Wave analysis in one dimensional structures with a wavelet finite element model and precise integration method

Numerical simulation of ultrasonic wave propagation provides an efficient tool for crack identification in structures, while it requires a high resolution and expensive time calculation cost in both time integration and spatial discretization. Wavelet finite element model provides a highorder finite element model and gives a higher accuracy on spatial discretization, B-Spline wavelet interval (BSWI) has been proved to be one of the most commonly used wavelet finite element model with the advantage of getting the same accuracy but with fewer element so that the calculation cost is much lower than traditional finite element method and other high-order element methods. Precise Integration Method provides a higher resolution in time integration and has been proved to be a stable time integration method with a much lower cut-off error for same and even smaller time step. In this paper, a wavelet finite element model combined with precise integration method is presented for the numerical simulation of ultrasonic wave propagation and crack identification in 1D structures. Firstly, the wavelet finite element based on BSWI is constructed for rod and beam structures. Then Precise Integrated Method is introduced with application for the wave propagation in 1D structures. Finally, numerical examples of ultrasonic wave propagation in rod and beam structures are conducted for verification. Moreover, crack identification in both rod and beam structures are studied based on the new model. Introduction The safety, durability, and sustainability are essential requirements for civil structures in the longtermservice. The performance and the safety of civil structures would be gradually weakened due to possible damage from artificial factors or natural disasters. Thus, damage detection have attracted plenty of attention in different industry areas, i..e, aerospace, mechanical, materials science and civil engineering {Cha 2015 & 2017; Cole, 2004; Song, 2006; Song, 2010; Wandowski 2015; Zhang, 2017, Cheng}. The effective detection methods have been proved to be an effective way to prevent structure failure. Different approaches have been developed for damage detection in structures, such as vibration-based damage detection approaches{Yam, 2003;Chen, 2007 ;Adewuyi, 2011;Bagheri, 2009}, test method{An, 2015}, model-free method {Li, 2008}, wrapping method{Nicknam, 2011}, axial strain method{Blachowski, 2017}, deep response surface method{Zhou, 2013}, learning method{Cha, 2017} and so on. Nondestructive testing to detect small damages is still a challenging problem since it requires very high-frequency excitation guided wave signal. Numerical simulation of guided wave propagation in structures has been studied for a long time and numerous models has been applied for the simulation of guided wave propagation. A lot of numerical models have been developed for simulation of wave propagation in 1D structures{Kudela, 2007; Harari, 1995, Anderson, 2006; Seemann, 1996; Xiang, 2007}. Finite Element Method{Marfurt, 1984; Moser, 1999} is most widely used method because of its adaptivity for different shapes and dimensions in structure. However, it requires very fine mesh for structure, i.e., at least ten nodes per wavelength are required to obtain an accurate simulation result{Chen, 2012}. Hence the element size should be small enough for ultrasonic waves with high frequencies in FEM-based simulation. To improve the computation efficiency, wave propagation problems are usually solved in frequency domain and then convert the solution to time domain. FFT based spectral element method{Doyle, 1989} is one solution because it could transform the governing PDEs to a set of ODEs, which are much easier to solve, with constant coefficients. Then the solution of ODEs in frequency domain is transformed into time domain with inverse transform (IFFT). However, FFT based SEM is only applicable to solve problems of infinite rods, semi-finite rods, and beams due to the assumption of the periodic FFT. Laplace transform is subsequently applied by different researchers to replace FFT to overcome such problems{Igawa, 1999}. Thus, the time domain SEM{Kudela, 2007} is much more effective than FFT based spectral element method, especially in guided wave nondestructive testing with small damages, because each element in SEM has more than one node and could get the same accuracy with fewer elements. Wavelet transform analysis has been widely used in engineering applications due to its timefrequency analysis features, wavelet-based finite element method has also got its great success in modeling guided wave propagation. There are also two different kinds of wavelet finite element methods, one of which is time domain wavelet finite element method { Ma, 2003; Xiang, 2007; Amiri, 2015; Chen, 2012}, which is very similar with the traditional finite element method in time domain but with high spatial resolution, while the other one is frequency domain finite element method{Mitra, 2005}. Both methods have been applied in numerical simulation