单纯形积分的递推公式

石根华提出的单纯形积分是一种精确积分，被广泛应用于非连续变形分析和数值流形法中。基于Kronecker积、Hadamard积和拉直等矩阵特殊运算，将单项式函数在单纯形上的积分表示为矩阵形式，提出了单纯形积分的递推公式。与石根华的单纯形积分公式相比，递推公式计算量大为减少，而且在计算高阶函数积分的同时，还附带获得所有低阶函数的积分。

Recursive Formula for Simplex Integration

The simplex integration formula presented by Genhua Shi gives analytical solution and is widely used in the discontinuous deformation analysis and the numerical manifold method. Based on the special matrix operations including Kronecker product, Hadamard product and vectorization, the integration of monomial integrand on simplex domain is expressed in matrix form and a recursive formula for the simplex integration is presented. Compared with Shi’s integration formula, the recursive formula requires much less computation and incidentally obtains the integrals of all lower order integrands at the same time of computing integration of high order integrand. 