# Adjoint Variable Method for the Study of Combined Active and Passive Magnetic Shielding

Authors: Peter Sergeant; Ivan Cimrák; Valdemar Melicher; Luc Dupré; Roger Van Keer
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

For shielding applications that cannot sufficiently be shielded by only a passive shield, it is useful to combine a passive and an active shield. Indeed, the latter does the “finetuning” of the field reduction that is mainly caused by the passive shield. The design requires the optimization of the geometry of the passive shield, the position of all coils of the active shield, and the real and imaginary components of the currents (when working in the frequency domain). As there are many variables, the computational effort for the optimization becomes huge. An optimization using genetic algorithms is compared with a classical gradient optimization and with a design sensitivity approach that uses an adjoint system. Several types of active and/or passive shields with constraints are designed. For each type, the optimization was carried out by all three techniques in order to compare them concerning CPU time and accuracy.


---

## Body

## 1. Introduction

As the computational cost has decreased significantly
during the last years, more and more commercial software packages offer
standard algorithms for optimization in order to solve design problems. Many
researchers in industry and in academia now use commercial finite-element
software in combination with commercial optimization routines, such as genetic
algorithms. This approach is usually successful but very time consuming. For
example, the design of a combined active and passive shield in [1] required between 8 and 14
days of CPU  time. In spite of the increased computational power of computers,
the conventional combination of genetic algorithm
and finite elements remains a computational challenge.This paper is devoted to designing a passive and
active shield for the same axisymmetric induction heater discussed in [1] and shown in Figure 1 in
much less time. Therefore, not only another algorithm is explored (adjoint
variable method, explained in Section 3), but also the optimization problem
itself is changed in order to take advantage of the adjoint system. The
optimization technique is similar to [2, 3] and allows the designer to use commercial software.
The numerical model is quasistatic (and not static like in [2, 3]), which results in complex
equations in the frequency domain (instead of real equations), and is based on
Maxwell's equations. Both currents and geometrical shapes are optimized.
Moreover, the optimization is multiobjective, and all of the five contributions
to the cost are modelled in the same adjoint system. After validating the
adjoint variable approach in Section 4, the method is compared in Section 5
with a classical gradient algorithm and with a genetic algorithm regarding the
computation time and the efficiency of the shield.Figure 1
The axisymmetric induction heater
application with shields discussed in [1].


