Edge-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 improved MITC3 element for vibration response analysis of piezoelectric functionally graded porous plates

The main goal of this paper is to improve the mixed interpolation of tensorial components triangular (MITC3) by using the edge-based smoothed finite element method (ES-FEM), so-called ES-MITC3, for analyzing the vibration of piezoelectric functionally graded porous (p-FGP) plates subjected to dynamic loading. The material properties of the FGP core vary through thickness with uneven porosity distribution. Besides, the linear relationship between the electric potential and the thickness of the piezoelectric sublayer is taken into account. A closed-loop control algorithm is employed to actively control the vibration of p-FGP plates, through feedback from displacement and velocity. The performance of the proposed method is verified through comparative examples. Finally, the authors hope that the present method can be effectively applied to many smart material models in a multiphysics environment and contribute to understanding texture control by piezoelectric materials through numerical results.

An edge-based smoothed finite element method for semi-implicit coupling of unsteady viscoelastic fluid–structure interaction

In this paper, a new partitioned semi-implicit coupling algorithm is developed for unsteady viscoelastic fluid–structure interaction in association with an edge-based smoothed finite element method (ESFEM). Stabilization techniques tailored for the viscoelastic fluid subproblem are naturally integrated into the hybrid explicit/implicit coupling framework to achieve the enhanced stability. In particular, a mass source term derived from the modified continuity equation in the finite-element context enables the ESFEM to fulfill the geometric conservation law on variable-node smoothing cells during the subiterations. Moreover, both multi-field equations and interface conditions are easily discretized with the ESFEM which enjoys high flexibility of smoothed Galerkin weak-form integral. Desirable numerical results are given for two benchmark problems.

Coupling edge-based smoothed finite element method with incompressible smoothed particle hydrodynamics for thermal fluid structure interaction problems

Numerical simulation of thermal fluid structure interaction (TFSI) problems with incompressible flows is one of the most challenging topics in computational fluid dynamics. In this paper, the recently developed edge-based smoothed finite element method (ES-FEM) is coupled with incompressible smoothed particle hydrodynamics (ISPH) for modeling TFSI problems, where the fluid domain is simulated by ISPH and the structure domain is simulated by ES-FEM method. A ghost particle based algorithm to couple the ES-FEM elements and ISPH particles is applied to implement the modeling of TFSI problems. The numerical simulation of TFSI problems using the ES-FEM-ISPH coupling method under pure Lagrangian scheme is very attractive because it can naturally handle the moving interface and free surface. The effectiveness of the ES-FEM-ISPH coupling method developed in this paper is verified by a number of numerical simulation examples with TFSI problems. From the obtained numerical results, we can conclude that the ES-FEM-ISPH coupling method is effective to simulate TFSI problems.

Dynamic response of functionally graded porous-core sandwich plates subjected to blast load using ES-MITC3 element

The main objective of this paper is to use the combination of the edge-based smoothed finite element method (ES-FEM) and mixed interpolation of tensorial components triangular (MITC3) element. As a result, the so-called ES-MITC3 element is proposed for the dynamic analysis of sandwich plates with a functionally graded porous (FGP) core subjected to blast load. Sandwich plates include a bottom layer made of homogeneous metal, an all-ceramic top layer and a FGP core layer with uneven porosity distribution. The ES-MITC3 element is created to enhance the accuracy and convergence of the MITC3 element as well as the traditional triangular element. Employing Hamilton’s principle, governing equations of sandwich plates are inferred. Then, solving linear differential equations is conducted by using the Newmark-beta approach. A few studies are conducted to confirm the performance of the current method. Finally, the new numerical results on the vibration characteristic of sandwich plates are provided.

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.

