JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 3, pp. 445-460, Warsaw 2004

FUZZY-NEURAL AND EVOLUTIONARY COMPUTATION

IN IDENTIFICATION OF DEFECTS

Tadeusz Burczyński

Department for Strength of Materials and Computational Mechanics, Silesian University of

Technology, Gliwice, Poland

Institute of Computer Modelling, Cracow University of Technology, Kraków, Poland

e-mail: tadeusz.burczynski@polsl.pl

Piotr Orantek
Antoni Skrobol

Department for Strength of Materials and Computational Mechanics, Silesian University of

Technology, Gliwice, Poland

e-mail: piotr.orantek@polsl.pl; antoni.skrobol@polsl.pl

It is known that an elastic body contains some internal defects such as voids,
cracks, additional masses, etc. This paper is devoted to a method based
on computational intelligence for non-destructive defect identification. In
the presented paper, an elastic body loaded statically is considered. The
body contains an unknown number of internal defects. There are a lot of
applicationsbasedonnon-destructivemethods.TheEvolutionaryAlgorithm
(EA) with the Boundary ElementMethod (BEM) is a very effective tool in
the identification of internal defects. In this method, the fitness function
is calculated for each chromosome in each generation by the BEM. The
number of chromosomes in each generation is quite large, and the number of
generations is also large, so the time needed to carry out the identification
is very long.
Methods based on Artificial Neural Networks (ANN) find the position and
shape of internal defects in a very short time. BecauseANNs are usually tra-
ined using gradientmethods, the risk that the solution is in a local optimum
is one of disadvantages of such amethod. There is also a problemwhen the
ANNhas to identify two ormore different kinds of defects (cracks, voids and
additional masses) in one body.
In the presentedmethod, an EA is connected with the ANN in one system.
This operation allows to avoidmain disadvantages of these methods and to
use their advantages. The evolutionary algorithm is applied to identify the
number of defects and their parameters (position and size).



446 T.Burczyński et al.

The identification of a defect in the body is performed by minimizing the
fitness function which is calculated as a difference between measured and
computed displacements in some sensor points on the boundary of the in-
vestigated structure. The fitness function is computed using an Artificial
Neural Network (ANN).

Key words: fuzzy neural network, evolutionary algorithm, defect, identifica-
tion, boundary element method

1. Introduction

The main target of this paper is to present a computational intelligence
system in identification of defects in the form of cracks and voids in two-
dimensional elastic systems. The computational intelligence system is compo-
sedof coupledanevolutionary algorithm(EA)withanartificial neuralnetwork
(ANN) (Rutkowska, 1997). The identification process is realized on thebasis of
knowledge about displacements in some sensor points on the boundary of the
body. There are several approaches to identification problems. One group of
methods is based on sensitivity analysis (Bonnet et al., 2002). This approach,
frommathematical point of view, is very elegant and strict but sometimes fails
because theminimization of identification functions leads to a localminimum.
Another group of methods is based on techniques which try to simulate

(or imitate) biological systems. One approach which belongs to this group
concerns artificial neural networks. The ANN has been used to identification
problems by (Waszczyszyn and Ziemiański, 2001, 2003; Piątkowski and Zie-
miański, 2003; Ziemiański and Piątkowski, 2000). In such amethod there is a
problemwith the identification of a large numberof differentdefects, especially
when the number of defect is a unknown. The second very common approach
is making use of evolutionary algorithms in identification tasks (Burczyński,
2002; Burczyński et al., 2000; Nowakowski, 2000). AnEA enables to findmul-
tiple defects. It can distinguish different kinds of defects as voids and cracks,
and the number of defects canbe considered as a design variable. AnEAmini-
mizes a fitness function which is formulated as a difference betweenmeasured
displacements at sensor points xi, i=1,2, ...,n on the boundary of the inve-
stigated body and displacements computed for the assumed numerical model
with defects

min
ch

F(ch) F(ch)=
1
2

n∑

i=1

[û(xi)−u(xi)]
2 (1.1)



Fuzzy-neural and evolutionary computation... 447

where û(xi) denotes the measured displacements at the sensor point xi,
u(xi) are computed displacements for themodel in the same point xi, ch is a
vector of defect parameters which plays the role of a chromosome in the EA.

Usually, these computations need to solve a boundary-value problemusing
the boundary element method (BEM) or the finite element method (FEM) as
it is shown in Fig.1.