## 2. Magnetic Shielding Problem

A magnetic shield is developed to reduce the magnetic
stray field of an axisymmetric induction heating device for the heat treatment
of aluminum discs. After adding the shield, the shielded device should comply
with the reference levels of ICNIRP [4] or the European Community [5]. Next to the field reduction
in a predefined “target area,” several other constraints are involved with a
magnetic shielding problem: limits on the resistive heating in the shields and
on the influence of the heating process, development costs, and geometrical
constraints to guarantee the accessibility of the shielded device. In the
literature, many papers are published concerning passive shields [6], studying the effect of
dimensions, permeability (including nonlinearity, hysteresis), and conductivity
of the metal sheets on the shielding effectiveness. A number of papers were
written about active shielding [7–9]: field reduction caused by a counter field that is
produced by controlling the currents in a number of compensation coils.Passive shields can be very efficient especially at
frequencies above 1 kHz where induced currents become effective, especially if
the passive shields are closed, that is, if they completely enclose the source
like Faraday's cage. Active shields can be more attractive at lower frequencies
especially if the compensation coils are close to the source, because the
generator of the compensation current can have a lower voltage and current
rating. However, a lot of devices cannot easily be shielded by only a passive
or only an active shield because of geometrical constraints. It is shown in
[1] that a combination
of both types of shielding may improve the performance significantly. Here, the
passive shield should be rather close to the source. By flux shunting or by
induced currents, the passive shield causes a significant field reduction in
the target area that is usually not homogeneous: a high field reduction is
achieved in parts of the target region, but still high fields occur in other
parts. Here, active shields can additionally reduce the field in these regions.The equations to solve are Maxwell's equations in the
frequency domain:(2.1)∇×E=−jωB,∇×H=Je+Ji=Je+σE,where the total current density
consists of the external current density Je and the induced
current density Ji due to the
conductivity σ and the
electric field E. The system of (2.1) is solved together
with the constitutive law B=μH and with the
following boundary conditions (see Figure 2):(2.2)E×n=0onΓ1,∇×E×n=0onΓ2,expressing the fact that the
electric field should be perpendicular to Γ1 and that the
magnetic field should be perpendicular to Γ2. Introducing the vector potential so that B=∇×A leads
to(2.3)∇×E=−jω(∇×A),∇×[1μ(∇×A)]=Je+σE.Substitution of the electric
field yields the equation to be solved by a finite-element model (FEM)
[10]:(2.4)∇×[1μ(∇×A)]+jωσA=Jewhich is written in weak
formulation with test variable φ:(2.5)(1μ(p)∇×A(p),∇×φ)+(jωσ(p)A(p),φ)=(Je(p),φ),taking into account the boundary
conditions. The variable p contains all parameters to be optimized: the position rp and height hp of the passive
shield with thickness tp in steel, and
the (complex) currents Iri+jIii and positions (ri,zi), i=1⋯n of the n active shield
coils in the axisymmetric problem. Symbol (⋅,⋅) denotes the
standard scalar product defined by (u,v)=∫Ωu·vdΩ. Notice that μ and σ depend on the
position and size of the passive shield. Therefore, they both depend on p. The weak formulation has the advantage that μ and σ can be
discontinuous quantities.Figure 2
DomainΩ. Γ2 is the bottom
horizontal border of Ω and Γ1 is the rest of
the border of Ω; ΩW is the
workpiece; ΩP is the passive
shield; ΩTA is the target
area; ΩCi are the
compensation coils of the active shield.The shielding application that we study is
axisymmetric so that the vector potential can be written asA=01r+A1ϕ+01z. Here, A is the
azimuthal component of the vector potential of which the r- and z-components are
zero. Maxwell's equations in the frequency domain (2.5) are in the
axisymmetric case:(2.6)(1μ(p)∇A(p),∇φ)+(1rμ(p)A(p),∂φ∂r)+(jωσ(p)A(p),φ)=(Je(p),φ).On the boundary (see Figure 2)
we prescribe(2.7)A=0onΓ1,∇A⋅n=0onΓ2,where ∇A:=(∂A/∂r,∂A/∂z).The objective value is a function of the scalar
potentialA and of ∇A. First of all, it takes into account the magnetic
field in the target area (FB): the main objective is to minimize the magnetic
induction B=∇×A in the
reference region ΩTA. This can be
achieved by minimizing the cost functional(2.8)F(p)=12∫ΩTA|B|2dΩ=12∫ΩTA|∇×A|2dΩ=12∫ΩTA(|∇A|2+2Ar∂A∂r+A2r2)dΩ.In order to reduce the
complexity of the cost function, we ignore the last terms and choose (1/2)∫ΩTA|∇A|2dΩ as cost value.
Other objective values in the multiobjective optimization are the dissipation
in the passive shield (FP), the dissipation in the active shield (FA), the change of the heating of the workpiece by the
adding of shields (FW), and the volume of the passive shield as an
investment cost (FV):(2.9)F(p)=w1FB+w2FP+w3FA+w4FW+w5FV,wherein(2.10)FB=12∥∇A(p)∥ΩTA,2(2.11)FP=12∥Ji⋅E∥ΩP=12σω2∥A(p)∥ΩP,2(2.12)FA=122πrρSA∑i=1n(Ir,i2+Ii,i2),(2.13)FW=PW,0−12∥Ji⋅E∥ΩW=PW,0−12σω2∥A(p)∥ΩW,2(2.14)FV=2πrphptp.Herein, SA is the
cross-section of an active shield coil and ρ is the
resistivity of the coil material. The constant PW,0 is the power
dissipated in the workpiece without shields present. The weighting factors wk, k=1,…,5 should be
chosen as explained in Section 4.


