Fourier Series Multiscale Method For Computational Analysis In Science And Engineering

In recent years, considerable effort has been spent for the solution of the multiscale problems in science and engineering. These multiscale problems are characterized by the existence of the boundary and/or internal layers, where sharp gradients may appear due to numeric values of the geometrical and/or physical parameters differing in several orders of magnitudes. As to the multiscale problems, traditional analysis methods find difficulties such as low precision, high cost or even are of no effect for their inherent limitations. Therefore, by properly modifying the traditional analysis methods, the flexible, accurate and efficient multiscale analysis techniques are to be developed, which forms the research direction for computational science in the next decade or even in a longer period.Nowadays, by successful modification of some traditional numeric methods, a series of multiscale analysis methods, for instance stabilized finite element method, bubble method, wavelet based finite element method, meshfree method, finite increment calculus based multiscale method and variational multiscale method, have been proposed, which bring to a dramatic breakthrough for the current multiscale analysis methods. But it is worthy of note that all the research is restricted within the theoretical framework of numeric methods and the accordingly obtained multiscale analysis methods inevitably have such disadvantages as high costs of computation, low precision of higher derivates of the solution, and difficulties in analysis of computational parameters'effects on computational results. By contrast with this usual approach, in this dissertation the theoretical framework of analytic methods is adopted, on the basis of which new type of multiscale method is to be developed and properly applied.Herein academic achievements of the Fourier series method, a traditional analytic method, are taken as a beginning of the research of analytic methods for the multiscale problems. By deriving general formulas of higher (partial) derivatives of Fourier series, a theoretical foundation of multiscale method research has been laid firstly. And with this theoretical foundation, the composite Fourier series method for combined approach of functions and their higher (partial) derivatives is proposed. Then within the theoretical framework of the composite Fourier series method, a concise, efficient multiscale analysis procedure, by name the Fourier series multiscale method, is obtained. The research in this dissertation is composed of three parts.The first part is focused on the theoretical foundation of the Fourier series multiscale method. In chapter 2, the Stokes transform is employed and the iterative relations between the Fourier coffecients in Fourier series of different order (partial) derivatives of the functions and ulteriorly the general formulas for the Fourier series of higher order (partial) derivatives of the functions are acquired. Sets of coefficients concerned in the higher order (partial) derivatives and linear differential operators with constant coefficients of the functions are derived. And accordingly the sufficiency conditions for term-by-term differentitation of Fourier series of the functions are put forward. Distribution of coefficients concerned during the course of higher order differentitation of Fourier series of the functions, such as Fourier coefficients of the functions, boundary Fourier coefficients of the functions, boundary (or end) values of the functions and sometimes the corner values of the functions, are thoroughly analyzed, which makes the complexity of the higher order differentitation of Fourier series of the functions understandable and rectifys the mistakes in professor Chaudhuri's research. In Chapter 3, on the basis of the requirements of the 2r (r is a positive integer) times term-by-term differentitation of Fourier series of the functions, desired decomposition structures of the functions are settled and the methodology of combined approach of the functions with composite Fourier series is proposed. And specifically for the algebraic polynomial interpolation based composite Fourier series method, theoretical analysis of algebraic polynomial regeneration is carried out and numeric examples are demonstrated. This newly proposed composite Fourier series method is verified as an improvement of the Fourier series method with supplementary terms, which is feasible for functions with varied boundary conditions, has excellent uniform and combined convergence of functions and their (partial) derivatives up to 2r order, and strikes a proper balance in the use of different kinds of basis functions in approach function series.The second part is focused on the computational procedure of the Fourier series multiscale method. In chapter 4, as to the analytic analysis of multiscale phenomena inherent in the 2r order linear differential equations with constant coefficients, limitations of the algebraic polynomial interpolation based composite Fourier series method are discussed and a new solution pattern, where homogeneous solutions of the differential equations are adopted as interpolation functions in the composite Fourier series method, is developed. The theoretical framework of the Fouriel series multiscale method is consequently established, in which decomposition structures of solutions of the differential equations are specified and practical schemes for application are detailed. The Fourier series multiscale method has not only made full use of academic achievements of the Fourier series method, but also given prominence to the fundamental position of structural decomposition of solutions of the differential equations, which results in perfect integration of invariance and flexibility of the Fourier series solution.The third part is focused on the practical application of the Fourier series multiscale method. In chapter 5 and chapter 6, one-dimensional and two-dimensional convection-diffusion-reaction equations and elastic bending of a thick plate on biparametric foundation are analyzed successively by the Fourier series multiscale method, where the specific Fourier series multiscale solutions are derived, convergence characteristics of the Fourier series multiscale solutions are investigated by numeric examples, schemes for application of the Fourier series multiscale method are optimized, and multiscale properties of the convection-diffusion-reaction equations and the bending problem of a thick plate on biparametric foundation are demonstrated. In chapter 7, the Fourier series multiscale method is applied to the analysis of wave propagation in an infinite rectangular beam. Initially, by solving the three-dimensional elastodynamic equations a Fourier series multiscale solution is derived for wave motion within the beam. And then implementation procedures of symmetric decomposition of different kinds of waves propagating in a rectangular beam as well as acquisition and disposal of the frequency equation are presented. Finally numeric examples are given in illustration of the convergence characteristics of Fourier series multiscale solution in a rectangular beam, along with propagation characteristics and multiscale behaviors of elastic waves in a square beam. The three case studies above provide detailed schemes for application of the Fourier series multiscale method to science and engineering, arrive at a fusion of the Fourier series multiscale solution and the derivation techniques of discrete systems, and demonstrate the merit of the Fourier series multiscale method which yields stabilized and accurate numeric results for all range of computational parameters and boundary conditions.