spectral element modelling of wave propagation with boundary and structural discontinuity...

1. INTRODUCTIONGuided wave (GW) is stress waves propagating along apath defined by the material boundaries of a structure.When there are damages in the path, part of thepropagating wave will be reflected and part transmitted,resulting in the changes of the received waves.Recently, GW based method has been found as aneffective and efficient way to detect incipient damages(Raghavan and Cesnik 2007). Proper use of thistechnique requires good knowledge of the effects of

Advances in Structural Engineering Vol. 15 No. 5 2012 855

Spectral Element Modelling of Wave Propagation

with Boundary and Structural Discontinuity

Reflections

Ying Wang1, *, Hong Hao1, Xinqun Zhu1 and Jinping Ou2

1School of Civil and Resource Engineering, the University of Western Australia, WA 6009, Australia2Dalian University of Technology, Dalian 116024, China

Abstract: Spectral element method is very efficient in modelling high-frequencystress wave propagation because it works in the frequency domain. It does not need touse very fine meshes in order to capture high frequency wave energy as the timedomain methods do, such as finite element method. However, the conventionalspectral element method requires a throw-off element to be added to the structuralboundaries to act as a conduit for energy to transmit out of the system. This makes themethod difficult to model wave reflection at boundaries. To overcome this limitation,imaginary spectral elements are proposed in this study, which are combined with thereal structural elements to model wave reflections at structural boundaries. Theefficiency and accuracy of this proposed approach is verified by comparing the numerical simulation results with measured results of one dimensional stress wavepropagation in a steel bar. The method is also applied to model wave propagation in asteel bar with not only boundary reflection, but also reflections from single andmultiple cracks. The reflection and transmission coefficients, which are obtained fromthe discrete spring model, are adopted to quantify the discontinuities. Experimentaltests of wave propagation in a steel bar with one crack of different depths are alsocarried out. Numerical simulations and experimental results show that the proposedmethod is effective and reliable in modelling wave propagation in one-dimensionalwaveguides with reflections from boundary and structural discontinuities. Theproposed method can be applied to effectively model stress wave propagation forstructural damage detection.

Key words: spectral element, wave propagation, boundary reflection, structural discontinuity.

damage on the wave characteristics. Basically, it needsaccurate and computationally efficient modelling forwave propagation in the structures. Different numericalcomputational techniques have been developed for theanalysis of wave propagation, such as finite elementmethod (Moser et al. 1999) and spectral element method(Doyle 1997). Finite element method is normally usedto model wave propagation in the time domain (Moseret al. 1999; Wu and Chang 2006). It may provideaccurate dynamic characteristics of a structure if the

*Corresponding author. Email address: [email protected]; Fax: +61-03-5227-2028; Tel: +61-03-5227-2106.

wavelength is large as compared to the mesh size.However, the finite element solutions becomeincreasingly inaccurate and expensive as the wavefrequency increases. This is because the wavelengthsare small at high frequencies, which requires a very finefinite element mesh size for a reliable modelling. Hence,the finite element formulation for wave propagationproblems requires a large system size to capture the highfrequency waves, which are usually required fordetecting small damages in a structure. This problemcan be solved by the spectral element method (SEM)(Doyle 1997) in the frequency domain. In SEM, thegoverning partial differential equations (PDEs) aretransferred to ordinary differential equations (ODEs) inspatial dimension by using FFT. These ODEs are thensolved exactly, and used as the interpolation functionsfor the spectral element formulation. The use of exactsolution in the element formulation ensures the exactmass and stiffness distribution. It means that only onespectral element needs be used for modelling wavepropagation in a very large structure component, underthe condition that there is no damage or discontinuity init. Therefore, the required number of spectral elementsis much less in comparison to the conventional finiteelement formulation.

However, an issue comes out in the analysis ofwave propagation in structures based on theconventional SEM, for it is difficult to simulateboundary reflection. Usually, wave propagation in astructure with boundary reflection needs be modelled,especially for structural health monitoring sinceboundaries, joints and places with changinggeometries of a structure are the locations of stressconcentrations and therefore are the most likelydamage locations. Moreover, to cover a wide range ina structural health monitoring system, the actuatorand/or sensor of an active sensing system might beplaced near the structural boundaries. After excitationby actuator, waves are generated and propagate in thestructure and are reflected by the structuralboundaries. Reflected waves are sometimes combinedwith the incoming waves. This complicates therecorded wave forms and may change the wavecharacteristics. Therefore it is very important tounderstand and to accurately model the effect ofboundary reflection for reliable structural damagedetections. Unfortunately, due to the inherentlimitation of FFT used in SEM for frequency domainmodelling, a throw-off element needs be added to theoriginal SEM (Doyle 1997). This makes it difficult toinclude the boundary reflection in the SEMsimulation. Doyle (1997) gave a detailed descriptionabout the treatment in SEM to model wave reflection

from different kinds of boundaries, including elastic,fixed, free, spring, dashpot and mass. However, themethod requires that the amplitude of the incidentwave at the boundary location is known as a priori.This makes the use of the measured wave history fordetecting damages in a structure difficult becauseseparating the incoming and the reflected waves fromstructural boundaries is not possible. This is probablywhy most of the current studies reported in theliterature that use SEM to model wave propagation forstructural condition monitoring assume that thestructure is sufficiently long so that the boundaryreflection can be neglected (e.g. Palacz et al. 2005;Krawczuk et al. 2006a, b; Wang et al. 2009). Thislargely limits the applicability of using SEM to modelwave propagation in a structure for structural healthmonitoring (SHM).

Several researchers have proposed their solutions tothis problem of modelling boundary reflections. Igawaet al. (2004) introduced a new SEM by using theLaplace transform. Mitra and Gopalakrishnan (2005,2006) developed a wavelet-based spectral finiteelement for studying elastic wave propagation in 1-Dconnected waveguides. Alternatively, Kudela et al.(2007) and Rucka (2010) resort to time domain SEMproposed by Patera (1984) to simulate wavepropagation with non-periodic boundaries. However,all the aforementioned solutions need a new time-frequency transform algorithm or even a new elementformulation method, which introduce furthercomplications in SEM. In this study, an imaginaryspectral element method to model boundary reflectionwith only a minor modification of the original FFT-based SEM is suggested. The proposed imaginaryspectral elements are combined with the originalstructural elements to formulate dynamic stiffnessmatrices for simulating one-dimensional wavepropagation with boundary reflections. The accuracyof the proposed method is verified by comparing thenumerical simulation results with measured wavepropagation in experimental studies carried out on asteel bar of a finite length.

