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In this paper, three different smoothed finite element method (SFEM), viz., node-based smoothed finite element method (NS-FEM), face-based smoothed finite element method (FS-FEM) and α -finite element method (α -FEM) are adopted for 3D solids undergoing large deformation. The common feature of all these techniques is the introduction of smoothed strain which is written as a weighted average of the compatible strain field over smoothing domains. The choice of smoothing domain is what differentiates them. The spatial discretization can be based on the simplest and automatically genera-table four-node tetrahedral elements and aforementioned techniques have shown to yield accurate results even on a coarser discretization. To take the advantages of the SFEM, it is beneficial to the FEM community to have it implemented in the widely used Abaqus ® software. Such an implementation is challenging because the neighboring SFEM elements are interconnected in the smoothed strain matrices in the elemental level. In this work, the above-mentioned SFEM models are implemented in the commercial software Abaqus using the softwares' user element (UEL) feature. The challenges during the definition and the assembly of the smoothing domains are effectively addressed in this work. The developed UEL and the associated files can be downloaded from https://github.com/nsundar/3DSFEM. The implementation is validated against benchmark examples and the robustness is demonstrated with complicated real-life problems, viz., tire patch contact with road and simulation of human thumb. [ABSTRACT FROM AUTHOR]

Our research introduces a novel advanced method for fluid computation in complicated domain. It makes use of an advanced polygonal finite element applying equal-order scheme; within each element. Both velocity and pressure fields are represented by a polygonal basis shape function system. This technique is based on a combination of the equal-order mixed scheme method, the polygon finite element method (PFEM) and the adapted local projection method that is used to stabilize the pressure. In order to simulate the fluid flow, MATLAB is utilized for coding and programming. A numerical example of an incompressible fluid flow governed by Stokes system in curved domain is performed to demonstrate the applicability of our developed polygon element. Besides, through the numerical analysis, we show the high flexibility of PFEM for curved domains. [ABSTRACT FROM AUTHOR]

This study adapts the flexible characteristic of meshfree method in analyzing three-dimensional (3D) complex geometry structures, which are the interlocking concrete blocks of step seawall. The elastostatic behavior of the block is analysed by solving the Galerkin weak form formulation over local support domain. The 3D moving least square (MLS) approximation is applied to build the interpolation functions of unknowns. The pre-defined number of nodes in an integration domain ranging from 10 to 60 nodes is also investigated for their effect on the studied results. The accuracy and efficiency of the studied method on 3D elastostatic responses are validated through the comparison with the solutions of standard finite element method (FEM) using linear shape functions on tetrahedral elements and the well-known commercial software, ANSYS. The results show that elastostatic responses of studied concrete block obtained by meshfree method converge faster and are more accurate than those of standard FEM. The studied meshfree method is effective in the analysis of static responses of complex geometry structures. The amount of discretised nodes within the integration domain used in building MLS shape functions should be in the range from 30 to 60 nodes and should not be less than 20 nodes. [ABSTRACT FROM AUTHOR]

• 2D behaviors of multi-directional FG plates are analyzed by using GSDT and IGA. • NURBS function describes material distribution in all three directions of FG plates. • Deep neural network is utilized to directly predict behaviors of the FG plates. • Optimal material distributions of tri-directional FG plates are found. • An isogeometric multimesh design is used to save significant computational cost. The paper is aimed at enhancing computational performance for optimizing the material distribution of tri-directional functionally graded (FG) plates. We exploit advantages of using a non-uniform rational B-spline (NURBS) basis function for describing material distribution varying through all three directions of functionally graded (FG) plates. Two-dimensional free vibration and buckling behaviors of multi-directional (1D, 2D and 3D) FG plates analyzed by using a combination of generalized shear deformation theory (GSDT) and isogeometric analysis (IGA) is first proposed. This approach can help to save a significant amount of computational cost while still ensure the accuracy of the solutions. The effectiveness and reliability of the present method are demonstrated by comparing it to other methods in the literature. The obtained results are in excellent agreement with the reference ones. More importantly, data sets consisting of input-output pairs are randomly generated from the analysis process through iterations for the training process in deep neural networks (DNN). DNN is utilized as an analysis tool to supplant finite element analysis to reduce computational cost. By using DNN, behaviors of the multi-directional FG plates are directly predicted from those material distributions. Optimal material distributions of tri-directional FG plates under free vibration or compression in various volume fraction constraints are found by using modified symbiotic organisms search (mSOS) algorithm for the first time. Moreover, an isogeometric multimesh design technique is also used to diminish a large number of design variables in optimization. Optimal results obtained by DNN are compared with those of IGA to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper presents an adapted stabilisation method for the equal-order mixed scheme of finite elements on convex polygonal meshes to analyse the high velocity and pressure gradient of incompressible fluid flows that are governed by Stokes equations system. This technique is constructed by a local pressure projection which is extremely simple, yet effective, to eliminate the poor or even non-convergence as well as the instability of equal-order mixed polygonal technique. In this research, some numerical examples of incompressible Stokes fluid flow that is coded and programmed by MATLAB will be presented to examine the effectiveness of the proposed stabilised method. [ABSTRACT FROM AUTHOR]

