配点型无网格法理论和研究进展

Theory and Research Progress of the Collocation-Type Meshfree Methods

有限元法是当前工程科学领域应用最广泛的数值计算方法之一,但是其在求解极端大变形、高速碰撞等一些复杂问题时,容易出现网格畸变和网格敏感性,从而导致计算结果精度低和不收敛的问题.为了避免网格带来的问题,出现并兴起了各种无网格法.无网格法不仅建模简便,而且收敛速度更快、计算精度更高,可用于求解有限元等网格类方法难以求解乃至尚未触及的问题.本文首先阐述了无网格法的分类以及具有代表性的方法.目前限制无网格法发展的主要问题是效率偏低.伽辽金型无网格法效率较低,而配点型无网格法效率较高,在复杂问题的高效高精度数值模拟中具有更大潜力.因此本文详细介绍了配点型无网格法的起源和研究进展,归纳了其常用的近似函数和离散方法,最后对无网格法的发展做出了总结和展望.无网格法的研究和改进,为复杂问题的高效高精度数值模拟开辟了新的途径.

The finite element method is one of the most widely used numerical methods in the engineering applications. However,it is prone to grid distortion and grid sensitivity when solving some complex problems such as the extreme large deformations and high-speed impacts, which lead to the low accuracy and non-convergence. In order to avoid the troubles caused by the grid, the meshfree methods were born and emerged. The meshfree methods not only possess the easier modeling, but also has faster convergence and higher accuracy, which can be used to solve the tough problems that are hard to be handled by the mesh-based methods such as the finite element method. In this work, the classification of the meshfree methods and their representative methods are introduced firstly. Currently, the main disadvantage that restricts the development of the meshfree methods is the low efficiency. Although the Galerkin-type meshfree methods are not efficient, the collocation-type meshfree methods have much higher efficiency, which are the potential methods for the high-efficiency and high-accuracy numerical simulation of the complex problems. Therefore, the origin and the research progress of the collocation-type meshfree methods are presented in details in this work. After that, the commonly used approximation functions and the methods for discretization are exhibited, and the remarks and developments of the meshfree method are summarized at last. The improvements and developments of the meshfree methods can open up a new way for the high-efficient and high-accurate numerical simulation of the complex problems.