Gradient Smoothing Technique Research Articles

A modified smoothed finite element method (M-SFEM) for analyzing the band gap in phononic crystals

The present work proposes a novel modified smoothed finite element method (M-SFEM) to calculate the band structures of two-dimensional in-plane elastic waves in phononic crystals (PCs). Using the gradient smoothing technique over the cell-based smoothing domains, the cell-based gradient smoothing operation can offer ‘proper softening effects’ in SFEM modeling. According to the generalized integration rules, by simply shifting integration points to an unconventional location in the mass matrix, the accuracy of the M-SFEM model can be further improved. Numerical examples are presented for PCs using the proposed method for the computation of band gap (BG) frequency regions. The accuracy and efficiency of the modified SFEM are compared with those of the corresponding FEM and SFEM. The advantages of the modified SFEM for computing the BGs in PCs are discussed as compared to the conventional FEM. The results show the performance improvement of the M-SFEM compared to the FEM and SFEM.

Stochastic interval analysis for structural natural frequencies based on stochastic hybrid perturbation edge-based smoothing finite element method

Realistically calculating of natural frequencies is crucial for studying dynamical characteristics of structures in engineering. In this paper, the gradient smoothing technique is first applied into the conventional finite element method (named ES-FEM), to improve the accuracy of the deterministic calculation of natural frequencies. Then, the hybrid random and interval uncertain parameters are introduced into the proposed ES-FEM based on hybrid stochastic and interval perturbation method. Expressions for the mean value and variance of natural frequencies are derived by combining the random interval perturbation method and the random interval moment method. The upper and lower bounds of mean value and variance of natural frequencies are calculated based on hybrid perturbation vertex method. The proposed method (denoted ESHPVM) is compared with the different methods. The high accuracy and efficiency of the proposed methods are verified by two numerical examples.

Coupled Thermal–Electrical–Mechanical Inhomogeneous Cell-Based Smoothed Finite Element Method for Transient Responses of Functionally Graded Piezoelectric Structures to Dynamic Loadings

A coupled thermal–electrical–mechanical inhomogeneous cell-based smoothed finite element method (CICS-FEM) is presented for the multi-physics coupling problems, the displacements, the electrical potential and the temperature are obtained by combining the modified Wilson-[Formula: see text] method. By introducing the gradient smoothing technique into the FE model, the system stiffness of the model is reduced. In addition, due to the absence of mapping, CICS-FEM is insensitive to mesh distortion. Curves and contour plots of displacements, electrical potential and temperature of three FGP structures are given in the article. The results shows that CICS-FEM possesses several advantages: (i) insensitive to mesh distortion; (ii) reduce the system stiffness; (iii) convergent and accuracy; (iv) efficient than FEM when the results are at the same accuracy.

A cell-based smoothed radial point interpolation method using condensed shape functions for free and forced vibration analysis of solids

In this work, a novel cell-based smoothed radial point interpolation method (CS-RPIM) has been developed for free and forced vibration analysis of solids by using condensed radial point interpolation method (RPIM) shape functions with virtual points. In the scheme of the present method, the generalized gradient smoothing technique (GST) is used for constructing smoothed strains and the ingenious six virtual nodes are selected together with real nodes for creating condensed RPIM shape functions, which helps the proposed method avoid not only the overly-stiff problem existing in traditional finite element method (FEM) but also the overly-soft issue of original CS-RPIM models. Thanks to the well-constructed system stiffness, the proposed method has shown very good performance in terms of accuracy, efficiency and stability for the examples studied. As three-nodes triangular elements, that can be automatically generated for complex problem domains, are used as background mesh, the proposed method is supposed to possess good potential for solving practical engineering problems.

