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.
The method of constructing any scale wavelet finite element(WFE)based on the one-dimensional or two-dimensional Daubechies scaling functions was presented,and the corresponding WFE adaptive lifting algorithm was given.In order to obtain the nested increasing approximate subspaces of multiscale finite element,the Daubechies scaling functions with the properties of multi-resolution analysis were employed as the finite ele-ment interpolating functions.Thus,the WFE could adaptively mesh the singularity domain caused by local cracks,which resulted in better approximate solutions than the traditional finite element methods.The calculations of natural frequencies of cracked beam were used to check the accuracy of given methods.In addition,the results of cracked cantilever beam and engineering application were satisfied.So,the current methods can provide effective tools in the numerical modeling of the fault prognosis of incipient crack.
