ORIGINAL ARTICLE

Two-level refined direct optimization scheme using intermediatesurrogate models for electromagnetic optimization of a switchedreluctance motor

Guillaume Crevecoeur • Ahmed Abou-Elyazied Abdallh •

Ivo Couckuyt • Luc Dupre • Tom Dhaene

Received: 2 February 2011 / Accepted: 21 June 2011 / Published online: 8 July 2011

� Springer-Verlag London Limited 2011

Abstract Electromagnetic optimization procedures require

a large number of evaluations in numerical forward

models. These computer models simulate complex prob-

lems through the use of numerical techniques, e.g. finite

elements. Hence, the evaluations need a large computa-

tional time. Two-level methods such as space mapping

have been developed that include a second model so as to

accelerate the inverse procedures. Contrary to existing two-

level methods, we propose a scheme that enables acceler-

ation when the second model is based on the initial

numerical model with coarse discretizations. This paper

validates the proposed refined direct optimization method

onto algebraic test functions. Moreover, we applied

the methodology onto the geometrical optimization of the

magnetic circuit of a switched reluctance motor. The

obtained numerical results show the efficiency of the opti-

mization algorithm with respect to the computational time

and the accuracy.

Keywords Switched reluctance motor � Optimal design �Finite elements � Geometrical optimization �Surrogate models � Kriging

1 Introduction

Electromagnetic rotating machines are indispensable in

industry. Specifically, switched reluctance motors (SRMs)

are widely used due to their simple working principle. In

order to optimally design such machines, optimal design

procedures with high-fidelity computer models, e.g. finite

element (FE) models, are commonly utilized [1–3]. The

motor under study is a 6/4 SRM, see Fig. 1, where we aim

at optimizing the geometry of the magnetic circuit of the

motor so as to obtain an average torque profile that is as

high as possible.

In a general electromagnetic optimization framework,

the electromagnetic field computations are performed by

solving the classical equations of Maxwell, specifying the

geometry, materials, and sources. Using efficient numerical

techniques, e.g. finite element, finite difference, boundary

element methods, etc. computer models of the electro-

magnetic device can be built. These computer models solve

the so-called forward problems with high solution accu-

racy. However, these computer models are generally CPU

time-consuming. So that the traditional direct optimization

approaches, which strive towards the minimization of a

predefined cost function iteratively by the use of the for-

ward model, become very time demanding, difficult, and

impractical.

In this perspective, so-called two-level optimization

methods, e.g. space mapping [4, 5], manifold mapping [6],

response and parameter mapping [7], etc., were presented.

In these two-level optimization methods, the optimization

procedure is accelerated by incorporating, next to the high-

fidelity ‘‘fine model’’, an additional low-fidelity ‘‘coarse

model’’. Indeed, these two-level optimization methods

were successfully applied onto different electromagnetic

devices. For example, the space mapping technique was

G. Crevecoeur (&) � A. A.-E. Abdallh � L. Dupre

Department of Electrical Energy, Systems and Automation,

Ghent University, Sint-Pietersnieuwstraat 41,

9000 Ghent, Belgium

e-mail: Guillaume.Crevecoeur@ugent.be

I. Couckuyt � T. Dhaene

Department of Information Technology (INTEC),

Ghent University-IBBT, Sint-Pietersnieuwstraat 41,

9000 Ghent, Belgium

123

Engineering with Computers (2012) 28:199–207

DOI 10.1007/s00366-011-0239-5

applied onto the efficient optimal design of electromagnetic

actuators [8, 9], optimal design of a SRM [10], a trans-

former [11] etc. In [8–10], the used coarse models were

mostly analytical models, i.e. lumped magnetic reluctance

network, where approximations were made with respect to

geometry, materials, and sources. However, the construc-

tion of such fast coarse models can be also time demanding

and difficult, especially when dealing with complex for-

ward models. Moreover, these coarse models have to be

sufficiently faster than the fine models; otherwise the

optimization procedure is not remarkably accelerated [12].

Therefore, a two-level optimization method based on a

relatively easier to build coarse models is needed. We

propose a novel alternative two-level optimization scheme

that enables to solve optimization problems on the fly in a

more efficient way when including a coarse model that is

directly derived from the fine numerical model with coarse

discretizations. The reason for considering such class of

coarse model in two-level procedures is because they are

much easier to build than analytical models and because

they enable better approximation of the system under study

where the physics of the model are more accurately

incorporated, e.g. for the case geometrical details are

important in the forward solution or in case the nonlinearity

of the material model is important to the forward solution.

