Polygonal elements with Richardson-extrapolation based numerical integration schemes for hyperelastic large deformation analysis

This paper presents a novel approach based on polygonal finite elements (PFEM) for hyperelastic large deformation analysis. Compared to the mesh of quadratic 8-node quadrangular finite elements (FEM-Q8), which is usually used in this problem type, a mesh of polygonal element usually requires less number of nodes (and thus less number of degrees of freedom), given the same number of elements. Traditionally, numerical integration in PFEM involves two steps: i) sub-division of polygonal (element) domain into triangular cells, then ii) Dunavant's integration scheme is conducted in the triangular cells. Each n-gonal domain could be divided into n triangular cells. Since the problem is non-linear, three integration points per triangular cells are typically taken. As a result, 3n integration points are required for each n-gonal element. Alternatively, the Richardson extrapolation is combined with the one-point rule, resulting in the so-called $(n+1)$-point integration scheme. Due to the less number of integration points, faster computation could be achieved, yet accuracy is preserved. Here, the PFEM being incorporated with the $(n+1)$-point integration scheme, namely RE-PFEM, is employed for hyperelastic large deformation analysis. Performance of the developed RE-PFEM approach in this type of nonlinear analysis, which has not been reported in the literature before, is illustrated and assessed through several numerical examples. Comparison is conducted with PFEM and FEM-Q8, as well as available reference solutions obtained by other methods.