Besides developing the imaginary element to modelboundary reflection in SEM, another effort of this paper isto model wave propagation in a structure with localstructural damages or discontinuities for SHM. Normally,there are two kinds of models to describe damage in one-dimensional waveguide, namely the global and localdamage models. For a homogeneous structure, most ofglobal damages can be easily modelled by the reduction ofstructural stiffness. For local damages or discontinuities,such as cracks, the modelling methods generally fall intothree groups: local stiffness reduction, discrete spring

856 Advances in Structural Engineering Vol. 15 No. 5 2012

Spectral Element Modelling of Wave Propagation with Boundary and Structural Discontinuity Reflections

models, and complex models in two or three dimensions(Friswell and Penny 2002). Based on fracture mechanicstheory, Dimarogonas and Paipetis (1983) proposed adiscrete spring to model crack effects. Using this method,Palacz and Krawczuk (2002) presented a rod spectralfinite element with a transverse non-propagation crack andexamined the influence of crack growth on wavepropagation. Ostachowicz (2008) presented aninformative review of the principles, equations andapplications of damage modelling and elastic wavepropagation in a cracked rod, a beam and a plate. Wangand Rose (2003) used higher order plate theories todetermine wave reflections and transmissions of adamaged region in a beam. Lee et al. (2007) used thegeneralised wave approach to study wave reflection andtransmission at a discontinuity when GW propagates in anon-uniform structure. Muggleton et al. (2007) presenteda method to estimate the reflection and transmission (RT)coefficients of discontinuities in waveguides from themeasured data. These methods can be used to model wavetransmission and reflection at local discontinuities. SinceSHM analysis is an inverse problem, the relations betweenthe reflection and transmission coefficients and the crackdepth and location need be defined first as references sothat by determining the reflection and transmissioncoefficients from the recorded wave history the structuraldamages can be identified.

In this paper, the method using the RT coefficientsare adopted to model the effect of structuraldiscontinuities on GW propagation. The discrete springmodel, based on fracture mechanics, is used as abaseline to perform parametric studies to determine therelations of RT coefficients with the crack depths,locations and the number of cracks on GW wavepropagation. These relations can be used as referencesin SHM analysis to identify crack damage in a structure.Experimental tests of wave propagation on a crackedsteel bar is also carried out to verify the crack modelbased on the RT coefficients. It is found that theproposed method can effectively model GWpropagation in a steel bar with boundary anddiscontinuity reflections.

2. IMAGINARY SPECTRAL ELEMENT FORBOUNDARY REFLECTION

For an intact structure or one with global damages, thedynamic stiffness matrices can be obtained based on thedisplacement and strain continuity principle, assummarised in Appendix A. More information regardingthe SEM formulation can be found in Doyle (1997).

When boundary reflection is taken intoconsideration, incoming wave and reflected wave needbe modelled separately using SEM. As illustrated in

Figure 1, the excitation source generates waves thatpropagate in both the positive and negative x-direction.The one that propagates along the positive direction isnamed wave 1 as it reaches the sensor first. Afterpassing through the sensor, this wave hits the rightboundary and is reflected and then propagates in thenegative direction. This wave is named wave 2 when itarrives at the senor. Another wave that propagatesalong the negative direction of the x-axis is alsoreflected by the left boundary and then propagates inthe positive x-direction and reaches the sensor. Thiswave is named wave 3. Wave 3 continues to propagatein the positive x-direction and is reflected by the rightboundary. When it arrives at the sensor again, it isnamed wave 4. These wave propagation and boundaryreflections will continue until the wave energy isconsumed by the damping. Only the first four wavesare shown in the figure.

Doyle (1997) has discussed about the wave reflectionby different kinds of boundaries. However, the result isbased on the solution of a discrete point in the structure.Under this condition, only the simplest situation can besolved. Specifically, the amplitude of the incident waveat the boundary location must be known. In practice, thisneeds solving the incident wave first. Due to theinherent limitation of FFT, a throw-off element isusually added to the last spectral element in order to getthe amplitude of the incident wave, which acts as aconduit to allow the propagation of the trapped energyout of the system. In calculation, this throw-off elementacts as a transmitting boundary and eliminates the waveenergy into this element. A comparison of the resultsbetween the simplest one-element rod models with andwithout throw-off element is shown in Figure 2. Thedynamic stiffness of this model can be found in manyreferences, such as Doyle (1997). As shown, if there isno throw-off element in the model, the wave energy istrapped in the element. Although a damping ratio of1.5% is considered in the analysis, the solution makesno sense due to the trapping of the wave energy in theelement. Therefore the throw-off element need be used.However, to use the throw-off element, the boundaryreflected waves can only be modelled provided that theincoming wave is known.


Ying Wang, Hong Hao, Xinqun Zhu and Jinping Ou

Excitation location Response locationWave 1Wave 2

Wave 3Wave 4

x

Figure 1. Illustration of original wave and reflected waves

To overcome this problem, in this study, animaginary element is added to the original element tomodel the boundary reflection, as illustrated in Figure 3.The boundary can be modelled as a discontinuitybetween the original and imaginary spectral element.In reality, the amplitudes of actuated waves are sosmall that the end boundaries should be regarded asfixed ends. Because the “waveguides” before and afterreflection are the same, further simplification can bemade. In comparison with the wave propagation theoryin geometrical optics, the principle of wave reflectionis applied here to modify the conventional SEM. Thus,a wave can “propagate” forward into the imaginaryelement until it reaches the testing point in the “mirror”(therefore upside down). In the calculation, theimaginary element is combined with the originalelement and a throw-off element. Then, a solution forone reflection can be obtained using the model with animaginary element. The dynamic stiffness matrix foran imaginary element can be similarly obtained andone reflected wave can be simulated using SEM. Fourreceived waves in Figure 1 can be simulated by fourmodels illustrated in Figure 4. The whole wavepropagation and reflection process can be obtained bysuperposition of these models. Due to its manifestcharacteristics, this method can be easily extended tohigher order waveguides. However, this is out of thescope of this paper and therefore is not discussed here.

