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A new complex variable meshless method for transient heat conduction problems 被引量:5
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作者 王健菲 程玉民 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第12期42-50,共9页
In this paper, based on the improved complex variable moving least-square (ICVMLS) approximation, a new complex variable meshless method (CVMM) for two-dimensional (2D) transient heat conduction problems is pres... In this paper, based on the improved complex variable moving least-square (ICVMLS) approximation, a new complex variable meshless method (CVMM) for two-dimensional (2D) transient heat conduction problems is presented. The variational method is employed to obtain the discrete equations, and the essential boundary conditions are imposed by the penalty method. As the transient heat conduction problems are related to time, the Crank-Nicolson difference scheme for two-point boundary value problems is selected for the time discretization. Then the corresponding formulae of the CVMM for 2D heat conduction problems are obtained. In order to demonstrate the applicability of the proposed method, numerical examples are given to show the high convergence rate, good accuracy, and high efficiency of the CVMM presented in this paper. 展开更多
关键词 meshless method improved complex variable moving least-square approximation com-plex variable meshless method transient heat conduction problem
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New complex variable meshless method for advection-diffusion problems 被引量:1
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作者 王健菲 程玉民 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第3期92-98,共7页
In this paper,an improved complex variable meshless method(ICVMM) for two-dimensional advection-diffusion problems is developed based on improved complex variable moving least-square(ICVMLS) approximation.The equi... In this paper,an improved complex variable meshless method(ICVMM) for two-dimensional advection-diffusion problems is developed based on improved complex variable moving least-square(ICVMLS) approximation.The equivalent functional of two-dimensional advection-diffusion problems is formed,the variation method is used to obtain the equation system,and the penalty method is employed to impose the essential boundary conditions.The difference method for twopoint boundary value problems is used to obtain the discrete equations.Then the corresponding formulas of the ICVMM for advection-diffusion problems are presented.Two numerical examples with different node distributions are used to validate and investigate the accuracy and efficiency of the new method in this paper.It is shown that ICVMM is very effective for advection-diffusion problems,and has a good convergent character,accuracy,and computational efficiency. 展开更多
关键词 meshless method improved complex variable moving least-square approximation improved complex variable meshless method advection-diffusion problem
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A meshless method for the nonlinear generalized regularized long wave equation
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作者 王聚丰 白福浓 程玉民 《Chinese Physics B》 SCIE EI CAS CSCD 2011年第3期35-42,共8页
This paper presents a meshless method for the nonlinear generalized regularized long wave (GRLW) equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtain... This paper presents a meshless method for the nonlinear generalized regularized long wave (GRLW) equation based on the moving least-squares approximation. The nonlinear discrete scheme of the GRLW equation is obtained and is solved using the iteration method. A theorem on the convergence of the iterative process is presented and proved using theorems of the infinity norm. Compared with numerical methods based on mesh, the meshless method for the GRLW equation only requires the.scattered nodes instead of meshing the domain of the problem. Some examples, such as the propagation of single soliton and the interaction of two solitary waves, are given to show the effectiveness of the meshless method. 展开更多
关键词 generalized regularized long wave equation meshless method moving least-squares approximation CONVERGENCE
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A STUDY ON THE WEIGHT FUNCTION OF THE MOVING LEAST SQUARE APPROXIMATION IN THE LOCAL BOUNDARY INTEGRAL EQUATION METHOD 被引量:4
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作者 Long Shuyao Hu De’an (Department of Engineering Mechanics,Hunan University,Changsha 410082,China) 《Acta Mechanica Solida Sinica》 SCIE EI 2003年第3期276-282,共7页
The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the bas... The meshless method is a new numerical technique presented in recent years.It uses the moving least square(MLS)approximation as a shape function.The smoothness of the MLS approximation is determined by that of the basic function and of the weight function,and is mainly determined by that of the weight function.Therefore,the weight function greatly affects the accuracy of results obtained.Different kinds of weight functions,such as the spline function, the Gauss function and so on,are proposed recently by many researchers.In the present work,the features of various weight functions are illustrated through solving elasto-static problems using the local boundary integral equation method.The effect of various weight functions on the accuracy, convergence and stability of results obtained is also discussed.Examples show that the weight function proposed by Zhou Weiyuan and Gauss and the quartic spline weight function are better than the others if parameters c and α in Gauss and exponential weight functions are in the range of reasonable values,respectively,and the higher the smoothness of the weight function,the better the features of the solutions. 展开更多
关键词 weight function meshless methods local boundary integral equation method moving least square approximation
