结构动力分析的新型准凸无网格法研究

The study of super-accurate numerical methods is an important topic in the field of structural dynamics. In this proposal, a set of relaxed consistency conditions are proposed by introducing nodal attached functions into the conventional reproducing kernel meshfree consistency conditions in order to resolve the conundrum between the desired convex approximation property and the completeness requirement of meshfree shape functions. Subsequently arbitrary high order novel reproducing kernel meshfree shape functions are rationally developed based upon the relaxed consistency conditions, which are then employed together with the sub-domain stabilized conforming integration to formulate a bran-new and highly accurate quasi-convex meshfree method. The quasi-convex meshfree shape functions share the same the computational structure as the standard moving least square or reproducing kernel shape functions and thus they are very computationally efficient and extension to multi-dimensional higher order formulations is straightforward. This approach overcomes the severe issues associated with the existing max-entropy convex meshfree scheme, i.e., the construction difficulty for high order shape functions as well as the computational complexity and extremely low efficiency. Meanwhile, the proposed quasi-convex meshfree shape functions inherit the major advantages of convex approximation and they are nearly positive over the problem domain except near the boundary. The proposed quasi-convex meshfree method succeeds all the advantages of the conventional meshfree methods, i.e., simple model generation, flexible local model refinement and high order smoothness and compatibility. More importantly, the present method possesses much more superior dynamic frequency spectrum and remarkably excellent computational accuracy, which provides a very valuable way for the dynamic analysis of large scale complex structures and is of great importance from both academic research and engineering application points of view.

高精度动力计算方法是结构分析领域的一个重要研究内容。本项目通过在再生核无网格完备性条件中引入节点附加函数，提出能够解决任意高阶形函数凸近似特性和完备性要求之间根本矛盾的松弛完备性条件，进而构造任意高阶准凸无网格形函数，并结合稳定子域积分方法建立结构动力分析的高精度新型准凸无网格法。准凸无网格法的形函数具有与传统移动最小二乘或再生核无网格形函数相同的架构，形式简单、计算高效，可直接推广到任意高阶和多维情况，能够有效解决最大熵凸近似无网格法不能构建高阶形函数且需要迭代计算、繁琐低效的问题。同时准凸无网格形函数具有凸近似的主要性质，在除了边界附近的整个离散区域内均为正值。准凸无网格法除了具有整体建模简洁、局部模型细化方便、形函数高阶协调等优点外，与传统无网格法相比具有更加优异的频谱特性和十分显著的精度优势，可为大型复杂结构的动力分析提供更为精确高效的计算方法，有重要的理论研究和工程应用价值。

本项目较系统地建立了满足任意高阶完备性条件的准凸无网格形函数构造方法，并发展了配套的结构动力分析准凸无网格法。该准凸无网格形函数仍然具有与传统再生核无网格形函数相似的构造形式，数值实现比较便捷，并且其多项式再生条件具有准确的修正系数，无需引入额外的人工节点松弛参数。与传统无网格形函数相比，准凸无网格形函数的负值部分和数值震荡明显减少，具有良好的频谱精度。同时，发展了具有二阶精度的无网格嵌套子域数值积分方法，提高了无网格法的计算效率。本项目还进一步研究了无网格形函数和等几何基函数的耦合方法及位移光滑无网格法，提升了无网格法的计算精度，拓展了其应用范围。此外，对于结构振动分析问题，发展了具有超收敛特性的等几何与有限元频率计算方法。项目执行期内发表SCI期刊论文26篇，包括计算力学领域国际一流期刊Computer Methods in Applied Mechanics and Engineering 6篇，Computational Mechanics 3篇。项目负责人获聘为国际计算力学学会理事，2016年和2018年分别获得国际华人计算力学学会计算力学奖和ICACM奖。