薄壁结构静、动力屈曲分析的新型非线性有限元降阶方法研究

The standard finite element method(FEM) is commonly used to do the static and dynamic buckling analysis of structures where geometric nonlinearities are considered. This method usually takes a large amount of analysis time, which influences the design cycle and cost of the thin-walled structures used in aerospace engineering. The reduced-order methods based on the traditional Koiter perturbation theory are computationally efficient, however the completely nonlinear response of the structure cannot be achieved using these methods, due to the perturbation expansion is applied only once at the bifurcation point. Therefore, this project intends to propose a novel nonlinearly reduced-order method in the framework of FEM, by reducing the number of degrees of freedom in the finite element model based on the improved Koiter perturbation theory and then using Newton method to correct the nonlinear prediction produced by the reduced order model. First, the method is constructed to solve the complex static buckling problems, such as the structure with a huge number of closely-spaced buckling modes, or with geometric imperfections, or with complicated bifurcation branches. Then, the method is reconstructed to be capable for dynamic buckling analysis by extending the improved perturbation theory into structural dynamic problems. Finally, the method is further enhanced for engineering application and is validated using several numerical and experimental tests. This project plans to break the current system of nonlinear numerical methods by constructing a self-contained novel reduced-order method, which possesses both high computational accuracy and high computational efficiency as a unique advantage. The achievements are expected to be directly used in the engineering field, exhibiting great scientific and practical value.

考虑几何非线性开展结构静、动力屈曲计算的常规有限元法由于计算规模大、求解效率低，极大影响了飞行器薄壁结构研发的周期和成本。而基于传统Koiter摄动理论的降阶方法虽然提高了分析效率，但由于仅能在屈曲分支点处单次摄动展开，因而不能获得结构完整的非线性响应。为此，本课题拟基于改进的Koiter摄动理论来缩减有限元模型的自由度数，并用Newton法修正降阶模型的非线性预测，进而提出一种新型非线性有限元降阶方法。首先，实现该方法对复杂静力屈曲问题的分析功能，可分析含大量密集屈曲模态、几何形状缺陷及复杂屈曲分支路径的结构。然后，对改进的摄动理论进行动力学实现，使其具备动力屈曲分析能力。最后，增强其实际工程应用能力并加以实例验证。本课题以构造系统完备的新型降阶方法的思路寻求对现行非线性数值算法体系的突破，具有兼顾高计算精度和高计算效率的独特优势，成果可望迅速应用于工程实际，具有重要科学意义和应用价值。

薄壁结构在各种复杂载荷作用下极易发生静、动力屈曲失稳，其受载变形同时有着明显的几何非线性效应。为准确评估此类结构的极限承载能力，结构非线性屈曲响应分析将是轻质薄壁结构设计中必不可少的重要技术环节。大规模数值计算与工程设计相结合是现代结构设计的最显著技术特征与最有效的方法途径，基于Newton增量迭代策略的非线性有限元数值计算是当前求解结构非线性屈曲问题最为常用的方法，然而其庞大计算量及对复杂响应的数值稳定性和精度问题已严重阻碍结构非线性屈曲计算在高端装备设计研发中的深入应用。为此，本项目研发了薄壁结构静、动力屈曲分析的新型非线性有限元降阶方法，攻克了当前方法普遍存在的数值计算效率与精度问题。本项目首先构造了改进的Koiter摄动理论，建立了可以完全基于降阶模型来跟踪结构非线性平衡路径的静力学非线性有限元降阶方法；其次，给出了能有效降低降阶模型建立繁琐程度的关键模态挑选准则，实现了对含大量密集屈曲模态结构的高效、高精度分析；然后，成功将结构的几何形状缺陷场投射到降阶模型所在子空间，大幅提高了基于不同缺陷的结构重分析效率；接下来，提出了探测及跟踪屈曲分支路径的数值扰动策略，准确评估了分支点位于极值点前的结构的实际承载能力；最后，发展了基于模态空间的动力学模型降阶方法，揭示了结构非线性刚度及内力的物理特性和显示表达，形成了数值高效的非线性动力学有限元降阶方法。通过一系列数值算例和工程应用检验，本项目所提出的新型非线性分析方法对复杂非线性问题的计算精度大幅提高，并且非线性屈曲分析的计算量只有常规方法的20-30%。由此可见，该方法不但具备高度的原创性和自主可控能力，而且其性能明显优于当前国内、外已有方法。该项目研究工作瞄准航空航天装备设计对结构非线性屈曲分析的迫切需求，有利促进了结构精细非线性计算在高端装备设计研发中的深入应用，同时为国产CAE结构分析软件的研发夯实基础。