基于矩阵特殊运算的高阶流形单元分析

Element Analysis of High-order Numerical Manifold Method based on Special Matrix Operations

在数值流形法的单元分析中广泛采用单纯形积分进行精确积分,然而在高阶数值流形法中采用单纯形积分并不容易,因为要求被积函数为显式多项式函数。采用Kronecker积、Hadamard积和拉直等矩阵特殊运算进行高阶流形单元分析,使得单元矩阵的推导过程简单且被积函数易表示为多项式形式。在此基础上,开发了三维弹性连续体静力分析的高阶数值流形法程序。通过实例验证了公式和程序的正确性。

The simplex integration is commonly used in the numerical manifold method （NMM） for accurate integration. But a difficulty is encountered in adopting the simplex integration in high-order NMM since all integrands have to be explicitly expressed in polynomial functions. To determine the explicit form of integrands, the formulation of element matrices of high-order NMM is derived by using the special matrix operations including Kronecker product, Hadamard product and vectorization, resulting in a convenience of the derivation of the element matrices and the explicit expression of integrands. Based on the derived formulae, a program of 3-D NMM with high-order for elastic static anlysis of continuum is developed and verified by examples.