China, Email: Baiyu berkeley. Rod structure has been widely used in aerospace engineering and civil engineering. Nondestructive testing is a very important method applied to detect unseen flaws in structures, ultrasonic wave nondestructive testing has been used in many areas. Finite Element Method is one of the most widely used numerical methods but would have a high cost when doing simulation on ultrasonic wave due to the requirement of small time interval and element size.
Wavelet based finite element method could improve the spatial resolution with fewer elements needed but still needs very small time interval. Laplace transform could easily convert the time domain into frequency and then inverse to time domain. This paper presents an innovative method combining Laplace transform and B-spline wavelet on interval BSWI finite element method, which could not only decrease the element number but also increase the time integration interval. Moreover, this innovative method is applied to simulate the ultrasonic wave propagation in 1D rod structure as well as used for nondestructive testing of damages in rod structures.
Thus structure design and nondestructive testing has caught a lot of attention in Aerospace, Mechanical structures and civil engineering, like Zhang et al. Structure design is the first step to avoid disaster of happening, and nondestructive testing would help the structure to work safely like Hu and Pratt and Hu et al. Moreover, both experimental and numerical methods have been applied for nondestructive testing, including vibration-based damage detection approaches, like Yam et al.
As more and more engineering problems could be solved and simulated on computer with numerical methods like Hu , Hu et al. Machine learning has been a hot topic in recent years and has also been used in engineering problems like Liu et al. A large amount of numerical methods have been developed for elastic wave propagation simulation in structures, such as finite element method FEM applied by Marfurt and Moser et al. Obviously, the mesh size and temporal interval could not only determine the accuracy of numerical simulation results, especially due to its high frequency property but also could determine the running time of numerical simulation time.
Thus, a good numerical simulation method could not only get accurate results but also could be time efficient. A lot of numerical models have been developed for numerical simulation of wave propagation in rod structures, like Harari and Turkel , Seemann and so on.
Finite element method is one of the most widely used one and has been applied in simulation of ultrasonic wave propagation by many researchers like Tang and Yu and Tang et al. Hence the element size must be very small especially when the frequency of ultrasonic wave is very high.
Wavelet finite element method is a relatively new numerical simulation method developed in recent years. It should be noticed that there are two kinds of methods are developed based on wavelet transform, the other one method developed by Mitra and Gopalakrishnan is in frequency domain. This is because wavelet is a good tool for both time and frequency analysis.
The B-splined wavelet finite element models used by Chen et al. The strongest advantage of time domain WFEM is that only very few elements are needed for accurate analysis. However, the time interval is still required to be very small to find desired solutions, so the cost for calculation is still high. Another option to solve these problems is to solve the dynamic problems in frequency domain and then convert the solution back to time domain for visualization.
Most importantly, usually one single element is sufficient to handle a whole rod structure of any length under the case that there is no discontinuity, so the time cost is much more efficient than traditional FEM. Igawa et al. In this paper, a novel method that combines Laplace transform with B-splined WFEM is proposed to simulate the wave propagation in bar structures. LWFEM would combine the strength of WFEM which would build a high accurate finite element model with the strength of Laplace transform which will transform the time domain to frequency domain for faster calculation.
In such a way LWFEM could be applied in wave propagation as well ultrasonic damage detection with high accuracy and small calculation cost.
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Besides, numerical simulation of damage detection in rod structures will be applied with this novel method. The governing equations for ultrasonic wave propagation in isotropic structure is shown below based on Navier equations:. There are usually three different modes in cylindrical waveguides, namely, longitudinal modes, torsion modes, and flexural modes. The longitudinal modes have two types of displacement radial and axial displacement that propagate in the rod structure.
Dispersion curves are usually used to depict the relationships between frequency and eigenvalue, as well as phase velocity and group velocity. The relationship between wave velocity and frequency is usually depicted by Pochhammer frequency equation, which is shown in equation 2 :. Selecting the appropriate excitation frequency based on the dispersion curves of UGW in rod structure could help excite the ideal UGW mode, so the dispersion curve is really important. A typical dispersion curve for a steel rebar is shown in figure 1 shown in Li et al The specific procedure of plotting the dispersion curves could be found in Li et al B-spline wavelet function is built with piecewise polynomial by joining different knots together on the interval.
In order to have at least one inner wavelet on the interval [0 1], for any picked scale j , the dimension of the mth order B-spline scaling function must satisfy the following equation:. Wavelet transform would have two corresponding functions, one of which is mother wavelet function also called wavelet function , and the other is father wavelet function also called scaling function.
In this paper, the 4 scale 3 rd order scaling functions are selected to build the B-spline wavelet on interval finite element, also shown in Shen et al. For one dimensional classical rod structure, by transforming any subdomain [ a,b ] to basic BSWI wavelet subdomain [ 0 1 ], where the basic rod element is shown below,. The displacement as a function of scaling function and wavelet coefficients could be expressed as,. But the FEM is based on the nodal displacement so that the new displacement equation must be established.
In which N e is the shape function. By substituting the displacement equation into the potential energy U e and kinetic energy T e which are functions of displacement shown below. Since wave propagation in rod structure is a dynamics process, by building the stiffness and mass matrix of BSWI rod element, we could get the final WFEM based wave propagation equation in rod structure:. Where u t is the displacement vector in time domain, f t is the time domain excitation force vector. Laplace transform could convert the time domain equation into frequency domain equation shown as Eq.
