A topology optimization of on-chip planar
inductor based on evolutional on/off method and
CMA-ES
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言語: English
出版者: Emerald Publishing Limited
公開日: 2024-08-06
キーワード (Ja):
キーワード (En): Covariance matrix adaptation
evolution strategy, planar inductor, on-chip inductor,
topology optimization.
作成者: 佐藤, 孝洋, 渡邊, 浩太
メールアドレス:
所属: 室蘭工業大学, 室蘭工業大学

URL

http://hdl.handle.net/10258/0002000239
This work is licensed under a Creative Commons
Attribution-NonCommercial 4.0 International
License.

COMPELA TOPOLOGY OPTIMIZATION OF ON-CHIP PLANAR INDUCTORBASED ON EVOLUTIONAL ON/OFF METHOD AND CMA-ES
Takahiro Sato*, Kota Watanabe** Graduate School of Engineering, Muroran Institute of Technology, Japan,
27-1 Mizumoto-cho, Muroran, Hokkaido, 050-8585, JAPANE-mail: t-sato@muroran-it.ac.jp
Abstract. This paper presents a topology optimization of on-chip planar inductor based on the normalized
Gaussian network-based evolutional on/off topology optimization method (NGnet-TO) and the covariancematrix adaptation evolution strategy (CMA-ES). This method is relatively suitable for the conductor
optimization problems in which connectivity of the shapes is strongly required because NGnet-TO caneasily find spatially smooth shapes in comparison with classical on/off-based topology optimization
methods. This paper also introduces a multi-objective NGnet-TO based on a modified CMA-ES for multiobjective problems. These methods are applied to a conductor design problem with an on-chip inductor.
From the single- and multi-objective optimization results of the on-chip inductor, it is shown that theconnected conductor geometry can be obtained. Moreover, performance of NGnet-TO for conductor design
problems is discussed.
Keywords: Covariance matrix adaptation evolution strategy, planar inductor, on-chip inductor, topology
optimization.

INTRODUCTION
Switching-mode power supply circuits are one of the important parts for various electrical and electronicsystems. In such power supply circuits, inductors are key components to determine device sizes. Because theswitching frequency of the power supply circuits has been being continued to increase in order to reduce volume
and improve efficiency, compact inductors have aggressively been developed. On-chip planar inductors are oneof the suitable design approaches to realize compact inductors [1]-[5]. For example, spiral-shaped planar inductors
have been developed [6],[7]. As for these inductor designs, in general, a basic coil geometry, that is geometricparameters of the conductor, is defined by designers in advance, and those parameters are then optimized to satisfy
various design specifications, such as inductance, size, dc-superposition current, and so on. For this reason, todevelop some innovative inductor designs, expert designers have to carefully consider various conductor
geometries; it is very difficult and time-consuming.
To automatically obtain innovative designs, topology optimization techniques [8] have been received
recent attentions and various numerical methods have been proposed. For example, some evolutional on/off
methods, which does not use differential information of objectives, have successfully been applied to magnetic
core designs of permanent magnet motors [9]-[11]. However, it is rarely reported that the evolutional topology
optimization methods are applied to conductor geometry design problems. One of this reason is that one has togenerate connected conductor shapes to form a current-flow path. This constraint is too restrictive for theevolutional optimization methods to effectively seek for better solutions, because most candidate solutions
generated during optimizations result in infeasible shapes.
In this work, we present an evolutional topology optimization of on-chip planar inductors. As an
evolutional on/off-based method, the normalized Gaussian network-based topology optimization method (NGnetTO) [9],[10],[12],[13] is employed, because this method relatively easily finds smooth and connected shapes incomparison with classical on/off-based methods. In addition, as an evolutionary algorithm, the covariance matrix
adaptation evolution strategy (CMA-ES) [14], which is one of the state-of-the-arts algorithms for single-objective
problems, is applied. It is shown that connected conductor shapes of an on-chip inductor can be optimized by the
NGnet-TO with CMA-ES. Moreover, a modified CMA-ES for multi-objective problems, called MOEA/DD-CMA[15], is newly combined with NGnet-TO, and resultant Pareto solutions are compared with the results obtained by
the single-objective optimization. From these optimization results, the performance of NGnet-TO for conductor
design problems is discussed.

