论文标题
傅立叶系列（基于）科学与工程中计算分析的多尺度方法：VI。傅立叶串联多尺度解决方案，用于带有矩形横截面的光束中的波传播
Fourier series (based) multiscale method for computational analysis in science and engineering: VI. Fourier series multiscale solution for wave propagation in a beam with rectangular cross section
论文作者
论文摘要
傅立叶系列多尺度方法是一种简洁有效的多尺度计算方法，将根据这一系列论文开发。在第六篇论文中，对具有矩形横截面的光束中的波传播的精确分析扩展到对模态函数的完全耦合的二阶线性差分方程的系统进行彻底的多尺度分析，并在其中规定了一般边界条件。为此，模态函数每个函数都表示为角函数的线性组合，两个边界函数和内部函数，以确保串联表达式获得均匀收敛的术语和术语可区分到二阶。同时，角函数和内部函数的总和对应于特定的解决方案，两个边界函数对应于满足方程式同质形式的一般解决方案。由于通用解决方案已适当地解释了微分方程的含义，因此预计方程解的空间特征将在单独的方向上更好地捕获。借助角函数，两个边界函数和内部函数特别选择为多项式，沿X2（或X1）方向沿X2（或X1）方向的一维全距离傅立叶序列，以及二维全距傅立叶级数，均带有矩形横截面的梁中的波浪传播的傅立叶级数多尺度序列。对于具有各种边界条件的光束，进行了计算和分析，并显示了梁中弹性波的传播特性。新提出的准确波模型为在光束中同时控制耦合波和建立引导波NDE技术的耦合波提供了坚实的基础。
Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. In the sixth paper, exact analysis of the wave propagation in a beam with rectangular cross section is extended to a thorough multiscale analysis for a system of completely coupled second order linear differential equations for modal functions, where general boundary conditions are prescribed. For this purpose, the modal function each is expressed as a linear combination of the corner function, the two boundary functions and the internal function, to ensure the series expressions obtained uniformly convergent and termwise differentiable up to second order. Meanwhile, the sum of the corner function and the internal function corresponds to the particular solution, and the two boundary functions correspond to the general solutions which satisfy the homogeneous form of the equations. Since the general solutions have appropriately interpreted the meaning of the differential equations, the spatial characteristics of the solution of the equations are expected to be better captured in separate directions. With the corner function, the two boundary functions and the internal function selected specifically as polynomials, one-dimensional full-range Fourier series along the x2 (or x1)-direction, and two-dimensional full-range Fourier series, the Fourier series multiscale solution of the wave propagation in a beam with rectangular cross section is derived. For the beam with various boundary conditions, computation and analysis are performed, and the propagation characteristics of elastic waves in the beam are presented. The newly proposed accurate wave model has laid a solid foundation for simultaneous control of coupled waves in the beam and establishment of guided wave NDE techniques.
