Composites: Part B 54 (2013) 1–10

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Composites: Part B

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Non-reflecting boundary condition for Lamb wave propagation problemsin honeycomb and CFRP plates using dashpot elements

1359-8368/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compositesb.2013.04.061

⇑ Corresponding author. Tel.: +49 3916711723.E-mail addresses: rsg.931@gmail.com (S.M.H. Hosseini), sascha.duczek@st.

ovgu.de (S. Duczek), ulrich.gabbert@ovgu.de (U. Gabbert).

Seyed Mohammad Hossein Hosseini ⇑, Sascha Duczek, Ulrich GabbertInstitute of Numerical Mechanics, Department of Mechanical Engineering, Otto-von-Guericke-University Magdeburg, Universitätsplatz 2, 39016 Magdeburg, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 January 2013Received in revised form 8 April 2013Accepted 15 April 2013Available online 29 April 2013

Keywords:Non-reflecting boundary conditionD. UltrasonicsA. HoneycombA. Carbon fibreC. Finite element analysis (FEA)

The paper’s objective is to introduce a new non-reflecting boundary condition using dashpot elements.This is an useful tool to efficiently simulate Lamb wave propagation within composite structures, suchas honeycomb and CFRP plates. Due to the steadily increasing interest in applying Lamb waves in modernonline structural health monitoring techniques, several numerical and experimental studies have beencarried out recently. The proposed boundary condition poses the advantage of reducing the computa-tional costs required to simulate the wave propagation in heterogenous materials. Different parameterswhich can influence the functionality of such an artificial boundary are discussed and several applicationsare presented. Finally, the results are also experimentally validated.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Structural health monitoring (SHM) in composite structuresusing guided Lamb waves is a new technology in modern indus-tries such as aviation and transportation. Piezoelectric (PZT) actu-ators and sensors are used to excite and to receive waves withincomplicated composite structures [1,2]. This approach for SHMapplications is an interesting technique because of the low costsof the required equipment, the possibility of an online monitoringand the high sensitivity to detect small structural damages [1].

1.1. Lamb wave propagation in composite plates

Lamb wave propagation in composite plates has been studied inseveral Refs. [1,3–12]. guided waves were used to detect sub-inter-face damages in foam core sandwich structures in [3]. A suitablefrequency was found that offers the highest sensitivity to detectskin/foam core delaminations and to facilitate the interpretationof the measured waveforms. Finally, delaminations were locatedand characterized using an adapted signal processing. The numer-ical simulations were validated experimentally. Wave propagationin light-weight plates with truss-like cores was investigated in [4].It has been shown, that the vibrational behavior can be reduced toequivalent plate models in the low frequency region where globalplate waves are dominant. An application example of a train floor

section was tested to validate the theoretical dispersion character-istics. The Lamb wave propagation in particle reinforced compositeplates was studied in [5]. It has been reported that the volume frac-tion and the stiffness to density ratio of the particles are the mainparameters to affect the Lamb wave propagation properties in suchmaterials. In addition, a homogenization method was used to sim-plify the models. A reasonable agreement between the complexmodel, incorporating many details of the real structure and thesimplified model in terms of the group velocity and the wavelengthhas been observed, while tremendous savings in computationalcosts were archived. Localized phase velocities in the frequencyrange of 5–50 kHz were measured in honeycomb plates in [6]. Ithas been reported that the proposed method is suitable to detectdelamination between the cover plate and the core in honeycombsandwich panels. In another study, a homogenization techniquewas formulated for the analysis of vibration and the wave propaga-tion problems in a honeycomb-like slender skeleton [7]. In addi-tion, the effect of the cell size on the overall dynamic behavior ofa composite solid was characterized. The effect of the geometryof the unit cells on the dynamics of the propagation of elasticwaves within the structure was studied in [8] using a two-dimen-sional finite element model and the theory of periodic structures.The desired transmissibility levels in specified directions wereinvestigated for an optimal design configurations to obtain effi-cient vibration isolation capabilities. Debonding in sandwich CF/EP composite structures with a honeycomb core was detectedusing the anti-symmetric (A0) Lamb mode in [9] and the finite ele-ment modeling approach was validated with experimental results.In a similar study theoretical, numerical and experimental

2 S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10

approaches were used to detect the delamination between the skinand the honeycomb of a composite helicopter rotor blade in [10].In addition, the Lamb wave propagation in honeycomb sandwichpanels was studied using a three-dimensional finite element ap-proach in [1,11]. The results were compared with a simplifiedmodel, where the core layer was replaced by a heterogeneousmaterial with a simple cubic geometry. The received results werevalidated by experimental results in [1]. Continuous mode conver-sation of the Lamb wave propagation in CFRP composites was stud-ied in [12] using numerical and experimental approaches.

These studies show the steadily growing research activities inthe field of structural health monitoring and highlight the needfor computationally efficient numerical tools to gain a deeperunderstanding of the underlying physics of Lamb wave propaga-tion in heterogenous materials.

1.2. Non-reflecting boundary condition

The application of a non-reflecting boundary condition wasstudied in several references in order to reduce the computationalefforts [13–18]. A non-reflecting boundary condition was describedfor the scalar wave equation in [13]. It has been mentioned thatthis method is only applicable for wave propagation when uniformmedium at the boundary (any inhomogeneities should be avoided)exist. In addition, the proposed method offers few choices for theshape of an artificial non-reflecting boundary. Infinite elementsin Abaqus were used to design a non-reflecting boundary in [14].However, it has been shown in [15] that this method is not satis-factory. In [15] a finite element approach for the analysis of thewave propagation in an infinitely long plate was presented. Toavoid any spurious reflections generated by the finite boundaryof the finite element model a non-reflecting boundary conditionwith a gradually damped artificial boundary was designed. Thelength of the damping section was considered to be long enoughfor gradual changes of the damping factor to avoid any spuriousreflection from any sudden damping. The proposed method wasimplemented using the available finite element packages. In addi-tion, the results in a plate with a horizontal crack were comparedwith the strip element method and a good agreement has been re-ported. A similar design of non-reflecting boundary was introducedin [16] using frequency domain analysis and absorbing regions. Inthis paper the longest wavelength was suggested as a measure forthe length of the absorption region. It has been indicated in [15]that the proposed non-reflecting boundary condition in [15,16]can be computationally expensive depending on the model beinganalyzed. Furthermore, the perfectly matched layer (PML) is intro-duced as a flexible and accurate method to simulate the wavepropagation in unbounded structures [17]. However, it is men-tioned that this approach can be computational expensive [18].Many additional unknowns insert in the standard PML formula-tions because the required wave equations stated in their standardsecond-order form to be reformulated as first-order systems.