A moving Kriging interpolation (MKI) meshfree method based on naturally stabilized nodal integration (NSNI) scheme is presented to study static, free vibration and buckling behaviors of isotropic Reissner–Mindlin plates. Gradient strains are directly computed at nodes similar to the direct nodal integration (DNI). Outstanding features of the current approach are to alleviate instability solutions in the DNI and to decrease computational cost significantly when compared with the traditional high-order Gauss quadrature scheme. The NSNI is a naturally implicit gradient expansion and does not employ a divergence theorem for strain fields as addressed in the stabilized conforming nodal integration method. The present formulation is derived from the Galerkin weak form and avoids a naturally shear-locking phenomenon without using any other techniques. Thanks to satisfied Kronecker delta function property of MKI shape function, essential boundary conditions (BCs) are easily and directly enforced similar to the finite element method. A variety of numerical examples with various geometries, stiffness ratios and BCs are studied to verify the effectiveness of the present approach. [ABSTRACT FROM AUTHOR]

This paper provides a comprehensive overview of a phase-field model of fracture in solid mechanics setting. We start reviewing the potential energy governing the whole process of fracture including crack initiation, branching or merging. Then, a discretization of system of equation is derived, in which the key aspect is that for the correctness of fracture phenomena, a split into tensile and compressive terms of the strain energy is performed, which allows crack to occur in tension, not in compression. For numerical analysis, standard finite element shape functions are used for both primary fields including displacements and phase field. A staggered scheme which solves the two fields of the problem separately is utilized for solution step and illustrated with a segment of Python code. [ABSTRACT FROM AUTHOR]

Abstract We in this study present a mixed solid-shell finite element formulation for laminated shell analysis. With the commonly adopted truncations applied to the strain components and the Jacobian determinant, a generalized stiffness matrix is established. Discrete shear gap (DSG) technique together with assumed natural strain (ANS) method is employed in order to alleviate shear locking and trapezoidal locking. Stiffness matrix formulation is analytically calculated based on the edge-based cell backgrounds, which result in simple and efficient computation. The proposed formulation does not require using cross-diagonal meshes, which can be an obstacle for several existing three-node degenerated shell elements. This aspect makes the present method much more appealing than others through examples tested. Numerical validations show that the present method performs well for both structured and unstructured triangular meshes. [ABSTRACT FROM AUTHOR]

Abstract This paper presents an isogeometric finite element method based on Lagrange extraction in combination with a primal-dual algorithm for estimating limit and shakedown loads of pressure vessel components. The primal-dual algorithm is used to determinate simultaneously both upper and quasi-lower bounds of the limit load multiplier. The Lagrange extraction is integrated into isogeometric analysis. This way, which directly connects a C 0 Lagrange basis with smooth NURBS basis and offers an alternative implementation of isogeometric analysis in the standard finite element method code, allows computations in the same manner as the standard finite element, but the present approach possesses the distinguishing advantages of using higher-order basis approximations. Numerical results of plane and axisymmetric problems are compared with analytical or other available solutions to prove the reliability and efficiency of the present approach. The application of shakedown analysis in pressure vessel engineering is illustrated by some examples. Highlights • A novel numerical approach for evaluating limit and shakedown load factors of pressure vessel components. • An isogeometric analysis based on Lagrange extraction in association with primal-dual algorithm is proposed. • Both quasi-lower and upper bounds of load factors is obtained simultaneously. • Numerical results demonstrate high accuracy of present method with moderate number of degrees of freedom. [ABSTRACT FROM AUTHOR]

A new solid-shell finite element formulation is investigated in this paper. Departing from the S-FEM concept of node-based stress smoothing scheme, a new pattern that introduces new stress variables linked with an extra smoothing region inside the element is proposed. This finding preserves all the advantageous features of the standard NS-FEM but it makes approximate solutions much more accurate. The assumed natural strain (ANS) and the Discrete shear gap (DSG) techniques are employed to alleviate trapezoidal and shear locking. The simple assumptions made allow the analytical computation of the discrete operators based on the node-based backgrounds making the model particularly efficient. Popular benchmark tests are successfully solved for structured and unstructured triangular meshes in linear and buckling analysis. Numerical results show that the proposed method avoids cross-diagonal meshes, which reduces the generality of three-node degenerated shell elements proposed in the literature. This aspect makes the present formulation much more appealing than the published methods available in the literature. Moreover, in the context of the S-FEMs it is the first time in which geometric nonlinear contributions are presented. • New stress variables linked with an extra smoothing region inside the element are proposed. • Discrete FE solid-shell matrices are analytically computed. • Trapezoidal and shear locking are solved. • High accuracy and robust convergence in solutions are proved. [ABSTRACT FROM AUTHOR]

We propose a simple and efficient scheme based on adaptive finite elements over conforming quadtree meshes for collapse plastic analysis of structures. Our main interest in kinematic limit analysis is concerned with both purely cohesive-frictional and cohesive materials. It is shown that the most computational efficiency for collapse plastic problems is to employ an adaptive mesh strategy on quadtree meshes. However, a major difficulty in finite element formulations is the appearance of hanging nodes during adaptive process. This can be resolved by a definition of conforming quadtree meshes in the context of polygonal elements. Piecewise-linear shape functions enhanced with generalized bubble functions in barycentric coordinates are used to approximate the velocity field. Numerical results prove the reliability and benefit of the present approach. [ABSTRACT FROM AUTHOR]