Stochastic hybrid perturbation technique-based smoothed finite element-statistical energy method for mid-frequency analysis of structure–acoustic systems with parametric and nonparametric uncertainties

At present, the most widely used mid-frequency method, i.e., the hybrid finite element-statistical energy analysis method (FE/SEA), still holds potential flaws, e.g., low accuracy and ignoring uncertain factors. In order to eliminate these problems, a generalized smoothing finite element statistical energy analysis method (SFE/SEA) is first proposed in this work, which is constructed based on the edge-based gradient smoothing technique, thus reducing the dispersion error in the deterministic calculations. Second, on the basis of the proposed SFE/SEA, the parametric uncertainty models will be introduced, combining the second-order random interval perturbation method and the random interval moment method, etc.; thus, a mid-frequency analysis method (SHPSFEM/SEA) considering both parametric uncertainties (both the random and interval variables) and nonparametric uncertainties is proposed, to effectively analyze the probability and interval characteristics of the random structural–acoustic coupled systems. Moreover, for the interval parameters with wide variation range, the subinterval perturbation method (S-SHPSFEM/SEA) is introduced by dividing the large interval parameters into small interval parameters. The proposed methods are compared with the Monte Carlo simulations of the hybrid FE/SEA model. The high accuracy and efficiency of the proposed methods are verified by two numerical examples.

A coupled FE-meshfree method for Helmholtz problems using point interpolation shape functions and edge-based gradient smoothing technique

In this paper, the point interpolation shape functions from a meshfree technique is combined with the generalized gradient smoothing technique (GGST) based on mesh grids to give a coupled FE-meshfree method for analyzing acoustic problems addressed by the Helmholtz equation. It is well-known that the numerical results of Helmholtz problems from several classical numerical approaches, such as the finite element approach, are not always sufficient to provide satisfying accuracy because of the pollution effect for relatively high wave numbers. The pollution effect may generally come from the “overly-stiff” property of the numerical models. Owing to the appropriate softening feature resulting from the edge-based gradient smoothing technique (EGST), the present coupled FE-meshfree method can effectively depress the pollution error effect of the numerical solutions. To avoid the singular issue of the moment matrix for the point interpolation shape functions used, several cell-based T-schemes are utilized for choosing nodes in the interpolation. Numerical results have validated that the present method could not only provide much more accurate solutions compared with the finite element method with the same node distribution, particularly at the relatively high frequency range, but also have higher computational efficiency with certain node selection schemes.

An inhomogeneous cell-based smoothed finite element method for the nonlinear transient response of functionally graded magneto-electro-elastic structures with damping factors

In this article, an inhomogeneous cell-based smoothed finite element method (ICS-FEM) was proposed to overcome the over-stiffness of finite element method in calculating transient responses of functionally graded magneto-electro-elastic structures. The ICS-FEM equations were derived by introducing gradient smoothing technique into the standard finite element model; a close-to-exact system stiffness was also obtained. In addition, ICS-FEM could be carried out with user-defined sub-routines in the business software now available conveniently. In ICS-FEM, the parameters at Gaussian integration point were adopted directly in the creation of shape functions; the computation process is simplified, for the mapping procedure in standard finite element method is not required; this also gives permission to utilize poor quality elements and few mesh distortions during large deformation. Combining with the improved Newmark scheme, several numerical examples were used to prove the accuracy, convergence, and efficiency of ICS-FEM. Results showed that ICS-FEM could provide solutions with higher accuracy and reliability than finite element method in analyzing models with Rayleigh damping. Such method is also applied to complex structures such as typical micro-electro-mechanical system–based functionally graded magneto-electro-elastic energy harvester. Hence, ICS-FEM can be a powerful tool for transient problems of functionally graded magneto-electro-elastic models with damping which is of great value in designing intelligence structures.

A valid inhomogeneous cell-based smoothed finite element model for the transient characteristics of functionally graded magneto-electro-elastic structures

For the sake of surmounting defect of over-stiffness of finite element model (FEM) and accurately solving the transient response problems of structures comprise functionally graded magneto-electro-elastic (FGMEE) materials, we put forward an inhomogeneous cell-based smoothed finite element model (ICS-FEM) and a modified Newmark method. By employing the inhomogeneous gradient smoothing technique into FEM, the mass matrix M and the equivalent stiffness matrix Keq are derived, ICS-FEM that provides a stiffness coinciding with the actual condition is also obtained. Moreover, this model can be carried out with user-defined subroutines in the existing FEM software. Several numerical examples including cantilever beams, a layered FGMEE sensor and an FGMEE energy harvester are analyzed, which prove that ICS-FEM could achieve results with higher accuracy and reliability than FEM. ICS-FEM are applied to more complex structures such as FGMEE layered sensor and energy harvester. Therefore, such method to solve the transient characteristics of FGMEE structures can be a reference for the design of smart structures.

