Modified basis functions for MPHT-splines

Abstract

By allowing split-in-half in mesh refinement, MPHT-splines (modified polynomial splines over hierarchical T-meshes) were recently introduced to increase the flexibility of PHT-splines. However, some basis functions of MPHT-splines may severely decay as the basis functions of PHT-splines during the process of certain refinement of hierarchical T-meshes. This decay phenomenon may cause the isogeometric analysis matrices assembled by these basis functions to be ill-conditioned. To overcome this defect, we present a method to modify the basis functions of MPHT-splines when the supports of the original truncated basis functions are rectangular domains. Hence, the truncation operations are less used, which alleviates the decay phenomenon of MPHT-splines. In addition, the modified basis functions preserve the good properties of the original MPHT-spline basis functions, e.g., the partition of unity, local support and linear independence. Numerical examples show that the condition numbers of isogeometric analysis matrices significantly decrease when new basis functions are applied.