General complete Lagrange interpolations with applications to three-dimensional finite element analysis

In this paper, we have first derived the interpolation polynomials for the general serendipity solid elements of rectangular shape which allow arbitrarily placed nodes along the edges. We have then presented a method to determine the interpolation functions for the general complete Lagrange elements which allow arbitrarily placed nodes. Explicit expressions for interpolation functions of the serendipity and complete Lagrange families with uniform spacing of nodes over the element domain are derived for elements of orders 4-10. We have also modified the shape functions of complete Lagrange family to correctly interpolate the complete polynomials in the global space for angular distortions of quadrilaterals over the six planar facets of linear solid hexahedron elements. © 2001 Elsevier Science B.V. All rights reserved

Integration of polynomials over linear polyhedra in Euclidean three-dimensional space

This paper is concerned with explicit formulas and algorithms for computing integrals of polynomials over a linear polyhedron in Euclidean three-dimensional space. Symbolic formulas for surface and volume integration are given. Two different approaches are discussed: The first algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric line integral while the second algorithm is obtained by transforming a volume integral into a surface integral and then into a parametric double integral. These algorithms and formulas are followed by an application-example for which we have explained the detailed computational scheme. The symbolic results presented in this paper may lead to an easy incorporation of global geometric properties of solid objects, for example, the volume, centre of mass, moments of inertia, required in the engineering design process. © 1995

Boundary integration of polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space

This paper is concerned with explicit integration formulas and algorithms for computing integrals of trivariate polynomials over an arbitrary linear tetrahedron in Euclidean three-dimensional space. This basic three-dimensional integral governing the problem is transformed to surface integrals by use of the divergence theorem. The resulting two-dimensional integrals are then transformed into convenient and computationally efficient line integrals. These algorithms and explicit finite integration formulas are followed by an application - example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with previous works. Further, it is shown that the present algorithms are much simpler and more economical as well, in terms of arithmetic operations. The symbolic finite integration formulas presented in this paper may lead to an easy incorporation of geometric properties of solid objects, for example, the centre of mass, moment of inertia, etc. required in the engineering design process as well as several applications of numerical analysis where integration is required, for example in the finite element and boundary integral equation methods

Boundary integration of polynomials over an arbitrary linear hexahedron in Euclidean three-dimensional space

This paper is concerned with explicit integration formulas and algorithms for computing volume integrals of trivariate polynomials over an arbitrary linear hexahedron in Euclidean three-dimensional space. Three different approaches are discussed. The first algorithm is obtained by transforming a volume integral into a sum of surface integrals and then into convenient and computationally efficient line integrals. The second algorithm is obtained by transforming a volume integral into a sum of surface integrals over the boundary quadrilaterals. The third algorithm is obtained by transforming a volume integral into a sum of surface integrals over the triangulation of boundary. These algorithms and finite integration formulas are then followed by an application example, for which we have explained the detailed computational scheme. The symbolic finite integration formulas presented in this paper may lead to efficient and easy incorporation of integral properties of arbitrary linear polyhedra required in the engineering design process. © 1998 Elsevier Science S.A. All rights reserved

An Explicit Finite Element Integration Scheme for Linear Eight Node Convex Quadrilaterals Using Automatic Mesh Generation Technique over Plane Regions

This paper presents an explicit integration scheme to compute the stiffness matrix of an eight node linear convex quadrilateral element for plane problems using symbolic mathematics and an automatic generation of all quadrilateral mesh technique , In finite element analysis, the boundary 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 bivariates over an eight node 2-square (-1 ).In this paper,we have computed these integrals in exact and digital forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme is illustrated by computing the Prandtl stress function values and the torisonal constant for the square cross section by using the eight node linear convex quadrilateral finite elements.An automatic all quadrilateral mesh generation techniques for the eight node linear convex quadrilaterals is also developed for this purpose.We have presented a complete program which automatically discritises the arbitrary triangular domain into all eight node linear convex quadrilaterals and applies the so generated nodal coordinate and element connection data to the above mentioned torsion problem. Key words: Explicit Integration, Gauss Legendre Quadrature, Quadrilateral Element, Prandtl’s Stress Function for torsion, Symbolic mathematics,all quadrilateral mesh generation technique

Integration of polynomials over N-dimensional linear polyhedra

This paper is concerned with explicit integration formulae for computing integrals of n-variate polynomials over linear polyhedra in n-dimensional space ℝn. Two different approaches are discussed; the first set of formulae is obtained by mapping the polyhedron in n-dimensional space ℝn into a standard n-simplex in ℝn, while the second set of formulae is obtained by reducing the n-dimensional integral to a sum of n - 1 dimensional integrals which are n + 1 in number. These formulae are followed by an application example for which we have explained the detailed computational scheme. The symbolic integration formulae presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, such as, for example, volume, centre of mass, moments of inertia etc., required in engineering design problems. © 1997 Elsevier Science Ltd

