Generalized Gaussian quadrature rules over two-dimensional regions with linear sides

Jayan, Sarada; Nagaraja, K.V.

doi:10.1016/j.amc.2010.12.039

Cited by 15 publications

(7 citation statements)

References 13 publications

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“…The above integrals in Eq. (6) can be calculated by using Gauss quadrature rules over triangle, as in Rathod et al [20][21][22] and Sarada and Nagaraja [23,24]. After calculating the matrices for each element, assembling is performed to take the effect of all the elements into account and, imposing the boundary condition, we get Eq.…”

Section: Finite Element Methods Over Regular and Irregular Domainsmentioning

confidence: 99%

Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements

Nagaraja

Naidu

Siddheshwar

2014

International Journal for Computational Methods in Engineering

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Section: Finite Element Methods Over Regular and Irregular Domainsmentioning

confidence: 99%

Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements

Nagaraja

Naidu

Siddheshwar

2014

International Journal for Computational Methods in Engineering

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“…Multi-variate integration has attracted a lot of attention in the past decades, see e.g. [21,24,27], and the encyclopedia of cubature rules [10] and the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Gaussian quadrature for C1 cubic Clough–Tocher macro-triangles

Kosinka

Bartoň

2019

Journal of Computational and Applied Mathematics

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A numerical integration rule for multivariate cubic polynomials over n-dimensional simplices was designed by Hammer and Stroud [14]. The quadrature rule requires n + 2 quadrature points: the barycentre of the simplex and n + 1 points that lie on the connecting lines between the barycentre and the vertices of the simplex. In the planar case, this particular rule belongs to a two-parameter family of quadrature rules that admit exact integration of bivariate polynomials of total degree three over triangles. We prove that this rule is exact for a larger space, namely the C 1 cubic Clough-Tocher spline space over macro-triangles if and only if the split-point is the barycentre. This results into a factor of three reduction in the number of quadrature points needed to integrate the Clough-Tocher spline space exactly.

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“…In this section, all examples have taken from the recent paper 14 which are shown in Tables 2-4. Table 3.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…e-mail: mdshafiqul_mat@du.ac.bd Recently, a rigorous and elaborate survey has been reported by Sarada and Nagaraja 14 . In this paper, they have derived some formulas for limited shapes of triangles and quadrilaterals, and then generalized their process for any arbitrary polygon.…”

Section: Introductionmentioning

confidence: 99%

“…In contrast to this study, we propose to develop a general composite integration formula over any arbitrary polygon by decomposing the polygon into arbitrary shapes of triangles, described in section II, using lower order (e.g., 3, 5 and 10) Gauss points to get more accuracy. Numerical examples are taken from the recent paper 14 , and thus are compared. The subsequent formulations are developed by Mathematica.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Composite Numerical Integration Rule Over a Polygon Using Gaussian Quadrature

Hossain¹,

Islam

2015

Dhaka Univ. J. Sci.

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The aim of this paper is to evaluate double integrals over a polygon exploiting coordinate transformation. At first any polygon with m-sides is decomposed into triangles. Then each triangle is transformed into a standard triangular finite element using the basis functions in local space. Then the standard triangle is decomposed into right isosceles triangles with side lengths , and thus composite numerical integration is employed. In addition, the affine transformation over each decomposed triangle and the use of linearity property of integrals are applied. Finally, each isosceles triangle is transformed into a 2-sqare finite element to compute new sampling points and corresponding weight coefficients, using2 s s point's conventional Gaussian quadrature, which are applied again to evaluate the double integral. We demonstrate some numerical examples through the proposed method.

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Generalized Gaussian quadrature rules over two-dimensional regions with linear sides

Cited by 15 publications

References 13 publications

Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements

Optimal Subparametric Finite Elements for Elliptic Partial Differential Equations Using Higher-Order Curved Triangular Elements

Gaussian quadrature for C1 cubic Clough–Tocher macro-triangles

Generalized Composite Numerical Integration Rule Over a Polygon Using Gaussian Quadrature

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