A hierarchical approach to adaptive local refinement in isogeometric analysis

Abstract

Adaptive local refinement is one of the key issues in Isogeometric Analysis. In this article we present an adaptive local refinement technique for Isogeometric Analysis based on extensions of hierarchical B-splines. We investigate the theoretical properties of the spline space to ensure fundamental properties like linear independence and partition of unity. Furthermore, we use concepts well-established in finite element analysis to fully integrate hierarchical spline spaces into the isogeometric setting. This also allows us to access a posteriori error estimation techniques. Numerical results for several different examples are given and they turn out to be very promising.

THB-splines: The truncated basis for hierarchical splines

“…In addition, the property of local linear independence and the nested nature of the spline spaces are inherited from the background normalized tensor-product B-splines -see Lemmas 2 and 3 in [20]. The partition of unity property, instead, is lost during the hierarchical construction.…”

“…These restrictions are not required anymore by Definition 1. In particular, the modified support definition (2) -firstly considered in [20] -allows for refinement close to ∂Ω 0 , which was not covered by the original construction presented in [11]. Hierarchical B-splines, however, do not satisfy the partition of unity property.…”

“…Kraft developed a quasi-interpolation algorithm based on hierarchical B-splines in [11]. There is an increasing interest in hierarchical splines coming from recent studies related to isogeometric analysis [20].…”

“…In order to construct a basis of the hierarchical spline space, a specific selection mechanism was proposed in [11] and extended in [20]. More precisely, the so-called hierarchical B-splines (HBsplines)…”

IGA: A Simplified Introduction and Implementation Details for Finite Element Users

“…Apart from the inclusion of NURBS basis function for the modelling and analysis in IGA, other CAD tools such as T-spline [10], Polynomials splines over Hierarchical Tmeshes (PHT-spline) [16], Locally Refined (LR) splines [17] and Hierarchical spline [18] have also been employed for the advancement of this technique. A technique based on Bézier extraction operator that transforms the isogeometric element in standard C 0 -continuous element through their projection into Bézier element is devised in [19], which employs the IGA elements into finite element (FE) structure.…”