Smoothing Domains Research Articles

A smoothed finite element method for octree-based polyhedral meshes with large number of hanging nodes and irregular elements

Techniques are now available to generate high-quality polyhedral elements offering a good alternative way for accurate stress analysis for solids. However, distorted polyhedral elements cannot be effectively used in standard finite element methods (FEM) without extra complicated treatments, due to the hanging nodes on the element edges and complex shape of polyhedral elements. On the contrary, the hanging nodes and complex shape of the elements can be naturally dealt with in the smoothed finite element method (S-FEM). This paper, utilizes the octree method to automatically and efficiently generate unstructured polyhedral meshes, using the standard tessellation language (STL). We develop three S-FEM models for octree meshes, using all existing node-based, edge-based and face-based smoothing domains. A subdomain smoothing technique is developed to re-cut the smoothing domain (SD), which simplifies the process of smoothing domain formation. The shapes of the subdomains have some common geometries that are much easier to handle, making use of the good features offered by the S-FEM formulation. The introduction of subdomains does not affect the results of the calculations, and the properties of the three S-FEMs are still preserved. Several demonstration examples are presented to verify the efficiency of the present S-FEM models.

A local Lagrangian gradient smoothing method for fluids and fluid-like solids: A novel particle-like method

The Lagrangian Gradient Smoothing Method (L-GSM) proposed in our earlier work overcomes effectively the ‘tensile instability’ problem inherently existed in the widely used smoothed particle hydrodynamics (SPH) method. However, the employment of background grid in L-GSM for the construction of gradient smoothing domains leads some drawbacks to the L-GSM framework: low computational efficiency, complex implementation procedure and technical challenges for parallel computing algorithm development. Accordingly, this study proposes a novel particle-like method, termed as local L-GSM (LL-GSM), through constructing gradient smoothing domains locally for LL-GSM particles. The present LL-GSM consists of three unique ingredients: (1) Only locally constructed gradient smoothing domains are used; (2) an efficient localized neighbor-searching algorithm is developed for the search of supporting particles; (3) a simple and effective free surface technique is adopted for accurate application of free surface effect. The accuracy, stability and efficiency of the newly proposed LL-GSM framework are then investigated comprehensively by conducting thorough theoretical and numerical analyses. At last, benchmark problems including two incompressible flows and two free surface flows are used to verify the capability of LL-GSM in handling large deformation of fluids and fluid-like solids. The LL-GSM results are evaluated carefully by comparing with experimental results, theoretical solutions, and numerical solutions. Results comparison demonstrates that the proposed LL-GSM method can give very accurate numerical solutions in all these problems with a much better computational efficiency and easier implementation. It is also evidenced that, same to L-GSM, the LL-GSM is free from the ‘tensile instability’ issue.

The Equilibrium Cell-Based Smooth Finite Element Method for Shakedown Analysis of Structures

This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cell-based smoothing technique based on elements of discretized mesh. Fictitious elastic stresses are also determined within the framework of finite element method (CS-FEM)-based Galerkin procedure, and equilibrium equations for residual stresses are satisfied in an average sense at every cell-based smoothing cell. All constrains are imposed at only one point in the smoothing domains, instead of Gauss points as in a standard FEM-based procedure. The resulting optimization problem is then handled using the highly efficient solvers. Various numerical examples are investigated, and obtained solutions are compared with available results in the literature.

An Edge-Based Smoothed Finite Element Method for Analyzing Stiffened Plates

In this paper, an edge-based smoothed finite element method with the discrete shear gap using triangular elements (ES-DSG3) is presented for static, free vibration and sound radiation analyses of plates stiffened by eccentric and concentric stiffeners. In the present model, the ES-DSG3 for the plate element with the isoparametric thick-beam element is employed to formulate stiffened plate structures. The deflections and rotations of the plates and the stiffeners are connected at tying positions. By using Rayleigh integral, sound radiation of stiffened plates subjected to a point load can be obtained. The edge-based gradient smoothing technique is employed to perform the related numerical integrations over the edge-based smoothing domains. Compared with the original DSG3 model, the present ES-DSG3 model is relatively softer as a result of the edge-based gradient smoothing technique. From several numerical examples, it is observed that the ES-DSG3 can produce more accurate numerical solutions than the original DSG3 for stiffened plates.

