Face-based Smoothed Finite Element Method Research Articles

Investigation on smoothed finite element methods using linear tetrahedral elements for high-speed train hollow axle strength analysis

PurposeThis paper aims to investigate the performance of two novel numerical methods, the face-based smoothed finite element method (FS-FEM) and the edge-based smoothed finite element method (ES-FEM), which employ linear tetrahedral elements, for the purpose of strength assessment of a high-speed train hollow axle.Design/methodology/approachThe calculation of stress for the wheelset, comprising an axle and two wheels, is facilitated through the application of the European axle strength design standard. This standard assists in the implementation of loading and boundary conditions and is exemplified by the typical CRH2 high-speed train wheelset. To evaluate the performance of these two methods, a hollow cylinder cantilever beam is first used as a benchmark to compare the present methods with other existing methods. Then, the strength analysis of a real wheelset model with a hollow axle is performed using different numerical methods.FindingsThe results of deflection and stress show that FS-FEM and ES-FEM offer higher accuracy and better convergence than FEM using linear tetrahedral elements. ES-FEM exhibits a superior performance to that of FS-FEM using linear tetrahedral elements, showing accuracy and convergence close to FEM using hexahedral elements.Originality/valueThis study channels the novel methods (FS-FEM and ES-FEM) in the static stress analysis of a railway wheelset. Based on the careful testing of FS-FEM and ES-FEM, both methods hold promise as more efficient tools for the strength analysis of complex railway structures.

An SFEM Abaqus UEL for Nonlinear Analysis of Solids

In this paper, three different smoothed finite element method (SFEM), viz., node-based smoothed finite element method (NS-FEM), face-based smoothed finite element method (FS-FEM) and [Formula: see text]-finite element method ([Formula: see text]-FEM) are adopted for 3D solids undergoing large deformation. The common feature of all these techniques is the introduction of smoothed strain which is written as a weighted average of the compatible strain field over smoothing domains. The choice of smoothing domain is what differentiates them. The spatial discretization can be based on the simplest and automatically genera-table four-node tetrahedral elements and aforementioned techniques have shown to yield accurate results even on a coarser discretization. To take the advantages of the SFEM, it is beneficial to the FEM community to have it implemented in the widely used Abaqus[Formula: see text] software. Such an implementation is challenging because the neighboring SFEM elements are interconnected in the smoothed strain matrices in the elemental level. In this work, the above-mentioned SFEM models are implemented in the commercial software Abaqus using the softwares’ user element (UEL) feature. The challenges during the definition and the assembly of the smoothing domains are effectively addressed in this work. The developed UEL and the associated files can be downloaded from https://github.com/nsundar/3DSFEM. The implementation is validated against benchmark examples and the robustness is demonstrated with complicated real-life problems, viz., tire patch contact with road and simulation of human thumb.

Solution Bounds and Nearly Exact Solutions for 3D Nonlinear Problems of Large Deformation of Solids Using S-Fem

In this work, a three-dimensional (3D) nonlinear smoothed finite element method (S-FEM) solver is developed for large deformation problems. Node-based and face-based S-FEM using automatically generable four-noded tetrahedral elements (NS-FEM-Te4 and FS-FEM-Te4) are adopted to find the solution bounds in strain energy. The lower bound solutions are obtained using FEM-Te4 and FS-FEM-Te4, while the upper bound solutions are obtained using NS-FEM-Te4. A combined [Formula: see text]S-FEM-Te4 with a scaling factor [Formula: see text] that controls the combination is constructed to find nearly exact solutions for the nonlinear solids mechanics problems through adjusting [Formula: see text]. This is achieved using the property that a successive change of scaling factor [Formula: see text] can make the model transform from “overly-stiff” to “overly-soft”. Considering the properties of FS-FEM and NS-FEM, a selective FS/NS-FEM-TE4 is also used to solve 3D nonlinear large deformation problems, which produces a lower bound in strain energy. Hyperelastic Mooney–Rivlin and Ogden materials are both used in this study. Numerical examples reveal that S-FEM-Te4 is an effective method for obtaining solution bounds together with the standard FEM, and the FS-FEM-Te4, NS-FEM-Te4 and selective FS/NS-FEM-TE4 are robust with the high accuracy and computational efficiency for large deformation nonlinear problems.

A Concept of Cell-Based Smoothed Finite Element Method Using 10-Node Tetrahedral Elements (CS-FEM-T10) for Large Deformation Problems of Nearly Incompressible Solids