Additionally, it has been mentioned in [18] that local absorbingboundary conditions among the available non-reflecting boundaryconditions are known as the simplest and most flexible approachwith a reasonable computational cost. They do not require any spe-cial functions and are capable to be coupled with standard finitedifference or finite element methods. Therefore, the aim of thepresent paper is to introduce a novel local non-reflecting boundarycondition using dashpot elements which can reduce the computa-tional efforts tremendously. The finite element method is known tobe a versatile and efficient numerical tool for a vast variety of engi-neering problems. Thus we decided to employ FEM to model theguided wave propagation in heterogenous medium instead of uti-lizing other numerical approaches such as finite differences or theboundary element method [19,20]. The proposed non-reflecting

boundary condition can be applied using all commercial finite ele-ment packages. In the following sections numerical and experi-mental results are presented. Different parameters are consideredto design an efficient non-reflecting boundary condition designedby dashpot elements. Afterward, the capability of the proposedmethod is compared with the approach of gradually damped arti-ficial boundaries. Thereafter, the ability of dashpot non-reflectingboundary condition is demonstrated to reduce the computationalcosts. To this end, the model size is reduced in such a way that onlythe minimum required solid medium between actuator and thesensor is considered for the Lamb wave propagation analysis in astructure. Finally, the results are validated numerically as well asexperimentally.

2. Finite element modeling

A widely accepted approach is to use PZT patches to generatethe Lamb waves within a structure, cf. Fig. 1. As time dependentexcitation signal (Vin) a three and half-cycle narrow banded toneburst [1] is applied as given in the following equation:

Vin ¼ V ½HðtÞ � Hðt � 3:5=fcÞ� 1� cos2pfct3:5

� �sin 2pfct: ð1Þ

There t is the time, fc is central frequency and H(t) is the Heavisidestep function. A zero voltage is applied to the bottom surfaces of thesensors and the actuator. Symmetric boundary conditions are ap-plied to the inner borders of the plate to reduce the model sizeand the computational costs, cf. Fig. 1. The piezoelectric sensor is at-tached parallel to the boundaries and located at a distance of180 mm from the actuator.

Dashpot elements are used to damp the wave reflections fromborders of a structure, cf. Fig. 1. The viscous behavior of dashpotsin which the damping force (F) is proportional to the velocity, pro-vides the ability of energy dissipation during cyclic loading [21].

F ¼ Cð _u2 � _u1Þ; ð2Þ

F represents the force generated by the dashpot, C is the dampingfactor, _u1 and _u2 are the velocity of two ends of the dashpot element(in our case _u1 ¼ 0). To apply the new non-reflecting boundary con-dition, dashpot elements are connected only to one row and columnof the outer elements as a primary choice, cf. Fig. 1.

In this paper, each numerical model includes an ‘‘inner part’’(shown by dashed lines in Fig. 1) which represents and capturesthe main features of the micro- and macrostructure of the propa-gating medium. The sensor is considered to be attached to theplate in this region. The rest of the structure (apart from the innerpart) is considered as outer borders of the plate. The definition ofthe ‘‘frame’’ is used to indicate number of outer rows and columnswhich are used in the border of a numerical model. A sketch isshown in Fig. 1, each frame of elements includes a set of rowsand columns of elements from the outer borders of the plate. How-ever, each frame may consist of several number of rows and col-umns (in this study number of rows and columns in each frameis considered to be equal to the number of rows in the inner part).The frame definition is also used to describe the model size and thegradually increasing damping boundary condition. As an exampleFig. 1 shows a model with three columns and rows as frame size.One can see the outer element frames (shown by gradually chang-ing colors) in a numerical model in Fig. 2. It has to be mentionedthat each frame consists of all the nodes and elements across thestructure. For instance, in order to damp the waves in a honeycombsandwich plate, dashpot elements which are applied on each frameare connected to all the nodes over the entire thickness of thesandwich structure including the nodes on the cover plates andthe nodes in the honeycomb core.

Fig. 1. A schematic representing of dashpot elements connected to a three frame-size plate. For the sake of visualization, the number of elements shown in the sketch of themodel has been reduced. Each element in the figure represents four elements in x and y directions for the original FE-model.

Z

YXActuator

Symmetric boundary condition

Sensor topFEMAPMaterial2:asdmaterial7FEMAPMaterial2:asd_1FEMAPMaterial2:asd_2FEMAPMaterial2:asd_3FEMAPMaterial2:asd_4FEMAPMaterial2:asd_5FEMAPMaterial2:asd_6FEMAPMaterial2:asd_7FEMAPMaterial2:asd_8FEMAPMaterial2:asd_9FEMAPMaterial2:asd_10FEMAPMaterial2:asd_11FEMAPMaterial2:asd_12FEMAPMaterial2:asd_13FEMAPMaterial2:asd_14FEMAPMaterial2:asd_15FEMAPMaterial2:asd_16FEMAPMaterial2:asd_17FEMAPMaterial2:asd_18FEMAPMaterial2:asd_19FEMAPMaterial2:asd_20FEMAPMaterial2:asd_21FEMAPMaterial2:asd_22FEMAPMaterial2:asd_23FEMAPMaterial2:asd_24FEMAPMaterial2:asd_25FEMAPMaterial2:asd_26FEMAPMaterial2:asd_27

Sensor bottomFEMAPMaterial2:asdmaterial7FEMAPMaterial2:asd_1FEMAPMaterial2:asd_2FEMAPMaterial2:asd_3FEMAPMaterial2:asd_4FEMAPMaterial2:asd_5FEMAPMaterial2:asd_6FEMAPMaterial2:asd_7FEMAPMaterial2:asd_8FEMAPMaterial2:asd_9FEMAPMaterial2:asd_10FEMAPMaterial2:asd_11FEMAPMaterial2:asd_12FEMAPMaterial2:asd_13FEMAPMaterial2:asd_14

Fig. 2. Schematic representation of the outer element frames in numerical model.The PZT elements’ orientation and the symmetric boundary conditions are alsoshown. The PZT actuator and sensors are modeled by SOLID5, coupled fieldelements with displacement and voltage degree of freedoms in ANSYS� 11.0.