1

TOPOLOGY OPTIMIZATION OF ON-CHIP INDUCTOR
On-chip Inductor for Power Supply Circuits
The on-chip planar inductors play an important role in the power supply circuits. Because of the highswitching frequency of the power supply circuits, inductance of the on-chip inductors required for the power supply
circuits is very small (under 100nH). To make whole circuit size small, the inductor size must be compact (about
under 0.1mm3). For example, spiral-shaped planar inductors have been developed [6],[7].
In this work, as a target inductor, an on-chip planar inductor with low-permeability magnetic core isconsidered. The schematic view of the target inductor is shown in Fig. 1. As a winding, a thin copper film is printed,
and its conductor film is covered by a magnetic core. Thus, relatively higher inductance can be obtained in spiteof the small volume compared with air-core inductors. However, we have to consider dc-superposition current
characteristics due to the magnetic saturation. To make the inductance and size of the inductors larger and smaller,
respectively, the planar conductor shape must effectively be designed. Hence, it would have a potential to
drastically improve the performance of the inductors through topology optimizations.
When modifying the conductor of the on-chip inductors by topology optimizations, a conductor generated
by the topology optimizations must form a current-flow path from an in-flow point to an out-flow one. If we use
the evolutional optimization approaches, it is difficult to generate connected conductor shapes, because theevolutionary algorithms mainly randomly generate candidate solutions in the early stage of the optimizations. In
other words, optimization problems of the conductor designs of the on-chip inductors are considered as constraint
satisfaction ones; it is incompatible with the evolutional methods.
On-chip inductor(our target)

Circuit board

Control ID
with switching devices

magnetic corewinding(thin conductor film)

Figure 1. Overview of on-chip planar inductro for power supply circuits.

NGnet-TO with CMA-ES [12],[13]To apply the topology optimizations to the conductor designs of the on-chip inductors, we employ theevolutional on/off method based on the normalized Gaussian network (NGnet), which is here called NGnet-TO
for simplicity. Thanks to the evolutional approach, NGnet-TO can effectively consider various objectives andconstraints which are difficult to evaluate the differentiability with respect to design variables. Moreover, theresultant shapes obtained by NGnet-TO tend to become relatively simple and smooth in comparison with classicalon/off methods, because the device shapes are generated through the spatially smooth function called NGnet.
Therefore, it is expected that NGnet-TO is suitable for the optimizations of the on-chip inductors.
As for NGnet-TO, a design region is divided into small cells, and the cell states are determined on thebasis of NGnet. The output of NGnet is obtained as a weighted summation of the normalized Gaussians:𝑁𝑦(𝒙) = ∑ 𝑤𝑖 𝑏𝑖 (𝒙),

(1)

𝑖=1

where x is an input vector, N is the number of Gaussians. In addition, bi(x) is a normalized Gaussian, and wi is aweight for i-th Gaussian. The state of a cell e is determined through y(x) as follows:
conductor
𝑦(𝒙𝑒 ) ≥ 0(2)
𝐼𝑒 = {,
magnetic core otherwise
where xe is the center of a cell e. From these equations, it is clear that the cell states are determined only by wi, if
we set appropriate Gaussians. For this reason, the topology optimizations using NGnet-TO result in the parameter
optimizations of w={wi| i=1,…,N}, and any evolutionary algorithms for real-valued parameters can be employed.
In this work, as an evolutionary algorithm to optimize w, CMA-ES is employed. The CMA-ES generates
candidate solutions using the multivariate normal distribution, and its distribution parameters are adapted on thebasis of evaluation results of candidate solutions. It has been shown in some references [12],[13] that NGnet-TO
with CMA-ES has overwhelming performance in comparison with NGnet-TO based on some conventional genetic
algorithms. As for NGnet-TO with CMA-ES, M-candidate solutions, that is wm (m=1,…M), are generated as:

