Correspondence ∗N. Sukumar, Department of Civil and Environmental Engineering, University of California, One Shields Avenue, Davis, CA 95616, USA Email: nsukumar@ucdavis.edu In this paper, we construct new high-order numerical integration schemes for tetrahedra, with positive weights and integration points that are in the interior of the domain. The construction of cubature rules is a challenging problem, which requires the solution of strongly nonlinear algebraic (moment) equations with side conditions given by affine inequality constraints. We present a robust algorithm based on a sequence of three modified Newton procedures to solve the constrained minimization problem. In the literature, numerical integration rules for the tetrahedron are available up to order p = 15. We obtain integration rules for the tetrahedron from p = 2 to p = 20, which are computed using multi-precision arithmetic. For p ≤ 15, our approach provides integration rules that have the same or fewer number of integration points than existing rules; for p = 16 to p = 20, our rules are new. Numerical tests are presented that verify the polynomial-precision of the cubature rules. Convergence studies are performed for the integration of exponential, rational, weakly singular and trigonometric test functions over tetrahedra with flat and curved faces. In all tests, improvements in accuracy is realized as p is increased, though in some cases nonmonotonic convergence is observed.

中文翻译：

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四面体的高阶体积规则

信函*N。Sukumar，加利福尼亚大学土木与环境工程系，One Shields Avenue, Davis, CA 95616, USA 电子邮件：nsukumar@ucdavis.edu 在本文中，我们为四面体构建了新的高阶数值积分方案，具有正权重和域内部的集成点。培养规则的构建是一个具有挑战性的问题，它需要求解具有仿射不等式约束给出的边条件的强非线性代数（矩）方程。我们提出了一种基于三个改进牛顿程序序列的稳健算法来解决约束最小化问题。在文献中，四面体的数值积分规则最高可达 p = 15。我们获得了四面体从 p = 2 到 p = 20 的积分规则，使用多精度算术计算。对于 p ≤ 15，我们的方法提供的积分规则与现有规则的积分点数相同或更少；对于 p = 16 到 p = 20，我们的规则是新的。给出了验证体积规则的多项式精度的数值测试。对具有平面和曲面的四面体上的指数、有理、弱奇异和三角测试函数的积分进行收敛研究。在所有测试中，随着 p 的增加，精度会得到提高，尽管在某些情况下会观察到非单调收敛。给出了验证体积规则的多项式精度的数值测试。对具有平面和曲面的四面体上的指数、有理、弱奇异和三角测试函数的积分进行收敛研究。在所有测试中，随着 p 的增加，准确性的提高都实现了，尽管在某些情况下会观察到非单调收敛。给出了验证体积规则的多项式精度的数值测试。对具有平面和曲面的四面体上的指数、有理、弱奇异和三角测试函数的积分进行收敛研究。在所有测试中，随着 p 的增加，精度会得到提高，尽管在某些情况下会观察到非单调收敛。