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Global interpolating meshless shape function based on generalized moving least-square for structural dynamic analysis 被引量:2
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作者 Dan XIE Kailin JIAN Weibin WEN 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2016年第9期1153-1176,共24页
A global interpolating meshless shape function based on the generalized moving least-square (GMLS) is formulated by the transformation technique. Both the shape function and its derivatives meet the Kronecker delta ... A global interpolating meshless shape function based on the generalized moving least-square (GMLS) is formulated by the transformation technique. Both the shape function and its derivatives meet the Kronecker delta function property. With the interpolating GMLS (IGMLS) shape function, an improved element-free Galerkin (EFG) method is proposed for the structural dynamic analysis. Compared with the conven- tional EFG method, the obvious advantage of the proposed method is that the essential boundary conditions including both displacements and derivatives can be imposed by the straightforward way. Meanwhile, it can greatly improve the ill-condition feature of the standard GMLS approximation, and provide good accuracy at low cost. The dynamic analyses of the Euler beam and Kirchhoff plate are performed to demonstrate the feasi- bility and effectiveness of the improved method. The comparison between the numerical results of the conventional method and the improved method shows that the proposed method has better stability, higher accuracy, and less time consumption. 展开更多
关键词 structural dynamics meshless method element-free Galerkin (EFG)method generalized moving least-square (GMLS)
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A meshless algorithm with moving least square approximations for elliptic Signorini problems 被引量:1
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作者 王延冲 李小林 《Chinese Physics B》 SCIE EI CAS CSCD 2014年第9期35-42,共8页
Based on the moving least square (MLS) approximations and the boundary integral equations (BIEs), a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection ope... Based on the moving least square (MLS) approximations and the boundary integral equations (BIEs), a meshless algorithm is presented in this paper for elliptic Signorini problems. In the algorithm, a projection operator is used to tackle the nonlinear boundary inequality conditions. The Signorini problem is then reformulated as BIEs and the unknown boundary variables are approximated by the MLS approximations. Accordingly, only a nodal data structure on the boundary of a domain is required. The convergence of the algorithm is proven. Numerical examples are given to show the high convergence rate and high computational efficiency of the presented algorithm. 展开更多
关键词 meshless method Signorini problem moving least square approximations CONVERGENCE
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STUDY ON PARAMETERS FOR TOPOLOGICAL VARIABLES FIELD INTERPOLATED BY MOVING LEAST SQUARE APPROXIMATION 被引量:3
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作者 Rehan H.Zuberi 《Acta Mechanica Solida Sinica》 SCIE EI 2009年第2期180-188,共9页
This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the materi... This paper presents a new approach to the structural topology optimization of continuum structures. Material-point independent variables are presented to illustrate the existence condition,or inexistence of the material points and their vicinity instead of elements or nodes in popular topology optimization methods. Topological variables field is constructed by moving least square approximation which is used as a shape function in the meshless method. Combined with finite element analyses,not only checkerboard patterns and mesh-dependence phenomena are overcome by this continuous and smooth topological variables field,but also the locations and numbers of topological variables can be arbitrary. Parameters including the number of quadrature points,scaling parameter,weight function and so on upon optimum topological configurations are discussed. Two classic topology optimization problems are solved successfully by the proposed method. The method is found robust and no numerical instabilities are found with proper parameters. 展开更多
关键词 topological optimization continuum structure meshless method moving least square approximation checkerboard pattern mesh-dependence
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A complex variable meshless method for fracture problems 被引量:16
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作者 CHENG Yumin1 & LI Jiuhong2 1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China 2. Department of Building Engineering, Xi’an University of Technology, Xi’an 710048, China 《Science China(Physics,Mechanics & Astronomy)》 SCIE EI CAS 2006年第1期46-59,共14页
Based on the moving least-square (MLS) approximation, the complex variable moving least-square approximation (CVMLS) is discussed in this paper. The complex variable moving least-square approximation cannot form ill-c... Based on the moving least-square (MLS) approximation, the complex variable moving least-square approximation (CVMLS) is discussed in this paper. The complex variable moving least-square approximation cannot form ill-conditioned equations, and has greater precision and computational efficiency. Using the analytical solution near the tip of a crack, the trial functions in the complex variable moving least-square approxi- mation are extended, and the corresponding approximation function is obtained. And from the minimum potential energy principle, a complex variable meshless method for fracture problems is presented, and the formulae of the complex variable meshless method are obtained. The complex variable meshless method in this paper has greater precision and computational efficiency than the conventional meshless method. Some examples are given. 展开更多
关键词 moving least-square approximation COMPLEX VARIABLE moving least-square approximation meshless method COMPLEX VARIABLE meshless method fracture.
