Wavelet-based Finite Element Method Research Articles

Investigation of approximate mode shape and transition velocity of pipe conveying fluid in failure analysis

Structures dynamic characteristics and their responses can change due to variations in system parameters. With modal characteristics of the structures, their dynamic responses can be identified. Mode shape remains vital in dynamic analysis of the structures. It can be utilized in failure analysis, and the dynamic interaction between structures and their supports to circumvent abrupt failure. Conversely, unlike empty pipes, the mode shapes for pipes conveying fluid are tough to obtain due to the intricacy of the eigenvectors. Unfortunately, fluid pipes can be found in practice in various engineering applications. Thus, due to their global functions, their dynamic and failure analyses are necessary for monitoring their reliability to avert catastrophic failures. In this work, three techniques for obtaining approximate mode shapes (AMSs) of composite pipes conveying fluid, their transition velocity and relevance in failure analysis were investigated. Hamilton’s principle was employed to model the pipe and discretized using the wavelet-based finite element method. The complex modal characteristics of the composite pipe conveying fluid were obtained by solving the generalized eigenvalue problem and the mode shapes needed for failure analysis were computed. The proposed methods were validated, applied to failure analysis, and some vital results were presented to highlight their effectiveness.

Modeling and analysis of in-plane and out-of-plane elastic wave propagation in a phononic-crystal circular beam

An efficient formulation of a circular Timoshenko beam is developed for static, vibration, and wave propagation problems. The B-spline wavelet on the interval (BSWI) interpolation functions are used to construct wavelet-based elements, which have analytical expressions at all levels and sufficient continuity. This paper derives static equilibrium, vibration, and wave propagation wavelet-based finite element models of in-plane and out-of-plane motions of circular beams according to the Hamilton's principle. Wave propagation characteristics of in-plane and out-of-plane waves in a phononic-crystal (PC) circular beam are analyzed using the wavelet-based finite element method (WFEM) based on the Bloch's theorem. Effects of geometric and material parameters on in-plane and out-of-plane wave propagation characteristics of the PC circular beam are discussed. Numerical simulations show that the WFEM is more effective for static, vibration, and wave propagation problems than the traditional finite element method (TFEM). The WFEM can achieve the same accuracy as the TFEM with much fewer elements and degrees of freedom. It is shown that both in-plane and out-of-plane elastic wave band gaps exist in the PC circular beam, which exhibits some interesting phenomena due to coupling effects. This study can provide good support for wave filtering and vibration control of PC circular beam structures.

Effect of internal surface damage on vibration behavior of a composite pipe conveying fluid

Composite pipes have become a viable alternative to metallic pipes in several applications. Flow-accelerated erosion and internal surface attack often result in thickness thinning that may compromise the structural integrity of the pipe and lead to its failure. In this paper, we investigate how the internal surface damage is reflected on the vibration behavior of a composite pipe conveying fluid. The defected pipe-fluid system is modeled using the extended Hamilton’s approach and discretized using the wavelet-based finite element method. The modal characteristics of the pipe vibrations have been obtained by solving the generalized eigenvalue problem. The developed model was validated and some benchmark solutions are presented to highlight the effects of the internal wall-thinning on the vibrational behavior of composite pipes conveying fluid. The obtained results facilitate the future research by revealing the potential of using the vibration signature as a basis for detection of erosion-induced internal defects in pipelines.

Band Structures Analysis Method of Two-Dimensional Phononic Crystals Using Wavelet-Based Elements

A wavelet-based finite element method (WFEM) is developed to calculate the elastic band structures of two-dimensional phononic crystals (2DPCs), which are composed of square lattices of solid cuboids in a solid matrix. In a unit cell, a new model of band-gap calculation of 2DPCs is constructed using plane elastomechanical elements based on a B-spline wavelet on the interval (BSWI). Substituting the periodic boundary conditions (BCs) and interface conditions, a linear eigenvalue problem dependent on the Bloch wave vector is derived. Numerical examples show that the proposed method performs well for band structure problems when compared with those calculated by traditional FEM. This study also illustrates that filling fractions, material parameters, and incline angles of a 2DPC structure can cause band-gap width and location changes.

