独立覆盖流形法的本质边界条件施加方法

Application of Essential Boundary Conditions in Numerical Manifold Method Based on Independent Covers

目前,数值流形方法、无网格法等新的数值计算方法存在本质边界条件不易严格施加的问题。针对笔者前期提出的独立覆盖流形法,通过一个悬臂梁的例子,系统地分析了本质边界条件施加问题。采用多项式覆盖函数,提出了改进边界覆盖函数和直接设定独立覆盖函数2种方法,不仅严格满足边界条件,而且能保证边界附近的近似函数逼近真实解。这2种方法避免了常用罚函数法中的罚数取值对计算结果和方程性态的影响问题,而且只需令部分自由度不参与计算就能实现,操作简单。通过设置覆盖函数来施加边界条件的方式可供其他新方法借鉴。

At present,some new numerical methods such as Numerical Manifold Method( NMM) and Meshless Method are facing with the difficulty of strictly applying essential boundary conditions. Through a case study of a cantilever beam,the application of essential boundary conditions is systematically analyzed in NMM based on independent covers previously proposed by the authors. On the basis of polynomial cover functions,two methods are presented: one is the improved method of boundary cover functions; and the other is the method of setting independent cover functions. The boundary conditions are strictly satisfied,and the approximate functions near the boundaries are guaranteed to approach the real solutions. The proposed methods are refrained from the influence of penalty number in common penalty method on computational results and linear equation conditions. Moreover,the implementation is very simple,for it just needs some degrees of freedom not involved in the computation. The proposed approach of applying boundary conditions by setting cover functions has a reference value for other new methods.