A Reissner-Mindlin plate formulation on arbitrary polygonal meshes is proposed for plate analysis. We consider four barycentric shape function types named Wachspress, mean-value, Laplace and piecewise-linear and show its properties in numerical computation for Reissner-Mindlin plate problems. We then generalize an assumed strain field along sides of polygons under the enforcement of the Timoshenko’s beam assumption. The present approach is numerically verified by the bending patch test. It offers a general yet simple form for Reissner-Mindlin plate elements that is not only implementable for arbitrary polygonal meshes but also avoids transverse shear locking phenomenon at thin plate limit. The performance of the proposed elements is found through numerical examples. Our key contribution to this work is that the present formulation is established in a compact (or unified) form which is valid for triangular, quadrilateral and arbitrary polygonal meshes. [ABSTRACT FROM AUTHOR]

We propose a highly effective approach using a novel adaptive methodology to perform topology optimization with polygonal meshes, called polytree meshes. Polytree is a hierarchical data structure based on the principle of recursive spatial decomposition of each polygonal element with n nodes into ( n + 1) arbitrary new polygonal elements; enabling more efficient utilization of unstructured meshes and arbitrary design domains in topology optimization. In order to treat hanging nodes after each optimization loop, we define the Wachspress coordinate on a reference element and then utilize an affine map to obtain shape functions and their gradients on arbitrary polygons with n vertices and m hanging nodes, called side-nodes, as the polygonal element with ( n + m) vertices. The polytree meshes do not only improve the boundary description quality of the optimal result but also reduce the computational cost of optimization process in comparison with the use of uniformly fine meshes. Several numerical examples are investigated to show the high effectiveness of the proposed method. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We in this paper present a novel adaptive finite element scheme for limit analysis of cracked structures. The key idea is to develop a general refinement algorithm based on a so-called polytree mesh structure. The method is well suited for arbitrary polygonal elements and furthermore traditional triangular and quadrilateral ones, which are considered as special cases. Also, polytree meshes are conforming and can be regarded as a generalization of quadtree meshes. For the aim of this paper, we restrict our main interest in plane-strain limit analysis to von Mises-type materials, yet its extension to a wide class of other solid mechanics problems and materials is completely possible. To avoid volumetric locking, we propose an approximate velocity field enriched with bubble functions using Wachspress coordinates on a primal-mesh and design carefully strain rates on a dual-mesh level. An adaptive mesh refinement process is guided by an L 2 -norm-based indicator of strain rates. Through numerical validations, we show that the present method reaches high accuracy with low computational cost. This allows us to perform large-scale limit analysis problems favorably. [ABSTRACT FROM AUTHOR]

In this study, we propose a novel deep learning model named as the Finite-element-informed neural network (FEI-NN), inspired from finite element method (FEM) for parametric simulation of static problems in structural mechanics. The approach trains neural networks in the supervised manner, in which parametric variables of structures are considered as input features of network and spatial ones are implicitly embedded into the loss function based on a soft constraint called by finite element analysis (FEA) loss. The training process simultaneously minimizes the empirical risk function and partially respects the mechanical behaviors via the FEA loss defined as a residual calculated from the weak form of the surrogate system scaled from the actual corresponding structure. Besides, a technique developed from batch matrix multiplication is proposed to significantly reduce the time complexity for estimating the FEA loss. The method applies to some typical systems in structural mechanics including truss, beam and plate structures. Through several experiments we statistically demonstrate the superiority of the approach in terms of faster convergence and producing better DNN models in comparison to the traditional data-driven approach concerning both generalization and extrapolation performance. • A finite-element-informed neural network (FEI-NN) for parametric simulation in structural mechanics is introduced. • FEI-NN is trained based on optimizing empirical risk function while partially respecting structural responses. • Roles of spatial and parametric variables are separately determined. • Enhanced techniques including characteristic scaling and batch matrix implementation are developed. • The superiority of FEI-NN regarding generalization and extrapolation performance, and convergence and anti-noise capabilities are confirmed through practical problems. [ABSTRACT FROM AUTHOR]