3. EXPERIMENTAL VERIFICATIONS FORTHE PROPOSED METHOD

To verify the proposed approach in modelling wavepropagation in steel bars with boundary reflections, theexperimental study is carried out on a rectangular steelbar in laboratory. In section 5.3, further experimentalstudies on the same steel bar with crack will be conducted



Original element Imaginary element

Boundary: mirror

Figure 3. Imaginary element for boundary reflection simulations

(a) Illustration

Original model

Model 2

Model 3

Model 4

Model 1

Original elementImaginary element after reflecting onceImaginary element after reflecting twiceImaginary mirror

L2 L1 L3

(b) Model 1

(c) Model 2

L1

BoundaryImaginaryelement

L1 L3 L3

(d) Model 3

BoundaryImaginaryelement

L1L2L2

(e) Model 4

Boundary Imaginaryelement

L2 L2 L1 L3 L3

0 5 10 15 20Time (ms)

Sca

led

ampl

itude

3

2

1

0

−1

−2

−3

Without throw-off elementWith throw-off element

× 10−5

Figure 2. Wave propagation time history using the models with

and without throw-off element

Figure 4. Illustration of imaginary SEM models

and the results will be compared with numerical resultsby using the method described in section 4.

3.1. Experimental SetupThe experimental system, shown in Figure 5, includestwo parts: a) the actuating part to provide the excitationor input of the system. It includes the actuator based onpiezoelectric strips and the power amplifier thatprovides the power supply of the actuator; b) the piezosensing element to measure the response. This partincludes the piezo film element and its charge amplifier.

The actuators were mounted on the surface of the steelreinforcing bar with Araldite Kit K138. The strip actuatorsfrom APC International, Ltd. were selected as actuators inthis study. The actuator includes two thin strips ofpiezoelectric ceramic that are bonded together, with thedirection of polarisation coinciding with each other andare electrically connected in parallel. When electricalinput is applied, one ceramic layer expands and the othercontracts, causing the actuator to flex. In this study, onlyone ceramic layer was applied with the electrical input sothat it would generate the longitudinal wave. NI USB-6251 was used to provide the short-time Morlet waveletfor actuating the structure by a linear power amplifier. Thefrequency and the number of cycles can be adjusted tooptimise the wave propagation along the steel bars. Thewaves were generated according to the following equation

(1)

where Aw is the amplitude, f is the centre waveletfrequency, σ is the bandwidth parameter that controlsthe shape of the basic wavelet, and ϕ is the phase.Normally, the waveform is defined in ± 3σ, and thenumber of cycles is n = 6σ f. 6σ is the time duration ofthe waveform. Because wave signals decay quickly, theinput on the actuator should be as strong as possible togenerate readable output at the sensors. A linear poweramplifier was made to amplify the signals from theoutput of NI USB 6251.

ψ π ϕσ( ) cos( )t A e ftw

t

= +−

2

22 2

The DT1 series piezo film elements from MeasurementSpecialties, Inc. were selected as the sensors. The sensorswere also glued to the steel bars with Araldite Kit K138.Signals from sensors were collected by a data acquisitionsystem made with NI PCI-6133. The sampling frequencyof the system is up to 2 MHz. A program based onLabview was developed to control the NI USB-6251 andmake the two NI PCI-6133 cards work simultaneously.

3.2. Experimental Verification for WavePropagation with Boundary Reflection

The wave propagation tests were carried out on arectangular steel bar with a 1.500 m length and0.025*0.025 m cross section, shown in Figure 6. Thematerial parameters of the steel bar are: Young’smodulus 206 GPa, and mass density 7850 kg/m3. Theactuator and sensor were mounted on the steel bar at0.250 m from the left and right ends of the bar,respectively. The distance between the actuator and thesensor is 1.000 m. In the test, the frequency ofthe generated waves is 40 kHz and the number of the



Scaled m

odel

Linear poweramplifier

Excitationsignal

Embedded actuator

Embeddedsensors:

piezoelectric filmelements

Chargeamplifier

Responsesignal

NI 6133:A/D

NI USB 6251:D/A

Computersystem

Figure 5. Experimental system scheme

Actuator

1.000 m

Sensor

0.250 m

0.250 m Crack location

(a) Plot

(b) Picture: overview

(c) Picture: crack

0.250 m

Figure 6. Test specimen

wave cycles is 5. Other excitation frequencies between30 kHz and 60 kHz were also used in the tests. Similarresults, which are not shown here, are obtained. Thesampling frequency for data record is 2 MHz. Figure 7shows the results from three repeated tests on the wavepropagation along the steel bar. As shown, the responsesfrom the repeated tests are almost identical to each otherwhen the input signal is the same, indicating the systemis reliable and stable to carry out the experimental studyfor wave propagation. There are mainly five waves in therecorded time history. Compared with the excitationsignal, the time intervals between the excitation signaland the five waves are 0.1985 ms, 0.2975 ms, 0.3945 ms,0.4560 ms and 0.6115 ms, respectively. Wave A1 is theincident wave from the excitation source, and Wave A2is the combination of the reflected waves from the rightand left ends of the steel bar as the propagation lengthsfor these two reflected waves are the same. Wave A3 isthe reflected waves first by the left and then the right end.The wave propagation distances of these three wavesfrom the actuator are 1.000 m, 1.500 m and 2.000 m,respectively. So the wave speeds from these waves areestimated to be 5038 m/s, 5042 m/s and 5070 m/s,respectively. They are approximately the same as thelongitudinal wave propagation speed in the steel mediumof 5048 m/s. By examining the arrival time, it is foundthat wave X1 and X2 are transverse waves. The wavepropagation distance of these two waves are 1.500 m and2.000 m, respectively, and the corresponding wave speedis 3289 m/s and 3271 m/s. In this study, the actuator onlygenerates longitudinal waves, and there is no incidenttransverse wave is recorded in the experimental results.Thus, it can be concluded that the transverse waves areinduced by mode conversion after boundary reflections.In practice, these waves can be easily identified for they

keep nearly unchanged under different damagescenarios, as will be discussed in Section 5.3. As shownin Figure 7, the other recorded waves are reflected wavesfrom the boundary after waves propagating in the steelbar back and forth a few times. They arrive at the sensorat a further delayed time and can be easily separatedfrom the first three waves. It should be noted that in mostprevious studies (Krawczuk et al. 2006a, b; Palacz et al.2005; Wang et al. 2009), only the incident (first) waveand the damage induced waves are modelled. Theboundary reflected and transverse waves are notconsidered. However, as demonstrated in theexperimental results, the incident wave may be difficultto be separated from the first two boundary-reflectedwaves when the actuator/sensor is placed close to thestructural boundary as in this experimental study. In fact,it is always desirable to place the sensors close to thestructural boundary in structural damage detection tohave a wider coverage of the sensing domain. Therefore,modelling boundary reflection is very important.