A new concept of smoothed finite element method (S-FEM) using 10-node tetrahedral (T10) elements, CS-FEM-T10, is proposed. CS-FEM-T10 is a kind of cell-based S-FEM (CS-FEM) and thus it smooths the strain only within each T10 element. Unlike the other types of S-FEMs [node-based S-FEM (NS-FEM), edge-based S-FEM (ES-FEM), and face-based S-FEM (FS-FEM)], CS-FEM can be implemented in general finite element (FE) codes as a piece of the element library. Therefore, CS-FEM-T10 is also compatible with general FE codes as a T10 element. A concrete example of CS-FEM-T10 named SelectiveCS-FEM-T10 is introduced for large deformation problems of nearly incompressible solids. SelectiveCS-FEM-T10 subdivides each T10 element into 12 four-node tetrahedral (T4) subelements with an additional dummy node at the element center. Two types of strain smoothing are conducted for the deviatoric and hydrostatic stress evaluations and the selective reduced integration (SRI) technique is utilized for the stress integration. As a result, SelectiveCS-FEM-T10 avoids the shear/volumetric locking, pressure checkerboarding, and reaction force oscillation in nearly incompressible solids. In addition, SelectiveCS-FEM-T10 has a relatively long-lasting property in large deformation problems. A few examples of large deformation analyses of a hyperelastic material confirm the good accuracy and robustness of SelectiveCS-FEM-T10. Moreover, an implementation of SelectiveCS-FEM-T10 in the FE code ABAQUS as a user-defined element (UEL) is conducted and its capability is discussed.

Coupled Analysis of Structural–Acoustic Problems Using the Cell-Based Smoothed Three-Node Mindlin Plate Element

The cell-based smoothed finite element method (CS-FEM) using the original three-node Mindlin plate element (MIN3) has recently established competitive advantages for analysis of solid mechanics problems. The three-node configuration of the MIN3 is achieved from the initial, complete quadratic deflection via ‘continuous’ shear edge constraints. In this paper, the proposed CS-FEM-MIN3 is firstly combined with the face-based smoothed finite element method (FS-FEM) to extend the range of application to analyze acoustic fluid–structure interaction problems. As both the CS-FEM and FS-FEM are based on the linear equations, the coupled method is only effective for linear problems. The cell-based smoothed operations are implemented over the two-dimensional (2D) structure domain discretized by triangular elements, while the face-based operations are implemented over the three-dimensional (3D) fluid domain discretized by tetrahedral elements. The gradient smoothing technique can properly soften the stiffness which is overly stiff in the standard FEM model. As a result, the solution accuracy of the coupled system can be significantly improved. Several superior properties of the coupled CS-FEM-MIN3/FS-FEM model are illustrated through a number of numerical examples.

THERMO-MECHANICAL ANALYSES OF COMPOSITE STRUCTURES USING FACE-BASED SMOOTHED FINITE ELEMENT METHOD

A face-based smoothed finite element method (FS-FEM) is extended to deal with the transient thermal mechanical analyses of composite structures. For this method, the problem domain is first discretized into a set of tetrahedral elements, and the face-based smoothing domains are further formed along the faces of the tetrahedral meshes. The smoothed Galerkin weak form is employed to obtain discretized system equations, and the face-based smoothing domains are used to perform the smoothing operation. After applying these approaches, the numerical integration becomes a simple summation over each smoothing domain. Several composite structures with different kinds of boundary conditions are studied in this paper. Compared with the 4-node tetrahedral FEM, the FS-FEM can achieve much better accuracy and higher convergence when using the same tetrahedral mesh.

Transient thermal mechanical analyses using a face-based smoothed finite element method (FS-FEM)

A face-based smoothed finite element method (FS-FEM) is formulated for transient thermal mechanical analyses of 3-D solids with nonlinearity. For this face-based smoothed finite element method, the problem domain is first discretized into a set of tetrahedral elements, and the face-based smoothing domains are further formed along the faces of the tetrahedral meshes. The smoothing operations are performed in these smoothing domains. Several numerical examples with different kinds of boundary conditions are investigated in this paper. Compared with the 4-node tetrahedral FEM, the FS-FEM can achieve much better accuracy and higher convergence when dealing with the thermal mechanical analyses of 3-D solids.

An edge-based smoothed tetrahedron finite element method (ES-T-FEM) for thermomechanical problems

This paper presents an edge-based smoothed tetrahedron finite element method (ES-T-FEM) to improve the accuracy of the finite element method for three-dimensional thermomechanical problems. In this approach, the smoothed Galerkin weak form is then used to construct discretized system equations using smoothing domains constructed based on edges. The model created by ES-T-FEM is softer than the standard finite element method (FEM) and face-based smoothed finite element method (FS-FEM). In addition, a novel mixed formulation of FS-FEM and FEM is proposed in solid mechanical model for thermoelastic problem. Numerical results demonstrate that the proposed method possesses a close-to-exact stiffness of the continuous system and gives better results than both FEM and FS-FEM using tetrahedron mesh. The proposed method is an innovative and unique numerical method with its distinct features, which possesses strong potentials in the successful applications thermomechanical problems.

An improved modal analysis for three-dimensional problems using face-based smoothed finite element method

In this work, we further extended the face-based smoothed finite element method (FS-FEM) for modal analysis of three-dimensional solids using four-node tetrahedron elements. The FS-FEM is formulated based on the smoothed Galerkin weak form which employs smoothed strains obtained using the gradient smoothing operation on face-based smoothing domains. This strain smoothing operation can provide softening effect to the system stiffness and make the FS-FEM provide more accurate eigenfrequency prediction than the FEM does. Numerical studies have verified this attractive property of FS-FEM as well as its ability and effectiveness on providing reliable eigenfrequency and eigenmode prediction in practical engineering application.

