Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 2, pp. 481-496, Warsaw 2017
DOI: 10.15632/jtam-pl.55.2.481

CRACK IDENTIFICATION IN PLATES USING 1-D DISCRETE

WAVELET TRANSFORM

Anna Knitter-Piątkowska, Michał Guminiak

Poznan University of Technology, Poznań, Poland

e-mail: anna.knitter-piatkowska@put.poznan.pl; michal.guminiak@put.poznan.pl

George Hloupis

Technological Educational Institute of Athens, Athens, Greece

e-mail: hloupis@teiath.gr

In the present work, the defect detection while using DiscreteWavelet Transform in rectan-
gular plate structures is investigated.The plate bending is described by using the Boundary
ElementMethodwithboundary integral equations formulated inamodified simplifiedappro-
ach.The boundary elements of a constant type in a non-singular approachare implemented.
Defects are introduced by additional edges forming slots or holes in relation to the basic
plate domain. Estimation of the defect position is performedwhile usingwavelet coefficients
of curvature and deformation signals aswell as a newly proposedmoving variance estimator.

Keywords: damage detection, crack, plates, wavelet transform, Boundary ElementMethod

1. Introduction

Damage generally can be defined as a change in the material which impairs functioning of the
structural element at a given moment or in the future. As a result, it may lead to destruction
of the element or, in the worst case, even of the whole structure. Defects usually have the form
of delaminations, cracks, local material damage due to corrosion or fatigue; they may also take
the form of voids or undesired inclusions. The issue of early detection, locating and quantifying
structural damage is one of themost important engineering problems because it is closely linked
to safety and durability of the object.
For years, many researchers have developed different methods of identifying defects with a

focus on non-destructive testing (NDT). The essence of NDT is to locate failure in the structu-
ral element without changing its properties and functionality. Further analysis and subsequent
stages of identification include: classification and severity assessment of damage, determination
of its position, forecasting of probability of an element or the whole structure destruction and
estimating the remaining potential lifetime of the structure.
Starting from the simplest NDT technique, namely visual inspection, one can mention me-

thods basing on information from e.g. acoustic emission (Rogers, 2005), X-rays (Shinoba et al.,
2004), eddy current (Gros, 1995), ultrasonography (Zhang et al., 2004),magnetic field (Lee et al.,
2004) or soft computingmethods such as artificial neural networks (Waszczyszyn and Ziemiań-
ski, 2001) and evolutionary algorithms (Burczyński et al., 2004). The traditional approach, based
on the analysis of natural frequencies (Dems and Mróz, 2001) and modal shapes of structure
vibrations (Ostachowicz andKaczmarczyk, 2001), is still used and further developed. However,
the global static or dynamic structural response is rather insensitive to localized damage.
A promising tool in structural identification is Wavelet Transform (WT) which can surpri-

singly well extract the desired detailed information from numerous data representing the global
response of a defective structure. The application ofWT for identification of cracks in structures



482 A. Knitter-Piątkowska et al.

received a great attention over the last decade. Douka et al. (2003), Quek et al. (2001), Gentile
andMessina (2003), Garstecki et al. (2004) as well as Kim andMelhem (2004) used theWT for
crack detection in beams. The effectiveness of the discrete wavelet transform (DWT) combined
with the inverse analysis for damage identification in beamswas discussed inKnitter-Piatkowska
andGarbowski (2013). In plate structures, a 2DWTused by Loutridis et al. (2005) and Rucka
and Wilde (2004); Douka et al. (2004) used 1D WT to analytically determined mode shapes
along several perpendicular lines while Chang and Chen (2004) used spatial wavelet analysis
to estimate the crack size. Damage detection while using 1D and 2D DWT of a temperature
field recorded on the surface of a plate structure was discussed in Ziopaja et al. (2011). More
complex structures such as frames were analysed by Ovanesova and Suarez (2004) and trusses
by Knitter-Piątkowska et al. (2014). The previous studies clearly demonstrate that the use of
WT in search for structural damage is a promising and developing field of investigation.
Our intention in the present work is to examine performance of DWT as amethod for crack

identification in plates. The attention is focused on defects propagated from the external edge
of a plate, which contributes new elements to the established knowledge. The position of the
crack is determined as the absolute maximum of a moving variance estimator which uses the
wavelet transformed response signals as the input. The feasibility of the proposed method is
demonstrated by means of numerical examples. The effect of added noise on the performance
of themethod is investigated via noise immunity tests. The contribution of the current study is
based on a simple fact which provides accurate results (even with high portions of added noise)
and its implementation could be computationally efficient.