The proposed approach with imaginary spectralelements is used to model longitudinal wave propagationin the steel bar. The steel bar is simulated by four models,including one original model and three models withimaginary elements to model boundary reflection, asshown in Figures 4(b) to (e). Here, L1 is the distancebetween actuator and sensor. L2 is the distance betweenleft end and actuator and L3 is the distance between sensorand right end. Figure 8 shows the comparison betweenthe experimental and numerical results, where the wavesA1, A2 and A3 and further boundary reflected wavesobtained from numerical simulation are very close to theexperimental results. The only discrepancy is that themode conversion is not included in numerical simulation.It demonstrates that the proposed approach is effective



2

1

0

−1

−2

Time (ms)

0.20.10 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.1

Am

plitu

de

(a) Excitation

1st2nd3rd

1st2nd3rd

Wave A1 Wave A3Wave X1

Wave X2

Wave A2

(b) Response

Other reflected wave

1

0.5

0

−0.5

−1

Time (ms)

0.20.10 0.3 0.4 0.5 0.6 0.7 0.8 0.9−0.1

Am

plitu

de

Figure 7. Wave propagation along the bare steel bar

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Time (ms)

6

4

2

0

−2

−4

−6

Am

plitu

de

×10−6

Experimental result

Numerical simulation

Wave A1

Wave A2

Wave A3Wave X1Wave X2

Figure 8. Comparison between numerical and experimental results

for wave propagation in a steel bar

and accurate in modelling the longitudinal wavepropagation and boundary reflections in steel bars. Itshould be noted that, in the following numericalsimulations, only the incident longitudinal wave and firsttwo reflected waves are modelled. This simplification isjustifiable if the method is used for structural conditionmonitoring because the transverse waves (waves X1 andX2) and the further reflected waves can be easilyidentified and separated from the first arrived longitudinalwaves, and separated from the crack induced waves sincethese waves always appear before the further boundaryreflected waves.

4. SPECTRAL ELEMENT MODELLINGFOR A STRUCTURE WITH LOCALDISCONTINUITIES

4.1. RT Coefficient MethodWhen there is a discontinuity in a waveguide, a criticalstep in the formulation of SEM is to obtain thecorresponding element boundary matrix, namely EqnA2 in Appendix A. As shown in Figure 9, therelationship of the wave amplitudes at the discontinuitycan be expressed as (Muggleton et al. 2007)

(2)

where B and C are the amplitudes of the forward andbackward waves respectively; subscripts a and b denotedifferent locations; R and T are matrices of RTcoefficients. The relationship between two elementsbefore and after the discontinuity can be written as:

where Fa = e�jkLa, Fb = e�jkLb, defined as the propagationmatrix in Lee et al. (2007).

(3)

C

B

R T

T R

F

F

a

b

aa ab

ba bb

a

b

=

+

−

0

0

=

+ −

+ −

B

C

R F T F

T F R F

a

b

aa a ab b

ba a bb b

B

Ca

b

C

B

R T

T R

B

Ca

b

aa ab

ba bb

a

b

=

′

′

According to the reversibility, Tab = Tba = t and Raa =Rbb = r. For a conservative discontinuity, the magnitudesof the RT coefficients at a discontinuity should have

(4)

It should be noted that RT coefficient method is atechnique that allows for an easy simulation of structuraldiscontinuities. Its main shortcoming is that in practice,the relationship between the RT coefficients and thedamage severity, such as crack depth, is usually unknownand needs be determined by test results on theexperimental models with different crack depths or fromtheoretical derivations. In this study, parametric analysis iscarried out to quantify the RT coefficient with respect todifferent crack depths. In order to consider both boundaryreflections and discontinuity reflections, a model shown inFigure 10 is proposed. Here, L1 and L2 are the distancebetween actuator and discontinuity and that betweendiscontinuity and sensor. L3 and L4 are the lengths of twoimaginary elements, respectively. The displacement andforce boundary conditions are summarised in Appendix B.

r t2 2 1+ =



Element a Element b

Discontinuity

LaCa

C ′bLb

Ba BbB ′a Cb

Figure 9. Elements with discontinuity

L1 L2

Discontinuity

(a) Model 1

L1 L2 L3 L3

Discontinuity

Discontinuity

Imaginaryelement

(b) Model 2

(c) Model 3

(d) Model 4

ImaginaryelementBoundary

Boundary

L4 L4 L1 L2

Boundary DiscontinuityImaginaryelement

L4 L4 L1 L2 L3L3

Figure 10. Spectral element model with both boundary reflections

and discontinuity reflections

4.2. Discrete Spring ModelIn this study, discrete spring model, based on fracturemechanics theory (Dimarogonas and Paipetis 1983) isemployed to determine the r and t coefficients withrespect to the crack depth. It has been successfullyapplied to spectral element modelling of a cracked rodwithout boundary reflections (Krawczuk et al. 2006b).