A New Approach to Automatic Generation of an all Pentagonal Finite Element Mesh for Numerical Computations over Convex Polygonal Domains

A new method is presented for subdividing a large class of solid objects into topologically simple subregionssuitablefor automatic finite element meshing withpentagonalelements. It is known that one can improve the accuracy of the finite element solutionby uniformly refining a triangulation or uniformly refining a quadrangulation.Recently a refinement scheme of pentagonal partition was introduced in [31,32,33]. It is demonstrated that the numerical solutionbased on the pentagonal refinement scheme outperforms the solutions based on the traditional triangulation refinement scheme as well as quadrangulation refinement scheme. It is natural to ask if one can create a hexagonal refinement or general polygonal refinement schemes with a hope to offer even further improvement. It is shown in literature that one cannot refine a hexagon using hexagons of smaller size. In general, one can only refine an n-gon by n-gons of smaller size if n = 5. Furthermore, we introduce a refinement scheme of a generalpolygon based on the pentagon scheme. This paper first presents a pentagonalization (or pentagonal conversion) scheme that can create a pentagonal mesh from any arbitrary mesh structure. We also introduce a pentagonal preservation scheme that can create a pentagonal mesh from any pentagonal mesh

An explicit integration scheme based on recursion for the curved triangular finite elements

This paper is concerned with explicit formulae for computing integrals of rational functions of bivariate polynomial numerator with linear denominator over a unit triangle {(ξ,η)|0⩽ξ,η⩽1,ξ+η⩽1} in the local parametric space (ξ,η). These integrals arise in a variety of applications governing the second order linear partial differential equations. Explicit evaluation of these integrals produce algebraic finite element relations, provided that the original element geometry is restricted to a triangle with two straight sides and one curved side. For a rational integral of nth degree bivariate polynomial numerator with a linear denominator there are {(n+1)(n+2)/2} rational integrals of monomial numerators with the same linear denominator. By an expansion it is shown that these {(n+1)(n+2)/2} integrals can be computed in two ways. We have presented a recursive scheme to compute {n(n+1)/2} such integrals efficiently whenever (n+1) integrals of order 0 (zero) to n in one of the variates are known by explicit integration formulae. The analytical quadrature formulae from zeroth to sextic order due to different element geometry are, for clarity and reference, summarized in tabular forms. An integration formula is also presented for the evaluation of integrals that occur in the calculation of stiffness matrices when straight sided triangular finite elements are considered. Finally we have considered two application examples: First example is to compute the Prandtl stress function values and the torsional constant k for an elliptic cross-section with varying ratios of major and minor distances. The other example is concerned with numerical solution for the Poisson equation in 2D. It is observed in the calculation of components of element matrices by using the Gauss quadrature rule (e.g. 7-point and 13-point) a remarkable discrepancy occurs if the element geometry contains a concave curved side. A computer code based on the present integration scheme to obtain the element stiffness matrices for both the curved and straight sided elements is appended

Integration of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space

In this paper, we present explicit integration formulas and algorithms for computing integrals of polynomials over an arbitrary tetrahedron in Euclidean three-dimensional space. Two different approaches are discussed: the first algorithm/formula is obtained by mapping the arbitrary tetrahedron into a unit orthogonal tetrahedron, while the second algorithm/formula computes the required integral as a sum of four integrals over the unit triangle. These algorithms/formulas are followed by an example for which we have explained the detailed computational scheme. The numerical result thus found is in complete agreement with the previous work. Further, it is shown that the present algorithms are much simpler and more economical as well in terms of arithmetic operations

Some pre-computed universal numeric arrays for linear convex quadrilateral finite elements

In this paper we investigate the finite element analysis for the solution of linear partial differential equations using linear convex quadrilateral elements. We show that a linear polygonal domain can be discretized into a set of special linear convex quadrilateral elements, which generate the same expression for the determinant of the Jacobian matrix under the isoparametric coordinate transformation. Analytical integration of the above element matrices is shown to depend on certain 'universal pre-computed numeric arrays', i.e., the arrays which are computed once, stored on a permanent file and then reused in all subsequent applications of the program. We have constructed such arrays for the five commonly used linear quadrilateral elements: Q4, Q8, Q9, Q12 and Q16. One speciality of these pre-computed arrays is that the arrays for lower order elements are already contained in the arrays for higher order elements. The performance of the proposed method is demonstrated by means of a numerical example. © 2001 Elsevier Science B.V. All rights reserved