A novel FEM by scaling the gradient of strains with factor α (αFEM)

Liu, G. R.; Nguyen-Thoi, T.; Lam, Kwok‐Yan

doi:10.1007/s00466-008-0311-1

Cited by 60 publications

(41 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the other front of the development of numerical methods, Liu and Nguyen Thoi [39] have integrated the strain smoothing technique [40] into the finite element method (FEM) to create a series of smoothed FEMs (S-FEMs) such as cell/element-based smoothed FEM (CS-FEM) [41][42][43], node-based smoothed FEM (NS-FEM) [44][45][46], edge-based smoothed FEM (ES-FEM) [47,48], face-based smoothed FEM (FS-FEM) [49], and a group of alpha-FEM [50][51][52][53]. Each of these smoothed FEMs has different properties and has been used to produce desired solutions for a wide class of benchmark and practical mechanics problems.…”

mentioning

confidence: 99%

RETRACTED ARTICLE: A limit analysis of Mindlin plates using the cell-based smoothed triangular element CS-MIN3 and second-order cone programming (SOCP)

Nguyen-Thoi

Phung‐Van

Canh

2014

Asia Pac. J. Comput. Engin.

View full text Add to dashboard Cite

Background: The paper presents a numerical procedure for kinematic limit analysis of Mindlin plate governed by von Mises criterion. Methods: The cell-based smoothed three-node Mindlin plate element (CS-MIN3) is combined with a second-order cone optimization programming (SOCP) to determine the upper bound limit load of the Mindlin plates. In the CS-MIN3, each triangular element will be divided into three sub-triangles, and in each sub-triangle, the gradient matrices of MIN3 is used to compute the strain rates. Then the gradient smoothing technique on whole the triangular element is used to smooth the strain rates on these three sub-triangles. The limit analysis problem of Mindlin plates is formulated by minimizing the dissipation power subjected to a set of constraints of boundary conditions and unitary external work. For Mindlin plates, the dissipation power is computed on both the middle plane and thickness of the plate. This minimization problem then can be transformed into a form suitable for the optimum solution using the SOCP. Results and Conclusions: The numerical results of some benchmark problems show that the proposal procedure can provide the reliable upper bound collapse multipliers for both thick and thin plates.

show abstract

mentioning

confidence: 99%

RETRACTED ARTICLE: A limit analysis of Mindlin plates using the cell-based smoothed triangular element CS-MIN3 and second-order cone programming (SOCP)

Nguyen-Thoi

Phung‐Van

Canh

2014

Asia Pac. J. Comput. Engin.

View full text Add to dashboard Cite

show abstract

“…Note the fact that, the solution of the VC FEM with = 0.0 (or FEM) is always an underestimation of that using the VC FEM with = 1± reg and the exact solution [4,29,59]. Combining these results with Remarks 1-5 and Property 1, we have two following important properties on the producibility of the exact solution by the VC FEM.…”

Section: Determination Of Overestimation or Underestimation Problemsmentioning

confidence: 55%

“…In practical application, we need to make sure that a set of meshes with the same aspect ratio is used to locate exact . Some details of using the same aspect ratio of subsequent meshes for the VC FEM can be found in Liu et al [59,60].…”

Section: Property 6 (Location Of the Exact Solution)mentioning

confidence: 99%

See 1 more Smart Citation

A variationally consistent αFEM (VCαFEM) for solution bounds and nearly exact solution to solid mechanics problems using quadrilateral elements

Liu

Nguyen‐Xuan

Nguyen-Thoi

2010

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

SUMMARYA variationally consistent alpha finite element method (VC FEM) for quadrilateral isoparametric elements is presented by constructing an assumed strain field in which the gradient of the compatible strain field is scaled with a free parameter . The assumed strain field satisfies the orthogonal condition and the Hellinger-Reissner variational principle is used to establish the discretized system of equations. It is shown that the strain energy is a second-order continuous function of , and the VC FEM can produce both lower and upper bounds to the exact solution in the strain energy for all elasticity problems by choosing a proper ∈ [0, upper ]. Based on this bound property, an exact-approach has been devised to give an ultra-accurate solution that is very close to the exact one in the strain energy. Furthermore, the exactapproach also works well for volumetric locking problems, by simply replacing the strain gradient matrix by a stabilization matrix. In addition, a regularization-approach has also been suggested to overcome possible hourglass instability. Intensive numerical studies have been conducted to confirm the properties of the present VC FEM, and a very good performance has been found in comparing to a large number of existing FEM models.

show abstract

“…and compatible method, so-called -FEM-T3/T4 [40,41], was proposed combining the overestimation property of FEM and the underestimation property of NS-FEM-T3/T4. By scaling the strain field reproduced by FEM-T3/T4 and NS-FEM-T3/T4 with a continuous factor ∈[0, 1], an underestimated discrete model constructed by -FEM ( = 0) can be continuously shifted to an overestimated discrete model ( = 1).…”

mentioning

confidence: 99%

Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

Zhang

Liu

2010

Numerical Meth Engineering

Self Cite

View full text Add to dashboard Cite

SUMMARYNode-based smoothed finite element method (NS-FEM) using triangular type of elements has been found capable to produce upper bound solutions (to the exact solutions) for force driving static solid mechanics problems due to its monotonic 'soft' behavior. This paper aims to formulate an NS-FEM for lower bounds of the natural frequencies for free vibration problems. To make the NS-FEM temporally stable, an -FEM is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by ∈[0, 1], so that both the properties of stiff FEM and the monotonically soft NS-FEM models can be properly combined for a desired purpose. For temporally stabilizing NS-FEM, is chosen small so that it acts like a 'regularization parameter' making the NS-FEM stable, but still with sufficient softness ensuring lower bounds for natural frequency solution. Our numerical studies demonstrate that (1) using a proper , the spurious non-zero energy modes can be removed and the NS-FEM becomes temporally stable; (2) the stabilized NS-FEM becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum; (3) -FEM can even be tuned for obtaining nearly exact natural frequencies.

show abstract

A novel FEM by scaling the gradient of strains with factor α (αFEM)

Cited by 60 publications

References 39 publications

RETRACTED ARTICLE: A limit analysis of Mindlin plates using the cell-based smoothed triangular element CS-MIN3 and second-order cone programming (SOCP)

RETRACTED ARTICLE: A limit analysis of Mindlin plates using the cell-based smoothed triangular element CS-MIN3 and second-order cone programming (SOCP)

A variationally consistent αFEM (VCαFEM) for solution bounds and nearly exact solution to solid mechanics problems using quadrilateral elements

Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

Contact Info

Product

Resources

About