An Explicit Integration Scheme Based on Recursion and Matrix Multiplication for the Linear Convex Quadrilateral Elements

Finit element solution of Poisson Equation over Polygonal Domains using a novel auto mesh generation technique and an explicit integration scheme for linear convex quadrilaterals of cubic order Serendipity and Lagrange families

“…(13)(14) and from Fig 1e , Fig 1f ; we see that there is a one to one correspondence between (( 1(1)16) and (( ( ) for cubic Lagrange element.…”

“…Torsion of a square cross section modeled as a right isoscles triangle:R ,where R={ x,y)|0 nnode=9,Nine node -linear convex quadrilaterals of Lagrange family elememts nnode=12,Twelve node -linear convex quadrilaterals of Serendipity family elememts nnode=16,Sixteen node -linear convex quadrilaterals of Lagrange family elememts exact solution of torisonal constant= 0.140577014955156) nnode=number of nodes in the triangular region R nel=number of elements in the region R - ----------------------------------------------------------------------------------------------------------------------------------------------------------- [3]quadrilateral_mesh4MOINEX_q12.m [4]quadrilateral_mesh4MOINEX_q16LG.m [5]D2LaplaceEquationQ12Ex3automeshgenNewContour.m√ [6]D2LaplaceEquationQ12Ex3automeshgenNewPolygonContour.m√ [7]D2LaplaceEquationQ16Ex3automeshgenNewContour.m√ [8]D2LaplaceEquationQ16Ex3automeshgenNewPolygonContour.m√ [9]polygonal_domain_coordinates_3rd_orderLG.m [10]polygonal_domain_coordinates_3rd_order.m [11]coordinate_special_quadrilaterals_in_stdtriangle_3rd_orderLAGR.m [12]coordinate_special_quadrilaterals_in_stdtriangle_3rd_order.m [13]integral_valuesof_localderivative_products.m [14] 1:nnode,1)=gcoord(1:nnode,1); ycoord(1:nnode,1)=gcoord(1:nnode,2); %extract coordinates for each element for i=1:nel for j=1:nnel x(1,j)=xcoord(nodes(i,j),1); y(1,j)=ycoord(nodes(i,j),1); end;%j loop xvec (1,1:5)=[x(1,1),x(1,2),x(1,3),x(1,4),x(1,1)]; yvec(1,1:5)=[y(1,1),y(1,2),y(1,3),y(1,4),y(1,1) (i),'\bf)']); end %*********************************** %***************************** end;%i loop switch tri case 1 xlabel('\bfx axis') ylabel('\bfy axis') st1='\bfone eigth (1/8)square cross section '; st2=' using '; st3='12-node cubic serendipity '; st4='quadriateral'; st5=' elements' title([st1,st2,st3,st4,st5]) case 2 xlabel('\bfx axis') ylabel('\bfy axis') st1='\bfone eigth (1/8)square cross section '; st2=' using '; st3='12-node cubic serendipity '; st4='quadriateral'; st5=' elements' title([st1,st2,st3,st4,st5]) case 3 xlabel('\bfx axis') ylabel('\bfy axis') st1='\bfequilateral triangular cross section '; st2=' using '; st3='12-node cubic serendipity '; st4='quadriateral'; st5=' elements' title([st1,st2,st3,st4,st5]) case 4 xlabel('\bfx axis') ylabel('\bfy axis') st1='\bfequilateral triangular cross section '; st2=' using '; st3='12-node cubic serendipity '; st4='quadriateral'; st5=' elements' title ([st1,st2,st3,st4,st5]) end %************************************** %********************************* scatter(x(1,1:nnel),y(1,1:nnel),15,'filled','g') %************************************************* end%for nmesh-the number of meshes …”

“…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 to 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.…”

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

“…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.…”