## 3. Adjoint System

In order to use any gradient-based optimization
technique, one has to evaluate the derivative ofF(p) with respect to p in some
direction ζ. For simplicity of the exposition, we use the symbol δ defined
as(3.1)δu=∂u∂pζ=limϵ→0u(p+ϵζ)−u(p)ϵ.Formal derivation of the cost
functional gives us(3.2)δF=w1δFB+w2δFP+w3δFA+w4δFW+w5δFV,where particular derivatives
read as(3.3)δFB=12[(∇δA*,∇A(p))ΩTA+(∇δA,∇A*(p))ΩTA],δFP=12δσω2∥A(p)∥ΩP2+12σω2[(δA*,A(p))ΩP+(δA,A*(p))ΩP]+12σω2∫δΩPA(p)⋅A*(p)d(δΩ),δFA=2πrρSA∑i=1nδIr,iIr,i+2πrρSA∑i=1nδIi,iIi,i+δrπρSA∑i=1n(Ir,i2+Ii,i2),δFW=−12δσω2∥A(p)∥ΩW2−12σω2[(δA*,A(p))ΩW+(δA,A*(p))ΩW],δFV=2πrptpδhp.The symbol “*" denotes the
complex conjugate. The extraboundary integral in δFP is necessary
because the domain ΩP, on which FP is evaluated,
changes in size when the height of the passive shield changes. Here, we have to
evaluate the topological derivative, which depends on concrete settings. In our
case of a rectangular passive shield domain between r-coordinates (a,b) and z-coordinates (0,hp), where we change just one dimension, height hp, we have to evaluate(3.4)limϵ→0(∫0h+ϵ∫abf(x,y)dxdy−∫0h∫abf(x,y)dxdy)ϵ=∫ablimϵ→01ϵ∫hh+ϵf(x,y)dydx=∫abf(x,h)dx.We used here the mean value
theorem. Thus in our case,(3.5)∫δΩPA(p)·A*(p)dδΩ=∫ΓPA(p)·A*(p)dδΩ,where ΓP is the upper
boundary of ΩP.The derivativeδσ is used to
handle the variation of domain size, considering a transition region. For δσ, one can use an explicit expression: instead of
modelling one solid domain with high conductivity (dashed line in Figure 3), we
model next to the domain with high conductivity a small transition domain (the
domains 2-3-4-5 in solid line in Figure 3) that surrounds the domain with high σ. In this transition zone, the conductivity decreases
linearly from σ to zero. The
derivative δσ differs from
zero in this transition domain only.Figure 3
Layout of a domain of which the position or shape can
change.ForδA, however, one needs to set up corresponding PDE. For
clarity, the dependence of p is not
explicitly written any more in the following equations. Differentiation of
(2.6) with
respect to p results
in(3.6)((μ)−1∇δA,∇φ)+(jωσδA,φ)+((μr)−1δA,∂φ∂r)=(δJe,φ)−(δ(μ−1)∇A,∇φ)−(jωδσA,φ)−(r−1δ(μ−1)A,∂φ∂r).We obtain a PDE for δA. This is, however, not so practical. For the
implementation, one needs to know a partial derivative in one component of p. Imagine p=(a,b,c) consists of
three components. So for computations we need to know three values ∂A(p)/∂a, ∂A(p)/∂b, and ∂A(p)/∂c, that is, we need to compute δA for three
different values of ζ, namely, ζ1=(1,0,0), ζ2=(0,1,0), and ζ3=(0,0,1). This is,
however, time-consuming since every computation of δA means solving
one PDE.To solve this problem, we apply the so-called adjoint
variable method. This method was successfully used in the optimal design problem in
micromagnetism involving the shape optimization of the ferromagnetic core in MRAM memories [11]. We set up a dual problem to (3.6). Find ξ∈H0,Γ1(Ω) such that for
all φ∈H0,Γ1(Ω), the following equality holds:(3.7)((μ)−1∇φ,∇ξ)+(jωσφ,ξ)+((μr)−1φ,∂ξ∂r)=a(φ),where the functional a(φ) is composed of
those terms from δF that contain δA, namely,(3.8)a(φ)=w12[(∇φ*,∇A)ΩTA+(∇φ,∇A*)ΩTA]+w22σω2[(φ*,A)ΩP+(φ,A*)ΩP]−w42σω2[(φ*,A)ΩW+(φ,A*)ΩW].The functional a(φ) serves as a
pseudosource for the adjoint problem derived from the cost functional. Since
the terms in the cost functional are nonzero only on the corresponding
subdomains ΩTA,ΩP, and ΩW, so is the pseudosource. Notice that the cost terms
of the active shield dissipation FA and the volume
of the passive shield (FV) in (2.9) do not result in a pseudosource in the dual problem.
Although (3.7)
and (3.8)
represent only one adjoint system, its solution requires two finite-element
evaluations. Indeed, the test functions φ are real-valued
functions in the commercial software we use while they should be complex in
order to solve (3.7) and (3.8). In practice, the adjoint system is solved a first
time using only the real parts of the pseudosources. Afterwards, it is solved a
second time using the imaginary parts of the pseudosources. This means that in
total three finite-element calculations are needed to obtain the objective
value and all gradients. Possibly, as the two adjoint calculations are
independent from each other, they can be executed in parallel on appropriate
computer architectures.Knowing the solutionξ of the dual
problem (3.7),
one can settle the following result: by setting φ=δA in (3.7) and by setting φ=ξ in (3.6) we arrive
at(3.9)a(δA)=(δJe,ξ)−(δ(μ−1)∇A,∇ξ)−(jωδσA,ξ)−(r−1δ(μ−1)A,∂ξ∂r).When comparing a(δA) obtained by
setting φ=δA in (3.8) with the
derivative of the cost (3.3), it can be observed that all terms occurring in
(3.8) also
occur in (3.3). By adding the missing terms, we obtain the explicit
expression for the derivative of the cost:(3.10)δF=(δJe,ξ)−(δ(μ−1)∇A,∇ξ)−(jωδσA,ξ)−(r−1δ(μ−1)A,∂ξ∂r)+w22δσω2∥A∥ΩP2+w22σω2∫δΩPA⋅A*d(δΩ)+w32πrρSA∑i=1n(δIr,i)Ir,i+w32πrρSA∑i=1n(δIi,i)Ii,i+δrw3πρSA∑i=1n(Ir,i2+Ii,i2)−w42δσω2∥A∥ΩW2+w52πrptpδhp.With this explicit expression
for the derivative of F(p) with respect to p, we are able to evaluate δF for any ζ. We just need to solve the PDE for ξ.The actual minimization algorithm starts from some
initial guess, for example,p0=0 or other value
obtained, for example, from a genetic algorithm.The
gradient-adjoint optimization algorithm can in principle be seen as a classical
gradient algorithm that tries to minimize a cost function by iteratively
evaluating a finite-element model (step 2). There is, however, one major
difference: the gradients are obtained by solving another (adjoint) finite-element
model (step 3) and by using the solution in an expression for the gradient
(step 4). In the classical algorithm, the gradients are determined by applying
perturbations in the parameters.