2. The discrete wavelet transform – theoretical foundations

In the current study, the WT will be implemented in which, for the representation of a signal
f(t), a linear combination of wavelet functions is applied. The theory of the WT has been
presented in many publications, e.g. Meyer (1992) and Daubechies (1992). The foundations of
thewavelet transformationwill bementioned below.As amatter of fact, the continuous wavelet
transform of the signal f(t) in the time and frequency domain can be defined as

Wf(a,b)=

∞∫

−∞

f(t)ψa,b(t) dt (2.1)

where the overbar denotes the complex conjugate of the function under it. The function ψ(t) is
called thewavelet (mother) function, it belongs to theL2(R) field andmust satisfy the condition
of admissibility (Mallat, 1989) which leads to the inequality

∞∫

0

|Ψ(ω)|2

ω
dω < ∞ (2.2)

where Ψ(ω) is the Fourier transform of ψ(t), and it is defined as

Ψ(ω)=

∞∫

−∞

ψ(t)e−iωt dt (2.3)

In this case it becomes oscillatory because its average value is equal to zero. The character of
mother function may be real or complex-valued. In the considered cases, real-valued family of
wavelets is applied. The set of wavelets is obtained by scaling and translating of the function ψ,
which leads to the relation

ψa,b =
1
√
|a|
ψ
(t− b

a

)
(2.4)



Crack identification in plates using 1-D Discrete Wavelet Transform 483

where t denotes time or a space coordinate, a is the scale parameter and b the translation
parameter. The parameters a and b take real values (a,b ∈ R) and additionally a 6= 0. The
element

√
|a| is the scale factor which ensures constant wavelet energy regardless of the scale.

It means that ‖ψa,b‖= ‖ψ‖=1.
In the present numerical approach, the leading role will be taken by DWT. DWT requires

neither integration nor explicit knowledge of the scaling and wavelet function. The family of
discrete wavelet functions can be obtained on the assumption that a = 1/2j, b = k/2j and
substitution of them into Eq. (2.4). It leads to the following relation

ψj,k(t)= 2
j

2ψ(2jt−k) (2.5)

in which k and j are scale and translation parameters, respectively. The meaning of these
parameters can be clearly illustrated for the simplest Haar wavelet (Fig. 1).

Fig. 1. Haar wavelet family: (a) mother wavelet j =0, k =0, (b) wavelet with parameters j =0, k =1,
(c) j =2, k =2, (d) j =2, k =4

Themain requirement (which is the one of contributions of the current study) is the use of
a WT implementation algorithm that is computationally efficient in terms of required memory
and processing power. The classical implementation of DWT consists of a pair of finite impulse
response filters (FIR) (high-pass and low-pass) which are applied to the signal in parallel. The
pyramidalgorithmbyMallat (1989) computes the1Dconvolution basedWTatdifferent levels of
resolution. The produced coefficients are the result of a recursive convolution between the signal
and corresponding filter for preselected values of j and k. This convolution-basedDWT requires
a large number of arithmetic computations andmemory allocations leading to quite demanding
computing scheme.Towards to the direction of the computational efficient algorithmic approach,
a light-weight implementation for performing theWTcanbeadopted.This is theLiftingScheme
(Daubechies and Sweldens, 1998; Sweldens, 1996) which requires fewer computations (half of
those needed for the convolution based DWT). In general, it consists of three steps: split, lift
and scale. The basic idea is to initially compute a trivial wavelet by splitting the analyzed signal
into odd and even subsequences and then modifying these subsequences by consecutive predict
and update steps.



