The Smoothed Extended Finite Element Method (original) (raw)
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An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics
Finite Elements in Analysis and Design, 2007
Smoothed finite element method (SFEM) using quadrilateral elements was recently proposed by Liu et al. [A smoothed finite element method for mechanics problems, Comput. Mech. 39 (2007) 859-877; Free and forced vibration analysis using the smoothed finite element method (SFEM), J. Sound Vib. 301 (2007) 803-820; Theoretical aspects of the smoothed finite element method (SFEM), Int. J. Numer. Methods Eng. (2006), in press] to improve the accuracy and convergence rate of the existing standard four-node finite element method (FEM). In this paper the SFEM is further extended to a more general case, n-sided polygonal smoothed finite elements (nSFEM), in which the problem domain can be discretized by a set of polygons, each with an arbitrary number of sides. Stability condition is examined for this type of new elements and some criteria are provided to avoid the presence of spurious zero-energy modes. Approach to constructing nSFEM shape functions are also suggested with emphasis on a novel and simple averaging method. Selective integration scheme is recommended to overcome volumetric locking for nearly incompressible materials. Several numerical examples are investigated and the present results are in good agreement with exact solutions or FEM results. It is found that the present method gives very accurate stresses and desirable convergence rate as compared with FEM. In addition problem domain can be discretized in a very flexible manner as demonstrated in the examples.
A Smoothed Finite Element Method for Mechanics Problems
Computational Mechanics, 2006
In the finite element method (FEM), a necessary condition for a four-node isoparametric element is that no interior angle is greater than 180 • and the positivity of Jacobian determinant should be ensured in numerical implementation. In this paper, we incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems. It is found that a quadrilateral element divided into four smoothing cells can avoid spurious modes and gives stable results for integration over the element. Compared with original FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. More importantly, as no mapping or coordinate transformation is involved in the SFEM, its element is allowed to be of arbitrary shape. Hence the restriction on the shape bilinear isoparametric elements can be removed and problem domain can be discretized in more flexible ways, as demonstrated in the example problems.
An n-sided polygonal edge-based smoothed finite element method (nES-FEM) for solid mechanics
International Journal for Numerical Methods in Biomedical Engineering, 2011
An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper the ES-FEM is further extended to a more general case, n-sided polygonal edge-based smoothed finite element method (nES-FEM), in which the problem domain can be discretized by a set of polygons, each with an arbitrary number of sides. The simple averaging point interpolation method is suggested to construct nES-FEM shape functions. In addition, a novel domain-based selective scheme of a combined nES/NS-FEM model is also proposed to avoid volumetric locking. Several numerical examples are investigated and the results of the nES-FEM are found to agree well with exact solutions and are much better than those of others existing methods. Copyright
A cell-based smoothed finite-element method for gradient elasticity
Engineering With Computers, 2022
In this paper, the cell-based smoothed finite-element method (CS-FEM) is proposed for solving boundary value problems of gradient elasticity in two and three dimensions. The salient features of the CS-FEM are: it does not require an explicit form of the shape functions and alleviates the need for iso-parametric mapping. The main idea is to sub-divide the element into simplicial sub-cells and to use a constant smoothing function in each cell to compute the gradients. This new gradient is then used to compute the bilinear/linear form. The robustness of the method is demonstrated with problems involving smooth and singular solutions in both two and three dimensions. Numerical results show that the proposed framework is able to yield accurate results. The influence of the internal length scale on the stress concentration is studied systematically for a case of a plate with a hole and a plate with an edge crack in two and three dimensions.
A cell based strain smoothed extended finite element method for fracture mechanics problems
This paper presents the effect of cell based strain smoothing applied to enriched finite element method such as extended finite element method for solving two dimensional fracture mechanics problems. Brief XFEM formulations and concept of strain smoothing applied to XFEM (SmXFEM) are presented. Strain smoothing simplifies the numerical integration of element stiffness matrix in the extended finite element method by converting the area integration into boundary integration and hence one dimensional gaussian quadrature integration can be used to evaluate element stiffness matrix. Fracture analysis of structural components has been carried out and stress intensity factor (SIF), computed using domain form of interaction integral, obtained from strain smoothed extended finite element method (SmXFEM) and classical finite element method are compared with the analytical solutions available in the literature. Though the SIF obtained from SmXFEM and XFEM are in good agreement with that of SIF available in the literature, SmXFEM results depend upon the type of enrichment functions, number of subcells used for strain smoothing and order of one dimensional gauss quadrature used for boundary integration.
Smoothed Finite Element Method
In this paper, the smoothed finite element method (SFEM) is proposed for 2D elastic problems by incorporation of the cell-wise strain smoothing operation into the conventional finite elements. When a constant smoothing function is chosen, area integration becomes line integration along cell boundaries and no derivative of shape functions is needed in computing the field gradients. Both static and dynamic numerical examples are analyzed in the paper. Compared with the conventional FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. In addition, as no mapping or coordinate transformation is performed in the SFEM, the element is allowed to be of arbitrary shape. Hence the well-known issue of the shape distortion of isoparametric elements can be resolved. Index Terms-finite element method (FEM), Gauss quadrature, isoparametric element, smoothed finite element method (SFEM), strain smoothing.
2009
A node-based smoothed finite element method (NS-FEM) for solving solid mechanics problems using a mesh of general polygonal elements was recently proposed. In the NS-FEM, the system stiffness matrix is computed using the smoothed strains over the smoothing domains associated with nodes of element mesh, and a number of important properties have been found, such as the upper bound property and free from the volumetric locking. The examination was performed only for two-dimensional (2D) problems. In this paper, we (1) extend the NS-FEM to three-dimensional (3D) problems using tetrahedral elements (NS-FEM-T4), (2) reconfirm the upper bound and free from the volumetric locking properties for 3D problems, and (3) explore further other properties of NS-FEM for both 2D and 3D problems. In addition, our examinations will be thorough and performed fully using the error norms in both energy and displacement. The results in this work revealed that NS-FEM possesses two additional interesting properties that quite similar to the equilibrium FEM model such as: (1) super accuracy and superconvergence of stress solutions; (2) similar accuracy of displacement solutions compared to the standard FEM model.
Smoothed Finite Element Methods for Nonlinear Solid Mechanics Problems: 2D and 3D Case Studies
2016
The Smoothed Finite Element Method (SFEM) is presented as an edge-based and a facebased techniques for 2D and 3D boundary value problems, respectively. SFEMs avoid shortcomings of the standard Finite Element Method (FEM) with lower order elements such as overly stiff behavior, poor stress solution, and locking effects. Based on the idea of averaging spatially the standard strain field of the FEM over so-called smoothing domains SFEM calculates the stiffness matrix for the same number of degrees of freedom (DOFs) as those of the FEM. However, the SFEMs significantly improve accuracy and convergence even for distorted meshes and/or nearly incompressible materials. Numerical results of the SFEMs for a cardiac tissue membrane (thin plate inflation) and an artery (tension of 3D tube) show clearly their advantageous properties in improving accuracy particularly for the distorted meshes and avoiding shear locking effects.