An enriched Bernstein–Bézier finite element method for problems with sharp gradients or singularities. (March 2022) - Peng, Xuan; Lian, Haojie; Shen, Gang; Yang, Yong; Zheng, Chao
Highlights: A least-squares based enriched Bernstein-Bézier finite element is developed. The enrichment is realized without introducing extra degrees of freedom. Support points of least-squares basis are obtained from projection of control points. Influence of the size of the support domain on accuracy and convergence is studied. The method results in well-conditioned system matrix and exact geometry is preserved. Abstract: We develop an extended Bernstein-Bézier finite element based on least-squares enrichment, inspired by the idea of improved XFEM by Tian and Wen (2015). The contribution of this paper includes: (1) A projection approach is detailed to relate the support points of least-squares basis and control points, to ensure the former are located on the body of the domain; (2) A rule to identify the size of the support domain is outlined for enriched Bézier elements, as well as the accuracy and convergence rate are explored for enriched higher-order basis. Numerical examples with sharp gradient or various singularities in the solutions reveal that, the accuracy of enriched Bézier elements can be improved by two orders of magnitude, nevertheless degradation in convergence is observed. The proposed method not only inherits some advantages of improved XFEM such as well-conditioned system matrix and extra-dof-free enrichment, but also is capable of preserving exact geometry due to the utilization of (rational) Bernstein-Bézier basis.