A finite element framework based on bivariate simplex splines on triangle configurations

Abstract Recently, triangle configuration based bivariate simplex splines (referred to as TCB-spline) have been introduced to the geometric computing community. TCB-splines retain many attractive theoretic properties of classical B-splines, such as partition of unity, local support, polynomial reproduction and automatic inbuilt high-order smoothness. In this paper, we propose a computational framework for isogeometric analysis using TCB-splines. The centroidal Voronoi tessellation method is used to generate a set of knots that are distributed evenly over the domain. Then, knot subsets are carefully selected by a so-called link triangulation procedure (LTP), on which shape functions are defined in a recursive manner. To achieve high-precision numerical integration, triangle faces served as background integration cells are obtained by triangulating the entire domain restricted to all knot lines, i.e., line segments defined by any two knots in a knot subset. Various numerical examples are carried out to demonstrate the efficiency, flexibility and optimal convergence rates of the proposed method.