B-Spline Wavelet on Interval Finite Element Method for Static and Vibration Analysis of Stiffened Flexible Thin Plate

A new wavelet finite element method(WFEM)is constructed in this paper and two elements for bending and free vibration problems of a stiffened plate are analyzed.By means of generalized potential energy function and virtual work principle,the formulations of the bending and free vibration problems of the stiffened plate are derived separately.Then,the scaling functions of the B-spline wavelet on the interval(BSWI)are introduced to discrete the solving field variables instead of conventional polynomial interpolation.Finally,the corresponding two problems can be resolved following the traditional finite element frame.There are some advantages of the constructed elements in structural analysis.Due to the excellent features of the wavelet,such as multi-scale and localization characteristics,and the excellent numerical approximation property of the BSWI,the precise and efficient analysis can be achieved.Besides,transformation matrix is used to translate the meaningless wavelet coefficients into physical space,thus the resolving process is simplified.In order to verify the superiority of the constructed method in stiffened plate analysis,several numerical examples are given in the end.

Finite Elements Based on Deslauriers-Dubuc Wavelets for Wave Propagation Problems

This paper presents the formulation of finite elements based on Deslauriers-Dubuc interpolating scaling functions, also known as Interpolets, for their use in wave propagation modeling. Unlike other wavelet families like Daubechies, Interpolets possess rational filter coefficients, are smooth, symmetric and therefore more suitable for use in numerical methods. Expressions for stiffness and mass matrices are developed based on connection coefficients, which are inner products of basis functions and their derivatives. An example in 1-D was formulated using Central Difference and Newmark schemes for time differentiation. Encouraging results were obtained even for large time steps. Results obtained in 2-D are compared with the standard Finite Difference Method for validation.

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.

ALGORITHM DESIGN AND IMPLEMENTATION OF FINITE WAVELET GROW TREE

Based on grow tree composite model, Finite Field Wavelet Grow Tree (FW-GT) was proposed in this paper. FW-GT is a novel framework to be used in data encryption enhancing data security. It is implemented by replacement operator and wavelet operator. Forward integration and inverse decomposition of FW-GT are performed by replacement, inverse wavelets and its corresponding replacement, wavelet transforms. Replacement operator joined nonlinear factor, wavelet operator completed data transformation between lower dimensional space and higher dimensional space. FW-GT security relies on the difficulty of solving nonlinear equations over finite fields. By using FW-GT, high security of data could be obtained at the cost of low computational complexity. It proved FW-GT algorithm’s correctness in this paper. The experimental result and theory analysis shows the excellent performance of the algorithm.

THE CONSTRUCTION OF WAVELET-BASED TRUNCATED CONICAL SHELL ELEMENT USING B-SPLINE WAVELET ON THE INTERVAL

Based on B-spline wavelet on the interval （BSWI）, two classes of truncated conical shell elements were constructed to solve axisymmetric problems, i.e. BSWI thin truncated conical shell element and BSWI moderately thick truncated conical shell element with independent slopedeformation interpolation. In the construction of wavelet-based element, instead of traditional polynomial interpolation, the scaling functions of BSWI were employed to form the shape functions through the constructed elemental transformation matrix, and then construct BSWI element via the variational principle. Unlike the process of direct wavelets adding in the wavelet Galerkin method, the elemental displacement field represented by the coefficients of wavelets expansion was transformed into edges and internal modes via the constructed transformation matrix. BSWI element combines the accuracy of B-spline function approximation and various wavelet-based elements for structural analysis. Some static and dynamic numerical examples of conical shells were studied to demonstrate the present element with higher efficiency and precision than the traditional element.

Second Generation Wavelet Finite Element and Rotor Cracks Quantitative Identification Method

The presence of cracks in the rotor is one of the most dangerous and critical defects for rotating machinery. Defect of fatigue cracks may lead to long out-of-service periods, heavy damages of machines and severe economic consequences. With the method of finite element, vibration behavior of cracked rotors and crack detection was received considerable attention in the academic and engineering field. Various researchers studied the response of a cracked rotor and most of them are focused on the crack detection based on vibration behavior of cracked rotors. But it is often difficult to identify the crack parameters quantitatively. Second generation wavelets （SGW） finite element has good ability in modal analysis for singularity problems like a cracked rotor. Based on the fact that the feature of SGW could be designed depending on applications, a multiresolution finite element method is presented. The new model of SGW beam element is constructed. The first three natural frequencies of the rotor with different crack location and size were solved with SGW beam elements, and the database for crack diagnosis is obtained. The first three metrical natural frequencies are employed as inputs of the database and the intersection of the three frequencies contour lines predicted the normalized crack location and size. With the Bently RK4 rotor test rig, rotors with different crack location and size are tested and diagnosed. The experimental results denote the cracks quantitative identification method has higher identification precision. With SGW finite element method, a novel method is presented that has higher precision and faster computing speed to identify the crack location and size.

