Strain Smoothing Technique Research Articles

A cell-based smoothed three-node Mindlin plate element (CS-MIN3) for static and free vibration analyses of plates

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.

Improvement of the performance of a tetrahedral element with rotational DOFs using strain smoothing techniques

In this paper, the performance of a 4-node tetrahedral element with rotational DOFs is improved using strain smoothing techniques. A partition-of-unity based approximation is used to construct the tetrahedral element with rotational DOFs. To overcome the stiff behavior of the tetrahedral element with rotational DOFs, a face-based strain smoothing technique is used. The stiffness matrix is expressed explicitly and the element performance is evaluated through numerical examples, which demonstrate that the proposed strain smoothed tetrahedral elements with rotational DOFs satisfy the patch test and have a better performance than other tetrahedral elements with rotational DOFs.

Fatigue analysis using the singular ES-FEM

A novel procedure of singular edge-based smoothed finite element method (singular ES-FEM) has been customized for the fatigue crack growth analysis. For this method, a layer of crack-tip elements with special shape functions was adopted to capture the theoretical singularity of the stress field. In addition, a ‘strain smoothing technique’ on the smoothing domains associated with the elements’ edges was implemented that significantly improves the accuracy of results compared to the existing finite element method. A comparison of results with the experimental observations reported by others reveals an excellent agreement between them implying that singular ES-FEM accurately simulates the fatigue behavior.

Smoothed Galerkin methods using cell-wise strain smoothing technique

A smoothed Galerkin method (SGM) using cell-wise strain smoothing operation is formulated in this paper. In present method, the field nodes can be divided into two types: boundary field nodes and interior field nodes. The background cells are divided into SC smoothing cells, and the strains in each smoothing cell are obtained using a gradient smoothing technique which can avoid evaluating derivatives of shape functions at integration point. The field variables of boundary points are approximated using linear interpolation of neighbour boundary field nodes, and the shape functions possess the Kronecker Delta property and facilitate the impositions of essential boundary conditions. The field variables of interior points are approximated using moving least-squares approximation using the support field nodes around them. A number of numerical examples are studied and confirm the significant features of the present methods: (1) can pass the standard patch test; (2) can easily impose essential boundary conditions as those in finite element method; (3) can avoid evaluating derivatives of shape functions; (4) no numerical parameter is required.

A COMBINED EXTENDED AND EDGE-BASED SMOOTHED FINITE ELEMENT METHOD (ES-XFEM) FOR FRACTURE ANALYSIS OF 2D ELASTICITY

This study combines the edge-based smoothed finite element method (ES-FEM) and the extended finite element method (XFEM) to develop a new simulation technique (ES-XFEM) for fracture analysis of 2D elasticity. In the XFEM, the need for the mesh alignment with the crack and remeshing, as the crack evolves, is eliminated because of the use of partition of unity. The ES-FEM uses the generalized smoothing operation over smoothing domain associated with edges of simplex meshes, and produces a softening effect leading to a close-to-exact stiffness, "super-convergence" and "ultra-accurate" solutions for the numerical model. Taking advantage of both ES-FEM and XFEM, the present method introduces the edge-based strain smoothing technique into the context of XFEM. Thanks to strain smoothing, the necessity of sub-division in elements cut by discontinuities is suppressed via transforming interior integration into boundary integration. Hence, it simplifies the numerical integration procedure in the XFEM. Numerical examples showed that the proposed method improves significantly the accuracy of stress intensity factors and achieves a quasi optimal convergence rate in the energy norm without geometrical enrichment or blending correction.

Analysis of functionally graded plates using an edge-based smoothed finite element method

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.

A Node-Based Smoothed eXtended Finite Element Method (NS-XFEM) for Fracture Analysis

This paper aims to incorporate the node-based smoothed finite element method (NS-FEM) into the extended finite element method (XFEM) to form a novel numerical method (NS-XFEM) for analyzing fracture problems of 2D elasticity. NS-FEM uses the strain smoothing technique over the smoothing domains associated with nodes to compute the system stiffness matrix, which leads to the line integrations using directly the shape function values along the boundaries of the smoothing domains. As a result, we avoid integration of the stress singularity at the crack tip. It is not necessary to divide elements cut by cracks when we replace interior integration by boundary integration, simplifying integration of the discontinuous approximation. The key advantage of the NS-XFEM is that it provides more accurate solutions compared to the XFEM-T3 element. We will show for two numerical examples that the NS-XFEM significantly improves the results in the energy norm and the stress intensity factors. For the examples studied, we obtain super-convergent results

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

AbstractBy using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25:137–156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6):859–877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near‐tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non‐polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright © 2011 John Wiley & Sons, Ltd.

Buckling and vibration analysis of laminated composite plate/shell structures via a smoothed quadrilateral flat shell element with in-plane rotations

This paper presents buckling and free vibration analysis of composite plate/shell structures of various shapes, modulus ratios, span-to-thickness ratios, boundary conditions and lay-up sequences via a novel smoothed quadrilateral flat element. The element is developed by incorporating a strain smoothing technique into a flat shell approach. As a result, the evaluation of membrane, bending and geometric stiffness matrices are based on integration along the boundary of smoothing elements, which leads to accurate numerical solutions even with badly-shaped elements. Numerical examples and comparison with other existing solutions show that the present element is efficient, accurate and free of locking.