(3)𝒘𝑚 = 𝒎𝑔 + 𝜎𝑔 √C𝑔 𝒛𝑚 ,where mg, σg, Cg are the mean, step size, covariance matrix at g-th iteration, respectively, which are the distribution
parameters. In addition, zm is generated from N-dimensional standard normal distribution:(4)
𝑧𝑚 ~𝑁(𝟎, I),
where I is N×N identity matrix. Then, the device shapes are determined from wm and those shapes are evaluatedby some numerical methods. On the basis of the evaluation results, the distribution parameters are updated. Please
see the references [14] for more derailed procedure of CMA-ES.
NGnet-TO with multi-objective CMA-ES
Although CMA-ES has good performance, it is only applicable to single-objective optimization problems.
Hence, some modified CMA-ES have been proposed in order to apply CMA-ES to multi-objective problems. Oneof the simplest but effective approaches is that a multi-objective problem is decomposed into several singleobjective subproblems and CMA-ES with different distribution parameters are applied to those subproblems. Suchmodified CMA-ES is called MOEA/D-CMA [16] or MOEA/DD-CMA [15]. We here combine NGnet-TO and
MOEA/DD-CMA.
To decompose a multi-objective problem, multiple objectives are scalarized on the basis of weighting
coefficient vectors as follows:𝑓(𝒇, 𝒌𝑘 ) =‖𝒌𝑘 T (𝒇 − 𝒛)‖𝒌𝑘 ‖𝒌𝑘 T (𝒇 − 𝒛)‖+ 𝜃𝑃𝐵𝐼 ‖𝒇 − (𝒛 +
)‖ ,
‖𝒌𝑘 ‖
‖𝒌𝑘 ‖2(5)where f is an objective vector, kk is a weighting coefficient vector, and z is the ideal point of f. In addition, θPBI is
an user-defined penalty parameter, which is usually set to 5. This scalarized function is called the penalty-based
boundary intersection (PBI) function. Using K-different weighting coefficient vectors, a multi-objective problem
is decomposed into K single-objective subproblems. Then, CMA-ESs with different distribution parameters areapplied to them. The candidate solutions generated by each CMA-ES are stored in a same temporal archive, andthe non-dominant sort and niching selection [17] are applied to its archive. As a result, non-dominant candidates
stored in the archive are obtained as the Pareto solutions. The basic flow of the NGnet-TO with MOEA/DD-CMA
is summarized in Fig. 2. Note here that the original MOEA/DD-CMA is partly simplified because the numericalcost of the optimization algorithm itself is sufficiently low in comparison with the that of the numerical evaluations
for the candidate solutions. The advantage of this method is that we can utilize the good search performance of
CMA-ES in multi-objective problems. However, the drawback is a heavy computational cost because the numberof candidate solutions in each algorithmic iteration simply becomes K times that of the original CMA-ES.
start
Generate K-different weighing vectors:
k1,…,kK
Initialize K-different distribution parameters(m0,1, σ0,1, C0,1),…, (m0,K, σ0,K, C0,K)
g 0, Qk
Generate M-candidate solutions:wm N(mg,k, σg,kCg,k) (m=1,…,M)

Apply non-dominant sort to Q

Generate M-device shapes based on wm

Select NP candidates from Q
based on niching selection

Evaluate f, and update z

min(f)
g g+1

Evaluate f (f, kk) of M-shapes
Update k-th distribution parameters: (mg,k, σg,k, Cg,k)Q

Q

wm for m=1,…,M

k=K

terminatecondition issatisfied
end

k k+1
otherwiseotherwise

yes

Figure 2. Flow of NGnet-TO with MOEA/DD-CMA.

NUMERICAL EXPERIMENTS
Target inductor model
An on-chip planar inductor model optimized in this work is shown in Fig. 3. A low-permeability magnetic
core with sizes of 1mm×0.5mm×0.5mm is assumed. As a magnetic material of this core, soft magnetic composite
can be assumed. At the center of the core, the in-flow and out-flow areas are set. A conductor shape between these
two areas is optimized by NGnet-TO. Note that the thickness of the conductor is assumed to be 40μm, while thewidth of the conductor on the in- and out-flow points are not fixed and follow to the optimization results. Referringsome commercial on-chip air-core inductors with similar sizes, their dc resistance is about in the hundreds of milliohms [18]. Hence, considering the fact that the target inductor has the magnetic core, it would be said that the
proposed approach is valid to a certain level, if the resultant dc resistance is less than 100mΩ.
The design region of the conductor is set on the xz-plane at the center of the core. Considering thesymmetry, a half area of the center plane is defined as a design region, which contains 2704 elements, as shownin Fig. 3. In this region, 360 Gaussians whose radius is 15.7μm are deployed as shown in Fig. 4, thus, the numberof design variables in this optimization is set to 360.
To analyze the performance of the inductor, finite element method (FEM) is applied. Because of therectangle geometry, the nonconforming voxel-FEM [19] is used to effectively discretize the region around thecores and other air region into fine and coarse elements, respectively.
xz-plane on y=0
Design region

Design region

In-flow area

480μm

480μm

Current
out-flow

480μm

zCurrent
in-flow

40μm
y

z

960μm
x

Soft magnetic composite core: μr=18

480μm

x

Figure 3. Target on-chip inductor model for optimization.

zx
Figure 4. Gaussian deployment in design region.

