Wave Motion Analysis in Plane via Hermitian Cubic Spline Wavelet Finite Element Method

A plane Hermitian wavelet finite element method is presented in this paper. Wave motion can be used to analyze plane structures with small defects such as cracks and obtain results. By using the tensor product of modified Hermitian wavelet shape functions, the plane Hermitian wavelet shape functions are constructed. Scale functions of Hermitian wavelet shape functions can replace the polynomial shape functions to construct new wavelet plane elements. As the scale of the shape functions increases, the precision of the new wavelet plane element will be improved.,e newHermitian wavelet finite element method which can be used to simulate wave motion analysis can reveal the law of the wave motion in plane. By using the results of transmitted and reflected wave motion, the cracks can be easily identified in plane. ,e results show that the new Hermitian plane wavelet finite element method can use the fewer elements to simulate the plane structure effectively and accurately and detect the cracks in plane.


Introduction
e application of plate structure in industry is very important [1]. Plane wave motion analysis has been researched in mechanical engineering in recent years [2]. Wave motion analysis remains an open research field in real engineering though mathematical principles are well developed [3]. Many numerical methods are used to analyze the motion of elastic waves area [4]. Manktelow et al. [5] proposed a perturbation analysis by using discretization finite element method to research wave motion in continuous and periodic structures. Wang and Sett [6] used the stochastic Galerkin method, where material parameters and mechanical functions are uncertain to solve the solid mechanics partial differential equation. Gravenkamp et al. [7] resolved the results of high frequencies wave motion by using the following methods: scaled boundary finite element method and the nonuniform rational B-splines. Komijani and Gracie [8] proposed the global enrichment method which adopts the harmonic functions. e merit of global enrichment and the generalized finite element method is that they can analyze wave motion in plane with cracks by using the phantom node method. Pamel et al. [9] derive the finite element equations which can solve the problem of three-dimensional elastic dynamic wave scattering motion. e fundamental characteristics of wave scattering in attenuation and dispersion are used to be studied. For solving the problem of plane vibration in engineering, most methods are difficult to identify small defects such as cracks. e reason is that small defects are related to high frequency effects. e latest technique for identifying small defects is shown by using the result of elastic wave motion in high frequency propagation and reflection. Stawiarski et al. [10] adopt the elastic wave to detect the initiation of the fatigue damage in isotropic plate. e propagation and reflection of waves indicate a bright path for nondestructive testing of structures. Dubuc et al. [11] used the three-dimensional numerical model to analyze guided wave motion with tensile and shear cracks in isotropic plane. Komijani et al. [12] adopted enhanced finite element models to solve dynamic crack propagation and wave propagation.
Numerical simulation has become the key factor of product design in engineering for the latest decades. To reduce the costs and product development time in engineering, many researchers adopt numerical simulation techniques to guide their design process, which is called conceptual design [13]. With the development of commercial software and computer resources, many engineers can use the numerical models to analyze more complex structures for fault diagnosis and for identification at higher frequencies [14]. At present, the most popular numerical modeling technique for high frequency wave motion is finite element method. Manktelow et al. [5] used integrated commercial software which can explore and optimize the complex structure to analyze nonlinear wave dispersion. e finite element method is used to accurately simulate the high frequency wave motion analysis.
e method needs at least 20 nodes which are generated by a lot of elements to simulate a wavelength that leads to a huge computational cost. Spatial discretization problem is the main problem that high frequency wave motion faced in engineering. By fine and accurate processing, the space structure, the wave motion dispersion, and response of the structure can be obtained. Nanda et al. [15] present a spectral finite element method which can adopt the efficient and accurate layerwise theory to analyze the nonuniform composite and sandwich structure beams. However, spectral finite element methods require modifying two-dimensional or three-dimensional problems, geometrical complexities, nonperiodic boundary conditions, and so on. Guo et al. [16] solved linear wave motion equations by adopting a fully discrete element method.
Due to the uncertainty of modeling, even a small error can cause a large error of high frequency fluctuation. erefore, the accuracy of structural models is very important. Park et al. [17] proposed a generalized multiscale finite element method to simulate the question of fluid flows. is paper considered a coupled factor of two equations, which can fine grid in order to improve accuracy. Wavelets have been widely used in many physics and engineering problems in recent years [18]. Due to the fact that the wavelet has the characteristic of multiresolution analysis, it provides a new mechanism that can decompose the solution into a series of coefficients. e numerical method of wavelet functions can be regarded as finite element shape functions. It seems like the shape functions of signal or image processing. Basu et al. [19] pointed out that the finite element, the boundary element, and the meshless methods have already replaced the finite difference methods and Ritz methods. ose methods may be replaced by the wavelet numerical methods in the near future. Chen and Ma [20,21] constructed beam and plane Daubechies wavelet finite element methods. ose can solve the Euler beam and thin plate bending problem. Because B-spline wavelet functions have the characteristics of display expression and high precision and efficiency, many scholars research the finite element method which adopts B-spline wavelet functions as shape functions [22,23]. Further, Xiang et al. [24] proposed Hermite cubic spline wavelet to solve intensity factors. Xiang indicated that the Hermitian scale and wavelets functions should be truncated. Xue et al. [25] presented a modified Hermitian interpolation wavelet base by adding the appropriate wavelet function to solve the wave motion and load identification. However, these interpolation functions satisfy the elements of C 0 , whose functions can be interpolated for the displacement. e method of interpolating the rotation by calculating the first derivative of displacement makes the accuracy of Hermitian spline wavelets be limited. Moreover, these interpolation functions are very complex, which contain more nodes in a wavelet element. Although the accuracy of the wavelet is very high, the amount of calculation is very large.
e new effective wavelet finite element methods that adopt Hermitian wavelet functions as shape functions are presented in this paper. e new Hermitian wavelet plane element is called Hermitian spline wavelet on interval (HSWI) element.
e new Hermitian wavelet functions satisfy the elements of C 1, whose functions can be interpolated for the displacement and rotation at the same time. Moreover, this new plane element has a very small number of nodes. e accuracy is higher than that of the Hermitian element constructed by the authors in [26]. e wavelet functions are orthogonal under the condition of given inner product. ese new shape functions can decouple totally or partially for the Hermitian plane wavelet element. e precision of element can rise by improving the scale and nesting the approximation space.