The surrounding cell method based on the S-FEM for analysis of FSI problems dealing with an immersed solid

In this paper, a numerical method based on the cell-based smoothed finite element method is proposed to analyze three-dimensional fluid–solid interaction problems. Applying the gradient smoothing technique to the fluid and solid domain, the fully-coupled FSI formulation is achieved, and the analysis is performed by means of polyhedral smoothed finite elements. The element shape function based on the linear point interpolation leads to simple element formulation and a good geometric adaptability. Due to the properties of the smoothed finite element method, it is possible to connect non-matching meshes by polyhedral elements and provide seamless connection without overlapping or gap satisfying the interfacial conditions of continuity and force equilibrium, which is verified through the patch test. In addition, a treatment for dealing with moving boundaries in FSI problems with an immersed solid is proposed in the framework of local remeshing. By adopting a surrounding cell around a submerged solid, ill-shaped meshes occurring due to solid motion is reconstructed effectively. The effectiveness of the proposed scheme is illustrated through numerical examples in comparison with some previous works.

An effective cell-based smoothed finite element model for the transient responses of magneto-electro-elastic structures

To overcome the over-stiffness and the imprecise magneto-electro-elastic coupling effects of finite element model, we presented a cell-based smoothed finite element model to more accurately simulate the transient responses of magneto-electro-elastic structures. In the cell-based smoothed finite element model, the gradient smoothing technique was introduced into a magneto-electro-elastic multi-physical-field finite element model. The cell-based smoothed finite element model can achieve a close-to-exact stiffness of the continuum structures which could automatically discrete elements for complicated regions more readily and thus remarkably reduced the numerical errors. In addition, the modified Wilson- θ method was presented for solving the motion equation of magneto-electro-elastic structures. Several numerical examples were investigated and exhibited that the cell-based smoothed finite element model could receive more accurate and reliable simulation results than the standard finite element model. Besides, the cell-based smoothed finite element model was employed to calculate transient responses of magneto-electro-elastic sensor and typical micro-electro-mechanical systems–based magneto-electro-elastic energy harvester. Therefore, the cell-based smoothed finite element model can be adopted to tackle the practical magneto-electro-elastic problems such as smart vibration transducers, magnetic field sensors, and energy harvester devices in intelligent magneto-electro-elastic structures systems.

Application of Smoothed Finite Element Method to Two-Dimensional Exterior Problems of Acoustic Radiation

In this work, the smoothed finite element method using four-node quadrilateral elements (SFEM-Q4) is employed to resolve underwater acoustic radiation problems. The SFEM-Q4 can be regarded as a combination of the standard finite element method (FEM) and the gradient smoothing technique (GST) from the meshfree methods. In the SFEM-Q4, only the values of shape functions (not the derivatives) at the quadrature points are needed and the traditional requirement of coordinate transformation procedure is not necessary to implement the numerical integration. Consequently, no additional degrees of freedom are required as compared with the original FEM. In addition, the original “overly-stiff” FEM model for acoustic problems (governed by the Helmholtz equation) is properly softened due to the gradient smoothing operations implemented over the smoothing domains and the present SFEM-Q4 possesses a relatively appropriate stiffness of the continuous system. Therefore, the well-known numerical dispersion error for Helmholtz equation is decreased significantly and very accurate numerical solutions can be obtained by using relatively coarse meshes. In order to truncate the unbounded domains and employ the domain-based numerical method to tackle the acoustic radiation in unbounded domains, the Dirichlet-to-Neumann (DtN) map is used to ensure that there are no spurious reflections from the far field. The numerical results from several numerical examples demonstrate that the present SFEM-Q4 is quite effective to handle acoustic radiation problems and can produce more accurate numerical results than the standard FEM.

