A one point integration rule over star convex polytopes. (15th April 2019) - Francis, Amrita; Natarajan, Sundararajan; Atroshchenko, Elena; Lévy, Bruno; Bordas, Stéphane P.A.
Highlights: Uses Voronoi approach to generate polyhedral meshes for complex shapes. Extends the linearly consistent one point integration for meshfree approximation to arbitrary polytopes. Requires no sub-division of the polytopes into triangles/tetrahedra for the purpose of numerical integration. Yields optimal convergence rates in both the L 2 -norm and the H 1 -seminorm. Abstract: In this paper, the recently proposed linearly consistent one point integration rule for the meshfree methods is extended to arbitrary polytopes. The salient feature of the proposed technique is that it requires only one integration point within each n -sided polytope as opposed to 3 n in Francis et al. (2017) and 13 n integration points in the conventional approach for numerically integrating the weak form in two dimensions. The essence of the proposed technique is to approximate the compatible strain by a linear smoothing function and evaluate the smoothed nodal derivatives by the discrete form of the divergence theorem at the geometric center. This is done by Taylor's expansion of the weak form which facilitates the use of the smoothed nodal derivatives acting as the stabilization term. This translates to 50% and 30% reduction in the overall computational time in the two and three dimensions, respectively, whilst preserving the accuracy and the convergence rates. The convergence properties, the accuracy and the efficacy of the one point integration scheme are discussed by solving few benchmark problems in elastostatics.