We present a selective edge-based smoothed finite element method (sES-FEM) of kinematic theorem for predicting the plastic limit loads in structures. The basic idea in this method is to use two levels of mesh repartitioning for the finite element limit analysis. The master level begins with an adaptive primal-mesh strategy guided by a dissipation-based indicator. The slave level consists of further subdividing each triangle into three sub-triangles and creating a dual mesh through a careful selection of a pair of two sub-triangles shared by the corresponding common element edge. By applying a strain smoothing projection operator to the strain rates on the dual mesh, the flow rule constraint is enforced over the edge-based strain smoothing domains, and likewise everywhere in the problem domain. This numerical procedure is performed for a cohesive-frictional material. This numerical procedure is also performed necessarily to avoid the volumetric locking problem for a purely cohesive material. The optimization formulation of limit analysis is next presented by the form of a second-order cone programming (SOCP) for the purpose of exploiting the efficiency of interior–point solvers. The present method uses linear triangular elements and handles a low number of optimization variables. This leads to a convenient way to design and solve the large-scale optimization problems effectively. Several numerical examples are given to demonstrate the simplicity and effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This article proposes an efficiently computational tool based on an isogeometric finite element formulation of three-dimensional (3-D) elasticity for static and dynamic response analysis of functionally graded material (FGM) plates. The material properties of FGM plate structures are continuously assumed to vary according to the four-parameter power-law distributions in the thickness direction. The weak form is discretized and approximated by NURBS-based isogeometric approach. The method allows us to preserve exact geometry and is flexible to employ higher-order approximations for the desired accuracy. Through numerical tests, a quartic NURBS element is strongly recommended for eliminating naturally shear locking. Moreover, this element can produce an ultra-accurate solution with a low number of degrees of freedom. Numerical validation shows excellent performance of the present method in comparison with other available methods. [ABSTRACT FROM AUTHOR]

This paper deals with critical roles of cubic bubble functions for the edge-based finite element method (ES-FEM) formulation within the framework of the kinematic theorem for predicting the plastic collapse loads of structures. We show that the bubble function can be scaled by a non-zero coefficient alpha, while the volumetric locking is entirely eliminated. Based on this finding, the present method is designed, in a very small value, with a bubble function that is maximum at the center of the element. This can lead to the loss of the partition of unity. For easy reference, the present method is termed as α bES-FEM. The significant advantage of the present method lies in the rigorous treatment of the volumetric locking-free that can occur in the fully plastic range. The α bES-FEM works well with high efficiency by using triangular meshes. It achieves both simplicity and computational efficiency for implementation into packages of plastic limit analyses. In case of α bES-FEM, the locking issue can be solved conveniently by two schemes: (1) the usual piecewise linear displacements enriched with a cubic bubble function on a primal mesh of triangular elements and (2) a projection operator of strain rates through a dual mesh associated with the edges in the mesh. The optimization formulation of finite element limit analysis is written in the form of a second-order cone programming (SOCP). The well-established interior-point solvers can be exploited efficiently. The α bES-FEM using a small number of integration points enables to solve the large-scale optimization problems efficiently. In addition, an adaptive meshing procedure based on an alternative indicator of dissipation is also derived to further enhance the quality of the solution without increasing significantly the number of degrees of freedom of the model. Numerical results show the robustness of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: We present in this paper a simple and effective approach that incorporates isogeometric finite element analysis (IGA) with a refined plate theory (RPT) for static, free vibration and buckling analysis of functionally graded material (FGM) plates. A new inverse tangent distributed function through the plate thickness is proposed. The RPT enables us to describe the non-linear distribution of shear stresses through the plate thickness without any requirement of shear correction factors (SCF). IGA utilizes basis functions namely B-splines or non-uniform rational B-splines (NURBS) which reach easily the smoothness of any arbitrary order. It hence satisfies the C 1 requirement of the RPT model. The present method approximates the displacement field with four degrees of freedom per each control point allowing an efficient solution process. [Copyright &y& Elsevier]

SUMMARY A cell-based smoothed discrete shear gap method (CS-FEM-DSG3) based on the first-order shear deformation theory (FSDT) was recently proposed for static and dynamic analyses of Mindlin plates. In this paper, the CS-FEM-DSG3 is extended to the C0-type higher-order shear deformation plate theory (C0-HSDT) and is incorporated with damping-spring systems for dynamic responses of Mindlin plates on viscoelastic foundations subjected to a moving sprung vehicle. At each time step of dynamic analysis, one four-step procedure is performed including the following: (1) transformation of the weight of a four-wheel vehicle into the sprung masses at wheels; (2) dynamic analysis of the sprung mass of wheels to determine the contact forces; (3) transformation of the contact forces into loads at nodes of plate elements; and (4) dynamic analysis of the plate elements on viscoelastic foundations. The accuracy and reliability of the proposed method are verified by comparing its numerical solutions with those of other available numerical results. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: A novel and efficient approach which is based on the framework of isogeometric analysis for elliptic homogenization problems is proposed. These problems possess highly oscillating coefficients leading to extremely high computational expenses while using traditional finite element methods. The isogeometric analysis heterogeneous multiscale method (IGA-HMM) investigated in this paper is regarded as an alternative approach to the standard finite element heterogeneous multiscale method (FE-HMM) which is currently an effective framework to solve these problems. The method utilizes non-uniform rational -splines (NURBS) in both macro and micro levels instead of standard Lagrange basis. Besides the ability to describe exactly the geometry, it tremendously facilitates high-order macroscopic/microscopic discretizations thanks to the flexibility of refinement and degree elevation with an arbitrary continuity level provided by NURBS basis functions. Furthermore, the nearly optimal quadrature rule for IGA (Auricchio et al., 2012) introduced recently is utilized to reduce significantly the number of micro problems, which is the main factor contributing into the computational cost in heterogeneous multiscale method. A priori error estimates of the discretization error coming from macro and micro meshes and optimal micro refinement strategies for macro/micro NURBS basis functions of arbitrary orders are derived. An efficient coupling between degrees of macro and micro basis functions is introduced. Numerical results show the excellent performance of the proposed method. [Copyright &y& Elsevier]

