A spline wavelets element method for frame structures vibration

Chen, W.-H.; Wu, Che-Wei

doi:10.1007/s004660050044

Cited by 7 publications

(9 citation statements)

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“…A general issue with wavelets is that it can be difficult to impose boundary conditions since they have no natural interpolation property that would make boundary condition handling simpler. Another problem that can arise is that many wavelets have no closed-form description, for example, Daubechies wavelets that are described as the result of the iteration of operator O in (39). This can make their application to PDE more difficult, for example, when computing scalar products involved in weighted residual and other methods.…”

Section: Some Examples Of Application Of Wavelets To Pdementioning

confidence: 99%

“…Another important advantage of splines is that a simple closed-form expression is known. Examples of spline applications can be found in [38][39][40]. Of special interest is the application of Hermite cubic splines (HCS), a kind of multiwavelet [41] that shows promise in handling in a numerically robust way boundary conditions.…”

Section: Some Schemes From the Literaturementioning

confidence: 99%

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Wavelets for Differential Equations and Numerical Operator Calculus

Bernardini¹

2019

Wavelet Transform and Complexity

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Differential equations are commonplace in engineering, and lots of research have been carried out in developing methods, both efficient and precise, for their numerical solution. Nowadays the numerical practitioner can rely on a wide range of tools for solving differential equations: finite difference methods, finite element methods, meshless, and so on. Wavelets, since their appearance in the early 1990s, have attracted attention for their multiresolution nature that allows them to act as a "mathematical zoom," a characteristic that promises to describe efficiently the functions involved in the differential equation, especially in the presence of singularities. The objective of this chapter is to introduce the main concepts of wavelets and differential equation, allowing the reader to apply wavelets to the solution of differential equations and in numerical operator calculus.

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Section: Some Examples Of Application Of Wavelets To Pdementioning

confidence: 99%

Section: Some Schemes From the Literaturementioning

confidence: 99%

Wavelets for Differential Equations and Numerical Operator Calculus

Bernardini¹

2019

Wavelet Transform and Complexity

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“…Therefore, the key problem is to calculate connection coefficients [11,12] when the Daubechies scaling and wavelet functions are employed to construct wavelet-based element. Since the connection coefficients derivation can only be obtained for integration in global coordinates, it will fail when the integrand involves variant Jacobians [6,7]. Moreover, it is a complicated process to calculate the connection coefficients, which will increase the costs of coding work.…”

Section: Introductionmentioning

confidence: 99%

“…By means of "two-scale relations" of scaling functions, we can change the wavelet scale freely to improve computational precision and meet analytical requirements. So wavelet numerical methods are well argued by many researchers not only in numerical analysis domains [2][3][4][5] but also in structural analysis fields [6][7][8][9]. Basu indicated that the finite difference and Ritz type methods of the pre-computer era had largely been replaced in the computer era by FEM, boundary element method (BEM), Meshless method, and in the near future it might be the turn for wavelet method [10].…”

Section: Introductionmentioning

confidence: 99%

The construction of 1D wavelet finite elements for structural analysis

Xiang

Chen

et al. 2006

Comput Mech

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Adopting the scaling functions of B-spline wavelet on the interval (BSWI) as trial functions, a new finite element method (FEM) of BSWI is presented. Instead of traditional polynomial interpolation, scaling functions at the certain scale have been adopted to form the shape functions and construct wavelet-based elements. Unlike the process of wavelets added directly in the other wavelet numerical methods, the element displacement field represented by the coefficients of wavelets expansions is transformed from wavelet space to physical space via the corresponding transformation matrix. The transformation matrix is the key to construct wavelet-based elements freely as long as we can ensure its non-singularity. Then, classes of C 0 and C 1 type elements are constructed. And the lifting scheme of BSWI elements is also discussed. The numerical examples indicate that the BSWI elements have higher efficiency and precision than traditional finite element method in solving 1D structural problems especially for geometric nonlinear, variable cross-section and loading cases.

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“…Chen et al (1995Chen et al ( , 1996 solved the truss and membrane vibration problems by using the element constructed by the spline wavelets, and derived the lifting algorithm that takes an advantage of the "two-scale relation" of wavelets. By introducing a transformation matrix that transforms the element deflection field represented by the coefficients of wavelet expansions from wavelet space to physical space, Ma et al (2003) and Chen et al (2004) constructed the wavelet beam element based on Daubechies wavelet and B-spline wavelet, respectively.…”

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confidence: 99%

Adaptive Trigonometric Wavelet Composite Beam Element Method

Ren

2013

Advances in Structural Engineering

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It is always an issue to use as few finite elements as possible to reduce a computational load with the precision assured to solve engineering problems. In this paper, the trigonometric wavelet function and polynomial function are combined together to be the displacement interpolation function of beam element. Accordingly, the trigonometric wavelet composite beam element is formulated based on the principle of potential energy minimum to carry out the bending, free vibration and buckling analysis of beam structures. Such a trigonometric wavelet composite beam element utilizes the advantages of both conventional finite element and trigonometric wavelet finite element. As the order of wavelet function can be enhanced easily and the multi-resolution can be achieved by the multi-scales of wavelet function, the trigonometric wavelet composite finite element provides an alternate way to realize the adaptive analysis. Numerical examples have illustrated that the proposed trigonometric wavelet composite beam element is effective and the solution accuracy can be improved adaptively with enhancing the scale order of wavelet.

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A spline wavelets element method for frame structures vibration

Cited by 7 publications

References 0 publications

Wavelets for Differential Equations and Numerical Operator Calculus

Wavelets for Differential Equations and Numerical Operator Calculus

The construction of 1D wavelet finite elements for structural analysis

Adaptive Trigonometric Wavelet Composite Beam Element Method

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