Wavelet Finite Element Method Research Articles

A semi-analytical wavelet finite element method for wave propagation in rectangular rods

Prior knowledge of the dispersion curves and mode shapes of guided waves provides valuable information for wave mode selection and excitation in the field of non-destructive evaluation (NDE) and structural health monitoring (SHM). They are typically computed by the matrix methods, the finite element (FE) and semi-analytical finite element (SAFE) methods. However, the former is prone to numerical instability, and the latter two are limited by the refinement level of the FE mesh. In this paper, a semi-analytical wavelet finite element (SAWFE) method is presented to characterize wave propagation in rectangular rods. The piecewise polynomial interpolation functions of the SAFE method are replaced by two-dimensional scaling functions of the B-spline wavelet on the interval (BSWI). To demonstrate the accuracy of the proposed SAWFE technique, the propagation of guided waves in an aluminium plate is studied first. Then, the propagation of guided waves in rectangular rods of arbitrary aspect ratio is investigated. The results of this work clearly show that the SAWFE method presented here has higher accuracy and efficiency than the SAFE method.

Free Vibration Analysis of Laminated Sandwich Plates Using Wavelet Finite Element Method

The accuracy of a wavelet depends on the choice of the mother wavelet adopted. The present work aims to predict the free vibration behavior of laminated sandwich plates using wavelet finite element (WFE). Different kinds of mother wavelets, namely, B-spline wavelet on the interval (BSWI), Gaussian, Haar, Daubechies 6 (db6), Biorthogonal 3.7 (bior3.7), Coiflet5 (coif5), Symlets (sym8), Morlet, Mexican hat (Mh), and Meyer mother wavelets, are employed in WFE for predicting the frequencies. Both symmetric and unsymmetric laminates are studied using the proposed approaches. A wide range of problems, including the influence of the geometric and material properties and end conditions on the free vibration behavior of the laminated sandwich plates, are solved. The effectiveness of the WFE over the conventional finite element method in terms of computational efficiency is discussed. In conclusion, BSWI-based WFE method (WFEM) is found to be the most accurate and computationally efficient in predicting the free vibration behavior of laminated sandwich plates. The accuracy of the WFEM depends widely on the type of mother wavelet adopted.

Mid-frequency vibration analysis of built-up structures using WFE-SEA method

The idea of classifying built-up structures into hybrid deterministic-statistical subsystems is formed as a framework to deal with the dynamical response in the mid-frequency region. This paper explores the potential combination of the wavelet finite element method and SEA, resulting in the hybrid WFE-SEA method, to achieve a solution with reduced computational burdens and increased accuracy. The development of the hybrid WFE-SEA method commences with the formulation of a 3-D B-spline wavelet theory, utilizing the semi-orthogonality of B-spline wavelet. A multi-resolution analysis of this theory facilitates the creation of a 3-D B-spline wavelet on an interval element model, incorporating nodes distributed on both the surface and inside of the hexahedron. Subsequently, the 3-D B-spline wavelet on an interval element is integrated with SEA theory, producing in a hybrid WFE-SEA framework suitable for examining dynamic problems of built-up structures. The energy-balance equation is derived by coupling the 3-D B-spline wavelet theory with diffuse field reciprocity relation to estimate the mid-frequency dynamic response. Furthermore, the validity of the proposed approach is rigorously assessed by comparing it against detailed mid-frequency analyses, including the hybrid FE-SEA method and the hybrid ES-FE-SEA method. The conclusion highlights the substantial agreement of the present hybrid method with other approaches, achieved at a small cost in terms of degrees of freedom.

