A numerical integration scheme for finite element method based on symbolic manipulation

Yagawa, Genki; Ye, G.-W.; Yoshimura, Shinobu

doi:10.1002/nme.1620290711

Cited by 40 publications

(11 citation statements)

References 12 publications

(3 reference statements)

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“…Yagawa et al [6] reported CPU time savings of 15% using a combined technique involving both numerical and analytical integrations of stiffness in plane elasticity. Kikuchi [7] used the Reduce package to obtain explicit formulas for an isoparametric 4-node FE, showing accurate results for distorted elements.…”

Section: Introductionmentioning

confidence: 99%

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

Griffiths¹,

Huang²,

Schiermeyer³

2008

Commun. Numer. Meth. Engng.

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SUMMARYMost finite element (FE) software uses numerical integration to compute FE stiffness matrices. In the case of straight-sided isoparametric triangular elements, numerical integration is exact, provided a sufficient number of integration points are used. In this paper, the same integrations are performed analytically in closed form with the help of computer algebra software for 3-, 6-, 10-and 15-noded triangles. The resulting analytical expressions have been converted into Fortran 95 subroutines and compared with the conventional numerical integration methods. The analytical routines produce exactly the same results as their numerical counterparts, but run significantly faster. All Fortran 95 code developed in this study is available on the first author's Web site.

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Section: Introductionmentioning

confidence: 99%

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

Griffiths¹,

Huang²,

Schiermeyer³

2008

Commun. Numer. Meth. Engng.

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“…Among various numerical integration schemes, Gauss Legendre quadrature, which can evaluate exactly the (2n -1) th degree polynomial with 'n' Gaussian integration points, is mostly used in view of the accuracy and efficiency of calculation. However, the integrands of global derivative products in stiffness matrix computations of practical applications are not always simple polynomials but rational expressions which the Gaussian quadrature cannot evaluate exactly [7][8][9][10][11][12][13][14][15]. The integration points have to be increased in order improve the integration accuracy but it is also desirable to make these evaluations by using as few Gaussian points as possible, from the point of view of the computational efficiency.…”

Section: Introductionmentioning

confidence: 99%

Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for nine node linear convex quadrilateral of Lagrange family

Rathod

islsm

H.Y.Shrivalli³

et al. 2017

ijecs

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:This paper presents an explicit integration scheme to compute the stiffness matrix of a nine node linear convex quadrilateral element of Lagrange family using symbolic mathematics and discretisation of polygonal domain by such finite elements using a novel auto mesh generation technique , In finite element analysis, the boundary value problems governed by second order linear partial differential equations, the element stiffness matrices are expressed as integrals of the product of global derivatives over the linear convex quadrilateral region. These matrices can be shown to depend on the material properties and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+ ) in the bivariates over a nine node 2-square (-1 ) with nodes at the points {(-1,-1),(1,-1),(1,1),(-1,1),(0,-1),(1,0).(0,1),(-1,0),(0,0)} in local parametric space. In this paper, we have computed these integrals in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be applied to solve boundary value problems in continuum mechanics over convex polygonal domains. We have also developed a novel auto mesh generation technique of all nine node linear convex quadrilaterals for a polygonal domain which provides the nodal coordinates and element connectivity. We have used the explicit integration scheme and this novel auto mesh generation technique to solve the Poisson equation u ,where u is unknown physical variable and in with given Dirichlet boundary conditions over the given convex polygonal domain.

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“…Yagawa et al [9] reported CPU time savings of 15% using a combined technique involving both numerical and analytical integration of stiffness in plane elasticity. Bardel [10] showed that higher-order polynomial expressions appearing in p-adaptive FEM can be obtained using symbolic integration, giving large CPU time savings.…”

Section: Introductionmentioning

confidence: 99%

Exact integration of the stiffness matrix of an 8‐node plane elastic finite element by symbolic computation

Videla

Baloa

Griffiths

et al. 2007

Numerical Methods Partial

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Computer algebra systems (CAS) are powerful tools for obtaining analytical expressions for many engineering applications in both academic and industrial environments.CAS have been used in this paper to generate exact expressions for the stiffness matrix of an 8-node plane elastic finite element. The Maple software system was used to identify six basic formulas from which all the terms of the stiffness matrix could be obtained. The formulas are functions of the Cartesian coordinates of the corner nodes of the element, and elastic parameters Young's modulus and Poisson's ratio.Many algebraic manipulations were performed on the formulas to optimize their efficiency. The redaction in CPU time using the exact expressions as opposed to the classical Gauss-Legendre numerical integration approach was over 50%. In an additional study of accuracy, it was shown that the numerical approach could lead to quite significant errors as compared with the exact approach, especially as element distortion was increased.

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A numerical integration scheme for finite element method based on symbolic manipulation

Cited by 40 publications

References 12 publications

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

Elastic stiffness of straight‐sided triangular finite elements by analytical and numerical integration

Finite element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for nine node linear convex quadrilateral of Lagrange family

Exact integration of the stiffness matrix of an 8‐node plane elastic finite element by symbolic computation

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