Multi-degree smooth polar splines: A framework for geometric modeling and isogeometric analysis

• DOI: 10.1016/j.cma.2016.11.009
• Publication Date: 2016-11-16T00:28:03Z
• Authors: Deepesh Toshniwal; Hendrik Speleers; René R. Hiemstra; Thomas J.R. Hughes
• Keywords: Smooth hole-filling; Isogeometric analysis; Multi-degree spline; 0202 electrical engineering, electronic engineering, information engineering; 02 engineering and technology; Settore MAT/08 - ANALISI NUMERICA; Polar spline; Free-form design; Multi-degree splines; Polar splines; Free-form design; Smooth hole-filling; Isogeometric analysis

We develop a multi-degree polar spline framework with applications to both geometric modeling and isogeometric analysis. First, multi-degree splines are introduced as piecewise non-uniform rational B-splines (NURBS) of non-uniform or variable polynomial degree, and a simple algorithm for their construction is presented. Then, an extension to two-dimensional polar configurations is provided by means of a tensor-product construction with a collapsed edge. Suitable combinations of these basis functions, encoded in a so-called isogeometric analysis suitable extraction operator, yield CkCk smooth polar splines for any k≥0k≥0. We show that it is always possible to construct a set of smooth polar spline basis functions that form a convex partition of unity and possess locality. Explicit constructions for k∈{0,1,2}k∈{0,1,2} are presented. Optimal approximation behavior is observed numerically, and examples of applications to free-form design, smooth hole-filling, and high-order partial differential equations demonstrate the applicability of the developed framework.