Quadratic splines on quad-tri meshes: Construction and an application to simulations on watertight reconstructions of trimmed surfaces

Given an unstructured mesh consisting of quadrilaterals and triangles (we allow both planar and non-planar meshes of arbitrary topology), we present the construction of quadratic splines of mixed smoothness — C¹ smooth away from the unstructured regions of T and C⁰ smooth otherwise. The splines have several useful B-spline-like properties – partition of unity, non-negativity, local support and linear independence – and allow for straightforward imposition of boundary conditions. We propose a non-nested refinement process for the splines with multiple advantages — a simple computer implementation, reduction in the footprint of C⁰ smoothness, boundary preservation, and excellent approximation behaviour in simulations. Furthermore, the refinement process leaves the splines invariant on the mesh boundary. Numerical tests indicate that the spline spaces demonstrate optimal approximation behaviour in the L² and H¹ norms under mesh refinement, and provide a viable approach to simulations on watertight reconstructions of trimmed surfaces.