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.

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.

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.

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.

An efficient semi-analytical static and free vibration analysis of laminated and sandwich beams based on linear elasticity theory

2021 ◽  
pp. 030932472110626
Author(s):  
Zhao Yin ◽  
Hangduo Gao ◽  
Gao Lin
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Mechanical Load ◽  
Geometric Model ◽  
Free Vibration Analysis ◽  
Sandwich Beams ◽  
Beam Structure ◽  
Precise Integration ◽  
Element Method

Based on the two-dimensional (2D) elastic theory without enforcing any beam assumption, an efficient semi-analytical scaled boundary finite element method (SBFEM) is proposed to solve the bending and free vibration responses of composite laminated and sandwich beams under the mechanical load. The scaled center is placed at infinity, which produces the accurate result by discretizing only the longitudinal direction of the beam structure treated as a one-dimensional (1D) discretization problem. A new kind of 1D high-order spectral element shape functions with the advantages of high accuracy and superior convergence is introduced in SBFEM coordinate system to approximate the geometric model and corresponding variables. The principle of weighted residual in conjunction with the Green’s theorem are applied to obtain the SBFEM governing equation of each layer with respect to radial displacement fields. The solution of equation is indicated analytically by a matrix exponential function, which can be accurately solved by using the precise integration technique (PIT). Finally, an effective and simple stiffness matrix is obtained. By comparing two examples with the solutions based on the finite element method (FEM), the results show that the proposed method has good accuracy and rapid convergence with only a few meshes. The numerical examples are given to investigate the parametric effects of the stacking sequence, thickness ratio, boundary condition, and load form on the variation of the displacement, stress and natural frequency. The results validate that the present technique is also applicable to the complex beam structure with softcore layer inside.

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