Guided wave propagation analysis in stiffened panel using time-domain spectral finite element method

Zexing YU, Cho XU,b,*, Jiying SUN, Fei DU

a School of Astronautics, Northwestern Polytechnical University, Xi’an 710072, China

b Qingdao R&D Institute, Northwestern Polytechnical University, Qingdao 266200, China

KEYWORDS Absorbing layers with increasing damping;Guided waves;Stiffened Panel;Time-domain spectral finite element;Wave propagation

Abstract Stiffened panels have been widely utilized in fuselages and wings as critical load-bearing components. These structures are prone to be damaged under long-term and extreme loads, and their health monitoring has been a common concern. The guided wave-based monitoring method is regarded as an efficient approach to detect the damage in stiffened plates because of its wide monitoring range and high sensitivity to micro-damage. Efficient simulation of wave propagation can theoretically demonstrate the detection mechanism of the method. In this study, a Time-Domain Spectral Finite Element Method (TD-SFEM) is adopted to study the wavefield in stiffened plates,

1. Introduction

Stiffened structures have been widely utilized in aerospace engineering due to their advanced structural performance,such as excellent load-carrying capability and lightweight design.1-2For example, stiffened structures play the role of critical load-bearing components in wings and fuselages.Under long-term or extreme loads, these structures will inevitably experience degradation or damage. Hence, health monitoring for stiffened structures has attracted more attention.3In recent years, Structural Health Monitoring (SHM) based on guided waves has been a promising approach to detect damage in stiffened structures because waves can propagate over a long distance and carry damage information without much attenuation.4-5Generally, the monitoring procedure involves complex inverse problems, i.e., the backward characterization of damage detection based on measured data.Therefore,it is necessary to develop an efficient numerical method to simulate wave propagation and enhance the understanding of the mechanism, which also plays an important role in many SHM procedures, such as sensor configuration and damage indicator designation.6

For this purpose, a number of studies can be found in the Refs.5-9Mead studied wave propagation in a periodic stiffened plate analytically.7In their governing equation, the stiffener was formulated by the flexural stiffness, the St. Venant torsional stiffness and the torsion-bending stiffness.The propagation constants were studied over different frequency ranges in their study.Williams et al.8derived the wave motion equations of different modes in stiffened plates based on Fourier transform, in which stiffeners were modeled by Lagrangian multipliers. They established a dynamic stiffness matrix as a transcendental function of frequency, and the passband free wave was calculated based on propagation constants. Xin and Lu9studied the wave propagation in a rib-stiffened sandwich plate. They modeled the sandwich plate by defining two dynamic governing equations standing for the top and bottom layers, respectively. The contribution of the stiffener was expressed in terms of tensional forces, bending moments and torsional moments. In their study, the influence of factors,inertial of the stiffener, excitation position and stiffener spacings,on wave pressure level was studied.Haider et al.5adopted complex modes expansion with vector projection method to investigate wave propagation in a cracked stiffened aluminum plate.In their study,the two discontinuities,cracks and stiffeners,were considered,and the transmission and reflection of different modes waves at 50-350 kHz were given by defining the scattering coefficients. These analytical or semi-analytical methods are efficient and accurate while they are subject to many restrictions, such as infinite structure, simple support boundaries, harmonic excitation, the orthogonal and uniform cross-section of the stiffener.10