Fig. 1. The evolutionary identification using BEM or FEM to compute the fitness
function

This part of the identification process is very time consuming because the
fitness function has to be computed for each chromosome in every genera-
tion. The second disadvantage of such an approach is that the time needed for
solving the identification problem depends on geometry of the model (Nowa-
kowski, 2000). Themore complicated shape of the examined object the longer
time for computation is needed.

Oneway to speedup the identification process is to improve the evaluation
of the fitness function. It can be done by replacing the BEM or FEM solution
to the boundary-value problem by an approximate solution ehich is obtained
by using anANN.As a result of coupling the EAwithANN, a computational
intelligence system is obtained (Fig.2).

It can be said that the artificial neural network is an approximator of a
boundary-value problem for different kinds and positions of defects. The EA
will find the number, shapes and positions of internal defects based on the
results obtained using the ANN.



448 T.Burczyński et al.

Fig. 2. The computational intelligence system for defect identification

To approximate the boundary-value problem, a fuzzy neural network
(FNN) is chosen.

2. Fuzzy modelling

A fuzzy system is a system that uses a collection of fuzzy membership
functions and rules, instead of conventional (Boolean) logic, to reason about
data. Usually, the form of rules is following (Osowski, 1996)

IF x1 =A1 AND x2 =A2 AND xn =An THEN y=B (2.1)

where xi is the input variable, y is the output variable, Ai is the fuzzy subset
of rules premise, B is the fuzzy set of rules conclusion. The rules are collected
in one set called the rule base or knowledge base.



Fuzzy-neural and evolutionary computation... 449

A typical fuzzy system consists of four parts (Rutkowska, 1997) as it is
shown in Fig.3.

Fig. 3. A scheme of a typical fuzzy system

The inference process proceeds in three steps: fuzzification, inference and
defuzzification.

FUZZIFICATION block – the degree ofmembership of every input varia-
ble for each rule premise is determined.

INFERENCE block – the membership degrees are applied to the conclu-
sion part of each rule and the one fuzzy subset for each rule is obtained.

In the presented fuzzy system, the fuzzy subset A is calculated by the
following formula

µA(x)=µA1(x1)µA2(x2) . . .µAn(xn) (2.2)

where µA(x) is themembership function of the conclusion of the rule A
for input vector x, µAi(xi) is the degree of membership of every input
variable for A rule premise.

After that all fuzzy subsets are combined together to create one fuzzy
set.

DEFUZZIFICATION block – the output fuzzy set is converted to a crisp
number. In this paper, the centroidmethod is considered. In thismethod
the output value (the crisp value) is computed by finding the value of
the centre of gravity of the membership function of the output set

y=

M∑
l=1
clµA(l)(x)

M∑
l=1
µ
A(l)
(x)

(2.3)

where cl is the centre of the output set for the rule A
(l), µ

(l)
A
(x) is the

membership function calculated in the inference step, l=1,2, . . . ,M is
the rule number.



450 T.Burczyński et al.

Applying the above described methods an arbitrary continuous function
can be represented. Using (2.2) and (2.3) the following formula is obtained

f(x)=

M∑
l=1
cl

(N∏
i=1
µ
A
(l)
i

(xi)
)

M∑
l=1

N∏
i=1
µ
A
(l)
i

(xi)
(2.4)

where l=1,2, . . . ,M is the number of rule, I =1,2, . . . ,N is the number of
input, cl is the centre of the fuzzy output set.

3. Fuzzy neural network

To approximate the fitness value, a fuzzy neural network (FNN) is con-
sidered. The FNN should realize a multi-variable function using the sum of
single-variable fuzzy functions. These fuzzy functions are characterized by the
membership function µ(x).TheGaussiandescription of themembership func-
tion for every input in every rule is assumed

µA(x;c,σ)= exp
(
−

[
x−c

σ

]2)
(3.1)

In this case, formula (2.4) can be presented as follows

f(x)=

M∑
l=1
Wl

N∏
i=1
exp
(
−

[
xi−c

(l)
i

σ
(l)
i

]2)

M∑
l=1

N∏
i=1
exp
(
−

[
xi−c

(l)
i

σ
(l)
i

]2) (3.2)

where Wl corresponds to the centre cl in equation (2.3). In this formula, c
(l)
i

and σ
(l)
i
are centres andwidths of part ”IF” in each rule, and Wl is the centre

of part ”THEN” in each rule.
This function can be described by making use of a multi-layer structure

called the fuzzy neural network (Fig.4).
During the training process the parameters Wl, c

(l)
i
and σ

(l)
i
should be

found. In a gradient optimisation the learning process depends on the mini-
mization of the square error which can be presented as follows

E =
1
2
[f(x)−d]2 (3.3)