484 A. Knitter-Piątkowska et al.

For the selection of an appropriate wavelet, we adopt general recommendations byOvaneso-
va and Suarez (2004) for the selection of wavelets that can perform a Fast Wavelet Transform
(FWT) and satisfy symmetry and exact reconstruction. The candidate wavelets are the bior-
thogonal and theHaar. Fortunately, bothwavelet families are supported by the Lifting Scheme.
The last criterion that leads us to the choice of biorthogonal (bior.) wavelet (Cohen et al., 1992)
in the current study, is that the Haar is an irregular wavelet (while the biorthogonal is not)
(Meyer, 1992). This property has been proved significant for crack detection (Ovanesova and
Suarez, 2004).
The biorthogonal wavelets relax the assumption of a single orthogonal basis (such as the

Haar) andbelong to the family ofwavelets that are derived frombases that are semi-orthogonal,
biorthogonal or non-orthogonal. Instead, they have primary and dual scaling (ϕ,ϕ̃) andwavelet
(ψ,ψ̃) functions – a characteristic that provides more flexibility to the construction of wavelet
base.
The scaling and wavelet are related as

∫
ψ̃j,k(x)ψj′k′(x) dx =0 (2.6)

as soon as j 6= j′ or k 6= k′ and even

∫
ϕ̃0,k(x)ϕ0k′(x) dx =0 (2.7)

as soon as k 6= k′.
One wavelet (ψ̃) used for analysis of the signal s produces coefficients cj,k as below

c̃j,k =

∫
s(x)ψ̃j,k(x) dx (2.8)

while the other wavelet (ψ) is used for synthesis of the signal s from wavelet coefficients as
follows

s =
∑

j,k

c̃j,kψj,k (2.9)

Naming the biorthogonal wavelets follows the convention biorNr.Nd, where Nr is the number of
the order of the wavelet or scaling functions used for signal synthesis, while Nd is the order of
functions used for signal analysis. In our study, we adopt the bior2.2 wavelet which is presented
in Fig. 2.

3. Problem formulation and numerical analysis

The aim of this work is to detect location of a defect provided that the defect (damage) exists
in the considered plate structure. Numerical investigation is based on signal analysis of the
structural static response. The plate material is assumed as linear-elastic. The plate bending is
described and solved by the Boundary Element Method. The boundary integral equations are
derived from a non-singular approach. Rectangular plates simply-supported along the edges are
considered.Theanalysis of the structural response is conductedwith the use of signal processing
tool, namely the wavelet transformation in its discrete form. Defects in plates are modeled as
slots near the plate boundary and introduced as a set of free edges. An example of the plate
with a defected edge is illustrated in Fig. 3. The plate is loaded by a single concentrated P
force moving along the indicated line, for example 1), 2) or 3). The force P can have a static
or dynamic character. At the selected D point measured are: deflection w, angle of rotation in



Crack identification in plates using 1-D Discrete Wavelet Transform 485

Fig. 2. Biorthogonal 2.2 wavelet: (a) analysis scaling function, (b) analysis wavelet function,
(c) synthesis scaling function, (d) synthesis wavelet function

Fig. 3. Defective plate structure

arbitrary direction ϕ, curvatures κ or internal forces such as bending and twisting moments or
transverse forces as the response of the structure.

Themeasured responseparameters have a character of the influence lines in its discrete form.
The signal of the structural response defined in this way is processed while using DWT whose
basis is described in Section 2.

The plate bending is described as the Boundary Element Method in a simplified modified
approach where there is no need to introduce concentrated forces at the plate corners and
equivalent shear forces at the plate continuous edges. This approach was widely described for
static, dynamic and stability analysis in Guminiak (2007, 2014) and Guminiak and Sygulski
(2007). The boundary integral equations are derived from Betti’s theorem. For static analysis
of a plate subjected by an external distributed load q and concentrated force P, the governing
equations have the form



486 A. Knitter-Piątkowska et al.

c(x)w(x)+

∫

Γ

[
T∗n(y,x)w(y)−M

∗

ns(y,x)
dw(y)

ds
−M∗n(y,x)ϕn(y)

]
dΓ(y)

