Arbitrary polygon mesh for elastic and elastoplastic analysis of solids using smoothed finite element method

Abstract

This work extends the smoothed finite elements method (S-FEM) to the use of arbitrarily complicated meshes, including nonconvex polygonal meshes for the first time. A novel formulation for arbitrarily polygonal elements based on the S-FEM is presented for analyses of elastic and elastoplastic problems. This enable the use of polygonal elements that are of conventional quadrilateral, convex polygonal, concave polygon elements and n-sides concave polygonal elements. This is achieved by making use of the exceptional unique features of S-FEM in terms of stability, accommodation to mesh distortion, and free from differentiation in strain field computation. The ear clipping technique is used to automatically generate smoothing domains without any dummy nodes. The stability in using such a complicated mesh is ensured by the gradient smoothing technique in S-FEM, and no additional stability control measure is required. In addition, domain integration is converted to boundary integral along sub-triangular domain, which requires only the shape function values at the field nodes. This avoiding coordinate mapping, and hence it is a Jacobian free formulation. The von Mises elastoplastic model is implemented in our S-FEM by decomposing the tangent stiffness matrix into three parts: elastic stiffness matrix, strain–displacement matrix and the derivative of the stress–strain matrix. The first two matrices are calculated only once at the initial stage, and only the derivative of the stress–strain matrix is updated in each iteration for capture the nonlinearity. The numerical results show that the proposed method can handle the general non-convex elements effectively with good accuracy and robustness. It offers an effective tool in dealing with polycrystalline metallic materials with crystal grains of arbitrary shapes.