Thirdly, the standard Gauss-Legendre quadrature rule is adopted to evaluate these reduced 1D edge integrals and the two remaining 2D surface integrals. Finally, a spherical shell model with an available analytical solution for gravitational fields has been tested to verify our new formulae, the ...
When using Gauss integration points for a 2D quadrilateral element, we need to consider both axes. To account for the 2D nature of the element, the Gauss quadrature has the form (4.65)∫−11∫−11f(ξ,η)dξdη≈∑i=1n∑j=1mWi×Wj×f(ξi,ηj). For a 3D, 1-point Gauss qua...
even though this nomenclature, strictly speaking, is correct only for integration points defined by theGaussian quadraturemethod. In COMSOL Multiphysics, true Gaussian quadrature is used for integration in 1D, quadrilateral elements in 2D, and
2D: Seven-point rule is sixth-order accurate3D: Six-point (non-Gaussian) rule is fourth-order accurate (The quadrature points are located at the centers of the six faces of the cube, and the weights are all 4/3)3D: A 14-point rule is sixth-order accurate4.3 高斯积分ReferencesB.M. ...
withhigherorderelementsorformorecomplexdistortedelementstheintegralsbecometoocom- plicatedforanalyticalintegrationandthenumericalintegrationisessential,amongvariousintegration schemes,GaussLegendrequadraturewhichcanevaluateexactlythe(2nÀ1)thorderpolynomialwith n-Gaussianpointsismostcommonlyusedinviewoftheaccuracyandefficie...
Although Smolyak's quadrature rule can successfully generate sparse cubature points for high dimensional integral, it has a potential drawback that some cubature points generated by Smolyak's rule have negative weights, which may result in instability for the computation. A relative-weight-ratio ...
By using the IGBEM for potential problems, the effect of Gauss quadrature on the accuracy of each term arising in the IGBEM is studied for smooth geometry under constant boundary conditions. The results show that the method of computing singular integrals in the IGBEM is effic...
Gauss-Legendre quadratureIntegral theoremSpherical coordinateIn this study, we present a new method to compute the gravity field and gravity gradient tensor caused by tesseroid mass bodies in the spherical coordinate system. Using this new method, we are able to study continental scale and global ...
Gauss-Hermite quadrature together with a finite difference method is used to solve numerically jump-diffusion two-asset option pricing problem consisting in a partial integro-differential equation.doi:10.1007/978-3-319-59387-6_14L. JódarM. Fakharany...
Gauss-Jacobi quadrature ruleFractional stiffness matrixThough the finite element method has been widely used in solving fractional differential equations, the effects of the Gaussian quadrature rule on the numerical results have rarely been considered. Since the fractional derivatives of the basis functions...