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ANALYSIS OF 2-D THIN STRUCTURES BY THE MESHLESS REGULAR HYBRID BOUNDARY NODE METHOD 被引量:8
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作者 Zhang Jianming Yao Zhenhan 《Acta Mechanica Solida Sinica》 SCIE EI 2002年第1期36-44,共9页
Thin structures are generally solved by the Finite Element Method(FEM), using plate or shell finite elements which have manylimitations in applications, such as numerical locking, edge effects,length scaling and the c... Thin structures are generally solved by the Finite Element Method(FEM), using plate or shell finite elements which have manylimitations in applications, such as numerical locking, edge effects,length scaling and the cnvergence problem. Recently, by proposing anew approach to tranting the nearly- singular integrals, Liu et al.developed a BEM to successfully solve thin structures with thethickness-to- length ratios in the micro-or nano-scales. On the otherhand, the meshless Regular Hybrid Boundary Node Method (RHBNM), whichis proposed by the current authors and based on a modified functionaland the Moving Least-Square (MLS) approximation, has very promisingapplications for engineering problems owing To its meshless natureand dimension-reduction advantage, and not involving any singular ornearly-singular Integrals. Test examples show that the RHBNM can alsobe applied readily to thin structures with high accu- Racy withoutany modification. 展开更多
关键词 shell-like structures meshless moving least squares approximation hybridboundary node method
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Meshless analysis of an improved element-free Galerkin method for linear and nonlinear elliptic problems 被引量:2
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作者 唐耀宗 李小林 《Chinese Physics B》 SCIE EI CAS CSCD 2017年第3期215-225,共11页
We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin... We first give a stabilized improved moving least squares (IMLS) approximation, which has better computational stability and precision than the IMLS approximation. Then, analysis of the improved element-free Galerkin method is provided theoretically for both linear and nonlinear elliptic boundary value problems. Finally, numerical examples are given to verify the theoretical analysis. 展开更多
关键词 meshless method moving least squares approximation element-free Galerkin method error esti-mate
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The complex variable meshless local Petrov-Galerkin method of solving two-dimensional potential problems 被引量:1
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作者 杨秀丽 戴保东 张伟伟 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第10期49-55,共7页
Based on the complex variable moving least-square(CVMLS) approximation and a local symmetric weak form,the complex variable meshless local Petrov-Galerkin(CVMLPG) method of solving two-dimensional potential proble... Based on the complex variable moving least-square(CVMLS) approximation and a local symmetric weak form,the complex variable meshless local Petrov-Galerkin(CVMLPG) method of solving two-dimensional potential problems is presented in this paper.In the present formulation,the trial function of a two-dimensional problem is formed with a one-dimensional basis function.The number of unknown coefficients in the trial function of the CVMLS approximation is less than that in the trial function of the moving least-square(MLS) approximation.The essential boundary conditions are imposed by the penalty method.The main advantage of this approach over the conventional meshless local Petrov-Galerkin(MLPG) method is its computational efficiency.Several numerical examples are presented to illustrate the implementation and performance of the present CVMLPG method. 展开更多
关键词 meshless method complex variable moving least-square method complex variable meshless local Petrov-Galerkin method potential problems
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A meshless local Petrov–Galerkin method for solving the neutron diffusion equation
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作者 Shima Tayefi Ali Pazirandeh Mohsen Kheradmand Saadi 《Nuclear Science and Techniques》 SCIE CAS CSCD 2018年第11期304-322,共19页
The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshl... The goal of this study is to solve the neutron diffusion equation by using a meshless method and evaluate its performance compared to traditional methods. This paper proposes a novel method based on coupling the meshless local Petrov–Galerkin approach and the moving least squares approximation. This computational procedure consists of two main steps. The first involved applying the moving least squares approximation to construct the shape function based on the problem domain. Then, the obtained shape function was used in the meshless local Petrov–Galerkin method to solve the neutron diffusion equation.Because the meshless method is based on eliminating the mesh-based topologies, the problem domain was represented by a set of arbitrarily distributed nodes. There is no need to use meshes or elements for field variable interpolation. The process of node generation is simply and fully automated, which can save time. As this method is a local weak form, it does not require any background integration cells and all integrations are performed locally over small quadrature domains. To evaluate the proposed method,several problems were considered. The results were compared with those obtained from the analytical solution and a Galerkin finite element method. In addition, the proposed method was used to solve neutronic calculations in thesmall modular reactor. The results were compared with those of the citation code and reference values. The accuracy and precision of the proposed method were acceptable. Additionally, adding the number of nodes and selecting an appropriate weight function improved the performance of the meshless local Petrov–Galerkin method. Therefore, the proposed method represents an accurate and alternative method for calculating core neutronic parameters. 展开更多
