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

Abstract

This paper concerns with analytical integration of trivariate polynomials over linear polyhedra in Euclidean three-dimensional space. The volume integration of trivariate polynomials over linear polyhedra is computed as sum of surface integrals in R 3 on application of the well known Gauss's divergence theorem and by using triangulation of the linear polyhedral boundary. The surface integrals in R 3 over an arbitrary triangle are connected to surface integrals of bivariate polynomials in R 2 . The surface integrals in R 2 over a simple polygon or over an arbitrary triangle are computed by two different approaches. The first algorithm is obtained by transforming the surface integrals in R 2 into a sum of line integrals in a one-parameter space, while the second algorithm is obtained by transforming the surface integrals in R 2 over an arbitrary triangle into a parametric double integral over a unit triangle. It is shown that the volume integration of trivariate polynomials over linear polyhedra can be obtained as a sum of surface integrals of bivariate polynomials in R 2 . The computation of surface integrals is proposed in the beginning of this paper and these are contained in Lemmas 1-6. These algorithms (Lemmas 1-6) and the theorem on volume integration are then followed by an example for which the detailed computational scheme has been explained. The symbolic integration formulas presented in this paper may lead to an easy and systematic incorporation of global properties of solid objects, for example, the volume, centre of mass, moments of inertia etc., required in engineering design processes.

A New Approach to Automatic Generation of An All Pentagonal Finite Element Mesh For Numerical Computations Over Convex Polygonal Domains

“…We also introduce a pentagonal preservation scheme that can create a pentagonal mesh from any pentagonal mesh. This paper then presents a new numerical integration technique proposed earlier by the first author and co-workers, known as boundary integration method [34][35][36][37][38][39][40] is now applied to arbitrary polygonal domains using pentagonal finite element mesh. Numerical results presented is tested on examples of complicated integrals over convex polygons in the context of pentagonal domains with composite numerical integration scheme of triangular finite elements which can be easily created by joining the centre point of pentagons,this shows that the proposed method yields accurate results even for low order Gauss Legendre Quadrature rules.…”

Numerical Integration Over A Linear Convex Polyhedron Using An All Hexahedral Discretisation And Gauss Legendre Formulas

“…We can transform each of these hexahedrons in physical space (x,y ,z) into a standard 2-cube in a parametric space (r,s,t) by using the following coordinate transformations: =∑ ; =∑ ; =∑ ..........(10) Where, (r,s,t) are the nodal shape functions for a the standard 2-cube , in the parametric space and are the nodal coordinates of the hexahedron in the Cartesian space (x,y,z) and the shape functions (r,s,t) are: (r,s,t)= (r,s,t)= (r,s,t)= (r,s,t)= (r,s,t)= (r,s,t)= (r,s,t)= (r,s,t)= ....... (10) The integrals over the hexahedrons (i=1,2,3,4) in Cartesian space can be now expressed from the above transformations of eqn (9) Hence from eqns (5) and (11),we obtain In eqn(11) above = (r,s,t), = (r,s,t), = (r,s,t) are the coordinate transformations the hexahedrons and they can be computed using eqns (5)(6)(7)(8)(9)(10).Then we compute the Jacobian which can be obtained by computing the following equation …”

“…In the previous section, we have explained the division of each hexahedron into eight hexahedrons (j=1, 2,3,4,5,6,7,8). The tetrahedron consists of four hexahedrons (i=1, 2,3,4) and so this procedure divides the tetrahedron into 32 hexahedrons .In this section ,each hexahedron is further divided into eight hexahedrons (k=1,2,3,4,5,6,7,8).…”

“…In several earlier works [1][2][3][4][5][6][7][8][9][10][11][12], numerical integration rules for tetrahedron are already established. In a studies [13] , composites integration with all tetrahedron decomposition is also proposed.…”

“…function[]=integration_tetrahedron_hexahedron(N1,NC,N2,FN1,FN2) %integration_tetrahedron_hexahedron (5,5,40,7,10) (5,40,7,10) %integration_6pyramids_hexahedron (4,4,32,7,10) IFN=0; for FN=FN1:FN2%7:10FOR UNIT CUBE IFN=IFN+1; switch FN case 1 %TETRAHEDRA HTR PAPER disp('FF=x^2*y;') case 2%UNIT ORTHOGONAL TETRAHEDRON %FF=x^2*y^2; disp('FF=(x+y+z)^(1/2);') %0.142857142857143 case 3%UNIT ORTHOGONAL TETRAHEDRON % FF=x^4*y^4; disp('FF=(x+y+z)^(-1/2);') %0.2 case 4%UNIT ORTHOGONAL TETRAHEDRON disp('FF=x^3*sin(pi*y)*sin(pi*z);') %0.0011802154211291 case 5%UNIT ORTHOGONAL TETRAHEDRON disp(' FF=sin(x+2*y+4*z);') %0.13190232689018 case 6%UNIT ORTHOGONAL TETRAHEDRON disp('FF=1/((1+x+y+z)^4);') %0.020833333333333 case 7%unit cube or %UNIT ORTHOGONAL TETRAHEDRON disp('FF=x^3*sin(pi*y)*sin(pi*z);% 1/(pi^2)= 1.013211836423378e-001') case 8%unit cube or %UNIT ORTHOGONAL TETRAHEDRON disp(' FF=sin(pi*x)*sin(pi*y)*sin(pi*z); %ii=8/pi^3 %ii =.25801227546559591347537642150851 OR') case 9%unit cube or %UNIT ORTHOGONAL TETRAHEDRON disp(' FF=exp(-((x-.5)^2+(y-.5)^2+(z-.5)^2));%ii= 7.852115961743685e-001 for GL ORDER=20 OR') case 10%unit cube or %UNIT ORTHOGONAL TETRAHEDRON disp('FF=(27/8)*sqrt(1-abs(2*x-1))*sqrt(1-abs(2*y-1))*sqrt(1-abs(2*z-1));') case 11%%TETRAHEDRA F. (5,40,7,10) %integration_6pyramids_hexahedron (4,4,32,7,10) %integration_6pyramidsminustetrahedron_hexahedron (4,4,32,7,10) %integration_irregularheptahedron_hexahedron (4,4,32,7,10 …”