Gaussian quadrature forC1cubic Clough–Tocher macro-triangles (original) (raw)
Related papers
Several new quadrature formulas for polynomial integration in the triangle
2005
We present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that the number of quadrature points NNN be equal to the dimension of a lower dimensional polynomial space. Quadrature forumulas are presented for up to degree d=25d=25d=25, all which have positive weights and contain
Numerical integration based on trivariate C 2 quartic spline quasi-interpolants
BIT Numerical Mathematics, 2013
In this paper we consider the space generated by the scaled translates of the trivariate C 2 quartic box spline B defined by a set X of seven directions, that forms a regular partition of the space into tetrahedra. Then, we construct new cubature rules for 3D integrals, based on spline quasi-interpolants expressed as linear combinations of scaled translates of B and local linear functionals. We give weights and nodes of the above rules and we analyse their properties. Finally, some numerical tests and comparisons with other known integration formulas are presented.
A Cardinal Function Algorithm for Computing Multivariate Quadrature Points
SIAM Journal on Numerical Analysis, 2007
We present a new algorithm for numerically computing quadrature formulas for arbitrary domains which exactly integrate a given polynomial space. An effective method for constructing quadrature formulas has been to numerically solve a nonlinear set of equations for the quadrature points and their associated weights. Symmetry conditions are often used to reduce the number of equations and unknowns. Our algorithm instead relies on the construction of cardinal functions and thus requires that the number of quadrature points N be equal to the dimension of a prescribed lower dimensional polynomial space. The cardinal functions allow us to treat the quadrature weights as dependent variables and remove them, as well as an equivalent number of equations, from the numerical optimization procedure. We give results for the triangle, where for all degree d ≤ 25, we find quadrature formulas of this form which have positive weights and contain no points outside the triangle. Seven of these quadrature formulas improve on previously known results.
Integration of polynomials over n-dimensional polyhedra
Computer Aided Design, 1991
Integrating an arbitrary polynomial function f of degree D over a general simplex in dimension n is well-known in the state of the art to be NP-hard when D and n are allowed to vary, but it is time-polynomial when D or n are fixed. This paper presents an efficient algorithm to compute the exact value of this integral. The proposed algorithm has a time-polynomial complexity when D or n are fixed, and it requires a reasonable time when the values of D and n are less than 10 using widely available standard calculators such as desktops.
BIT Numerical Mathematics, 2018
In this paper, we consider the problem of approximating a definite integral of a given function f when, rather than its values at some points, a number of integrals of f over some hyperplane sections of simplices in a triangulation of a polytope P in R d are only available. We present several new families of "extended" integration formulas, all of which are a weighted sum of integrals over some hyperplane sections of simplices, and which contain in a special case of our result multivariate analogues of the midpoint rule, the trapezoidal rule and the Simpson's rule. Along with an efficient algorithm for their implementations, several illustrative numerical examples are provided comparing these cubature formulas among themselves. The paper also presents the best possible explicit constants for their approximation errors. We perform numerical tests which allow the comparison of the new cubature formulas. Finally, we will discuss a conjecture suggested by the numerical results.
Gauss-Legendre Quadrature Over Triangles: A practical Approach
WCMNA2019, 2019
It is presented the 1D Gauss-Legendre quadrature and it is extended to 2D triangular domain. The main objective of the present paper is to develop a practical and simple algorithm for numerical integration over triangular domain by using the well-established Gauss-Legendre quadrature. It is also demonstrated the effectiveness of the above algorithm by applying it to some typical integrals as shown in this work.
Consistent structures of invariant quadrature rules for the nnn-simplex
Mathematics of Computation, 1995
In this paper we develop a technique to obtain, in a systematic way, the consistency conditions for the «-dimensional simplex T" for any dimension n and degree of precision d. The introduction of a convenient basis of invariant polynomials provides a powerful tool to analyze and obtain consistent structures. We also present tables listing the optimal consistent structures for dimensions n = 2, ... ,8 and degree of precision up to d = 23. This paper is devoted only to structures. No quadrature rules are presented here.
Appropriate Gaussian quadrature formulae for triangles
International Journal of Applied Mathematics and Computation, 2012
This paper mainly presents higher order Gaussian quadrature formulae for numerical integration over the triangular surfaces. In order to show the exactness and efficiency of such derived quadrature formulae, it also shows first the effective use of available Gaussian quadrature for square domain integrals to evaluate the triangular domain integrals. Finally, it presents n × n points and n(n+1) 2 − 1 points (for n > 1) Gaussian quadrature formulae for triangle utilizing n-point one-dimensional Gaussian quadrature. By use of simple but straightforward algorithms, Gaussian points and corresponding weights are calculated and presented for clarity and reference. The proposed n(n+1) 2 − 1 points formulae completely avoids the crowding of Gaussian points and overcomes all the drawbacks in view of accuracy and efficiency for the numerical evaluation of the triangular domain integrals of any arbitrary functions encountered in the realm of science and engineering.