0.2

Time (ms)

0.1

0.05

0

-0.05

-0.1

-0.15

-0.2

Loc

atio

n (m

m)

0.150.10 0.05

Symmetric mode

Reflections

Anti-symmetric mode

Fig. 3. The propagation of the Lamb modes and reflection waves are shown in a B-scan diagram. The Lamb wave propagation is considered in a honeycomb sandwichplate. The geometrical properties of the plate are presented in Table 2, the centralfrequency of the loading signal is 250 kHz.

S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10 3

To execute these simulations the commercial FEM softwareANSYS� 11.0 has been used. In this software, the longitudinal

spring-damper option is provided by COMBIN14 elements. Theseuniaxial tension–compression elements are available with up tothree degrees of freedom (i.e. x, y and z directions) at each node.

3. Methodology

The Lamb waves propagate along the medium with differentwave forms, which are known as modes. Each mode can be eithera symmetrical (S) mode or an anti-symmetrical (A) mode. By sub-tracting (or adding) the signals on the top and bottom surfaces of aplate one can identify the different modes propagating inside thestructure. However, this method is not suitable for thick sandwichpanels, where the arrival of the modes on the top and the bottomsurfaces differs. In this paper different modes and their reflectionsare identified based on differences in the group velocity, the wave-length and the amplitude [11]. In addition, to verify the mode split-ting, using B-scan images is also an alternative method to identifydifferent modes and reflections [22,11], cf. Fig. 3. The displace-ments of the nodes (located along the wave propagation direction)in the time domain are shown in B-scan.

The energy transmission caused by the reflected and propa-gated waves is measured to show the functionality of the proposednon-reflecting boundary condition. The transmitted energy is de-fined within this paper as the integral over the squared signal [23].

Etrans ¼Z tend

tstart

V2ðtÞ � dt: ð3Þ

To show the efficiency of the non-reflecting boundary, the ratio be-tween the transmitted energy of the reflected waves and the Lambwaves (Ereflected/ELamb) is calculated. As an example, one can see theLamb waves and the reflection in Fig. 5 which are separated with adashed line. The Ereflected and ELamb stand for the energy transmis-sion caused by reflections and the Lamb waves (including both S0

and A0 modes), respectively. In this study an equal period of timewhich is needed for propagation of the Lamb modes (which de-pends on the central frequency of the excitation wave) is consideredas a minimum period to measure the reflected waves.

The value of less than 1 for this ratio, shows an attenuation inreflected waves and therefore one can distinguish the reflectedwaves from the propagated Lamb modes. The value of bigger than1 indicates that non-reflecting boundary is too weak and the re-flected waves are sensed several times with sensor. As the ratiotends to zero a better non-reflection boundary condition is indi-cated. In this study, the ratio of 0.009 and less is considered as

4 S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10

an ideal non-reflecting boundary condition. The post-processingcalculations described in this paper have been performed usingMATLAB�.

4. Design parameters

Following parameters are considered to design an efficientboundary condition using dashpot elements to avoid reflectionfrom borders.

� Damping factor.� Direction of dashpot elements.� Number of dashpot elements.

4.1. Damping factor

Eq. (2) indicates that the magnitude of the generated force withthe dashpot elements is related to the damping factor and the no-dal displacement. Therefore, the damping factor multiplied by thegroup velocity (C � Vg) is used to show the capabilities of the dash-pot elements as non-reflecting boundary. Fig. 4 shows the energytransmission values of the reflected waves over C � Vg. Finding anefficient value of C � Vg and knowing the group velocity of the Lambwaves (which depends on the central frequency of the excitationsignal, the material properties of the propagating medium andthe thickness of the plate) one can indicate the efficient dampingfactor. Fig. 4 demonstrates that very small values of damping factordo not generate enough force to avoid propagation of the reflectedwaves effectively. On the other hand, the nodal displacements arenot big enough to move dashpot elements with very high dampingfactors. A range of 28,000–280,000 of C � Vg is found for the mini-mum value of the ratio between the transmitted energy of the re-flected waves and the Lamb waves (Ereflected/ELamb). In this examplean aluminum plate with the thickness of 2 mm is considered andthe Lamb wave with central frequency of 200 kHz is generated.The lowest group velocity belongs to the A0 mode and is approxi-mately 2800 m/s [11], subsequently a damping factor between10 and 100 (N s/m) is suggested.

4.2. Direction of dashpot elements

Different direction of dashpot elements according to Eq. (2) areconsidered. It is observed that the dashpot elements in direction ofwave propagation have the best effect to reduce the reflectionwaves. In this case one side of the dashpots are connected to the

0.00

0.05

0.10

0.15

Ere

flec

ted / E

Lam

b (-

)

0.6

0.8

1.0

1.2

0·105

Damping factor times the group velocity, C·Vg (N)

fc = 200 kHzELamb = 7.44e-8 J.Ω

aluminum plate thickness: 2 mm

Model size: 4 frames

Dashpots: 1 frame

1·105 2·105 3·105 4·105 5·105 6·105

Fig. 4. The ratio between the energy transmission of the reflected waves and theLamb waves for different damping factors. A model of an aluminum plate withthickness of 2 mm is considered, and the Lamb wave is excited with a centralfrequency of 200 kHz.

nodes on the plate and the other sides are fully fixed (with C � Vg

equal to 2.8e5 N). In a particular example, an aluminum plate of0.5 mm thickness is considered as an initial model, where thenodes on the outer borders are connected to the dashpot elementsin direction of wave propagation (x). The Lamb wave is excitedwith a central frequency of 150 kHz, cf. Fig. 1. In this case the trans-mitted energy by the reflected waves is 1.13 � 10�7 J X and the ra-tio between the transmitted energy of the reflected waves and theLamb waves (Ereflected/ELamb) is equal to 0.7. In the first case thedirection of dashpot elements are changed and set to the y direc-tion. Subsequently, the energy transmission of the reflected wavesincrease to 166% with the initial model (Ereflected/ELamb = 2.0). In thesecond case the dashpot elements are to be considered to thedirection of z. It is observed that the energy transmission of the re-flected waves increase by 240% in comparison to the initial model(Ereflected/ELamb = 2.6). Fig. 5 compares the reflected waves in mod-els with dashpot elements in x and z directions.

