This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. Because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from its strengths. To solve these issues, we propose a novel strategy called B-spline based SFEM. To improve the accuracy of numerical integration, we employed cubic B-spline basis functions with $C^2$-continuity across element boundaries as the global basis functions. To avoid matrix singularity, we applied different basis functions to different meshes. Specifically, we employed the Lagrange basis functions as local basis functions. The numerical results indicate that using the proposed method, numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional SFEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional SFEM.

翻译：本文提出了一种从根本上解决传统s版本有限元方法问题的策略。由于s版本有限元方法可通过叠加不同空间分辨率的网格合理模拟分析域，因此具有局部精度高、计算时间短及网格剖分过程简单等固有优势。然而，该方法存在数值积分精度和矩阵奇异性等缺陷。尽管已有多种附加技术被提出以缓解这些局限性，但这类方法要么计算代价高昂，要么具有特殊针对性，反而削弱了其原有优势。为解决上述问题，我们提出了一种名为“基于B样条的s版本有限元方法”的新策略。为提高数值积分精度，我们采用在单元边界上具有$C^2$连续性的三次B样条基函数作为全局基函数；为避免矩阵奇异性，我们对不同网格采用不同的基函数——具体而言，采用拉格朗日基函数作为局部基函数。数值结果表明，采用所提方法无需传统s版本有限元方法中的任何附加技术即可实现足够精度的数值积分计算。此外，该方法避免了矩阵奇异性，且在求解线性方程组的收敛性方面优于传统方法。因此，所提方法在保持与传统s版本有限元方法相当精度的同时，具有缩短计算时间的潜力。