=

∫

Γ

[T̃n(y)w
∗(y,x)−Mn(y)ϕ

∗

n(y,x)] dΓ(y)+

∫

Ω

q(y)w∗(y,x) dΩ(y)+P(i)w∗(i,x)

c(x)ϕn(x)+

∫

Γ

[
Tn∗(y,x)w(y)−M

∗

ns(y,x)
dw(y)

ds
−M

∗

n(y,x)ϕn(y)
]
dΓ(y)

=

∫

Γ

[T̃n(y)w
∗(y,x)−Mn(y)ϕ

∗

n(y,x)] dΓ(y)+

∫

Ω

q(y)w∗(y,x) dΩ(y)+P(i)w∗(i,x)

(3.1)

where the fundamental solution of the biharmonic equation

∇4w∗(y,x) =
1

D
δ(y,x) (3.2)

is given as Green’s function

w∗(y,x) =
1

8πD
r2 lnr (3.3)

for a thin isotropic plate, r = |y−x|, δ is the Dirac delta, x is the source point and y – field
point, D = Eh3/[12(1−ν2)] is the plate stiffness, h – plate thickness, E and ν are the Young’s
modulus and Poisson’s ratio. The coefficient c(x) is taken as c(x) = 1 when x is located inside
the plate domain; c(x)= 0.5whenx is located on the smooth boundary and c(x) = 0whenx is
located outside the plate domain. The second boundary integral equation (2.9) can be obtained
by replacing the unit concentrated force P∗ = 1 and the unit concentrated moment M∗n = 1.
Such a replacement is equivalent to differentiation of the first boundary integral equation (3.1)1
with respect to the coordinate n at a point x belonging to the plate domain and letting this
point approach the boundary and taking n coincide with the normal to it. The expression
T̃n(y) denotes the shear force for clamped and for simply-supported edges, T̃n(y) = Vn(y) (an
equivalent shear force) on the boundary far from the corner in the case of a simply supported
edge or T̃n(y) = Rn(y) (distributed reaction force) on a small fragment of the boundary close
to the corner. As the concentrated force at the corner is used only to satisfy the differential
biharmonic equation of the thin plate, one can assume that it could be distributed along a plate
edge segment close to the corner (Guminiak and Sygulski, 2007). The relation between ϕs(y)
and the deflection is specified by a simple relation ϕs(y) = dw(y)/ds, hence the angle of rotation
in the tangent direction ϕs(y) can be evaluated using a finite difference scheme of the deflection
with two or more adjacent nodal values. In the present analysis, the employed finite difference
scheme includes the deflections of two adjacent nodes.

4. Procedure for crack identification

4.1. Wavelet selection

The fundamental question that arises before the application of wavelet analysis in any kind
of time series is about the selection of the most appropriate wavelet. In our case, a preliminary
selection has beenmade in accordance to the available selections in theMatlab Toolbox (Misiti
et al., 2000) since this is the software in which the proposedmethod is implemented. Following
the recommendations from Section 2, we have explored several biorthogonal wavelets before we
concluded that the biorthogonal 2.2 wavelet performed better.



Crack identification in plates using 1-D Discrete Wavelet Transform 487

4.2. Proposed methodology

The inspired idea behind the use of wavelets for crack identification lies in the fact that
the existence of cracks introduces discontinuities in the structural response (Ovanesova and
Suarez, 2004; Mallat, 1989; Meyer, 1992). It is not rare that these discontinuities cannot be
visually observed from response signals (i.e. amplitudes of displacement), but they can be easily
identified when these signals are projected onto the wavelet domain, especially from the detail
coefficients. Thus, our novel approach focuses on the identification and enhancement of the
information that we can derive from wavelet detail coefficients and can be directly correlated
with crack location. The original procedure for crack identification can be summarized as below:

• Calculate (or measure in the case of real experiments) the signal that can be directly
associated to the structural response. In the current study, we use curvatures κx, κy and
vertical displacement signals w.

• Compute the low level (up to scale 4) detail coefficients. This low level selection was
successfully applied in previous studies as well (Ovanesova and Suarez, 2004).

• The crack location can be estimated as the local maximum (LM) of the absolute value in
the detail coefficient.