To tackle these above- mentioned drawbacks, many numerical methods have been reported.10-15Orrenius and Finnveden11meshed the stiffened plate using the Finite Element Method (FEM) and calculated the nodal displacement and dispersion of waves in the frequency domain. Their model was developed based on the cross-section of the waveguide, so it is hard to simulate wave propagation in a variable cross-section stiffener. Reusser et al.12studied wave propagation in stiffened plates with different modes by the boundary element method. They used the force balance at the junction of the stiffener and the skin to define the transmission coefficient by introducing the generalized impedance.They focused on the changes of wavefields before and after the stiffener and investigated the wave scattering phenomenon with different mode waves, which was also validated by the experimental results. Zhang et al.13proposed a triangular composite stiffened plate/shell element to discrete the stiffened structures using Mindlin shear deformation theory. In their model, displacement compatibility was imposed on the interface of the stiffener and skin. Ramadas et al.14studied wave propagation in a stiffened composite plate with a T-joint in the FEM and experiment. In their work, the pure A0 mode waves were adopted as the excitation, and the turning modes of waves were captured near the junction corner, which may improve the SHM of complex structures. Ricci et al.15investigated wave propagation in a stiffened composite plate by FEM to localize the delamination of the plates. Their research rendered that the fundamental A0 mode wave was a preferred option because it can carry important damage information. Overall, the FEM is a frequently-used approach for simulating wave propagation in stiffened structures while the main drawback of FEM is the calculation scale will dramatically increase when structures become complex.10

Recently, the Time-Domain Spectral Finite Element Method (TD-SFEM) has been used to simulate wave propagation in different scenarios, which was firstly proposed by Patera in fluid dynamics.16-18This method can be viewed as a particular type of FEM and bears some similarity with the p-version FE method.19The main idea of this method is the use of a particular high-order interpolation function. In each spectral finite element, the inner interpolation nodes are collocated non-uniformly at the Gauss-Lobatto-Legendre (GLL) points, whose coordinates are relative to the first derivative of the nth Legendre polynomial.20The shape function is determined by the Lagrangian interpolation polynomials passing through the GLL points. In this respect, the significant drawback of high-order FEM in solving wave propagation called Runge’s phenomenon can be effectively suppressed.21In addition, the GLL quadrature rules are applied during the deduction of mass and stiffness matrices of spectral elements, which results in the diagonal mass matrix due to the orthogonality between integral weighting factors and shape functions. As a consequence,high accuracy of numerical interpolation is achieved. Many studies have been published to prove that the TD-SFEM is superior to the traditional FE method in simulating wavefields.22-24On the one hand, TD-SFEM can simulate wave propagation in complex structures and boundary conditions at a low cost. On the other hand, the proposed method can obtain higher accuracy in the case of A0 mode simulation.22However, to the best of the authors’ knowledge, few studies in stiffened plates related to TD-SFEM except Schulte et al.,in which the structure is meshed by the plate elements in their work, and more detailed propagation behaviors needs to be further studied.25Hence, it is necessary to develop an efficient TD-SFEM to study wave propagation in stiffened structures and reveal the effects of the stiffener and its geometric parameters on guided waves.

This paper was conducted to develop a strategy that combining the TD-SFEM and Absorbing Layers with Increasing Damping (ALID) to simulate the wave propagation in stiffened plates. The proposed method has the ability to simulate wave propagation in complex structures with high efficiency and a low memory requirement. Meanwhile, compared with the traditional approach,the updated ALID can better absorb the boundary reflection to study the reflection and transmission of waves. Besides, in this work, the effects of the parameters of the stiffener are also investigated. This paper is organized as follows. In Section 2, the theoretical background of TD-SFEM and ALID are introduced. In Section 3, the developed model is validated by comparing the results with those of the commercial FEM software.In Section 4,a specific model is proposed to study the wave propagation in the stiffened plates,and the parameter study is also conducted.In Section 5, some key conclusions are drawn.

2. TD-SFEM with ALID

As per the aforementioned studies, the simulation of wave propagation excited by the linear sensor array in stiffened plates can be considered as a 2D plane strain problem, which is validated in numerical and experimental methods.5,13Hence,the Legendre polynomials-based 2D spectral finite element is introduced to obtain the wavefield in this section. Moreover,the absorbing layers with increasing damping are combined to eliminate the reflected waves caused by the geometric boundary.