Fuzzy-neural and evolutionary computation... 451

Fig. 4. The scheme of the fuzzy neural network with one output

where x is the input vector, f(x) is the value approximated by the fuzzy
neuralnetworkand d is thedesirable answerof theFNNfor the inputvector x.
When the training process is carried out by making use of the gradient

method, theknowledge about the gradient vector ∇E is very important.When
the function f(x) is in the formshown in formula (3.2) and the error is defined
as it is presented in (3.3), the gradient vector ∇E has three components

∂E

∂Wl
= [f(x)−d]

yl

f2

∂E

∂c
(l)
i

=2
f(x)−d
f2

yl[Wl−f(x)]
xi− c

(l)
i

(σ
(l)
i
)2

(3.4)

∂E

∂σ
(l)
i

=2
f(x)−d
f2

yl[Wl−f(x)]
(xi− c

(l)
i
)2

(σ
(l)
i
)3

for every input i=1,2, . . . ,N and each rule l=1,2, . . . ,M.



452 T.Burczyński et al.

The change of parameters is proceeding according to the method shown
below

p(s+1)= p(s)−η
∂E

∂p
+α∆p(s−1) (3.5)

where p is the parameter put to optimisation, s – number of the iteration
step, η – learning rate value, α – momentum rate, ∆p(s− 1) – parameter
increment in the (s−1) step.

4. Formulation of an identification problem

A two dimensional elastic body with n internal defects in the form of
circular holes is considered. The EA should identify the number of defects
and their parameters based on information about displacements in m sensor
points on the boundary of the body. The unknown parameters of a defect are
coordinates of the hole centre (Xi,Yi) and its size Ri, i=1,2, . . . ,n.
The defects are specified by a chromosome

ch= [X1,Y1,R1,X2,Y2,R2, . . . ,Xi,Yi,Ri, . . . ,Xn,Yn,Rn] (4.1)

where Xi,Yi and Ri, i=1,2, . . . ,n, play the role of genes, n is the number of
a defect. The evolutionary algorithm sends the chromosome with suggestion
values of positions and radii of defects to the approximation block (Fig.2). In
the case when Ri <Rmin, the program assumes that the genes Xi, Yi,Ri are
inactive genes and

Ri =0 ∀Ri <Rmin (4.2)

Condition (4.2) controls the number of defects. The number of input values,
which are sent from the EA to the fuzzy neural networks, depends on the
number of active genes. Thus, in the approximation block there are several
fuzzy neural networks with different numbers of input neurons (Fig.5). Every
FNN is responsible for approximation of displacements on the boundaryof the
model with a different number of internal defects.
Geneswith information about the position and shape of defects are sent to

the inputs of FNNs. The number of active genes defines indirectly the number
of internal defects. Approximated displacements in several sensor points on
the boundary of the model are obtained on the outputs of FNNs. They are
sent back to EAwhere the fitness function of each chromosome is computed.



Fuzzy-neural and evolutionary computation... 453

Fig. 5. Approximation of displacements for one, two or more internal defects

5. Numerical tests

A two-dimensional elastic rectangle in the plane stress under statical load
is considered. The body contains one or two defects in the form of a circular
hole. The considered structure with one defect is presented in Fig.6a, the
structure with two defects is shown in Fig.6b. One should find the number,
position and size of internal defects. To solve the problem, an evolutionary
algorithm coupled with a fuzzy neural network is applied. The fuzzy neural
network is chosen because of its good approximation abilities (Osowski, 1996)
and the short time needed for learning (seeTable 3). The learning time of such
a network is much shorter than the time needed to learn BPNN (Burczyński
et al., 2003).
In both cases the defects are described by a chromosome with six genes

ch= [X1,Y1,R1,X2,Y2,R2] (5.1)

The evolutionary algorithmsends the chromosomewith suggestion of positions
and values of radii of two defects: Xi, Yi, Ri, i = 1,2 to the approximation
block (Fig.2). In the event when one of the values of radii is less than Rmin
this value equals zero. In such a case the three input values are sent to fuzzy
neural networks with three inputs. When both R1 and R2 are bigger than
Rmin then the input vector with six elements is sent to other fuzzy neural
networks with six inputs.



