On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region

SUMMARYAn assumed-strain finite element technique is presented for linear, elastic small-deformation models. Weighted residual method (reminiscent of the strain-displacement functional) is used to weakly enforce the balance equation with the natural boundary condition and the kinematic equation (the strain-displacement relationship). A priori satisfaction of the kinematic weighted residual serves as a condition from which strain-displacement operators are derived via nodal integration. A variety of element shapes is treated: linear triangles, quadrilaterals, tetrahedra, hexahedra, and quadratic (six-node) triangles and (27-node) hexahedra. The degrees of freedom are only the primitive variables (displacements at the nodes). The formulation allows for general anisotropic materials. A straightforward constraint count can partially explain the insensitivity of the resulting finite element models to locking in the incompressible limit. Furthermore, the numerical inf-sup test is applied in select problems and several variants of the proposed formulations (linear triangles, quadrilaterals, tetrahedra, hexahedra, and 27-node hexahedra) pass the test. Examples are used to illustrate the performance with respect to sensitivity to shape distortion and the ability to resist volumetric locking.

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“…For instance, the rule designed in Reference [8] associates negative weights with the corners; see also additional information in Reference [4].…”

Section: Second-order Elementsmentioning

confidence: 99%

“…Tetrahedral quadrature rules that would utilize integration points at the nodes exist, but are apparently not suitable. For instance, the rule designed in Reference [8] associates negative weights with the corners; see also additional information in Reference [4].…”

Section: Second-order Elementsmentioning

confidence: 99%

Locking‐free continuum displacement finite elements with nodal integration

Krysl

Zhu

2008

Numerical Meth Engineering

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“…There has been considerable interest in the area of numerical integration schemes over triangles and tetrahedra [4][5][6][7] . Recently, Rathod [4] et al derived various orders of extended numerical integration rules based on classical Gauss-Legendre quadrature over a triangle.…”

Section: Introductionmentioning

confidence: 99%

Correction of Gauss Legendre quadrature over a triangle

Pan

2011

2011 International Conference on Multimedia Technology

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Based on the remainder term for Gauss-Legendre quadrature rule, a correction formula for numerical integration over a triangle is proposed. The new formula increases the algebraic accuracy at least two-order in comparison with the original Gauss-Legendra quadrature rules, which were recently derived by Rathod et al. Results obtained with the correction formulae are compared with the existing formulae. It is shown that the correction formula has higher accuracy than the existing formulae. Thus it is of great use in many engineering applications.

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“…The precision of these formulae is limited to polynomials of degree seven; this is because the weights and roots of Jacobi polynomials are not tabulated in standard texts for sufficiently higher degree polynomials. The product formulae proposed in this paper and in the recent works [1,[10][11][12][13] are based only on the roots and weights of Gauss Legendre polynomials. The integral I of Eq.…”

Section: Formulation Of Integrals Over a Triangular Areamentioning

confidence: 99%

Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space

Rathod

Nagaraja²,

Venkatesudu³

2007

Applied Mathematics and Computation

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In this paper it is proposed to compute the volume integral of certain functions whose antiderivates with respect to one of the variates (say either x or y or z) is available. Then by use of the well known Gauss Divergence theorem, it can be shown that the volume integral of such a function is expressible as sum of four integrals over the unit triangle. The present method can also evaluate the triple integrals of trivariate polynomials over an arbitrary tetrahedron as a special case. It is also demonstrated that certain integrals which are nonpolynomial functions of trivariates x; y; z can be computed by the proposed method. We have applied Gauss Legendre Quadrature rules which were recently derived by Rathod et al.

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On the application of two Gauss–Legendre quadrature rules for composite numerical integration over a tetrahedral region

Cited by 6 publications

References 19 publications

Locking‐free continuum displacement finite elements with nodal integration

Locking‐free continuum displacement finite elements with nodal integration

Correction of Gauss Legendre quadrature over a triangle

Numerical integration of some functions over an arbitrary linear tetrahedra in Euclidean three-dimensional space

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