Abstract: We present in this paper a simple and effective formulation based on a fifth-order shear deformation theory (FiSDT) in combination with isogeometric finite element analysis (IGA) for composite sandwich plates. The FiSDT yields non-linear distribution of the transverse shear stresses through the plate thickness and ensures a prior tangential stress-free boundary condition. The IGA uses same basis functions, namely B-splines or non-uniform rational B-splines (NURBS), for preserving the precisely geometric representation and providing the numerical solution. It enables to achieve easily the smoothness with arbitrary continuity order and in the present method the C 1-continuity requirement of higher order shear deformation models is fulfilled. The static, dynamic and buckling analysis of rectangular and circular plates is investigated for different boundary conditions. Numerical examples are given to show high accuracy of the proposed method. [Copyright &y& Elsevier]

Highlights: [•] We propose an improved formulation of the edge-based smoothed element method (ES-FEM). [•] The original ES-FEM formulation is supplemented with a cubic bubble function in triangular elements. [•] The stiffness matrix is evaluated using smoothed strains over edge-based smoothing domains. [•] The method is free from both volumetric and bending locking and works well with extremely distorted mesh. [•] Numerical results show the highly effective performance of the present method. [Copyright &y& Elsevier]

An edge-based smoothed finite element method (ES-FEM-T3) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the solid mechanics analyses. In this paper, the ES-FEM-T3 is further extended to the dynamic analysis of 2D fluid-solid interaction problems based on the pressure-displacement formulation. In the present coupled method, both solid and fluid domain is discretized by triangular elements. In the fluid domain, the standard FEM is used, while in the solid domain, we use the ES-FEM-T3 in which the gradient smoothing technique based on the smoothing domains associated with the edges of triangles is used to smooth the gradient of displacement. This gradient smoothing technique can provide proper softening effect, and thus improve significantly the solution of coupled system. Some numerical examples have been presented to illustrate the effectiveness of the proposed coupled method compared with some existing methods for 2D fluid-solid interaction problems. [ABSTRACT FROM AUTHOR]

This paper presents a novel numerical procedure for limit analysis of plane problems using edge-based smoothed finite element method (ES-FEM) in combination with second-order cone programming. In the ES-FEM, the discrete weak form is obtained based on the strain smoothing technique over smoothing domains associated with the edges of the elements. Using constant smoothing functions, the incompressibility condition only needs to be enforced at one point in each smoothing domain, and only one Gaussian point is required, ensuring that the size of the resulting optimization problem is kept to a minimum. The discretization problem is transformed into the form of a second-order cone programming problem which can be solved using highly efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to various benchmark plane stress and strain problems. [ABSTRACT FROM AUTHOR]

A n-sided polygonal cell-based smoothed finite element method (nCS-FEM) was recently proposed to analyze the elastic solid mechanics problems, in which the problem domain can be discretized by a set of polygons with an arbitrary number of sides. In this paper, the nCS-FEM is further extended to the free and forced vibration analyses of two-dimensional (2D) dynamic problems. A simple lump mass matrix is proposed and hence the complicated integrations related to computing the consistent mass matrix can be avoided in the nCS-FEM. Several numerical examples are investigated and the results found of the nCS-FEM agree well with exact solutions and with those of others FEM. [ABSTRACT FROM AUTHOR]

This paper promotes a novel numerical approach to static, free vibration and buckling analyses of laminated composite plates by an edge-based smoothed finite method (ES-FEM). In the present ES-FEM formulation, the system stiffness matrix is established by using the strain smoothing technique over the smoothing domains associated with the edges of the triangular elements. A discrete shear gap {DSG3) technique without shear locking is combined into the ES-FEM to give a so-called edge-based smoothed discrete shear gap method (ES-DSG3) for analysis of laminated composite plates. The present method uses only linear interpolations and its implementation into finite element programs is quite simple. Numerical results for analysis of laminated composite plates show that the ES-DSG3 performs quite well compared to several other published approaches in the literature. [ABSTRACT FROM AUTHOR]

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

Abstract: This paper presents a singular edge-based smoothed finite element method (sES-FEM) for mechanics problems with singular stress fields of arbitrary order. The sES-FEM uses a basic mesh of three-noded linear triangular (T3) elements and a special layer of five-noded singular triangular elements (sT5) connected to the singular-point of the stress field. The sT5 element has an additional node on each of the two edges connected to the singular-point. It allows us to represent simple and efficient enrichment with desired terms for the displacement field near the singular-point with the satisfaction of partition-of-unity property. The stiffness matrix of the discretized system is then obtained using the assumed displacement values (not the derivatives) over smoothing domains associated with the edges of elements. An adaptive procedure for the sES-FEM is proposed to enhance the quality of the solution with minimized number of nodes. Several numerical examples are provided to validate the reliability of the present sES-FEM method. [Copyright &y& Elsevier]