In this model, the displacement boundary conditionfor wave propagation in a cracked one-dimensionalwaveguide can be described as:

The crack is simulated as a dimensionless spring withflexibility θ, which can be derived by using the fracturemechanics approach, detailed in the following:

(6)

where (7)

in which E is elastic modulus of the material; α is aparameter for integration; a is the crack depth; b and hare the width and the height of the steel bar, respectively.The parameters of the steel bar at the crack location areillustrated in Figure 11. is a

correction function with regard to different crack types.Given by Tada et al. (1973), the correction function FCfor crack in a rectangular cross section is:

aa

h hFC= =; ; (*)α

α

fEb

FCa

= ∫2 2

0α α α( )d

θ = Ebhf

(5)

ˆ ˆ

ˆ

u L L u Li pp i

N

i pp

i

+= + =

−

−

=

∂

∑ ∑11 1

θ

uu L

x

u L L

x

i pp

i

i pp i

N

=

+= +

∑

∑

∂

∂ −

∂

1

11

ˆ

−−

∂

∂=

=∑u L

x

i pp

i

10

(8)

5. WAVE PROPAGATION IN A STRUCTUREWITH BOUNDARY AND DISCONTINUITYREFLECTIONS: PARAMETRIC STUDIESAND EXPERIMENTAL VERIFICATIONS

5.1. Steel Bar with One Crack: Parametric StudyIn this section, a 1.500 m long steel bar is investigated indetail. The material parameters of the steel bar are:Young’s modulus 210 GPa, and mass density 7850 kg/m3.The actuator and sensor were mounted on each steel bar at0.250 m from the two ends of the bar, respectively. Thedistance between the actuator and the sensor is 1.000 m.The cross section of the steel bar is 0.025*0.025 m. Thefrequency of the transmitted waves is 50 kHz and thenumber of the wave cycles is 5.

5.1.1. RT coefficient methodBased on the proposed model shown in Figure 10,numerical simulations are conducted for wavepropagation in the steel rod with a crack at differentlocations and with different RT coefficients. In thecalculation, the crack location (L1) is assumed at0.100 m to 0.900 m from the actuator, with a 0.100 mincrement. In the calculation, the reflection coefficientr is taken as 0.001, 0.05 and then 0.1 to 0.8, with anincrement of 0.1.

To begin with, the results are compared with those ofthe conventional spectral element method. Figures 12(a)and (b) show some of the received waves with andwithout boundary reflections, respectively, for thecases when the crack is at the midpoint of the bar withdifferent reflection coefficients. In Figure 12(a), WaveA1 is the first incident wave from the actuator directly.Wave A2 consists of the waves reflected from the leftand the right ends. These two waves are mixedtogether as one single wave due to the same pathlength. Wave A3 is the third wave reflected from boththe left and right ends. The illustrative wavepropagation process is shown in Figure 1. Waves Band C are induced owing to the reflections at the cracklocation in the steel bar. When the crack depthincreases, the amplitudes of Waves A1, A2 and A3change only insignificantly, while those of Waves Band C increase obviously. As for the arrival time, thereis no apparent change in all the received waves. Asshown, boundary reflections generate more waves andmake the recorded time series more complex. It should

FC( )tan( / )

/

. . . [ sin(

απα

πα

α

=

×+ + −

2

2

0 752 2 02 0 37 1 ππαπα

/ )]

cos( / )

2

2

3



h

a

α

dα

b

Figure 11. Cross section of the steel bar with crack

be noted that Waves D and E also appear in Figure 12.However, these waves may combine with Waves A1,A2 and A3 when the crack location varies. Further, theinformation provided by Waves A1, B and/or C isenough for damage identification. Therefore, in thisstudy, only this information is adopted, while Waves Dand E are not considered.

Figure 13 shows the received waves with andwithout boundary reflection for the cases when r is 0.2and crack is located at 0.2 m, 0.5 m and 0.8 mrespectively. It can be found that the arrival time of thereflected waves by crack varies with the crack location.However, the amplitude ratio remains a constant whenr is the same. If boundary reflection is taken intoconsideration, received wave time history becomesmore complex.

From the above analysis, the amplitude ratiobetween waves B and A1 (B/A1) can be used as anindicator of damage severity, such as crack depth. Intheory, this amplitude ratio is independent of the cracklocation as it is only related to the amount of wave

energies that are transmitted and reflected by a crack.However, minor differences between the amplitude ratioswhen crack locations vary are observed. Figures 14(a)and (b) show the relationships between the averageamplitude ratio and the reflection coefficient r withrespect to the different crack locations with andwithout boundary reflections, respectively. Therelations can be perfectly fitted to a linear function.The fitted curve agrees well with the average results.As shown, when the crack location is taken intoconsideration, the coefficient of variation (COV) ofamplitude ratios with respect to the different cracklocations is about 1.9–2.4% when boundary reflectionis not considered. In contrast, when boundaryreflection is considered, COV for amplitude ratiosvaries between 2–12% at different crack locations (thebiggest one is when r = 0.8). The results indicate theinfluence of the boundary reflections. The wavesreflected by the crack and by the boundary mayinterfere with each other, and this interference changesthe defined amplitude ratio. If no boundary reflection,



(a) With boundary reflections

(b) Without boundary reflections

r = 0.2r = 0.4r = 0.6r = 0.8

Time (ms)0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1

Reflected waves from crackWave A1

Sca

led

ampl

itude

Sca

led

ampl

itude

8

6

4

2

0

−2

−4

−6

−8

× 10−6

Time (ms)0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−1.5

−1

−0.5

0

0.5

1.5

1

× 10−5

Wave A1Wave C

Wave A2 and A3(reflected waves from boundaries)

Wave DWave E

Wave B

r = 0.2r = 0.4r = 0.6r = 0.8

Figure 12. Numerical results based on different reflection

coefficients (Crack location: 0.5 m)


Time (ms)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

8

6

4

2

0

−2

−1.5

−1

−0.5

0

0.5

1

1.5

−4

−6

−8

Sca

led

ampl

itude

Sca

led

ampl

itude

×10−6


Time (ms)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

×10−5

Crack location: 0.2 mCrack location: 0.5 mCrack location: 0.8 m

Crack location: 0.2 mCrack location: 0.5 mCrack location: 0.8 m

Figure 13. Numerical results for different crack locations

(Reflection coefficient: 0.2)

the amplitude ratio is independent of the cracklocation, as expected. The COV of 1.9–2.4% might beattributed to numerical errors. These observationsindicate again the importance of considering the effectof boundary reflections. Since the received wavesbecome more complex when considering boundaryreflections, model updating method, besides directcomparison of the recorded waves, needs be used toidentify and quantify the damage. Therefore anefficient numerical model that accurately simulateswave propagation is essential.

