Edge-based Smoothed Finite Element Research Articles

A Topology Optimization Method Based on the Edge-Based Smoothed Finite Element Method

In this paper, we combined the edge-based smoothed finite element method (ES-FEM) with topology optimization. The edge-based gradient smoothing operation was introduced to overcome the accuracy-loss of the classical finite element method raised by the coarse mesh and “overly stiff” phenomenon. By employing the ES-FEM, design variables can be related to the smoothed edge, thus more design variables can be adaptively obtained without additional remeshing. Two classical topology optimization problems were considered, namely compliance minimization and stress-constrained topology optimization. We presented several numerical examples, among which the compliance minimization examples illustrated the potential of the proposed method, and the advantages of applying such a numerical method in topology optimization were demonstrated through the stress-constrained topology optimization.

The crack detection of acoustic metamaterials using a weighted mode shape-wavelet-based strategy

With the extensive application of acoustic metamaterial (AM) in the mechanical industry, the crack detection of AM becomes increasingly crucial. In this paper, a weighted mode shape-wavelet-based strategy coupled with edge-based smoothed finite element (ES-FEM) is presented for damage detection of AM. Firstly, the multi-order mode shapes of cracked AM are computed by ES-FEM, and the obtained mode shape data is further interpolated and extended. Then, the multi-order detail coefficients of smoothed mode shapes are calculated through the discrete wavelet transform (DWT). Finally, the damage index and crack width estimator are established from the weighted multi-order detail coefficients to locate the crack and predict crack width, respectively. Numerical results demonstrate that the traditional modal-wavelet based crack detection strategy is very ambiguous while the proposed ES-FEM coupled weighted crack detection strategy is very effective for crack location and crack size prediction of AM.

Phase-field model for brittle fracture based on the inner-element edge-based smoothed finite method (IES-FEM)

In this work, a phase field fracture model based on the Inner-based Edge-based Smoothed Finite Element Method (IES-FEM) is implemented to simulate the brittle fracture in ABAQUS through subroutine UEL. The specific application of IES-FEM on phase field method (PFM) is explained. Through the single-edge notched tension/shear test and symmetric double notched tensile test, the differences of effects on numerical simulation of different length scale parameters, different mesh size and different displacement increment were discussed between IES-FEM-PFM and FEM-PFM. The coupled governing equations were solved alternately by decoupling the phase and displacement field. The results showed that the maximum reaction force of the load–displacement curve of IES-FEM-PFM is always smaller than that calculated by FEM-PFM, which was closer to the results of monolithic scheme. The stress and phase field variable at the same step time were also smaller than those of FEM-PFM, which is closer to the real stress field. After that some complex models were implemented to simulate the crack growth path in the phase field fracture model of IES-FEM. The results showed that IES-FEM did not change the characteristics of the phase field fracture model and the applicability and accuracy of IES-FEM in the phase field fracture model was verified effectively.

An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle

An Edge-Based Smoothed Finite Element Method with TBC for the Elastic Wave Scattering by an Obstacle

Strategy to Improve Edge-Based Smoothed Finite Element Solutions Using Enriched 2D Solid Finite Elements

In this paper, we present an automatic procedure that enhances the solution accuracy of edge-based smoothed 2D solid finite elements (three-node triangular and four-node quadrilateral elements). To obtain an enhanced solution, an adaptive enrichment scheme that uses enriched 2D solid finite elements and can effectively improve solution accuracy by applying cover functions adaptively without mesh-refinement is adopted in this procedure. First, the error of the edge-based finite element solution is estimated using a devised error estimation method, and appropriate cover functions are assigned for each node. While the edge-based smoothed finite elements provide piecewise constant strain fields, the proposed enrichment scheme uses the enriched finite elements to obtain a higher order strain field within the finite elements. Through various numerical examples, we demonstrate the accuracy improvement and efficiency achieved.

