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An efficient adaptive analysis procedure using the edge-based smoothed point interpolation method (ES-PIM) for 2D and 3D problems

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2012年第9期36卷 1424-1443页

作者： Tang, Q. Zhang, G. Y. Liu, G. R. Zhong, Z. H. He, Z. C. Univ Western Australia Sch Mech & Chem Engn Nedlands WA 6009 Australia Hunan Univ State Key Lab Adv Design & Mfg Vehicle Body Changsha 410082 Hunan Peoples R China Natl Univ Singapore Dept Mech Engn Singapore 117576 Singapore Univ Cincinnati Sch Aerosp Syst Cincinnati OH 45221 USA

In this paper, an efficient adaptive analysis procedure is proposed using the newly developed edge-based smoothed point interpolation method (ES-PIM) for both two dimensional (2D) and three dimensional (3D) elasticity problems. The ES-PIM works well with three-node triangular and four-node tetrahedral meshes, is easy to be implemented for complicated geometry, and can obtain numerical results of much better accuracy and higher convergence rate than the standard finite element method (FEM) with the same set of meshes. All these important features make it an ideal candidate for adaptive analysis. In the present adaptive procedure, a novel error indicator is devised for ES-PIM settings, which evaluates the maximum difference of strain energy values among the vertexes of each background cell. A simple h-type local refinement scheme is adopted together with a mesh generator based on Delaunay technology. Intensive numerical studies of 20 and 3D examples indicate that the proposed adaptive procedure can effectively capture the stress concentration and solution singularities, carry out local refinement automatically, and hence achieve much higher convergence for the solutions in strain energy norm compared to the general uniform refinement. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Meshfree methods Error indicator Adaptive analysis point interpolation method Delaunay triangulation

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A linearly conforming point interpolation method (LC-PIM) for three-dimensional elasticity problems

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2007年第13期72卷 1524-1543页

作者： Zhang, G. Y. Liu, G. R. Wang, Y. Y. Huang, H. T. Zhong, Z. H. Li, G. Y. Han, X. Natl Univ Singapore Ctr Adv Computat Engn Sci ACES Dept Mech Engn Singapore 117576 Singapore SMA Singapore 117576 Singapore Inst High Performance Comp Singapore Singapore Six Tee Engn Grp Pte Ltd Singapore Singapore Hunan Univ State Key Lab Adv Technol Vechicle Body Desigh Mf Hunan Peoples R China

Linearly conforming point interpolation method (LC-PIM) is formulated for three-dimensional elasticity problems. In this method, shape functions are generated using point interpolation method by adopting polynomial basis functions and local supporting nodes are selected based on the background cells. The shape functions so constructed have the Kronecker delta functions property and it allows straightforward imposition of point essential boundary conditions. Galerkin weak form is used for creating discretized system equations, and a nodal integration scheme with strain- smoothing operation is used to perform the numerical integration. The present LC-PIM can guarantee linear exactness and monotonic convergence for the numerical results. Numerical examples are used to examine the present method in terms of accuracy, convergence, and efficiency. Compared with the finite element method using linear elements, the LC-PIM can achieve better efficiency, and higher accuracy especially for stresses. Copyright (c) 2007 John Wiley & Sons, Ltd.

关键词： mesh free linearly conforming point interpolation method nodal integration three dimensional

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A superconvergent point interpolation method (SC-PIM) with piecewise linear strain field using triangular mesh

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2009年第10期77卷 1439-1467页

作者： Liu, G. R. Xu, X. Zhang, G. Y. Nguyen-Thoi, T. Beijing Inst Technol Dept Mech Sch Sci Beijing 100081 Peoples R China Natl Univ Singapore Ctr Adv Computat Engn Sci ACES Dept Mech Engn Singapore 117576 Singapore Singapore MIT Alliance SMA Singapore 117576 Singapore Jilin Univ Coll Math Changchun 130012 Peoples R China

