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

2009 ◽  
Vol 87 (1-2) ◽  
pp. 14-26 ◽  
Author(s):  
G.R. Liu ◽  
T. Nguyen-Thoi ◽  
H. Nguyen-Xuan ◽  
K.Y. Lam
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Upper Bound ◽  
Solid Mechanics ◽  
Smoothed Finite Element Method ◽  
Element Method

ADDITIONAL PROPERTIES OF THE NODE-BASED SMOOTHED FINITE ELEMENT METHOD (NS-FEM) FOR SOLID MECHANICS PROBLEMS

2009 ◽  
Vol 06 (04) ◽  
pp. 633-666 ◽  
Author(s):  
T. NGUYEN-THOI ◽  
G. R. LIU ◽  
H. NGUYEN-XUAN
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Upper Bound ◽  
Solid Mechanics ◽  
Three Dimensional ◽  
Element Mesh ◽  
Fem Model ◽  
Volumetric Locking ◽  
Smoothed Finite Element Method ◽  
Element Method

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.

Topology Optimization Using Node-Based Smoothed Finite Element Method

2015 ◽  
Vol 07 (06) ◽  
pp. 1550085 ◽  
Author(s):  
Z. C. He ◽  
G. Y. Zhang ◽  
L. Deng ◽  
Eric Li ◽  
G. R. Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Topology Optimization ◽  
Optimization Design ◽  
Solid Mechanics ◽  
Optimality Criteria ◽  
Topology Optimization Problem ◽  
Design Variables ◽  
Smoothed Finite Element Method ◽  
Element Method

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

Coupled Analysis of Structural–Acoustic Problems Using the Cell-Based Smoothed Three-Node Mindlin Plate Element

2016 ◽  
Vol 13 (02) ◽  
pp. 1640007 ◽  
Author(s):  
Z. X. Gong ◽  
Y. B. Chai ◽  
W. Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Solid Mechanics ◽  
Mindlin Plate ◽  
Coupled Analysis ◽  
Plate Element ◽  
Fem Model ◽  
The Face ◽  
Smoothed Finite Element Method ◽  
Element Method

The cell-based smoothed finite element method (CS-FEM) using the original three-node Mindlin plate element (MIN3) has recently established competitive advantages for analysis of solid mechanics problems. The three-node configuration of the MIN3 is achieved from the initial, complete quadratic deflection via ‘continuous’ shear edge constraints. In this paper, the proposed CS-FEM-MIN3 is firstly combined with the face-based smoothed finite element method (FS-FEM) to extend the range of application to analyze acoustic fluid–structure interaction problems. As both the CS-FEM and FS-FEM are based on the linear equations, the coupled method is only effective for linear problems. The cell-based smoothed operations are implemented over the two-dimensional (2D) structure domain discretized by triangular elements, while the face-based operations are implemented over the three-dimensional (3D) fluid domain discretized by tetrahedral elements. The gradient smoothing technique can properly soften the stiffness which is overly stiff in the standard FEM model. As a result, the solution accuracy of the coupled system can be significantly improved. Several superior properties of the coupled CS-FEM-MIN3/FS-FEM model are illustrated through a number of numerical examples.

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

2013 ◽  
Vol 10 (01) ◽  
pp. 1340011 ◽  
Author(s):  
XU XU ◽  
YUANTONG GU ◽  
GUIRONG LIU
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Strain Energy ◽  
Solid Mechanics ◽  
Weighted Average ◽  
Accurate Solution ◽  
Numerical Studies ◽  
Smoothed Finite Element Method ◽  
Theoretical Results ◽  
Element Method

In this paper, a hybrid smoothed finite element method (H-SFEM) is developed for solid mechanics problems by combining techniques of finite element method (FEM) and node-based smoothed finite element method (NS-FEM) using a triangular mesh. A parameter α is equipped into H-SFEM, and the strain field is further assumed to be the weighted average between compatible stains from FEM and smoothed strains from NS-FEM. We prove theoretically that the strain energy obtained from the H-SFEM solution lies in between those from the compatible FEM solution and the NS-FEM solution, which guarantees the convergence of H-SFEM. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper- and lower-bound solutions can always be obtained by adjusting α; (2) there exists a preferable α at which the H-SFEM can produce the ultrasonic accurate solution.

