unique solution of the integral equation

λ(x)

2+

∑λ(y)∂nxE(x − y)ds(y) = −∇Φ0.n (63)

If ω = B(0, 1) one proves that δa = −2π∇Φ(0)∇V (0) + k2mes(ω)Φ(0)V (0)where V is solution to adjoint problem (66). For the functional we have

Jε(Φε) − J0(Φ) =

∫Ωε

|Φε − Φd|2 −∫

Ω

|Φ − Φd|2

=

∫ωε

|Φ − Φd|2

Using Taylor expansion of Φ and a change of variables we obtain

∫ωε

|Φ − Φd|2 = −πεn|Φ(0) − Φd(0)|2 + εn

Thus we have

Jε(Φε) − J0(Φ) = εn(−π|Φ(0) − Φd(0)|2) + o(εn) (64)

Then δJ = −π|Φ(0) − Φd(0)|2. We can proof easily that δl = 0

Theorem 3.2 Let Jε the functional defined by (47) where Φε is solution to(46) with ∂Φε

∂n= 0 on ∂ωε. The following asymptotic expansion holds

Jε(Φε) − J(Φ) = ε3(−2π∇Φ(0)∇U(0) + k2mes(ω)Φ(0)U(0) − π|Φ(0) − Φd(0)|2) + o(εn)(65)

where Φ is solution to (45) and U is solution to the adjoint state

{ −ΔU = 2(−→rot

−→rot(

−→Φ ) − Bd) in Ω

∂U∂n

= 0 on ∂Ω.(66)

Remark 2 In the case for Diriclet condition on ∂ωε and ω = B(0, 1), thetopological derivative reads G(x0) = −2πΦ(x0)U(x0) where Φ is solution to(45) and V is solution to the adjoint problem

{ −ΔU = 2(−→rot

−→rot(

−→A ) − Bd) in Ω

U = 0 on ∂Ω.(67)


3.5 Permanent Case

Let us consider the initial problem

Δ−→A − 1

c2

∂2−→A∂t2

= −μ0Jtot (68)

in which we have a Dirichlet or Neumann condition BΩ in Ω, where−→A = (A1, A2, A3) (69)

design the potential vector. In the permanent case ∂∂t

= 0, we obtain{ −Δ−→A = −μ0Jtot in Ω

BΩ−→A = g on ∂Ω(70)

In Ωε ⎧⎨⎩

−ΔAε = −μ0Jtot in Ωε

BΩAε = g on ∂ΩAε = 0 on ∂ωε

(71)

Multiplying (71) by a test function v and integrating we get∫Ωε

∇Aε : ∇vdx −∫

∂Ω

∂Aε

∂n.v −

∫∂ωε

∂Aε

∂nv =

∫Ωε

μ0Jtotvdx (72)

Consider the functional space Vp defined by

Vp = {v ∈ H1(Ωε); v = 0 on ∂Ω, v = 0 on ωε},then the bilinear form associated to the operator is the following

aε(Aε, v) =

∫Ωε

∇Aε : ∇vdx

and the linear form is given by

lε =

∫Ωε

μ0Jtotvdx

In the same way multiplying (70) by a test function v ∈ (H10 (Ω))3 and inte-

grating in Ω we have

a0(A, v) = l0(v) ∀v ∈ (H10 (Ω))3 (73)

where

a0(A, v) =

∫Ω

∇A : ∇v, l0(v) =

∫Ω

μ0Jtotvdx (74)

Consider the exterior problem defined in Rn\ω by{ −ΔA = μ0Jtot in R

n\ωA = 0 on ∂ω

(75)

Using the potential theory the problems (70),(71) and (75) admit solutions see[7]


3.5.1 Polarization matrix and topological derivative

Consider the basis of homogeneous polynomials in R3 of degrees inferior or

equal to 1 satisfying ΔP = 0. Let U1, . . . , UN the basis of such polynomials.The polynomials satisfy{

U i(zx) = zsiU i(x) with si the degree of U i∑Ni= Uk

j Uhj = δhk

(76)

As we have a Dirichlet condition on ∂ω, N = 3 then we can take U i(x) = ej

where ej is the vectors of the canonical basis of R3. Thus si = 0, i = 1, 2, 3. Let

Φi the fundamental matrix associated to the vectorial Laplacian ie satisfying

ΔΦj = ejδx, j = 1, 2, 3 (77)

where δx is the dirac mass concentrated at x, then Φj(x) = 1K3|x|e

j, j = 1, 2, 3.

We define the vectors U−i, i = 1, 2, 3 by the relation

U−i(x) =3∑

j=1

U ij(x)Φj = Φi (78)

Using the theory developed in [22, 23] and considering the following prob-lem { −Δζj = 0 in R

3\ωζj = 0 on ∂ω

(79)

the solution ζj of (79) can be written as follows

ζj = U j + zj = U j +3∑

j=1

mωjkU

−j + zj

where zj is solution to problem{ −Δzj = 0 in R3\ω

zj = −U j on ∂ω(80)

where zj is the remainder of the Taylor development of zj at the origin. Theterms mω

jk are the coefficients of the polarization matrix.In the following we give a theorem which give the topological derivative.

Theorem 3.3 Let Aε be the solution to (71) and Jε the functional defined by(47). Then Jε admits the following asymptotic expansion

Jε(Aε) − J0(A) = −εAAωU + o(f(ε)) (81)


where A is solution to problem (70) and U the solution to the adjoint problemdefined by { −ΔU = 2

−→rot

−→A (

−→rot

−→A −−→

B d) in ΩU = 0 on ∂Ω

(82)

Aω denotes the polarization matrix.

Proof 1 The proof of theorem 3.3 is standard in the literature see [23, 9, 13]

4 Numerical simulations

In this section we consider a reference magnetic field−→B d in a square Ω =

[−1, 1]× [−1, 1]. Our aim is to compare the magnetic field−→B =

−→rot(

−→A ) where−→

A is solution to (45) with the reference magnetic field−→B d in different cases.

Let ν be the frequency the pulsation is given by 2πν. In the case of atomicphysics the frequency ν is between 32768Hz and 9, 192631770GHz. we take aparticular value ν = 40000Hz. In the of geophysic the frequency ν is between0, 01 and 10Hz and we take a particular value ν = 1Hz.The reference magnetic field is given by Bd = (2x, 2x + y, 2x + 2)

4.1 Neumann condition

We simulate the topological derivative given in theorem 3.2 at a point x0 by

−2π(∇Φ(0)∇U(x0) + k2Φ(0)U(x0) − |Φ(x0) − Φd(x0)|2)where Φ is solution to (45) and U the adjoint state solution to (66). Thetopological derivative in both case is given by figure 1.

4.2 Dirichlet condition on the bounds of ωε

The topological derivative is given by in Remark 3.3 by G(xx0) = −2πΦ.Uwhere Φ is solution to (45) and U the adjoint state given by (67). The topo-logical derivative is given in both cases by figure 2.

4.3 Permanent Case

We simulate the topological derivative G(x0) given by theorem 3.3 where A issolution to (70) and U solution to (82). The topological gradient is given byfigure 3.

Remark 3 Note that in all figures where the topological derivative approacheszero, the magnetic field

−→B is close to the reference magnetic field Bd. When

they are very far away it means that the two fields do not approach.


Figure 1: topological derivative in atomic physic at the top and in geophysicat the bottom


Figure 2: topological derivative in atomic physic at the top and in geophysicat the bottom


Figure 3: topological derivative in physic atomic at the top and in geophysicat the bottom


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Received: July, 2010

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