Abstract:
Hierarchical B-splines, a type of locally refined spline, are widely utilized in geometric modeling, shaping, and isogeometric analysis due to their local stability, efficient evaluation, and standard data structure in numerical calculations. To address the lack of uniform refinement property in hierarchical B-splines, a weighted hierarchical B-spline basis function is introduced, incorporating the grid's topological constraints. Initially, the sum of the linear combination of hierarchical B-spline basis functions equals 1. Linear equations are then formulated and solved using special point values or the first- and second-order derivative values. The resulting weighted hierarchical B-spline basis function is derived from the product of these coefficients and their corresponding hierarchical B-spline basis functions. This weighted hierarchical B-spline maintains the desirable properties of non-negativity and local support. Data fitting experiments and numerical solutions of partial differential equations demonstrate its effectiveness in geometric figure fitting and physical equation analysis.