Hermitian Plane Wavelet Shape Functions.
e scale functions ϕ 1,k and wavelet functions ψ 1,k of Hermitian wavelet are shown in Figure 1. e equations of scale functions ϕ 1,k are e wavelet functions ϕ 1,k of Hermitian are when k is even, , when k is odd, where e shape functions of finite element method should satisfy the necessary condition that the sum of shape functions at any node is 1; the original scale functions of Hermitian wavelet cannot satisfy the conditions at the boundary node, which is called boundary problem. Stretching and translating can make some of Hermitian functions meet the requirements of Lagrange interpolation and the other meet the requirements of Hermite interpolation functions.
e new Hermitian interpolation functions are constructed by the Hermitian scale and wavelet functions. e equations are , when k is even.
ese functions can be interpolated for the displacement when k � 1, 3, 5, . . ., 2 j+1 + 1, as well as for rotation when k � 2, 4, 6, . . ., 2 j+1 + 2. When k is odd number, the interpolation functions satisfy the condition of element C 0 , and the displacement can be interpolated. When k is even number, the interpolation functions satisfy the C 1 type condition and can interpolate the rotation. Figure 2 is the graph of modified Hermitian wavelet shape functions. e wavelet space H 1 0 (0, 1) can be generated by the scale and wavelet functions; the equations of decomposition are where _ + is the symbol of direct sum, V 1 represents the initial scale space, the wavelet space is W j , and j is different level of wavelet.
e equations of Hermitian wavelet shape function are Shock and Vibration e Hermitian wavelet functions have an excellent characteristic: the first derivative has good continuity, which makes modified Hermitian wavelet functions meet interpolation condition of C 1 . ese features are more suitable for the research of beam, plane, and so on. With finite element shape functions, HSWI can solve the engineering problems with high precision. Figure 2 shows the new scale functions ϕ j,k , the corresponding approximation space is H 1 0 (0, 1) , and the intervals of HSWI shape functions are [0, 1]. Using the tensor product, the Hermitian plane wavelet approximation space H 1 0 (0, 1) is constructed. e initial scale space is V 1 , the wavelet space is W j , and j is different level of wavelet. So the subspace which generates tensor product is φ j � Φ j ⊗ Φ j , the scale functions are φ 1 � Φ 1 ⊗ Φ 1 at j � 1, the scale functions are φ 2 � Φ 2 ⊗ Φ 2 at j � 2, and so on. Figure 3 shows the tensor product of HSWI plane elements where the scale is j � 1, 2. e plane Hermitian wavelet shape functions can replace the traditional finite element shape functions. Using the new shape functions, stiffness matrix and mass matrix can be solved.
e Newmark time integration can calculate the results of high frequency wave motion by the stiffness matrix and mass matrix.