B-Spline Wavelet-Based Finite Element Vibration Analysis of Composite Pipes with Internal Surface Defects of Different Geometries

The vibration analysis of composite pipes with internal wall defects due to erosion-induced surface degradation is investigated. The surface defects are treated as discontinuities. The geometry of the discontinuity is permitted to vary within the cross-section both in the angular and radial directions, and to occupy any length of the pipe span. A B-spline wavelet-based finite element method (BWFEM) that takes advantage of the localization properties of wavelets is invoked; thus utilizing its effectiveness in modeling of crack problems and local damages. The composite pipe was treated as beam elements that obey the Euler–Bernoulli beam theory. Unlike the conventional finite element method (FEM), the developed BWFEM uses fewer elements without compromising the accuracy. Numerical simulations are performed to demonstrate the accuracy and efficiency of the developed element through comparison with available results in the literature, as well as results obtained using ANSYS. Some benchmark solutions are obtained for the composite pipe with internal surface defects of different geometries.

Wavelet-based finite element method for multilevel local plate analysis

In this paper, an efficient multilevel method is presented for local static analysis of plates based on the coupling of finite element method and discrete wavelet transform (FEM–DWT). The problem is discretized using finite element method and the corresponding governing equation is transformed into a localized one by applying the discrete Haar wavelet. Then the obtained governing equation is reduced using special averaging and reduction algorithms. Two types of the localization approach are used, the first is localization with respect to nodes and it is rather efficient for plates with localized parameters (such as concentrated loads or stress concentration). Another approach is the localization with respect to the degrees of freedom of each node and it is rather efficient for the evaluation of the effect of the degrees of freedom on the plate. The numerical results indicate that the proposed method provides accurate results for the selected regions, with respect to corresponding FEM solution, with a considerable reduction in the size of the problem. Moreover, in problems which itself have a localized properties, the efficiency of the method is considerably increased.

Vibration analysis of an elastically restrained microcantilever beam under electrostatic loading using wavelet‐based finite element method

An electro-mechanical analysis of a microcantilever beam considering the effect of size dependence and flexible supports is presented. Both static and dynamic analyses are performed to show the coupled effect of flexible support and electrical voltage on the static and dynamic performance of the microcantilever beam. The wavelet-based finite element method (FEM) is used to derive the elastodynamic model of the microcantilever beam. The energy expressions are derived using couple stress theory, while additional energy terms are introduced to represent the flexible boundaries and fringing fields. Numerical simulations and comparisons with experimental data show the validity of the developed wavelet-based finite element model and its potential in tackling practical problems in the design and evaluation of micro-devices.

Modal frequencies of fiber-reinforced polymer pipes with wall-thinning using a wavelet-based finite element model

Wall-thinning due to chemical reactions, heat, erosion, or a combination of such influences is the most dominant type of internal surface damage in piping systems. In order to examine the effect of wall-thinning on the natural frequencies, the elastodynamic model of the fiber-reinforced polymer pipe is formulated using a wavelet-based finite element method. In this context, a new set of Hermite shape functions is developed. The generalized eigen value problem is solved and the natural frequencies are obtained for an fiber-reinforced polymer pipe with different depths and locations of the wall-thinning. Moreover, the effect of wall-thinning on the modal frequencies of the pipe was verified experimentally. Both the analytical and experimental results demonstrate the potential of using vibration signature to detect internal surface damage in fiber-reinforced polymer pipes.

A second-generation wavelet-based finite element method for the solution of partial differential equations

A new second-generation wavelet (SGW)-based finite element method is proposed for solving partial differential equations (PDEs). An important property of SGWs is that they can be custom designed by selecting appropriate lifting coefficients depending on the application. As a typical problem of SGW algorithm, the calculation of the connection coefficients is described, based on the equivalent filters of SGWs. The formulation of SGW-based finite element equations is derived and a multiscale lifting algorithm for the SGW-based finite element method is developed. Numerical examples demonstrate that the proposed method is an accurate and effective tool for the solution of PDEs, especially ones with singularities.

