Numerical integration over a disc. A new Gaussian quadrature formula

1998 ◽  
Vol 80 (1) ◽  
pp. 39-59 ◽  
Author(s):  
Borislav Bojanov ◽  
Guergana Petrova
Keyword(s):  
Numerical Integration ◽  
Quadrature Formula ◽  
Gaussian Quadrature ◽  
Gaussian Quadrature Formula

scholarly journals Gauss-Simpson Quadrature Algorithm for Calcaulting Additional Stress in Foundation Soils

2021 ◽  
Author(s):  
Shuai WANG ◽  
Chao WANG ◽  
Chenliang XING
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Elasticity Theory ◽  
Quadrature Formula ◽  
Gaussian Quadrature ◽  
Additional Stress ◽  
The Finite Element Method ◽  
Integral Sign ◽  
Gaussian Quadrature Formula ◽  
Element Method

Abstract The additional pressure at the bottom of a building’s foundation produces an additional stress in the foundation soils under the building’s foundation. In order to overcome the limitations of traditional elastic theory methods and the finite element method when calculating the additional stress in foundation soils, we use the Gauss-Simpson formula to derive the Gauss-Simpson Quadrature Algorithm based on the elasticity theory. The Gauss-Simpson Quadrature Algorithm is a method designed to calculate the additional stress in foundation soils under an irregularly shaped foundation and an irregular load distribution. This new method is based on the fact that the Gaussian quadrature formula and the Simpson formula are independent of the specific type of integrand. The finite element method with n interpolation points can only achieve an algebraic accuracy of n. The interpolation points of the Gaussian quadrature formula are n zeros of orthogonal polynomials, which can achieve an algebraic accuracy of 2n + 1. Moreover, the weights of the nodes in the quadrature formula are all positive, and thus, it has a high numerical stability. In the proposed method, the Simpson formula is necessary. The Simpson formula is used to transform the implicit additional stress formula with the integral sign into an explicit cumulative integral, which can be considered similar to the rectangular domain case to obtain an explicit analytical algebraic formula for solving the additional stress approximation. In engineering applications, we only need to provide the field engineers with the locations of the interpolation points of the Gauss-Legendre formula, the interpolated weight coefficients, and the specific type of Simpson's formula, and then, the results of the additional stress can be calculated manually, which is nearly impossible using the traditional methods and finite element methods. From the point of view of academic rigor and theoretical completeness, it is possible to use the compound Gauss-Simpson Quadrature Algorithm in conjunction with the looping function in computer programs. Under standard conditions, the proposed Gauss-Simpson Quadrature Algorithm is in good agreement with the results of the traditional elasticity theory.

The Finite-Element Method—Part II

1989 ◽  
Author(s):  
Shiro Kobayashi ◽  
Soo-Ik Oh ◽  
Taylan Altan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Integration ◽  
Gaussian Quadrature ◽  
The Finite Element Method ◽  
Gaussian Quadrature Formula ◽  
Stiffness Matrices ◽  
Minimum Number ◽  
Element Technique ◽  
Element Method

Numerical integration is an important part of the finite-element technique. As seen in Section 6.5 of Chap. 6, volume integrations as well as surface integrations should be carried out in order to represent the elemental stiffness equations in a simple matrix form. In deriving the variational principle, it is implicitly assumed that these integrations are exact. However, exact integrations of the terms included in the element matrices are not always possible. In the finite-element method, further approximations are made in the procedure for integration, which is summarized in this section. Numerical integration requires, in general, that the integrand be evaluated at a finite number of points, called Integration points, within the integration limits. The number of integration points can be reduced, while achieving the same degree of accuracy, by determining the locations of integration points selectively. In evaluating integration in the stiffness matrices, it is necessary to use an integration formula that requires the least number of integrand evaluations. Since the Gaussian quadrature is known to require the minimum number of integration points, we use the Gaussian quadrature formula almost exclusively to carry out the numerical integrations.

scholarly journals On the error in a certain interpolation formula and in the Gaussian integration formula

1975 ◽  
Vol 19 (2) ◽  
pp. 196-209 ◽  
Author(s):  
Yudell L. Luke
Keyword(s):  
Quadrature Formula ◽  
Gaussian Quadrature ◽  
Interpolation Formula ◽  
Orthogonality Relation ◽  
Integration Formula ◽  
Basic Theorem ◽  
Gaussian Quadrature Formula ◽  
Gaussian Integration ◽  
Orthogonality Relations ◽  
Error Coefficients

AbstractWe first prove a basic theorem that if a set of polynomials satisfies an orthogonality relation with respect to integration, the set also satisfies an orthogonality relation with respect to summation. This result is then used to derive the Gaussian quadrature formula. The orthogonality relations give rise to interpolation formulas and a connection between the coefficients in these interpolation formulas is established. Finally, the analysis is used to get an estimate of the error in the Gaussian quadrature formula. Some error coefficients are evaluated in the cases where the orthogonal polynomials are those of Jacobi, Laguerre, Hermite and Bessel.

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