2.1. TD-SFEM

A comparison of the high-order finite element and spectral element in the local coordinate system is given in Fig.1.The difference is that the interpolation nodes for the finite element are linearly located at each edge while they are non-uniformed distributed in the spectral element. This feature helps the TDSFEM avoid the Runge’s phenomenon, which indicates a problem of oscillation near the edges of an interval that occurs when using polynomial interpolation with polynomials of the high degree over a set of nodes spaced in a uniform grid.26Therefore, when compared with the FEM, the TD-SFEM has the ability to simulate the wave propagation with faster convergence and higher accuracy27, where the coordinates of interpolation nodes are determined by Gauss-Lobatto-Legendre polynomial:

Like the FEM, the displacement field of the spectral element can be expressed by the Lagrange polynomial interpolation and nodal displacement. The element matrices, such as stiffness matrix Keand mass matrix Me, are derived by the Hamilton principle with the GLL integration rules.27The weight factor ωiis determined as:

where Nn(ξi) is the element of shape function and the subscripts n, m indicate the different items. This property promises that the mass matrix is in diagonal form. In this way,the matrices inversion and high requirement of memory are avoided in the solving process.

2.2. ALID with spectral finite element

This paper aims to study the influence of the stiffener on wave propagation. The reflected waves caused by geometric boundary will complicate the wavefields and time history responses,which dramatically increase the difficulty of propagation analysis.Hence,it is necessary to carry out a kind of non-reflection operation to eliminate the undesirable factors except for the stiffeners.

According to the literature, there are mainly three kinds of methods for eliminating the waves reflected by boundaries:nonreflecting boundary conditions28, infinite elements and absorbing layer methods.29Two absorbing layer methods,the Perfectly Matched Layers (PML) and the ALID, are utilized to absorb the reflected wave successfully. In the case of the PML, the impedances of absorbing elements and host structure elements are perfectly matched. Hence there is no reflection.30For the ALID, the impedance in absorbing materials gradually increases and is mismatched with the host structure.Compared with the PML,the ALID is easier to integrate with the TD-SFEM code.31-32Hence, the ALID is adopted in the following study. This method was first proposed by Israel and Orszag in the 1980s and developed by many researchers.33As shown in Fig.2(a),the ALID generally is set up at the edge of host structures. The incident waves will pass through the edge of host structures and continue to propagate towards the ALID. In absorbing layers, the energy of waves is gradually attenuated by structural damping until it can be ignored.In this way, the waves are absorbed, and no reflected wave occurs. Besides, there are no abrupt changes of damping in the ALID so that the mechanical impedance matching is approximately achieved. The main drawback of this approach is that the absorbing layers increase the size of the structure.The calculation scales significantly go up because of the strict requirement of the minimum mesh size for the FEM when simulating the wave propagation.34Furthermore,the ALID fitted with TD-SFEM is proposed to reduce the obstacle of modeling complexity and cost, and the reflected waves can be absorbed at a low cost.

Generally, the damping in the ALID is expressed as Raleigh damping:

where α and β are damping coefficients for Meand Ke,respectively.In practice,the coefficient β is usually set to zero.31The reason is that the introduction of β will decrease the stability in the explicit scheme and result in a low computation efficiency.32Only mass-related component α is sufficient to absorb the reflected waves,which leads to a diagonal damping matrix and increases the computation efficiency. Besides, the results shown in the following Section 4 prove that ignoring β has no obvious effect on the accuracy of the results.35

In the classic numerical analysis procedure, to use the ALID method, the attached absorbing layer region has to be divided into many substructures for getting a smooth increase of damping.The damping coefficient α is constant in each substructure layer and varies between different substructures. In the case of complex structures with multiple edges, it is necessary to set the ALID at each boundary and divide it into substructures, and each substructure has to be defined with different material properties, which results in a cumbersome modeling step and dramatically decrease the efficiency in convergence analysis.In addition,as shown in Fig.2(a),due to the resulted discontinuous damping properties, there are still apparent interfaces of damping between different substructures.

In this study, a simple continuous ALID strategy is proposed by taking the advantage of spectral element analysis.As shown in Fig.2(b),the coefficient α is considered as a function of space coordinates:

where αmaxis the maximum value of the damping coefficient and L stands for the length of the ALID.x and x0are coordinates of the local point of the ALID and the edge of the host structure, respectively. The coefficient α varies as an exponent function, in which the n normally equals 2 or 3. The damping values of ALID are calculated at each element integration point, defined as

in which α(ξ ) uses the form of a power function. N and Jeare shape function and the Jacobian matrix, respectively. The superscript T is the transpose operator. Thus, damping values are varied.

