The Weighted Extended Spline Method Based on B-Spline Weighted Function

To address the problem of solving equations on complex computational domains, a method is proposed by adopting a combined approach integrating the weighted extended B-splines method with the collocation method, in which both the construction of the weighted function and the equation solving are based on the same set of spline basis functions. First, the distance function is introduced, and the discrete signed distance field is generated based on a set of points on the boundary of the computational domain to construct the appropriate weighted function. Then, the approximate phase field function is constructed, and the computational domain is discretized. Finally, B-splines are used to approximate the phase field function. Adopting the uniform B-splines that are used to construct the weighted function as the basis functions of the numerical solution can enable the reuse of the basis function values and their derivatives at the collocation points during the process of solving the equation. Experimental results on the computation time for the example equations reveal that reusing the basis function values and derivatives reduces the solution time by approximately 35% compared to the non-reuse case, improving computational efficiency.