Node-based Smoothed Finite Element Method Research Articles

A coupling approach of state-based peridynamics with node-based smoothed finite element method

In this work, a novel approach to couple ordinary state-based peridynamics (OSPD) with node-based smoothed finite element method (NS-FEM) is proposed. In present method, the solution domain is partitioned into two regions, one is discretized by OSPD, the other by NS-FEM, and more importantly, no transition region is introduced. The physical information is transmitted mutually from local to non-local regions, which is governed by the unified coupling equations of motion. The coupling takes full advantage of the generality of OSPD and the efficiency of NS-FEM. The parts of regions where damage and fracture either exist or are expected to propagate are described by OSPD, and the rest of regions are described by NS-FEM to reduce the computational cost and surface effect. Additionally, the critical bond work in OSPD is assumed to depend on the bond length, which is derived by the relation with the critical energy release rate in this study. Several numerical examples involving crack propagation are investigated under either dynamic or quasi-static conditions and satisfactory results have been obtained demonstrating the validity and efficiency of the proposed coupling approach.

Coupled-field simulation of electromagnetic tube forming process using a stable nodal integration method

This paper proposes a stable nodal integration method for analyzing coupling problems of electromagnetic field and mechanical field using linear triangular mesh. For transient electromagnetic field analysis, the node-based smoothed solution is firstly formulated to compute system matrices, and then the equivalent smoothing domain and temporal integration points are constructed to achieve a stable nodal integration method (SNIM), whose coefficient matrix is finally computed using the smoothed shape function derivatives together with the variance terms over the equivalent smoothing domain associated with nodes of the mesh. For dynamic large-deformation analysis, the node-based smoothed finite element method (NS-FEM) is utilized after testifying its stability and effectiveness in thin-walled structures of electromagnetic forming (EMF) process. Furthermore, by treating the morphing region as a static mechanical problem, a weighted elastomer method is presented to accomplish the mesh updating in sequential coupling simulation. Finally, it turns out that the proposed methodologies can be applied successfully to simulate EMF process, and numerical examples for both tube bulging and tube compression are investigated to demonstrate the validity, accuracy and efficiency of the proposed methods.

Lower Bound of Vibration Modes Using the Node-Based Smoothed Finite Element Method (NS-FEM)

The smoothed finite element method (S-FEM) has been recently developed as an effective solver for solid mechanics problems. This paper represents an effective approach to compute the lower bounds of vibration modes or eigenvalues of elasto-dynamic problems, by making use of the important softening effects of node-based S-FEM (NS-FEM). We first use NS-FEM, FEM and the analytic approach to compute the eigenvalues of transverse free vibration in strings and membranes. It is found that eigenvalues by NS-FEM are always smaller than those by FEM and the analytic method. However, NS-FEM produces spurious unphysical modes because of overly soft behavior. A technique is then proposed to remove them by analyzing their vibration shapes (eigenvectors). It is observed that spurious modes with excessively large wave numbers, which are unrelated to the physical deflection shapes but related to the discretization density, therefore can be easily removed. The final results of NS-FEM become the lower bound of eigenvalues and the accuracy can be improved via mesh refinement. And NS-FEM solutions (softer) are more reliable, because the large wave number component can be used as an indicator, which is available in FEM (stiffer), on the quality of the numerical solutions. The proposed NS-FEM procedure offers a viable and practical computational means to effectively compute the lower bounds of eigenvalues for solid mechanics problems.

Stability of twin circular tunnels in cohesive-frictional soil using the node-based smoothed finite element method (NS-FEM)

This paper presents an upper bound limit analysis procedure using the node-based smoothed finite element method (NS-FEM) and second order cone programming (SOCP) to evaluate the stability of twin circular tunnels in cohesive-frictional soils subjected to surcharge loading. At first stage, kinematically admissible displacement fields of the tunnel problems are approximated by NS-FEM using triangular elements (NS-FEM-T3). Next, commercial software Mosek is employed to deal with the optimization problems, which are formulated as second order cone. Collapse loads as well as failure mechanisms of plane strain tunnels are obtained directly by solving the optimization problems. For twin circular tunnels, the distance between centers of two parallel tunnels is the major parameter used to determine the stability. In this study, the effects of mechanical soil properties and the ratio of tunnel diameter and the depth to the tunnel stability are investigated. Numerical results are verified with those available to demonstrate the accuracy of the proposed method.

