On the Unification of Schemes and Software for Wavelets on the Interval

We revisit the construction of wavelets on the interval with various degrees of polynomial exactness, and explain how existing schemes for orthogonal- and Spline wavelets can be extended to compactly supported delay-normalized wavelets. The contribution differs substantially from previous ones in how results are stated and deduced: linear algebra notation is exploited more heavily, and the use of sums and complicated index notation is reduced. This extended use of linear algebra eases translation to software, and a general open source implementation, which uses the deductions in this paper as a reference, has been developed. Key features of this implementation is its flexibility w.r.t. the length of the input, as well as its generality regarding the wavelet transform.

Hermitian Mindlin Plate Wavelet Finite Element Method for Load Identification

A new Hermitian Mindlin plate wavelet element is proposed. The two-dimensional Hermitian cubic spline interpolation wavelet is substituted into finite element functions to construct frequency response function (FRF). It uses a system’s FRF and response spectrums to calculate load spectrums and then derives loads in the time domain via the inverse fast Fourier transform. By simulating different excitation cases, Hermitian cubic spline wavelets on the interval (HCSWI) finite elements are used to reverse load identification in the Mindlin plate. The singular value decomposition (SVD) method is adopted to solve the ill-posed inverse problem. Compared with ANSYS results, HCSWI Mindlin plate element can accurately identify the applied load. Numerical results show that the algorithm of HCSWI Mindlin plate element is effective. The accuracy of HCSWI can be verified by comparing the FRF of HCSWI and ANSYS elements with the experiment data. The experiment proves that the load identification of HCSWI Mindlin plate is effective and precise by using the FRF and response spectrums to calculate the loads.

Direct method for Solving Nonlinear Variational Problems by Using Hermite Wavelets

In this work, we first construct Hermite wavelets on the interval [0,1) with it’s product, Operational matrix of integration 2^k M×2^k M is derived, and used it for solving nonlinear Variational problems with reduced it to a system of algebric equations and aid of direct method. Finally, some examples are given to illustrate the efficiency and performance of presented method.

SPLINE WAVELETS WITH BOUNDARY VALUES AND VANISHING MOMENTS

This paper deals with the construction of spline wavelets on the interval [0, 1], which have zero boundary values and vanishing moments. One begins with some primal scaling functions and their biorthogonal duals. Then desired biorthogonal wavelets are given by the method of Dahmen, Kunoth and Urban. Since the structure of those wavelets looks complicated, one tries to construct those types of wavelets only in primal side finally, without using any dual informations. Some numerical experiments show good effects, although the uniform stability remains to be proved theoretically.

A Class of Wavelet-Based Rayleigh-Euler Beam Element for Analyzing Rotating Shafts

A class of wavelet-based Rayleigh-Euler rotating beam element using B-spline wavelets on the interval (BSWI) is developed to analyze rotor-bearing system. The effects of translational and rotary inertia, torsion moment, axial displacement, cross-coupled stiffness and damping coefficients of bearings, hysteric and viscous internal damping, gyroscopic moments and bending deformation of the system are included in the computational model. In order to get a generalized formulation of wavelet-based element, each boundary node is collocated six degrees of freedom (DOFs): three translations and three rotations; whereas, each inner node has only three translations. Typical numerical examples are presented to show the accuracy and efficiency of the presented method.

A numerical study using Hermitian cubic spline wavelets for the analysis of shafts

Shafts are the main component of rotating machinery. The key problem for designing this kind of machine is how to accurately compute its dynamic characteristic. By using Hermite cubic spline wavelets on the interval, a multi-scale wavelet-based numerical method is proposed. For the orthogonal characteristic of the wavelet bases with respect to the given inner product, the corresponding multi-scale equations will be decoupled across scales partially and suit for nesting approximation using the lifting scheme of the wavelet numerical method. The present method is verified by some examples in the literatures. It shows that the proposed method has high precision and can be applied to analyse dynamic characteristic of shafts.