Volumetric parameterization with truncated hierarchical B-splines for isogeometric analysis

Abstract

Constructing spline parameterizations for computational domains is one of the fundamental problems in isogeometric analysis. In this paper, we present an efficient method for parameterizing volumetric computational domains using truncated hierarchical B-splines (THB). To ensure the injectivity of the parameterization, we utilize a collocation-based framework where the foldovers are eliminated on an increasing collocation set until the injectivity can be verified. A local–global solver is employed to eliminate any foldover effectively by projecting the mapping onto a bounded conformal distortion space. Local refinement of THB splines is also performed to enlarge the injectivity space. Various sufficient conditions are proposed to check the injectivity of the mapping efficiently. The quality of the bijective mapping is further improved by optimizing a geometrical energy, which is also embedded in the collocation framework to ensure injectivity. A coarse-to-fine variant of our method, which is more robust, is also presented. THB splines not only reduce the number of degrees of freedom in the parametric representation, but also increase the possibility of producing a bijective parameterization by local refinement. Some examples are provided to demonstrate the efficiency, effectiveness and robustness of our method.