在结构工程领域中广泛存在的自由振动问题和弹性稳定问题，都可以归结为常微分方程（Ordinary Differential Equation，简称ODE）或常微分方程组（简称ODE组）特征值问题，因此对特征值问题高效可靠的数值求解对力学研究和工程计算均具有重要的意义。本文针对一维和二维特征值问题，基于单元能量投影（Element Energy Projection，简称EEP）法，提出了一套高效稳定、精确可靠、统一通用的自适应求解算法。全文主要工作如下： 1. 提出了一套适用于各类一维ODE和ODE组特征值问题的有限元法（Finite Element Method，简称FEM）自适应求解策略。该法在当前FEM网格下进行常规计算得到常规FEM解，进而得到该网格下与原特征值问题同FEM解的线性问题，将线性问题中业已成熟的EEP自适应分析技术直接引入进行误差估计和网格细分，从而无需对特征值问题本身单独建立超收敛公式及其自适应算法，形成了一套统一、通用的自适应求解策略。最终得到的特征值和特征函数的精度均满足用户给定的误差限。 2. 采用上述FEM自适应分析策略成功求解了二阶和四阶ODE特征值问题，建立了相应的求解方案、实施策略及具体算法。对多种Euler梁的轴向、横向自由振动和弹性稳定问题以及Sturm-Liouville问题（简称SL问题）的大量数值算例表明，该法的求解速度、解答精度、误差控制均优于现行通用软件SLEDGE、SLEUTH；能够得到满足精度要求的特征值和按最大模度量逐点满足用户事先给定误差限的特征函数，且误差分布均匀、精度冗余较小。 3. 将上述FEM自适应基本策略拓宽推广到二阶、四阶ODE组特征值问题的自适应求解，从而形成了一个高效可靠的ODE特征值问题求解器(ODE Eigen-Solver)。数值算例表明本文算法对ODE组问题同样高效可靠，具有广泛的适用范围和广阔的使用前景。 4. 对二维特征值问题建立了有限元线法（Finite Element Method of Lines，简称FEMOL）的自适应求解方法并成功求解。FEMOL将二维特征值问题半离散为ODE组特征值问题，使用前述的ODE Eigen-Solver，并直接引入二维FEMOL线性问题的EEP自适应技术，构建了二维特征值问题的自适应FEMOL求解算法，文中成功求解了弹性力学平面问题、中厚板、中厚扁壳、空间轴对称问题的自由振动或弹性稳定等一系列问题。
The widely-applied free vibration and elastic stability analyses in structural engineering can be attributed to the mathematical eigenproblems in ordinary differential equation (ODE) or ordinary differential equations (ODEs). In this regard the effective and reliable numerical solution of eigenproblems plays an important role in both the mechanics research and the engineering practice. For one- and two-dimensional eigenproblems, this paper proposed an efficient, robust, accurate, reliable, general and unified self-adaptive procedure based on Element Energy Projection (EEP) method. The main work of this dissertation is as follows: 1. A self-adaptive finite element (FE) strategy for various kinds of one-dimensional ODE and ODEs eigenproblems was proposed. In this strategy, after the conventional FE solution on a given mesh has been obtained, the eigenproblem on the same mesh is equivalently regarded as a linear problem with the same FE solution, and then the well-developed self-adaptive strategy based on EEP method for linear ODEs is applied directly to the solution of eigenproblem for error estimation and mesh refinement without the need for constructing super-convergent formulae for each specific and individual eigenproblem. As a result, a general and unified self-adaptive algorism has been developed with both the eigenvalues and eigenfunctions from the final FE solutions fully satisfying the user-preset error tolerance. 2. The proposed self-adaptive FE strategy was successfully applied to the ODE eigenproblems of second- and fourth-order with corresponding adaptive scheme and implementing algorithm well developed. Numerous numerical examples for the analysis of axial and flexural free vibration, elastic stability of Euler members and the Sturm-Liouville (SL) problems are given in the paper to show that the proposed procedure performs much better in both solution speed and accuracy than the general codes SLEDGE and SLEUTH. The present procedure yields eigenvalues matching accuracy requirements and eigenfunctions satisfying the user-preset error tolerance point-wisely by maximum norm with uniform error distribution and little accuracy redundancy. 3. The proposed self-adaptive FE strategy was extended to the solution of second- and fourth-order ODEs eigenproblems, as a result of which an efficient and reliable ODE Eigen-Solver has been developed. The numerical examples demonstrate that the proposed method is again highly efficient and reliable for ODEs eigenproblems, and can be effectively applied to a wide range of various eigenproblems. 4. A self-adaptive method for Finite Element Method of Lines (FEMOL) was proposed for two-dimensional eigenproblems with full success. The two-dimensional eigenproblems are semi-discretized into ODEs eigenproblems by FEMOL, and the developed ODE Eigen-Solver and the earlier successful self-adaptive strategy in two-dimensional linear FEMOL analysis are both incorporated to form a self-adaptive analysis algorithm, which has succeeded in a series of free vibration and elastic stability problems for plane elasticity problems, moderately thick plates, moderately thick shallow shells and three-dimensional axisymmetric problems.
