基于弱形式求积元法的加筋壳结构非线性屈曲分析方法

As one of the most common structural failure modes, buckling deformation deserves the highest attention in structural analysis and design for its swiftness and the sudden decrease of bearing capacity accompanied by large deformation. The complexity of buckling deformation also makes it a pioneering orientation in computational structural mechanics. By utilizing the weak form quadrature element method accompanied with the geometrically exact model, this project focuses on the formulation of the nonlinear triangular quadrature element and a series of new shell elements for the fulfillment of the analysis of nonlinear buckling for the widely-used stiffened composite shell structures. The algorithmic optimization would be carried out along with the introduction of material plasticity. The implementation to obtain accurate results of configuration, bearing capacity and stresses for arbitrary stiffened shells under large deformation is scheduled to propose to present an efficient solving procedure with high accuracy for these problems. The weak form quadrature element method is a new numerical method for solving differential equations, which shows superiority in structural analysis and overcomes many problems caused by the defects of finite element method. The geometrically exact model, which exactly describes spatial rotation, is the most effective model for nonlinear structural analysis. The innovative formulation based on the quadrature element method and geometrically exact model has enormous potential in engineering application and can serve as references for structural analysis and design.

屈曲失稳是工程结构最常见的失效模式之一，其发生的突然性及伴随结构变形的承载力急剧下降使其在结构设计分析中需得到最高度重视。屈曲破坏的复杂性也使其成为计算结构力学领域的难点之一。本项目拟采用弱形式求积元法结合几何精确模型，构造具有高度创新性的三角形求积元单元及一系列新型壳单元，并以此为基础对工程中应用广泛的复合材料加筋壳进行屈曲全过程分析，考虑结构大变形及材料塑性并进行算法优化，对任意加筋壳结构失稳情况下的位形、承载力、应力及塑性破坏给出准确计算结果，为此类问题提供更加准确、可靠、高效的数值分析方法。弱形式求积元法是一种求解微分方程的新型数值方法，在结构分析尤其是几何非线性分析中能克服有限元存在的诸多缺陷，具有很大优势。几何精确模型精确描述了转动量，是模拟结构大变形最准确有效的模型。利用弱形式求积元法及几何精确模型进行结构屈曲全过程分析具有独创性，在工程设计具有广泛的应用价值和借鉴意义。

本项目面向具有广泛工程应用背景的复杂壳体及加筋壳结构的受力变形、破坏过程分析及结构设计需求，构造相应的非线性屈曲数值分析方法。研究内容及重要成果包括：提出了以边界转动量重构替代位移C1连续，以满足薄壳边界连续性及协调性条件的构想，并以此构造了基于薄壳理论的几何精确求积壳单元，克服了过往大变形薄壳单元自由度过多，计算效率低下的缺陷；提出了基于修正变形梯度张量极分解引入壳面外扭转分量的新方案，并推导了其相对应的新型等效约束关系，以此构造了包含面外扭转及壳厚度变化的几何精确壳模型及相应的求积元单元，利用弱形式求积元的高阶特性结合拉氏乘子法，避免了低阶有限元无法精确保证约束条件成立的弊端，圆满解决了一系列面外扭矩驱动壳体大变形等现有有限元单元几乎无法实现的高难度问题；基于哈密顿系统动量与能量守恒律提出了同时满足时域离散后一致性与正交性关系的混合离散导数法，为完整约束条件下系统的守恒动力算法实现提供了全新的方案，并利用其构建了含面外扭转项壳的动量-能量守恒动力计算格式；将弱形式求积元法应用于复合材料层合壳及加筋壳的非线性屈曲分析，研究了不同条件下加筋壳的分叉屈曲路径，并建立了加筋壳非线性动力屈曲计算格式。本项目所完成的上述工作为复杂板壳结构及加筋壳结构大变形分析及非线性动力分析提供了一套基于新型弱形式求积元法的高效、准确、通用的计算模拟方案，在计算结构力学领域具有较重要的科学意义。