High‐order cubature rules for tetrahedra

Abstract

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 multiprecision 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. K E Y W O R D Sleast squares, numerical integration, polynomial basis, positive weights, strictly interior integration points, tetrahedral domain 1 2418

High‐order symmetric cubature rules for tetrahedra and pyramids

“…4,20,21,28 In these contributions, symmetry is applied to the nodal positions and the basis functions, so that in the reduced-scale problem only the unknowns in s (fewer than in z) need to be determined. In this article, we follow the approach in Jaśkowiec and Sukumar, 1 and apply preconditioning to the Jacobian matrix. This operation destroys the symmetry of the basis functions.…”

“…However, to improve the convergence rate in the third phase, the optimal can be determined as presented in Jaśkowiec and Sukumar. 1 The Jacobian matrix J k = f k / z needs to be calculated at every iteration step. The pair (J k , f k ) is calculated using the monomial basis, which if used as is would lead to ill-conditioning of the Jacobian matrix as p is increased.…”

“…where the pair (J k , f k ) is the preprocessed pair (J k , f k ) using MGJ. 1 The vector Δz calculated in (18) belongs to S 0 c , which is established using Lemma 3.…”

“…Note that for p = 14, we obtain fewer number of integration points than Zhang et al 21 In Table 4, we compare our results for the number of integration points in the symmetric tetrahedral cubature scheme to the nonsymmetric rules presented in Jaśkowiec and Sukumar. 1 The lower bound estimate for the number of integration points is also indicated. It is seen that placing the symmetry requirements results in greater number of nodes.…”

“…We include full symmetries via additional equality constraints in the algorithm to realize symmetric cubature rules. This article is self-contained; however, to avoid needless repetitions, where appropriate we point the reader to Jaśkowiec and Sukumar 1 for some of the details. Finite element (FE) methods require numerical integration to be performed over two-and three-dimensional element shapes.…”