Strain smoothing for compressible and nearly-incompressible finite elasticity

We present a robust and efficient form of the smoothed finite element method (S-FEM) to simulate hyperelastic bodies with compressible and nearly-incompressible neo-Hookean behaviour. The resulting method is stable, free from volumetric locking and robust on highly distorted meshes. To ensure inf-sup stability of our method we add a cubic bubble function to each element. The weak form for the smoothed hyperelastic problem is derived analogously to that of smoothed linear elastic problem. Smoothed strains and smoothed deformation gradients are evaluated on sub-domains selected by either edge information (edge-based S-FEM, ES-FEM) or nodal information (node-based S-FEM, NS-FEM). Numerical examples are shown that demonstrate the efficiency and reliability of the proposed approach in the nearly-incompressible limit and on highly distorted meshes. We conclude that, strain smoothing is at least as accurate and stable, as the MINI element, for an equivalent problem size.

Smoothed finite element method for topology optimization involving incompressible materials

It is well known that the finite element method (FEM) suffers severely from the volumetric locking problem for incompressible materials in topology optimization owing to its numerical ‘overly stiff’ property. In this article, two typical smoothed FEMs with a certain softened effect, namely the node-based smoothed finite element method (NS-FEM) and the cell-based smoothed finite element method, are formulated to model the compressible and incompressible materials for topology optimization. Numerical examples have demonstrated that the NS-FEM with an ‘overly soft’ property is fairly effective in tackling the volumetric locking problem in topology optimization when both compressible and incompressible materials are involved.

Hybrid smoothed finite element method for two-dimensional underwater acoustic scattering problems

It is well known that the standard finite element method (FEM) is unreliable to solve acoustic problems governed by the Helmholtz equation with large wave numbers due to the “overly-stiff” nature of the FEM. In order to overcome this shortcoming, the hybrid smoothed finite element method (HS-FEM) using triangular elements is presented for the two-dimensional underwater acoustic scattering problems. In the HS-FEM, a scale factor α∈ [0, 1] is introduced to establish the area-weighted gradient field that contains contributions from both the standard FEM and the node-based smoothed finite element method (NS-FEM). The HS-FEM can provide a close-to-exact stiffness of the continuous system, thus the numerical dispersion error can be significantly decreased. To handle the underwater acoustic scattering problems in an infinite fluid medium, the bounded computational domain is obtained by introducing an artificial boundary on which the Dirichlet-to-Neumann (DtN) condition is imposed. Several numerical examples are investigated and the results showed that HS-FEM can provide more accurate solutions than the standard FEM. Therefore, the present method can be applied to practical underwater acoustic scattering problems such as sonar mine-hunting and sonar detection in ocean acoustics.

Topology Optimization Using Node-Based Smoothed Finite Element Method

The node-based smoothed finite element method (NS-FEM) proposed recently has shown very good properties in solid mechanics, such as providing much better gradient solutions. In this paper, the topology optimization design of the continuum structures under static load is formulated on the basis of NS-FEM. As the node-based smoothing domain is the sub-unit of assembling stiffness matrix in the NS-FEM, the relative density of node-based smoothing domains serves as design variables. In this formulation, the compliance minimization is considered as an objective function, and the topology optimization model is developed using the solid isotropic material with penalization (SIMP) interpolation scheme. The topology optimization problem is then solved by the optimality criteria (OC) method. Finally, the feasibility and efficiency of the proposed method are illustrated with both 2D and 3D examples that are widely used in the topology optimization design.

Hybrid smoothed finite element method for two dimensional acoustic radiation problems

It is well known that one key difficulty of solving the boundary-value problems governed by the Helmholtz equation using standard finite element method (FEM) is the loss of accuracy with increasing wave number due to the “numerical dispersion error”. In order to overcome this issue, the hybrid smoothed finite element method (HS-FEM) using linear triangular elements is presented to analyze two dimensional radiation problems. An important feature of HS-FEM is the introduction of a scale factor α∈ [0, 1] which is designed to establish the area-weighted strain field that contains contributions from both the standard FEM and the node-based smoothed finite element method (NS-FEM). The gradient smoothing technique used in the HS-FEM guarantees the numerical model can provide a close-to-exact stiffness to the continuous system and hence significantly reduces the numerical dispersion error. To solve the acoustic radiation problems in an infinite fluid domain, the HS-FEM is combined with the Dirichlet-to-Neumann (DtN) boundary condition to give a HS-FEM-DtN model for two dimensional acoustic radiation problems. Several numerical examples are given and it is found that HS-FEM can provide more accurate results than FEM with the same mesh.

