Reproducing kernel triangular B-spline-based FEM for solving PDEs

• A novel way to deal with instabilities of the triangular B-spline is developed.
• The reproducing kernel approximation technique is taken.
• The kernel correction term is calculated locally.
• We numerically compare the improved triangular B-spline with the conventional one.

We propose a reproducing kernel triangular B-spline-based finite element method (FEM) as an improvement to the conventional triangular B-spline element for solving partial differential equations (PDEs). In the latter, unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the B-spline. The performance therefore becomes unstable and cannot be controlled in a desirable way. To address this issue, the proposed improvement adopts the reproducing kernel approximation in the calculation of B-spline kernel function. Three types of PDE problems are tested to validate the present element and compare against the conventional triangular B-spline. It has been shown that the improved triangular B-spline satisfies the partition of unity condition even for extreme conditions including corners and holes.