In this way, the damping successively increases in each element, and the redundant operation for substructures division can be circumvented. The ALID is a whole part without any interfaces of damping or substructures. A comparison of the two kinds of ALID is carried out in the following study.

3. Model validation

Based on the wave propagation in a stiffened plate, the feasibility and effectiveness of the proposed TD-SFEM model with the ALID is validated by comparing with the FEM analysis.The dimensions of the structure are graphed in Fig. 3, where the shadow area implies the ALID zone. The properties of the adopted material are Young’s modulus E=7×1010Pa,mass density ρ=2700 kg/m3and Poisson’s ratio ν=0.3.The excitation is applied at x= 100 mm of the plate in the x-direction, which is generated by a Hanning windowed sinusoidal signal with frequency 200 kHz:

where n= 5 is the number of tunnels and f stands for the central frequency. To generate a pure S0 mode of Lamb waves,the center mode shape technique is carried out.36Two monitoring points M and N are pick up to analyze responses, and the coordinates are shown in Fig. 3. The schematic view of the center mode shape is illustrated in Fig. 4. The legends uxand uyare displacement in two main directions, respectively.The vertical axis represents the thickness of the plate,in which the upper layer is +d and bottom layer is -d. The force is applied at the nodes along the line x=100 mm. To simulate the mode shape of specific S0 mode wave, the input profile at each node is determined by the center mode shape method based on the local y-coordinate.

In ALID,the length of the absorbing layer L and the maximum of the damping value αmaxare two critical parameters affecting the absorbing effect.Hence,the convergence analysis for the parameters should be conducted before the wave propagation simulation.In this part,an Absorbing Indicator(AI)is defined as:

where Preand Pindenote magnitudes of reflected waves and incident wave, respectively. As shown in Fig. 5, the different AIs are listed for several scenarios. The results demonstrate that the AI is less than 1% in the case of L=60 mm,αmax=107. In this case, reflected waves will not exist in the wavefield and this is achieved at a low cost.Therefore,the following study is performed with this ALID setup.

Meanwhile, commercial FEM software is utilized to analyze the same work case. Four-node bilinear quadrilateral plane strain elements are adopted to mesh the structure,where the recommendation of element size is:

where leis the element length and λminis the interested shortest wavelength. In the FE model, the model is shown in Fig. 6.The length of the plate is large enough to avoid the reflected waves. The responses of two monitoring points, which stand for the districts before and behind the stiffener, are acquired to compare the performance of the two methods. As shown in Fig. 7, it is evident that a good agreement between the two methods is performed for both two points. Besides, the computation scales of the proposed method and FEM are listed in Table 1, which only refers to the mesh of the host structure. It is obvious that the TD-SFEM with ALID can simulate the wave propagation correctly without reflected waves. Moreover, when compared with FEM, the proposed approach has the ability to obtain reasonable results at less cost.

In addition, a comparison of the substructures ALID and successive ALID is carried out to demonstrate the virtues of the proposed method. The displacement responses at Point M for the two strategies with the same parameters(L= 60 mm,αmax=107)are depicted in Fig.8.It is noted that in comparing with the responses in Fig.7(a),these results contain an extra wave packet, which is related to the waves reflected by the left edge of the structure. In the legend, the numbers represent the quantities of substructures for the ALID. As per the amplitude of the reflected waves, it is clear that the more the quantities of the substructures, the better the absorption effect. In the case of eight-layer substructures,the AI is 2.8%, and the reflected waves can be ignored while the effect is still worse than that of the proposed ALID. In addition, after t=10-4s, the responses of the two methods diff in phase. Combining Figs. 7 and 8, It indicates that the proposed method has higher accuracy.

