Study On Wavelet Stochastic Finite Element Method

The aim of this study is to develop a Wavelet Stochastic Finite Element Method to be applied in solving partial difference equations. This work includes 5 sections as follows:A. The properties of Quasi shannon wavelet was studied in this paper, and a wavelet collocation method for partial differential equations was conducted. The extrapolation was used in this method for improving efficiency and accuracy, and the Quasi shannon interval wavelet collocation method was constructed based on the concept of interpolation wavelet transform. This method can handle the problems of complex boundary conditions and improve computation accuracy greatly.B. Based on the Precise Integration Method for structural dynamic systems proposed by Zhong Wanxie, an adaptive precise integration method for nonlinear time-invariant structural dynamic systems was proposed in this paper. This method was conducted by combining extrapolation method with precise time-integration method. In this method the nonlinear equations can be discreted in time domain adaptively. It should be pointed out that the discrete method in time domain is identical between the precise computation of the matrix exponential and the extrapolation, so that the combination of them would not increase the computation time. Hence this method can improve accuracy and efficiency of the calculation.C. Based on these work upwards, an adaptively wavelet precise time-invariant integration method was proposed in this paper. In this method, an adaptive multilevel interpolation wavelet collocation method for partial difference equations (PDEs) was conducted, in which the time complexity is less than Oleg V's method, and then the adaptive precise integration method was combined with, so that in this method the adaptively discretes both in time domain and physical domain were realized. The sub-domain precise integration method is not necessary in this method due to the collocation points can be reduced greatly in this method. So the difficulties of looking for sub-domain are avoided, and the stability of the method is ensured..D. As numerical examples, the Quasi-shannon wavelet collocation for FPK equations was conducted based on the theory of this method. The specific procedures for solving stationary FPK equation, the analysis of the problems raised in this method and the remedies of the problems were presented in this paper. Otherwise, this method was applied into solving Burgers Equation, and the numerical results show that it is a prospective numerical method for nonlinear partial differential equation.E. Lastly, the wavelet stochastic finite element method for a stochastic model of soil erosion in a rill was proposed. In this method, a new perturbation technique called linearization-correction method was used to linearize the nonlinear equations in the model, and then the wavelet precise integration method was used to calculate the sensitivity of the response. At last the stochastic perturbation method is used to analyse the variance and expectation of sediment concentration, rate and depth of flow. The calculated result was high agreement with that result of Monte Carlo method.