In the case inwhich the aforementioned estimation is not clear (i.e. for noisy signals or when
detail coefficients present more than one nearby LM) apply an appropriate estimator to isolate
the LMM. In the current study, a moving variance estimator (MVE) is used as below

MVE=
1

n−1

n∑

i=1

(xi−xi)
2 (4.1)

where n is number of samples in the moving window. Small values of n provide a more rapid
detection of sharp changes (increased sensitivity) but may lead to false estimations due to
outliers (decreased selectivity). On the contrary, higher values of n are not prone to sharp peaks
or outliers but they introduce significant delays between the real and the detected peak. In our
case, n = 3 is selected as the best compromise between sensitivity and selectivity. This value
corresponds to a moving window with length around 5% of the total number of measurements
(which are N =64).
The LM from detail coefficients are projected as the maximum of the moving variance esti-

mator.

5. Numerical results

The aim of this work is to detect localization of a defect provided that the damage (crack)
exists in the considered plate structure. Numerical investigation is conducted basing on signal
analysis of the structural static response. A rectangular plate structure, simply-supported on
the boundary is considered. The Boundary Element Method is applied to solve a thin plate
bending problem. Each plate edge is divided into 30 boundary elements of the constant type.
The collocation point is located slightly outside the plate edge, which is estimated by parameter
ε = δ/d, where δ is the real distance of the collocation point from the plate edge, and d is
the element length (Guminiak, 2014). For each example, ε = 0.001 is assumed. The diagonal
boundary terms in the characteristic matrix are calculated analytically, and the rest of them
using12-pointGauss quadrature.Theplates properties are:E =205.0GPa, ν =0.3,h =0.02m.
All plates are loaded statically. A static concentrated external load P = 1000N is replaced
by an equivalent constant distributed loading q acting over the square surface of dimensions
0.05m×0.05m. Plate defects are introduced by the additional boundaries (free edges) forming



488 A. Knitter-Piątkowska et al.

a hole in relation to the basic plate domain. The static concentrated load is applied at selected
points along the direction parallel to one plate dimension.As the structural response, deflections
and curvatures are taken into account. Curvatures κx, κy can be identified using displacements
given in three points: left, central and right, where the left and right points are located in
direct vicinity of the central point. The curvature κxy can be found while using displacements
of four points located in direct vicinity of the central point. Subsequently, the curvatures can be
calculated using classic difference operators.Themeasurementpoint is located near thedamaged
area of the considered plate. The minimum number of measurements is equal to 32 (Knitter-
-Piątkowska et al., 2014). In the considered examples, a number of 64 measurements has been
applied.

5.1. Plate with middle crack

The plate loaded by a static concentrated P force is considered and presented in Fig. 4. The
coordinates of the measurement point A are: xA = 2.15m and yA = 0.25m. The introduced
plate defect is described by the parameter e =0.005m.

Fig. 4. The considered plate structure with a middle crack

The response signals that are used for the crack identification are: curvatureκx, curvatureκy
and displacement w, which are presented in Fig. 5, respectively.

For each above signal, a DWT using bior.2.2 wavelet has been performed. The obtained
results are shown in Fig. 6 for detail 1 coefficients. The detail coefficients from the remaining
scales of each transform are not presented since the significant information can be derived from
coefficients of scale 1 only. At each figure, location of the crack is depicted as a vertical dotted
line. A common pattern is unfolded for all three signals: the LMM clearly indicates the crack
position. More specificly, for signal κx, the crack position is identified from the maximum of
detail 1 coefficient while for κy and displacement signals, the crack position is identified from
the minimum. The importance of the previous observation is based on the fact that a simple
threshold detector (applied to detail 1 coefficients) is used, and as a result, we can identify
location of the crack with a significant accuracy.

A criticism to the previous findings can be raised if we assume that one of the response
signals κy (and probably other signals that are contaminated by noise), presents nearby LMM
(in form of subordinate peaks) that may affect the accuracy of threshold detection that we
propose earlier. For this reason, we applied MVE to detail 1 coefficients in order to enhance
information of the crack location and thus to enhance the accuracy of the proposed threshold
detector. Figure 7 shows the results where the crack location is presentedwith a very clear peak
of MVE. This event eliminates any uncertainty about the crack identification procedure and
provides a clear answer to the question “if there is a crack and where it is located”.