4. Guided wave propagation analysis

In this section, the proposed method is used to study the guided wave propagation in a stiffened plate. Many different scenarios are conducted to investigate the effects of the stiffener on wavefields. The model and load case are the same as the example described in Section 3.

4.1. Effects of the stiffener on wave propagation

As shown in Fig. 9, a plate without stiffener is considered as the control group with the same size as the stiffened plate.Based on the convergence analysis, the structures are divided by spectral elements with four-order interpolation (element size = 5 mm) and calculation time interval Δt=1×10-8s.The responses are acquired at the monitoring points described in Fig. 9, where points A (x= 200 mm) and B (x= 500 mm)are on the upper surface, and points A’(x= 200 mm) and B’(x= 500 mm) are on the bottom surface. The monitoring points A and B (A’ and B’) are symmetrical about the axis of the stiffener.

Table 1 Computation scales of two methods.

The displacement responses at monitoring points for the control group are illustrated in Fig. 10. There are no discontinuities of structure, such as the stiffener. Hence, the time histories only contain one wave packet for all points, and no reflection occurs. In x-direction, the responses of points A and A’ are almost the same, while the responses are opposite in y-direction. The same phenomenon can be seen in the case of points B and B’. As per the cut-off frequency of the dispersion curve and the mode shape of Lamb waves,37it is evident that all the wave packets captured in Fig. 10 are S0 mode.

In addition, Fig. 11 illustrates wavefields at several typical moments.After t=26×10-6s of excitation,the waves propagate from the applied points x= 0.1 m to both sides. At t=50×10-6s, the backward waves reach the left edge, no reflection occurs due to the ALID,and the forward waves continue to propagate.When t=130×10-6s,the forward waves reach the right edge and are absorbed by the ALID.After this,there will be no wave propagating in the plate.

In the case of the stiffened plate, the responses of monitoring points are plotted in Fig. 12, which contain noticeable reflected waves caused by the stiffener. At Points A and A’,there are five wave packets in the responses for both main directions. The excitation generates the first wave packet,and its amplitude and phase are the same as the responses in Fig. 10(a). The last four packets are reflected waves. Based on the displacement in x and y directions, wave Packets I and II are identified as S0 modes, Packets III and V are A0 modes, and Packet IV contains both S0 and A0 modes. Compared with the results of the control group, the mode conversion can be obviously observed. At Points B and B’, the responses contain four wave packets. The first packet is the transmitted wave generated by the excitation, and its phase is consistent with the results of the control group, but the amplitude is smaller. The last three packets are caused by the reflection of the stiffener. In the responses, Packet I, II and IV are S0, A0 and A0 modes. For all monitoring points,the wave packet located between t=120×10-6s and t=150×10-6s contains mixed modes of waves.

Fig. 13 shows the wavefield near the stiffener region at different moments. At t=45×10-6s, the wave propagates to the vicinity of the stiffener. After t=35×10-6s, the wave is divided into three parts.First,a part of waves continues propagating forward, and the waves are defined as transmitted waves. Some other waves are reflected by the stiffener and propagate backward,which is named reflected waves.The last part propagates into the stiffener. Because of the difference in wave velocity between S0 and A0 modes, the two modes of waves in the plate can be clearly distinguished in xdisplacement contour. The S0 mode propagates fast and has the same displacement pattern on the upper and lower surfaces. In comparison, A0 waves are slower and have opposite patterns on the upper and lower surfaces. The results are also supported by dispersion curves and the mode shape of Lamb waves.Hence,the mode conversion phenomenon is easily captured in the contour. In the stiffener, the waves, including S0 and A0, propagate upward until they reach the top surface.The width of the stiffener is 4 mm, and the frequencythickness product is 800 kHz mm, where the gap between the velocity of S0 and A0 waves gets narrow compared with the case for the plate.Therefore,at t=100×10-6s,just after the S0 waves propagate into the plate, the A0 waves are also about to enter the plate. After this, the S0 waves induced by A0 waves are faster than A0 waves induced by S0 waves.The two waves interfere at the monitoring region when t=120×10-6s,which results in wave packet located between t=120×10-6s and t=150×10-6s containing mixed modes of waves. Because the energy of S0 waves in the stiffener and their induced waves is relatively low, the subsequent wave packets can be considered unaffected.

