Continuity and Convergence in Rational Triangular Bézier Spline Based Isogeometric Analysis

This paper presents a method for isogeometric analysis using rational Triangular Bézier Splines (rTBS) where optimal convergence rates are achieved. In this method, both the geometry and the physical field are represented by bivariate splines in Bernstein Bézier form over the triangulation of a domain. From a given physical domain bounded by NURBS curves, a parametric domain and its triangulation are constructed. By imposing continuity constraints on Bézier ordinates, we obtain a set of global C smooth basis functions. Convergence analysis shows that isogeometric analysis with such C rTBS basis can deliver the optimal rate of convergence provided that the C geometric map remains unchanged during the refinement process. This condition can be satisfied by constructing a pre-refinement geometric map that is sufficiently smooth. Numerical experiments verify that optimal rates of convergence are achieved for Poisson and linear elasticity problems.