454 T.Burczyński et al.

Fig. 6. The structure with (a) one internal defect (b) two defects

Input neurons get three or six values – the radius of the hole, X and Y co-
ordinates of thehole centre for one or two defects.Thenumber of sensor points
is m=20, so the output values are twenty displacements in the OX direction
and twenty displacements in the OY direction. These are the displacements
in sensor points on the boundary of themodel with hole parameters proposed
by the chromosome. Because the fuzzy neural network with only one input
is used, in order to approximate displacements in two directions in 20 sensor
points on the boundary of the body a set of forty fuzzy neural networkswith 3
inputs and one output and the set of 40 FNN’s with 6 inputs and one output
has to be built. Each FNN is responsible for displacement approximation in
one direction in only one sensor point.
The artificial neural network was learned and tested on values obtained

bymaking use of the boundary element method for the 2D problem of elasto-
statics (Burczyński, 1995). The set of fuzzy-neural networks with three input
neurons (FNN-3) was trained by 2374 vector pairs, and for 231 pairs was veri-
fied. The set of FNNs with six input neurons (FNN-6) was trained using 5032
vector pairs, and for 184 was verified. The procedure of training by the back
propagation method with momentum was applied. The error was computed
in the following way

Er=
1
2T

T∑

t=1

U∑

u=1

(f(u)
(xt)
−d
(u)
t
) (5.2)

where T is the number of training pairs, U is the number of outputs, f(x) is
the value given by the FNN and d is the desirable answer for the input vec-
tor xt.
The error of training set (Erl) and testing set (Ert) for different fuzzy

neural network sets with different numbers of rules are given in Table 1 for



Fuzzy-neural and evolutionary computation... 455

the set of fuzzy neural networks with three inputs (FNN-3), and in Table 2
for the set of fuzzy neural networks with six inputs (FNN-6).

Table 1. Error values for FNN-3 with different number of rules in the
experimental training

FNN-3

No. of
3 5 7 10 13

rules
Erl 0.00052 0.00052 0.00052 0.00052 0.00052
Ert 0.00619 0.00586 0.00568 0.00582 0.00555
No. of

20 20 20 20 10
iterations

No. of
15 17 20 25 40

rules
Erl 0.00052 0.00053 0.00053 0.00052 0.00053
Ert 0.00557 0.00559 0.00549 0.00526 0.00513
No. of

10 10 10 10 10
iterations

Table 2. Error values for FNN-6 with different number of rules in the
experimental training

FNN-6

No. of
3 5 7 10 15

rules
Erl 0.00030 0.00030 0.00030 0.00030 0.00030
Ert 0.00640 0.00582 0.00643 0.00685 0.00619
No. of

10 10 10 10 10
iterations

No. of
17 20 25 35 40

rules
Erl 0.00030 0.00030 0.00030 0.00031 0.00030
Ert 0.00599 0.00631 0.00602 0.00580 0.00599
No. of

10 10 10 10 10
iterations

The startingparameters W , c,σwere randomvalues.BasedonTable 1and
Table 2, two fuzzy neural network architectures and two sets of the starting
parameters W , c, σ were chosen for further training. Finally, the following
fuzzy neural networks were obtained (Table 3).



456 T.Burczyński et al.

Table 3.Architecture and training parameters

FNN-3 FNN-6

Architecture

No. of inputs 3 6
No. of outputs 40 40
No. of rules per input 40 35

Training parameters

No. of iterations 9 14
Erl 0.00052 0.00030
Ert 0.00504 0.00571
Learning rate value (η) 0.2 0.2
Momentum rate (α) 0.9 0.9
Time of training [s] 35 29
The number of learning pairs 2374 5032
The number of testing pairs 231 184

The following evolutionary parameters were applied (Table 4).

Table 4.The parameters of the evolutionary algorithm

Number of chromosomes 300
Number of iterations 100
Number of design parameters 6
Probability of uniformmutation 0.25
Probability of arithmetic crossover 0.25
Probability of cloning 0.05
Selection coefficient 0.75

In this paper, only two examples are presented.The geometrical andmate-
rial parameters of the bodywith one and two defects are described inTable 5.

The actual and founddefects using theEAwithBEMare shown in Fig.7a
(one defect) and in Fig.7b (two defects). The deffects determined by the com-
putational intelligence system are presented in Fig.7c (one defect) and in
Fig.7d (two defects). It is seen that in both cases the evolutionary algorithms
have found actual numbers of defects.
The evolutionary algorithm, with fitness function values approximated by

the FNN, found the best solution in 2min. 20sec. in the case of the bodywith
two internal defects and in 2min. 25sec. for the body with one defect. In the