In this part,frequency domain analyses are carried out.The amplitude-frequency curves for each captured wave packet at Point A are illustrated in Fig.15 and Fig.16 for two structures,respectively.In the control group,the wave packet frequency is mainly concentrated around 200 kHz,which is the same as the excitation frequency. The frequency results of the five packets are given in Fig. 16. It is worth noting that the frequency is also near the excitation frequency for every packet. Hence,the stiffener alters the frequency-thickness product on the propagation path, which results in the mode conversion phenomenon,but does not affect the frequency of waves.In addition, the 2D FFT is applied to study the wave components in the plates.In Fig.17,the relationship of wave number and frequency is plotted for the control group, which possesses only one hump region. Meanwhile, the relationship of S0 mode is calculated based on the dispersion curve. The results render that this hump indicates the S0 mode wave, and the wavefield in the control group only contains S0 mode waves.In the case of the stiffened plate, the result shown in Fig. 18 contains two humps. According to the dispersion curve, the higher hump represents the S0 mode waves and another one is the A0 mode waves. The conclusions for the two wavefields are consistent with the results discussed in time domain.

By comparing the wavefields in the two structures,the influences of the stiffener on wave propagation are studied in time and frequency domains. First, the stiffener significantly complicates the wavefield in the plate. On the one hand, due to the existence of the stiffener, Lamb waves will be reflected and transmitted after they propagate near the stiffener. On the other hand, waves are also reflected several times in the stiffener and then re-incident into the plate. Therefore, the wavefield in the stiffened plate can be regarded as a multisource field. Second, the stiffener changes the frequencythickness product of the propagation path, causing a mode conversion. This allows the multiple waves with different velocities to propagate in the structure, resulting in a complex interference phenomenon. The propagation behavior is illustrated explicitly in Fig. 14.

4.2. Parametric study

As discussed in the previous part,the stiffener has a significant influence on wave propagation. The heights and widths of the stiffener will change the wave propagation and result in totally different wavefields.Hence,in this part,the different scenarios are conducted to investigate the effect of the parameters of the stiffener.

In addition, a monitoring Point C (0.35,0.022) on the stiffener is pick up and the displacements are illustrated in Fig.20.Based on responses, it is noted that the height of the stiffener has few influences on the magnitude of the response.The peak values of the x-direction displacements are 5×10-12m for three cases.The difference is that the waves are reflected more quickly in the shorter stiffener.In the case of height 40 mm,the results show that when the first wave packet of the response has not entirely passed through the monitoring point,the subsequent reflected wave packets have already reached the point.The two wave packets interfere at the position,so it is difficult to identify the two packets based on the time-domain response clearly. When the height increases to 60 mm, the propagation path in the stiffener becomes longer so that the reflected waves only just reach the point after the first wave packet has completely passed that point. For the 80 mm case, the reflected waves arrive at the point after the first wave packet has passed through it for a while. Hence, the two wave packets are easily distinguished in this scenario.

Besides, wave propagation in three different widths of the stiffener is studied in this part. The widths are configured as 2 mm, 4 mm and 6 mm, respectively. The displacement responses at Point A are plotted in Fig. 21. In the xdirection, the responses of the three cases are basically consistent at the first wave packet. In the second packet, the phases of all responses are also consistent, while it is noticed that the amplitudes are obviously different. The lowest value of the response is captured in the case of Width 2 mm. Besides, as the width of the rib increases, the amplitude of the wave packet also increases. There is a similar phenomenon in the third wave packet. The result renders that the width of the stiffener has few influences on the velocity of waves S0prand A0pr. The differences among the three responses get obvious after t=120×10-6s.The reason is that in the different width stiffeners, the frequency-thickness product for wave propagation has changed,resulting in a difference in velocity of waves induced by the reflected waves in the stiffener, which finally causes the different responses.The conclusion is also available in the case of the y-direction responses.