关键词 Neutron diffusion equation meshless LOCAL Petrov–Galerkin(MLPG) moving least SQUARES approximation(MLSA) meshless methods
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A complex variable meshless local Petrov-Galerkin method for transient heat conduction problems
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作者 王启防 戴保东 栗振锋 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第8期238-244,共7页
On the basis of the complex variable moving least-square (CVMLS) approximation, a complex variable meshless local Petrov-Galerkin (CVMLPG) method is presented for transient heat conduction problems. The method is ... On the basis of the complex variable moving least-square (CVMLS) approximation, a complex variable meshless local Petrov-Galerkin (CVMLPG) method is presented for transient heat conduction problems. The method is developed based on the CVMLS approximation for constructing shape functions at scattered points, and the Heaviside step function is used as a test function in each sub-domain to avoid the need for a domain integral in symmetric weak form. In the construction of the well-performed shape function, the trial function of a two-dimensional (2D) problem is formed with a one-dimensional (1D) basis function, thus improving computational efficiency. The numerical results are compared with the exact solutions of the problems and the finite element method (FEM). This comparison illustrates the accuracy as well as the capability of the CVMLPG method. 展开更多
关键词 meshless method complex variable moving least-square method complex variable meshless localPetrov-Galerkin method transient heat conduction problems
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Elastoplastic Large Deformation Using Meshless Integral Method
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作者 Jianfeng Ma X. J. Xin 《World Journal of Mechanics》 2012年第6期339-360,共22页
In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integ... In this paper, the meshless integral method based on the regularized boundary integral equation [1] has been extended to analyze the large deformation of elastoplastic materials. The updated Lagrangian governing integral equation is obtained from the weak form of elastoplasticity based on Green-Naghdi’s theory over a local sub-domain, and the moving least-squares approximation is used for meshless function approximation. Green-Naghdi’s theory starts with the additive decomposition of the Green-Lagrange strain into elastic and plastic parts and considers aJ2elastoplastic constitutive law that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. A simple, generalized collocation method is proposed to enforce essential boundary conditions straightforwardly and accurately, while natural boundary conditions are incorporated in the system governing equations and require no special handling. The solution algorithm for large deformation analysis is discussed in detail. Numerical examples show that meshless integral method with large deformation is accurate and robust. 展开更多
关键词 meshless method Large Deformation Local Boundary Integral Equation moving least-squareS approximation SUBTRACTION method SINGULARITY Removal Elastoplasticity Green-Naghdi’s Theory
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A new complex variable element-free Galerkin method for two-dimensional potential problems 被引量:4
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作者 程玉民 王健菲 白福浓 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第9期43-52,共10页
In this paper, based on the element-free Galerkin (EFG) method and the improved complex variable moving least- square (ICVMLS) approximation, a new meshless method, which is the improved complex variable element-f... In this paper, based on the element-free Galerkin (EFG) method and the improved complex variable moving least- square (ICVMLS) approximation, a new meshless method, which is the improved complex variable element-free Galerkin (ICVEFG) method for two-dimensional potential problems, is presented. In the method, the integral weak form of control equations is employed, and the Lagrange multiplier is used to apply the essential boundary conditions. Then the corresponding formulas of the ICVEFG method for two-dimensional potential problems are obtained. Compared with the complex variable moving least-square (CVMLS) approximation proposed by Cheng, the functional in the ICVMLS approximation has an explicit physical meaning. Furthermore, the ICVEFG method has greater computational precision and efficiency. Three numerical examples are given to show the validity of the proposed method. 展开更多
关键词 meshless method improved complex variable moving least-square approximation im- proved complex variable element-free Galerkin method potential problem
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Complex variable element-free Galerkin method for viscoelasticity problems 被引量:2
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作者 程玉民 李荣鑫 彭妙娟 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第9期60-71,共12页
Based on the complex variable moving least-square (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional viscoelasticity problems under the creep condition is presente... Based on the complex variable moving least-square (CVMLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional viscoelasticity problems under the creep condition is presented in this paper. The Galerkin weak form is employed to obtain the equation system, and the penalty method is used to apply the essential boundary conditions, then the corresponding formulae of the CVEFG method for two-dimensional viscoelasticity problems under the creep condition are obtained. Compared with the element-free Galerkin (EFG) method, with the same node distribution, the CVEFG method has higher precision, and to obtain the similar precision, the CVEFG method has greater computational efficiency. Some numerical examples are given to demonstrate the validity and the efficiency of the method. 展开更多
