Ebook A spline wavelet finite-element method in structural mechanics
Finite-element method (FEM) based on energy variational principle and discrete interpolation has been developed since the middle of 20th century, which successfully combines the advantages of conventional energy method and finite difference method. FEM is now a powerful tool to solve structural mechanics problems in every aspect. In the development of FEM, the energy variational principle provides a theoretical foundation to develop varieties of finite element formulations. Trefftz presented his method in 1926 as an alternative to the Ritz method.
The basic idea of Trefftz method is to use trial functions which satisfy the governing differential equations. Trefftz applied the additional trial functions to the common boundaries of the sub-domains, which already demonstrates an important idea of contemporary FEM. Many scholars had developed different techniques to formulate Trefftz-based elements and to deal with different problems such as the solution of coupling boundary, cracks, holes or sharp corners. Melosh established the finite-element displacement formulation using the potential energy principle, whereas Pian built up an intercross stress (hybrid) finite-element formulation utilizing the complementary energy principle. Elias proposed a finite-element stress method by applying the complementary energy principle. Herrmann constructed a mixed finite-element formulation. Shen et al. and Prenter proposed a spline finite-element formulation based on the Hellingger–Reissner general variational principle and applied their formulation to solve the problems in structural mechanics.
Wavelet theory is a mathematical tool that was developed recently. It can be considered as an important breakthrough in Mathematics. Recently, wavelet theory has drawn more and more attentions in engineering fields, such as signal processing, images processing, pattern and phonetic recognition, quantum physics, earthquake reconnaissance, fluid mechanics, electromagnetic field, CT imagery, diagnosis and monitoring of machinery defects, etc.
The wavelet analysis based on the wavelet transform is a perfect combination of functional analysis, Fourier transform, harmonic analysis and numerical analysis. In recent years, several wavelet based FEMs that combine the finite element concept with wavelet theory have been proposed to solve various engineering problems. Wei and Zhang developed a wavelet-decomposed method to solve the Navier–Stokes equations. Wells and Zhou advised a wavelet Galerkin method to deal with the Dirichlet problem. Zhou and Wang presented a numerical algorithm of higher-order Daubechies wavelet derivative to solve the differential equations of beam and plate.
Authors presented a multivariable wavelet-based FEM to resolve the bending problems of thick plates. However, most of current wavelet-based finite-element formulations are often constructed in a wavelet space, in which the field variables like displacements are expressed by a product of wavelet functions and wavelet coefficients. In such a way, when a complex structural problem is analysed, it becomes very difficult to deal with the interface between elements. In addition, the boundary conditions cannot be easily treated as in the case of conventional FEMs. These shortcomings limit a wide application of wavelet-based FEMs in structural analysis.
In this paper, a new wavelet-based FEM in structural mechanics is proposed by using the spline wavelet functions. To overcome the shortcomings of current wavelet-based finite elements, the proposed spline wavelet finite-element formulation is constructed in a similar way of conventional displacement-based FEMs. The spline wavelet functions are used as the displacement interpolation functions and the shape functions are expressed by wavelets. By adopting the rational local co-ordinates and co-ordinate transformation, the spline wavelet finite element formulations of plane beam element, two-dimensional in-plane elements and three dimensional solid elements are derived. The numerical examples illustrate that the proposed spline wavelet FEM has a high accuracy and fast convergent rate to solve the problems in structural mechanics.
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