An edge‐based smoothed finite element framework for partitioned simulation of vortex‐induced vibration problems

AbstractThis article proposes an edge‐based smoothed finite element method (ESFEM) for predicting vortex‐induced vibration (VIV) of multiple rigid and elastic structures. The ESFEM is applied to discretize the Navier–Stokes and elastodynamic equations with three‐node triangular (T3) element. The fluid excitation is also evaluated in the edge‐based notion. New integration points are arranged in local smoothing domains to ease the resultant approximation. Dynamic grids are moved by an efficient two‐level mesh scheme. The fluid‐structure mechanical system is formulated under the arbitrary Lagrangian–Eulerian description which enables tight coupling of interacting fields in a partitioned way. Especially, the geometric conservation law is respected for the ESFEM through a mass source term constructed in the T3 element context. The developed technique is validated against previously published data for three low‐Reynolds‐number VIV problems. Flow features and structural responses have been correctly identified therein as a result of the numerical prediction.

Effects of partially supported elastic foundation on free vibration of FGP plates using ES-MITC3 elements

The main goal of this article further extends the ES-MITC3 element based on first-order shear deformation theory (FSDT) for the free vibration analysis of functionally graded porous (FGP) plates resting on the partially supported elastic foundation (PSEF). The ES-MITC3 element is the union between the mixed interpolation of tensorial components (MITC) technique and an edge-based smoothed finite element method (ES-FEM). The elastic foundation (EF) is a Winkler-Pasternak’s model with two parameters consisting of Winkler-stiffness ( k 1 ) and Pasternak-stiffness ( k 2 ). The porosity distribution in plates is assumed to vary according to the even distribution (case 1) and uneven distribution (case 2) rules. The governing equation is derived from Hamilton's principle. Numerical examples are performed to verify the accuracy and reliability of the present method. In addition, the effects of types of PSEF, material properties and geometry characteristics on the free vibration behavior of FGP plates are fully investigated.

Modeling fluid–structure interaction with the edge-based smoothed finite element method

We address in this work the numerical modeling of fluid–structure interaction (FSI) with the edge-based smoothed finite element method (ESFEM). We prove that the smoothed Galerkin weak-form integral of the Navier–Stokes equations is very flexible in the ESFEM. A simple integration scheme is accordingly proposed to facilitate the weak-form approximation. The ESFEM is subsequently used for the elastodynamics equation and interface conditions as a matter of course. The FSI system is completely formulated in the edge-based notion and then is iteratively solved in a partitioned way. Numerical results obtained with the ESFEM are convincing in various FSI examples, indicating that the method can be a general solution approach to not only FSI but also many other real-world phenomena.

On the edge‐based smoothed finite element approximation of viscoelastic fluid flows

AbstractThis article discusses the numerical simulation of viscoelastic fluid flows using the edge‐based smoothed finite element method (ESFEM). The incompressible Navier–Stokes equations coupled with the Oldroyd‐B constitutive relation are decoupled via the characteristic‐based split scheme that enables the use of equal‐order interpolation for the triple primitive variables. For this reason, the spatial discretization is implemented with linear three‐node triangular elements which work very well with the ESFEM. Edge‐based smoothing cells (SCs) are constructed on grounds of existing elements, directly underpinning the essential gradient smoothing procedure. New integration points are proposed within local SCs since the ESFEM enjoys the tremendous flexibility of the smoothed Galerkin weak‐form integral. The discrete viscoelastic system is entirely formulated in the edge‐based notion such that all gradient‐related terms are readily smoothed cell‐by‐cell. Finally, several numerical examples are presented to demonstrate the desirable capability of the ESFEM.