Furthermore, the combination of dashpot elements in all threedirections (x, y and z) is considered and only 3% reduction of theenergy transmission of the reflected waves (Ereflected/ELamb = 0.68)in comparison to the initial model is observed. These results canbe explained by the fact that in a reduced-size model (cf. Fig. 1)there are more nodes which are connected to the dashpot elementsin the direction of wave propagation (x) in comparison to the otherdirections of y and z.

4.3. Number of dashpot elements

The influence of number of dashpot elements on the reflectingwave is also considered. In a particular example an aluminum platewith the thickness of 2 mm is considered and the Lamb wave withcentral frequency of 200 kHz is excited. Initially 617 dashpot ele-ments (with C � Vg equal to 2.8e5 N and the dashpots are in x direc-tion) are used to be connected to the first outer element row andcolumn from the outer borders of the plate (one frame of outer ele-ments, cf. Fig. 1). In this case the transmitted energy by the re-flected waves is 2.05 � 10�9 J X, where the energy of the Lambwaves is 70.6 � 10�9 J X (Ereflected/ELamb = 0.03).

To show the influence of dashpot element number on the re-flected waves, it is increased in two steps and the results are com-pared with the initial model. In the first case the number ofdashpot elements is increased to 1234 to be connected to two out-er rows and columns (two frames of outer elements). In the secondcase the dashpot elements are further increased to 2468 elements,which are connected to four outer rows and columns (four frames

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Vol

tage

(V

)

Time (s)

Dashpot in x direction

Dashpot in z direction

Reflections

fc = 150 kHz

C · Vg = 2.8e5 N

aluminum plate thickness: 0.5 mm

S0

A0

Lamb waves

Model size: 4 frames Dashpots: 1 frame

0.5·10-4 1.0·10-4 1.5·10-4 2.0·10-4

Fig. 5. Reflected waves in models with different dashpot element directions. Amodel of an aluminum plate with thickness of 0.5 mm is considered, and the Lambwave is excited with a central frequency of 150 kHz.

Fig. 6. Increasing number of dashpot elements is shown schematically in a six-frame size model.

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Vol

tage

(V

)

0.5·10-4

Time (s)

617 dashpots (1 frame)

2468 dashpots (4 frames )fc = 200 kHz

Model size: 6 frames aluminum plate thickness: 2 mm

Reflections

S0

A0

Lamb waves

C · Vg = 2.8e5 N

1.0·10-4 1.5·10-4 2.0·10-4

Fig. 7. Reflected waves in models with different number of dashpot elements. Amodel of an aluminum plate with thickness of 2 mm is considered, and the Lambwave is excited with central frequency of 200 kHz.

0

1

2

3

4

5

6

Ere

flec

ted / E

Lam

b (-

)

100 150 200 250 300 350 400

Frequency (kHz)

Damping

Dashpot

Top surface honeycomb

Model size: 4 frames

Dashpots: 1 frame

C · Vg = 2.8e5 N

Fig. 8. The values of energy transmission of the reflected waves are plotted over thecentral frequency of the excitation signal. Two different kinds of non-reflectingboundary conditions are considered. First a non-reflecting boundary conditionbased on gradually damped artificial boundary with four frames of dampingmaterials on the borders is considered, labeled damping. Secondly a model withdashpot elements on the borders is taken into account, labeled dashpot. Ahoneycomb sandwich panel is considered, cf. Table 2 for the geometrical propertiesand cf. Table 3 for the materials properties.

S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10 5

of outer elements). Fig. 6 represents increasing number of dashpotelements schematically.

The ratio between the transmitted energy of the reflected wavesand the Lamb waves (Ereflected/ELamb) decreases in the first and thesecond cases to 0.012 and 0.011, respectively. For all three casesthe obtained ratio between the transmitted energy of the reflectedwaves and the Lamb waves is in an acceptable range and fairly farfrom 1, cf. Section 3. Fig. 7 compares the reflected waves in a modelwith 617 dashpot elements and a model with 2468 dashpotelements.

The results in Fig. 7 and the Ereflected/ELamb values indicates thatthe additional number of dashpot elements does not improve theefficiency of the proposed boundary condition significantly. Inaddition, considering the fact that the most of added dashpot ele-ments (in the first and second cases) are connected to the nodesalong the outer rows one can conclude that the most of reflectedwaves which are sensed with the sensor are reflected from the out-er columns (shown on the right hand-side of the actuator in Fig. 1).

5. Applications

5.1. Influence of different central frequencies

The influence of changes in the central frequency of the excita-tion signal on the wave reflection in a honeycomb sandwich plateusing a non-reflecting boundary condition with dashpot elements(labeled dashpot) is compared with the gradually damped artificialboundary which is introduced in [15] (labeled damping). Thesetwo approaches can be realized in commercial finite element soft-ware without difficulties. The geometrical properties and thematerials properties are given in Tables 2 and 3, respectively.Fig. 8 shows that, as the central frequency increases less energyis transmitted by the reflected waves from the non-reflecting

boundary condition with gradually damped artificial boundary. Asimilar trend can be observed for the non-reflecting boundary withdashpot elements. This phenomenon can be explained by the factthat as the central frequency of the excitation Lamb wave in-creases, the energy which is transmitted by the Lamb waves, de-creases. However, it is clear that the non-reflecting boundarywith dashpot elements is less sensitive to the changes in centralfrequency of the exciting signal and it works enough good evenin lower frequency ranges. Table 1 shows the absolute values oftransmitted energy by the Lamb waves with different centralfrequencies.