Single-objective optimization results
First, a single-objective optimization problem with the inductor is defined. Although various conditions
are required for on-chip inductors for the high-frequency dc/dc converters actually, e.g., ac-resistance and dcsuperposition current, we here consider the simplest design conditions, i.e., only dc resistance is considered andlinear permeability is assumed, because the main purpose of this work is to reveal whether conductor shapes canbe optimized by the evolutional topology optimization method or not. Therefore, we define the optimization
problem to realize a target inductance value with minimum dc resistance as:(6)
𝑓𝑠 = 𝑅𝑑𝑐 + 𝑐𝑁𝑐 → min. ,subject to(7)
0.99𝐿T ≤ 𝐿 ≤ 1.05𝐿T ,where Rdc is the dc resistance of the conductor from in- to out-flow area, and L is the inductance. These are
evaluated by the magnetostatic analysis based on the nonconforming voxel-FEM considering linear permeability.
In addition, LT is the target inductance. Moreover, Nc is the number of cells attributed as the conductor and
c=1/2704 is a weight, where 2704 is the number of cells in the design region. The term, cNc, is introduced to reduce
meaningless area of the conductor, because fs becomes worse if there are some meaningless conductors that do notaffect Rdc and L. Note here that, when non-connected conductor shapes are generated, those shapes are treated as
infeasible solutions. Namely, a sufficient large penalty value, which is here set to 106, is added to fs, and Rdc and L
are set to 100.
The optimization is performed over 500 algorithmic iterations under the condition that LT is set to 15nH.
The number of candidate solutions in each iteration is set to 180. The constraint condition is considered on thebasis of the total ranking-based approach [20] (please see the appendix). The other settings of CMA-ES are thesame with the defaults [14]. The optimization is performed three times with different random series. The changesof a modified objective function, that is a weighted sum of fs and residual of the constraint, are evaluated to show
the progress of the optimizations on the basis of the oracle penalty method [21]. The modified objective function
is given as
𝐿 2
(8)
) ,
𝐿T
where Ωp is a user-defined hyper-parameter, and α is a connection weight which is adaptively determined between
the balance of fs and (1−L/LT). Note that fmod is not used in the optimizations and is just used to illustrate theprogress of the optimizations considering both the objective and constraint. Figure 5 shows the changes in themodified objective function. From this result, we can see that the resultant values strongly depend on the random
series. The three resultant shapes are shown in Fig. 6, from which it can be seen that the resultant resistance issmaller than the guideline: 100mΩ. It also can be seen that these shapes are clearly different. This dependency on
the random series would be because of the difficulty in finding connected conductor shapes. When one connectedshape is found, the algorithm would tend to search for its neighborhood. Consequently, the final objective function
values are drastically changed with each optimization. In other words, there are many local optimum solutions inthis problem, and a neighborhood of the search space around one connected shape that is intensely searched for.
Moreover, it can be seen from Fig. 6 that these shapes seem to have meaningless conductor regions in which thecurrent would not flow. This fact also indicates that the algorithm cannot even find a local optimum solution,
because meaningless conductors are ideally deleted thanks to the term of cNc in (6). This would also due to theconstrain about the connectivity, because the conductor may be separated when the algorithm tries to reduce Nc.
modified objective function

𝑓mod = 𝛼|𝑓𝑠 − Ωp | + (1 − 𝛼) (1 −

5

#1
#2
4

#3

3

2
0

100

200
300
num. of iterations

400

500

Figure 5. Changes in objective function against algorithmic iterations.

(a) #1, L=14.72nH, Rdc=75.82mΩ

(b) #2, L=14.71nH, Rdc=55.61mΩ

(c) #3, L=14.71nH, Rdc=60.09mΩ

Figure 6. Resultant conductor shapes.

