基于精确几何缺陷描述的壳体概率屈曲研究

The buckling problems of thin-walled shell structures received considerable amount of attention in the past century. The sensitivity of shell structures to its geometric imperfections is one of the key points of shell buckling research. In traditional finite element framework, the geometries of imperfect shell structures are approximated using facet faces which will lead to discretization errors, and these errors sometimes have great influences on the final buckling loads of shell structures..In this research project, we propose a NURBS-based surface reconstruction method to fit the point cloud data of real shell geometric imperfections, by which the geometric discretization errors of shells can be naturally avoided. The reconstructed real shell surfaces will be utilized directly in the framework of isogeometric analysis without further need of meshing. The high-order and high-continuity NURBS basis will be fully exploited to the study of buckling of shell structures with real geometric imperfections. An arc-length path following technique will be developed to trace both the pre- and post-buckling phenomena of such shell structures, and the sensitivity of shell buckling to the geometric imperfections will be carried out. In addition, by introducing the concept of random fields, a set of shell samples with random geometric imperfections can be generated. Then the probability features of shell buckling loads can be obtained using direct Monte Carlo simulation, based on which, the reliability analysis can be conducted..Based on the accurate reconstruction and spectra representation of random geometric imperfections of shell structures, more reliable and accurate buckling loads can be obtained using direct Monte Carlo simulation. The proposed research framework could provide the shell designers with a better guideline than the well-known NASA SP-8007, and resulting in a more robust and accurate shell design.

薄壁壳体的屈曲问题一直受到广泛关注，其对于几何缺陷的敏感性一直是壳体稳定性研究的重点。使用传统有限元方法对壳体真实几何缺陷表面进行面片化离散时，会不可避免地产生误差，从而导致所预测的屈曲载荷值的较大偏差。.本项目拟结合逆向工程、等几何方法及随机分析研究具有精确几何缺陷描述的壳体概率屈曲问题。主要内容为：采用基于NURBS的曲面重构方法对壳体真实的几何缺陷点云数据进行逆向建模，建立精确的壳体几何模型，消除了几何离散误差；巧妙地结合等几何方法研究具有精确几何缺陷描述的壳体前、后屈曲规律以及对缺陷的敏感度；建立壳体几何缺陷随机场模型，结合蒙特卡洛模拟，统计屈曲载荷值的分布特征，并进行壳体屈曲的可靠性分析。.本项目预期得到更为可靠的屈曲载荷，揭示精确几何缺陷描述对壳体屈曲的影响及规律，为设计人员提供优于NASA SP-8007准则的参考，提升壳体屈曲设计的精度，具有较为重要的科学研究价值。

薄壁圆柱壳的屈曲对于几何缺陷非常敏感，且传统有限元会产生几何离散误差，本研究结合光滑连续的NURBS样条曲面对各项同性圆柱壳的真实几何缺陷进行重构，并巧妙地结合了基于NURBS的等几何方法，研究了薄壁壳体的非线性屈曲问题，在此基础上，对各项同性圆柱壳体进行了概率屈曲分析。此外，本研究针对复杂薄壁结构建立了一整套分析方法，包括针对剪裁曲面的有限单胞法，以及针对多片曲面耦合的Nitsche方法等。本研究通过一系列数值算例验证了所提方法的精确性以及可靠性等。首先，通过实例验证了等几何壳单元以及非线性求解器的精度，结果表明在相同自由度下，该壳单元比有限元精度更高。其次，通过一些复杂算例验证了有限单胞法以及Nitsche多片耦合方法的精度，其可以实现剪裁曲面耦合界面上的内力及弯矩的连续，并且能准确捕捉其屈曲载荷以及位移载荷曲线。再次，本研究对各项同性圆柱壳的实测几何缺陷进行NURBS重构，并引入等几何分析模型，结果表明，几何缺陷对其屈曲载荷影响较大，相比于理论屈曲值，其降低了约20%。最后，结合几何缺陷随机场的谱表示方法，生成了大量随机样本，并进行了蒙特卡洛分析，得到了圆柱壳的屈曲载荷的分布形式，其均值与实验均值较为接近，表明了本研究的有效性。