IMPROVEMENT OF A PHYSICAL FIELD IN FEA BY APPLYING A SMOOTHING METHOD

Abstract

Due to its reliability and efficacy, the finite element method (FEM) is considered to be one of the most successful numerical methods, which is widely applied in science and engineering for structural analysis. Besides its wide application in engineering practice, the standard FEM has certain limitations. Linear triangular and tetrahedral finite elements result in very stiff behavior and a very low rate of convergence which stems from the constant stress field. In addition, the FEM model exhibits sensitivity to distorted isoparametric elements, during the use of which the Jacobian matrix is badly conditioned, leading to poor solutions and even the breakdown of the computation process. Moreover, considering the C0 continuity of shape functions, the stress field across the element boundaries is discontinuous. Great efforts have been invested to develop numerical strategies that may be used to overcome the abovementioned problems. In this manner, the smoothed finite element method (S-FEM) [1] is created, and it is used to modify the compatible strain field, by applying the gradient smoothing approach in the FEM, originally employed to stabilize the nodal integrated Galerkin meshfree methods [2]. Depending on the applied scheme for creating smoothing domains on top of the element mesh, a series of S-FEM methods has been developed [1]. Integration in the S-FEM methods has significantly been simplified by applying Gauss’s divergence theorem, by means of which the domain integration is transformed into the boundary integration. The formulation of S-FEM methods requires the shape function values only, not their derivatives. Therefore, the Jacobian matrix is not needed, due to which isoparametric mapping is avoided. Significant properties of the S-FEM methods complete with their diverse application were discussed thoroughly in [1]. Bearing in mind all the advantages of the S-FEM methods, including the ones related to enhancing the convergence behaviour of 3-node triangular and 4-node tetrahedral finite elements [1], the only elements the mesh of which can be automatically generated even when it comes to extremely complex geometry, the S-FEM methods are expanded to solve problems involving discontinuous and singular physical fields. Coupling the strain smoothing with the partition of unity enrichment led to the formation of a series of new methods [3,4], which exploit the advantages of not only the S-FEM method, but the extended finite element method (X-FEM) as well. By transforming the internal integration into the boundary integration, the subdivision of elements intersected by discontinuities was not found to be necessary [3]. In addition, since the derivatives of shape functions in the S-FEM method are replaced by the shape functions multiplied by the component of the outward unit vector on the boundary, the singular term integration 1/√r, which occurs in the derivatives of near-tip enrichment functions (branch functions), is eliminated during the computation of the stiffness matrix in the fracture mechanics. To improve the accuracy of 3-node triangular and 4-node tetrahedral finite elements, a new strain smoothing method, coined the strain-smoothed element (SSE) method [5], has recently been proposed. The strain field of such elements is constant, whereas by applying the S-FEM methods the piecewise constant strain fields are constructed through smoothing domains. In the context of the SSE method, a linear strain field is constructed in the elements themselves, by utilizing the constant strains of the adjacent elements. A significant characteristic of the SSE method is the lack of creation of special smoothing domains. Therefore, the standard FEM framework is maintained. To the best of the authors’ knowledge, the SSE method has been successfully developed for the polygonal finite elements at last [6]. A numerical analysis was conducted within this particular research in order to provide a comparative presentation of the accuracy of linear and quadratic tetrahedral finite elements, implemented within the PAK and Nastran program packages, compared to the 4-node tetrahedral finite element of the corrected strain field [5]. The 3D problem, known as the Lame problem in the literature [5], was analyzed. The obtained results demonstrated that the corrected elements, even in case of extremely coarse mesh, were achieving a considerably higher level of accuracy compared to the linear elements. When comparing the quadratic and corrected tetrahedral finite elements, a slightly enhanced convergency of the quadratic element was observed. Since the shape functions of these two elements were not in the same range, the elements were described as the ones with various degrees of freedom, meaning that it was completely expected to see the quadratic element converging in a slightly faster manner towards the reference solution. The results of the program packages were well-aligned.