For such class of coarse models, it is better to use a

numerical model.

This paper describes the proposed refined direct opti-

mization (RDO) scheme in detail and implements the

scheme within the widely used Nelder–Mead simplex

(NMS) method and nonlinear least squares methods. The

scheme is based on the framework presented in [13, 14]

and the two-level genetic algorithm [15], which employ

surrogate models. For details concerning the surrogate

models, we refer to [16]. Acceleration of the optimal

design is obtained by optimizing only once a surrogate

model that is based on the coarse model with coarse dis-

cretizations. This surrogate model is corrected by an

interpolation model calibrated with the fine model with fine

discretizations. In order to validate the method, we apply

the method onto algebraic test functions. In a next stage,

we apply the method for the optimal geometrical design of

a 6/4 SRM and compare the results with the space mapping

technique and the traditional direct optimization technique.

2 Two-level minimization methods

Minimization methods that include next to the initial

computer model, a second model, are so-called two-level

minimization methods. In forward modelling problems, the

initial ‘‘fine’’ model is mostly based on numerical tech-

niques such as the finite element method (FEM), finite

difference method (FDM), etc. This model has a high level

of accuracy that requires a large computational time. The

second ‘‘coarse’’ model has a lower level of fidelity and is

computationally fast. We denote the fine and coarse model

as f(x) and c(x) respectively with input parameter vector x.

Metamodels, denoted here as models that interpolate

input–output data and which do not solve the physics of the

problem, e.g. response surface models [17], Kriging mod-

els [18, 19], can act as ‘‘coarse’’ models in two-level

minimization methods [14, 20–22] and are constructed by

interpolating response data, obtained by evaluating the

Fig. 2 Refined direct optimization scheme

Dse

Dsi

Dg

Dre

tsp

Dri

trp

x.

. x

Fig. 1 Geometry of the 6/4 SRM under study. d is the air gap at

alignment condition between stator and rotor poles

200 Engineering with Computers (2012) 28:199–207

123

model fðxiÞ for a certain set of sample points xi; i ¼1; . . .;Nd in the design space, where Nd is the number of

design points. When dealing with complex problems, Nd

needs to be large in order to obtain a sufficiently accurate

metamodel. Metamodels can be used within optimization

schemes, i.e. metamodel-assisted optimization (MAO) (e.g.

[23]) and surrogate-based optimization (SBO) (e.g. [24]),

with or without additional evaluations of the fine model for

refinement of the metamodel. The efficient global optimi-

zation (EGO) algorithm [25] is an example of a minimi-

zation method that enables refinement of the metamodel

(Kriging model) during the optimization procedure by

performing additional fine model evaluations. Indeed,

EGO/expected improvement provides a balance between

exploration, i.e. enhancing the accuracy of the metamodel,

and exploitation, i.e. refining the metamodel solely in the

region of the current optimum. The main drawback when

using metamodels is that it becomes difficult to determine

an accurate metamodel when dealing with high-dimen-

sional parameters and with highly nonlinear forward

models (‘‘curse of dimensionality’’) [26]. The number of

evaluations that need to be carried out in the fine model for

building the metamodel can increase to a large extend.

A second type of coarse models can be physics-based

where assumptions are made with respect to the geometry,

sources, materials, etc. Space mapping (SM) and manifold

mapping mostly include such models. In the most basic

methods, e.g. the aggressive space mapping (ASM) algo-

rithm [5], P forward model evaluations are carried out as

well as P minimizations of the coarse model for different

objectives. P is the total number of iterations in the two-

level algorithm. The total time for minimizing the objective

function equals:

TSM ¼ PTf þ PNcTc ð1Þ

with Tf and Tc being the needed computational time for

carrying out one evaluation in the fine and coarse forward

model, respectively. Nc is the average number of

evaluations that need to be carried out for minimizing the

coarse model, given certain objective(s). If we assume that

the traditional ‘‘one-level’’ (1L) minimization method

needs Nf & Nc evaluations in the fine model, then we

can calculate the total time as T1L = Nf Tf. Acceleration of

space mapping with respect to traditional minimization

methods can be defined as:

A1 ¼T1L

TSM

� NcTf

PTf þ PNcTc: ð2Þ

This acceleration depends on the ratio s = Tf/Tc and

A1 [ 1 is obtained when Nc Tf [ PTf ? PNcTc or

s[PNc

Nc � Pð3Þ

where we can assume Nc � P so that s needs to be larger

than the number of iterations in space mapping or manifold

mapping. It is not always possible to build a sufficiently

fast coarse model, e.g. coarse models that are numerical

models with coarse discretizations.

