Adaptive isogeometric analysis with hierarchical box splines
For researchers in isogeometric analysis, this provides a novel method for local refinement using box splines, though it is an incremental extension of existing hierarchical B-spline techniques.
This paper extends isogeometric analysis to hierarchical box splines, enabling local refinement and achieving optimal convergence rates for PDE solutions with weak boundary conditions.
Isogeometric analysis is a recently developed framework based on finite element analysis, where the simple building blocks in geometry and solution space are replaced by more complex and geometrically-oriented compounds. Box splines are an established tool to model complex geometry, and form an intermediate approach between classical tensor-product B-splines and splines over triangulations. Local refinement can be achieved by considering hierarchically nested sequences of box spline spaces. Since box splines do not offer special elements to impose boundary conditions for the numerical solution of partial differential equations (PDEs), we discuss a weak treatment of such boundary conditions. Along the domain boundary, an appropriate domain strip is introduced to enforce the boundary conditions in a weak sense. The thickness of the strip is adaptively defined in order to avoid unnecessary computations. Numerical examples show the optimal convergence rate of box splines and their hierarchical variants for the solution of PDEs.