5.2. Reduced-size model

The major benefit to use dashpot elements is the possibility toreduce the model size. This reduction in the model size results indecreased computational costs. Within this study a non-reflectingboundary condition which results in an attenuation of the reflectedwaves is considered to be acceptable (Ereflected/ELamb < 1). Thisattenuation behavior will help to distinguish the reflected wavesfrom the propagated modes in signal processing for structuralheath monitoring applications [11]. Fig. 9 shows how the modelreduction can effect the energy transmission of the reflected wavesfrom the borders. It is clear that the attenuation of the reflectedwaves completely depends on the model size. This can be ex-plained by the fact, that the damping factor of such a non-reflect-ing boundary is increasing gradually. Consequently, a bigger modelhas a greater attenuation effect on the reflected waves.

However, the models with dashpots show less dependency onthe model size (in this example only one frame of outer elementsare connected to dashpot elements). This phenomena can also beexplained by the fact that number of dashpot elements does notchange the efficiency of the proposed boundary condition signifi-cantly, cf. Section 4.3 (by decreasing the model size, the numberof dashpot elements is also decrease).

5.3. Comparison with an ‘‘ideal’’ non-reflecting boundary

In order to verify the proposed non-reflecting boundary andmodel reduction, Fig. 10 compares the Lamb wave propagation ina model using an ‘‘ideal’’ non-reflecting boundary condition with

Table 1Absolute ELamb values for different central excitation frequencies.

Centralexcitationfrequency(kHz)

100 150 200 250 300 350 400

ELamb (J X)(�10�6)

0.060 0.046 0.036 0.014 0.0048 0.0017 0.00057

0.0

0.5

1.0

1.5

2.0

2.5

Ere

flec

ted / E

Lam

b (-

)

0 4 8 12 16 20

Number of frames (model size)

Damping

Dashpot

Aluminum plate thickness: 2 mm

Dashpots: 1 frame

fc = 200 kHzC ·Vg = 2.8e5 NELamb = 7.44e-8 J.W

Fig. 9. The values of energy transmission of reflected waves plotted over the modelsize, cf. Fig. 1. Different non-reflecting boundaries including (a) damping materialsand (b) dashpot elements are compared. An aluminum plate with thickness of2 mm is considered, and the Lamb wave are excited with a central frequency of200 kHz.

-0.100

-0.075

-0.050

-0.025

0.000

0.025

0.050

0.075

0.100

Vol

tage

(V

)

0·10-4

Time (s)

Ideal non-reflecting boundary

Dashpot non-reflecting boundary

Top surface fc = 40 kHz

aluminum plate thickness: 2 mm

C · Vg = 2.8e5 N

1·10-4 2·10-4 3·10-4 4·10-4

Fig. 10. Comparison of the Lamb wave propagation in a model using an ‘‘ideal’’ non-reflecting boundary condition with twenty damping frames of gradually dampedartificial boundary (dashed line) and a reduced-size model with four frames andone frame of dashpot elements (solid line). The excited Lamb wave is propagatingwith a central frequency of 40 kHz in an aluminum plate of 2 mm thickness.

Table 2The geometrical properties of the honey comb sandwich panel and the PZT actuator/sensor (units are in mm).

Skin plate (28 frames) Honeycomb cell PZT actuator and sensor

Length 290 Height 15 Radius 3.17Width 124 Thickness 0.5 Thickness 0.7Thickness 2 Core size 4.8 actuator/sensor distance 180

Z

X Y

Honeycomb height

Honeycomb thickness

Honeycomb core size

Fig. 11. Schematic representation geometrical properties of a honeycomb sandwichpanel, cf. Table 2.

Table 3Material properties of the plate and honeycomb cells [1].

Young’smodulus (GPa)

Poisson’s ratio (–) Density (kg m�3)

Skin plate (Aluminum alloy T6061)70 0.33 2700

Honeycomb cell (HRH-36-1/8-3.0)Ex = Ey

(GPa)Ez

(GPa)txy &tyz = txz (–) Gxy

(GPa)Gyz= Gxz

(GPa)Density(kg m�3)

2.46 3.4 0.3 0.94615 1.154 50

Table 4Material properties of the twill CFRP (0�/90�) plate [12].

Ex 127.5 (GPa) txy 0.273 (–) Gxy 5.58 (GPa)Ey 7.9 (GPa) tyz 0.348 (–) Gyz 2.93 (GPa)Ez 7.9 (GPa) txz 0.017 (–) Gxz 2.93 (GPa)

6 S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10

twenty damping frames of gradually damped artificial boundary(Ereflected/ELamb is equal to 0.0027) and a reduced-size model (withfour frames) with one frame of dashpot elements (Ereflected/ELamb isequal to 0.17). The time-of-flight is the same for both signals. Butthe A0 mode in the model with dashpot elements has relativelyhigher amplitude which can be explained by the amplification ef-fect of reflections from the borders in a reduced-size model.

5.4. Combination of damping materials and dashpot elements

Furthermore, a combination of gradually damped artificialboundary [15] and a dashpot boundary is considered, labeled com-bined boundary. The energy transmission by the reflected waves in

an aluminum plate (thickness of 2 mm) from borders of the com-bined boundary increases by 23% in comparison to the model inwhich only dashpot elements are used (where the transmitted en-ergy by the reflected waves is 2.05 � 10�9 J X). The ratio betweenthe transmitted energy of the reflected waves and the Lamb waves(Ereflected/ELamb) for the models with dashpot elements and com-bined boundary are 0.02 and 0.03, respectively. This increase canbe explained due to the fact that the material damping results ina reduction in the nodal velocity and subsequently the dashpotforce decreases.

5.5. Wave propagation in composite structures

In this section two examples of application of the non-reflectingboundary condition on the wave propagation in a honeycomb anda CFRP plates are presented. Finally, the experimental results arepresented.