3 Refined direct optimization (RDO) scheme

3.1 Iterative scheme

As mentioned in the previous section, existing two-level

schemes cannot accelerate the procedure when the coarse

model is not sufficiently fast or when a large number of

evaluations are needed for building a metamodel. In this

paper, we carry out only one optimization of a surrogate-

based model [P = 1 in (1)] that is iteratively refined during

the optimization itself (increased number of fine model

evaluations). This surrogate model is based on the coarse

model and tries to approximate the fine model through the

use of iteratively refined metamodels. The specific feature

of this scheme is that acceleration is possible even for

relatively small s.

The basic idea of the RDO scheme is to alter the opti-

mization of the cost Y; e.g. least-squares difference

between targets and simulations, of the fine model:

x�f ¼ arg minxYðfðxÞÞ ð4Þ

to the optimization of the cost of the surrogate model s(x):

x�s ¼ arg minxYðsðxÞÞ ð5Þ

where we want that near x�s ; sðxÞ well-approximates f(x), so

that x�s is close to x�f : Here, we use metamodels for

interpolating the coarse model response data to the fine

model response data. The relation between coarse model

response and fine model response can become less complex

and less difficult to determine. In this way, Nd can be

reduced. The surrogate model, used in the RDO scheme,

has the following form:

sðxÞ ¼ cðxÞ þ eðxÞ ð6Þ

with error function e(x) that is determined using meta-

models. Notice that (6) can also be of the following form:

s(x) = c(x)e(x). In this paper, we use the Kriging meta-

model, see e.g. [18], for building the error function. The

surrogate model s(x) is refined during the optimization

procedure by performing a limited number of fine model

evaluations. For more details concerning the use of Kriging

within the RDO scheme, see Sect. 3.2.

Notice that when using a coarse model that is based on the

fine model with coarse discretizations that we have to be

sure that the error model e(x) is not modelling the numerical

Engineering with Computers (2012) 28:199–207 201

123

noise but that due to the use of coarse discretizations, i.e.

some physics are not fully included in the fine model. For

example, the level of discretizations near an air gap in

electromagnetic devices can be modelled in a coarse way so

that the physics of the coarse model are not modelled with a

high fidelity near an air gap, i.e. neglecting fringing effects.

The proposed method has the same features as the tra-

ditional direct optimization method, i.e. start value, stop-

ping criteria, etc., where the internal parameters of the

RDO method are self-tunable. The method uses a trust-

region strategy for updating the surrogate model. An out-

line is given:

Step 1: An initial set of Ninit samples is generated by an

optimal maximin Latin hypercube design (LHD; [27])

around start value xð0Þ within the trust region radius

Dð0Þ : xð0Þi with i ¼ 1; . . .;Ninit: Evaluations are then

made in the coarse and fine model:

F ¼ f xð0Þ1

� �; . . .; f x

ðNÞ1

� �h ið7Þ

C ¼ c xð0Þ1

� �; . . .; c x

ðNÞ1

� �h i: ð8Þ

Step 2: Construction of surrogate model sð0ÞðxÞ by

determining eð0ÞðxÞ in (6) by interpolating xð0Þi with

F–C. We initialize m = 1.

Step 3: Partial run of direct minimization method (NMS,

gradient method, etc.) using surrogate model sðm�1ÞðxÞwith start value xð0Þ: Updates xðkÞ; k ¼ 1; . . .;K are

carried out, depending on the used direct optimization

method. The partial run of direct optimization method is

stopped when xðKÞ is near to the trust region boundary or

when the stopping criteria of the direct minimization

method are fulfilled. xðmÞ becomes xðKÞ:Step 4: Determine the accuracy of the surrogate model in

order to determine the new trust region DðmÞ:The accuracy

of the previous surrogate model sðm�1Þ; depends on the

fidelity of the coarse model c(x) relatively to the fine

model and on the accuracy of the error function e(x). The

accuracy is determined as follows [13]:

qðmÞ ¼Y f xðm�1Þ� �� �

� Y f xðmÞ� �� �

Y sðm�1Þ xðm�1Þð Þð Þ � Y sðm�1Þ xðmÞð Þð Þ : ð9Þ

On the basis of q(m), we determine DðmÞ; similar to [14].

