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

Abstract

This paper presents an explicit integration scheme to compute the stiffness matrix of twelve node and sixteen node linear convex quadrilateral finite elements of Serendipity and Lagrange families using an explicit integration scheme 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 matrices and the matrix of integrals with integrands as rational functions with polynomial numerator and the linear denominator (4+) in the bivariates over a 2-square (-1 ) with the nodes on the boundary and in the interior of this simple domain. The finite elements up to cubic order have nodes only on the boundary for Serendipity family and the finite elements with boundary as well as some interior nodes belong to the Lagrange family. The first order element is the bilinear convex quadrilateral finite element which is an exception and it belongs to both the families. We have for the present ,the cubic order finite elements which havee 12 boundary nodes at the nodal coordinates {(-1,-1),(1,-1),(1,1),(-1,1),(-1/3,-1), (1/3,-1),(1,-1/3),(1,1/3),(1/3,1),(-1/3,1),(-1,1/3),(-1,-1/3)} and the four interoior nodal coordinates at the points (-1/3,-1/3),(1/3,-1/3),(1/3,1/3),(-1/3,1/3)} in the local parametric space ( In this paper, we have computed the integrals of local derivative products with linear denominator (4+) in exact forms using the symbolic mathematics capabilities of MATLAB. The proposed explicit finite element integration scheme can be then 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 12-node and 16-node linear(straight edge) convex quadrilaterals for a polygonal domain which provides the nodal coordinates and the 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 an unknown physical variable and in with Dirichlet boundary conditions over the convex polygonal domain.