Adaptive isogeometric analysis with hierarchical box splines

TL;DR

This paper introduces an adaptive isogeometric analysis framework using hierarchical box splines, enabling efficient and accurate PDE solutions with optimal convergence rates through weak boundary condition enforcement.

Contribution

It presents a novel adaptive approach for isogeometric analysis with hierarchical box splines, including a weak boundary condition treatment and adaptive domain strip thickness.

Findings

Optimal convergence rates demonstrated for PDE solutions.

Adaptive boundary treatment improves computational efficiency.

Hierarchical box splines effectively model complex geometries.

Abstract

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.