科学与工程计算中的Fourier级数多尺度方法
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摘要
近年来,科学与工程中的多尺度问题引起了计算科学工作者的重视。多尺度问题中区域几何参数或材料物理参数在多个量级上变化,导致求解区域中具有大梯度特性的边界层或间断层出现。对于多尺度问题而言,传统的计算分析方法逐步暴露出精度不高、计算量大的缺点,甚至由于求解模式的内在局限而难以应用。因此,如何改进传统分析方法,建立灵活、精确、高效的多尺度计算手段构成未来10年、乃至更长一段时期内计算科学的热点研究方向。
其中,通过对传统的数值计算方法进行改进,已先后提出稳定化有限元方法、泡函数方法、小波有限元方法、无网格方法、基于有限增量微积分的多尺度方法、变分多尺度方法等一批多尺度方法,掀起了多尺度方法研究的一次小高潮。但由于研究工作仍局限在数值计算方法的理论框架内,相应的研究成果不可避免地具有数值计算方法的计算成本高、场变量高阶导数计算精度低、计算参数对计算结果影响分析困难等不足。为此,本文将从数值计算方法的理论框架中走出来,及时进入解析计算方法的理论框架中去,着力拓展多尺度方法理论研究与运用研究。
本文在传统解析计算方法——Fourier级数方法已有研究成果的基础上,开展多尺度问题解析求解方法研究工作。通过推导函数高阶(偏)导数的Fourier级数的一般表达式奠定方法研究的理论基础；在统一的理论基础上,发展函数及其高阶(偏)导数联合逼近的复合Fourier级数方法理论体系；并结合新的理论体系,逐步形成多尺度计算中简明、高效的解析计算手段——Fourier级数多尺度方法。
本文研究工作主要由以下三部分组成：
第一部分为理论基础研究部分。其中,第2章运用Stokes变换技巧,获得了函数不同阶次(偏)导数的Fourier级数中Fourier系数之间的迭代关系以及关于函数高阶(偏)导数的Fourier级数的一般表述结果,构建出函数高阶(偏)导数以及函数常系数线性微分算子中系数集合,并进一步明确了函数Fourier级数逐项可导的充分条件。关于函数Fourier级数高阶求导过程中函数的Fourier系数、函数的边界Fourier系数、函数的边界(端点)值或函数的角点值等系数分布规律的理论分析深化了函数Fourier级数的高阶(偏)导数运算的复杂性认识,并纠正了Chaudhuri的理论错误。第3章基于Fourier级数逐项2r次(r为正整数)可导的技术要求,确立了函数的分解结构,构建了复合Fourier级数联合逼近方法框架体系,完成了基于代数多项式插值的复合Fourier级数联合逼近方法的完备代数多项式再生性理论分析以及逼近精度算例验证。复合Fourier级数方法是带补充项的Fourier级数方法理论体系的完善,不仅摆脱了函数边界条件的强烈依赖性,具备函数及其2r阶(偏)导数一致逼近、联合逼近的能力,而且全面实现了逼近函数序列中不同性质函数的均衡使用、有机融合。
第二部分为计算方法研究部分。第4章针对具有一般边界条件的2r阶常系数线性微分方程中多尺度现象的解析求解问题,分析了基于代数多项式插值的复合Fourier级数联合逼近方法的局限性,发展了基于微分方程齐次解插值的复合Fourier级数方法的新型求解模式,确立了Fourier级数多尺度方法的理论框架,并在此基础上明确了微分方程解函数的分解结构,细化了Fourier级数多尺度方法中技术环节的实施方法。Fourier级数多尺度方法既充分利用已有Fourier级数方法的成就,又突出解函数结构分解的基础地位,实现了Fourier级数解形式的确定性、灵活性的完美结合。
第三部分为应用案例研究部分。其中,第5章、第6章分别针对Fourier级数多尺度方法在一维、二维对流扩散反应方程以及双参数地基上厚板弹性弯曲问题中的运用问题,导出了具体的Fourier级数多尺度解形式,利用数值算例分析了Fourier级数多尺度解的收敛特性,完成了Fourier级数多尺度计算方案的优化设置,并揭示了对流扩散反应方程和双参数地基上厚板弹性弯曲问题的多尺度性态。第7章针对Fourier级数多尺度方法在矩形截面梁中波传播问题中的运用问题,导出了基于三维弹性动力学方程的矩形截面梁中波传播问题的Fourier级数多尺度解形式,明确了矩形截面梁中波型的对称性分解以及频率方程获取、求解的实施方法,并利用数值算例分析了矩形截面梁中Fourier级数多尺度解收敛特性、弹性波在方形截面梁中的传播特性及其多尺度表现形式。关于Fourier级数多尺度方法的算例验证规范了Fourier级数多尺度方法的运用过程,实现了Fourier级数多尺度解形式与离散系统导出技术的有效融合,充分体现出边界条件、乃至计算参数大范围变动情况下Fourier级数多尺度计算方案的稳定性、可靠性等优越性质。
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.
