A meshfree weak-strong (MWS) form method for time dependent problems

Gu, YuanTong; Liu, Guirong

doi:10.1007/s00466-004-0610-0

Cited by 57 publications

(22 citation statements)

References 15 publications

(25 reference statements)

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“…This agrees well with those derived from other methods, e.g., the MK (Bui et al, 2011a,b), MWS (Gu and Liu, 2005), or LoKriging (Li et al, 2004).…”

Section: Tablesupporting

confidence: 91%

“…We have also extended their applicability to study free vibrations of piezoelectric structures for which the electric and the mechanical fields are coupled. It is shown that the numerical results obtained with the CQ4 element agree well with those reported in the literature and obtained by such methods as the boundary element method (Manolis and Beskos, 1988), the FEM (Petyt, 2010;Miranda et al, 2008;Dai and Liu, 2007), mesh-free methods (Bui et al, 2011a,b;Bui and Nguyen, 2011;Kosta and Tsukanov, 2014;Sadeghirad et al, 2009;Cui et al, 2010;Gu and Liu, 2005), and the isogeometric analysis (Valizadeh et al, 2013;Cottrell et al, 2006).…”

Section: Introductionsupporting

confidence: 82%

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Analysis of 2-dimensional transient problems for linear elastic and piezoelectric structures using the consecutive-interpolation quadrilateral element (CQ4)

Bui

Nguyen

Zhang

et al. 2016

European Journal of Mechanics - A/Solids

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a b s t r a c tThe recently developed consecutive-interpolation procedure (CIP) based 4-node quadrilateral element (CQ4) is used to study free and forced 2-dimensional vibrations of elastic and piezoelectric structures of complex geometries. For some of these problems results have also been computed with the standard quadrilateral element under the constraint that the two approaches have the same number of degrees of freedom. In each case it is found that the numerical solution using the CQ4 element has better accuracy than that found with the standard quadrilateral element, and these solutions agree well with results available in the literature. The main difference between the CIP basis functions and those for the traditional 4-node quadrilateral element is that the former incorporate both nodal values and averaged gradients of the trial solution at the nodes and the latter only the nodal values. However, for both sets of basis functions only nodal values appear as unknowns in the matrix formulation of the problem.

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“…This agrees well with those derived from other methods, e.g., the MK (Bui et al, 2011a,b), MWS (Gu and Liu, 2005), or LoKriging (Li et al, 2004).…”

Section: Tablesupporting

confidence: 91%

Section: Introductionsupporting

confidence: 82%

Analysis of 2-dimensional transient problems for linear elastic and piezoelectric structures using the consecutive-interpolation quadrilateral element (CQ4)

Bui

Nguyen

Zhang

et al. 2016

European Journal of Mechanics - A/Solids

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show abstract

“…Some research work has been done to get rid of the shear locking phenomena, e.g., adding the transverse shear strain as another variable in MLPG [6]. The shear locking phenomena could be easily avoided by simple using the high order meshless shape function (more than four interpolation nodes) in the meshless method [12,17]. There is no difficulty to satisfy this requirement in 1-D problem.…”

Section: Validation Example: Static Analysis Of a Fixed-fixed Thin Beammentioning

confidence: 98%

Analysis of Microelectromechanical Systems (MEMS) Devices by the Meshless Point Weighted Least-squares Method

Wang

Lam

2006

Comput Mech

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In this paper, the characteristics of microelectromechanical systems (MEMS) devices are analyzed by a meshless method-point weighted least-squares (PWLS) method. In the present meshless method, field nodes and collocation points are adopted. The field nodes are used to construct the trial functions based on locally supported interpolation domains. The collocation points that can be independent of the field nodes are adopted to evaluate the total residuals of the problem domain and its boundaries. The least-squares technique is used to obtain the solution of the problem by minimizing the functional of the summation of weighted residuals. The present meshless method possesses some advantages compared with the conventional collocation methods, e.g., it is very stable for both regularly or irregularly nodal distributions; the displacement and derivative boundary conditions can be easily enforced; and the final coefficient matrix is symmetric. Several one-dimensional and twodimensional MEMS devices that are governed by the nonlinear equations are studied by the present PWLS method. The simulated results are compared with those obtained by other simulation approaches and experimental results. It is shown that the PWLS method is very efficient and accurate for the analysis of MEMS devices.

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“…Onate et al [14] introduced a stabilization technique by adding artificial terms in both governing equations and Neumann boundary conditions, however, these terms only serve the stabilization purpose and their suitability is restricted to some special problems. Liu and Gu [15,16] proposed a meshfree weak-strong-form method, in which the weak form is applied to the subdomain concerned with Neumann boundary conditions and strong form to the one with Dirichlet boundary conditions. Pan et al [17] presented meshless Galerkin least-squares method by making use of Galerkin method in the boundary domain and least-squares method in the interior domain.…”

Section: Introductionmentioning

confidence: 99%

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

Mai‐Duy

Tran-Cong

et al. 2009

Numerical Meth Engineering

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In this paper, high order systems are reformulated as first order systems which are then numerically solved by a collocation method. The collocation method is based on Cartesian discretisation with 1D-integrated radial basis function networks (1D-IRBFN) [1]. The present method is enhanced by a new boundary interpolation technique based on 1D-IRBFN which is introduced to obtain variable approximation at irregular points in irregular domains. The proposed method is well suited to problems with mixed boundary conditions on both regular and irregular domains. The main results obtained are (a) the boundary conditions for the reformulated problem are of Dirichlet type only; (b) the integrated RBFN approximation avoids the well known reduction of convergence rate associated with differential formulations; (c) the primary variable (e.g. displacement, temperature) and the dual variable (e.g. stress, temperature gradient) have similar convergence order; (d) the volumetric locking effects associated with incompressible materials in solid mechanics are alleviated. Numerical experiments show that the proposed method achieves very good accuracy and high convergence rates.

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A meshfree weak-strong (MWS) form method for time dependent problems

Cited by 57 publications

References 15 publications

Analysis of 2-dimensional transient problems for linear elastic and piezoelectric structures using the consecutive-interpolation quadrilateral element (CQ4)

Analysis of 2-dimensional transient problems for linear elastic and piezoelectric structures using the consecutive-interpolation quadrilateral element (CQ4)

Analysis of Microelectromechanical Systems (MEMS) Devices by the Meshless Point Weighted Least-squares Method

A Cartesian‐grid collocation technique with integrated radial basis functions for mixed boundary value problems

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