The cell-based strain smoothing technique is combined with the well-known three-node Mindlin plate element (MIN3) to give a so-called the cell-based smoothed MIN3 (CS-MIN3) for static and free vibration analyses of plates. In the process of formulating the system stiffness matrix of the CS-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the stabilized MIN3 is used to compute the strains and to avoid the transverse shear locking. Then the strain smoothing technique on whole the triangular element is used to smooth the strains on these three sub-triangles. The numerical examples demonstrated that the CS-MIN3 is free of shear locking, passes the patch test and shows four superior properties such as: (1) be a strong competitor to many existing three-node triangular plate elements in the static analysis, (2) can give high accurate solutions for problems with skew geometries in the static analysis, (3) can give high accurate solutions in free vibration analysis, (4) can provide accurately the values of high frequencies of plates by using only coarse meshes. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents an improved finite element approach in which a node-based strain smoothing is merged into shear-locking-free triangular plate elements. The formulation uses only linear approximations and its implementation into finite element programs is quite simple and efficient. The method is then applied for static, free vibration and mechanical/thermal buckling problems of functionally graded material (FGM) plates. In the FGM plates, the material properties are assumed to vary across the thickness direction by a simple power rule of the volume fractions of the constituents. The behavior of FGM plates under mechanical and thermal loads is numerically analyzed in detail through a list of benchmark problems. The numerical results show high reliability and accuracy of the present method compared with other published solutions in the literature. [Copyright &y& Elsevier]

Abstract: A novel singular seven-node crack-tip triangular element with edge-based strain smoothing is formulated for modeling crack growth in solids. The present singular ES-FEM method uses a basic mesh of linear triangular elements and a special layer of seven-noded crack-tip elements connected to the crack-tip. The seven-noded crack-tip element has two additional nodes on each of the edges connected to the crack-tip, allowing efficient enrichment of the displacement field near the crack-tip without loss of partitions-of-unity property. The stiffness matrix of the discretized system is then obtained using the assumed displacement values (not the derivatives) over smoothing domains associated with the edges of elements. For crack propagation simulation, the Delaunay triangulation procedure associated with the Laplacian smoothing technique is used to automatically generate meshes. Several benchmark problems are used to show the accuracy and robustness of the proposed method. [Copyright &y& Elsevier]

Abstract: An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap (DSG) technique using triangular meshes (ES-DSG) was recently proposed to enhance the accuracy of the existing FEM with the DSG for analysis of isotropic Reissner/Mindlin plates. In this paper, the ES-DSG is further formulated for static, free vibration and buckling analyses of functionally graded material (FGM) plates. The thermal and mechanical properties of FGM plates are assumed to vary across the thickness of the plate by a simple power rule of the volume fractions of the constituents. In the ES-DSG, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains associated with the edges of the elements. The present formulation uses only linear approximations and its implementation into finite element programs is quite simple. Several numerical examples are given to demonstrate the performance of the present formulation for FGM plates. [Copyright &y& Elsevier]

An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper the ES-FEM is further extended to a more general case, n-sided polygonal edge-based smoothed finite element method ( nES-FEM), in which the problem domain can be discretized by a set of polygons, each with an arbitrary number of sides. The simple averaging point interpolation method is suggested to construct nES-FEM shape functions. In addition, a novel domain-based selective scheme of a combined nES/NS-FEM model is also proposed to avoid volumetric locking. Several numerical examples are investigated and the results of the nES-FEM are found to agree well with exact solutions and are much better than those of others existing methods. Copyright © 2010 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: Isogeometric analysis has become a powerful alternative to standard finite elements due to its flexibility in handling complex geometries. One of the major drawbacks of NURBS-based isogeometric finite elements is the inefficiency of local refinement. In this study, we present an alternative to NURBS-based isogeometric analysis that allows for local refinement. The idea is based on polynomial splines and exploits the flexibility of T-meshes for local refinement. The shape functions satisfy important properties such as non-negativity, local support and partition of unity. Several numerical examples are used to demonstrate the reliability of the present method. [Copyright &y& Elsevier]

The paper presents an adaptive analysis within the framework of the node-based smoothed finite element method (NS-FEM) using triangular elements. An error indicator based on the recovery strain is used and shown to be asymptotically exact by an effectivity index and numerical results. A simple refinement strategy using the newest node bisection is briefly presented. The numerical results of some benchmark problems show that the present adaptive procedure can accurately catch the appearance of the steep gradient of stresses and the occurrence of refinement is concentrated properly. The energy error norms of adaptive models for both NS-FEM and FEM obtain higher convergence rate compared with the uniformly refined models, but the results of NS-FEM are better and achieve higher convergence rate than those of FEM. The effectivity index of NS-FEM is also closer and approaches to unity faster than that of FEM. The upper bound property in the strain energy of NS-FEM is always verified during the adaptive procedure. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