Predicting dynamic characteristics of a submerged vertical cylinder in the wide-frequency domain based on the wavelet fluid-solid interaction method (WFSIM)

Due to the huge computational cost and dispersion errors, wide-frequency and fluid-structure interaction dynamic analysis of submerged cylindrical shells is difficult to carry out in the design phase. To solve this problem, the wavelet fluid-solid interaction method (WFSIM) is proposed in this paper to avoid the huge computational cost and dispersion errors when predicting the dynamic characteristics of submerged vertical cylinders in a wide-frequency domain. For this purpose, the formulas of the wavelet fluid-solid coupling algorithm to couple the wavelet finite element method and the theory of fluid-structure interaction are derived and introduced carefully first. Then, the formulas to build the numerical model of submerged vertical cylinders are given and introduced in this paper. To verify the effectiveness and efficiency of the proposed method, an experimental study is adopted for predicting the dynamic characteristics of the scaling model of a submarine cabin in a wide-frequency domain. The experimental results indicate that the WFSIM can be adopted for predicting the dynamic characteristics of submerged vertical cylinders in the wide-frequency domain within a few seconds and that the relative errors are less than 10%. Additionally, the wavelet fluid-solid coupling algorithm can also be introduced to model other submerged vertical cylinders for predicting the dynamic characteristics in the wide-frequency domain.

Study on Propagation Characteristics of Guided Waves in Plate Structure via Semianalytical Wavelet Finite Element Method

The propagation characteristics of guided wave in waveguide structure plays an indispensable role in guided wave inspection. However, the analytical matrix method fails to solve the problems of wave propagation in waveguides with complex cross-section. The finite element method (FEM) has high computational cost for guided wave propagation problems. To overcome those shortcomings, a novel semi-analytical wavelet finite element (SAWFE) method is presented in this paper to study the guided wave propagation characteristics of plate structures. With the advantage of multi-scale and multi-resolution, the B-spline wavelet on the interval (BSWI) scale functions are adopted to construct one-dimensional BSWI elements and form the shape function. The element displacement field represented by the coefficients of wavelet expansions is transformed from wavelet space to physical space through the corresponding transformation matrix. Then, the constructed one-dimensional BSWI elements are employed to discretize the cross-section of waveguide structure, meanwhile, the wave displacement field in the wave propagation direction is described in an analytical way as a harmonic exponential function. The propagation characteristics of guided wave in aluminum plate and composite laminate are studied by using the proposed method, and the results show that the SAWFE method has high accuracy and efficiency in calculating dispersion curves and analyzing wave structure. The simulation analysis and experimental validation results further verify the effectiveness of the proposed method.

A three-dimensional B-spline wavelet finite element method for structural vibration analysis

In this study, a three-dimensional (3-D) B-spline wavelet finite element method (WFEM) is proposed by combining the wavelet theory and 3-D finite element method (FEM). Owing to the semi-orthogonality and multi-resolution analysis of spline wavelet over the solution domain, a cuboid with uniform distribution nodes is established for the 3-D B-spline wavelet on the interval (BSWI) element. The governing equations of vibration for the elastic structures are derived by introducing the 3-D B-spline wavelet theory into the displacement components of each node. A procedure for assembling the BSWI elements is presented by redefining the index element node functions. Numerical studies including the vibration analysis of solid and hollow beams and plates under different boundary conditions are performed to show the performance of the present method. A comprehensive comparison of the predicted vibration characteristics obtained by the present method, traditional FEM, and reference data is presented to verify the accuracy and efficiency of the 3-D B-spline WFEM. The core of the present method is involving the 3-D B-spline wavelet to replace the polynomial interpolation functions in the traditional FEM, which enhances computational accuracy during the vibration analysis of the elastic structures.

Thermo-mechanical coupling analysis of laminated composite plates using wavelet finite element method

This paper proposes a novel high performance wavelet finite element formulation for thermo-mechanical coupling analysis of laminated composite plates. The Reddy’s higher-order shear deformation theory (HSDT) adopted for theoretical formulation represents parabolic variation of transverse shear stresses through thickness to take into account the effects of transverse shears and traction-free boundary conditions for laminated composite plates. The governing equations involving displacement field and temperature gradient field in term of the thickness variable are derived by implementing the principle of virtual work. Then the B-spline wavelet on the interval (BSWI) wavelet-based element is constructed by means of scaling functions as approximating functions instead of polynomial interpolations used in traditional finite element method (FEM). The convergence, accuracy and efficiency of proposed BSWI wavelet-based method are verified for mechanical and thermal analysis of laminated composite plates with fewer degrees of freedom. Then numerical examples concerning various length-to-thickness ratios, aspect ratios and stacking sequences are investigated for thermo-mechanical coupling analysis of symmetric and anti-symmetric laminated composite plates. These comparison examples demonstrate the accuracy and reliability of the proposed BSWI wavelet-based method comparing with the three-dimensional elasticity solutions and referential numerical solutions available in literatures.

