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加权层次B样条基函数

Weighted HB-spline Basis

  • 摘要: 层次B样条作为一类可局部细分样条, 因其在数值计算方面具有局部稳定性、高效的求值性和标准的数据结构, 广泛应用于几何建模与造型以及等几何分析中. 为了解决层次B样条没有单位剖分性性质的问题, 通过给出网格的拓扑限制, 定义一种加权层次B样条基函数. 首先用层次B样条基函数的线性组合求和等于1, 然后取特殊点点值或者一阶、二阶导数值构造线性方程组并求解, 求得的线性组合系数与对应层次B样条基函数的乘积, 即为加权层次B样条基函数. 加权层次B样条还继承了层次B样条的非负性、局部支撑性等性质. 数据拟合实验和偏微分方程求数值解的结果表明, 加权层次B样条基函数可以应用于几何图形拟合以及物理方程求解分析.

     

    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.

     

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