A generalized gradient smoothing technique and a smoothed bilinear form of Galerkin weak form have been recently proposed by Liu et al. to create a wide class of efficient smoothed point interpolation methods (PIMs) using the background mesh of triangular cells. In these methods, displacement fields are constructed by polynomial or radial basis shape functions and strains are smoothed over the smoothed domain associated with the nodes or the edges of the triangular cells. This paper summarizes and assesses bound property, convergence rate and computational efficiency for these methods. It is found that: (1) the incorporation of the PIMs with the node-based strain smoothing operation allows us to obtain an upper bound to the exact solution in the strain energy; (2) the incorporation of the PIMs with the edge-based strain smoothing operation using triangular background mesh can produce a solution of 'ultra-accuracy' and 'super-convergence'; (3) the edge-based strain smoothing operation together with the linear interpolation can provide better computational efficiency compared with other smoothed PIMs and the finite element method when the same triangular mesh is used. These conclusions have been examined and confirmed by intensive examples. Copyright © 2009 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: An edge-based smoothed finite element method (ES-FEM) for static, free vibration and buckling analyses of Reissner–Mindlin plates using 3-node triangular elements is studied in this paper. The calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements. In order to avoid the transverse shear locking and to improve the accuracy of the present formulation, the ES-FEM is incorporated with the discrete shear gap (DSG) method together with a stabilization technique to give a so-called edge-based smoothed stabilized discrete shear gap method (ES-DSG). The numerical examples demonstrated that the present ES-DSG method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature. [Copyright &y& Elsevier]

Abstract: This paper presents two novel numerical procedures to determine upper and lower bounds on the actual collapse load multiplier for plates in bending. The conforming Hsieh–Clough–Tocher (HCT) and enhanced Morley (EM) elements are used to discrete the problem fields. A Morley element with enhanced moment fields is used. The constant moment fields is added a quadratic mode in which the pressure is equilibrated by corner loads only, ensuring that exact equilibrium relations associated with a uniform pressure can be obtained. Once the displacement or moment fields are approximated and the bound theorems applied, limit analysis becomes a problem of optimization. In this paper, the optimization problems are formulated in the form of a standard second-order cone programming which can be solved using highly efficient interior point solvers. The procedures are tested by applying it to several benchmark plate problems and are found good agreement between the present upper and lower bound solutions and results in the literature. [Copyright &y& Elsevier]

An edge-based smoothed finite element method (ES-FEM) using triangular elements was recently proposed to improve the accuracy and convergence rate of the existing standard finite element method (FEM) for the elastic solid mechanics problems. In this paper, the ES-FEM is extended to more complicated visco-elastoplastic analyses using the von-Mises yield function and the Prandtl–Reuss flow rule. The material behavior includes perfect visco-elastoplasticity and visco-elastoplasticity with isotropic and linear kinematic hardening. The formulation shows that the bandwidth of stiffness matrix of the ES-FEM is larger than that of the FEM, and hence the computational cost of the ES-FEM in numerical examples is larger than that of the FEM for the same mesh. However, when the efficiency of computation (computation time for the same accuracy) in terms of a posteriori error estimation is considered, the ES-FEM is more efficient than the FEM. [ABSTRACT FROM AUTHOR]

A node-based smoothed finite element method (NS-FEM) for solving solid mechanics problems using a mesh of general polygonal elements was recently proposed. In the NS-FEM, the system stiffness matrix is computed using the smoothed strains over the smoothing domains associated with nodes of element mesh, and a number of important properties have been found, such as the upper bound property and free from the volumetric locking. The examination was performed only for two-dimensional (2D) problems. In this paper, we (1) extend the NS-FEM to three-dimensional (3D) problems using tetrahedral elements (NS-FEM-T4), (2) reconfirm the upper bound and free from the volumetric locking properties for 3D problems, and (3) explore further other properties of NS-FEM for both 2D and 3D problems. In addition, our examinations will be thorough and performed fully using the error norms in both energy and displacement. The results in this work revealed that NS-FEM possesses two additional interesting properties that quite similar to the equilibrium FEM model such as: (1) super accuracy and super-convergence of stress solutions; (2) similar accuracy of displacement solutions compared to the standard FEM model. [ABSTRACT FROM AUTHOR]

A free vibration analysis of Mindlin–Reissner plates using the stabilized smoothed finite element method is studied. The bending strains of the MITC4 and STAB elements are incorporated with a cell-wise smoothing operation to give new proposed elements, the mixed interpolation and smoothed curvatures (MISCk) and SMISCk elements. The corresponding bending stiffness matrix is computed along the boundaries of the smoothing elements (smoothing cells). Note that shearing strains and the shearing stiffness matrix of the proposed elements are unchanged from the original elements, the MITC4 and STAB elements. It is confirmed by numerical tests that the present method is free of shear locking and has the marginal improvements compared with the original elements. Copyright © 2008 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: A carefully designed procedure is presented to modify the piecewise constant strain field of linear triangular FEM models, and to reconstruct a strain field with an adjustable parameter α. A novel Galerkin-like weakform derived from the Hellinger–Reissner variational principle is proposed for establishing the discretized system equations. The new weak form is very simple, possesses the same good properties of the standard Galerkin weakform, and works particularly well for strain construction methods. A superconvergent alpha finite element method (SαFEM) is then formulated by using the constructed strain field and the Galerkin-like weakform for solid mechanics problems. The implementation of the SαFEM is straightforward and no additional parameters are used. We prove theoretically and show numerically that the SαFEM always achieves more accurate and higher convergence rate than the standard FEM of triangular elements (T3) and even more accurate than the four-node quadrilateral elements (Q4) when the same sets of nodes are used. The SαFEM can always produce both lower and upper bounds to the exact solution in the energy norm for all elasticity problems by properly choosing an α. In addition, a preferable-α approach has also been devised to produce very accurate solutions for both displacement and energy norms and a superconvergent rate in the energy error norm. Furthermore, a model-based selective scheme is proposed to formulate a combined SαFEM/NS-FEM model that handily overcomes the volumetric locking problems. Intensive numerical studies including singularity problems have been conducted to confirm the theory and properties of the SαFEM. [Copyright &y& Elsevier]

