快速多极虚边界元法对含圆孔薄板有效弹性模量的模拟分析

针对虚边界元法,引入快速多极展开和广义极小残值法(GMRES)的思想,以形成快速多极虚边界元法的求解思想,并将此方法用于含圆孔薄板有效弹性模量的模拟分析.由于本文方法采用了"源点"多极展开和"场点"局部展开的组合处理方案,从而使得原问题方程组求解的计算耗时量和储存量降至与所求问题的计算自由度数成线性比例.本文工作的研究目的在于:提高虚边界元法在普通台式机上的运算能力和拓宽虚边界元法对大规模复杂问题的求解(或数值模拟).文中给出了均布圆孔的正方形薄板和之字形分布圆孔薄板二个算例,以验证该方法的可行性,计算精度和计算效率.

Fast multipole VBEM for analyzing the effective elastic moduli of a sheet containing circular holes

To the virtual boundary element method (VBEM), generalized minimal residual algorithm (GMRES) and fast multipole method (FMM) are used to form the idea of fast multipole VBEM, which is applied to analyze the effective elastic moduli of a plate containing circular holes. Owing to the combined scheme of multipole expansion at the collocation point and local expansion at the integration point, the complexities of operation and memory about solution of the equations would be made to be of linear proportion to the freedoms of the problem. The purpose of the present study is to improve the operational capability of VBEM on a laptop computer and extend the numerical simulation of large scale complex problem. Two numerical examples about a square plate with uniformly distributed holes and a sheet containing zig-zag distributed holes are presented to demonstrate the feasibility, accuracy and efficiency of the method.