In practice, when the defined amplitude ratio isused to quantify the crack depth, the crack locationshould also be taken as a parameter because of theinevitable boundary reflections. Given a structuralmaterial, the wave speed is usually known. Then thetime difference between Waves A1 and B can be usedto determine the crack location in damage detection.The crack depth can be determined by matching thenumerically simulated waves with the measured datato determine the reflection coefficient r. The

correlation between the reflection coefficient r and thecrack depth can be pre-determined by conducting aseries of laboratory tests, or numerical analyses ofwave propagation based on the discrete spring modeland RT coefficient approach as discussed below.

5.1.2. Discrete spring modelIn this section, the model described in section 4.2 isemployed to determine the r and t coefficients withrespect to the crack depth. The crack depth Cr is takenas 1.0 mm, 3.0 mm, 6.0 mm, 9.0 mm and 12.5 mmrespectively; which correspond to 4%, 12%, 24%,36% and 50% of the depth of the example steel barcross section. Under this condition, θ is calculated andequal to 5.55 × 10−5, 6.08 × 10−4, 3.32 × 10−3, 1.06 ×10−2 and 3.27 × 10−2 respectively. It should be notedthat these crack depths are selected here todemonstrate the effect of crack on stress wavepropagation. The developed method has no limitationin modelling large or small cracks by choosing anappropriate parameter.

Figure 15(a) shows the received waves with boundaryreflections when the crack is located at0.500 m from the actuator. Figure 15(b) shows thedetailed waveform for crack induced waves when thecrack is 1 mm. Figure 15(c) shows the received wavewithout boundary reflections under the same conditionas that in Figure 15(a). From the figure, the amplitudesof the first three received waves decrease when crackdepth increases, and the amplitudes of the reflectedwaves caused by the crack increase correspondingly. Ifthere is no boundary reflection, the received wavesconsist of only one main wave plus several reflectedwaves from the crack.

Figures 16(a) and (b) show the received waves withand without boundary reflections when the crackdepths are 0.2 mm and are located at 0.200 m, 0.500 mand 0.800 m from the actuator respectively. From theresults, it is obvious that the time difference betweenWave A1 and Waves reflected by crack is dependenton the crack location, while the amplitudes of thereflected waves from the crack are related to the crackdepth. These observations are consistent with theresults using RT coefficient method. However, theamplitude ratio varies more significantly, as comparedto the results obtained above. The relation between thecrack depth and the average amplitude ratio is given inFigure 17, fitted to two power functions. Figure 17(a)is based on the model with consideration of boundaryreflections. The COV for the data at different locationis very big, varies from 29.6% to 59.2%. Figure 17(b)is based on that without considering boundary



0 0.2 0.4 0.6 0.8 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

−0.1

0

Relationship between reflection coefficient and amplitude ratio

Reflection coefficient

(a) With boundary reflection

Am

plitu

de r

atio

: B/A

1

y = 0.9921x − 0.0064R 2 = 0.9995

y = 1.0065x − 0.0056R 2 = 0.9998

0 0.2 0.4 0.6 0.8 1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

−0.1

0

Relationship between reflection coefficient and amplitude ratio

Reflection coefficient

(b) Without boundary reflection

Am

plitu

de r

atio

: B/A

1

Crack location: 0.1 mCrack location: 0.4 mCrack location: 0.7 mAverageLinear (Average)

Crack location: 0.2 mCrack location: 0.5 mCrack location: 0.8 mAverageLinear (Average)

Figure 14. Relationship between amplitude ratio and r value

reflection. The COV is also big, with the maximumvalue of about 32.1% when the crack depth is 1 mm.For cracks deeper than 3 mm, the COVs becomesmaller (below 5%). These results indicate that thefitted function works well when crack depth is bigger.It should be noted that although there is a large COVof the RT value obtained in this study when crack is

small, indicating small crack depth might not beaccurately identified, the method is reliable to identifydamage existence from the crack reflected waves eventhe crack depth is small. If more accurateidentification results are needed, higher actuatingfrequency and more sophisticate testing equipmentswill be required, which will be impracticable for largecivil structures.

Based on the results with boundary reflections,when the crack depth is 4%, 12%, 24%, 36% and 50%of the cross section, the amplitude ratios betweenwaves B and A1 are estimated to be 0.0013, 0.0166,0.0976, 0.3030 and 0.6757. Based on the equationshown in Figure 14(a), the corresponding r values arecalculated as 0.0077, 0.0232, 0.1048, 0.3119 and0.6875. Figure 18 shows the relation between thecrack depth and r value. Based on this relation, thecrack effect on wave propagation can be simulatedwith the RT coefficient method. Figure 19 shows thesimulated stress waves when there is a 9.0 mm crackat different locations in the steel bar by two methods.



1.5

1

0.5

0

−0.5

−1

−1.5

−6

−4

−2

0

2

4

6

8

Sca

led

ampl

itude

×10−5

0.7 0.75 0.8 0.90.85 0.95 1

0.30.2 0.4 0.5 0.6 0.7 0.8 0.9 1Time (ms)


Wave D Wave E Wave B Wave C

Wave A2 and A3(Reflected waves from boundaries)

Wave A1

Crack depth: 1.0 mm

Crack depth: 1.0 mm

Crack depth: 3.0 mmCrack depth: 6.0 mmCrack depth: 9.0 mmCrack depth: 12.5 mm

Crack depth: 1.0 mmCrack depth: 3.0 mmCrack depth: 6.0 mmCrack depth: 9.0 mmCrack depth: 12.5 mm

Wave B

Wave C

Time (ms)

(b) Detailed crack induced waves for small crack

Sca

led

ampl

itude

×10−9

−6

−8

−4

−2

0

2

4

6

8

Sca

led

ampl

itude

×10−6

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time (ms)

(c) Without boundary reflections

Waves reflected from crackWave A1

Figure 15. Numerical simulation for wave propagation in a steel

bar with one crack at 0.5 m from actuator

Waves reflected by carck

Sca

led

ampl

itude

6

8

4

2

0

−2

−4

−6

−8

Time (ms)


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (ms)


Sca

led

ampl

itude

1

1.5

Wave A1

Wave A2

Wave A3

Waves reflected by carck

Crack location: 0.2 m


Crack location: 0.5 mCrack location: 0.8 m

10-5

0.5

0

0.2

−0.5

−1

−1.50.3 0.4 0.5 0.6 0.7 0.8 0.9 1

×

Crack location: 0.5 mCrack location: 0.8 m

Figure 16. Numerical results for different crack locations

(Crack depth: 9 mm)

The results agree well. These results indicate that byupdating the r values to match the recorded stresswave, the crack locations and depth can be determinedin practice to identify the damage.