Time-Domain Hybrid Global-Local Prediction of Guided Waves Interaction with Damage
Thus, we could obtain the displacement u s in Laplace domain when obtaining the accurate equivalent Laplace domain stiffness matrix. Then the time domain displacement could be obtained by applying inverse Laplace transform. In order to achieve the numerical results in an easy way, the numerical Laplace method is used in this paper, as we know that the nodal displacement in time domain could be achieved by substituting the frequency domain solution and inverse Laplace transform, which is shown below,. By applying substitution rule of variables and change the integration variable s to be w , the following equation could be achieved,.
As we all know that Laplace transform is a symbol operation and is very difficult to get the accurate solution in for matrix operation, while Fast Fourier Transform is very easy to achieve in MATLAB, so it would be excellent if we could find a way to build a relationship between Laplace transform and Fourier transform. In a similar way, the Laplace transform of activation force is defined as,. Thus, we could use the fast Fourier transform to replace the symbol operation of Laplace transform. Two rod element models are proposed, classical rod element with 11 nodes for BSWI43 element is shown in Fig.
The classical rod element theory has been proposed in part 2. In Rayleigh-Love rod theory, the lateral motion that holds a significant role for large diameter rods or high frequency problems for each node is considered. The displacement field of Rayleigh-Love rod is shown by. The three DOFs for each node on the Rayleigh-Love rod model are dependent on each other, we could only the longitudinal displacement as independent variable.
Considering the lateral inertia, the kinetic energy T e of Rayleigh-Love rod element is.
Where, I p is the polar moment of the inertia of the cross section. Due to the influence of axial force, the opening crack mainly occurs in the axial rod. The spring used to simulate the crack in rod only has axial stiffness, and the axial flexibility of spring c a can be calculated based on Castigliano's theorem shown in Przemieniecki and Tada et al. Several numerical examples of wave propagation simulation in rod structures are proposed to validate the Laplace based wavelet finite element method, a uniform rod is used in the numerical simulation, the geometry parameter and material properties are shown in Table 1.
An excitation signal with 5-cycle sinusoidal tone burst is picked for wave propagation simulation in rod structure, the single central frequency of which is kHz, and the largest frequency is khz as shown in Fig. The excitation signal in time domain is listed in Eq. Also, the plots of the excitation signal in time domain and wavelet time-frequency spectrum are shown in Fig.
Guided Waves in Structures for SHM. The Time - domain Spectral Element Method
Since the element size and time step increment are dependent on the wavelength of the excitation signal, which is related with the maximum frequency. Firstly, a comparison of ultrasonic wave propagation in rod between Laplace based BSWI method and theoretical group velocity is compared in Fig. As we could see from Fig.
And we would like to study the advantages and disadvantages of this method. For comparison, conventional FEM would also be applied to compare the advantages and disadvantages between the two methods. From the time frequency analysis of the wave propagation in rod structure, the central frequency is moving along the rod as the wave propagate along the rod, which also proves the validity of our method. The displacement response at the middle point of the rod is picked for comparison for these two different numerical methods.
Also the EPW is different for different simulation methods. In comparison, in the Laplace based wavelet finite element method, we will also set four different values for the number of elements per wavelength, which are set as 0. As we could see from this plot, the results would converge very quickly when EPW is bigger than 0. This means that we only need 0.
Another important factor in finite element method for time integration is the setup of time step, selecting a good value for time interval is so important that it could influence the accuracy of the results as well as the time cost during computation. So choose an appropriate time interval value which could both ensure the accuracy of results and not let the time cost be too high. Here we would like to come up with a concept of number of integration steps per period denoted as SPP.
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And the comparison results for the two methods are shown in Fig. As shown in Fig. Wavelet transform could provide more information on ultrasonic wave propagation in rod structure. Since wavelet transform is a very good time-frequency analysis tool, so we would like to study the time-frequency properties of ultrasonic wave propagation in rod structure.
A new excitation signal is proposed to study the velocity dispersion of guided waves in rod structure, where double center frequencies Khz and Khz are included in this excitation signal. Also, the equation of this new excitation signal is shown in Eq. Both rod elements proposed in previous sections are applied to find the ultrasonic wave propagation response at the middle of the rod by LWFEM.
Firstly, the wave propagation response and the velocity dispersion in classical rod structure and in Rayleigh-love rod are shown in Fig. Figure 10 a, b , because the waves of each frequency component propagate at the same rod speed. However, for the Rayleigh-Love rod theory, it is can be seen from Fig. The group velocities of waves in the vicinity of Hz change slowly, while those of in the vicinity of Hz have lower speeds and change more quickly. When the ratio of the cross-section size to the wavelength is less than 0. Otherwise, it is necessary to develop and apply the complex multi-dimension theory.
Ge, L. Ghajari, M. Ha, S. Haywood, J. Hollandsworth, P. Hossain, M. Katunin, A. Kazemi, M. Khoo, S.
Time-Domain Hybrid Global-Local Prediction of Guided Waves Interaction with Damage
Kudela, P. Mahdavi, S. Martin, M. Ostachowicz, W. Patera, A. Perry, M. Sanchez, N. Xu, Q. Xu, C. Yan, G. Personalised recommendations. Cite article How to cite?
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