Construction of a C 1 $$ {C}^1 $$ Polygonal Spline Element Based on the Scaled Boundary Coordinates

We construct a new polygonal $C^1$ spline finite element method based on the scaled boundary coordinates to address the plate bending problems in the Kirchhoff-love formulation. The Bernstein interpolations are utilized in both radial and circumferential directions in the scaled boundary coordinates. Firstly, the $C^1$ continuity conditions inside an S-domain and normal derivatives constraining conditions are imposed by a simple linear system on the S-net coefficients. Secondly, to satisfy the $C^1$ connection between different polygonal elements, we construct the Hermite interpolation by equivalently transforming part of the S-net coefficients to proper boundary degrees of freedom, namely, three degrees of freedom at each vertex and a normal derivative at the midpoint of each edge. Moreover, we discuss the convergence analysis of the proposed element over convex meshes by finding the necessary and sufficient geometric conditions, where the corresponding unisolvency theorem is proved by studying the dimension of the spline space $S_{4,3}^{1,\ast }\left({𝒯}_S\right)$. This proposed spline element base have explicit expressions, and the computation of the stiffness matrix can be greatly simplified by using the S-net coefficients. Some numerical tests verify the cubic polynomial completeness, the optimal 4th-order convergence rate, and the continuity of the derivatives. It also shows other good properties like superconvergence in the square mesh and insensitivity to the mesh distortion.