A stable nodal integration method with strain gradient for static and dynamic analysis of solid mechanics

A stable nodal integration method with strain gradient (SNIM-SG) for curing the temporal instability of node-based smoothed finite element method (NS-FEM) is proposed for dynamic problems using linear triangular and tetrahedron element. In each smoothing domain, except for considering the smoothed strain into the calculation of potential energy functional as NS-FEM, a term related to strain gradient is taken into account as a stabilization term. The proposed SNIM-SG can achieve appropriate system stiffness in strain energy between FEM and NS-FEM solutions and obtains quite favorable results in elastic and dynamic analysis. The accuracy and stability of SNIM-SG solution are studied through detailed analyzes of benchmark cases and practical engineering problems. In elastic-static analysis, it is found that SNIM-SG can provide higher accuracy in displacement field than the reference approaches do. In free vibration analysis, the spurious non-zero energy modes can be eliminated effectively owing to the fact that SNIM-SG solution strengths the original relatively soft NS-FEM, and SNIM-SG is confirmed to obtain fairly accurate natural frequency values in various examples. All in all, SNIM-SG cures the flaws of NS-FEM and enhances the dominant of nodal integration. Thus, the efficacy of the presented formulation in solving solid mechanics problems is well represented and clarified.

A stable node-based smoothed finite element method for acoustic problems

It is well-known that the classical “overly-soft” node-based smoothed finite element method (NS-FEM) fails to provide reliable results to the Helmholtz equation due to the “temporal instability”. To cure the fatal drawback of NS-FEM and reduce the dispersion error in computational acoustics, this paper proposed a stable node-based smoothed finite element method (SNS-FEM) for analyzing acoustic problems using linear triangular (for 2D space) and tetrahedral (for 3D space) elements that can be generated automatically for any complicated configurations. In the present formulation, the system stiffness matrix is computed using the smoothed acoustic pressure gradients together with the gradient variance items over the smoothing domains associated with nodes of element mesh. It turns out the addition of stabilization term makes the SNS-FEM possess an ideal stiffness, thus successfully cures the temporal instability and significantly reduces the dispersion error in acoustic problems. Numerical examples, including both benchmark cases and practical engineering problems, demonstrate that the SNS-FEM possesses the following important properties: (1) temporal stability; (2) super accuracy and super convergence; (3) higher computational efficiency; (4) insensitive to mesh distortion.

A combined scheme of edge-based and node-based smoothed finite element methods for Reissner–Mindlin flat shells

In this paper, a combined scheme of edge-based smoothed finite element method (ES-FEM) and node-based smoothed finite element method (NS-FEM) for triangular Reissner---Mindlin flat shells is developed to improve the accuracy of numerical results. The present method, named edge/node-based S-FEM (ENS-FEM), uses a gradient smoothing technique over smoothing domains based on a combination of ES-FEM and NS-FEM. A discrete shear gap technique is incorporated into ENS-FEM to avoid shear-locking phenomenon in Reissner---Mindlin flat shell elements. For all practical purpose, we propose an average combination (aENS-FEM) of ES-FEM and NS-FEM for shell structural problems. We compare numerical results obtained using aENS-FEM with other existing methods in the literature to show the effectiveness of the present method.

Hybrid smoothed finite element method for acoustic problems

It is well known that “overly-stiff” finite element (FE) model fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known “pollution error” caused by the numerical dispersion. In this paper, the hybrid smoothed finite element method (HS-FEM) using triangular (2D) and tetrahedron (3D) elements that can be generated automatically for any complicated domain is formulated to solve acoustic problems in order to overcome the shortcomings of standard finite element method (FEM). In the formulation, a parameter α is equipped into HS-FEM, and the strain field is further assumed to be the weighted average between compatible strains from standard FEM and smoothed strains from node-based smoothed finite element method (NS-FEM). The smoothing technique can provide a softening effect to the model and have a very close-to-exact stiffness of the continuous system and hence significantly improve the accuracy of solution in both structural and acoustic analyses. Intensive numerical studies have been conducted here to demonstrate the effectiveness of the HS-FEM.

An ultra-accurate hybrid smoothed finite element method for piezoelectric problem

An ultra-accurate hybrid smoothed finite element method (HS-FEM) is presented for the analysis of piezoelectric structures, in which the electrostatic equations governing piezoelectric problem are solved numerically with simplest triangular elements in 2D and tetrahedral elements in 3D. In the present method, the strain field is assumed to be the weighted average between compatible strains from finite element method (FEM) and smoothed strains from node-based smoothed finite element method (NS-FEM). Numerical results demonstrate that the proposed method possesses a novel bound solution in terms of strain energy and eigenfrequencies, which is very important for safety and reliability assessments of piezoelectric structural properties. In addition, the numerical results obtained from HS-FEM are much more accurate than the standard finite element method using the same of nodes. Furthermore, the computational efficiency of HS-FEM is much better than the FEM.