An efficient and accurate numerical method for the heat conduction problems of thermal metamaterials based on edge-based smoothed finite element method

Thermal metamaterials are a class of artificial composite structures with special thermal conduction properties. Due to the existence of complex multi-phase materials arrangement and interfaces in the thermal metamaterials, an efficient and accurate numerical method is on-demand to predict their temperature distribution and performances. In this paper, we studied the edge-based smoothed finite element method (ES-FEM) for simulating the heat conduction of thermal metamaterials in both steady-state and transient circumstances. Benefit from the adopted smoothing technique, the numerical errors caused by the complex interface are further eliminated. The most significant advantage of the proposed method is that one can directly use the automatically generated coarse triangle mesh to reduce the computational costs while ensuring accuracy. Two numerical examples are presented, in which two typical thermal metamaterials, the thermal cloak and thermal concentration device, are considered respectively in both steady-state and transient cases. The results illustrate that the ES-FEM based approach has good efficiency and accuracy in the heat conduction simulation of the thermal metamaterials.

Static Analysis of Stiffened Shells Using an Edge-Based Smoothed MITC3 (ES-MITC3) Method

A novel approach for solving the stiffened shell structures by using an edge-based smoothed MITC3 finite element method (ES-MITC3) is presented in this paper. The ES-MITC3 method is an efficient finite element method by combining the edge-based smoothed finite element method (ES-FEM) with the original MITC3 triangular element to not only significantly improve the accuracy but also overcome the shear-locking phenomenon in the Reissner–Mindlin shell analysis. In this study, the ES-MITC3 method is applied for shell structures and then reinforced by stiffeners based on the Timoshenko beam theory to achieve more durability and strength structures. The transverse displacements of the shell structures and stiffeners at the contact positions are assumed compatible. Numerical results of the ES-MITC3 element are compared with those of available other numerical results to demonstrate a good convergence and accuracy of the present method.

Fracture simulations using edge-based smoothed finite element method for isotropic damage model via Delaunay/Voronoi dual tessellations

In this study, an effective numerical framework for fracture simulations is proposed using the edge-based smoothed finite element method (ES-FEM) and isotropic damage model. The duality between the Delaunay triangulation and Voronoi tessellation is utilized for the mesh construction and the compatible use of the finite element solution with the Voronoi-cell lattice geometry. The mesh irregularity is introduced to avoid calculating the biased crack path by adding random variation in the nodal coordinates, and the ES-FEM elements are defined along the Delaunay edges. With the Voronoi tessellation, each nodal mass is calculated and the fractured surfaces are visualized along the Voronoi edges. The rotational degrees of freedom are implemented for each node by introducing the elemental formulation of the Voronoi-cell lattice model, and the accurate visualizations of the rotational motions in the Voronoi diagram are achieved. An isotropic damage model is newly incorporated into the ES-FEM formulation, and the equivalent elemental length is introduced with an additional geometric factor to simulate the consistent softening behaviors with reducing the mesh sensitivity. The full matrix form of the smoothed strain-displacement matrix is constructed for optimal use in the element-wise computations during explicit time integration, and parallel computing is implemented for the enhancement of the computational efficiency. The simulated results are compared with the theoretical solutions or experimental results, which demonstrates the effectiveness of the proposed methodology in the simulations of the quasi-brittle fractures.

An edge-based smoothed finite element for buckling analysis of functionally graded material variable-thickness plates

The paper aims to extend the ES-MITC3 element, which is an integration of the edge-based smoothed finite element method (ES-FEM) with the mixed interpolation of tensorial components technique for the three-node triangular element (MITC3 element), for the buckling analysis of the FGM variable-thickness plates subjected to mechanical loads. The proposed ES-MITC3 element is performed to eliminate the shear locking phenomenon and to enhance the accuracy of the existing MITC3 element. In the ES-MITC3 element, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains formed by two adjacent MITC3 triangular elements sharing the same edge. The numerical results demonstrated that the proposed method is reliable and more accurate than some other published solutions in the literature. The influences of some geometric parameters, material properties on the stability of FGM variable-thickness plates are examined in detail.