FREE AND FORCED VIBRATION ANALYSIS USING THE n-SIDED POLYGONAL CELL-BASED SMOOTHED FINITE ELEMENT METHOD (nCS-FEM)

2013 ◽  
Vol 10 (01) ◽  
pp. 1340008 ◽  
Author(s):  
T. NGUYEN-THOI ◽  
P. PHUNG-VAN ◽  
T. RABCZUK ◽  
H. NGUYEN-XUAN ◽  
C. LE-VAN
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Forced Vibration ◽  
Solid Mechanics ◽  
Mass Matrix ◽  
Elastic Solid ◽  
Polygonal Cell ◽  
Smoothed Finite Element Method ◽  
Forced Vibration Analysis ◽  
Element Method

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.

scholarly journals On stabilization of the node-based smoothed finite element method for free vibration problems

2010 ◽  
Vol 32 (3) ◽  
pp. 167-181
Author(s):  
Bui Xuan Thang ◽  
Nguyen Xuan Hung ◽  
Ngo Thanh Phong
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Solid Mechanics ◽  
High Reliability ◽  
Free Vibration Analysis ◽  
Zero Energy ◽  
Computational Effect ◽  
Smoothed Finite Element Method ◽  
Element Method

The node-based smoothed finite element method (NS-FEM) has been recently proposed by Liu et al to enhance the computational effect for solid mechanics problems. However, it is evident that the NS-FEM behaves "overly-soft" and so it may lead to instability for dynamic problems. The instability can be clearly shown as spurious non-zero energy modes in free vibration analysis. In this paper, we present a stabilization of the node-based smoothed finite element method (SN-FEM) that is stable (no spurious non-zero energy modes) and more effective than the standard finite element method (FEM). Three numerical illustrations are given to evince the high reliability of the proposed formulation.

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

2017 ◽  
Vol 14 (04) ◽  
pp. 1750036 ◽  
Author(s):  
Guirong Liu ◽  
Meng Chen ◽  
Ming Li
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Lower Bound ◽  
Lower Bounds ◽  
Solid Mechanics ◽  
Wave Number ◽  
Vibration Modes ◽  
Large Wave ◽  
Smoothed Finite Element Method ◽  
Element Method

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

A Modified Smoothed Finite Element Method for Static and Free Vibration Analysis of Solid Mechanics

2016 ◽  
Vol 13 (06) ◽  
pp. 1650043 ◽  
Author(s):  
Xiang Yang Cui ◽  
Xiao Bin Hu ◽  
Guang Yao Li ◽  
Gui Rong Liu
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Convergence Rate ◽  
Solid Mechanics ◽  
Weighted Average ◽  
Free Vibration Analysis ◽  
Average Value ◽  
Smoothed Finite Element Method ◽  
Element Method

The smoothed finite element method (S-FEM) proposed recently is more accurate and has higher convergence rate compared with standard four-node isoparametric finite element method (FEM). In this work, a modified S-FEM using four-node quadrilateral elements is proposed, which greatly reduces further the computation cost while maintaining the high accuracy and convergence rate. The key idea of the proposed modification is that the strain of the element is a weighted average value of the smoothed strains in the smoothing cells (SCs), which means that only one integration point is required to construct the stiffness matrix, similar to the single cell S-FEM. A stabilization item is proposed using the differences of the smoothed strains obtained in four SCs, which installs the stability of algorithm and increases the accuracy. To verify the efficiency, accuracy and stability of the present formulation, a number of numerical examples of static and free vibration problems, are studied in comparison with different existing numerical methods.

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