5.2. Steel Bar with Two or more Cracks:Numerical Study

In this section, the same steel bar as described insection 5.1 is studied, with two cracks at 0.200 m and0.600 m, respectively. Figure 20 shows the receivedwaves in the steel bar based on the discrete springmodel and RT coefficient model. The results with a



Relationship between crack depth and amplitude ratio0.8

y = 3.5728x2.4749

R 2 = 0.99870.7

0.6

0.5

0.4

0.3

0.2

0.1

00 0.1 0.2

Relative crack depth


Am

plitu

de r

atio

: B/A

1

0.3 0.4 0.5 0.6


Relationship between crack depth and amplitude ratio

y = 3.5728x2.5414

R 2 = 0.9984

0.7

0.6

0.5

0.4

0.3

0.2

0.1

00 0.1 0.2


Am

plitu

de r

atio

0.3 0.4 0.5 0.6




AveragePower (Average)




AveragePower (Average)

Figure 17. Relationship between amplitude ratio and relative

crack depth

Relationship between crack depth and R coefficient0.8

0.7y = 3.5815x2.4407

R 2 = 0.99870.6

0.5

0.4

0.3

0.2

0.1

00 0.1 0.2


R v

alue

0.3 0.4 0.5 0.6

Computed results

Power (computed results)

Figure 18. Relationship between amplitude ratio and relative

crack depth

Time (ms)

(a) Crack location: 0.2 m

Sca

led

ampl

itude

1

1.5

0.5

0

0.2

−0.5

−1

−1.50.3 0.4 0.5 0.6 0.7 0.8 0.9 1

×10−5

Time (ms)

(b) Crack location: 0.5 m

Sca

led

ampl

itude

1

1.5

0.5

0

0.2

−0.5

−1

−1.50.3 0.4 0.5 0.6 0.7 0.8 0.9 1

×10−5

Time (ms)

(c) Crack location: 0.8 m

Sca

led

ampl

itude

1

1.5

0.5

0

0.30.2

−0.5

−1

−1.50.4 0.5 0.6 0.7 0.8 0.9 1 1.1

×10−5

Discrete spring modelRT coefficient model



Figure 19. Comparison between two methods for wave

propagation in steel bar with 9 mm crack

single crack at 0.200 m and 0.600 m based on thediscrete spring model are also shown in the figure.The corresponding r values are 0.08 for the crack at0.200 m and 0.22 for that at 0.600 m, representing thecrack depths of 5.4 mm and 8.0 mm at the twolocations, respectively. The results show that the firstthree waves simulated from the two approaches areapproximately the same for these different cases. Thisis because these three waves are induced byreflections of the transmitted waves at two ends of thesteel bar.

Based on the results from the RT coefficient model,the effect of two cracks on wave propagation andreflection is almost equivalent to the superposition ofthe effect of two single cracks at the correspondinglocations. Compared with the result using the discretespring model, the first few waves generated by cracksare almost identical. However, there is a large differencebetween the arriving times of the last wave using thesetwo methods. This difference may result from the springmodel used in the latter method.

5.3. Steel Bar with One Crack: ExperimentalVerification

To verify the proposed method in simulating stresswave propagation with reflections from cracks andboundaries, the steel bar with one cut, as shown inFigure 6, is tested in laboratory. The crack locates at0.250 m from the actuator and 0.750 m from thesensor. The crack is cut to 3.0 mm, 6.0 mm, 9.0 mmand 12.5 mm respectively (12%, 24%, 36% and 50%of diameter). The test condition is exactly the same asthat in section 3.

Figure 21 shows the comparison of the recordedstress wave histories in the intact and cracked steel bar.Compared with those in the intact steel bar, there are

new waves after Waves A1, A2 and A3, and before andafter Wave X2. They are reflected waves by the crack.The time intervals between the two new waves and theexcitation wave are 0.5805 ms and 0.6785 msrespectively, implying that the length of wave path is3.0 m and 3.5 m. The amplitudes of the new waves aresmaller than other waves, because only part of thewave energy is reflected at the crack location. Itdemonstrates that the GW is sensitive to cracks andcan be used to detect the crack damage in steel bars.When crack depth increases, the amplitude of reflectedwave increases correspondingly. It should be notedthat there may be other crack induced waves.However, they are combined with the existingboundary reflection and/or transverse waves.Therefore, only the present information is used here.The problem to differentiate these two kinds of wavesis out of the scope of this study.

The RT coefficient method is used here to simulatewave propagation with both crack and boundaryreflections. Figure 22 shows the comparison of thenumerical and experimental results. As shown, therecorded and simulated longitudinal waves withboundary and crack induced reflections fit very wellexcept transverse waves which appear in the test butare not modelled in numerical simulation. This isbecause the proposed method is only used to model thelongitudinal waves as discussed above. The resultsshow that the proposed model is effective and accuratein simulating wave propagation in a cracked steel barwith crack and boundary reflections. Besidesproviding an effective numerical model for stress wavepropagation, the method can also be used in structuraldamage identification. The damage can be quantifiedin model updating by adjusting the reflectioncoefficient to match the numerical simulations with therecorded waves.



Time (ms)

Sca

led

ampl

itude

1

1.5

0.5

0

0.30.2

−0.5

−1

−1.50.4 0.5 0.6 0.7 0.8 0.9 1 1.1

× 10−5

RT coefficient methodCrack location: 0.2 mCrack location: 0.6 mCrack location: 0.2 m and 0.6 m

Figure 20. Comparison of wave propagation in steel bar with one

or two cracks

Sca

led

ampl

itude

6Wave A1 Wave A2

IntactCrack depth: 3.0 mCrack depth: 6.0 mCrack depth: 9.0 mCrack depth: 12.5 m

Wave X1

Waves reflected by crack

Wave X2Wave A34

2

0

−2

−4

−6

−8

Time (ms)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

×10−6

Figure 21. Comparison of experimental results for intact and

cracked steel bar

6. CONCLUSIONImaginary elements are proposed in spectral elementmethod to model wave propagation and boundaryreflection in one-dimensional waveguides. The reliabilityof the method is proven using laboratory test data.

