The construction of one-dimensional Daubechies wavelet-based finite elements for structural response analysis

The objective of this paper is to develop a family of wavelet-based finite elements for structural response analysis. First, independent wavelet bases are used to approximate displacement functions, unknown coefficients are determined through imposing the continuity, linear independence, completeness, and essential boundary conditions. A family of Daubechies wavelet-based shape functions are then developed, which are hierarchical due to multiresolution property of wavelet. Secondly, to construct wavelet-based finite elements, derivation of the shape functions for a subdomain is employed. Thus, the wavelet-based finite elements being presented are embodied with properties in adaptivity as well as locality. By wavelet preconditioning technology, the two difficulties involving imposition of boundary conditions and compatibility with the traditional finite element methods, which are gathered in the experiments of wavelet-Galerkin context, are well overcome. Numerical examples are used to illustrate the characteristics of the current elements and to assess their accuracy and efficiency.