Fuzzy-neural and evolutionary computation... 457

Table 5. Geometrical andmaterial parameters of the examined objects

Geometrical The structure The structure

andmaterial with one with two

parameters defect defects

l [m] 4.0 4.0

h [m] 2.0 2.0

q [N/m] 3750 3750

E [MPa] 2 ·105 2 ·105

ν 0.3 0.3

X1 [m] 0.92 0.7

Y1 [m] 1.54 1.1

R1 [m] 0.16 0.07

X2 [m] – 1.55

Y2 [m] – 0.6

R2 [m] – 0.07

Rmin [m] 0.0314

Fig. 7. Actual and found defects using EAwith BEM: (a) one defect (b) two defects;
using EAwith FNN: (c) one defect, (d) two defects

case of the evolutionary algorithmwithBEMtheCPUtimewas 22min. 25sec.
and 11min. 40sec., respectively. It can be said then that the evolutionary
algorithm with the fitness function approximated by using the fuzzy neural



458 T.Burczyński et al.

network is much faster than an evolutionary algorithm with the boundary
element method (Fig.8).

Fig. 8. CPU time using EAwith BEM and EAwith FNN for a body with (a) one
defect (b) two defects

6. Conclusions

The presented tests confirm that the evolutionary algorithmwith the arti-
ficial neural network identifies the number, positions and radii of circular holes
in a 2D body under static load.
This approach is less accurate but much faster than the evolutionary al-

gorithm with the boundary element method. In the case of identification of
two internal defects, the computing time using the computational intelligen-
ce system is about 90% shorter than the computing time consumed by the
EA with BEM. The more complicated geometry of the examined body the
longer time for the identification through EA with BEM is needed. In the
proposed approach the time of computations does not depend on geometry of
the body.
The advantage of employing theFNN instead ofBPNN is themuch shorter

timeneeded for theFNNtraining (Burczyński et al., 2003).When fuzzyneural
networks are applied, there is also a possibility of containing some knowledge
about a problem before the training process (Jang et al., 1997).
The time of computation with the EA and FNN used does not take into

account the time needed to learn the FNN and the time needed to prepare
the learning and testing sets. The computational intelligence system is worth
using when the defect identification has to be done in many structures with
the same shape. In such a case the time needed to prepare the learning and
testing sets and to train the FNN is not significant.



Fuzzy-neural and evolutionary computation... 459

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460 T.Burczyński et al.

Neuronowo-rozmyte oraz ewolucyjne obliczenia w identyfikacji defektów

Streszczenie

Obiekty techniczne jako układymechaniczne zawierają różne defektywewnętrzne
takie jak pustki, pęknięcia itp.Artykuł jest poświęconynieniszczącymmetodom iden-
tyfikacji defektów opartymna inteligencji obliczeniowej. Rozważane jest ciało spręży-
ste znajdujące się pod wpływem obciążenia statycznego zawierające nieznaną liczbę
defektów wewnętrznych. Istnieje wiele nieniszczących metod identyfikacji defektów
wewnętrznych. Jedną z nich jest metoda oparta naAlgorytmachEwolucyjnych (AE)
połączonych z Metodą Elementów Brzegowych (MEB). W tej metodzie dla każdego
chromosomu w każdym pokoleniu obliczana jest za pomocą MEB funkcja przysto-
sowania. Ponieważ liczba chromosomów w epoce oraz liczba epok jest dosyć duża,
zatem czas potrzebny do przeprowadzenia identyfikacji jest znaczący.
Metody bazujące na Sztucznych Sieciach Neuronowych (SSN) identyfikują poło-

żenie oraz kształt defektówwewnętrznychwbardzokrótkim czasie. SSN są zazwyczaj
uczone z wykorzystaniemmetod gradientowych. Istnieje zatem spore ryzyko, że uzy-
skane rozwiązanieutknęłowminimum lokalnym.Wykorzystując SSNnapotykamyna
spore trudności również w przypadku identyfikacji dwóch lub więcej różnych rodza-
jówdefektów (pęknięć, pustek itp.), którewystępują jednocześnie w identyfikowanym
układzie,
W metodzie opisywanej w niniejszym artykule połączono AE oraz SSN w jeden

system. Operacja ta pozwoli ustrzec się przed głównymi wadami i uwypuklić zalety
obydwu metod. AE identyfikuje liczbę, położenie oraz wymiary defektów. Identyfi-
kacja następuje przez minimalizację funkcji przystosowania, która jest mierzona jako
różnica pomiędzy zmierzonymi i obliczonymi przemieszczeniami na brzegu modelu
obiektu w punktach kontrolnych. Funkcja przystosowania jest obliczana z wykorzy-
staniem SSN.

Manuscript received March 3, 2004; accepted for print April 26, 2004