Application of the edge-based gradient smoothing technique to acoustic radiation and acoustic scattering from rigid and elastic structures in two dimensions

As is well-known to all, the conventional finite element method (FEM) is constrained by the “numerical dispersion error” issue for solving acoustic problems at high frequencies. In this paper, the gradient smoothing technique (GST) which is based on the edges of the elements is combined with the conventional FEM to construct a novel edge-based smoothed FEM (ES-FEM) for two dimensional exterior structural–acoustic problems. The smoothed gradient field is obtained by performing the GST over the obtained smoothing domains (SDs). The present ES-FEM is able to provide a relatively appropriate stiffness of the real system owing to the “softening effects” from the GST. Therefore, the accuracy of the numerical solutions can be significantly improved. To effectively handle the exterior Helmholtz equation in unbounded domains, a predefined artificial boundary B is employed to obtain a bounded computational domain and the well-known Dirichlet-to-Neumann (DtN) map is used to prevent the possible reflections from the far-field. Several supporting numerical examples indicated that the ES-FEM with DtN map was very effective for exterior structural–acoustic problems and could produce more accurate numerical results than the conventional FEM.

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method (NS-PIM), for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection. This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations, but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.

A Lagrangian gradient smoothing method for solid‐flow problems using simplicial mesh

SummaryA novel Lagrangian gradient smoothing method (L‐GSM) is developed to solve “solid‐flow” (flow media with material strength) problems governed by Lagrangian form of Navier‐Stokes equations. It is a particle‐like method, similar to the smoothed particle hydrodynamics (SPH) method but without the so‐called tensile instability that exists in the SPH since its birth. The L‐GSM uses gradient smoothing technique to approximate the gradient of the field variables, based on the standard GSM that was found working well with Euler grids for general fluids. The Delaunay triangulation algorithm is adopted to update the connectivity of the particles, so that supporting neighboring particles can be determined for accurate gradient approximations. Special techniques are also devised for treatments of 3 types of boundaries: no‐slip solid boundary, free‐surface boundary, and periodical boundary. An advanced GSM operation for better consistency condition is then developed. Tensile stability condition of L‐GSM is investigated through the von Neumann stability analysis as well as numerical tests. The proposed L‐GSM is validated by using benchmarking examples of incompressible flows, including the Couette flow, Poiseuille flow, and 2D shear‐driven cavity. It is then applied to solve a practical problem of solid flows: the natural failure process of soil and the resultant soil flows. The numerical results are compared with theoretical solutions, experimental data, and other numerical results by SPH and FDM to evaluate further L‐GSM performance. It shows that the L‐GSM scheme can give a very accurate result for all these examples. Both the theoretical analysis and the numerical testing results demonstrate that the proposed L‐GSM approach restores first‐order accuracy unconditionally and does not suffer from the tensile instability. It is also shown that the L‐GSM is much more computational efficient compared with SPH, especially when a large number of particles are employed in simulation.

A smoothed finite element method for exterior Helmholtz equation in two dimensions

In this work, a smoothed finite element method (SFEM), in which the gradient smoothing technique (GST) from meshfree methods is incorporated into the standard Galerkin variational equation, is proposed to handle the acoustic wave scattering by the obstacles immersed in water. In the SFEM model, only the values of shape functions, not the derivatives at the quadrature points, are required and no coordinate transformation is needed to perform the numerical integration. Due to the softening effects provided by the GST, the original “overly-stiff” FEM model has been properly softened and a more appropriate stiffness of the continuous system can be obtained, then the numerical dispersion error for the acoustic problems is decreased conspicuously and the quality of the numerical solutions can be improved significantly. To tackle the exterior Helmholtz equation in unbounded domains, we use the well-known Dirichlet-to-Neumann (DtN) map to guarantee that there are no spurious reflecting waves from the far field. Numerical tests show that the present SFEM cum DtN map (SFEM-DtN) works well for exterior Helmholtz equation and can provide better solutions than standard FEM.

