混合边界条件板弯曲问题的谱元法

Plates with mixed boundary conditions are key components in civil and mechanical engineering and industrial design. Although theoretical analysis is valuable for providing basic understanding, in general, there is no analytical solution to these problems. Therefore, numerical computation is one of the most important approaches for obtaining full solutions for theoretical analysis and engineering design. Spectral method as an important numerical method for solving differential equations, which has been increasingly widely used in many fields of science and engineering computing. The work will be dedicated to the spectral element method for the rectangular plate bending problems with mixed nonhomogeneous boundary conditions. We first propose the higher-order quasi-orthogonal projection approximation theory, and then develop C1 conforming and C0 nonconforming spectral element methods for rectangular plate bending problems with mixed nonhomogeneous boundary conditions, we also give the corresponding error estimates of numerical solutions and its efficient numerical algorithms. The theoretical analysis of the work will certainly further expand the application of spectral method and its numerical simulation for sovling the plate vibration and bending problems with more complex boundary conditions.

混合边界条件板是土木、机械工程和工业设计的重要组成部分。混合边界条件板振动和弯曲问题的理论分析相当重要，但是人们一般很难求出这类问题的解析解。因此，数值计算就成了获得此类问题理论分析和工程设计所需信息的重要途径。而谱方法作为求解微分方程的一种重要数值方法，已被广泛应用于科学和工程计算的众多领域。本项目将致力于研究混合非齐次边界条件矩形板弯曲问题的谱元方法。我们首先建立新的高阶拟正交投影逼近理论，继而发展混合非齐次边界条件矩形板弯曲问题的C1协调与C0非协调谱元法，并给出相应的误差估计及其高效数值算法。本项目的研究成果必将进一步拓展谱方法在求解带有更加复杂边界条件的板振动和弯曲问题的理论分析及数值模拟中的应用。

混合边界条件板振动和弯曲问题是土木、机械工程和工业设计中的重要问题。但是人们一般很难求出这类问题的解析解，因此，数值求解就成了获得此类问题理论分析和工程设计所需信息的重要途径。而谱方法作为求解微分方程的一种重要数值方法，已被广泛应用于科学和工程计算的众多领域。本项目首先建立了新的一维和二维高阶拟正交投影逼近理论，继而发展了混合非齐次边界条件矩形板弯曲问题的Petrov-Galerkin谱方法和高阶微分方程的新谱元法，并给出了相应数值格式的最优误差估计和高效数值算法。另外，该项目还分别研究了多比例时滞微分方程和一阶线性常微分方程初值问题的Legendre-Gauss-Radau和Legendre-Gauss谱配置方法。该研究成果必将进一步拓展谱方法在求解带有更加复杂边界条件的板振动和弯曲问题中的应用。