Computer Methods in Applied Mechanics and Engineering
A node-based smoothed finite element method (NS-FEM) for upper bound solution to visco-elastoplastic analyses of solids using triangular and tetrahedral meshes
Introduction
In numerical computation using the compatible finite element method (FEM), the 3-node linear triangular element (T3) in two-dimensional (2D) problems and 4-node linear tetrahedral element (T4) in three-dimensional (3D) problems are preferred by many engineers due to its simplicity, robustness, and efficiency of adaptive mesh refinements. However, these T3 and T4 elements still possess certain drawbacks: (1) they overestimate excessively the stiffness of the problem which leads to poor accuracy in both displacement and stress solutions; (2) they are subjected to locking in the problems with bending domination and incompressible materials.
In the other front of development of novel numerical methods, Liu et al. have combined the strain smoothing technique [1] used in meshfree methods [2], [3] into the FEM using quadrilateral elements to formulate a cell/element-based smoothed finite element method (SFEM or CS-FEM) [4], [5], [6]. Applying this strain smoothing technique on smoothing domains will help to soften the over-stiffness of the compatible FEM model, and hence can improve significantly the accuracy of solutions in both displacement and stress. In the CS-FEM, the smoothing domains are created based on elements, and each element can be subdivided into 1 or some quadrilateral smoothing domains as shown in Fig. 1. In practical applications, using four smoothing domains for each element (Fig. 1c) in the CS-FEM gives good accuracy and is easy to implement, and hence advised for all problems. The CS-FEM has also been studied further theoretically [4], [7], [8], [9], and extended to the general n-sided polygonal elements (nSFEM or nCS-FEM) [10], dynamic analyses [11], incompressible materials using selective integration [12], [13], plate and shell analyses [14], [15], [16], [17], [18], and fracture mechanics problems in 2D continuum and plates [19].
In the effort to overcome two above-mentioned drawbacks of T3 and T4 elements, Liu and Nguyen-Thoi et al. have then extended the concept of smoothing domains to formulate a series of smoothed FEM (S-FEM) models [37] with different applications such as the node-based S-FEM (NS-FEM) [20], [21], edge-based S-FEM (ES-FEM) [25], [26], [27], [28], [29], [30], face-based S-FEM (FS-FEM) [31], [32], and an alpha-FEM [33]. Similar to the standard FEM, these S-FEM models also uses a mesh of elements. However, being different from the standard FEM which evaluates the discrete weakform based on compatible strains over the elements, these S-FEM models evaluate the discrete weakform based on smoothed strains over smoothing domains. The smoothed strains are created by multiplying the compatible strains with a smoothing function, and the smoothing domains are created based on the entities of the element mesh such as nodes (Fig. 2), or edges (Fig. 3) or faces (Fig. 4). These smoothing domains are linear independent and hence ensure stability and convergence of the S-FEM models. They cover parts of adjacent elements, and therefore the number of supporting nodes in smoothing domains is larger than that in elements. This leads to the bandwidth of stiffness matrix in the S-FEM models to increase and the computational cost is hence higher than those of the FEM. However, also due to contributing of more supporting nodes in the smoothing domains, the S-FEM models often produce the solution that is much more accurate than that of the FEM. Therefore in general, when the efficiency of computation (computation time for the same accuracy) in terms of the error estimator versus computational cost is considered, the S-FEM models are more efficient than the counterpart FEM models [21], [25], [27], [31], [32]. It is clear that these S-FEM models have the features of both models: meshfree [2], [3] and FEM. The element mesh is still used but the smoothed strains bring the information beyond the concept of only one element in the FEM: they bring in the information from the neighboring elements.
Among these S-FEM models, the NS-FEM [20], [21] shows some important properties in the elastic solid mechanics such as: (1) it possesses the upper bound property in strain energy; (2) it is immune naturally from the volumetric locking; (3) it achieves super-accurate and super-convergent properties of stress solutions; (4) it can use linear triangular and tetrahedral elements; and (5) the stress at nodes can be computed directly from the displacement solution without using any post-processing. These five properties are very promising to apply the NS-FEM effectively in the more complicated non-linear problems. The NS-FEM was then extended to perform adaptive analysis [22], linear elastostatics and vibration problems [23] and the limit and shakedown analyses [24].
This paper attempts to further formulate the NS-FEM for more complicated visco-elastoplastic analyses of 2D and 3D solids using triangular and tetrahedral meshes, respectively. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic hardening and linear kinematic hardening. A dual formulation for the NS-FEM with displacements and stresses as the main variables is performed. The von-Mises yield function and the Prandtl–Reuss flow rule are used. In the numerical procedure, however, the stress variables are eliminated and the problem becomes only displacement-dependent. The numerical results show that the NS-FEM has higher computational cost than the FEM. However the NS-FEM is much more accurate than the FEM, and hence the NS-FEM is more efficient than the FEM. It is also observed from the numerical results that the NS-FEM possesses the upper bound property which is very meaningful for the visco-elastoplastic analyses which almost have not got the analytical solutions. This suggests that we can use two models, NS-FEM and FEM, to bound the solution, and can even estimate the global relative error of numerical solutions.
Section snippets
NS-FEM for visco-elastoplasticity: a dual formulation
In the engineering literature, the elastoplastic evolution problem is usually modeled based on the so-called primal or dual formulation. In the primal formulation [35], the strains are treated as the primary variables and a discretization is required for simultaneous approximations of both the displacement and plastic strain fields. In the dual formulation [35], the displacement and stress approximations are computed simultaneously with yield functions and flow rules written in terms of
A-posteriori error estimator
In next section, the numerical performances by the NS-FEM using triangular elements (NS-FEM-T3) and tetrahedral elements (NS-FEM-T4) are conducted. The results of NS-FEM-T3 will be compared with those of the standard FEM using triangular elements (FEM-T3) [34] and the edge-based smoothed FEM using triangular elements (ES-FEM-T3) [27]. The results of NS-FEM-T4 will be compared with those of the standard FEM using tetrahedral elements (FEM-T4) [34] and the face-based smoothed FEM using
Numerical examples
In this section, the properties of NS-FEM are observed through six numerical examples computed for three different visco-elastoplastic cases: perfect visco-elastoplasticity, visco-elastoplasticity with isotropic hardening and visco-elastoplasticity with linear kinematic hardening. Three first examples are for the 2D problems and the numerical results of NS-FEM-T3 will be compared with those of ES-FEM-T3 [27] and FEM-T3 [34]. Three last examples are for 3D problems and the numerical results of
Conclusion
This paper attempts to further formulate the NS-FEM for more complicated visco-elastoplastic analyses of 2D and 3D solids using triangular (NS-FEM-T3) and tetrahedral (NS-FEM-T4) meshes, respectively. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic hardening and linear kinematic hardening. A dual formulation for the NS-FEM-T3/NS-FEM-T4 with displacements and stresses as the main variables is performed. The von-Mises yield function and the
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