ADAPTIVE INTERVAL WAVELET PRECISE INTEGRATION METHOD FOR PARTIAL DIFFERENTIAL EQUATIONS

The quasi-Shannon interval wavelet is constructed based on the interpolation wavelet theory, and an adaptive precise integration method, which is based on extrapolation method is presented for nonlinear ordinary differential equations ( ODEs). And then, an adaptive interval wavelet precise integration method (AIWPIM) for nonlinear partial differential equations(PDEs) is proposed. The numerical results show that the computational precision of AIWPIM is higher than that of the method constructed by combining the wavelet and the 4th Runge-Kutta method, and the computational amounts of these two methods are almost equal. For convenience, the Burgers equation is taken as an example in introducing this method, which is also valid for more general cases.

COMPACTLY SUPPORTED NON-TENSOR PRODUCT FORM TWO-DIMENSION WAVELET FINITE ELEMENT

Some theorems of compactly supported non-tensor product form two-dimension Daubechies wavelet were analysed carefully. Compactly supported non-tensor product form two-dimension wavelet was constructed, then non-tensor product form two dimension wavelet finite element was used to solve the deflection problem of elastic thin plate. The error order was researched. A numerical example was given at last.

Second-generation wavelet finite element based on the lifting scheme for GPR simulation

Ground-penetrating radar(GPR)is a highly efficient,fast and non-destructive exploration method for shallow surfaces.High-precision numerical simulation method is employed to improve the interpretation precision of detection.Second-generation wavelet finite element is introduced into the forward modeling of the GPR.As the finite element basis function,the second-generation wavelet scaling function constructed by the scheme is characterized as having multiple scales and resolutions.The function can change the analytical scale arbitrarily according to actual needs.We can adopt a small analysis scale at a large gradient to improve the precision of analysis while adopting a large analytical scale at a small gradient to improve the efficiency of analysis.This approach is beneficial to capture the local mutation characteristics of the solution and improve the resolution without changing mesh subdivision to realize the efficient solution of the forward GPR problem.The algorithm is applied to the numerical simulation of line current radiation source and tunnel non-dense lining model with analytical solutions.Result show that the solution results of the secondgeneration wavelet finite element are in agreement with the analytical solutions and the conventional finite element solutions,thereby verifying the accuracy of the second-generation wavelet finite element algorithm.Furthermore,the second-generation wavelet finite element algorithm can change the analysis scale arbitrarily according to the actual problem without subdividing grids again.The adaptive algorithm is superior to traditional scheme in grid refinement and basis function order increase,which makes this algorithm suitable for solving complex GPR forward-modeling problems with large gradient and singularity.

Application of Wavelet Finite Element Method to Simulation of the Temperature Field of Copier Paper

Simulation of the temperature field of copier paper in copier fusing is very important for improving the fusing property of reprography. The temperature field of copier paper varies with a high gradient when the copier paper is moving through the fusing rollers. By means of conventional shaft elements, the high gradient temperature variety causes the oscillation of the numerical solution. Based on the Daubechies scaling functions, a kind of wavelet based element is constructed for the above problem. The temperature field of the copier paper moving through the fusing rollers is simulated using the two methods. Comparison of the results shows the advantages of the wavelet finite element method, which provides a new method for improving the copier properties.

4TH-ORDER SPLINE WAVELETS ON A BOUNDED INTERVAL

The 4th-order spline wavelets an a bounded interval are constructed by the 4th-order truncated B-spline functions. These wavelets consist of inner and boundary wavelets. They are bases of wavelet space with finite dimensions. Arty function on an interval will be expanded as the sum of finite items of the scaling functions and wavelets. It plays an important role for numerical analysis of partial differential equations, signal processes, and other similar problems.

Study on spline wavelet finite-element method in multi-scale analysis for foundation

A new finite element method （FEM） of B-spline wavelet on the interval （BSWI） is proposed. Through analyzing the scaling functions of BSWI in one dimension, the basic formula for 2D FEM of BSWI is deduced. The 2D FEM of 7 nodes and 10 nodes are constructed based on the basic formula. Using these proposed elements, the multiscale numerical model for foundation subjected to harmonic periodic load, the foundation model excited by external and internal dynamic load are studied. The results show the pro- posed finite elements have higher precision than the tradi- tional elements with 4 nodes. The proposed finite elements can describe the propagation of stress waves well whenever the foundation model excited by extemal or intemal dynamic load. The proposed finite elements can be also used to con- nect the multi-scale elements. And the proposed finite elements also have high precision to make multi-scale analysis for structure.

