基于蒙特卡罗积分的无网格伽辽金法及其在渗流分析中的应用

Element-free Galerkin method based on Monte Carlo integration and its application in seepage analysis

无网格法在处理有限元法难以解决的问题时具有显著的优势,且前后处理比较简单。无网格伽辽金法(Element-free Galerkin method,EFG)是无网格法的一种,它采用滑动最小二乘法近似场函数,计算精度较高,且有较好的稳定性,在结构和渗流分析中受到欢迎。但EFG方法需要背景积分网格进行高斯积分,离真正的"无网格"方法还有一定距离。对其积分方法进行改进,采用蒙特卡罗方法(Monte Carlo)进行积分运算,由此提出了基于蒙特卡罗积分的无网格法(MCEFG)。从而摆脱EFG方法对背景积分网格的依赖,使之成为真正意义上的无网格方法。基于MCEFG方法的渗流分析程序对有自由面的渗流问题进行了较好模拟。该方法可以推广应用到其他需要背景积分网格的方法中,从而使无网格法真正与网格脱离。

Meshless methods were preponderant in dealing with problems when it was difficult to employ finite element method. And they could simplify pre- and post-processings. Element-free Galerkin method （EFG）, based on moving least squares method （MLS）, was one of the meshless methods. Owing to its accuracy and stability, EFG method was widely applied in structure and seepage analysis. However, EFG method was not a ＂truly＂ meshless method, since mesh was still used to conduct numerical integration. A Monte Carlo-based EFG method （MCEFG） was proposed to do the integration with Monte Carlo method. The mesh or cell was not used in MCEFG method. So it was truly a meshless method. A program based on MCEFG method was compiled and applied to seepage analysis. It was shown that the method was suitable for simulating seepage with a free surface. The idea in this study could be generally extended to other methods with integral mesh, and transformed them to be truly meshless.