Hybrid gradient smoothing technique with discrete shear gap method for shell structures

In order to enhance the performance of the discrete shear gap technique (DSG) for shell structures, the coupling of hybrid gradient smoothing technique (H-GST) with DSG using triangular elements (HS-DSG3) is presented to solve the governing partial differential equations of shell structures. In the formulation HS-DSG3, we firstly employ the node-based gradient smoothing technique (N-GST) to obtain the node-based smoothed strain field, then a scale factor α∈[0,1] is used to reconstruct a new strain field which includes both the strain components from standard DGS3 and the strain components from node-based smoothed DSG3 (NS-DSG3). The HS-DSG3 takes advantage of the “overly-soft” NS-DSG3 model and the “overly-stiff” DSG3 model, and has a relatively appropriate stiffness of the continuous system. Therefore, the degree of the solution accuracy can be improved significantly. Several typical benchmark numerical examples have been investigated and it is demonstrated that the present HS-DSG3 can provide better numerical solutions than the original DSG3 for shell structures.

Application of smoothed finite element method to acoustic scattering from underwater elastic objects

In this work, the smoothed finite element method (S-FEM) is employed to solve the acoustic scattering from underwater elastic objects. The S-FEM, which can be regarded as a combination of the standard finite element method (FEM) and the gradient smoothing technique (GST) from the meshless methods, was initially proposed for solid mechanics problems and has been demonstrated to possess several superior properties. In the S-FEM, the smoothed gradient fields are acquired by performing the GST over the obtained smoothing domains. Due to the proper softening effects provided by the gradient smoothing operations, the original “overly-stiff” FEM model is softened and the present S-FEM possesses a relatively appropriate stiffness of the continuous system. Therefore, the quality of the numerical results can be significantly improved. The numerical results from several typical numerical examples demonstrate that the S-FEM is quite effective to handle acoustic scattering from underwater elastic objects and can provide more accurate numerical results than the standard FEM.

Polygonal type variable-node elements by means of the smoothed finite element method for analysis of two-dimensional fluid-solid interaction problems in viscous incompressible flows

A cell-based smoothed finite element method is proposed to make a polygonal type variable-node element for analyzing two-dimensional FSI problems. The gradient smoothing technique for the velocity field provides a proper softening effect, which relieves the stiff behavior of the conventional FEM. In addition, by using polygonal elements, a seamless connection is achieved between non-matching meshes satisfying the interface conditions of continuity and compatibility. In order to prevent mesh distortion along the fluid-solid interface, a sliding polygonal mesh was developed using the cell-based smoothed finite element method. Some numerical examples were presented to assess the performance and the effectiveness of the present method. The smoothed finite element methods yielded better accuracy compared with the conventional FEM, and the sliding polygonal mesh method successfully treated mesh distortion.

Numerical investigation of the edge-based gradient smoothing technique for exterior Helmholtz equation in two dimensions

In order to improve the accuracy of the standard finite element method (FEM) solutions for Helmholtz equation and reduce the numerical error, an edge-based gradient smoothing technique (E-GST) is incorporated into the standard finite element method to give an ES-FEM for the exterior Helmholtz equation in two dimensions. In the ES-FEM, the smoothing domains are formed by sequentially linking the terminal points of each edge and the centers of the neighboring elements. Then the E-GST is used to smooth the gradient on these smoothing domains. Due to the softening effect of the E-GST, the “overly-stiff” property of the original finite element model can be properly softened and the accuracy of the solution will be improved significantly. With the aim to handle the exterior Helmholtz problems in infinite domain, we introduce an artificial boundary B to obtain a finite computational domain and the Dirichlet-to-Neumann (DtN) map is used to guarantee that all the waves on B are outgoing. Several typical numerical examples are considered to assess the effectivity and feasibility of the ES-FEM with the DtN map. It is shown that the present model works well for exterior Helmholtz equation and can produce better solutions than standard FEM.

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.