THE FINITE ELEMENT METHOD BASED ON INTERPOLATING WITH WAVELET BASIS FUNCTION

The compactly supported wavelet basis functions are introduced into the construction of interpolating function of traditional finite element method when analyzing the problems with high gradient, and the traditional interpolating method is modified. The numerical stability of the new interpolating pattern is discussed and the convergence of the new method is also discussed by patch test analysis. The additional freedom of the new interpolating pattern is eliminated by static condensation method. Finally, the wavelet finite element formulations based on variational principles are put forward.

A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media

In this paper, we consider numerical simulation of wave propagation in fluidsaturated porous media. A wavelet finite-difference method is proposed to solve the 2-D elastic wave equation. The algorithm combines flexibility and computational efficiency of wavelet multi-resolution method with easy implementation of the finite-difference method. The orthogonal wavelet basis provides a natural framework, which adapt spatial grids to local wavefield properties. Numerical results show usefulness of the approach as an accurate and stable tool for simulation of wave propagation in fluid-saturated porous media.

Steam turbine rotor crack detection using sifting process of EMD and B-spline wavelet on the interval element model

A high-precision identification method for steam turbine rotor crack is presented. By providing me nrst three measured natural frequencies, contours for the specified natural frequency are plotted in the same coordi- nate, and the intersection of the three curves predicts the crack location and size. The cracked rotor system is mod- eled using B-spline wavelet on the interval （BSWI） finite element method, and a method based on empirical mode decomposition （EMD） and Laplace wavelet is implemented to improve the identification precision of the first three measured natural frequencies. Compared with the classical nondestructive testing, the presented method shows its effectiveness and reliability. It is feasible to apply this method to the online health monitoring for rotor structure.

Seismic Wavelet Analysis Based on Finite Element Numerical Simulation

The practice of exploration and production has proved that explosives are excited in different surrounding rocks and the seismic wavelets collected have different characteristics. In this paper, by establishing a numerical model of the explosion in the well, using finite element analysis technology for numerical simulation, the simulation calculated the stress structure in the near-source area of the earthquake excitation, and extracted the seismic wavelet. The results show that the simulation seismic wavelet characteristics of different thin interbedded sand and mudstone structures have changed significantly. Through excitation simulation, the amplitude and spectrum information of seismic wavelets can be compared and analyzed, and the excitation parameters can be optimized. .

Wavelet Numerical Solutions for Weakly Singular Fredholm Integral Equations of the Second Kind

Daubechies interval cally weakly singular Fredholm kind. Utilizing the orthogonality equation is reduced into a linear wavelet is used to solve nurneriintegral equations of the second of the wavelet basis, the integral system of equations. The vanishing moments of the wavelet make the wavelet coefficient matrices sparse, while the continuity of the derivative functions of basis overcomes naturally the singular problem of the integral solution. The uniform convergence of the approximate solution by the wavelet method is proved and the error bound is given. Finally, numerical example is presented to show the application of the wavelet method.

A wavelet multiscale method for inversion of Maxwell equations

This paper is concerned with estimation of electrical conductivity in Maxwell equations. The primary difficulty lies in the presence of numerous local minima in the objective functional. A wavelet multiscale method is introduced and applied to the inversion of Maxwell equations. The inverse problem is decomposed into multiple scales with wavelet transform, and hence the original problem is reformulated to a set of sub-inverse problems corresponding to different scales, which can be solved successively according to the size of scale from the shortest to the longest. The stable and fast regularized Gauss-Newton method is applied to each scale. Numerical results show that the proposed method is effective, especially in terms of wide convergence, computational efficiency and precision.

Parallel-Computing Wavelet-Based FDTD Method for Modeling Nanoscale Optical Resonator

An efficient wavelet-based finite-difference time-domain(FDTD)method is implemented for analyzing nanoscale optical devices,especially optical resonator.Because of its highly linear numerical dispersion properties the high-spatial-order FDTD achieves significant reduction in the number of cells,i.e.used memory,while analyzing a high-index dielectric ring resonator working as an add/drop multiplexer.The main novelty is that the wavelet-based FDTD model is extended in a parallel computation environment to solve physical problems with large dimensions.To demonstrate the efficiency of the parallelized FDTD model,a mirrored cavity is analyzed.The analysis shows that the proposed model reduces computation time and memory cost,and the parallel computation result matches the theoretical model.

Matrix Expression of the Orthogonal Wavelet (Packets) Transform

Matrix expression of finite orthogonal wavelet transform of finite impulse response signal is more valuable for theoretical analysis and understanding. However, clear deduction for matrix expression has not been provided yet. In this paper, the formulation to generate the re-lated matrix is put forward and the theorem on the orthogonality of this matrix proved. This effort deploys a basis for more deeper and wider applications in chemical processes. *