Truncated hierarchical tricubic C0 spline construction on unstructured hexahedral meshes for isogeometric analysis applications

We present a new method for truncated hierarchical tricubic spline (TH-spline3D) construction to enable adaptive isogeometric analysis on unstructured hexahedral meshes. Taking the input unstructured hexahedral mesh as the control mesh, we first develop blending functions with the aid of tricubic Bernstein polynomials. This development is an extension of defining bicubic Bernstein polynomial blending functions over quadrilateral meshes. We further build the hierarchical structure and apply the truncation mechanism to the developed blending functions for highly localized refinement. During the refinement of TH-spline3D, high-level blending functions are added in the solution space, whereas certain low-level ones are discarded or truncated depending on the high-level subdomain. The blending functions are piecewise polynomials that form a partition of unity. Their support overlapping is also reduced due to the truncation mechanism, resulting in sparser stiffness matrices compared to the classical hierarchical refinement. TH-spline3D supports Bézier extraction such that it can be easily incorporated into existing finite element frameworks. Several examples are used to demonstrate the analysis suitability and efficiency of the proposed method.