Higher-continuity s-version of finite element method with B-spline functions

TL;DR

This paper introduces a B-spline based extension of the s-version finite element method, enhancing numerical integration accuracy and avoiding matrix singularity, thereby improving convergence and computational efficiency.

Contribution

It proposes a novel B-spline based SFEM that improves numerical integration accuracy and prevents matrix singularity, outperforming conventional SFEM.

Findings

Numerical integration accuracy is improved without additional techniques.

The method avoids matrix singularity issues.

Convergence for solving linear equations is enhanced.

Abstract

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.