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An advanced numerical modeling for Riesz space fractional advection-dispersion equations by a meshfree approach

APPLIED MATHEMATICAL MODELLING 2016年第17-18期40卷 7816-7829页

作者： Yuan, Z. B. Nie, Y. F. Liu, F. Turner, I. Zhang, G. Y. Gu, Y. T. Northwestern Polytech Univ Sch Sci Xian 710072 Peoples R China Queensland Univ Technol Sch Math Sci GPO Box 2434 Brisbane Qld 4001 Australia Dalian Univ Technol Sch Naval Architecture Dalian 116024 Peoples R China Queensland Univ Technol Chem Phys & Mech Engn Sch GPO Box 2434 Brisbane Qld 4001 Australia

The space fractional advection-dispersion equations (SFADE) have been found to be very adequate in describing anomalous transport and dispersion phenomena. Aside from enjoying huge advantage in modeling, we have to face severe challenge presented by the non-locality of space fractional order derivatives, which is difficult to be handled by the traditional finite difference method (FDM) especially on a complex domain with irregularly distributed nodes. Therefore, it is crucial to develop a powerful numerical method to overcome this barrier. In this paper, the point interpolation method (PIM), a meshfree method, is further developed to solve SFADE, where the polynomial point-interpolation functions and their fractional derivatives with explicit expressions are substituted into Galerkin weak form of SFADE to obtain the discrete approximation system. Adopting an expanding technique, we develop an extended point interpolation method for SFADE with zero Dirichlet boundary condition. This technique avoids singular integral and turns system matrix into Toeplitz matrix. An efficient iteration algorithm is also suggested to reduce computing time and storage. Numerical experiments is presented to validate the newly developed method and to investigate accuracy and efficiency. (C) 2016 Elsevier Inc. All rights reserved.

关键词： point interpolation method Space fractional advection-dispersion equations Riemann-Liouville fractional derivatives Toeplitz matrix

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Meshfree Sequentially Linear Analysis of Concrete

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JOURNAL OF COMPUTING IN CIVIL ENGINEERING 2016年第2期30卷 1-9页

作者： Al-Sabah, A. Salam Laefer, Debra F. Univ Coll Dublin Sch Civil Struct & Environm Engn Urban Modelling Grp Newstead Room G67 Dublin 4 Ireland Univ Coll Dublin Sch Civil Struct & Environm Engn Head Urban Modelling Grp Newstead Room G25 Dublin 4 Ireland

A new, meshfree method employing the node-based, smoothed point interpolation method (NS-PIM) is presented as an alternative to the nonlinear finite-element approach for concrete members. The nonlinear analysis is replaced by sequentially linear analyses (SLA), and a smeared, fixed concrete cracking model was used. A notched concrete beam was employed for validation. Using a crack band width factor of 2.0 and 10-mm nodal spacing, the peak load differed by only 3.5% from experimental results. Overall results were similar to experimental ones, as well as to those published by researchers using finite-element SLA. The approach provides two major advantages over finite-element-based SLA: (1)nodal distortion insensitivity, and (2)nodal spacing insensitivity. (C) 2015 American Society of Civil Engineers.

关键词： Mesh free method point interpolation method Sequentially linear analysis Concrete Cracking

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Construct 'FE-Meshfree' Quad4 using mean value coordinates

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2015年 59卷 78-88页

作者： Yang, Yongtao Tang, Xuhai Zheng, Hong Chinese Acad Sci Inst Rock & Soil Mech State Key Lab Geomech & Geotech Engn Wuhan Peoples R China Princeton Univ Dept Civil & Environm Engn Princeton NJ 08544 USA

The present work uses mean value coordinates to construct the shape functions of a hybrid 'FE-Meshfree' quadrilateral element, which is named as Quad4-MVC. This Quad4-MVC can be regarded as the development of the 'FE-Meshfree' quadrilateral element with radial-polynomial point interpolation (Quad4-RPIM). Similar to Quad4-RPIM, Quad4-MVC has Kronecker delta property on the boundaries of computational domain, so essential boundary conditions can be enforced as conveniently as in the finite element method (FEM). The novelty of the present work is to construct nodal approximations using mean value coordinates, instead of radial basis functions which are used in Quad4-RPIM. Compared to the radial basis functions, mean value coordinates does not utilize any uncertain parameters, which enhances stability of numerical results. Numerical tests in this paper show that the performance of Quad4-RPIM becomes even worse than four-node iso-parametric element (Quad4) when the parameters of radial basis functions are not chosen properly. However, the performance of Quad4-MVC is stably better than Quad4. Published by Elsevier Ltd.

关键词： Mean value coordinates Quad4-MVC FE-Meshfree point interpolation method Mesh distortion

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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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An extended Galerkin weak form and a point interpolation method with continuous strain field and superconvergence using triangular mesh

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COMPUTATIONAL MECHANICS 2009年第5期43卷 651-673页

作者： Liu, G. R. Xu, X. Zhang, G. Y. Gu, Y. T. Jilin Univ Coll Math Changchun 130012 Peoples R China Natl Univ Singapore Ctr Adv Computat Engn Sci Dept Mech Engn Singapore 117576 Singapore SMA Singapore 117576 Singapore Queensland Univ Technol Sch Engn Syst Brisbane Qld 4001 Australia