## 4. Validation of the Adjoint Variable Method

The studied application to design shields for is an
axisymmetric induction heating device with an excitation coil of 0.2012 m
radius carrying a current of 4000 A at 1 kHz. The workpiece of 10 mm height and
191 mm radius is made from aluminum.We now add a passive shield of 0.65 mm thickness at
radial position 0.3 m. Figure4 shows the several cost terms from (2.9) using the
weighting factors w1=4×109, w2 = 8 ×10−3, w3 = 2 ×10−4, w4 = 2 ×10−2, and w5=104 as a function of the passive
shield height. The choice of the weighting factors depends on the relative
importance that is assigned to each cost term. No active shield was considered
in this situation (FA=0). It can be seen that the term FB related to the
magnetic field in the target area decreases strongly with increasing height of
the passive shield, which is a purely conductive shield with μr=1 and σ=5.9×106 S/m. The losses
caused by induced currents in this shield FP increase with
the height for rather low shields with hp<0.1 m, but
remain almost constant for hp>0.1 m. The cost FW increases with
the height of the passive shield: as the shield is conductive, it reduces the
flux produced by the excitation current. Consequently, the heating induced in
the workpiece decreases, which causes the cost FW.Figure 4
Several contributions to the objective
valueF as a function
of the passive shield height (no active shield).Figure5 illustrates the derivatives to the parameter hp, calculated on the one hand by the adjoint method and
on the other hand by the conventional approach: applying a perturbation of 2 mm
in the height of the shield, re-evaluating the cost, and approximating the
derivative by dividing the subtraction of both cost values by the perturbation
step. The correspondence is good for this conductive shield. However, the
finite-element mesh in the 1.25 mm wide and 2 mm high transition domain where
the relevant terms of (3.10) are integrated had a maximal edge size of 0.1 mm:
much finer than needed to obtain accurate gradients by the conventional method
(0.25 mm).Figure 5
Conventional and adjoint gradient of
all cost terms separately as a function of the height of the passive shield forμr=1 and σ=5.9×106 S/m (no active
shield).In order to illustrate the limits of the adjoint
method, we compared the conventional and adjoint gradients for several
permeabilities in Figure6. In the expression for the adjoint gradients
(3.10), the
term (δ(μ−1)∇A,∇ξ) appears. For
high permeability in the shield, the coefficient δ(μ−1)=−μ−2dμ/dhp is very small
in the region of the transition zone near the shield because here the
permeability is high and μ−2 is very small.
In the region of the transition zone near the air, the relative permeability is
close to 1 and the term is much larger. The mesh refinement for sufficient
accuracy is here much more critical than for a conductive shield. As can be
observed in Figure 6 for μr=372, the
position at which the adjoint gradient crosses zero—important to find the
correct solution in optimization problems—is about 30% too low.Total
conventional and adjoint gradients as a function of the height of the passive
shield for severalμr and σ=5.9×106 S/m (no active
shield).
(a)(b)(c)Figure7 depicts the gradients as a function of the
current in an active shield coil and as a function of the position of such a
coil. For the amplitude of the current, the accuracy is comparable to the one
of classical derivatives, even for a coarse mesh. The first of the three main
reasons for the better accuracy compared to δhp of the passive
shield is that the shape is square in contrast to the shape of the passive
shield, which is a very small and high rectangle. Secondly, the term δJe is constant and
not quadratic like δ(μ−1). Thirdly, the quantities A and ξ to integrate on
the active shield coil transition domain are much more smooth than the
quantities ∇A and ∇ξ to integrate on
the passive shield transition domain. For the position, however, the difference
is larger (up to 20% as can be seen in Figure 7(b)). The reason for this is that the
calculation of the gradient requires the subtraction of integrals in two domains: the domains 2
and 4 in Figure 3 if the object is an active shield coil that moves in horizontal direction. The integrals have the same order of magnitude, causing the difference to be sensitive to numerical
noise.Total conventional and adjoint gradients as
a function (a) of the current in a coil and (b) as a function of the position
of a coil.
(a)(b)