Smoothed finite element method with exact solutions in heat transfer problems

Node-based smoothed finite element method (NS-FEM) with the triangular elements in 2D and tetrahedral elements in 3D has been found capable to produce upper bound solutions in terms of equivalent energy for heat transfer problems attributable to its monotonic ‘softened’ behavior. In this paper, a hybrid smoothed finite element method (HS-FEM) which combines the temperature gradient of NS-FEM and FEM is further extended to solve heat transfer problems. A parameter α that controls the weight of NS-FEM and FEM is equipped into HS-FEM to ensure the stability and accuracy. The theoretical analysis has proved that the exact equivalent energy in heat transfer problem obtained from HS-FEM lies in between those from the compatible FEM solution and the NS-FEM solution. The numerical results for 2D with triangular elements and 3D with tetrahedral elements confirm that the present method provides the exact solution in terms of equivalent energy using very coarse mesh. In addition, the accuracy of temperature distribution in HS-FEM model is much more accurate compared with the standard FEM with the same number of degrees of freedom.

A new hybrid smoothed FEM for static and free vibration analyses of Reissner–Mindlin Plates

A new smoothed finite element method (S-FEM) is proposed using hybrid smoothing operations based on nodes and edges of the mesh for static and free vibration analyses of plates governed by the Reissner---Mindlin plate theory. In the present approach, both the node-based smoothed finite element method (NS-FEM) and edge-based smoothed finite element method (ES-FEM) are utilized in a careful designed manner to overcome the shear locking. The formulations use 3-node triangular elements for easy automatic mesh creation, and linear interpolation functions are used for simplicity and robustness. The bending strains field are smoothed by the means of gradient smoothing technique over smoothing domains constructed by element edges, while the shear strains filed is smoothed based on the combination of NS-FEM and ES-FEM with a proper weightage controlled by a coefficient. A simple formula is developed for automatic selection of the coefficient by considering mesh size and thickness of the plate. For easy reference, the present technique is termed as NS+ES-FEM. The numerical examples demonstrate that the proposed method passes the shear-locking test and improves accuracy of the solution.

The verification of the quantities of interest based on energy norm of solutions by node-based smoothed finite element method

Verification of the quantities of interest computed with the finite element method (FEM) requires an upper bound on the strain energy, which is half of the energy norm of displacement solutions. Recently, a modified finite element method with strain smoothing, the node-based smoothed finite element method (NS-FEM), has been proposed to solve solid mechanics problems. It has been found in some cases that the energy norm formed by the smoothed strain of NS-FEM solutions bounds the energy norm of exact displacements from above. We analyze the bounding property of this method, give three kind of energy norms of solutions computed by FEM and NS-FEM, and extend them to the computation of an upper bound and a lower bound on the linear functional of displacements. By examining the bounding property of NS-FEM with different energy norms using some linear elastic problems, the advantages of NS-FEM over the traditional error estimate based methods is observed.

A temporal stable node-based smoothed finite element method for three-dimensional elasticity problems

A stabilized node-based smoothed finite element method (sNS-FEM) is formulated for three-dimensional (3-D) elastic-static analysis and free vibration analysis. In this method, shape functions are generated using finite element method by adopting four-node tetrahedron element. The smoothed Galerkin weak form is employed to create discretized system equations, and the node-based smoothing domains are used to perform the smoothing operation and the numerical integration. The stabilization term for 3-D problems is worked out, and then propose a strain energy based empirical rule to confirm the stabilization parameter in the formula. The accuracy and stability of the sNS-FEM solution are studied through detailed analyses of benchmark cases and actual elastic problems. In elastic-static analysis, it is found that sNS-FEM can provide higher accuracy in displacement and reach smoother stress results than the reference approaches do. And in free vibration analysis, the spurious non-zero energy modes can be eliminated effectively owing to the fact that sNS-FEM solution strengths the original relatively soft node-based smoothed finite element method (NS-FEM), and the natural frequency values provided by sNS-FEM are confirmed to be far more accurate than results given by traditional methods. Thus, the feasibility, accuracy and stability of sNS-FEM applied on 3-D solid are well represented and clarified.

A contact analysis approach based on linear complementarity formulation using smoothed finite element methods

Based on the subdomain parametric variational principle (SPVP), a contact analysis approach is formulated in the incremental form for 2D solid mechanics problems discretized using only triangular elements. The present approach is implemented for the newly developed node-based smoothed finite element method (NS-FEM), the edge-based smoothed finite element method (ES-FEM) as well as standard FEM models. In the approach, the contact interface equations are discretized by contact point-pairs using a modified Coulomb frictional contact model. For strictly imposing the contact constraints, the global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be readily solved using the Lemke method. This approach can simulate different contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is conducted to investigate the effects of various parameters and validate the proposed method. The numerical results have demonstrated the validity and efficiency of the present contact analysis approach as well as the good performance of the ES-FEM method, which provides solutions of about 10 times better accuracy and higher convergence rate than the FEM and NS-FEM methods. The results also indicate that the NS-FEM provides upper-bound solutions in energy norm, relative to the fact that FEM provides lower-bound solutions.

A HYBRID SMOOTHED FINITE ELEMENT METHOD (H-SFEM) TO SOLID MECHANICS PROBLEMS

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.