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

Vol

tage

(V

)

Time (s)

Damping

Dashpot

Top surface honeycomb fc = 200 kHz

Reflections

S0

A0

Lamb waves

Model size: 4 frames

Dashpots: 1 frame C · Vg = 2.8e5 N Maximum

0.5·10-40·10-4 1·10-4 1.5·10-4 2·10-4

Fig. 12. Reflected waves are compared for a honeycomb reduced-size model withfour frames of damping materials on borders, labeled damping. Secondly the samesize model is considered with dashpot elements on borders, labeled dashpot. Thegraph shows the voltage signal received from the sensor on the top surface of thehoneycomb model, the signal has a central frequency of 200 kHz, cf. Fig. 1 fordefinition of frames and the reduced model size.

4·10-9

Dis

plac

emen

t (m

)

0.5·10-4

Time (s)

Twill CFRP (0/90°)

fc = 150 kHzWithout dashpot

With dashpot

Model size: 4 frames

Dashpots: 1 frame

Reflections

S0

A0

Lamb waves

C · Vg = 2.8e5 N

Maximum

3·10-9

2·10-9

1·10-9

0·10-9

-1·10-9

-2·10-9

-3·10-9

-4·10-9

0·10-4 1·10-4 1.5·10-4 2·10-4

Fig. 13. Propagated Lamb wave (nodal displacement signal) in twill CFRP (0�/90�)plate (a) without non-reflection boundary and (b) with non-reflection boundaryusing dashpot elements. The excited Lamb wave is propagated with a centralfrequency of 150 kHz. The plate thickness is 1 mm, and the rest of geometricalproperties are presented in Table 2, cf. Table 4, also cf. Fig. 1 for definition of framesand the reduced model size.

Actuator Position

3D laser sanningvibrometer

Retro-reflective layer

Composite plate

Silicon for damping

Fig. 14. Setup for experimental test.

S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10 7

The geometrical properties of the honeycomb sandwich plateand the PZT transducers are presented in Table 2. Fig. 11 representsthe geometrical dimensions of a honeycomb sandwich plate. Thesame dimension is used for the other plates in the following sec-tions. The material properties of the honeycomb sandwich panelcomponents and the aluminum plate are provided in Table 3, whilethe material properties of the twill CFRP (0�/90�) plate are shownin Table 4. It has been reported in [12] that the finite element mod-el of a twill CFRP (0�/90�) plate without matrix provides resultswhich are in a better agreement with the experimental results.The fibers are modeled using the material coordinate option inthe finite element model.

The dielectric matrix [e] and the piezoelectric matrix [e] are,respectively [1]:

½e� ¼6:45 0 0

0 6:45 00 0 5:62

264

37510�9 ðC V�1 m�1Þ;

½e� ¼

0 0 �5:20 0 �5:20 0 15:10 0 00 12:7 00 12:7 0

2666666664

3777777775ðC m�2Þ;

and the stiffness matrix is

½c� ¼

13:9 6:78 7:43 0 0 013:9 7:43 0 0 0

11:5 0 0 0sym: 3:56 0 0

2:56 02:56

2666666664

3777777775

1010ðPaÞ:

The mass density of the PZT is 7700 kg m�3 [1] and the mass densityof the twill CFRP (0�/90�) is 1550 kg m�3 [12]. The skin plates of thehoneycomb sandwich plate are modeled using cubic 3D solid ele-ments while 2-D shell elements are used to model the honeycombcells. Fig. 11 shows the connection of 2-D shell elements (core) andthe 3-D elements (cover plate), however, the thickness of corestructure is shown for a better visualization and explanation of geo-metrical dimensions. An accurate modeling approach is needed toconsider the correct stresses and deformation in the connection re-gion where the solids elements links to the shells which is often theweakest area. In the Lamb wave propagation study, where we areonly interested in the displacement field of the plate, multi-pointconstraint equations are a reliable method to connect two portionsof a structure using solids and shells [24]. Constraint equations aredeployed to join the shell and solid elements within this study.

5.5.1. Honeycomb composite plateFig. 12 compares effects of different non-reflecting boundary

conditions (including artificial damping boundary and dashpotboundary) on the wave propagation in a reduced-size honeycombmodel. In the reduced model four frames of outer boundary ele-ments are used and the plate length is reduced to 220 mm andthe plates width is reduced to 24 mm (the dimensions of the origi-nal model are given in Table 2 which represents a model with

Time increment (-)

Scal

e (-

)

150 200 250 300 350 400 450 500

10

15

20

25

30

35

40S0

A0

Time of the flight for A0Time of the flight for S0

Maximum value of the CWT coefficients

Fig. 15. Schematic representation of the absolute values of the CWT coefficients based on the Daubechies wavelet D10 in a contour plot. The Lamb wave is excited with acentral frequency of 200 kHz on a honeycomb sandwich plate.

-20

-10

0

10

20

Dif

fere

nce

in g

roup

vel

ocity

(%

)

75 100 125 150 175 200

Frequency, fc (kHz)

Honeycomb top surface: twill CFRP (0/90°)

S0 mode

A0 mode

Fig. 16. The group velocity values which are obtained from the experimental testsare compared with simulation results at all tested frequencies. A honeycombsandwich plate (the honeycomb cell height is 8 mm and the cover plate is made oftwill CFRP (0�/90�) with 1 mm thicknesses, cf. Table 2 for the rest of geometricalproperties, cf. Table 3 and 4 for the material properties) is investigated.

8 S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10

twenty-eight frames of outer boundary elements, also cf. Fig. 1 fordefinition of frames and the reduced model size).

The attenuation behavior of the reflected waves in the modelwith non-reflecting boundary using dashpot elements is clearlyshown (cf. the solid line in Fig. 12). One can observe the amplifica-tion and attenuation effects of the reflected waves on the wavepropagation in reduced-size models with the gradually dampedartificial boundary. It is clear that the reflected waves in the modelwith gradually damped artificial boundaries have almost the sameamplitude as the propagated A0 which can cause difficulties toidentify modes in a later performed signal processing (cf. thedashed line in Fig. 12). The ratio between the transmitted energyof the reflected waves and the Lamb waves (Ereflected/ELamb) is equalto 1.1 (the acceptable range is less than 1) for the model with grad-ually damped artificial boundary.