关键词 meshless method complex variable moving least-square approximation complex variableelement-free Galerkin method VISCOELASTICITY
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Meshless acoustic analysis using a weakly singular Burton-Miller boundary integral formulation
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作者 Linchong CHEN Xiaolin LI 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2020年第12期1897-1914,共18页
The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method(CVBEFM)for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers.To regularize... The Burton-Miller boundary integral formulation is solved by a complex variable boundary element-free method(CVBEFM)for the boundary-only meshless analysis of acoustic problems with arbitrary wavenumbers.To regularize both strongly singular and hypersingular integrals and to avoid the computation of the solid angle and its normal derivative,a weakly singular Burton-Miller formulation is derived by considering the normal derivative of the solid angle and adopting the singularity subtraction procedures.To facilitate the implementation of the CVBEFM and the approximation of gradients of the boundary variables,a stabilized complex variable moving least-square approximation is selected in the meshless discretization procedure.The results show the accuracy and efficiency of the present CVBEFM and reveal that the method can produce satisfactory results for all wavenumbers,even for extremely large wavenumbers such as k=10000. 展开更多
关键词 meshless method complex variable moving least-square approximation boundary element-free method Burton-Miller formulation REGULARIZATION high frequency acoustic problem large wavenumber
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An improved interpolating element-free Galerkin method with a nonsingular weight function for two-dimensional potential problems 被引量:15
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作者 王聚丰 孙凤欣 程玉民 《Chinese Physics B》 SCIE EI CAS CSCD 2012年第9期53-59,共7页
In this paper, an improved interpolating moving least-square (IIMLS) method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the II... In this paper, an improved interpolating moving least-square (IIMLS) method is presented. The shape function of the IIMLS method satisfies the property of the Kronecker 5 function. The weight function used in the IIMLS method is nonsingular. Then the IIMLS method can overcome the difficulties caused by the singularity of the weight function in the IMLS method. The number of unknown coefficients in the trial function of the IIMLS method is less than that of the moving least-square (MLS) approximation. Then by combining the IIMLS method with the Galerkin weak form of the potential problem, the improved interpolating element-free Galerkin (IIEFG) method for two-dimensional potential problems is presented. Compared with the conventional element-free Galerkin (EFG) method, the IIEFG method can directly use the essential boundary conditions. Then the IIEFG method has higher accuracy. For demonstration, three numerical examples are solved using the IIEFG method. 展开更多
关键词 meshless method improved interpolating moving least-square method improved inter-polating element-free Galerkin method potential problem
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A 3D shell-like approach using element-free Galerkin method for analysis of thin and thick plate structures 被引量:6
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作者 Yu Yin Lin-Quan Yao Yang Cao 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2013年第1期85-98,共14页
A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analo... A new efficient meshless method based on the element-free Galerkin method is proposed to analyze the static deformation of thin and thick plate structures in this paper. Using the new 3D shell-like kinematics in analogy to the solid-shell concept of the finite element method, discretization is carried out by the nodes located on the upper and lower surfaces of the structures. The approximation of all unknown field variables is carried out by using the moving least squares (MLS) approximation scheme in the in-plane directions, while the linear interpolation is applied through the thickness direction. Thus, different boundary conditions are defined only using displacements and penalty method is used to enforce the essential boundary conditions. The constrained Galerkin weak form, which incorporates only dis- placement degrees of freedom (d.o.f.s), is derived. A modified 3D constitutive relationship is adopted in order to avoid or eliminate some self-locking effects. The numeric efficiency of the proposed meshless formulation is illustrated by the numeric examples. 展开更多
关键词 meshless methods 3D shell-like moving least squares approximation SELF-LOCKING Thin plate
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An improved boundary element-free method (IBEFM) for two-dimensional potential problems 被引量:8
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作者 任红萍 程玉民 张武 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第10期4065-4073,共9页
The interpolating moving least-squares (IMLS) method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method (B... The interpolating moving least-squares (IMLS) method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method (BEFM), combining the boundary integral equation (BIE) method with the IMLS method, the improved boundary element-free method (IBEFM) for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker ~ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker 5 function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method. 展开更多
关键词 moving least-squares approximation interpolating moving least-squares method mesh- less method improved boundary element-free method potential problem
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