Research On The Application Of Lifting Multi-wavelet In Finite Element

In recent decades, the second generation wavelet (SGW) which based on the firstgeneration wavelet, inherited the multiresolution property of it, and is constructed inspatial domain by the lifting scheme which is no longer defined as the dilation andtranslation of one fixed function. At the same time, the flex ibility property of the liftingscheme allows custom-design of the wavelet, and a new wavelet with excellentproperties can be constructured by adopting lifting scheme to improve the properties ofan initial wavelet so that to meet the need of solving the special engineering problems.Multi-wavelet based on lifting scheme have multiple scaling function and waveletfunction which own fast operation, high accuracyfast and so on. In this paper,the liftingMulti-wavelet is contructed based on the Hermite interpolation，and then taking themultiresolution analysis property of Multi-wavelet, combine Multi-wavelet theory andfinite element method, the displacement function of finite element method isconstructed which instead the traditional shape function in finite element method, oncethe wavelet function is derived from wavelet finite element formulation, the highgradient, singularity mutation of practical engineering problems can be convenientsolved while the traditional finite element method is difficult to do it.For one dimensional structure problem, firstly, This paper mainly study theproblem of one-dimensional beam, construct one-dimensional cubic Hermite waveletunit based on function cubic Hermite interpolation, and then solve numerical exampleof uniform beam bending problems and free vibration problems with the high precision and good effect when compared with the theoretical solution of the theoretical solutionand the simulation solution of ANSYS software.For two dimensional structure problem, a two-dimensional cubic Hermite liftingmultiwavelet scaling function is constructed by tensor product based onone-dimensional cubic Hermite interpolating scaling function,and then rectangular unitwith four node is given.According to the numerical example of the bending and freevibration problem of two-dimensional thin plate and inclined plate,this method needsless memory, fast operation, high accuracy when compared with the theoretical solutionof traditional finite element method based on the polynomial interpolation and thesimulation solution of ANSYS software.Finally, aiming at the practical engineering problems, our method is combined withANSYS software, and quantitative fault diagnosis problem of single cracked beam isstudied, predicting the existence of crack in beam structure and calculate the positionand depth of single cracked beam. Then fault diagnosis problem of double crackedbeam is qualitatively analyzed, which pave the way for early fault the diagnosis ofmultiple crack of beam structure, and also provides an effective method for the furtherstudy of more multiple crack problems in complex engineering structure and othercomplex problems.