Multi-objective optimization results
Next, NGnet-TO with MOEA/DD-CMA is applied. Like the single-objective problem, a simple twoobjective optimization problem is defined:(9)𝑓𝑎 = −𝐿 → min. ,(10)
𝑓𝑏 = 𝑅𝑑𝑐 → min. ,subject to(11)
𝐿 > 𝐿T .
The above constraint is imposed to prevent from obtaining inductors with too small inductance. Thus, LT is here
set to 10nH. This value is a little smaller than the value used in the single-objective problem. Namely, we determine
to use 10nH in order to obtain Pareto solutions around the resultant solution obtained by the single-objective
optimization. To treat the two-objective problem by CMA-ES, it is decomposed into five subproblems using fivedifferent weighing coefficient vectors; K is set to 5. The scalarized objective functions are defines as:(12)
𝑓𝑚 = 𝑓(𝒇, 𝒌𝑚 ) + 𝑐𝑁𝑐 → min. ,where f=(fa, fb)T. The settings of MOEA/DD-CMA are summarized in Table I, where the five weighing vectors
are set in equal space from 0 to 1. The other settings are the same with the single-objective problem.
The optimization is performed over 500 iterations. The resultant Pareto solutions are plotted in Fig. 7,
which shows that almost the same results obtained by the single-objective optimization are obtained by
MOEA/DD-CMA. On the other hand, the inductance range obtained by this optimization is from 10nH to about
15nH. This result means that the inductance larger than the single-objective optimization result cannot be obtained.
This would also be because of the difficulty in finding the connected conductors. To realize larger inductance,
more complicated conductor shapes are necessary. Such complex shapes, however, easily changes into
disconnected shapes even if a limited conductor part is changed into the core. Hence, it is quite difficult for theevolutional methods to find such complex shapes. The resultant shapes for the Pareto solutions marked in Fig. 7
are shown in Fig. 8. From this figure, we can see that the resultant shapes become more complex with increasing
the inductance. These resultant shapes support the above-mentioned hypothesis that the larger inductance over
15nH is not obtained.
Through these single- and multi-objective optimizations of the on-chip inductor, we can conclude that the
conductor shapes of the inductors can be optimized on the basis of NGnet-TO. However, it also can be said thatthe shape expression method of NGnet-TO would not perfectly match to find connected-conductor shapes in orderto realize large inductance.
Table I. Hyperparameter settings of NGnet-TO with MOEA/DD-CMA.
Number of weighting vectors: K
Weighting vectors: k1,…,k5
Number of selected candidates for archive: NP

5(0.95, 0.05), (0.25, 0.75), (0.5, 0.5), (0.75, 0.25), (0.05, 0.95)
100dc-resistance (mΩ)

60
55
50
45
40
35
30
25
20
15

(b)

(a)
(c)

MOEA/DD-CMA
Single-objective #2

9

10

11

12
13
14
inductance (nH)15

16

Figure 7. Resultant Pareto solutions.

(a) L=14.22nH, Rdc=53.45mΩ

(b) L=12.11nH, Rdc=40.41mΩ

(c) Ldc=10.00nH, R=21.34mΩ

Figure 8. Resultant conductor shapes marked in Fig. 7.

CONCLUSIONSIn this work, NGnet-TO with CMA-ES has been introduced to the conductor design problems of the onchip planar inductor. Moreover, the multi-objective NGnet-TO based on MOEA/DD-CMA has also been
introduced. The single- and multi-objective optimization problems are defined and NGnet-TO has been applied tothem. The optimization results have shown that NGnet-TO is applicable to the conductor design problems that theconnectivity of the shapes is strongly required. On the other hand, it has also been strongly indicated that NGnetTO is not perfectly match to the connected-conductor shape expression, because the shapes with larger inductanceare not obtained. In other words, complicated conductor shapes are difficult to be found by the NGnet-TO.
For the future works, it is neccesary to propose some new shape expression approaches to search forcomplicated conductor shapes. It would also be important to consider some effective search methods to find andkeep connectec conductor shapes in evolutional methods.
APPENDIX
In this work, constraint conditions are considered on the basis of the total ranking-based approach [20].
This approach has invariance to arbitrary element-wise increasing transformation of objectives and constraintfunctions, to affine transformation of the search space, and to redundant constraints. As for the total ranking-based
approach, the rankings of an objective value and constraint violations of a m-th candidate are evaluated as:
𝑀

𝑟O,𝑚 = ∑ 1{𝑓𝑙<𝑓𝑚 } ,

(13)

𝑙=1
𝑀

𝑟C,𝑚 = ∑ 1{𝐺𝑙 <𝐺𝑚} ,

(14)

𝑙=1

where rO,m and rC,m are the rankings of m-th candidate about an objective function and constraint conditions,
respectively. In short, those are the number of other candidates which are inferior to the m-th candidate with respect
to the objective function and constraint conditions, respectively. Moreover, Gm is a residual that is the summationof the degree of violation about constraints. For example, the residual of (7) can be evaluated as follows when theconstraint is not satisfied:
𝐺𝑚 = (1 −

𝐿 2
) .
𝐿T

(15)

Then, the re-defined objective function of m-th candidate, which is referred to update the distribution parametersin CMA-ES, is given as:(16)
𝑓𝑚 = 𝑟O,𝑚 + 𝛼𝑟C,𝑚 ,where α is a connection weight. Although α can be determined adaptively, we here set to M/2 empirically forsimplicity.
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