An alternative alpha finite element method with discrete shear gap technique for analysis of laminated composite plates

This paper presents an alternative alpha finite element method using triangular meshes (A αFEM) for static, free vibration and buckling analyses of laminated composite plates. In the A αFEM, an assumed strain field is carefully constructed by combining compatible strains and additional strains with an adjustable parameter α which can produce an effectively softer stiffness formulation compared to the linear triangular element. The stiffness matrices are obtained based on the strain smoothing technique over the smoothing domains and the constant strains on triangular sub-domains associated with the nodes of the elements. The discrete shear gap (DSG) method is incorporated into the A αFEM to eliminate transverse shear locking and an improved triangular element termed as A αDSG3 is proposed. Several numerical examples are then given to demonstrate the effectiveness of the A αDSG3.

A theoretical study on the smoothed FEM (S‐FEM) models: Properties, accuracy and convergence rates

AbstractIncorporating the strain smoothing technique of meshfree methods into the standard finite element method (FEM), Liu et al. have recently proposed a series of smoothed finite element methods (S‐FEM) for solid mechanics problems. In these S‐FEM models, the compatible strain fields are smoothed based on smoothing domains associated with entities of elements such as elements, nodes, edges or faces, and the smoothed Galerkin weak form based on these smoothing domains is then applied to compute the system stiffness matrix. We present in this paper a general and rigorous theoretical framework to show properties, accuracy and convergence rates of the S‐FEM models. First, an assumed strain field derived from the Hellinger–Reissner variational principle is shown to be identical to the smoothed strain field used in the S‐FEM models. We then define a smoothing projection operator to modify the compatible strain field and show a set of properties. We next establish a general error bound of the S‐FEM models. Some numerical examples are given to verify the theoretical properties established. Copyright © 2010 John Wiley & Sons, Ltd.

A cell‐based smoothed finite element method for kinematic limit analysis

AbstractThis paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell‐based smoothed finite element method with second‐order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non‐smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second‐order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged. Copyright © 2010 John Wiley & Sons, Ltd.

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

AbstractIt is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.

A node-based smoothed finite element method with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates

In this paper, a node-based smoothed finite element method (NS-FEM) using 3-node triangular elements is formulated for static, free vibration and buckling analyses of Reissner–Mindlin plates. The discrete weak form of the NS-FEM is obtained based on the strain smoothing technique over smoothing domains associated with the nodes of the elements. The discrete shear gap (DSG) method together with a stabilization technique is incorporated into the NS-FEM to eliminate transverse shear locking and to maintain stability of the present formulation. A so-called node-based smoothed stabilized discrete shear gap method (NS-DSG) is then proposed. Several numerical examples are used to illustrate the accuracy and effectiveness of the present method.

Free-vibration analysis of shells via a linearly conforming radial point interpolation method (LC-RPIM)

In this paper, free-vibration analysis of shells is carried out using a linearly conforming radial point interpolation method (LC-RPIM). The present formulation is based on the first-order shear deformation shell theory. A set of arbitrarily distributed nodes in local support domains is selected to perform interpolation, and radial and polynomial basis functions are employed to construct shape functions with Delta function property. Nodal integration, which is enforced by a strain-smoothing stabilization technique, is used to restore conformability and improve the accuracy of solutions. Comparisons of current solutions are made with existing results available in literature and good agreements are obtained. Numerical examples demonstrate that the present approach provides very stable and accurate results.

An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates

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.

An Improved Quadrilateral Flat Element with Drilling Degrees of Freedom for Shell Structural Analysis

This paper reports the development of a simple and efficient 4-node flat shell element with six degrees of freedom per node for the analysis of arbitrary shell structures. The element is developed by incorporating a strain smoothing technique into a flat shell finite element approach. The membrane part is formulated by applying the smoothing operation on a quadrilateral membrane element using Allman-type interpolation functions with drilling DOFs. The plate-bending component is established by a combination of the smoothed curvature and the substitute shear strain fields. As a result, the bending and a part of membrane stiffness matrices are computed on the boundaries of smoothing cells which leads to very accurate solutions, even with distorted meshes, and possible reduction in computational cost. The performance of the proposed element is validated and demonstrated through several numerical benchmark problems. Convergence studies and comparison with other existing solutions in the literature suggest that the present element is efficient, accurate and free of lockings.

About the edge-based smoothed finite element method for the Reissner-Mindlin plate-bending problem

The paper further develops the edge-based smoothed finite element method (ES-FEM) for analysis of Reissner-Mindlin plates using triangular meshes. The bending and shearing stiffness matrices are obtained using strain smoothing technique over the smoothing domains associated with edges of elements. Transverse shear locking can be avoided with help of the discrete shear gap (DSG) method. The numerical examples show that the present ES-FEM-DSG method obtains very accurate results compared to the exact solution and other existing elements.

A node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems

This paper presents a node-based smoothed finite element method (NS-FEM) for upper bound solutions to solid mechanics problems using a mesh of polygonal elements. The calculation of the system stiffness matrix is performed using strain smoothing technique over the smoothing cells associated with nodes, which leads to line integrations along the edges of the smoothing cells. The numerical results demonstrated that the NS-FEM possesses the following properties: (1) upper bound in the strain energy of the exact solution when a reasonably fine mesh is used; (2) well immune from the volumetric locking; (3) can use polygonal elements with an arbitrary number of sides; (4) insensitive to element distortion.