压电智能层合板壳结构的3-D无网格方法研究

After nearly twenty years of rapid development, meshless methods is a new numerical methods used to solve initial-boundary value problems of partial differential equation. This project mainly aims at creating the meshless new methods which is based on 3-D meshless discretization scheme for the numerical calculation and analysis of the piezoelectric laminated plate and shell structures. These methods can be applied to structures with wide ranges of thickness-to-length ratio, can eliminate various self-locking phenomena and can correctly reflect the transverse stresses distribution. They also have the advantages of fast convergence, well stability, superior accuracy, etc. Firstly, we focused on laminated plate and shell structures, adopted 3-D meshless discretization scheme and created 3-D meshless shape function to approach displacements. Secondly, based on the generalized variational principle for the geometry nonlinear deformation, buckling and dynamics problems, abandoned various traditional shear deformation theory assumptions, from the view of 3-D elastic theory, the space discrete control equations will be derived. And then, this research intended to systematically analyze the self-locking phenomena such as shear, thickness and poisson's ratio, and seek effective methods to overcome those phenomena. Again, after thoroughly consideration of the characteristics of piezoelectric materials, the meshless new methods is expanded to piezoelectric structures. Finally, based on a variety of 3-D meshless methods and numerical results created by diverse problems, this investigation analyzed the order of error and made a universal applicable error estimation method.

无网格法是近二十多年来迅速发展起来的求解偏微分方程初边值问题的新型数值方法。本项目采用3-D无网格离散方案，构造无网格新方法对压电层合板壳结构进行数值计算和分析。新方法具有：适合大范围的厚-跨比结构、能消除自锁现象、能正确反映横向应力分布规律以及收敛快、稳定性好和精度高等优点。首先，以层合板壳结构为研究对象，针对3-D节点离散方案构造无网格的位移近似形函数。其次，基于广义变分原理对几何非线性形变、屈曲及动力学问题，放弃传统的各种剪切形变理论假设，从三维弹性理论出发，导出它们的空间离散控制方程。再次，系统地分析剪切、厚度、泊松比等自锁现象，寻求克服自锁现象的有效方法。然后，在充分考虑压电层的机电耦合性、非均匀性等特性，将层合板壳结构的3-D无网格方法推广到压电层合板壳结构中。最后，通过对不同研究问题所构造的3-D无网格方法和数值结果，进行误差阶的估计，给出具有普适意义的误差估计方法。

本项目主要针对层合板、压电材料结构、功能梯度材料、纳米材料等结构的形变、振动等力学问题利用无网格等数值方法进行深入研究，研究成果主要体现在以下五个方面：第一，针对复合材料层合板弯曲和振动问题，构造了一类3-D无网格离散方法；第二，针对2D弹性力学问题和压电结构构造了一类新型无网格方法（RBF-PUM）；第三，基于层合板理论和高阶剪切理论对压电功能梯度材料板弯曲和振动进行数值分析；第四，基于非经典连续理论特别是非局部理论，结合各类数值方法研究了纳米结构的若干力学性能；第五，针对无网格方法离散压电方程、弹性力学问题所得到的大型稀疏鞍点问题设计出高效的预处理子并进行了理论分析。