Parametric mesh regularization for interpolatory shape design and isogeometric analysis over a mesh of arbitrary topology

Isogeometric analysis (IGA) over a mesh of arbitrary topology is a challenging task since the concept of IGA was proposed. In the literature, one can find some results in IGA over an irregular mesh based on approximating schemes, while little work is reported so far using interpolatory schemes. This paper presents a parametric mesh regularization (PMR) method for IGA using mapped interpolatory basis functions over a mesh of arbitrary topology. Given a geometric model defined over a mesh of arbitrary topology, a global parametric mesh is first defined from the given control mesh using a Gravity Center Method (GCM) Yuan and Ma (2014). The resulting parametric mesh is further regularized such that it ensures absolute interpolatory modeling with mapped interpolatory basis functions. Necessary geometric rules are developed for the regularization process while maintaining the same mesh topological structure. The interpolatory basis function used in this paper is a kind of truncated interpolatory basic function (TIBF) studied in our early work Yuan and Ma (2013). It can also be applied to other mapped interpolatory basis functions. In addition to the capability of interpolatory modeling, the proposed PMR method can also be applied to approximating schemes, which further enriches the capability of such schemes in both modeling and isogeometric analysis of some models in a more appropriate and convenient manner. The proposed PMR method has also the potential in achieving higher order continuity at extraordinary vertices using higher order mapped interpolatory basis functions.