Exploring three-dimensional edge-based smoothed finite element method based on polyhedral mesh to study elastic mechanics problems

This paper established the three-dimensional edge-based smoothed finite element method(ES-FEM) based on polyhedral mesh, divided the smoothed domain, constructed the shape function and derived the geometric matrix and the stiffness matrix. The MATLAB software was used to prepare the corresponding computing programs, with which the paper studied the stress distribution of a hollow sphere model and a beam model under different numbers of polyhedral elements. The paper compared the calculation results from the conventional finite element methods(FEM) that use tetrahedral elements and hexahedral elements respectively in terms of stress relative error and energy relative error. The comparison results show that the three-dimensional ES-FEM based on polyhedral mesh has better precision and convergence than the conventional FEM and better adaptability to complex geometric structures.

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.

Free vibration of functionally graded porous non-uniform thickness annular-nanoplates resting on elastic foundation using ES-MITC3 element

In this paper, the free vibration of the functionally graded porous (FGP) non-uniform annular-nanoplates lying on Winkler foundation (WF) is studied by using the smoothed finite element method based on the first-order shear deformation theory (FSDT). The combination of the mixed interpolation of the tensorial components for the three-node triangular element (MITC3 element) and the edge-based smoothed finite element method (ES-FEM) creates the ES-MITC3 element. This element is employed to avoid the shear locking problem as well as to improve the accuracy of the original MITC3 element. The small-scale effect is considered based on the nonlocal theory. Applying Hamilton's principle, the governing equation of the FGP non-uniform thickness annular-nanoplate is derived. Material properties of the nanoplate are characterized by two parameters: power-law index (k) and maximum porosity distributions (Ω) in the forms of cosine functions. The results of the present work are compared with other published work to verify accuracy and reliability. Moreover, the effects of geometry parameters and material properties on the free vibration of FGP non-uniform annular-nanoplates are comprehensively investigated.

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

Coupling edge-based smoothed finite element method with smoothed particle hydrodynamics for fluid structure interaction problems

Numerical simulation of fluid structure interaction (FSI) problems is one of the most challenging topics in computational fluid dynamics. In this paper, coupling edge-based smoothed finite element method (ES-FEM) and smoothed particle hydrodynamics (SPH) method (ES-FEM-SPH) is proposed for solving FSI problems, where the edge-based smoothed finite element method is used to model the movement and deformation of structures, and the smoothed particle hydrodynamics is used to model the fluid flow. In ES-FEM, the gradient smoothing technique is applied over the smoothing domain and it can effectively overcome the ‘‘overly-stiff’’ effect in conventional FEM model. Some correction algorithms including density correction, kernel gradient correction and particle shift technique are integrated into the SPH method to improve computational stability and accuracy. A virtual particle coupling scheme is used to implement the coupling of ES-FEM and SPH with complex geometry interface. As ES-FEM is more accurate than conventional FEM, and it is expected that this ES-FEM-SPH coupling approach should be superior than existing FEM-SPH coupling approaches. A number of test examples with FSI are investigated with the presented ES-FEM-SPH, and compared with results from other approaches including FEM-SPH. From the obtained numerical results, we can conclude that the ES-FEM-SPH coupling approach is effective to simulate FSI problems.

An accurate edge‐based FEM for electromagnetic analysis with its applications to multiscale structures

AbstractThis paper introduces an accurate edge‐based smoothed finite element method (ES‐FEM) for electromagnetic analysis for both two‐dimensional cylindrical and three‐dimensional Cartesian systems, which shows much better performance in terms of accuracy and numerical stability for mesh distortion compared with the traditional FEM. Unlike the traditional FEM, the computational domain in ES‐FEM is divided into non‐overlapping smoothing domains associated with each edge of elements, triangles in two‐dimensional domain and tetrahedrons in three‐dimensional domain. Then, the gradient smoothing technique is used to smooth the gradient components in the stiff matrix of the FEM. Several numerical experiments are carried out to validate its accuracy and numerical stability. Results show that the ES‐FEM can obtain much more accurate results compared with the traditional FEM and is almost independent of mesh distortion.