A novel scheme for material updating in source distribution optimization of magnetic devices using sensitivity analysis
ABSTRACT A novel material updating scheme, which does not require intermediate states of a material used, is presented for source distribution optimization problems. A mutation factor to determine a degree of topological change in the next design stage on the basis of a current layout accelerates the convergence of an objective function. Easy implementation and fast convergence of the scheme are verified using two MRI design problems where current and permanent magnet distributions have been optimized, respectively.
- Amethodofcomputingthesensitivityofelectromag-netic quantitiesto changes in material and sources. 3415-3418..
1752IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 5, MAY 2005
A Novel Scheme for Material Updating in Source
Distribution Optimization of Magnetic Devices
Using Sensitivity Analysis
Dong-Hun Kim?, Jan K. Sykulski?, Senior Member, IEEE, and David A. Lowther?, Member, IEEE
School of Electrical Engineering and Computer Science, Kyungpook National University, Daegu 702-701, Korea
School of Electronics and Computer Science, Electrical Power
Engineering, University of Southampton, Southampton SO17 1BJ, U.K.
Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7 Canada
A novel material updating scheme, which does not require intermediate states of a material used, is presented for source distribution
optimization problems. A mutation factor to determine a degree of topological change in the next design stage on the basis of a current
layout accelerates the convergence of an objective function. Easy implementation and fast convergence of the scheme are verified using
two MRI design problems where current and permanent magnet distributions have been optimized, respectively.
Index Terms—Adjoint variable method, design sensitivity, topology optimization.
cessfully applied to finding the optimized material distribution
(OMD) and the optimized source distribution (OSD) in elec-
tromagnetic systems –. However, it is important to ap-
preciate that there exists a difference between applications in
electromagnetic and structural TO: In electromagnetic design,
the OSD additionally deals with forcing terms of the system
structural TO. Moreover, from the point of view of the adjoint
system of the sensitivity analysis, while the permeability of the
primary system strongly distorts the adjoint field distribution,
the magnetic sources do not affect the fields because they are
replaced by air regions. This implies that the magnetic sources
the material updating scheme—should be taken when solving
source distribution problems.
This paper presents a very fast and efficient TO algorithm
for optimizing source distribution in linear magnetostatic prob-
lems. In order to make a distinction between OMD and OSD,
a unified design sensitivity formula with respect to the system
parameters of magnetic material and sources is first derived
by exploiting the adjoint variable method (AVM) and the aug-
mented Lagrangian method. Then, the effects of the magnetic
properties on sensitivity coefficients are examined from the
viewpoint of the mathematical expression and the construction
of the adjoint system. This enables a novel material updating
algorithm—which does not require intermediate states, nor
any penalty functions of a material and an objective function
used—to be applied to the OSD method. The scheme has
been successfully combined with a commercial finite element
code OPERA and the validity and efficiency of the approach
OPOLOGY OPTIMIZATION (TO), which originated in
the structural mechanics community, has also been suc-
Digital Object Identifier 10.1109/TMAG.2005.846036
have been verified using two MRI design problems involving
searching for optimized current and permanent magnet distri-
II. DERIVATION OF A UNIFIED DESIGN SENSITIVITY
The Tellegen’s theorem, or mutual energy concept, in con-
junction with AVM has been utilized to obtain analytical sensi-
tivity formulae that contain the derivative information of an ob-
jective function to changes of material properties , . How-
ever, it is recognized that the meaning and physical interpreta-
and thus difficult to understand.