Hermitian Wavelet Finite Element Formula.
e elements are divided into two types in plane structure: one is plane stress element, and the other is plane strain element. Plane stress element is established based on the Hermitian wavelet shape functions. e plane strain element can use E/(1 − μ 2 ) and μ/(1 − μ) to replace E and μ, where E is Young's modulus, and μ is Poisson's ratio.
For the plane structure, the formula of potential energy is where h represents the thickness of element, f � f x , f y shows the body forces vector, the displacements vector is Assume that the material has the characteristic of being linearly elastic and isotropic; the stress equation is Adopting the scale j � 2 as an example, the arrangement of plane nodes is shown in Figure 4; the displacement functions in x direction and y direction are Φ represents the plane Hermitian wavelet shape functions, T shows the transformation matrix which is from wavelet space to physical space, the displacement in x direction is u, and in y direction it is v: e standard element domain can be gained by mapping the original element domain. Substituting equations (8) and (9) into equation (7), the principle of Galerkin variation can obtain the finite element equations: where e equation of consistent mass matrix is where the density is ρ , the plane area is A, G 00 � 1, 2) can be easily obtained. l ey and dη can replace l ex and dξ. e plane Hermitian wavelet shape functions are constructed by using the tensor product of modified Hermitian wavelet element in this paper. Substituting the plane Hermitian wavelet shape functions into the finite element formula, the stiffness matrix and mass matrix can be gained. e stiffness matrix can be converted from the wavelet domain to the physical domain. Similarly, the where K is the stiffness matrix, the mass matrix is M, C represents the damping matrix, the excitation force is F(t), and u, _ u, and € u are displacement, velocity, and acceleration, respectively.
e Rayleigh damping formula is used to calculate the damping matrix in this paper. e velocity and displacement which are in Newmark time integration are assumed as follows: where p represents the number of time steps and the time interval is Δt, which is from p − 1 to p step. Substituting equations (17) and (18) into equation (16), the response of wave motion equation is e wave motion in plane can be analyzed by equation (19). It is assumed that the material is homogenous and isotropic, and the displacements and velocities which are in the initial situation are zero.

Numerical Examples
e vibration structure especially for the high frequency vibration plays an important role in engineering. e high frequency vibration is also called wave motion. e effect of wave motion is becoming more and more important, and it has significant advantage in detecting small defects, especially cracks. Four working conditions are used to describe plane wave motion analysis in this paper. A sinusoidal signal has a frequency of 100 kHz, and the signal is added by Hanning window. e signal is used in this section as the excitation. Figure 5 shows that the excitation location is at point A. Time domain and frequency domain are shown in Figure 6, respectively. e HSWI finite element method is used to analyze wave motion of plane structure in this paper. e wave motion is assumed to be carried out under undamped conditions. Shock and Vibration e research object is a thin plate of plane, where length and width are 1 m and the thickness is 0.001 m. Aluminum is used for research in this paper; the material parameters are as follows: Young's modulus is 70 GPa, Poisson's ratio is 0.3, and the density is 2730 kg/m 3 . e computer used to calculate the wave motion was composed of Intel CPU, 1.7 GHz, and 35G memory. e Matlab software is used under the condition of Windows 10 operating system. Various small defects can be accurately detected by suitable high frequency excitation works on the mechanical structure.