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.

An Edge-Based Smoothed Discrete Shear Gap Method for Static and Free Vibration Analyses of FG-CNTRC Plates

In this paper, an edge-based smoothed stabilized discrete shear gap method (ES-DSG) is integrated with the C0-type high-order shear deformation plate theory (C0-HSDT) for free vibration and static analyses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates. The material properties of FG-CNTRC are assumed to be graded through the thickness direction according to several distributions of the volume fraction of carbon nanotubes (CNTs). The stiffness formulation of the ES-DSG based on C0-HSDT is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements. This hence does not require shear correction factors. The accuracy and reliability of the proposed method are confirmed in several numerical examples.

Fully coupled elastoplastic hydro‐mechanical analysis of unsaturated porous media using a meshfree method

SummaryThis paper presents the first application of an advanced meshfree method, ie, the edge‐based smoothed point interpolation method (ESPIM), in simulation of the coupled hydro‐mechanical behaviour of unsaturated porous media. In the proposed technique, the problem domain is spatially discretised using a triangular background mesh, and the polynomial point interpolation method combined with a simple node selection scheme is adopted for creating nodal shape functions. Smoothing domains are formed on top of the background mesh, and a constant smoothed strain, created by applying the smoothing operation over the smoothing domains, is assigned to each smoothing domain. The deformation and flow models are developed based on the equilibrium equation of the mixture, and linear momentum and mass balance equations of the fluid phases, respectively. The effective stress approach is followed to account for the coupling between the flow and deformation models. Further coupling among the phases is captured through a hysteretic soil water retention model that evolves with changes in void ratio. An advanced elastoplastic constitutive model within the context of the bounding surface plasticity theory is employed for predicting the nonlinear behaviour of soil skeleton. Time discretisation is performed by adopting a three‐point discretisation method with growing time steps to avoid temporal instabilities. A modified Newton‐Raphson framework is designed for dealing with nonlinearities of the discretised system of equations. The performance of the numerical model is examined through a number of numerical examples. The state‐of‐the‐art computational scheme developed is useful for simulation of geotechnical engineering problems involving unsaturated soils.

Highly accurate smoothed finite element methods based on simplified eight-noded hexahedron elements

Compared with the tetrahedron elements, hexahedron elements are preferred for their high accuracy. However, coordinate mapping required in the hexahedron elements of FEM formulation costs huge running time, leading to poor performance. Besides, the high quality of Jacobian matrix and mesh is required, which affects the accuracy of the strain results greatly. In order to solve these problems, we propose a novel simplified integration technique based on the smoothed finite element method (S-FEM) for the eight-noded hexahedron elements, where coordinate mapping is not demanded. The proposed new S-FEM-H8 models include simplified NS-FEM-H8 (using node-based smoothing domains) and simplified FS-FEM-H8 (using face-based smoothing domains). In the work, we divide a quadrilateral surface segment of a smoothing domain into two triangular sub-segments, so that the strain-displacement matrix can be calculated using a simple summation in the S-FEM theory instead of the integration in FEM. Then we conduct the Gauss integration scheme in each triangular surface sub-segment in order to avoid the coordinate mapping required in quadrilateral surface segments. The rest solving algorithm is the same as the standard S-FEM. Intensive numerical examples demonstrate that the simplified S-FEM-H8 possess the following features: (1) The strain energy of simplified NS-FEM-H8 is an upper bound of the exact solutions; (2) The simplified NS-FEM-H8 can overcome the volume locking problems for incompressible materials; (3) The method of dividing boundary surface into two triangular surfaces in smoothing domain keeps nearly the same accuracy as the standard S-FEM-H8.

