蒙特卡罗数值积分在自然单元法中的应用

自然单元法采用自然邻点插值方法在全域构造近似函数和试函数,该方法基于整个求解域内离散结点的Voronoi结构。当采用标准Galerkin法形成系统的平衡控制方程时,对弱形式的积分通常在Voronoi图的对偶图Delaunay三角形内进行,但由于自然邻接插值形函数的特性,自然单元法数值积分存在明显误差。分析了自然单元法数值积分产生误差的各种可能的原因,并提出使用蒙特卡罗方法解决这一问题。该方法权系数直接与精度相关,确定方法简单有效。采用Delaunay三角形内布积分点,使得这种概率积分结果接近数学期望。给出最少积分点数的确定方法,尽可能提高蒙特卡罗积分的计算效率。通过分片试验和悬臂梁等算例验证蒙特卡罗方法解决这些误差的可行性和有效性。

Application of Monte Carlo numerical integration in natural element method

In natural element method,the trial and test functions were constructed with the natural neighbor interpolation method.The interpolation was based on the Voronoi tessellation of the scattered nodes in the problem domain.The integration of the weak form was performed in the Delaunay triangles which were the dual diagram of the Voronoi tessellation when the Galerkin method was used to form the discrete system equation.But there was obvious error in the numerical integration of the natural element method due to the characteristics of natural neighbor interpolation function.Every factor that produced the error possibly was analyzed,and a method using Monte Carlo integration was proposed to solve this problem.The weight coefficient was directly related to the precision,and its determination was also simple and effective.Integral point was cast in the Delaunay triangle,so the result of probability integral was close to mathematical expectation.The definite method of the least integral points,was given so as to enhance computational efficiency of the Monte Carlo integral as far as possible.Finally,the numerical example of the patch experiment and the cantilever beam confirmed the validity and feasibility of using Monte Carlo integration to solve these errors.