of guided wave propagation and crack identification in structures, time domain wavelet finite element method has been widely used since its feature of wide adaptivity in space, another advantage is that it requires much less DOFs (almost save 75% of DOFs space). However, the small time interval is still required, so the computational cost for this method is still expensive. As we know that the element type plays a very important role in getting accurate results and improving the efficiency, and almost all these research papers focus on the element type instead of time integration method. In a consequence, the high-order elements such as Spectral element and Wavelet finite element have been widely used in numerical simulation of ultrasonic guided wave propagation in structures. It’s also very important for us to study the time integration method and find an efficient and stable integration method. Newmark-beta method has been widely used as time integration method due to its stable property. Recently, a Fourier transform based wavelet finite element model{Shen, 2017} has got great success due to its low requirement on the time integration step, but it still has the limitation that it requires the time samples are large enough to find the accurate solution. Precise integration method, which is proposed by Zhong, has been widely used in highly nonlinear heat transfer problems and vibrations due to its high stability and high accuracy. More importantly, it could always provide the precise numerical solution for certain cases, especially for ultrasonic guided waves with known excitation forces. So it’s also very important for guided wave propagation researchers to know this important work. This manuscript provides the wavelet finite element model combined with precise integration method, as wavelet finite element model provides high resolution in space coordinate while precise integration method will need very small time integration step to get the accurate results. Chapter two will introduce how to construct the wavelet finite element model for 1D structures; then chapter three will give a brief introduction on the precise integration method. Chapter four will provide numerical examples of ultrasonic guided wave propagation in 1D structures. Also the time integration step will be discussed for verification of the advantages of our model and method; Finally, crack identification in rod and beam structures will be presented for validation. 2. Numerical models 2.1 B-spline wavelet on interval finite element Unlike Fourier transform, wavelet transform has two different variables, scale a and translation τ, and the wavelet transform is defined as: W(a, τ) = 1 √a ∫ f(t) ∗ ψ( t − τ a )dt Where ψ(t) is a wavelet function. There is another important function called scaling function φ in wavelet transform. The scaling function is used to build the spaces in multi-resolution spaces: Vj = Span{ φj,k, j, k ∈ Z} There are many wavelet and scaling functions developed by researchers and engineers, in this manuscript, B-Spline scaling functions will be used to construct the wavelet finite element model. 0 scale mth order B-spline scaling function and wavelet function are developed by Goswami [20]. The 0 scale 2 order scaling functions (j=0, m=2) is expressed as: ∅2,−1 0 (ξ) = { 1 − ξ, ξ ∈ [0,1] 0, else (3) ∅2,0 0 (ξ) = { 1 − ξ, ξ ∈ [0,1] 2 − ξ, ξ ∈ [1,2] 0, else (4) All other scaling functions could be built based on the definition, here we take ∅4,k 3 (ξ), (k = −3, −2, ... ,7) to build destinated B-Spline wavelet on interval finite element model, specifically, the plot for the scaling functions for 4 order and 3 scale are shown in Fig. 1. Fig.1. Plot of B-Spline scaling functions The Wavelet finite element model for ultrasonic wave propagation is the same as the general finite element model: M?̈?(t) + Ku(t) = f(t) Where M is the mass matrix, K is the stiffness matrix, while u(t) is displacement as a function of time, f(t) is the load. Here we will show a brief introduction of the derivation of the mass matrix and stiffness matrix. Firstly, the nodal displacement could be expressed as a sum of the scaling functions and its coefficients. u(ξ) = ∑ am,k j 2j−1 k=−m+1 ∅m,k j (ξ) = ∑ a4,k 3 7 k=−3 ∅4,k 3 (ξ) = Φa Where Φ is a row vector with 11 scaling functions, and a is a column vector of 11 wavelet coefficients. For one element with 11 nodes, the nodal displacement matrix u could be expressed as: u = Φae = [R] −1 ae Where u = [u(ξ1) u(ξ2) ⋯ u(ξn)] T is the nodal displacement vector for the nodes in one element, [R] = Φ = [Φ(ξ1) Φ (ξ2) ⋯ Φ (ξn+1)], by substituting the solution of a e , we could get the displacement as a function of nodal DOFs, u(ξ) = Φae = ΦRue = Nue In which N = ΦR is the shape function. There are 11 shape functions in total, for convenience, here we would like to show the plots of the shape functions: Fig.2 Shape functions for BSWI FEM From the plots of different shape functions for BSWI FEM, there is much difference for BSWI FEM than traditional FEM that shape functions of BSWI FEM may be larger than 1. But what is the same as traditional FEM is that both satisfies a basic property: