Abstract: Based on the remainder term of Gauss-Legendre quadrature rule, the corresponding correction quadrature formulas, which increase the algebraic accuracy at least two-order, are proposed. Using the coordinate transformation to change the integral over the triangle and tetrahedral into an equivalent integral over the square and cube, the correction formula of Gaussian quadrature rule is extended to calculate the multiple integral over a multi-dimensional simplex. Compared with Hammer quadrature formulae over two-dimensional triangular elements and three-dimensional tetrahedral elements, it is shown that the corrected quadrature formula is of high accuracy, can quickly converge to the exact value of the integral. Thus it is of great use in many engineering applications.

Key words: simplex, Gauss quadrature, finite element, algebraic accuracy, integral remainder

摘要： 基于高斯-勒让德求积公式余项，给出相应的校正积分公式，提高了至少两阶代数精度。通过坐标变换将三角形、四面体区域变成正方形、立方体积分区域，把校正高斯求积公式推广到高维单纯形上多重积分的计算。通过与二维三角形单元和三维四面体单元上的Hammer求积公式比较发现，校正求积公式的精度非常高，能更快收敛到积分真值，在工程实际中具有较大的应用价值。

关键词: 单纯形, 高斯积分, 有限元, 代数精度, 积分余项