Step 5: Update of surrogate model: sðmÞðxÞ or error

model eðmÞ in the region DðmÞ: A limited number of

evaluations R are carried out in the fine and coarse model

so as to refine the surrogate model in the next trust

region. We add the simulations to the datasets (7), (8).

Step 6: If the termination criteria of the direct optimi-

zation method are not satisfied, then go to step 3, and set

m = m ? 1.

The computationally demanding steps 1 and 4 in the two-

level refined direct method can be parallelized so to

improve the acceleration of the procedure. The total time

equation of the RDO scheme (without parallelization) is

theoretically:

TRDO ¼ NinitðTf þ TcÞ þ QðKTc þ RTc þ RTf Þ ð10Þ

with Q the total number of iterations in the RDO scheme.

Remark that we can assume that QK & Nc with Nc from

equation (1), i.e. the total number of iterations for

minimizing the surrogate model is close to the total

number of iterations for minimizing the coarse model. As

long as Nf [ Ninit ? QR, acceleration with respect to the

traditional method:

A2 ¼T1L

TRDO

ð11Þ

is satisfied when

s\Ninit þ Nc þ RK

Nf � Ninit � QR: ð12Þ

3.2 Kriging in RDO scheme

Kriging is a popular technique to interpolate deterministic

noise-free data [20, 28]. These Gaussian process-based

surrogate models are compact and cheap to evaluate.

Kriging is applied in the RDO scheme for approximating

e(x) in equation (6). We elaborate here in a general way the

working of the Kriging modeling where a Kriging model is

made starting from a certain model m.

Let us consider the following NKr-dimensional base

(training) set

XB;Kr ¼ fxkr;1; xkr;2; . . .; xkr;NKrg ð13Þ

and

mB;Kr ¼ fmðxkr;1Þ;mðxkr;2Þ; . . .;mðxkr;NKrÞg ð14Þ

being the associated responses in the model m. Then, the

Kriging model mKrðxÞ with input vector x, is also known as

the best linear unbiased predictor (BLUP) that can be

obtained by

mKrðxÞ ¼Maþ rðxÞW�1ðmB;Kr � FaÞ ð15Þ

M and F are Vandermonde matrices of the test point x and

the base set XB;Kr; respectively. The coefficient vector a is

determined by generalized least squares (GLS). r(x) is an

1 9 NKr vector of correlations between the point x and the

base set XB;Kr; where the i-th element is given by

riðxÞ ¼ wðx; xkr;iÞ; i ¼ 1; . . .;NKr ð16Þ

W in (15) is a NKr 9 NKr correlation matrix, where the

entries are given by Wi;j ¼ wðxkr;i; xkr;jÞ:

202 Engineering with Computers (2012) 28:199–207

123

In this work, the correlation function is chosen

Gaussian:

wðxi; xjÞ ¼ expXn

k¼1

�hkkxi;k � xj;kk !

ð17Þ

where xi,k denotes the k-th component of vector xi and n the

dimension of the input vector x. The parameters hk; k ¼1; . . .; n of the correlation function are determined by

maximum likelihood estimation (MLE). The optimization

for MLE was performed using SQPLab [29]. The regres-

sion function is chosen constant, i.e., F ¼ ½1 1. . .1�T and

M the identity matrix.

In this RDO scheme, a Kriging interpolant is built in step

2. The base set is XB;Kr ¼ fxð0Þ1 ; . . .; xð0ÞN g that needs to be

interpolated with mB;Kr ¼ ffðxð0Þ1 Þ � cðxð0Þ1 Þ; . . .; fðxð0ÞN Þ�cðxð0ÞN Þg: In step 5 of the iterative scheme, the training set is

extended with R points yielding a more refined Kriging

interpolant.