The displacements at Point B are illustrated in Fig.22.In xdirection, the three responses have the same phase at the first two wave packets. After this, the phases of the responses are different because of the different frequency-thickness products.At the first packet, relating to waves S0pt, the amplitudes are negatively correlated with the thickness. It is the highest for Width 2 mm case and lowest in the case of Width 6 mm,which is opposite with the results of Point A. Hence, it is concluded that the width of the stiffener affects wave reflection and transmission. As the width of the stiffener increases, the energy of the reflected waves increases while the energy of the transmitted waves decreases. At the second packet, relating to waves A0pt, the relationship between the amplitude of responses and the width of the stiffener is opposite with the S0ptsituation.The width of the stiffener also influences mode conversion.With the increasement of the width of the stiffener, the more energy of S0 mode wave converts to A0 mode and the conversion phenomenon gets more pronounced, which is consistent with the results of the y-direction.

In a nutshell, the height of stiffeners affects the distance of the propagation path, thereby changing the arrival time of reflected waves.In the case of different heights,the interference of each wave packet is obviously distinct, which complicates wavefields and responses in the time domain.Hence,it is essential to select the matching structure size and excitation frequency. In this way, the reflected and transmitted waves of various modes are identified from the signals. Besides, the height has little effect on the peak value of each wave packet,and the mode conversion is not sensitive to the change of the height. The influence of width on wave propagation is mainly in wave reflection and transmission.As the width increases,the energy of the reflected wave becomes larger, and the transmitted wave gets lower. Meanwhile, the mode conversion phenomenon is more apparent. Besides, the width does not change the velocity of the reflected wave, but the frequencythickness product and velocity of the waves propagated in stiffeners will be different.

5. Concluding remarks and future research work

In this study, a time-domain spectral element method with a new proposed ALID is used to study the wave propagation in stiffened plates. The results render that the proposed method can simulate wave propagation with high efficiency compared with the other traditional approaches. The successive ALID has a better ability to eliminate the reflected waves,which also avoids complicated modeling.Based on the convergence analysis,the proposed method is validated by FEM software first. Then, the wave propagation in stiffened plates is studied by comparing with the non-stiffened plate. In this work, the wave scattering and mode conversion are analyzed in detail. Finally, the influence of geometric parameters of the stiffener on wave propagation is also investigated, where several widths and lengths of the stiffener are considered.Based on the numerical study in the paper, the following conclusions can be drawn.

(1) Based on the convergence analysis, the proposed TDSFEM with ALID has the ability to absorb the reflected waves from structural boundaries efficiently.The results have a good agreement with responses calculated by the commercial software, in which the dimensions of the model are big enough to avoid the reflected waves.

(2) The influences of stiffeners on wave propagation are emphasized as follows. First, because of the stiffener,Lamb waves will be reflected and transmitted near the stiffener. The re-incidence of waves from the stiffener to the plate significantly complicates the wavefield. Besides, the stiffener changes the frequencythickness product of the propagation path, which results in a mode conversion. Hence, in structure,multiple waves with different velocities propagate simultaneously, and a complex interference can be captured.

(3) In this study, the height of stiffeners mainly affects the distance of the propagation path. The different heights change the arrival time of each wave packet and result in distinct interference. The heights do not affect the energy of each wave packet, and the mode conversion is not sensitive to the change of heights. In the case of the width,it is concluded that the width is an important parameter affecting the reflection and transmission of waves.As the width of the stiffener increases,the energy of reflected waves gets larger and the mode conversion is also more apparent. The results render that the width will not change the velocity of the reflected wave in the plate. However, the frequency-thickness product and velocity of the waves propagated in stiffeners are different. This study facilitates the identification of wavefield changes caused by the stiffener or cracks and provides a guide for sensor configuration and damage indicator design.

In future work, the experimental research warrants further investigation,where the numerical study in this paper can promise a good design of the experiment setup.Common damage of stiffened planes, such as impact and air leak, could be considered.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This study was supported by the National Natural Science Foundation of China (Nos. 12072268 and 51705422).