An improved hydrodynamic/acoustic splitting method for fluid–structure interaction feedback with elastic boundaries

To simulate the noise caused by acoustic radiation, scattering, and flow, standard hydrodynamic/acoustic splitting methods have been extensively researched and developed. These methods decompose compressible Navier–Stokes equations into incompressible Navier–Stokes equations and acoustic perturbed equations but generally do not consider the influence of the fluid–structure interaction with elastic boundaries. In response to this defect, a wavelet finite element method is introduced to construct the elastic boundary static model, and a wavelet immersed boundary method is used to simulate the fluid–structure coupling incompressible flow field. In addition, a perfectly matched layer technique is applied to form non-reflective boundary conditions in the acoustic perturbed equations. Based on these approaches, an improved hydrodynamic/acoustic splitting method is presented for fluid–structure interaction feedback with elastic boundaries. Finally, several numerical examples were calculated, including acoustic radiations and scatterings, as well as flow-induced noise, to verify that the new method is a simple and efficient method for fluid–structure–sound interactions.

Heat transfer analysis of evaporator of air source heat pump (ASHP) in case of frost based on chirplet finite element method

To enhance the computing accuracy and reduce the computing time of heat transfer of evaporator in ASHP, the Chirplet finite element model is established. Firstly, heat transfer model of evaporator in case of frost based on large eddy simulation is established. Secondly, heat transfer Chirplet finite element model of evaporator considering frosting is established. And then frosting model of evaporator is confirmed. Finally, outlet temperature of refrigerant of evaporator is calculated by using Chirplet finite element method, B-spline wavelet finite element method, Gabor finite element method, Daubechies wavelet finite element method, traditional finite element method and experiment, analysis results illustrate that Chirplet finite element method can obtain highest computing accuracy and least computing time, which has wide application prospect in other fields.

3D Wavelet Finite-Element Modeling of Frequency-Domain Airborne EM Data Based on B-Spline Wavelet on the Interval Using Potentials

We present a wavelet finite-element method (WFEM) based on B-spline wavelets on the interval (BSWI) for three-dimensional (3D) frequency-domain airborne EM modeling using a secondary coupled-potential formulation. The BSWI, which is constructed on the interval (0, 1) by joining piecewise B-spline polynomials between nodes together, has proved to have excellent numerical properties of multiresolution and sparsity and thus is utilized as the basis function in our WFEM. Compared to conventional basis functions, the BSWI is able to provide higher interpolating accuracy and boundary stability. Furthermore, due to the sparsity of the wavelet, the coefficient matrix obtained by BSWI-based WFEM is sparser than that formed by general FEM, which can lead to shorter solution time for the linear equations system. To verify the accuracy and efficiency of our proposed method, we ran numerical experiments on a half-space model and a layered model and compared the results with one-dimensional (1D) semi-analytic solutions and those obtained from conventional FEM. We then studied a synthetic 3D model using different meshes and BSWI basis at different scales. The results show that our method depends less on the mesh, and the accuracy can be improved by both mesh refinement and scale enhancement.

Unified wavelet finite element formulation for static and vibration analysis of laminated composite shells

This study presents, for the first time, a unified wavelet finite element formulation for static and free vibration analysis of laminated composite shells, combining the wavelet finite element method and general shell theory. The governing equations of the proposed unified wavelet finite element formulation are derived from general shell theory and first-order shear deformation theory (FSDT) using the principle of minimum total potential energy. The formulated kinematics of laminated shells can be applied to flat plates, cylindrical shells, and doubly-curved shells with the selection of appropriate Lamé coefficients with surface metric tensor and radii of curvature parameters. With the excellent approximation of B-spline wavelets on the interval (BSWI) for structure analysis, BSWI scaling functions are used as interpolating functions to construct wavelet finite elements for laminated shells. The accuracy and efficiency of the proposed BSWI method are assessed through numerical comparisons with analytical 3-D elasticity solutions and other reference solutions in the literature for static and free vibration analysis of laminated composite plates, cylindrical shells, and doubly-curved shells.

