Isogeometric spline forests

• We introduce spline forests.
• Spline forests can represent complex surfaces and volumes.
• We describe adaptive h-refinement and coarsening algorithms.
• We develop a Bézier extraction framework for splines forests.
• Several transient advection–diffusion benchmark problems are solved.

In this paper we present isogeometric spline forests. An isogeometric spline forest is a hierarchical spline representation capable of representing surfaces or volumes of arbitrarily complex geometry and topological genus. Spline forests can accomodate arbitrary degree and smoothness in the underlying hierarchical basis as well as non-uniform knot interval configurations. We describe adaptive h-refinement and coarsening algorithms for isogeometric spline forests and develop a Bézier extraction framework which provides a simple and efficient single level finite element description of the complex multi-level, unstructured hierarchical spline basis. We then demonstrate the potential of spline forests as a basis for analysis in the context of transient advection–diffusion problems where fully integrated adaptivity is demonstrated for the first time in an isogeometric simulation. In all cases, the adaptive process remains local (even in the case of moving fronts) and preserves exact geometry at the coarsest level of the discretization. The accuracy and robustness of the approach is demonstrated in all cases.