刚柔流耦合动力学建模和高效无网格法数值分析研究

Large amplitude vibration caused by aerodynamic force and the variation of rotating speed in high speed rotating blade tends to induce the blade’s failure of operation, such as over deformation, buckling and even rupture etc.. The high speed rotating blade can be modeled as the structure in fixed axis rotation. Coupled modeling of the structural motion, deformation and the fluid flow and its numerical analysis are quite important for the dynamical analysis of the structure. Since remeshing is required in the analysis of large deformation and fluid-structure interaction for numerical methods based on mesh, in order to avoid the remeshing, other efficient numerical methods should be developed to solve the coupling dynamical equations. In this proposal, a coupling model is set up to simulate the coupling dynamics among structural motion, deformation and the fluid flow for linear and nonlinear beam and plate. The influences on the structural deformation and vibration from the rigid body motion and fluid flow, and the influences on the stability of the system and the fluid flow are investigated based on the coupling equations. The modeling and responses are compared with other existing modelings and their resulting responses. Meshfree Radial Basis Collocation Method is proposed to solve the coupling dynamical equations. Numerical integration method in the time domain for the time-variant problem is investigated. The accuracy, convergence and stability for the proposed method are also studied. Moreover, the numerical solutions are compared with the numerical results from other numerical methods, which examine and validate the applicability of the proposed numerical meshfree method. Based on these studies, a meshfree software is further developed.

高速旋转叶片中气动力和转速变化引起的大幅振动往往会导致其过度变形、屈曲乃至断裂等严重的失效行为。叶片可简化成做定轴转动的结构模型。结构的刚体运动、变形和周围流体流动完全耦合的刚柔流耦合动力学建模和数值分析对结构的振动性能研究至关重要。由于基于网格的数值算法在求解大变形和流固耦合问题中需要不断重新划分网格，新的高效数值算法研究也势在必行。本项目拟建立做高速定轴转动的线性梁板结构和考虑大变形的非线性梁板结构的刚柔流耦合动力学模型，分析结构的刚体运动和周围流场对结构变形和振动性能的影响，分析耦合效应对系统稳定性和周围流场的影响，并与传统建模方法及其响应进行比较；研发求解刚柔流耦合动力学方程的统一通用的高效无网格径向基函数算法，研发求解时变问题的时域积分方法，考察算法的精度、收敛性和稳定性，将所得数值结果与传统数值算法所得结果进行比较，考察研发的数值算法的适用性，并研制相应的无网格计算软件。

高速运动中周围流体对结构响应的影响非常显著，包括气动弹性耦合在内的各类流固耦合问题是自然界和工程领域中常见的现象，由于其非线性和多学科性，此类问题的研究具有很大挑战性。基于传统方法如有限元法直接求解Navier-Stokes 和非线性动力学的耦合方程往往会遇到难以处理大变形和多个模型的相互作用等问题，而无网格法因节点之间不受网格结构的限制，避免了大变形分析中的网格畸变和计算移动不连续问题时的网格重构带来的计算困难，因此在刚柔流耦合分析中具有明显优势。主要研究内容包括：（1）研发了无网格加权径向基函数配点法分析不可压流体动力学问题，这种方法无需引入人工可压缩系数，可以在同一时间步直接求解速度和压力，避免了压力振荡，从而提高了求解精度和稳定性，在边界和连续性方程上施加适当权重可以实现最佳收敛。能够很好地捕获流体的自由表面。（2）基于广义哈密尔顿原理建立了刚柔流耦合动力学模型，研发了半解析法、无网格径向基函数配点法、复模态法求解这类问题。刚柔流耦合动力学问题是一个时变问题，采用传统时不变模态的半解析法精度稍有欠缺。无网格径向基函数配点法求解此类问题的弥散误差小，能够获得较高精度。复模态法推导了时域连续的解析表达式，避免了时域离散带来的误差，可获得较高精度。数值结果分析了刚体运动角速度、流体流速和流动方向等对系统基频和振动响应的影响。（3）基于径向基函数配点法分析了面内梯度分布功能梯度材料薄板屈曲问题。（4）研发了分区径向基函数配点法并将其应用于大变形分析。（5）研发了稳定直接配点法分析波动反演问题。编写了统一通用的加权径向基函数配点法、分区径向基函数配点法、Hermite径向基函数配点法等相应的算法程序来求解时变动力学问题，开发的算法程序可为无网格法求解相关非线性问题和耦合动力学问题提供计算平台，刚柔流耦合动力学模拟为今后的动力学控制打下了基础。