High-precision Wavelet Numerical Methods And Applications To Nonlinear Structural Analysis

Wavelet numerical method is a class of newly developed numerical method during the last two decades. The application range of wavelet numerical method becomes much wider with its own development. And developing wavelet method for uniformly solving weak and strong nonlinear problems also has attracted more and more attention. Within the basic concept of wavelet closed method, we have extended wavelet methods efficiently to complex mechaincs problems with nonlinearity, singularity and differential-integral operator. In addition, novel high-precision wavelet methods for solving general initial and boundary value problems also have been proposed respectively by improving the accuracy of approximation and suggesting new solution procedure.Firstly orthonormal Coiflet bases of compactly supported wavelets and its quasi-interpolating formula are introduced, which are the theoretical foundation for wavelet closed method. Then we continue to introduce boundary extension technique in the construction of Coiflet-based approximation scheme for a square-integrable function over a bounded interval, which serve as the application foundation of wavelet methods. Numerical investigation shows that Coiflet with 6 vanishing moments is an optimized choice for the present wavelet method. By defining derivatives contained in the nonlinear term as new functions, we have extended the Coiflet based method to one and two dimensional quasi-linear differential equations. And by using formula of integration by part and variable transformation, we also have proposed an efficient wavelet method for nonlinear integral equations with singular kernels. A number of numerical examples and comparison with other existing methods demonstrates the advantages of these wavelet methods in accuracy and convergence.Large deflection and buckling problems of nonlinearly elastic rods and large deformation bending problem of thin rectangular plates are typical nonlinear structural problems in modern engineering, and specific adhesion between cells are is an elastic-stochastic coupled biological mechanic problem. The present wavelet methods in this study provide quantitative solution for these nonlinear problems. For the buckling analysis, the resulted discretized algebraic equation is concise, which is convenient for direct calculation of critical load points by solving the extended systems. For the large deformation analysis, the present wavelet method is more efficient than the traditional finite element method and can avoid the shear locking phenomena. For the adhesion problem, we have obtained the quantitative description of the nonlinear relation between the normalized force and mean surface separation. Meanwhile some characteristic of the wavelet methods in this study have been noticed, that they can handle nonlinear term of any form and they are not sensitive to nonlinear intensity of equation.At last, we have improved the accuracy of the Coiflet-based approximation scheme by deriving the numerical differential formula with an adjustable high order. Then a novel high-precision step-by-step time integrating wavelet approach is developed for the general nonlinear initial value problem, and combing the wavelet Galerkin method for spatial discretization we provide a simultaneous space-time wavelet method for nonlinear initial-boundary value problem. The accuracy and stability of this wavelet time integrating method have been estimated, which show that this approach has an Nth order accuracy. Numerical examples then show promising applications of this approach in dramatically changed spatiotemporal evolution problems with shock or soliton wave. In addition, we also developed a wavelet integral collocation method for nonlinear boundary value problems. Both theoretical analysis and numerical examples show the convergence rate of this wavelet approach is about O(2^-nN), in which n is the resolution level and N is the order of vanishing moment of Coiflet. Compared to the previous wavelet Galerkin method, the accuracy of the wavelet integral collocation method not only have been further justified, but also have been proved that it is independent of the order of differential equation to be solved.