A wavelet immersed boundary method for two-variable coupled fluid-structure interactions

In this paper, a wavelet immersed boundary (IB) method is proposed to solve fluid-structure interaction (FSI) problems with two-variable coupling, in which it is an interaction between fluid force and boundary deformation. This wavelet IB method is developed by introducing a wavelet finite element method to calculate the FSI force affected by the two-variable coupling. Furthermore, a boundary influence matrix and a series of B-spline wavelet delta functions are constructed to restrain the non-physical force oscillations. Finally, several FSI problems are simulated, which include flows past a fixed circular cylinder and a crosswise oscillating circular cylinder, as well as an in-line oscillating circular cylinder in a rest fluid. The numerical examples show that the new method is a simple and efficient method for two-variable coupled FSI problems.

Fuzzy Shannon wavelet finite element methodology of coupled heat transfer analysis for clearance leakage flow of single screw compressor

The Shannon wavelet function and its scale function are used as interpolating functions to establish the Shannon wavelet finite element. The temperature and velocity of every leakage path are calculated based on fuzzy Shannon wavelet finite element method, fuzzy Daubechies wavelet finite element model, fuzzy finite element method and experiment, and comparisons between numerical analysis results and experimental results show that fuzzy Shannon-cosine wavelet finite element method can get highest computing precision and accuracy. The coupling relationships between the flow and heat transfer of clearance leakage flow are analyzed based on fuzzy Shannon-cosine finite element method, and numerical results show that the Nusselt number of every leakage path decreases with increase of Mach number, the flow has great influence on heat transfer of clearance leakage flow of single screw compressor.

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. The new Hermitian 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. The 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.

Wavelet Element Modelling for Inviscid Fluid-Solid Coupling Problem based on Partitioned Approach.

To provide a simple numerical formulation based on fixed grids, a wavelet element method for fluid–solid modelling is introduced in this work. Compared with the classical wavelet finite element method, the presented method can potentially handle more complex shapes. Considering the differences between the solid and fluid regions, a damping-like interface based on wavelet elements is designed, in order to ensure consistency between the two parts. The inner regions are constructed with the same wavelet function in space. In the time and spatial domains, a partitioned approach based on Jacobi iteration is combined with the pseudo-parallel calculation method. Numerical convergence analyses show that the method can serve as an alternative choice for fluid–solid coupling modelling.

A stochastic B-spline wavelet on the interval finite element method for beams

The current paper presents the formulation of stochastic B-spline wavelet on the interval (BSWI) based wavelet finite element method (WFEM) for beams wherein, the spatial variation of modulus of elasticity is modelled as a homogeneous random field. Stochastic beam element formulations based on both Euler-Bernoulli beam theory and Timoshenko beam theory are proposed. BSWI scaling functions are used for the discretization of the random field and the response statistics are obtained using the perturbation approach. Numerical examples are solved and the results from perturbation approach are compared with that obtained from Monte Carlo simulation (MCS). A parametric study is also done to understand the effect of different coefficient of variation (CV) values and correlation length parameters on the response statistics. The study concludes that the proposed BSWI WFEM based perturbation approach for beams produce accurate response statistics for values of CV less than 15%. A comparative study is carried out between the results obtained from the proposed stochastic WFEM with stochastic finite element method (SFEM) wherein the random field discretization is done using Lagrange shape functions. Furthermore, normalized computational times for the execution of perturbation approach and MCS based on WFEM are evaluated and compared with those obtained for SFEM.

