An n-sided polygonal smoothed finite element method (nSFEM) for solid mechanics (original) (raw)

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

FREE AND FORCED VIBRATION ANALYSIS USING THE n-SIDED POLYGONAL CELL-BASED SMOOTHED FINITE ELEMENT METHOD (nCS-FEM)

International Journal of Computational Methods, 2013

A n-sided polygonal cell-based smoothed finite element method (nCS-FEM) was recently proposed to analyze the elastic solid mechanics problems, in which the problem domain can be discretized by a set of polygons with an arbitrary number of sides. In this paper, the nCS-FEM is further extended to the free and forced vibration analyses of twodimensional (2D) dynamic problems. A simple lump mass matrix is proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in the nCS-FEM. Several numerical examples are investigated and the results found of the nCS-FEM agree well with exact solutions and with those of others FEM.

Additional properties of the node-based smoothed finite element method (NS-FEM) for solid mechanics problems

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.

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.

A Hybrid Smoothed Finite Element Method (H-Sfem) to Solid Mechanics Problems

International Journal of Computational Methods, 2013

In this paper, a hybrid smoothed finite element method (H-SFEM) is developed for solid mechanics problems by combining techniques of finite element method (FEM) and node-based smoothed finite element method (NS-FEM) using a triangular mesh. A parameter α is equipped into H-SFEM, and the strain field is further assumed to be the weighted average between compatible stains from FEM and smoothed strains from NS-FEM. We prove theoretically that the strain energy obtained from the H-SFEM solution lies in between those from the compatible FEM solution and the NS-FEM solution, which guarantees the convergence of H-SFEM. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper- and lower-bound solutions can always be obtained by adjusting α; (2) there exists a preferable α at which the H-SFEM can produce the ultrasonic accurate solution.

A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems

Computers & Structures, 2009

Finite element method (FEM) Smoothed finite element method (SFEM) Upper bound Lower bound Global error Polygonal a b s t r a c t This paper presents a node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems using a mesh of polygonal elements. The calculation of the system stiffness matrix is performed using strain smoothing technique over the smoothing cells associated with nodes, which leads to line integrations along the edges of the smoothing cells. The numerical results demonstrated that the NS-FEM possesses the following properties: (1) upper bound in the strain energy of the exact solution when a reasonably fine mesh is used; (2) well immune from the volumetric locking;

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.

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.

Free and forced vibration analysis using the smoothed finite element method (SFEM)

Journal of Sound and Vibration, 2007

Smoothed finite element method (SFEM) was recently proposed by the authors to improve the accuracy and convergence rate of the existing standard four-node isoparametric finite element method (FEM) by 2 Â 2 quadrature in the static analysis. In this work, it is further extended to the free and forced analysis of two-dimensional (2D) dynamic problems. It is found that SFEM achieves more accurate results and higher convergence rates as compared with the corresponding finite elements in solving dynamic problems without increasing the computational cost. As only shape function itself is involved in computing the field gradients and no coordinate transformation is required, the method is very easy to implement. Nonlinear elastic forced vibration problem is also examined by using smoothed deformation gradients on each cell.