Some integration formulas for a four-node isoparametric element
Computer Methods in Applied Mechanics and Engineering
Proof of a conjectured asymptotic expansion for the approximation of surface integrals
Mathematics of Computation
Quadrature over curved surfaces by extrapolation
Mathematics of Computation
A local projection operator for quadrilateral finite elements
Mathematics of Computation
Analysis of Some Quadrilateral Nonconforming Elements for Incompressible Elasticity
SIAM Journal on Numerical Analysis
Explicit eight—noded quandrilateral elements
Finite Elements in Analysis and Design
Finite Element Modeling for Stress Analysis
Finite Element Modeling for Stress Analysis
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
A Stable Affine-Approximate Finite Element Method
SIAM Journal on Numerical Analysis
An encyclopaedia of cubature formulas
Journal of Complexity
Finite Elements in Analysis and Design
Hi-index | 7.29 |
For general quadrilateral or hexahedral meshes, the finite-element methods require evaluation of integrals of rational functions, instead of traditional polynomials. It remains as a challenge in mathematics to show the traditional Gauss quadratures would ensure the correct order of approximation for the numerical integration in general. However, in the case of nested refinement, the refined quadrilaterals and hexahedra converge to parallelograms and parallelepipeds, respectively. Based on this observation, the rational functions of inverse Jacobians can be approximated by the Taylor expansion with truncation. Then the Gauss quadrature of exact order can be adopted for the resulting integrals of polynomials, retaining the optimal order approximation of the finite-element methods. A theoretic justification and some numerical verification are provided in the paper.