Hierarchical p-version C¹ finite elements on quadrilateral and triangular domains with curved boundaries and their applications to Kirchhoff plates

This work focuses on the construction of p-version finite elements that have curve boundaries for C¹ problems. Both triangular and quadrilateral elements are constructed based on the C¹-version blending function interpolation methods that are developed in this work and in the literature. Orthogonal hierarchical bases are constructed and subsequently transformed into interpolative nodal bases to facilitate the imposition of boundary conditions and the implementation of C¹ conformity on curvilinear domains. Nodal collocation strategies are also studied for improving the numerical performance, and novel nonuniformly distributed nodes, namely, Gauss-Jacobi (GJ) points, are proposed. For parallelograms and straight-sided triangular elements, C¹ continuity is exactly satisfied between neighboring elements. The difficulty of C¹ conformity for elements that have curved boundaries is circumvented by interpolating the normal derivatives at Gauss-Lobatto nodes. Moreover, with the help of the blending function interpolation method, the bases on edges and the internal modes can differ in terms of approximation order. Therefore, local p-refinements can be easily performed by these elements. Numerical results demonstrated that these elements are computationally inexpensive and converge fast for problems with regular and irregular domains.