10
Shock and Vibration into the longitudinal wave (or major wave, P wave) and the transverse wave (or minor wave, S wave). e influence of wave under the different crack lengths is researched in this section. According to equations (20) and (21), the speed of P wave and S wave can be solved. e time of wave motion was estimated as 0.25 ms: Shock and Vibration 11 e number of time steps can be divided into 2500 steps in this paper, and the Newmark time integration method is chosen to analyze wave motion. e plane can be divided into 20 × 20 regular quadrilateral elements by using the HSWI elements. A HSWI element has 18 degrees of freedom (or 9 nodes), so the plane model has 3,362 degrees of freedom. e P wave and S wave are numerically simulated by the HSWI elements with 3362 degrees of freedom in plane. e calculation results are shown in Figures 7-10, where the displacement v refers to P wave and the displacement u refers to S wave. e graphs of wave motion are displayed at different time.
e difference about wave patterns is obvious with or without crack. e crack is easily identified in the graph of P wave and S wave. e effect of S wave is greater than that of P wave from Figures 7-10. e difference of P wave is also obvious with or without crack at 0.12 ms and 0.25 ms. As the length of crack is increased, the waveform of crack becomes more and more obvious. When the length of crack is consistent, the crack waveform becomes larger and larger with time development. e displacement response diagram at point A, point B, point C, and point D is shown in Figures 11-14. e diagram also shows the P wave and S wave at each point. e excitation signal is at point A. e displacement response diagram can clearly see whether there is crack or not, as well as the length of crack. As the length of the crack increases, the interval time between the reflected waves and the

Analysis of Different Crack
Locations. e cracks are distributed at three locations in plane: the first crack is 0.25 m at the top and right end, the second crack is in the middle of the plane, and the third crack is 0.25 m at the bottom and left end. Assume that the length of three cracks is 200 m, and the depth of them is 0.5 mm. e boundary condition is free for four edges in plane. Figure 15 is the plane distributed graph of the crack. HSWI elements are used to analyze high frequency wave motion in plane, and the material is aluminum. An excitation signal in the form of a force pulse signal with amplitude of 100 N is applied at point A. e response results are observed at point A, point B, point C, and point D in this section. e wave patterns of P wave and S wave are investigated at different crack positions.
e numerical modeling is established in Figure 15 by the use of HSWI elements. Figures 16-19 display the calculation results at different positions for cracks. It is worth mentioning that the waveform and displacement response graphs are Figures 9 and 13 for the second crack. e cracks under different positions are expanded on analysis at different time points. e propagation and reflection of waves will change greatly when the crack location changes. e crack location 2 is in the middle (Figure 9), and the influence of P waves is greater than S waves compared with the waveform of without crack (Figure 7). e crack locations 1 and 3 have great influence on P and S waves compared with the waveform without crack (Figures 16 and 17). Compared to the transmitted waves, the reflected crack waves are dominant in the P waves. So the v displacement response is worth researching. e wave motion graphs of P wave and S wave, where the excitation signal is applied at point A, were displayed in Figures 13, 18, and 19. However, the response signals are observed at point A, point B, point C, and point D. e wave pattern varies greatly for the cracks at different positions. When the crack is not in the middle, it can be seen from Figures 18 and 19 that there are additional S waves at point A and point D. But there was no S wave at point A and point D under the absence of crack ( Figure 11). When the crack is in the middle, there is no additional waveform at point A and point D. e wave propagation and wave reflection have a certain interval time. Due to the influence of crack position, the S wave pattern and amplitude have changed obviously at point B and point C.
is phenomenon can be used to identify the location of different cracks. e cracks will reduce the amplitude of waves at point D in the v direction displacement. e geometry is similar to literature [3], with 40 × 40 spectral elements, and a total of 80,802 degree of freedom are used in this literature. In this paper, the new   Hermitian wavelet elements have 20 × 20 elements, which have 3,362 degrees of freedom. When the number of Hermitian wavelet elements is further increased, the results of numerical modeling are converged. e numerical results show that the plane structure with crack can be accurately analyzed in this paper.

Conclusions
e new plane Hermitian wavelet shape functions are constructed, and the shape functions have been substituted into the finite element equations to calculate the new elements in this paper. e new elements have the characteristics of high precision and less calculation, which can save the calculation time.
e new elements are used to analyze the wave motion and to calculate the crack for the different length and location. As the length of crack is increased, the waveform of crack becomes more and more obvious. When the length of crack is consistent, the crack waveform amplitude becomes larger and larger with time development. ere are significant effects of wave motion for the crack of different length. As the length of the crack increases, the interval time between the reflected waves and the transmitted waves gets longer and longer. e propagation and reflection of waves will change greatly when the crack location changes. When the crack is not in the middle, the additional displacement response signals can be measured in the u displacement. Due to the influence of the cracks, the waveform and amplitude of waves have changed significantly. is phenomenon can be used to identify the location of different cracks. It is proved that the new elements of researching the wave motion are feasible and effective.

Data Availability
e data used to support the findings of this research are available from [3,25].

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.