Geometrically nonlinear analysis of functionally graded shells using an edge-based smoothed MITC3 (ES-MITC3) finite elements

An edge-based smoothed finite element method (ES-FEM) combined with the mixed interpolation of tensorial components technique (MITC) for triangular elements, named as ES-MITC3, was recently proposed to enhance the accuracy of the original MITC3 for analysis of shells. In this study, the ES-MITC3 is extended to the geometrically nonlinear analysis of functionally graded shells. In the ES-MITC3, the stiffness matrices are obtained using the strain smoothing technique over the smoothing domains that formed by two adjacent MITC3 triangular shell elements sharing an edge. The material properties of functionally graded (FG) shells are assumed to vary through the thickness direction by a power rule distribution of volume fractions of the constituents. The nonlinear finite element formulation based on the first-order shear deformation theory with the von Karman’s large deflection assumption is used to describe the large deformations of the FG shells. Several numerical examples are given to demonstrate the performance of the present approach in comparison with other existing methods.

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.

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This paper presents the results of the geometric nonlinear analysis of composite shell subjected to static load by using an edge-based smoothed finite elements (ES) and the mixed interpolation of tensorial components technique (MITC). The shell is decrete by three-node triangular elements inwhich the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains formed by two adjacent triangular elements sharing an edge. Equilibrium equation system of the composite shell is formed by the first order shear deformation theory with the geometrical nonlinearity in von Kármán strains. Numerical results show good accuracy and convergence as compared to other existing methods in the literature when employed to analyze the nonlinear static composite shells.

Implementing the Node Based Smoothed Finite Element Method as User Element in Abaqus for Linear and Nonlinear Elasticity

In this paper, the node based smoothed-strain Abaqus user element (UEL) in the framework of finite element method is introduced. The basic idea behind of the node based smoothed finite element (NSFEM) is that finite element cells are divided into subcells and subcells construct the smoothing domain associated with each node of a finite element cell [Liu, Dai and Nguyen-Thoi (2007)]. Therefore, the numerical integration is globally performed over smoothing domains. It is demonstrated that the proposed UEL retains all the advantages of the NSFEM, i.e., upper bound solution, overly soft stiffness and free from locking in compressible and nearly-incompressible media. In this work, the constant strain triangular (CST) elements are used to construct node based smoothing domains, since any complex two dimensional domains can be discretized using CST elements. This additional challenge is successfully addressed in this paper. The efficacy and robustness of the proposed work is obtained by several benchmark problems in both linear and nonlinear elasticity. The developed UEL and the associated files can be downloaded from https://github.com/nsundar/NSFEM.

A Gradient Stable Node‐Based Smoothed Finite Element Method for Solid Mechanics Problems

This paper presents a gradient stable node‐based smoothed finite element method (GS‐FEM) which resolves the temporal instability of the node‐based smoothed finite element method (NS‐FEM) while significantly improving its accuracy. In the GS‐FEM, the strain is expanded at the first order by Taylor expansion in a node‐supported domain, and the strain gradient is then smoothed within each smoothing domain. Therefore, the stiffness matrix includes stable terms derived by the gradient of the strain. The GS‐FEM model is softer than the FEM but stiffer than the NS‐FEM and yields far more accurate results than the FEM‐T3 or NS‐FEM. It even has comparative accuracy compared with those of the FEM‐Q4. The GS‐FEM owns no spurious nonzero‐energy modes and is thus temporally stable and well‐suited for dynamic analyses. Additionally, the GS‐FEM is demonstrated on static, free, and forced vibration example analyses of solids.

A Hybrid Smoothed Finite Element Method for Predicting the Sound Field in the Enclosure with High Wave Numbers

Wave‐based methods for acoustic simulations within enclosures suffer the numerical dispersion and then usually have evident dispersion error for problems with high wave numbers. To improve the upper limit of calculating frequency for 3D problems, a hybrid smoothed finite element method (hybrid SFEM) is proposed in this paper. This method employs the smoothing technique to realize the reduction of the numerical dispersion. By constructing a type of mixed smoothing domain, the traditional node‐based and face‐based smoothing techniques are mixed in the hybrid SFEM to give a more accurate stiffness matrix, which is widely believed to be the ultimate cause for the numerical dispersion error. The numerical examples demonstrate that the hybrid SFEM has better accuracy than the standard FEM and traditional smoothed FEMs under the condition of the same basic elements. Moreover, the hybrid SFEM also has good performance on the computational efficiency. A convergence experiment shows that it costs less time than other comparison methods to achieve the same computational accuracy.