Dynamic Analysis of Functionally Graded Porous Plates Resting on Elastic Foundation Taking into Mass subjected to Moving Loads Using an Edge-Based Smoothed Finite Element Method

The paper presents the extension of an edge-based smoothed finite element method using three-node triangular elements for dynamic analysis of the functionally graded porous (FGP) plates subjected to moving loads resting on the elastic foundation taking into mass (EFTIM). In this study, the edge-based smoothed technique is integrated with the mixed interpolation of the tensorial component technique for the three-node triangular element (MITC3) to give so-called ES-MITC3, which helps improve significantly the accuracy for the standard MITC3 element. The EFTIM model is formed by adding a mass parameter of foundation into the Winkler–Pasternak foundation model. Two parameters of the FGP materials, the power-law index (k) and the maximum porosity distributions (Ω), take forms of cosine functions. Some numerical results of the proposed method are compared with those of published works to verify the accuracy and reliability. Furthermore, the effects of geometric parameters and materials on forced vibration of the FGP plates resting on the EFTIM are also studied in detail.

Transient Wave Propagation Dynamics with Edge-Based Smoothed Finite Element Method and Bathe Time Integration Technique

In this work, the edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. The edge-based gradient smoothing technique (GST) can properly soften the “overly soft” system matrices from the standard finite element approach; then, the spatial numerical dispersion error of the calculated solutions for wave problems can be significantly reduced. To effectively solve the transient wave propagation problems, the Bathe time integration scheme is employed to perform the involved time integration. Due to the appropriate “numerical dissipation effects” from the Bathe time integration method, the spurious oscillations in the relatively large wave numbers (high frequencies) can be effectively suppressed; then, the temporal numerical dispersion error in the calculated solutions can also be notably controlled. A number of supporting numerical examples are considered to examine the capabilities of the present approach. The numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM.

An Edge-Based Smoothed Finite Element for Free Vibration Analysis of Functionally Graded Porous (FGP) Plates on Elastic Foundation Taking into Mass (EFTIM)

In this paper, free vibration analysis of the functionally graded porous (FGP) plates on the elastic foundation taking into mass (EFTIM) is presented. The fundamental equations of the FGP plate are derived using Hamilton’s principle. The mixed interpolation of the tensorial components (MITC) approach and the edge-based smoothed finite element method (ES-FEM) is employed to avoid the shear locking as well as to improve the accuracy for the triangular element. The EFTIM is a foundation model based on the two-parameter Winkler–Pasternak model but added a mass parameter of foundation. Materials of the plate are FGP with a power-law distribution and maximum porosity distributions in the forms of cosine functions. Some numerical examples are examined to demonstrate the accuracy and reliability of the proposed method in comparison with those available in the literature.

F-Bar Aided Edge-Based Smoothed Finite Element Method with 4-Node Tetrahedral Elements for Static Large Deformation Elastoplastic Problems

A new type of smoothed finite element method (S-FEM), F-barES-FEM-T4, is demonstrated in static large deformation elastoplastic cases. F-barES-FEM-T4 combines the edge-based S-FEM (ES-FEM) and the node-based S-FEM (NS-FEM) for 4-node tetrahedral (T4) elements with the aid of the F-bar method in order to resolve the major issues of Selective ES/NS-FEM-T4. As well as most of the other S-FEMs, F-barES-FEM-T4 inherits pure displacement-based formulation and thus has no increase in DOF. Moreover, the cyclic smoothing procedure introduced in F-barES-FEM-T4 is effective to adjust the smoothing level so that pressure checkerboarding (oscillation) is suppressed reasonably. Some examples of static large deformation analyses for elastoplastic materials proof the excellent performance of F-barES-FEM-T4 in contrast to the conventional hybrid T4 element formulation.