## 5. Comparison of Optimization Techniques for the Shielding Problem

The parameters to optimize are the height of the
passive shield (between 10 and 200 mm), the horizontal position of each compensation
coil (between 0.250 and 0.900 m andri+1>ri+0.06 m), and the
real and imaginary component of the compensation current in each coil (between
−400 and +400 A, which is 10% of the excitation current). The radius of the passive shield (0.3 m) and the vertical position of the compensation coils (1.15 m) were not optimized.
For all optimizations described below, genetic algorithms are compared with the conventional
gradient method and with the gradient method that uses the adjoint system. The efficiency of
several designed shields is also compared with the efficiency of the shields in [1] for the same
induction heating application.Genetic algorithms are stochastic optimization
routines that iteratively find a global optimum by applying selection,
mutation, and recombination (cross-over) techniques [12] on a “population.” The
initial population consists of individuals whose parameters are randomly chosen
within the boundaries.The classical gradient algorithm finds the solution of
the constrained nonlinear optimization problem by using sequential quadratic
programming (SQP) [13]. The minimum is found using numerical approximations
of the Jacobian and the Hessian. Each element of the Jacobian is determined by
applying a small perturbation in one parameter and re-evaluating the cost
value. In case of m parameters to
optimize, this procedure requires m+1 evaluations if
the gradient is approximated by [F(p+ϵ)−F(p)]/ϵ or 2m+1 if the gradient
is approximated by [F(p+ϵ)−F(p−ϵ)]/(2ϵ). For gradient algorithms, the starting value is
important in order to have convergence. The height of the passive shield was
chosen randomly to be 60 mm. The starting values for the active shield
parameters were chosen taken into account the
following issues: the coils should be positioned rather close to the
axisymmetric axis, because the field lines enter the target area from that
zone. The coil with the smallest radius should have the highest currents
because the field intensity decreases with increasing r. The ratio between the real and imaginary part of the
current was chosen to be equal to(5.1)∫TA[Re(Br)]2+[Re(Bz)]2da∫TA[Im(Br)]2+[Im(Bz)]2da=6.163μT3.742μT=1.65in case of a 90-mm high passive
shield with μr=372.The
design sensitivity approach also solves a constrained nonlinear optimization
problem without, however, the need to evaluate the modelm+1 or 2m+1 times in order
to determine the gradients. The gradients are retrieved from the solution of
the adjoint system, which requires two extra solutions (one for the real part
of the source term and one for the imaginary part), however, without the need
to generate a new geometry or mesh. Similar to the classical gradient
algorithm, the starting value is important and the solution may be a local
minimum.The following
optimizations are carried out each time using the three algorithms (see Table1).Table 1
Comparison of optimization techniques concerning
calculation time (CPU) and optimal cost valueFmin for several
shielding problems.
Optimized parametersGeneticGradient conv.Gradient adjointPas.Act.CPUFminCPUFminCPUFmin1ahp (μr=1)—1h 24' 29”4.39243' 15”4.39248' 10”4.39391bhp (μr=372)—1h 00' 39”2.95626' 28”2.95629' 24”3.13762—Ir1,Ii1,r12h 19' 33”2.90239' 31”2.916611' 20”2.91943ahp (μr=372)Ir1,Ii1,r12h 19' 45”2.90778' 40”2.928511' 59”3.01543b8' 13”2.91008' 15”2.96814—Iri,Iii,ri, i=1⋯32h 48' 01”2.86351h 16' 54”3.052316' 13”2.9832(1)Optimization of the Height of the Passive Shield (1 Parameter)
In optimization 1a, the passive shield has
conductivity5.9×106 S/m and permeability μr=1. All terms in the cost function and their derivatives
can be seen in Figures 4 and 5, respectively,
as well as the position of the optimum at 83.9 mm and its cost of
4.3924. Table 1 shows that the optimum is found by the conventional gradient
method and by the genetic algorithm. The adjoint method finds a very good
approximation (80.7 mm) of the optimum and a slightly higher objective value.
This can be explained by the fact that the gradients of the two methods almost
coincide in Figure 5. Evidently, for only one parameter to optimize, the