4 Optimal design of the magnetic circuit

of a switched reluctance motor

The forward problem for the optimal design of the magnetic

circuit of the SRM consists in determining the torque profile

of the SRM for a set of geometrical variables. The SRM is

excited from stator windings which are concentric coils

wound in series on diagonally opposite stator poles, see

Fig. 1. The rotor is brushless with no windings. The variable

geometrical parameters, as depicted in Fig. 1, are the width

of the stator pole tsp and the rotor pole trp, the internal

diameter of the stator yoke Ds,1 and the external diameter of

the rotor yoke Dr,0. The external diameter of the stator is

Ds,0 = 135 mm, the internal diameter of the rotor is

Dr,1 = 25 mm and the air gap width is d = 0.25 mm.

The motor can be analyzed using a magnetic equivalent

circuit [30]. For accurate prediction of the behavior of the

machine, i.e. correct simulation of the torque for different

rotor positions, numerical methods such as the Finite Ele-

ment Method (FEM) are more suitable for the accurate

simulation of the machine [10, 31]. The demand for servo-

type torque control requires the calculation of the instanta-

neous torque for each rotor angle [32]. The electromagnetic

torque can be computed through the following equation

Temðh0; I0Þ ¼o

ohWcoðh; I0Þjh¼h0

ð18Þ

for a certain given rotor angle h0 and excitation current I0.

Wco is the so-called co-energy defined as:

Wco ¼Z

Wdi ð19Þ

with W the flux linkage and i the current where the inte-

gration can be carried out in FEM directly through global

integration over the domain of the solution, see e.g. [32].

The forward problem can be solved using the following

Poisson’s equation:

r� ðl�1r� AÞ ¼ J ð20Þ

for the vector potential A and for a certain current density

J, which is related to the enforced current I0 in the

windings around the two opposite stator poles. Since the

currents J are perpendicularly oriented on the plane of

the magnetic circuit (J = Jz being the current density in

the z-direction), see Fig. 1, the magnetic induction and

field are also oriented in this plane. The vector potential has

thus a component perpendicular upon the plane of the

magnetic circuit: A = Az. The Poisson’s equation (20) can

in this way be reduced to the following equation in 2D:

r � ðl�1rAÞ ¼ �J ð21Þ

The FE calculations depend on the geometrical parameters

and on the specifications of the permeability l. For the

magnetic circuit, this permeability is nonlinear and we use

the following single-valued constitutive B - H relationship:

l ¼ B0

H0

1þ B

B0

� �m�1 !�1

ð22Þ

which is determined by the following three parameters

[H0, B0, m] and which originates from the following

equation [10, 33]:

H

H0

¼ B

B0

� �þ B

B0

� �m

: ð23Þ

The fine model consists of very fine discretizations of the

motor under study. The number of elements is approxi-

mately 250,000. The number of mesh elements in the

coarse model is approximately 10 times smaller. The tor-

que is calculated for the excitation current of I0 = 4A and

for 5 different rotor angles h0 = 25, 27.5, 30, 32.5, 35

mechanical degrees. During the optimization procedures,

the average torque is maximized for a fixed value of I0.

5 Results and discussion

5.1 RDO of algebraic test functions

In order to validate the RDO scheme, we applied the

method onto two different algebraic test functions and

compared the results with the traditional direct optimiza-

tion scheme. Firstly, the following algebraic function was

minimized:

Engineering with Computers (2012) 28:199–207 203

123

Y1ðf ðxÞÞ f ðxÞ ¼ � expð�ðx21 þ x2

2ÞÞ ð24Þ

with x�f ¼ ½0; 0�T : The coarse model is similar to the fine

model with altered output (Ac) and input (matrix Bc) space:

cðxÞ ¼ Acf ðBcxÞ ð25Þ

with optimal value x�c ¼ ½1;�1�T : Figure 3 shows the val-

ues of the iterates xðkÞ in the traditional NMS method for

minimizing the fine and the coarse model so as to obtain x�fand x�c , respectively. The figure additionally shows the

alternative path followed in the variable (design) space by

the RDO algorithm in order to achieve convergence to

x�s ¼ x�f : The internal parameters for the RDO algorithm

are the following: Ninit = 8, R = 4 with initial trust region

Dð0Þ ¼ 0:2D� where D� denotes the whole input space

region. The total number of iterations is Q = 8. The total

number of evaluations is in the fine model 40 and in the

coarse model 110.

