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 smooth Ck polar splines for any k≥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} 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.
Toshniwal, D., Speleers, H., Hiemstra, R.r., Hughes, T. (2017). Multi-degree smooth polar splines: a framework for geometric modeling and isogeometric analysis. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 316, 1005-1061 [10.1016/j.cma.2016.11.009].
Multi-degree smooth polar splines: a framework for geometric modeling and isogeometric analysis
Speleers H.;
2017-04-01
Abstract
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 smooth Ck polar splines for any k≥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} 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.File | Dimensione | Formato | |
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