Abstract: A quadrilateral element with smoothed curvatures for Mindlin–Reissner plates is proposed. The curvature at each point is obtained by a non-local approximation via a smoothing function. The bending stiffness matrix is calculated by a boundary integral along the boundaries of the smoothing elements (smoothing cells). Numerical results show that the proposed element is robust, computational inexpensive and simultaneously very accurate and free of locking, even for very thin plates. The most promising feature of our elements is their insensitivity to mesh distortion. [Copyright &y& Elsevier]

In this study, we propose a new design of prefabricated hollow blocks, which are made of pervious concrete. The effects of thickness of block walls are studied to optimize an adequate size. The comparison of performance of porous and solid blocks is studied through fluid-solid interaction models. A polyhedral finite element formulation (PFEM) is employed to analyse responses of the structure under equivalent static and dynamic wave load conditions. The effectiveness of PFEM is verified by comparing the obtained results with those of standard finite element method (FEM) using the commercial software ANSYS. Their accuracy is also proved via the validation model of a step-beam, in which the natural frequencies are used for comparison. In static analysis, the maximum displacement of block increases rapidly when porosity of concrete changes from 5% to 35%. The optimum porosity is found to be around 20%. The variable thickness of the block walls reasonably affects the displacement resistance. However, compared to porosity, the influence of thickness is not significant. The change of Poisson's ratio slightly affects the responses of the block. For dynamic analysis, the vibration of pervious concrete block types under the impact of wave load with small duration (<20 ms) lasts longer than the case attacked by long duration shock (around 50 ms). However, the amplitudes of displacement responses by long duration impact wave are greater than that of short impact. Study also showed that the porous concrete block could significantly reduce the run-down pressure acting on the structure compared to solid block. • Numerical investigation on using pervious concrete for seawalls. • Polyhedral finite element formulation (PFEM) is employed. • New design of prefabricated hollow blocks using pervious concrete with porosity changing from 5% to 35%. • The optimum porosity is found to be around 20%. [ABSTRACT FROM AUTHOR]

This paper presents a new high-order mixed scheme of polygonal finite element method (PFEM) for incompressible flow computation. The system of polygonal basis functions with bubble-enriched functions is applied to represent the fluid velocity field, while the pressure field is approximated by the set of polygonal basis functions. The new element is named as MINIPe. From a practical point of view, the present element naturally satisfies the well-known inf–sup condition. The numerical tests of the incompressible fluid flow problems governed by Stokes equation systems are presented to verify the performance quality of proposed method with respect to the stability and convergence rate. • A new high-order mixed scheme of polygonal finite element method (PFEM) for incompressible flow computation. • An enrichment of bubble functions is applied to represent the fluid velocity field. • The pressure field is approximated by the set of polygonal basis functions. • The present element naturally satisfies the well-known inf–sup condition. • High quality of proposed method with respect to the stability and convergence rate. [ABSTRACT FROM AUTHOR]

The polytree-based adaptive polygonal finite element method was recently proved to be very efficient for a wide class of mechanics problems. In this study, we further exploit its advantages for topology optimization of fluid-submerged breakwater interaction. Due to the inherent limitations of triangular and quadrilateral elements for topology optimization, an alternative formulation of using arbitrary polygons based on the Wachspress coordinate is employed for structural domain. In addition, we use a polytree-based adaptive technique to enhance the accuracy for topology problem in structural domain with a minimum number of degrees of freedom (DOF). We apply this method to fluid–structure interaction (FSI) problems and then finding an optimal shape of submerged breakwater (SBW), which is as a special kind of coastal structure located in front of the beach and underwater to protect the landside from the erosion. Several numerical results demonstrate the effectiveness of the present method. It is observed that the obtained results are in good agreement with reference solutions. [ABSTRACT FROM AUTHOR]

This study presents a polytree-based adaptive methodology for multi-material topology optimization (MMTOP). Polytree data structure is introduced as a general recursive multi-level mesh that is automatically refined in processing based on error analysis. In order to resolve hanging nodes in element edges, the Wachspress coordinate is employed on a reference element before using a mapping scheme to obtain shape functions and their derivatives for any polygons. A new definition of filter radius is also proposed to improve the efficiency of filters and optimized results. The combination of polytree meshes and adaptive filters not only clarifies the interfaces between material phases (including void phase), but also decreases the computing time of the overall process in comparison to using the regular fine meshes. Several benchmark and practical problems are considered to show distinct features of the proposed method. [ABSTRACT FROM AUTHOR]