Crack identification in plates using 1-D Discrete Wavelet Transform 489

Fig. 5. Response signals for the plate with a middle crack: (a) curvature κx, (b) curvature κy and
(c) displacement w; number of measurements N =64

Fig. 6. Detail coefficients at scale 1 of (a) curvature κx, (b) curvature κy and (c) displacement signals w
of the plate with a middle crack; vertical dotted line indicates the actual location of the crack

5.2. Plate with a corner crack

The plate loaded by a static concentrated P force is considered and presented in Fig. 8. The
coordinates of the measurement point D are: xA = 3.75m and yA = 0.35m. The introduced
plate defect is described by parameters: e1 = e2 = e3 =0.005m.

The same procedure as in Section 5.1 is applied to the plate with a corner crack. Figure 9
shows the response signals where the corresponding results from DWT are shown in Fig. 10.



490 A. Knitter-Piątkowska et al.

Fig. 7. MVE estimations for detail 1 coefficients of (a) curvature κx, (b) curvature κy and
(c) displacement signals w of the plate with a middle crack; vertical dotted line indicates the actual

location of the crack

Fig. 8. Considered plate structure with a corner crack

The used wavelet is the bior.2.2, and the detail coefficients at scale 4 are taken into account.
The vertical dotted line indicates the beginning of the corner crack. The results are clearer than
in the case of the plate with themiddle crack: the LMM can be easily detected as there are no
subordinate peaks in the detail coefficients. The LMMagain successfully points out the location
of the crack. Thus, the use of MVE in this case is not presented since it will not add significant
information to the crack identification procedure.

An important note, common for both cases, is regarding the detail coefficients amplitudes.
As we can observe, the amplitude of detail coefficients is 50 times smaller than the amplitude
or the corresponding response signal. This, in turn,means that the details coefficients are prone
to add with white noise. The level of noise influence is further examined in the next Section.

5.3. Noise immunity test

Due to unavailability of experimental data, the proposedmethodhas been tested for its noise
immunity by adding to the response signals independent realizations of artificialWhiteGaussian



Crack identification in plates using 1-D Discrete Wavelet Transform 491

Fig. 9. Response signals for the plate with a corner crack (a) curvature κx, (b) curvature κy,
(c) curvature κxy, and displacement w

Fig. 10. Detail coefficients at scale 4 of: (a) curvature κx, (b) curvature κy, (c) curvature κxy and
(d) displacement w signals of plate with corner crack; vertical dotted line indicates the actual location

of crack



492 A. Knitter-Piątkowska et al.

Noise (WGN). To ensure a large scale assessment, a range of 81 Signal-to-Noise (SNR) cases
are considered (for each plate) in order to represent several measurement conditions. The range
of the examined signals begins from 100db (almost noiseless) up to 20db (quite noisy signal)
in 1db decrements. Figure 11 presents the latter case (SNR=20db) for signals from the plate
with a middle crack, whereas in Fig. 12, the correspondingMVE results are presented.

Fig. 11. The noisy case (SNR=20db) of response signals for the plate with a middle crack:
(a) curvature κx, (b) curvature κy and (c) displacement w

Fig. 12. MVE estimations for detail 1 coefficients of (a) curvature κx, (b) curvature κy and
(c) displacement signals w with SNR=20db of the plate with a middle crack; vertical dotted line

indicates the actual location of the crack

FromFig. 12, it is obvious thatMVE fails to detect the crack location for every noisy signal.
This situation is observed in the plate with the corner crack as well. Such observations have led
us to the examination of the minimum SNRmin where the proposed method can successfully
estimate the crack location. For example, with signals having SNR=45db, it is still possible to



Crack identification in plates using 1-D Discrete Wavelet Transform 493

identify the location as presented in Fig. 13. The results from the aforementioned noise tests are
shown in Fig. 14. The results from Fig. 14 indicate that there is no common SNRmin, but it is
noticeable that forSNR­ 40db theproposedmethod can estimate the crack location accurately.
In addition, the results from noise tests for the plate with the corner crack reveal that the
performanceofMVEestimations forκx andκy signals is quite remarkable.This observationmust
be expected since (as shown inFig. 9) these two specific signals present intrinsic discontinuity at
the crack location (thus the DWT can easily detect it). The results from the curvature κxy and
displacement w signals are less prominent but their SNRmin remains below 40db. The detailed
results of the performance of MVE on each signal can be found in Table 1.

