Rational reparameterization of unstructured quadrilateral meshes for isogeometric analysis with optimal convergence

Abstract

Unstructured quadrilateral meshes are widely used in representing surfaces of arbitrary topology. However, in connection with isogeometric analysis, unstructured meshes suffer from sub-optimal convergence rates in extraordinary regions, especially for solving higher-order PDEs (partial differential equations). In this work, a rational reparameterization method is proposed to achieve optimal convergence rates for isogeometric analysis using Catmull-Clark surfaces. The scheme lifts the convergence of analysis solutions by tuning the weights of control vertices of the input geometry in extraordinary regions, which are often neglected in constructing analysis-suitable parameterizations. Preferred tuning parameters with optimal convergence rates in L2-norm are determined statistically based on a test suite composed of typical testing solutions with a range of parameters. Numerical experiments confirm the validity of the proposed method for PDEs up to the fourth-order. Comparisons with the original Catmull-Clark scheme and a state-of-the-art tuned subdivision scheme also show the superiority of the proposed method in terms of both optimal convergence and reduced absolute errors of resulting analysis solutions.