Formula omitted] Integration scheme for polygonal elements using Richardson extrapolation

Abstract

In this work, we propose a new n+1 integration scheme over arbitrary polygonal elements based on centroid approximation and Richardson extrapolation scheme. For the purpose of numerical integration, the polygonal element is divided into quadrilateral subcells by connecting the centroid of the polygon with the mid-point of the edges. The bilinear form is then computed in a two-stage approximation: as a first approximation, the bilinear form is computed at the centroid of the given polygonal element and in the second approximation, it is computed at the center of the quadrilateral cells. Both steps can be computed independently and thus parallelization is possible. When compared to commonly used approach, numerical integration based on sub-triangulation, the proposed scheme requires less computational time and fewer integration points. The accuracy, convergence properties and the efficiency are demonstrated with a few standard benchmark problems in two dimensional linear elasto-statics. From the systematic numerical study, it can be inferred that the proposed numerical scheme converges with an optimal rate in both L2 norm and H1 semi-norm at a fraction of computational time when compared to existing approaches, without compromising the accuracy.