In this paper, by exploiting the AVM and the augmented ob-
jective function, a new approach to TO yielding a better under-
standing of the adjoint system is proposed. First, consider an
, given by (1), which occurs when dealing
with TO of magnetostatic systems
magnetic vector potential
implicit functions of the system parameter vector
In order to deduce a design sensitivity formula and the adjoint
system equation systematically, the variational of Maxwell’s
equation—referred to as the primary system—is added to (1)
based on the augmented Lagrangian method
is a scalar function differentiable with respect to the
and, which are themselves
: The per-
vector interpreted as the adjoint variable.
is the reluctivity andmeans the Lagrange multiplier
0018-9464/$20.00 © 2005 IEEE
KIM et al.: NOVEL SCHEME FOR MATERIAL UPDATING1753
By taking the variation of both sides of (2) with respect to
of the system parameters, the augmented ob-
jective function can be developed as follows:
to the pseudo-sources of the adjoint system. For the simplicity
of (3), we can replace arbitrary variables,
on the right-hand side of (3), becomes zero because it
is relevant to , can be also set to zero as
ential of the objective function, play the roles of the magnetic
sources such as current density and permanent magnetization,
Therefore, a unified sensitivity formula applicable to TO of
magnetostatic problems is given by
where the three integrands are the sensitivity coefficients with
respect to system parameters and, respectively.
III. DISTINCTION BETWEEN OMD AND OSD
After solving the dual formulation, consisting of the primary
and the adjoint systems, the design sensitivity is easily obtained
from (5). However, a distinction should be made between OMD
and OSD type of problems. In order to explain the nature of this
difference, the mathematical expressions and effects of mate-
rial properties on the adjoint field distribution have been inves-
First, compare the sensitivity coefficients of OMD and OSD
in (5) with each other. The coefficient of OMD results from
multiplying the primary field
whereas that of OSD just relates to
and the adjoint field
. This implies that OMD
Fig. 1.Relationship between OMD, OSM, and dual system fields.
differences between OMD and OSM. (a) Presumed material distribution in the
primary system. (b) Adjoint system of OMD. (c) Adjoint system of OSD.
Effects of material properties on the adjoint field distributions showing
is more strongly coupled with the fields of the dual system than
OSD. Fig. 1 depicts the relation between OMD, OSD and the
dual system fields.
By examining the effects of material properties on the ad-
joint fields, the meaning of the relationships shown in Fig. 1
can be clarified. Fig. 2 illustrates such effects of the materials
on the adjoint fields. It is assumed that, in the primary system,
the materials occupy gray cells of rectangular design domains
at a certain design stage as in Fig. 2(a). The presence of the ma-
terials may contribute to the construction of the adjoint system
and affect the fields generated by the pseudo-sources, or not.
This depends only on the type of the design problem considered
as illustrated by fields in Fig. 2(b) and (c).
In the case of OMD, the permeability is still present in the
adjoint system, as well as in the primary one, and, thus, the
fields of both systems are strongly distorted. That causes the
increase of sensitivity coefficients over the cells already occu-
pied by materials. If the values of permeability in the gray cells
shown in Fig. 2(b) are greater by an order of magnitude than
those of the adjacent cells, the material layout cannot change
any more. In this case, the magnitude of the objective function
that indicates how far the current design is from the optimum is
not exactly reflected in the design sensitivity. This means that
an abrupt change of the permeability is liable to result in OMD
being trapped in local minima. This is the main reason why it is
essential to the OMD image process that the material is forced
to vary gradually from a void to solid state.
On the other hand, the current density and permanent mag-
netization of OSD are replaced in the adjoint system by air re-
gions as shown in Fig. 2(c). Thus, the adjoint field distribution
depends not on the relative value of distributed materials but
on the magnitude and location of the objective function eval-
uated. This means that OSD itself, unlike OMD, does not have
lots of local minima in the design space. Moreover, even if there
are abrupt changes of material values assigned to design cells,
OSD can easily converge to an optimum solution. These prop-
erties form the basis of a novel material updating algorithm for
OSD, which allows each design cell to have only one state, that
of a void or a solid.
1754IEEE TRANSACTIONS ON MAGNETICS, VOL. 41, NO. 5, MAY 2005
IV. NOVEL MATERIAL UPDATING SCHEME
According to the conventional density method, design vari-
ables can be represented as
and is the penalization factor
a solid (1) and
is set to 1 in the overall design procedure. As
a result, the scheme does not require the somewhat complicated
intermediate values of design variables to be either 0 or 1.
be initially given, the proposed iterative design process involves
the following steps.