Fig. 13. MVE estimations for detail 1 coefficients of (a) curvature κx, (b) curvature κy and
(c) displacement signals w with SNR=45db of the plate with a middle crack; vertical dotted line

indicates the actual location of the crack

Fig. 14.MVE errors in crack location estimation for response signals with the middle crack (a) and the
corner crack (b). The SNR is ranging from 20db to 100db (in 1db steps): curvature κx

(black solid line), curvature κy (black dashed line), displacement w (grey dotted line) and curvature κxy
(grey dashed line)

Stacking the previous results together, it is clear that the crack location identificationwill be
influenced (at least in one case) bynoise if the corresponding response signals have SNR < 40db.
This is the conservative point of view since there are response signals (the κx and κy) that can
performnearly perfect (i.e. exact identification of crack location) with SNRmin reduced down to
36db. The previous observations dictate that the proposed method is quite robust to additive



494 A. Knitter-Piątkowska et al.

Table 1.Noise immunity test results

Plate with middle crack Plate with corner crack

response signal κx κy w κx κy κxy w

SNRmin [db] 34 36 40 25 26 30 36

noise, and for this reason, further study with experimental data must take place in order to
validate the above conclusions.

6. Conclusions

The implementation of DWT for identification of signal discontinuity in the analysis of plate
structures is presented in the given paper. Bending of a thin plate (static analysis) is descri-
bed by boundary integral equations and solved using the BEM. Despite the fact, the studied
problem is two-dimensional from the point of view of deformation description. Application of
one-dimensional discrete DWT leads to satisfying results in defect detection. The proposed ap-
proach allows discovering small disturbances in the response signal of a defective structure and
does not require a reference to the signal from an undamaged structure (additional errors avo-
ided). The considered examples proved that DWT of the structural response signal expressed
in deflections or curvatures established at a selected domain point, quite correctly identifies the
presence and position of the defect. The data are gathered in one measurement point in equal
time intervals.Thedetection of defect position can easily be implemented byusing a simple thre-
shold detector applied to the corresponding DWT detail coefficients. In the case where a small
uncertainty exists, themakinguse of an additional detector (calledMovingVarianceEstimator –
MVE) is proposed.ThepurposeofMvE is to enhance the information thatmay remain “hidden”
(due to non-distinguishable peak) or “blurred” (due to subsequent peaks) near defect position,
and this is the original element in the paper. The results indicate that theMVE performs quite
satisfactory on deriving accurately the position of the defect, especially for deformation response
signals.

Since the current study is a numerical one, it is expected that there will be a contamination
of response signals by noise in real experiments. For this reason, additional research regarding
the performance of the proposedmethod over noisy signals has been carried out.More specificly,
we add independent realizations of artificialWhiteGaussianNoise to response signals generating
a dataset with SNR from 20db to 100db (in 1db increments). We have noticed that even with
the presence of noise there is a minimum SNR over which it is possible to reveal accurately the
position of the defect usingMVE. This minimum SNR has been estimated for both plates and
for every response signal.

The importance andnovelty of the current studydoes not lie only in the application ofDWT
for crack identification but goes a step further to the implementation of a real time system for
crack location. In a suchway, a prototype systemwill be highly benefited from the use of DWT
since it is computationally efficient regarding signal processing resources.Moreover, by using the
lifting approach implementation algorithm (Daubechies and Sewldens, 1998) we will be able to
adapt twomain advantages of the lifting approach against thewidely used polyphase algorithm:
doubling computation speed (Sweldens, 1996) and in-place computation of coefficients (without
allocating extra memory) (Meyer, 1992). The above properties may be mission-critical in real-
world installations where small memory buffers and very low-power microprocessors perform
necessary calculations in wireless sensor network implementation.



Crack identification in plates using 1-D Discrete Wavelet Transform 495

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Manuscript received April 26, 2016; accepted for print October 1, 2016