C 1 Natural Neighbor Interpolant based on the Extended Delaunay Triangulation

In the numerical methods of couple stress theory for elasticity, C 1 continuity on the shape functions is required, because the second-order derivatives of displacement appear in the variational equation. N. Sukumar constructed a C 1 natural neighbor interpolant by embedding Sibson’s interpolant in the Bernstein-Bézier surface representation of a cubic simplex and used it to solve the two-dimensional biharmonic problems of a circular plate under a biaxial state of stress. In this paper, all of the natural neighbors of the introduced point are determined based on the extended Delaunay triangulation which can be used to guarantee the uniqueness of the triangulation compared to traditional Delaunay triangulation, and the convex hull of the set of natural neighbors is defined to be the domain of the introduced point, so nodes outside of the domain have no influence on the introduced point. On account of the computational efficiency, the non-Sibson interpolant is calculated and introduced into a cubic Bernstein polynomials to obtain the Bernstein-Bézier basis function .According to the fact that the vertex Bézier ordinates are identical to the nodal function values, and the tangent Bézier ordinates and centre Bézier ordinates are related to the nodal gradient values, the relations between Bézier ordinates and the nodal function and gradient values are first presented, and the structure and computational algorithm are then outlined to construct the transformation matrix. When the construction of the transformation matrix is finished, a transformation matrix- Bernstein-Bézier basis function product is carried to compute the new C 1 natural neighbor interpolant and its derivatives. The new C 1 natural neighbor interpolant has quadratic completeness, interpolates to nodal function and nodal gradient values, and reduces to a cubic polynomial on the boundary of domain. The new C 1 natural element method based on Galerkin scheme is introduced to couple stress theory for elasticity to obtain the discrete form of governing equation. Numerical analyses of the method are presented, covering the influences of the couple stress on stress concentration of an infinite plate with central hole subjected to the uniaxial pull and biaxial pull respectively. Numerical results of typical problems show that when solving fourth-order elliptic problems such as those arising in couple stress theories, the C 1 natural neighbor interpolant based on the extended Delaunay triangulation is valid and feasible.