1) Solve the primary and adjoint systems, successively.
2) Compute design sensitivity using the analytical formula
3) Calculate a mutation factor
is the maximum value of the amount of topological
Inner loop: .
. However, as already
maybe forcedtobe only avoid(0) or
and its corresponding
according to transient values of
4) Enforce each design cell
5) Repeat the above procedure until the objective function
to be 0 or 1.
In a previous paper , the authors presented a program ar-
chitecture that allowed incorporation of the continuum design
without the need to modify the source code. In this paper, after
slightmodificationstoa partoftheoptimizationmoduleand the
command file, the same program architecture has been applied
to OSD problems. The design domain to be occupied by ma-
terials should be subdivided into multiple individual regions so
that material properties can be imposed in each region defined
prior to FE mesh generation. The individual regions correspond
to design cells and a linear static OPERA-2D solver was used
A. Optimized Current Distribution
The algorithm has been applied to a MRI test device with
magnetic iron shield ,  (refer to Fig. 5). The design goal
is to find an optimal configuration of coils, carrying a current of
1 A, which achieves uniform field distribution at 9 measuring
points. The objective function is given as
Fig. 3.Convergence of the objective function versus iterations.
(b) Three iterations. (c) Five iterations. (d) Nine iterations.
Changes of current distribution during optimization. (a) One iteration.
Fig. 5.Flux distribution after optimization.
intensity computed at the th measuring point and the desired
constant value of 36.26 A/m, respectively. The algorithm was
executed under initial conditions of
After only nine iterations, field intensities within 2% of the
desired value have been obtained as demonstrated in Fig. 3.
Fig. 4 gives examples of current distributions during optimiza-
tion, whereas the flux pattern after optimization is shown in
Fig.5. It is knownthat otherexistingmaterial updating schemes
usually require more than 100 iterations to obtain the optimized
coil and flux distributions presented here.
andare the -component of the magnetic field
% of all design
KIM et al.: NOVEL SCHEME FOR MATERIAL UPDATING1755
Fig. 6.Quarter model of an open permanent magnet-type MRI.
Fig. 7.Convergence of the objective function versus iterations.
B. Optimized Permanent Magnet Distribution
Fig. 6 shows a quarter of a model of a permanent magnet as-
sembly for an MRI device where the residual flux density of the
magnet is 1.21 T . Although the actual assembly is three di-
mensional, as it has two columns, here it has been simplified
to an axi-symmetric problem. The design goal is to find an op-
timized distribution of shimming magnets over the pole piece
surface to obtain homogenous field distribution in a 30-cm di-
ameter spherical volume (DSV). The shimming magnet has the
residual flux density of 0.22 T and thickness of 3 mm and the
domain is subdivided into 120 separate regions.
The objective function and design variable are defined as
quadrilateral regions along a 90 arc at a 300-mm radius), and
is the desired value. In (6),
according to the sign of the accumulated design
during optimization in order to take into account
the direction of
. The algorithm was executed under initial
% of all design cells and
The convergence of the objective function and the shimming
magnets distribution during the optimization are shown in
Figs. 7 and 8, respectively. Fig. 9 compares the
of magnetic fields over the surface of the DSV before and after
optimization, where the uniformity of the fields is improved
four times compared with the initial design.
is the -component of the magnetic field intensity
in each cell is forced
cell: ?? . Gray cell: ?? ). (a) One iteration. (b) Three iterations. (c) Seven
iterations. (d) Ten iterations.
Changes of shimming magnet distribution during optimization. Black
Fig. 9. Comparison of field distributions before and after optimization.
source distribution problems by emphasizing differences in re-
lation to optimized material distribution formulations. The pro-
posed material updating scheme, combined with commercial fi-
nite element software, offers advantages of easy implementa-
tion and fast convergence. The results demonstrate that opti-
mized magnetic source distributions can be found without the
need for invalid material states between a void and a solid that
lead to complicated implementations and unnecessarily lengthy
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Manuscript received June 8, 2004.