RT coefficient method is used to model the effect ofdiscontinuities on wave propagation. The relationshipbetween RT coefficients and amplitude ratio B/A1 isinvestigated. RT coefficients as a function of crackdepth can be experimentally obtained or numericallyderived by modelling the crack effects with the discretespring model. The proposed method is then used tomodel GW propagation with both reflections fromboundaries and structural discontinuities. The reliabilityof the proposed numerical approach is verified bylaboratory test results of GW propagation in a crackedsteel bar with boundary reflections.

Both the numerical and experimental resultsdemonstrated that the time difference between the firstwave and waves reflected from discontinuities changewith the crack location. The amplitude ratio B/A1 varieswith the crack depth. The proposed method correctly

simulates wave propagation process including these twoparameters, demonstrating that it can be used todetermine the crack location and extent in structuralhealth monitoring. Future efforts can be focused onextending the current method to higher orderwaveguide.

ACKNOWLEDGEMENTThe work described in this paper was supported byCIEAM through Project ID207.

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Rotor Dynamics, Applied Science Publishers, London, UK.

Doyle, J.F. (1997). Wave Propagation in Structures: Spectral

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(a) Crack depth: 3.0 mm

Am

plitu

de

4

6

2

0

0.2

−2

−4

−60.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Experimental resultNumerical simulation




1Time (ms)

(b) Crack depth: 6.0 mm

Am

plitu

de

5

0

Reflected waves from crack


0.2−5

0.3 0.4 0.5 0.6 0.7 0.8 0.9

×10−6

Time (ms)

(c) Crack depth: 9.0 mm

Am

plitu

de

4

3

2

1

0

−1

−2

−3

−4

−50.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

×10−6

×10−6


Sca

led

ampl

itude

−4

−3

−2

−1

0

1

2

3

4

Time (ms)

(d) Crack depth: 12.5 mm

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

×10−6

Figure 22. Comparison between numerical and experimental results for wave propagation in a cracked steel bar

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APPENDIX A: SPECTRAL ELEMENTFORMULATIONIn one-dimensional wave propagation, the spectralsolution of a general element can be written as:

(A1)

where L represents the length of the structure, Li and ki

are the length and the wavenumber of the ith elementrespectively, N is the number of elements.

Generally, for N elements, there are 2N unknowncoefficients G = {Bi, Ci, i = 1, 2, …, N}, which dependon the nodal spectral displacements. The displacementand force boundary condition can be respectivelyexpressed as:

(A2)

(A3)

where D is the displacement boundary matrix, Q is adisplacement vector for every node, H is the forceboundary matrix, and P is a force vector for every node.G can be expressed by the inversion of D, as

(A4)

Combining Eqns A3 and A4, a force-displacementrelationship can be obtained as

(A5)

where S = H · D−1. S is the dynamic stiffness matrix. Ifonly input and output waves at the locations of theactuator and sensor are considered, S can be simplifiedto a 2 × 2 matrix.

P H D Q S Q= ⋅ ⋅ = ⋅−1

G D Q= ⋅−1

H G P⋅ =

D G Q⋅ =

ˆ ( )u x B e C e

for x

i ijk x

i

jk L xi

i pp

i

= +

∈

−− ∑ −

=1

LL L Lpp i

N

pp

i

−

= =∑ ∑,

1



APPENDIX B: DISPLACEMENT AND FORCEBOUNDARY MATRIXThe boundary condition for a structure with onediscontinuity is summarized as follows.

D

e

r e t e

ik l l

ik l l ik

3

2

2

1 0 0

1 0

1 1 4

1 1 4

=⋅ − ⋅

− +

− + −

( )

( ) 11 2

1 1 4 1 1 4 1 22 20

l

ik l l ik l l ik lt e e r e⋅ − ⋅− + − + −( ) ( )

00 0 11 1 2 42e ik l l l− + +

( )

D

e

r e t e

t e

ik l

ik l ik l l

2

2

1 0 0

1 0

1 1

1 1 1 2 3

=⋅ − ⋅

⋅

−

− − +( )

−− − − +

− +

− ⋅ik l ik l ik l l

ik l l

e r e

e

1 1 1 1 1 2 3

1 1

0

0 0

2( )

( 22 32 1+

l )

D

e

r e t e

t e

ik l

ik l ik l

ik l1

1 0 0

1 0

1 1

1 1 1 2

1 1

=⋅ − ⋅

⋅

−

− −

− 00

0 0 1

1 1 1 2

1 1 2

− ⋅

− −

− +

e r e

e

ik l ik l

ik l l( )

where k1 represents the wavenumber of the spectralelement for round plain steel bar, li is defined as Figure 10.

H ik EAe

e

ik l l

ik l l l4 1

2

2

1 0 0

0 0

1 1 4

1 1 2

= ⋅−

−

− +

− + +

( )

( 33 42 1+

l )

H ik EAe

e

ik l l

ik l l l3 1

2

2

1 0 0

0 0

1 1 4

1 1 2

= ⋅−

−

− +

− + +

( )

( 44 1)

H ik EAe

e

ik l

ik l l l2 1 2

1 0 0

0 0 1

1 1

1 1 2 3

= ⋅−

−

−

− + +( )

H ik EAe

e

ik l

ik l l1 11 0 0

0 0 1

1 1

1 1 2

= ⋅−

−

−

− +( )

D

e

r e t e

ik l l

ik l l ik

4

2

2

1 0 0

1 0

1 1 4

1 1 4

=

⋅ − ⋅

− +

− + −

( )

( ) 11 2 3

1 1 4 1 1 4

2

2 20

( )

( ) ( )

l l

ik l l ik l lt e e r e

+

− + − +⋅ − ⋅ −− +

− + + +

ik l l

ik l l l le

1 2 3

1 1 2 3 4

2

2 20 0 1

( )

( )



spectral element modelling of wave propagation with boundary and structural discontinuity...

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