THB-splines: An effective mathematical technology for adaptive refinement in geometric design and isogeometric analysis

Carlotta Giannelli, Bert Jüttler, Stefan K. Kleiss, Angelos Mantzaflaris, Bernd Simeon, Jaka Špeh
2016 Computer Methods in Applied Mechanics and Engineering  
Local refinement with hierarchical B-spline structures is an active topic of research in the context of geometric modeling and isogeometric analysis. By exploiting a multilevel control structure, we show that truncated hierarchical B-spline (THB-spline) representations support interactive modeling tools, while simultaneously providing effective approximation schemes for the manipulation of complex data sets and the solution of partial differential equations via isogeometric analysis. A
more » ... of illustrative 2D and 3D numerical examples demonstrates the potential of the hierarchical framework. order to obtain nested spaces and to guarantee linear independence, the restricted class of analysissuitable T-splines was introduced [31] . Later, it was proposed to characterize them as dual-compatible T-splines [2, 3] . Recently, a refinement algorithm in the bivariate case for this class of T-splines with linear complexity has been described [33] . A classical approach to obtain local refinement in geometric modeling is provided by hierarchical B-splines [16, 20, 29] . The construction of the basis guarantees nested spaces and linear independence of the basis functions. The use of hierarchical constructions in isogeometric analysis is a very promising approach [14, 38, 43] . Unfortunately, the partition of unity property is not preserved by the standard hierarchical construction. For this reason, a new hierarchical basis -the truncated basis for hierarchical splines (THB-splines) -has recently been introduced [18] . THB-splines, defined as suitable linear combinations of refined B-splines, form a convex partition of unity, exhibit good stability and approximation properties [17, 42] , and are suitable for applications in computer aided design [28] . By providing a way to define an adaptive extension of the B-spline framework which is also suitable for geometric modeling applications, THB-splines satisfy both the demands of adaptive numerical simulation and geometric design, making them well suited for isogeometric analysis. The generalization of THB-splines to the more general context of generating systems and also to geometries with arbitrary topologies was recently addressed [46, 48, 49] . Polynomial splines over hierarchical T-meshes [30] are based on a different paradigm to construct bases for the entire space of piecewise polynomials with a given smoothness on a certain subdivision of the domain. Consequently, nested meshes automatically generate nested spaces. However, the construction of the basis -which is especially tailored to each specific case -either assumes reduced regularity [11] or the satisfaction of certain constraints on the admissible mesh configurations [47] . Applications in isogeometric analysis were reported in [34, 44] . Finally, locally refined (LR) splines rely on the idea of splitting basis functions, and resolve the issue of nested spaces but create difficulties with linear independence [12] that have been further investigated [4, 5] . The use of LR splines in the isogeometric setting was also discussed [23] . A comparison between hierarchical splines, THB-spline, and LR splines with respect to sparsity and condition numbers was presented in [24] . Even if LR splines have smaller supports than THB-splines, that comparison did not reveal significant advantages with respect to sparsity patterns and condition numbers of mass and stiffness matrices. The present paper is devoted to the truncated basis of hierarchical spline spaces. We show that THB-splines • possess a firm theoretical foundation with regard to basis construction, nested spaces, partition of unity, stability and approximation properties; 7
doi:10.1016/j.cma.2015.11.002 fatcat:saqayplxhfhxnjwnw74peza2bu