5.5.2. CFRP composite plateFig. 13 presents an application example of using dashpot ele-

ments as a non-reflecting boundary in a twill CFRP (0�/90�) plateas a complicated composite structures. The plate thickness is1 mm and the rest of geometrical properties are presented in Ta-ble 2 and cf. Table 4 for material properties. It is clearly shownhow dashpot elements can reduce the amplitude of the reflectedwaves from the borders (cf. the solid line in Fig. 13). In addition,it is shown that the reflected waves may influence the wave prop-

agation in reduced-size CFRP model (cf. the dashed line in Fig. 13).The ratio between the transmitted energy of the reflected wavesand the Lamb waves (Ereflected/ELamb) is equal to 1.2 (acceptablerange is less than 1) for the model without non-reflecting bound-ary condition.

5.5.3. Experimental validationIn addition to the numerical verification, results are also vali-

dated experimentally. The experimental setup is shown inFig. 14. The velocity of the nodes on the retro-reflective layer ismeasured with scanning a laser vibrometer to evaluate the waveproperties [22,12].

The flight velocity of the propagated waves are calculated in re-duced-size numerical models and compared with experimental re-sults. The nodal displacement signal in the vertical direction u(t) istransformed using the continuous wavelet transform (CWT) basedon the Daubechies wavelet D10 to evaluate the flight velocities(the bar indicates complex conjugation).

WTða; bÞ ¼ 1ffiffiffiap

Z þ1

�1uðtÞw t � a

b

� �dt: ð4Þ

The location of the maxima of the CWT coefficients gives the time-of-flight for each Lamb wave mode, cf. Fig. 15. Knowing the distanceand the time-of-flight one can calculate the group velocity [11]. Thetime-of-flight is measured between the excitation and the arrival attwo different points on the top and the bottom surface, in order toshow the influence of the sandwich plate thickness and the corematerial on the flight velocity of the wave propagation. Therefore,the flight velocity on the bottom surface includes the effect of platethickness and differs from the flight velocity on the top surfacewhich is known as the group velocity. In this paper we use ‘‘groupvelocity’’ to describe the flight velocity for both top and bottomsurfaces.

Because of the combination of the propagated modes and thereflected waves in the reduced-size models without non-reflectingboundary the location of the maxima of the CWT coefficients maychange up to 20% (in comparison to experimental results). Fig. 13shows how the group velocity of the A0 mode changes in a modelwithout non-reflecting boundary condition. One can observe thatthe maximum amplitude of the A0 mode occurs approximately0.2 ms earlier in the model without non-reflecting boundary whichcan be explained with amplification and attenuation effects of thereflected waves on the wave propagation in the reduced-size mod-el. A similar illustration is given in Fig. 12 where a weak non-reflecting boundary with gradually damped artificial in reducedsize model may effect the group velocity of the A0 mode. Therefore,

-15

-10

-5

0

5

10

15

Dif

fere

nce

in g

roup

vel

ocity

(%

)

100 150 200 250 300 350

-15

-10

-5

0

5

10

15

Dif

fere

nce

in g

roup

vel

ocity

(%

)

100 150 200 250 300 350

bottom surface

S0 modeA0 mode

Frequency, fc (kHz)Frequency, fc (kHz) Frequency, fc (kHz)Frequency, fc (kHz)

Aluminum honeycomb top surface

Fig. 17. The group velocity values which are obtained from the experimental tests are compared with simulation results for different excitation frequencies. The groupvelocities are compared on both sides of a honeycomb plate which is made of aluminum, cf. Table 3. The cover plate is 0.6 mm, the cell size is 6.4 mm and the honeycomb wallthickness is 0.0635 mm.

S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10 9

the agreement between the group velocities in the simulation andexperimental tests is considered to show the capabilities of theproposed reduced-size model using non-reflecting boundary basedon dashpot elements.

In the first example, the excited Lamb wave is propagated in ahoneycomb sandwich plate, where the honeycomb cell height is8 mm and the cover plate is made of twill CFRP (0�/90�) with1 mm thickness, cf. Table 2 for the rest of geometrical properties,cf. Table 3 for the material properties of the honeycomb cells,and cf. Table 4 for the material properties of the CFRP plate.Fig. 16 compares the group velocity values of different modeswhich are obtained experimentally and numerically, where thecentral frequency of the excitation signal is increased from50 kHz to 220 kHz. The relative difference in percent is calculatedas

Erel ¼ 100 � Gsim � Gexp

Gexp½%�: ð5Þ

Gsim represents the group velocity values which are obtained fromthe simulation test, and Gexp shows the group velocity values whichare obtained from the experimental investigations. For the pro-posed example we achieve a good agreement of the results. Theaverage of absolute differences (jErelj) is 2.30% and the absolutemaximum is 8.01%.

In another example the group velocity values of the propagatedwaves in a honeycomb sandwich palate are compared on bothsides of the structure, cf. Fig. 17. The honeycomb plate is fullymade of aluminum, cf. Table 3. The cover plate is 0.6 mm, the cellsize is 6.4 mm and the honeycomb wall thickness is 0.0635 mm.The relative error is not exceeding 11% and the average of absolutedifferences is 4.11%.

6. Summary

A non-reflecting boundary condition using dashpot elements isintroduced in order to reduce the reflections of the Lamb waves atboundaries. It has been shown that by applying the proposedboundary condition the computational costs for wave propagationsimulations can be reduced significantly. Different parametersincluding the damping factor, the direction of dashpot elementsand the number of dashpot elements were examined to designan efficient non-reflecting boundary. The influence of each param-eter is summarized as follows:

� Damping factor: It has been shown that very small damping fac-tors do not generate enough force to suppress the propagationof the reflected waves effectively. On the other hand, the nodaldisplacements are not large enough to move dashpot elements

with very high damping factors. A range of 28,000–280,000 ofC � Vg (damping factor times group velocity) is found as anappropriate choice to significantly reduce the propagation ofthe reflected waves.� Direction of dashpot elements: It has been observed that the

dashpot elements in direction of wave propagation are mosteffective in reducing the reflections.� Number of dashpot elements: To apply the new non-reflecting

boundary condition, dashpot elements are connected only toone row and column of the outer elements as a primary choice.It has been indicated that the additional number of dashpot ele-ments does not improve the efficiency of the proposed bound-ary condition significantly.