adjoint method is slower than the conventional gradient method. The genetic
algorithm is very slow.In optimization 1b, the shield has the same
conductivity5.9×106 S/m and also a permeability of μr=372. In Figure 6, it is observed that in this case the
accuracy of the adjoint method is much lower. This explains why the optimal
height found by the adjoint method is 71.7 mm—this is where the gradient of
the adjoint method crosses zero—instead of 100.1 mm. The difference in cost,
however, is very small (see Table 1) so that the solution obtained by the
adjoint method is still an acceptable solution. Notice that the shield with
high permeability has a lower cost value than the purely conductive one. The
average field in the target area is 7.76 μT versus 12.05 μT for the
non-ferromagnetic shield: a reduction compared to the average induction in the
unshielded case (28.9 μT) of 11.4 and
7.6 decibel, respectively. The optimal height of the non-ferromagnetic shield
is lower than the optimal height of the ferromagnetic one, because it has more
losses (FP=0.73 versus 0.51) and much more disturbance of the
heating process (FW=1.53 versus 0.61).(2)Optimization of an Active Shield Consisting of Only One Coil
The complex currentIr1+jIi1 and the
horizontal position r1 are optimized
(3 parameters). A passive shield with height hp = 90 mm is
present but not optimized. In order to test the gradient algorithms, a “bad”
starting value was chosen [Ir1,Ii1,r1] = [0,0,0.40]. The genetic algorithm found a very good solution [120.9,−67.4,0.503] while the
gradient algorithms found [143.3,−88.30,0.415] (conventional)
and [157.6,−90.3,0.400] (adjoint), both
having almost the same cost and requiring almost the same CPU time. The correct
position is not found by the gradient algorithms, but their solution has a cost
that is only 0.5% higher than the one of the genetic algorithm.(3)Optimization of Both the Passive and the Active Shield with One Coil
Next to the three parameters of case 2, also the
height of the passive shield is optimized (4 parameters in total). In case 3a,
the starting value was chosen according to the description above:[hp,Ir1,Ii1,r1]=[0.060,200,−121,0.300], resulting in a slightly higher cost for the
conventional gradient algorithm compared to the genetic algorithm that found
the optimal value: [0.1038,89.38,−103.62,0.342]. In case 3b, the starting values for the active
shield are chosen equal to the optimum of case 2. This time, the gradient
algorithm finds the global minimum. Both for 3a and 3b, the gradient method
using the adjoint system ends up with a higher cost because the height of the
passive shield is chosen too low for the same reason as in case 1b. The
calculation time of both gradient algorithms has the same magnitude.(4)Optimization of an Active Shield Consisting of 3 Coils
This optimization results in 9 optimization parameters (three times
real part of the current, imaginary part of the current, and horizontal
position). The starting positions were 0.3, 0.4, and 0.6 m, the starting values
for the currents wereIr1,0=[300,−100,50] and Ii1,0=−Ir1,0/1.65. None of both gradient algorithms found the low cost
value of the genetic algorithm (2.86) or the corresponding optimum [219.69, −72.442, 89.033, 17.53, −73.202, −6.1, 0.274, 0.404, 0.891>]. For 9 variables, the approach using the adjoint
variable is much faster than the conventional algorithm. Moreover, it finds a
lower cost value although the gradients do not deviate much from the
conventional ones.When comparing
these results to the ones from [1], it is seen that the calculation time of the genetic
algorithm is much faster than for the genetic algorithm in the cited paper.
This is because the objective function in the latter paper contained one
finite-element calculation for every compensation coil considered, as well as
an inner genetic algorithm nested in the main objective function, and up to
nine compensation coils. The resulting shield had a better efficiency than the
ones presented here (with less compensation coils), but the CPU time could be
reduced from several days to a few hours. The gradient techniques solve the
problem in less than one hour, but usually, their solution
has a slightly
higher cost value.