A point interpolation method (PIM) with continuous strain field (PIM-CS) is developed for mechanics problems using triangular background mesh, in which PIM shape functions are used to construct both displacement and strain fields. The strain field constructed is continuous in the entire problem domain, which is achieved by simple linear interpolations using locally smoothed strains around the nodes and points required for the interpolation. A general parameterized functional with a real adjustable parameter a are then proposed for establishing PIM-CS models of special property. We prove theoretically that the PIM-CS has an excellent bound property: strain energy obtained using PIM-CS lies in between those of the compatible FEM and NS-PIM models of the same mesh. Techniques and procedures are then presented to compute the upper and lower bound solutions using the PIM-CS. It is discovered that an extended Galerkin (x-Galerkin) model, as special case resulted from the extended parameterized functional with alpha = 1, is outstanding in terms of both performance and efficiency. Intensive numerical studies show that upper and lower bound solutions can always be obtained, there exist a values at which the solutions of PIM-CS are of superconvergence, and the x-Galerkin model is capable of producing superconvergent solutions of ultra accuracy that is about 10 times that of the FEM using the same mesh.

关键词： Finite element method Meshfree methods point interpolation method Solution bound Superconvergence x-Galerkin weak form

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A node-based smoothed point interpolation method (NS-PIM) for three-dimensional heat transfer problems

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INTERNATIONAL JOURNAL OF THERMAL SCIENCES 2009年第7期48卷 1367-1376页

作者： Wu, S. C. Liu, G. R. Zhang, H. O. Xu, X. Li, Z. R. Huazhong Univ Sci & Technol State Key Lab Digital Mfg & Equipment Technol Wuhan 430074 Peoples R China Natl Univ Singapore Dept Mech Engn Ctr Adv Computat Engn Sci Singapore 117576 Singapore Singapore MIT Alliance Singapore 117576 Singapore Jinlin Univ Coll Math Changchun 130012 Peoples R China

A node-based smoothed point interpolation method (NS-PIM) is formulated for three-dimensional (3D) heat transfer problems with complex geometries and complicated boundary conditions. Shape functions constructed here through PIM possess the delta function property and hence allow the straightforward enforcement of essential boundary conditions. The smoothed Galerkin weak form is employed to create discretized system equations, and the node-based smoothing domains are used to perform the smoothing operation and the numerical integration. The accuracy and efficiency of the NS-PIM solutions are studied through detailed analyses of actual 3D heat transfer problems. It is found that the NS-PIM can provide higher accuracy in temperature and its gradient than the reference approach does, in which very fine meshes are used in standard FEM code available with homogeneous essential boundary conditions. More importantly, the upper bound property of the NS-PIM is obtained using the same tetrahedral mesh. Together with the FEM, we now have a simple means to obtain both upper and lower bounds of the exact solution to heat transfer using the same type of mesh. Crown Copyright (C) 2008 Published by Elsevier Masson SAS. All rights reserved.

关键词： Meshfree Gradient smoothing Solution bounds point interpolation method Three-dimensional heat transfer

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A point interpolation method with least square strain field (PIM-LSS) for solution bounds and ultra-accurate solutions using triangular mesh

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2009年第17-20期198卷 1486-1499页

作者： Xu, X. Liu, G. R. Zhang, G. Y. Jilin Univ Coll Math Changchun 130012 Peoples R China Singapore MIT Alliance SMA Singapore 117576 Singapore Natl Univ Singapore Ctr Adv Computat Engn Sci ACES Dept Mech Engn Singapore 117576 Singapore

A point interpolation method with least square strain field (PIM-LSS) is developed for solid mechanics problems using triangular background mesh. In the PIM-LSS, PIM shape functions are used for displacement field construction that may or may not be compatible, and a least square fitting technique is adopted to construct the strain field. A strain constructed Galerkin (SC-Galerkin) weak formulation is then proposed for establishing discretized PIM-LSS models that have a number of special properties. We proved theoretically (1) the PIM-LSS provides a "softening" effect to the FEM model. and a "stiffing" effect to the node-based smoothed point interpolation method (NS-PIM) model;(2) the exact solution is bounded from both by PIM-LSS solutions with strain field of zero-order fitting and that with higher order fitting: (3) There exists a preferred order of fitting for the strain field such that ultra-accurate (one order higher accuracy) solution can be obtained using the PIM-LSS. These theorems and properties have been confirmed in numerical examples. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Numerical methods Finite element method Meshfree method point interpolation method Solution bound Strain constructed Galerkin weak form Least square method

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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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Upper bound solution to elasticity problems: A unique property of the linearly conforming point interpolation method (LC-PIM)

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2008年第7期74卷 1128-1161页

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

It is well known that the displacement-based fully compatible finite element method (FEM) provides a lower bound in energy norm for the exact solution to elasticity problems. It is, however, much more difficult to bound the solution from above for general problems in elasticity, and it has been a dream of many decades to find a systematical way to obtain such an upper bound. This paper presents a very important and unique property of the linearly conforming point interpolation method (LC-PIM): it provides a general means to obtain an upper bound solution in energy norm for elasticity problems. This paper conducts first a thorough theoretical study on the LC-PIM: we derive its weak form based on variational principles, study a number of properties of the LC-PIM, and prove that LC-PIM is variationally consistent and that it produces upper bound solutions. We then demonstrate these properties through intensive numerical studies with many examples of 1D, 2D, and 3D problems. Using the LC-PIM together with the FEM, we now have a systematical way to numerically obtain both upper and lower bounds of the exact solution to elasticity problems, as shown in these example problems. Copyright (c) 2007 John Wiley & Sons, Ltd.

关键词： numerical methods mesh-free methods point interpolation method nodal integration solution bound elasticity

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