A Unified Framework for Advanced Spline Constructions in Isogeometric Analysis
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Isogeometric Analysis (IGA) enables smooth bases in Finite Element discretisations. Due to the inherent smoothness of the spline basis, higher-order derivatives are available for finite element bases, and convergence of partial differential equation (PDE) discretizations increases. However, the higher-order continuity is only trivial for tensor-product bases. In case of multi-patch geometries or local refinement, alternative constructions are required. In this presentation, we present a unified framework for advanced spline constructions for isogeometric analysis. Our framework is based on [1] and [2] and enables both unstructured splines (i.e., spline constructions over multi-patches) as well as adaptive splines. For the latter, we show that the framework allows the construction of Truncated Hierarchical B-splines (THB-splines) [3], Locally Refined (LR) B-splines [4] on structured mesh configurations, and other constructions; giving rise to isotropic and anisotropic refinement capabilities in isogeometric analysis. We demonstrate our framework on a series of problems using unstructured splines, adaptive splines and combinations of those for isogeometric analysis, using the Kirchhoff--Love shell model. [1] Bressan, A. and Mokris, D. A Versatile Strategy for the Implementation of Adaptive Splines. In: Floater, M., Lyche, T., Mazure, ML., Mørken, K., Schumaker, L. (eds) Mathematical Methods for Curves and Surfaces. MMCS 2016. Lecture Notes in Computer Science, Vol. 10521, 2017 [2] Verhelst, H.M. and Weinmüller, P. and Mantzaflaris, A. and Takacs, T. and Toshniwal, T. A comparison of smooth basis constructions for isogeometric analysis. Comput. Meth. Appl. Mech. Eng., Vol. 419, pp. 116659, 2024 [3] Giannelli, C. and Jüttler, B. and Speleers, H. THB-splines: The truncated basis for hierarchical splines. Comput. Aided Geom. Design, Vol. 29, pp. 485-498, 2012 [4] Dokken, T. and Lyche, T. and Pettersen, K.F. Polynomial splines over locally refined box-partitions. Comput. Aided Geom. Design, Vol. 30, pp. 331-356, 2013