自然单元法数值积分方案研究

自然单元法采用自然相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节点的Voronoi结构。采用标准Galerkin法形成系统的平衡控制方程时,对弱形式的积分是在Voronoi图的对偶图Delaunay三角形内进行的;但自然单元法形函数其局部支撑域与背景积分域不一致,从而导致了相当的积分误差。单位分解积分方法利用无网格形函数满足单位分解的条件,从而将对弱形式的积分转化到形函数的紧支域内进行;但自然单元法中形函数紧支域的形状和大小由Voronoi结构所限定且形状较为复杂,实现单位分解积分算法需对其形函数紧支域进行分解映射,计算量较大。若在Delaunay三角形内采用基于三角形内平均应变的单点积分方案,利用散度定理可将三角形内平均应变的计算转化为在三角形边界环路上的积分,从而在形成系统矩阵的过程中无需形函数导数的计算,计算量小而精度高。采用单位分解积分方案,其计算精度和收敛性均好于基于平均应变的点积分方案,但综合计算精度和计算效率考虑,则基于平均应变的点积分方案较为理想。

STUDY ON THE NUMERICAL INTEGRATION SCHEME WITH NATURAL ELEMENT METHOD

In natural element method(NEM),the trial and test function are constructed with the natural neighbor interpolation(NNT) method. The interpolation is constructed with respect to the Voronoi tessellation of the scattered nodes in the problem domain. The integration of the weak form is performed in the Delaunay triangles, which is the dual diagram of the Voronoi tessellation when the Galerkin method is used to form the discrete system equation. But the integration domain does not align with the compacted support of the NEM shape function,which results in considerable integration error. The partition of unity quadrature method which employs the fact that the meshless shape function possesses the partition of unity is presented,and the integration can be done within the compacted support of the shape function. However,the shape and the size of NEM shape function are determined by the Voronoi tessellation and are somewhat complex. Therefore,the implementation of partition of unity quadrature in NEM involves the decomposition and mapping the support of NEM shape function. A single point quadrature rule is proposed based on the average strain of each Delaunay triangle,the average strain can be calculated by shift surface integral to contour integral via the divergence theorem,thus the derivatives of the shape function are unnecessary to form the global stiffness matrix. Numerical examples show both the accuracy and efficiency are improved. Considering both the accuracy and the efficiency,the point quadrature based on the average strain of Delaunay triangle is a better selection,though its convergence and accuracy are a little lower than the partition of unity quadrature method.