Non-uniform [formula omitted] patches around extraordinary points with applications to analysis-suitable unstructured T-splines

Abstract

This article extends the subset of analysis-suitable unstructured T-splines (ASUT-splines) that have C1 continuity in irregular regions to allow non-uniform knots at extraordinary points. That is, we remove the preceding restriction that required uniform spoke edge knot spans of ASUT-splines. To achieve this, the degenerate-patch (D-patch) framework, which has been demonstrated to be an effective approach for generating refinable spline spaces over unstructured quadrilateral meshes, is generalized to the non-uniform case. We prove that uniform smoothing matrices are also valid for non-uniform knots. However, we also show that nested refinements of ASUT-splines are not available if using these matrices in the non-uniform case. Therefore, a new idempotent smoothing matrix is introduced to overcome this limitation. The detailed construction of this smoothing matrix is given, which helps to study more kinds of smoothing matrices in the future. Then, the non-uniform D-patch framework is employed for ASUT-splines. We provide a thorough discussion of the differences between the uniform and non-uniform cases. A new refinement strategy for the design space of ASUT-splines is also presented. Numerical experiments show that our extension of ASUT-splines can construct non-uniform parameterization around extraordinary points while maintaining the C1 continuity of the surface and optimal convergence rates for isogeometric analysis of both the Poisson problem and the biharmonic problem. A car hood model with complex shape features is used to demonstrate the ability of the proposed method to design and analyze practical models within a seamlessly integrated framework.