Free Vibration Analysis of Functionally Graded Shells Using an Edge-Based Smoothed Finite Element Method

An edge-based smoothed finite element method (ES-FEM) combined with the mixed interpolation of tensorial components technique for triangular shell element (MITC3), called ES-MITC3, for free vibration analysis of functionally graded shells is investigated in this work. In the formulation of the ES-MITC3, the stiffness matrices are obtained by using the strain-smoothing technique over the smoothing domains that are formed by two adjacent MITC3 triangular shell elements sharing an edge. The strain-smoothing technique can improve significantly the accuracy and convergence of the original MITC3. The material properties of functionally graded shells are assumed to vary through the thickness direction by a power–rule distribution of volume fractions of the constituents. The numerical examples demonstrated that the present ES-MITC3method is free of shear locking and achieves the high accuracy compared to the reference solutions in the literature.

Evaluation of Fracture Parameters by Coupling the Edge-Based Smoothed Finite Element Method and the Scaled Boundary Finite Element Method

This paper presents a technique to evaluate the fracture parameters by combining the edge based smoothed finite element method (ESFEM) and the scaled boundary finite element method (SBFEM). A semi-analytical solution is sought in the region close to the vicinity of the crack tip using the SBFEM, whilst, the ESFEM is used for the rest of the domain. As both methods satisfy the partition of unity and the compatibility condition, the stiffness matrices obtained from both methods can be assembled as in the conventional finite element method. The stress intensity factors (SIFs) are computed directly from their definition. Numerical examples of linear elastic bodies with cracks are solved without requiring additional post-processing techniques. The SIFs computed using the proposed technique are in a good agreement with the reference solutions. A crack propagation study is also carried out with minimal local remeshing to show the robustness of the proposed technique. The maximum circumferential stress criterion is used to predict the direction of propagation.

A fully smoothed finite element method for analysis of axisymmetric problems

A fully smoothed finite element method is developed to model axisymmetric problems by incorporating a special integral into the Cell-based, Node-based and Edge-based Smoothed Finite Element Method (CS-FEM, NS-FEM, ES-FEM), respectively. The special integral is done by combining Gauss divergence theorem with the evaluation of an indefinite integral, which can be used for treatment of the axisymmetric term in strain matrix and shape function in mass matrix. Applying the special integral and smoothing technique, all the domain integrals in stiffness matrix and mass matrix can be smoothed and rewritten as boundary integrals of smoothing cells. Then the stiffness matrix and mass matrix of element are computed by a simple summation over the smoothing cells. In this work, the proposed method is extended to static and structure dynamic analysis of axisymmetric problems. Numerical examples show that the proposed method can yield good performance even for extremely irregular elements.

An edge-based smoothed finite element method (ES-FEM) for dynamic analysis of 2D Fluid-Solid interaction problems

The paper presents an extension of the Edge-based Smoothed Finite Element Method (ES-FEM-T3) using triangular elements for the dynamic response analysis of two-dimension fluid-solid interaction problems based on the pressure-displacement formulation. In the proposed method, both the displacement in the solid domain and the pressure in the fluid domain are smoothed by the gradient smoothing technique based on the smoothing domains associated with the edges of the triangular elements. Thanks to the softening effect of the gradient smoothing technique used in the ES-FEM-T3, the numerical solutions for the coupled systems by the ES-FEMT3 are improved significantly compared to those by some other existing FEM methods.

A coupled ES-BEM and FM-BEM for structural acoustic problems

In this paper, a coupled numerical method of the edge-based smoothed finite element (ES-FEM) with the fast multipole BEM (FM-BEM) is proposed to analyze structural acoustic problems. The vibrating structure is modeled using the so-called ES-FEM-DSG3 method, where the 3-node linear triangle plate elements based on the Reissner-Mindlin plate theory with the discrete shear gap (DSG) technique for overcoming the shear locking are applied. The edge-based gradient smoothing operations are applied to “soften” the “overly-stiff” behavior in the standard FEM, which significantly reduces the inherent numerical dispersion error. The normal velocities on the surface of the structure are imposed as boundary conditions for the acoustic domain which is modeled using the FM-BEM for both the interior and exterior acoustic domains. Comparing with the conventional BEM, the matrix vector multiplication and the memory requirement in the FM-BEM are reduced dramatically. The coupled ES-FEM/ FM-BEM method takes the advantages of both ES-FEM and FM-BEM, which can avoid drawbacks of the “overly-stiff” behavior in FEM and computational inefficiency in the conventional BEM. Two numerical examples are presented to verify and demonstrate the effectiveness of the combined method: one academic problem for studying in detail the accuracy and efficiency of the present method, and one application to a practical vehicle noise simulation.