Three Dimensional CS-FEM Phase-Field Modeling Technique for Brittle Fracture in Elastic Solids

The cell based smoothed finite element method (CS-FEM) was integrated with the phase-field technique to model brittle fracture in 3D elastic solids. The CS-FEM was used to model the mechanics behavior and the phase-field method was used for diffuse fracture modeling technique where the damage in a system was quantified by a scalar variable. The integrated CS-FEM phase-field approach provides an efficient technique to model complex crack topologies in three dimensions. The detailed formulation of our combined method is provided. It was implemented in the commercial software ABAQUS using its user-element (UEL) and user-material (UMAT) subroutines. The coupled system of equations were solved in a staggered fashion using the in-built non-linear Newton–Raphson solver in ABAQUS. Eight node hexahedral (H8) elements with eight smoothing domains were coded in CS-FEM. Several representative numerical examples are presented to demonstrate the capability of the method. We also discuss some of its limitations.

Eigenvalue analysis for plain-woven fabric structure using shell element and one smoothing cell in the smoothed finite element method

An efficient four-node quadrilateral (Q4) shell element based on the first-order shear deformation theory of plate (FSDT) and the strain smoothing technique in finite elements (referred as SFEM) was proposed for eigenvalue analysis of plain-woven fabric structure. A one smoothing domain (or cell) integration scheme in SFEM was proposed to evaluate the nodal train fields of Q4 shell elements. The numerical result of eigenvalue analysis, which was in the case of free vibration analysis, approximated to that one implemented in the finite element method (FEM) but gave a higher efficiency in computation in terms of central processing unit (CPU) time and numerical implementation.

A stable node-based smoothed finite element method for metal forming analysis

In this paper, a stable node-based smoothed finite element method (SNS-FEM) is presented for analyzing metal forming problems using linear triangular or tetrahedral elements. In present method, the numerical integration domains are approximately circular or spherical regions of the node-based smoothing domains generated by the node-based smoothed finite element method (NS-FEM). Four or six supplementary integration points, which are symmetrically located at the crossover points of the region and the coordinate axis, are employed for each node to form the stabilization items associated with the variance of smoothed shape function gradient. Through this operation without the introducing of any uncertain parameter, the SNS-FEM not only significantly cures the temporal instability of NS-FEM but also performs better in effectiveness and efficiency than FEM, which is well validated by several numerical examples containing benchmark cases. Additionally, a simple but effective contact algorithm including contact searching and contact force computation is presented.

A new strain smoothing method for triangular and tetrahedral finite elements

This paper presents a simple and effective strain smoothing method (the strain-smoothed element method) for 3-node triangular and 4-node tetrahedral solid finite elements. While piecewise constant strain fields are constructed through smoothing domains in previous strain smoothing methods, in the proposed method, linear strain fields are constructed within finite elements using constant strains of neighboring finite elements. When the new strain smoothing method is adopted, the triangular and tetrahedral solid finite elements pass all the basic tests; patch, isotropy and zero energy mode tests, and show improved predictive capability. The formulation of the proposed method is presented in detail. Through various numerical examples, we demonstrate the accuracy improvement achieved.

An edge-based smoothed finite element method for nonlinear magnetostatic and eddy current analysis

Abstract In this work, an edge-based smoothed finite element method (ES-FEM) is presented to solve electromagnetic field problems. First, the analysis domain is discretized into a set of tetrahedron cells that can be much easily generated automatically for complicated domains, and edge-based smoothing domains are further constructed associated with edges of tetrahedron elements. The Smoothed Galerkin Weak form is then evaluated based on edge-based smoothing domains. The ES-FEM is implemented here to solve nonlinear magnetostatic and eddy current problems, and to observe its properties using linear tetrahedral elements, in which relatively poor result is carried with standard finite element method (FEM). Numerical examples demonstrate that results obtained by the present ES-FEM are much more accurate than ones by standard FEM, and agree well with the experimental ones. It possesses potentials in the successful applications of electromagnetic problems.

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.