A superconvergent point interpolation method (SC-PIM) is developed for mechanics problems by combining techniques of finite element method (FEM) and linearly conforming point interpolation method (LC-PIM) using triangular mesh. In the SC-PIM, point interpolation methods (PIM) are used for shape functions construction;and a strain field with a parameter alpha is assumed to be a linear combination of compatible stains and smoothed strains from LC-PIM. We prove theoretically that SC-PIM has a very nice bound property: the strain energy obtained from the SC-PIM solution lies in between those from the compatible FEM solution and the LC-PIM Solution when the same mesh is used. We further provide a criterion for SC-PIM to obtain Upper and lower bound solutions. Intensive numerical studies are conducted to verify these theoretical results and show that (1) the upper and lower bound solutions can always be obtained using the present SC-PIM;(2) there exists an alpha(exact) is an element of (0, 1) at which the SC-PIM can produce the exact solution in the energy norm;(3) for any alpha is an element of (0, 1) the SC-PIM Solution is of superconvergence, and alpha=0 is an easy way to obtain a very accurate and superconvergent solution in both energy and displacement norms;(4) a procedure is devised to find a alpha(prefer) is an element of (0, 1) that produces a solution very close to the exact solution. Copyright (C) 2008 John Wiley & Sons, Ltd.

关键词： meshfree methods finite element method point interpolation method solution bound superconvergence Hellinger-Reissner's variational principle

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A NOVEL SCHEME OF STRAIN-CONSTRUCTED point interpolation method FOR STATIC AND DYNAMIC MECHANICS PROBLEMS

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INTERNATIONAL JOURNAL OF APPLIED MECHANICS 2009年第1期1卷 233-258页

作者： Liu, G. R. Zhang, G. Y. Natl Univ Singapore Dept Mech Engn Ctr Adv Computat Engn Sci ACES Singapore 117576 Singapore Singapore MIT Alliance SMA Singapore 117576 Singapore

This paper presents a new scheme of strain-constructed point interpolation method (SC-PIM) for static, free and forced vibration analysis of solids and structures using triangular cells. In the present scheme, displacement fields are assumed using shape functions created via the point interpolation method (PIM), which possess the Kronecker delta property facilitating the straightforward enforcement of displacement boundary conditions. Using the generalized gradient smoothing technique, the "smoothed" strains at the middle points of the cells edges are first obtained using the corresponding edge-based smoothing domains and the assumed displacement field. In each triangular background cell, the strains at the vertices are assigned using these smoothed strains in a proper manner, and then piecewisely linear strain fields are constructed by the linear interpolation for each sub-triangular cell using the edge-based "smoothed" strains. With the assumed displacements and constructed linear strain fields, the discretized system equations are created using the Strain Constructed Galerkin (SC-Galerkin) weak form. A number of benchmark numerical examples, including the standard patch test, static, free and forced vibration problems, have been studied and intensive numerical results have demonstrated that the present method possesses the following properties: (1) it works well with the simplest triangular mesh, no additional degrees of freedom and parameters are introduced and very easy to implement;(2) it is at least linearly conforming;(3) it possesses a close-to-exact stiffness: it is much stiffer than the "overly-soft" node-based smoothed point interpolation method (NS-PIM) and much softer than the "overly-stiff" FEM model;(4) the results of the present method are of superconvergence and ultra-accuracy: about one order of magnitude more accurate than those of the linear FEM;(5) there are no spurious non-zeros energy modes found and it is also temporally stable, hence the pr

关键词： Numerical method meshfree method point interpolation method weakened weak form gradient smoothing strain construction

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Numerical analysis for multi-group neutron-diffusion equation using Radial point interpolation method (RPIM)

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ANNALS OF NUCLEAR ENERGY 2017年 99卷 193-198页