A Wavelet-Based Finite Element Method for Modal Analysis of Beams

A new wavelet-based finite element method was proposed for analyzing modal parameters of beams. The Hermite cubic splines wavelet on the interval (HCSWI) was employed as the multi-scale interpolating basis for construct beam element. For the orthogonal characteristic of the wavelet basis with respect to given inner product, the corresponding multi-scale finite element equation will decoupled across scales totally or partially and suit for nesting approximation. Some numerical examples indicate that the proposed method have higher efficiency and precision.

Numerical solution of Poisson equation with wavelet bases of Hermite cubic splines on the interval

A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.

A novel wavelet-based finite element method for the analysis of rotor-bearing systems

The rotor dynamic theory, combined with finite element method, has been widely used over the last three decades in order to calculate the dynamic parameters in rotor-bearing systems. Since the wavelet-based elements offer multi-scale models, particularly in modeling complex systems, the wavelet-based rotating shaft elements are constructed to model rotor-bearing systems. The effects of translational and rotatory inertia, the gyroscopic moments, the transverse shear deformations, and the internal viscous and hysteretic dampings are considered in the present formulation. Numerical examples and experimental studies show the correctness of the wavelet-based finite element model of rotor-bearing system. Results of forward and backward whirl speeds and damped stability are presented and compared with other published works.

A Wavelet-Based Finite Element Method for the Self-Shielding Issue in Neutron Transport

This paper describes a new approach for treating the energy variable of the neutron transport equation in the resolved resonance energy range. The aim is to avoid recourse to a case-specific spatially dependent self-shielding calculation when considering a broad group structure. This method consists of a discontinuous Galerkin discretization of the energy using wavelet-based elements. A Σt-orthogonalization of the element basis is presented in order to make the approach tractable for spatially dependent problems.First numerical tests of this method are carried out in a limited framework under the Livolant-Jeanpierre hypotheses in an infinite homogeneous medium. They are mainly focused on the way to construct the wavelet-based element basis. Indeed, the prior selection of these wavelet functions by a thresholding strategy applied to the discrete wavelet transform of a given quantity is a key issue for the convergence rate of the method. The Canuto thresholding approach applied to an approximate flux is found to yield a nearly optimal convergence in many cases. In these tests, the capability of such a finite element discretization to represent the flux depression in a resonant region is demonstrated; a relative accuracy of 10–3 on the flux (in L2-norm) is reached with less than 100 wavelet coefficients per group.

A wavelet-based stochastic finite element method of thin plate bending

A wavelet-based stochastic finite element method is presented for the bending analysis of thin plates. The wavelet scaling functions of spline wavelets are selected to construct the displacement interpolation functions of a rectangular thin plate element and the displacement shape functions are expressed by the spline wavelets. A new wavelet-based finite element formulation of thin plate bending is developed by using the virtual work principle. A wavelet-based stochastic finite element method that combines the proposed wavelet-based finite element method with Monte Carlo method is further formulated. With the aid of the wavelet-based stochastic finite element method, the present paper can deal with the problem of thin plate response variability resulting from the spatial variability of the material properties when it is subjected to static loads of uncertain nature. Numerical examples of thin plate bending have demonstrated that the proposed wavelet-based stochastic finite element method can achieve a high numerical accuracy and converges fast.

A multivariable wavelet-based finite element method and its application to thick plates

A multivariable wavelet-based finite element method (FEM) is presented to resolve the bending problems of thick plates. The interpolating wavelet functions based on boundary conditions are constructed to represent the generalized field functions of thick plates. The formulation of multivariable wavelet-based FEM is derived by the Hellinger–Reissner generalized variational principle with two kinds of independent variables. The proposed formulation can be solved directly when the stress–strain relations and the differential calculations are not utilized in determining the variables. The applicability of the multivariable wavelet-based FEM is demonstrated by determining the bending solutions of a single thick plate and of an elastic foundation plate. Comparisons with corresponding analytical solutions are also presented. The wavelet-based approach is highly accurate and the wavelet-based finite element has potential to be used as a numerical method in analysis and design.