An edge-based smoothed tetrahedron finite element method (ES-T-FEM) for 3D static and dynamic problems

Strain smoothing operation has been recently adopted to soften the stiffness of the model created using tetrahedron mesh, such as the Face-based Smoothed Finite Element Method (FS-FEM), with the aim to improve solution accuracy and the applicability of low order tetrahedral elements. In this paper, a new method with strain smoothing operation based on the edge of four-node tetrahedron mesh is proposed, and the edge-based smoothing domain of tetrahedron mesh is serving as the assembly unit for computing the 3D stiffness matrix. Numerical results demonstrate that the proposed method possesses a close-to-exact stiffness of the continuous system and gives better results than both the FEM and FS-FEM using tetrahedron mesh or even the FEM using hexahedral mesh in the static and dynamic analysis. In addition, a novel domain-based selective scheme is proposed leading to a combined ES-T-/NS-FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed method is an innovative and unique numerical method with its distinct features, which possesses strong potentials in the successful applications for static and dynamics problems.

A coupled edge-/face-based smoothed finite element method for structural–acoustic problems

The edge-based smoothed finite element method (ES-FEM) and the face-based smoothed finite element method (FS-FEM) developed recently have shown great efficiency in solving solid mechanics problems with triangular and tetrahedral meshes. In this paper, a coupled ES-/FS-FEM model is extended to solve the structural–acoustic problems consisting of a plate structure interacting with the fluid medium. Three-node triangular elements and four-node tetrahedral elements are used to discretize the two-dimensional (2D) plate and three-dimensional (3D) fluid, respectively, as they can be generated easily and even automatically for complicated geometries. The field variable in each element is approximated using the linear shape functions, which is exactly the same as that in the standard FEM. The gradient field of the problem is obtained particularly using the gradient smoothing operation over the edge-based and face-based smoothing domains in 2D and 3D, respectively. The gradient smoothing technique can provide a proper softening effect to the model, effectively solve the problems caused by the well-known “overly-stiff” phenomenon existing in the standard FEM, and hence significantly improve the accuracy of the solution for the coupled systems. Intensive numerical studies have been conducted to verify the effectiveness of the coupled ES-/FS-FEM for structural–acoustic problems.

Simulation of Hyperthermia Treatment Using the Edge-Based Smoothed Finite-Element Method

Hyperthermia treatment is an effective tool of cancer therapy. The main feature of hyperthermia treatment is to use thermal energy to kill the cancer cells without minimum damage to surrounding tissue [1-3]. In such treatment, it is essential to predict the temperature distribution accurately for a given supply of heat at the cancer cells. In this article, an edge-based smoothed finite-element method (ES-FEM) in 2-D and face-based smoothed finite-element method (FS-FEM) in 3-D are presented to improve the accuracy of the finite-element method (FEM) without much change to the FEM setting. In the ES-FEM, the discretized equations are established using the smoothed Galerkin weak form with edge-based smoothing domains. Compared with the FEM that behaves overly-stiff, the ES-FEM model possesses a close-to-exact stiffness. Thus, ES-FEM can provide much more accurate result than the FEM using the same mesh. Numerical examples, including 2-D and 3-D of hyperthermia treatment have been analyzed using ES-FEM and FS-FEM, respectively. The results have demonstrated that ES-FEM (FS-FEM) significantly outperform the FEM using the same mesh.

A face-based smoothed finite element method (FS-FEM) for visco-elastoplastic analyses of 3D solids using tetrahedral mesh

A face-based smoothed finite element method (FS-FEM) using tetrahedral elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the solid mechanics problems. In this paper, the FS-FEM is further extended to more complicated visco-elastoplastic analyses of 3D solids using the von-Mises yield function and the Prandtl–Reuss flow rule. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic hardening and linear kinematic hardening. The formulation shows that the bandwidth of stiffness matrix of FS-FEM is larger than that of FEM, and hence the computational cost of FS-FEM in numerical examples is larger than that of FEM for the same mesh. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the FS-FEM is more efficient than the FEM.

A face‐based smoothed finite element method (FS‐FEM) for 3D linear and geometrically non‐linear solid mechanics problems using 4‐node tetrahedral elements

AbstractThis paper presents a novel face‐based smoothed finite element method (FS‐FEM) to improve the accuracy of the finite element method (FEM) for three‐dimensional (3D) problems. The FS‐FEM uses 4‐node tetrahedral elements that can be generated automatically for complicated domains. In the FS‐FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the faces of the tetrahedral elements. The results demonstrated that the FS‐FEM is significantly more accurate than the FEM using tetrahedral elements for both linear and geometrically non‐linear solid mechanics problems. In addition, a novel domain‐based selective scheme is proposed leading to a combined FS/NS‐FEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The implementation of the FS‐FEM is straightforward and no penalty parameters or additional degrees of freedom are used. The computational efficiency of the FS‐FEM is found better than that of the FEM. Copyright © 2008 John Wiley & Sons, Ltd.