Electromagnetic device modeling using a new adaptive wavelet finite element method

Wavelets provide sparse representations of integral and differential operators through a multiresolution approximation. This property is well-suited for adaptive implementation of numerical techniques, which is a common topic among researchers nowadays. The current state of the wavelet theory also allows flexible construction of transforms customized to certain applications, which is an important feature considering that the generalization of wavelet techniques to a large class of real-world numerical simulations is still limited. In this paper, it is proposed an adaptive wavelet finite element method for achieving operator-orthogonality between approximation spaces, a property which eliminates coupling among levels of resolution – a common shortcoming of adaptive methods. The resulting method is easily extended to any given finite element application and requires less computation power when compared to other adaptive techniques. These features are showcased by comparing the proposed technique to the popular h-refinement method using electromagnetic device simulations involving Dirichlet and Neumann boundary conditions, inhomogeneous and non-linear media, and irregular meshes.

A stochastic B-spline wavelet on the interval finite element method for problems in elasto-statics

The current paper presents the formulation of stochastic B-spline wavelet on the interval (BSWI) based wavelet finite element method (WFEM) for one dimensional (1D) and two dimensional (2D) problems in elasto-statics wherein, the spatial variation of modulus of elasticity is modelled as a homogeneous random field. Random field discretization is done using BSWI scaling functions. Perturbation approach is used to calculate the response statistics. The results are compared with that obtained from Monte Carlo simulation (MCS) and a parametric study is carried out by considering different coefficient of variation values by varying the standard deviation. For 1D problem in particular, results from proposed stochastic WFEM method are compared with those found using stochastic finite element method (FEM) wherein random field discretization is done using Lagrange shape functions. A comparative study of the computational time needed for the execution of perturbation approach and MCS based on WFEM and FEM is also performed.

Static and dynamic analysis of cylindrical shell by different kinds of B-spline wavelet finite elements on the interval

Cylindrical shell is a fundamental structure in the area of mechanical and architectural engineering. In the predesign stage, accurate analysis is a key step to guarantee the performance in application. Finite element method is a commonly used method in structural analysis. However, due to the limitations of interpolation functions, accuracy and efficiency are restricted. Wavelet finite element method is an advanced numerical method which uses wavelet functions to replace the traditional polynomial function to discrete the solving variables. Daubechies, B-spline wavelet on the interval (BSWI) etc. have been used to construct the elements. However, they are mainly focused on the elements with one kind of variable. That is, only the displacement variable is interpolated directly and the generalized stress and strain are calculated second. Multivariable wavelet finite element can deal with this problem, in which the three kinds of variables can be interpolated and solved directly, thus the calculation error can be avoiding. In this paper, the BSWI scaling functions are used to construct the wavelet finite elements for cylindrical shell, including BSWI element with one kind of variables (BSWI-WFE), BSWI element with two kinds of variables (BSWI-TwWFE) and BSWI element with three kinds of variables (BSWI-ThWFE). Several numerical examples for cylindrical shell are provided to analyze the performance of the constructed elements and compared with each other to indicate superiority and efficiency.

Thermo dynamic methodology of A type microreactor based on Contourlet finite element method

The thermo dynamical properties of A-type microreactor of hydrogen production are important factors for optimization of it, therefore the Contourlet finite element model is established through introducing Contourlet transform into traditional finite element method, which is used for thermo dynamical analysis of A-type microreactor. Firstly, the existing research results about heat and mass analysis for microreactor and wavelet finite element method are summarized, and the feasibility analysis on thermo dynamical analysis of A-type microreactor based on Contourlet finite element method is discussed. Secondly, the mathematical model and characteristics of Contourlet transform are studied. Thirdly, the thermo dynamical Contourlet finite element model is constructed, the Contourlet finite element model and boundary conditions are confirmed. Finally, the thermo dynamical characteristics of A-type microreactor is analyzed based on traditional finite element method, Curvelet finite element method, Contourlet finite element method and test respectively, and results show that the Contourlet finite element method has highest computing precision and efficiency, and the influence rules of main parameters and geometric sizes on thermo dynamical characteristics of A-type microreactor are obtained.