作者： Kim, Kyung-O Jeong, Hae Sun Jo, Daeseong Korea Atom Energy Res Inst 1045 Daedeok Daero Daejeon 305353 South Korea Kyungpook Natl Univ Sch Mech Engn 80 Daehak Ro Daegu South Korea

A mesh-free method is introduced to overcome the drawbacks (e.g., mesh generation and connectivity definition between the meshes) of mesh-based (nodal) methods such as the finite-element method and finite-difference method. In particular, the point interpolation method (PIM) using a radial basis function is employed in the numerical analysis for the multi-group neutron-diffusion equation. The benchmark calculations are performed for the 2D homogeneous and heterogeneous problems, and the Multiquadrics (MQ) and Gaussian (EXP) functions are employed to analyze the effect of the radial basis function on the numerical solution. Additionally, the effect of the dimensionless shape parameter in those functions on the calculation accuracy is evaluated. According to the results, the radial PIM (RPIM) can provide a highly accurate solution for the multiplication eigenvalue and the neutron flux distribution, and the numerical solution with the MQ radial basis function exhibits the stable accuracy with respect to the reference solutions compared with the other solution. The dimensionless shape parameter directly affects the calculation accuracy and computing time. Values between 1.87 and 3.0 for the benchmark problems considered in this study lead to the most accurate solution. The difference between the analytical and numerical results for the neutron flux is significantly increased in the edge of the problem geometry, even though the maximum difference is lower than 4%. This phenomenon seems to arise from the derivative boundary condition at (x,0) and (0,y) positions, and it may be necessary to introduce additional strategy (e.g., the method using fictitious points and Hermite-type collocation method) in order to solve this problem. On the basis of these results, it is expected that the mesh-free method including the RPIM can be sufficiently employed in numerical analysis for the multi-group neutron-diffusion equation and can be considered as an alternative numerical approach

关键词： Mesh-free method point interpolation method Radial point interpolation method Neutron-diffusion equation Multiplication eigenvalue

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A meshless method based on point interpolation method (PIM) for the space fractional diffusion equation

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APPLIED MATHEMATICS AND COMPUTATION 2015年 256卷 930-938页

作者： Liu, Q. Liu, F. Gu, Y. T. Zhuang, P. Chen, J. Turner, I. Xiamen Univ Sch Math Sci Xiamen Fujian Peoples R China Queensland Univ Technol Sch Math Sci Brisbane Qld 4001 Australia Queensland Univ Technol Sch Chem Phys & Mech Engn Brisbane Qld 4001 Australia Jimei Univ Sch Sci Xiamen Fujian Peoples R China Xiamen Univ Fujian Prov Key Lab Math Modeling & High Performa Xiamen 361005 Peoples R China

This paper aims to develop a meshless approach based on the point interpolation method (PIM) for numerical simulation of a space fractional diffusion equation. Two fully-discrete schemes for the one-dimensional space fractional diffusion equation are obtained by using the PIM and the strong-forms of the space diffusion equation. Numerical examples with different nodal distributions are studied to validate and investigate the accuracy and efficiency of the newly developed meshless approach. (C) 2015 Elsevier Inc. All rights reserved.

关键词： Fractional diffusion equation Meshless approach point interpolation method Numerical simulation

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APPLICATION OF point interpolation method IN JOINTED ROCK MASS

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2011年第1期8卷 41-55页

作者： Hajiazizi, M. Ashory, M. R. Semnan Univ Dept Civil Engn Semnan Iran Semnan Univ Sch Mech Engn Semnan Iran

In this paper, point interpolation method (PIM) is applied to analyze the nonlinear and elastoplastic behavior of jointed rock masses. PIM is advantageous over the other meshless methods that are based on moving least square (MLS) technique. On the other hand in this study, element-free Galerkin method (EFGM) has been applied to analyze a jointed rock mass for comparison purposes. The domain of jointed rock mass is represented by a set of arbitrary distributed nodes. The essential boundary conditions are imposed in EFGM using penalty method. Mohr-Coulomb failure criterion is used for analyzing the behavior of joint failure. Two numerical examples are considered to demonstrate the effectiveness of method in predicting the elastoplastic behavior of jointed rock mass. The results are compared with those of finite difference method (FDM).