Secondly, we applied the RDO onto the minimization of

the two-dimensional Rosenbrock test function, with fine

model:

Y2ðf ðxÞÞ ¼ 100 x2 � x21

� �2þ 1� x21

� �: ð26Þ

The coarse model is again seriously altered in input and

output space. Figure 4 compares the convergence history

of the traditional method (cost logðYðf ðxðkÞÞÞÞÞ with the

RDO method (cost logðYðsðxðkÞÞÞÞÞ in each k-th iteration.

The internal parameters are chosen as follows: Ninit = 10,

R = 5 with this time Dð0Þ ¼ D�: The trust region is reduced

during the minimization procedure. It can be observed from

Fig. 4 that the minimization procedure follows an alter-

nated path in the parameter space and that the iteratively

refined surrogate model is minimized. Near to the minimal

value of the cost function, the surrogate model sðQÞðxÞ is

close to the fine model f ðxÞ so that x�s � x�f : The value of

the trust region ratio DðkÞ

D� for each iteration is shown in

Fig. 5a and the minimal value of the cost function in the

fine model evaluated in (7) for the first iteration and in step

5 for the next iterations is shown in Fig. 5b. Convergence

is observed after Q = 8 iterations with 50 evaluations in

the fine model and 350 evaluations in the coarse model.

5.2 RDO scheme for the optimal design of a SRM

The computational time for one forward fine model is

Tf(np) = 21.2 min, for one coarse model is Tc = 8.1 min.

When using preconditioning of the fine model based on the

coarse model, i.e. solution of the fine model is obtained by

starting from the coarse model solution, then the total

computational time is Tf(wp) = 7.9 min. The superscript np

denotes that no preconditioning was performed, while the

superscript wp denotes that preconditioning was per-

formed. Preconditioning can be performed in steps 1 and 3.

The cost function that was implemented for the optimal

design calculates the maximum average torque for the rotor

angles:

−1.5 −1 −0.5 0 0.5 1 1.5−1.5

−1

−0.5

0

0.5

x1

x 2

x(k) in refined surrogate

x(k) in fine model

x(k) in coarse model

Fig. 3 Minimization of Y1 with the path followed in the parameter

space for the coarse model, fine model and surrogate model

0 20 40 60 80 100 120 140−30

−20

−10

0

10

Convergence history in fine model

Convergence historyin refined surrogate model

Fig. 4 Convergence history for the minimization of Y2 using NMS

and RDO scheme

Fig. 5 Minimization of Y2

using RDO with (a) trust region

in each iteration, (b) minimal

value of fine model evaluations

in step 5

204 Engineering with Computers (2012) 28:199–207

123

Y ¼ �Xh0

Temðh0; I0Þ ð27Þ

with the rotor angles h0 specified in Sect. 4.

The intermediate surrogate model (6) is modelled with

e(x) being a Kriging model [28]. In order to guarantee that

the numerical noise is not interpolated between the fine and

the coarse model, a co-Kriging model could be imple-

mented [34]. However, the numerical simulations showed

that a Kriging model was sufficient for obtaining a highly

accurate intermediate surrogate model.

The internal parameters taken for the RDO scheme are

the following: Ninit ¼ 24;Dð0Þ ¼ D�;R ¼ 12. The minimi-

zation in step 3 is carried out through sequential quadratic

programming (SQP) [35]. The identified optimal geomet-

rical parameters x�RDO are listed in Table 1. These optimal

parameters correspond well with the optimal parameters

x�SQP obtained using SQP with the fine model only.

Figure 6 depicts the minimal value of the fine model

evaluated in (7) and step 5. Convergence is obtained after

Q = 5 iterations. We also added the trust region radius in

this figure. Figure 7 shows the convergence history of the

surrogate model in the partial run of direct optimization

(step 3 of the RDO scheme).

The total number of fine model evaluations Nf and

coarse model evaluations in the RDO scheme and the one-

level direct optimization SQP method is given in Table 2.

The total computational time is also given where the total

time needed in RDO (np) is 1.6 times faster. We observe

that when using the preconditioned fine model evaluations

(RDO (wp)), the time needed in steps 1 and 5 can be

accelerated so that the total computational time for opti-

mization is approximately 2 times faster.

