柔性多体系统动力学后验误差估计及模型降阶策略研究

The posterior error estimation and the model order reduction (MOR) are two key factors for achieving the high-accuracy and high-efficiency numerical approximation for flexible multibody system dynamics (MSD). Up to now, there exists controversy as to how to describe flexible beam and plate\shell components undergoing overall rotation and lager deformation accurately in the field of MSD. Furthermore, the posterior error analysis for the space and time discretization of flexible MSD is still an open problem. Besides, the model order reduction for the small deformation problems in flexible multibody systems (MBS) has now achieved maturity to some extent. However, the development of reduction techniques for MBS considering the geometrically and material nonlinearities is still in the preliminary phase. Therefore, in order to develop the multibody system dynamics theory and improve the Advanced Manufacturing Technology (AMT) based on the achievement of MSD, the major studies of the project include the following aspects: the accurate modeling and the refinement strategy for flexible MBS under the framework of the Isogeometric Analysis (IGA), the goal-oriented posteriori error analysis and the adaptive procedure for flexible MSD, and the MOR technique for the parameterized MBS based on POD (Proper orthogonal decomposition) method. The project will focus on solving the following key problems: interpolating variables with non-linear constraints on manifolds, transferring physical variables from one mesh to another without loss of accuracy, and establishing and solving the associated dual problem of large deformation flexible MBS accurately and efficiently.

后验误差估计与模型降阶策略研究是实现高精度、高效率柔性多体系统动力学数值仿真的两个核心问题。目前，在多体系统动力学领域，如何精确描述具有大转动、大变形特性的梁与板\壳结构仍存在争议，柔性多体系统动力学时空离散后验误差估计几乎处于空白状态。此外，小变形柔性多体系统降阶方法已逐渐成熟，但考虑几何与材料非线性系统模型降阶方法研究仅处于起步阶段。因此，为推动柔性多体系统动力学理论的发展，促进本学科研究成果服务于提升我国先进制造业技术水平，本项目研究聚焦于：基于等几何分析的柔性多体系统建模方法以及细化策略研究、柔性多体系统目标导向的后验误差估计及自适应研究、基于POD的参数化柔性多体系统模型降阶方法研究。重点解决以下科学问题：(1)受非线性约束物理量参数插值问题；(2)自适应算法中不同网格间物理量精确投影问题；(3)大变形柔性多体系统对偶模型的建立与高效求解问题。

后验误差估计与模型降阶策略研究是实现高精度、高效率柔性多体系统动力学数值仿真的两个核心问题。针对上述两个问题，本项目主要研究成果如下：提出了一套李群局部标架（Local Frame of Lie Group, LFLG）多柔体系统动力学建模和计算方法体系。该方法能够规避刚体运动带来的几何非线性，保证柔性构件惯性力、弹性力及其雅可比矩阵满足刚体变换的不变性，可极大地提高多体系统动力学仿真效率；基于超收敛修复策略，提出了多体系统空间网格后验误差估计方法。基于层次B样条的细化与粗化算法，提出了一类多体系统动力学空间网格自适应求解算法，其中采用一种无积分的最小二乘拟合方法，在网格自适应过程中将场变量映射至新网格中，以保证计算精度；联合能量守恒网格采样及加权方法、贪心POD算法以及约束消去方法，提出了一种参数化多体系统的构件级POD降阶方法，通过奇异值分解方法获得系统中每个构件的降阶基矢量，并通过约束建立构件间的连接条件；基于有限元撕裂对接(FETI)区域分解技术，提出了一种柔性多体系统通用并行求解算法。其中，采用含预条件的共轭梯度(PCG)迭代算法并行求解区域分解模型的界面问题，从而能够高效地模拟百万自由度多体系统动力学行为。通过研究在著名期刊上共发表SCI文章5篇，EI论文5篇，完成了课题拟定的研究内容，实现了研究目标。