其中,通过对传统的数值计算方法进行改进,已先后提出稳定化有限元方法、泡函数方法、小波有限元方法、无网格方法、基于有限增量微积分的多尺度方法、变分多尺度方法等一批多尺度方法,掀起了多尺度方法研究的一次小高潮。但由于研究工作仍局限在数值计算方法的理论框架内,相应的研究成果不可避免地具有数值计算方法的计算成本高、场变量高阶导数计算精度低、计算参数对计算结果影响分析困难等不足。为此,本文将从数值计算方法的理论框架中走出来,及时进入解析计算方法的理论框架中去,着力拓展多尺度方法理论研究与运用研究。
本文在传统解析计算方法——Fourier级数方法已有研究成果的基础上,开展多尺度问题解析求解方法研究工作。通过推导函数高阶(偏)导数的Fourier级数的一般表达式奠定方法研究的理论基础；在统一的理论基础上,发展函数及其高阶(偏)导数联合逼近的复合Fourier级数方法理论体系；并结合新的理论体系,逐步形成多尺度计算中简明、高效的解析计算手段——Fourier级数多尺度方法。
本文研究工作主要由以下三部分组成：
第一部分为理论基础研究部分。其中,第2章运用Stokes变换技巧,获得了函数不同阶次(偏)导数的Fourier级数中Fourier系数之间的迭代关系以及关于函数高阶(偏)导数的Fourier级数的一般表述结果,构建出函数高阶(偏)导数以及函数常系数线性微分算子中系数集合,并进一步明确了函数Fourier级数逐项可导的充分条件。关于函数Fourier级数高阶求导过程中函数的Fourier系数、函数的边界Fourier系数、函数的边界(端点)值或函数的角点值等系数分布规律的理论分析深化了函数Fourier级数的高阶(偏)导数运算的复杂性认识,并纠正了Chaudhuri的理论错误。第3章基于Fourier级数逐项2r次(r为正整数)可导的技术要求,确立了函数的分解结构,构建了复合Fourier级数联合逼近方法框架体系,完成了基于代数多项式插值的复合Fourier级数联合逼近方法的完备代数多项式再生性理论分析以及逼近精度算例验证。复合Fourier级数方法是带补充项的Fourier级数方法理论体系的完善,不仅摆脱了函数边界条件的强烈依赖性,具备函数及其2r阶(偏)导数一致逼近、联合逼近的能力,而且全面实现了逼近函数序列中不同性质函数的均衡使用、有机融合。
第二部分为计算方法研究部分。第4章针对具有一般边界条件的2r阶常系数线性微分方程中多尺度现象的解析求解问题,分析了基于代数多项式插值的复合Fourier级数联合逼近方法的局限性,发展了基于微分方程齐次解插值的复合Fourier级数方法的新型求解模式,确立了Fourier级数多尺度方法的理论框架,并在此基础上明确了微分方程解函数的分解结构,细化了Fourier级数多尺度方法中技术环节的实施方法。Fourier级数多尺度方法既充分利用已有Fourier级数方法的成就,又突出解函数结构分解的基础地位,实现了Fourier级数解形式的确定性、灵活性的完美结合。
第三部分为应用案例研究部分。其中,第5章、第6章分别针对Fourier级数多尺度方法在一维、二维对流扩散反应方程以及双参数地基上厚板弹性弯曲问题中的运用问题,导出了具体的Fourier级数多尺度解形式,利用数值算例分析了Fourier级数多尺度解的收敛特性,完成了Fourier级数多尺度计算方案的优化设置,并揭示了对流扩散反应方程和双参数地基上厚板弹性弯曲问题的多尺度性态。第7章针对Fourier级数多尺度方法在矩形截面梁中波传播问题中的运用问题,导出了基于三维弹性动力学方程的矩形截面梁中波传播问题的Fourier级数多尺度解形式,明确了矩形截面梁中波型的对称性分解以及频率方程获取、求解的实施方法,并利用数值算例分析了矩形截面梁中Fourier级数多尺度解收敛特性、弹性波在方形截面梁中的传播特性及其多尺度表现形式。关于Fourier级数多尺度方法的算例验证规范了Fourier级数多尺度方法的运用过程,实现了Fourier级数多尺度解形式与离散系统导出技术的有效融合,充分体现出边界条件、乃至计算参数大范围变动情况下Fourier级数多尺度计算方案的稳定性、可靠性等优越性质。
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.
引文
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