关键词： point interpolation method elastoplastic analysis jointed rock mass

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A diagonal offset algorithm for the polynomial point interpolation method

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COMMUNICATIONS IN NUMERICAL methodS IN ENGINEERING 2008年第12期24卷 1909-1922页

作者： Kanber, B. Bozkurt, O. Y. Gaziantep Univ Dept Mech Engn TR-27310 Gaziantep Turkey

A diagonal offset algorithm is presented to overcome the singular moment matrix problem in the polynomial point interpolation method. The value of terms in the diagonal line of moment matrix is changed with different offsets. The effects of the offsets on the Kronecker delta function and partition of unity properties are investigated and an optimum offset range is proposed. The algorithm is validated solving some patch tests and various elasticity problems in 2-D domain. Their results demonstrate that the proposed algorithm is very easy to implement and completely solves the singular moment matrix problem. The accuracy of the method with proposed algorithm is investigated for regular and irregular local domains. Copyright (C) 2007 John Wiley & Sons, Ltd.

关键词： mesh-free methods point interpolation method basis functions polynomial interpolation stress analysis

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A CONFORMING point interpolation method (CPIM) BY SHAPE FUNCTION RECONSTRUCTION FOR ELASTICITY PROBLEMS

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2010年第3期7卷 369-395页

作者： Xu, X. Liu, G. R. Gu, Y. T. Zhang, G. Y. Jilin Univ Coll Math Changchun 130021 Peoples R China Singapore MIT Alliance SMA Singapore 117576 Singapore Queensland Univ Technol Sch Engn Syst Brisbane Qld 4001 Australia Natl Univ Singapore Ctr Adv Computat Engn Sci ACES Dept Mech Engn Singapore 117576 Singapore

A conforming point interpolation method (CPIM) is proposed based on the Galerkin formulation for 2D mechanics problems using triangular background cells. A technique for reconstructing the PIM shape functions is proposed to create a continuous displacement field over the whole problem domain, which guarantees the CPIM passing the standard patch test. We prove theoretically the existence and uniqueness of the CPIM solution, and conduct detailed analyses on the convergence rate;computational efficiency and band width of the stiffness matrix of CPIM. The CPIM does not introduce any additional degrees of freedoms compared to the linear FEM and original PIM;while convergence rate of quadratic CPIM is in between that of linear FEM and quadratic FEM which results in the high computational efficiency. Intensive numerical studies verify the properties of the CPIM.

关键词： Numerical method computational method finite element method meshfree methods point interpolation method convergence rates, computational efficiency, non-conforming

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A complex variable boundary point interpolation method for the nonlinear Signorini problem

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2020年第12期79卷 3297-3309页

作者： Li, Xiaolin Li, Shuling Chongqing Normal Univ Sch Math Sci Chongqing 400047 Peoples R China Chongqing Normal Univ Key Lab Optimizat & Control Minist Educ Chongqing 400047 Peoples R China

A complex variable boundary point interpolation method (CVBPIM) is presented for numerical analysis of the nonlinear Signorini problem. To reduce the computational cost and to improve the stability of the point interpolation method, a complex variable point interpolation method (CVPIM) is developed to construct meshless shape functions with interpolation properties. By using a projection technique to deal with the nonlinear inequality boundary conditions, and combining boundary integral equations with the CVPIM to establish discrete linear algebraic systems, the CVBPIM is an easy-to-implement and boundary-only meshless method for nonlinear Signorini problems. Numerical results are presented to show the accuracy and efficiency of the method. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： Meshless method Boundary integral equation point interpolation method Complex variable Nonlinear inequality boundary condition Signorini problem

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