Formulation and numerical evaluation of a novel four-node quadrilateral element with continuous nodal stress (Q4-CNS) are presented. Q4-CNS can be regarded as an improved hybrid FE-meshless four-node quadrilateral element (FE-LSPIM QUAD4), which is a hybrid FE-meshless method. Derivatives of Q4-CNS are continuous at nodes, so the continuous nodal stress can be obtained without any smoothing operation. It is found that, compared with the standard four-node quadrilateral element (QUAD4), Q4-CNS can achieve significantly better accuracy and higher convergence rate. It is also found that Q4-CNS exhibits high tolerance to mesh distortion. Moreover, since derivatives of Q4-CNS shape functions are continuous at nodes, Q4-CNS is potentially useful for the problem of bending plate and shell models.

A novel polygonal finite element method (PFEM) based on partition of unity is proposed, termed the virtual node method (VNM). To test the performance of the present method, numerical examples are given for solid mechanics problems. With a polynomial form, the VNM achieves better results than those of traditional PFEMs, including the Wachspress method and the mean value method in standard patch tests. Compared with the standard triangular FEM, the VNM can achieve better accuracy. With the ability to construct shape functions on polygonal elements, the VNM provides greater flexibility in mesh generation. Therefore, several fracture problems are studied to demonstrate the potential implementation. With the advantage of the VNM, the convenient refinement and remeshing strategy are applied.