Figure 8 shows the percentual error between the fine and

coarse model kF� Ck=kFk in step 1 of the RDO algorithm.

This figure shows the error for 5 different angles of the rotor

and shows that the error highly depends on the rotor angle.

This is because the discretization highly influences the

accuracy of the computer model for angles 25–30�. When

we compare this with Fig. 9 which is the percentual error

between the fine and surrogate model kF� Ck=kFk in step 5

of the RDO scheme then we observe that the surrogate

model has a relatively good quality.

Acceleration of the RDO scheme can possibly be

achieved by evaluating the fine model mainly in the region

where the error of the Kriging model is large, as in the

EGO algorithm.

It is difficult to provide a correct quantitative compari-

son between the space mapping methodology (e.g. ASM)

Table 1 Optimal parameters of SRM machine

Parameters tsp (mm) Dsi (mm) trp (mm) Dre (mm)

x�RDO 17.1 109.4 20.12 43.79

x�SQP 17.0 109.2 19.89 44.04

1 2 3 4 50

0.2

0.4

0.6

0.8

1Fig. 6 Fine model evaluations

in each iteration

0 5 10 15 20 25 30 35 40 45 500.3

0.4

0.5

0.6

0.7

0.8

0.9

1Fig. 7 Convergence history of

the first partial minimization of

the surrogate model

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with the here developed RDO methodology since the

efficiency of both methods strongly depends on the used

coarse model. Indeed, their convergence will depend on the

quality of the coarse model relatively to the fine model and

upon the computational time of the two models, i.e. their

ratio s = Tf/Tc. Theoretical work has been carried out in

[36] with respect to the influence of quality of the coarse

model upon the convergence of space mapping-based

methods. When using a class of coarse models that are

analytically built and that approximate very coarsely the

fine model or the system under study, s will be very high

and because of their relatively poor quality, they will have

a better convergence in the space mapping method rather

than in the RDO method. However, when using a class of

coarse models that are numerically built and that approx-

imate relatively accurate the fine model or the system

under study, the ratio s will be very low and because of

their relatively good quality, they will lead to a better

convergence in the RDO scheme rather than in ASM. This

can qualitatively be concluded by comparing the time

equations associated with the acceleration A1, see (2) in the

ASM, and acceleration A2, see (11) in the RDO. It is dif-

ficult to provide a quantitative comparison since the time

equations depend on some constants (i.e. P in ASM and

R, K, Q in RDO). These constant will also depend on the

quality of the used coarse model. We do not claim here that

the RDO will always be better than the ASM for the class

of coarse models that are numerically based. This could for

example be the case if this numerical model has a poor

quality because of using too coarse discretizations. It is

then possible that the ASM and RDO have a comparable

total time for solving the optimization problem.

6 Conclusion

This paper proposes a methodology where a coarsely dis-

cretized model can be included in the direct optimization

scheme. Intermediate surrogate models, here Kriging

models, were used so as to interpolate the so-called coarse

model (coarse mesh) and the fine model (fine mesh). The

refined direct optimization scheme was applied onto alge-

braic test functions for validation of the methodology.

The methodology was also applied onto the computation-

ally demanding optimal design of a switched reluctance

motor. We observed that the optimization procedure calcu-

lates accurately the optimal parameters because good corre-

spondence is obtained with the one-level direct optimization

Table 2 Number of iterations in forward models and the total com-

putational time

Algorithm Nf Nc Total time (h)

RDO (np) 42 242 47.5

RDO (wp) 42 242 38.2

SQP 220 0 77.7

25 27.5 30 32.5 350

10

20

30

40Fig. 8 Percentual error

between fine and coarse model

in first step for initial design of

experiment

25 27.5 30 32.5 350

2

4

6

8

10

12

Rotor angle (degrees)

Fig. 9 Percentual error

between F and S for points

specified in step 5

206 Engineering with Computers (2012) 28:199–207

123

method. The proposed methodology accelerates the opti-

mization procedure compared to the direct optimization

method with a factor of two.

Acknowledgments This work was supported by the GOA project

GOA07/GOA/006 and the IAP project IAP-P6/21. Ivo Couckuyt is

funded by the Institute for the Promotion of Innovation through

Science and Technology in Flanders (IWT-Vlaanderen). Guillaume

Crevecoeur is a postdoctoral researcher of the FWO.

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