## 6. Conclusion

The gradient algorithm using an adjoint system was
compared to a gradient method and a genetic algorithm for the design of a
passive and active shield. It is observed that the accuracy of the adjoint
variable method is good regarding the height of a conductive, non-ferromagnetic
shield and regarding the current in compensation coils. The accuracy is
somewhat less concerning the position of active shield coils. For a shield with
high permeability, the accuracy may be unacceptable. For one evaluation of the
objective function, the CPU time of the adjoint approach is about three times
longer (as it requires two extra finite-element calculations). The optimization
using the adjoint variables is slower than the conventional gradient method in
the case of less than three parameters to optimize, comparable in case of three
or four parameters, and faster in the case of more than four parameters.Algorithm 1:Gradient-adjoint algorithm.
1.i=0 and estimate
the initial value p0.2. ComputeA(pi) from the direct
problem (2.5)
setting p=pi.3. Computeξ from the dual
problem (3.7)
using A(pi) instead of A(p).4. Using explicit
expression (3.10) for the derivative, compute ∂F(pi)/∂p.5. By a line
search algorithm determine the optimal stepαi.6. Computepi+1=pi−αi(∂F(pi)/∂p).7. If convergence
is reached (controlling e.g., the differenceF(pi)−F(pi+1))then stop,
otherwise set i:=i+1 and go to step
2 of the algorithm.



---

*Source: 369125-2008-03-31.xml*