Structure And Research On The Element Based On Lifting Wavelet Finite Element

In recent decades, the second generation wavelet transform is the majorbreakthrough of wavelet analysis theory development. Compared with the firstgeneration wavelet transform, it has many obvious advantages. It is constructed in timedomain by using the lifting method, which is no longer dependent on a given waveletfunction’s dilation and translation. Due to the flexibility characteristics of the liftingmethod, the second generation wavelets not only have the characteristics ofmulti-resolution, but aslo have diversity and flexibility. Using the lifting method, theinitial wavelet’s performance can be improved. So the new wavelets with some specilcharacteristics which we want them to have could be constructed to meet the needs ofengineering practical problems. Making use of wavelet multi-resolution analysischaracteristics, a sequence of nested subspaces are defined. The multi-resolution finiteelement spaces could be established to approximate the exact function step by step. Ifthe constructed wavelet basis functions with decoupling, the decoupling finite elementapproximation space can be set up based on scale space’s decoupling, and on this basis,the corresponding adaptive algorithm could be constructed, which is suitable to solvelarge gradient, singularity mutation of engineering problems.For one dimensional structure problem, firstly, for the problem of axial force rod, aseries of wavelet elemnets are structured based on the Lagrange function; Secondly, inview of the euler beam structure, based on the cubic Hermite function, a series ofwavelet elements are constructed. The numerical examples of axial force rod and euler beam with uniform section are studied by the wavelet finite element analysis. Theaccuracy of the solution is very high. For two dimensional problem of thin plate,combined with tensor product knowledges, the two dimensional tensor product Hermiteinterpolation wavelet elements are constructed. The wavelet finite element analysis ofthin plate is realized to solve the problems of the thin plate’s bending and vibrationanalysis. The good analysis results are obtained.Next, based on the space lifting, the finite element multi-resolution analysis isrealized to solve the corresponding problems about displacement field function of axialforce rod,euler beam and thin plate. But the couplings exist beteween the low resolutionapproximation space and detail space, which leads to large amount of calculation andlow solving efficiency.Finally, for the euler beam problem, by using the relation between waveletdisappear moment and polynomial of orthogonal, the decoupling wavelet basisfunctions are structured based on cubic Hermite function. This kind of wavelet basisfunctions may eliminate the couplings between the low resolution space and detailspaces and that among the detail spaces in different scales. Then finite element solvingequation is decoupling according to the space scale. So the coarsest solution could beobtained in coarsest approximation space and each detail solution could be obtained inthe corresponding detail space independently. Based on the coarsest solution, the finerapproximation solution could be obtained in higher resolution space by adding the detailsolutions of detail spaces step by step to meet the accuracy requirement of result.Following, the beam element wavelet adaptive finite element analysis is constructed.Therefore, with the details of the space increased, any of the information can becaptured which you need. It provides a new research idea to further study morecomplicated structure of the adaptive finite element analysis.