A Phantom-Node Method with Edge-Based Strain Smoothing for Linear Elastic Fracture Mechanics

This paper presents a novel numerical procedure based on the combination of an edge-based smoothed finite element (ES-FEM) with a phantom-node method for 2D linear elastic fracture mechanics. In the standard phantom-node method, the cracks are formulated by adding phantom nodes, and the cracked element is replaced by two new superimposed elements. This approach is quite simple to implement into existing explicit finite element programs. The shape functions associated with discontinuous elements are similar to those of the standard finite elements, which leads to certain simplification with implementing in the existing codes. The phantom-node method allows modeling discontinuities at an arbitrary location in the mesh. The ES-FEM model owns a close-to-exact stiffness that is much softer than lower-order finite element methods (FEM). Taking advantage of both the ES-FEM and the phantom-node method, we introduce an edge-based strain smoothing technique for the phantom-node method. Numerical results show that the proposed method achieves high accuracy compared with the extended finite element method (XFEM) and other reference solutions.

An Edge-based Imbricate Finite Element Method (EI-FEM) with full and reduced integration

An Edge-based Imbricate Finite Element Method (EI-FEM), aiming at combining the ease of mesh generation using triangular elements and the improved accuracy of quadrilateral elements, is proposed. Two finite elements for 2D analyses that imbricate each other while being assembled are designed based upon the triangulation of the domain. By means of two numerical benchmark analyses, we show that similar improvements with respect to numerical accuracy as obtained by the recently proposed Edge-based Smoothed Finite Element Method (ES-FEM) can be obtained, without sacrificing the standard concepts of the Finite Element Method.

Ultra Accurate Edge-based Smoothed Finite Element Method of Tetrahedron for Visco-elastoplastic Analysis of 3D Solid

The tetrahedral mesh is good at building three-dimensional solid finite element discrete model because of its powerful geometric characterization,while it behaviors overly-stiff property leading to large errors,which greatly limit the widely use of the tetrahedral mesh.Since the tetrahedral mesh is an important issue in computational mechanics and is also a main focus.A new type of edge-based smoothed finite element method of tetrahedron mesh(ES-FEM-T) is extended to 3D solid mechanics visco-elastoplastic material.Numerical examples verified that computational efficiency of the ES-FEM-T is higher than the finite element method(FEM) and face-based smoothed finite element method.This method not only inherits the strong tetrahedral geometry characterization capabilities,but also improves the accuracy and efficiency of the tetrahedral mesh,which has a broad engineering application prospects.

Nonlinear 3D numerical computations for the square membrane versus experimental data

This paper presents a discussion on the three-dimensional (3D) mechanical model proposed by Shi et al. for the membrane deformation by comparing the experimental results, and the limitation of this model to predict the behaviors of the membrane structure with large deformations (i.e. geometric nonlinearity). Three nonlinear numerical models, all of which can avoid the limitation, are then established to simulate the membrane’s large deformations, including the membrane model with zero bending stiffness and the shell model with small but nonzero bending stiffness based on the latest Edge-based Smoothed Finite Element Method (ES-FEM) as well as the standard Finite Element Method (FEM). The effects of geometric nonlinearity on the numerical results are carefully checked by comparing the benchmark experimental results, and the effects of different models/methods on the numerical results are also quantitatively examined. Factors, e.g. pressure fluctuations in the experiment and boundary conditions in the numerical models, are discussed to illustrate the differences between the numerical and experimental results, so as to provide some further suggestions on the improvements of the corresponding numerical models.