The application of the proposed method is demonstrated for theLamb wave propagation within different heterogenous materialsincluding a honeycomb sandwich plate with twill CFRP (0�/90�)cover plate and an aluminum honeycomb sandwich plate. In addi-tion, results were validated experimentally. Furthermore, combi-nation of dashpot elements with gradually damped artificialboundary were considered and it has been shown that this combi-nation has only a minor effect on reflection of the waves from theborders and causes extra efforts in the modeling process.

7. Conclusions

The advantages of the proposed non-reflecting boundary condi-tion using dashpot elements to reduce the computational costs to-gether with the possibility of implementing the proposed schemein commercial FEM packages, provide an efficient tool for research-ers to numerically investigate the interaction of the Lamb waveswithin complicated structures such as CFRP and honeycomb com-posites even with ordinary personal computers. These investiga-tions are very important to design further health monitoringsystems using Lamb waves for composite structures. However, fur-ther investigations are required to extend the application of theproposed non-reflecting boundary condition to study wave propa-gation in other heterogenous structures.

Acknowledgments

By means of this, the authors acknowledge the German Re-search Foundation for the financial support (GA 480/13). We wouldalso like to thank C. Willberg for that was invaluable in the execu-tion of the experimental tests.

10 S.M.H. Hosseini et al. / Composites: Part B 54 (2013) 1–10

References

[1] Song F, Huang GL, Hudson K. Guided wave propagation in honeycombsandwich structures using a piezoelectric actuator/sensor system. SmartMater Struct 2009;18:125007–15.

[2] Ungethuem A, Lammering R. Impact and damage localization on carbon-fibre-reinforced plastic plates. In: Casciati F, Giordano M, editors. Proceedings 5theuropean workshop on structural health monitoring. Sorrento, Italy; 2010.

[3] Terrien N, Osmont D. Damage detection in foam core sandwich structuresusing guided waves. In: Leger MDA, editor. Ultrasonic wave propagation in nonhomogeneous media, springer proceedings in physics, vol. 128. Berlin,Heidelberg: Springer; 2009. p. 251–60.

[4] Kohr T, Petersson BAT. Wave beaming and wave propagation in lightweightplates with truss-like cores. J Sound Vib 2009;321:137–65.

[5] Weber R. Numerical simulation of the guided Lamb wave propagation inparticle reinforced composites excited by piezoelectric patch actuators.Master’s thesis, Institute of Numerical Mechanics, Department of MechanicalEngineering. Germany: Otto-von-Guericke-University Magdeburg; 2011.

[6] Thwaites S, Clark NH. Non-destructive testing of honeycomb sandwichstructures using elastic waves. J Sound Vib 1995;187(2):253–69.

[7] Wierzbicki E, Wozniak C. On the dynamic behavior of honeycomb basedcomposite solids. Acta Mech 2000;141:161–72.

[8] Ruzzene M, Scarpa F, Soranna F. Wave beaming effects in two-dimensionalcellular structures. Smart Mater Struct 2003;12:363–72.

[9] Mustapha S, Ye L, Wang D, Lu Y. Assessment of debonding in sandwich CF/EPcomposite beams using A0 Lamb. Compos Struct 2011;93:483–91.

[10] Qi X, Rose JL, Xu C. Ultrasonic guided wave nondestructive testing forhelicopter rotor blades. In: 17th World conference on nondestructive testing.Shanghai, China; 2008.

[11] Hosseini SMH, Gabbert U. Numerical simulation of the Lamb wavepropagation in honeycomb sandwich panels: a parametric study. ComposStruct 2012 [Accepted for publication].

[12] Willberg C, Mook G, Gabbert U, Pohl J. The phenomenon of continuous modeconversion of Lamb waves in CFRP plates. Key Eng Mater 2012;518:364–74.

[13] Alpert B, Greengard L, Hagstrom T. Nonreflecting boundary conditions for thetime-dependent wave equation. J Comput Phys 2002;180:270–96.

[14] Lysmer J, Kuhlemeyer R. Finite dynamic model for infinite media. J Eng MechDiv ASCE 1969;95:859–77.

[15] Liu GR, Jerry SQ. A non-reflecting boundary for analyzing wave propagationusing the finite element method. Finite Elem Anal Des 2003;39:403–17.

[16] Drozdz M, Moreau L, Castaings M, Lowe MJS, Cawley P. Efficient numericalmodelling of absorbing regions for boundaries of guided waves problems. In:AIP conference proceeding (820); 2006. p. 126.

[17] Bérenger JP. A perfectly matched layer for the absorption of electromagneticwaves. J Comput Phys 1994;114:185–200.

[18] Sim I. Nonreflecting boundary conditions for time-dependent wavepropagation. Ph D thesis, faculty of science. Switzerland: University ofBassel; 2010.

[19] Eymard RHR, Gallouët T. Handbook of numerical analysis Vk. Ch. finite volumemethods. North Holland; 2000. p. 713–1020.

[20] Cheng AH-D, Cheng DT. Heritage and early history of the boundary elementmethod. Eng Anal Bound Elem 2005;29:268–302.

[21] Bower A. Applied mechanics of solids. CRC Press; 2010.[22] Pohl J, Mook G, Szewieczek A, Hillger W, Schmidt D. Determination of Lamb

wave dispersion data for SHM. In: 5th European workshop of structural healthmonitoring; 2010.

[23] Song F, Huang G, Kim J, Haran S. On the study of surface wave propagation inconcrete structures using a piezoelectric actuator/sensor system. Smart MaterStruct 2008;17:055024–32.

[24] Suranat K. Transition finite elements for three-dimensional stress analysis. IntJ Numer Meth Eng 1980;15:991–1020.