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A radial boundary node method for two-dimensional elastic analysis

ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2003年第8期27卷 853-862页

作者： Xie, H Nogami, T Wang, JG Natl Univ Singapore Trop Marine Sci Inst Singapore 119260 Singapore Natl Univ Singapore Dept Civil Engn Singapore 119260 Singapore

The boundary node method (BNM) takes the advantages of both the boundary integral equation in dimension reduction and the moving least-square (MLS) approximation in elements elimination. However, the BNM inherits the deficiency of the MLS approximation, in which the shape functions lack the delta function property. As a result boundary conditions could not be exactly implemented for the BNM. In this paper, a radial boundary node method (RBNM) is proposed. The RBNM uses radial basis functions (RBFs) instead of the MLS to construct its interpolation. Consequently, the interpolation function could pass through nodes exactly, and the shape functions are of the delta function property. The exponential (EXP) and the multiquadric (MQ) RBFs are used in the current RBNM, and their shape parameters are studied in detail through some analyses of two-dimensional elastic problems. The suitable ranges of the shape parameters are proposed for both the EXP and the MQ basis functions. It is found that the RBNM is as accurate as or even more precise than the BEM. This suggests that the current RBNM could be robust and applicable. (C) 2003 Published by Elsevier Ltd.

关键词： boundary integral equation boundary node method point interpolation method radial basis function shape parameter

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A novel reduced-basis method with upper and lower bounds for real-time computation of linear elasticity problems

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COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2008年第2期198卷 269-279页

作者： Liu, G. R. Zaw, Khin Wang, Y. Y. Deng, B. Natl Univ Singapore Dept Mech Engn Ctr Adv Computat Engn Sci Singapore 117576 Singapore Singapore MIT Alliance SMA Singapore 117576 Singapore Inst High Performance Comp Singapore 138632 Singapore

This paper presents a novel reduced-basis method for analyzing problems of linear elasticity in a systematical, rapid and reliable fashion for solutions with both upper and lower bounds to the exact solution in the form of energy norm or compliance output. The lower bound of the solution output is obtained form the well-known reduced-basis method based on the Galerkin projection used in the finite element method, which is termed as GP_RBM. For the upper bound, a new reduced-basis approach is developed by the combination of the reduced-basis method and a smoothed Galerkin projection used in the linearly conforming point interpolation method, and it is thus termed as SGP_RBM. To examine the present SGP_RBM, we first conduct a theoretical study on the very important upper bound property. Reduced-basis models for both GP_RBM and SGP_RBM are constructed with the aid of an asymptotic error estimation and greedy adaptive procedure. The GP_RBM and the newly proposed SGP_RBM are applied to analyze a cantilever beam with an oblique crack to verify the proposed RBM technique in terms of accuracy, convergence, bound properties and computational savings. Both theoretical analysis and numerical results have demonstrated that the present method is a very efficient method for real-time solutions providing exact output bounds. (C) 2008 Elsevier B.V. All rights reserved.

关键词： Reduced-basis method Solution bound Asymptotic error estimation Linear elasticity point interpolation method Real-time computation

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ON G SPACE THEORY

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2009年第2期6卷 257-289页

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

This paper presents an examinations of the G space theory that was established recently for a unified formulation of compatible and incompatible models for mechanics problems using the finite element and meshfree settings. Using the generalized gradient smoothing technique, we first give a general definition for G spaces with more details on the G, space containing both continuous and discontinuous functions. The physical meanings and implications of various numerical treatments used in the G space theory are discussed in detail. Both normed and un-normed G spaces are discussed with emphases on the normed G spaces. Some important properties and a set of useful inequalities for the normed C spaces are proven in theory and analyzed in detail. Because discontinuous functions are allowed in a G space, much more types of function approximation methods and techniques can be used to create shape functions for numerical models. These models can be compatible and incomputable but are all stable and converge to the exact solution to the corresponding strong formulation as long as it is well-posed, based on the normed G space theory. methods based on normed G space theory does not use the derivatives of the displacement functions in the formulation and is known as the weakened weak (W-2) formulation that has a number of attractive properties such as conformability, softness, upper/lower bound, superconvergence, and ultra accuracy.

关键词： Numerical method meshfree method G space weakened weakform point interpolation method finite element method solution bound variational principle generalized smoothed Galerkin compatibility

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A novel PIM-based triangular element and its validations

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CONTINUUM MECHANICS AND THERMODYNAMICS 2020年第1期32卷 75-98页

作者： Huang, Zhecong Xue, Xinhua Sichuan Univ Coll Water Resource & Hydropower 24 South Sect 1Yihuan Rd Chengdu 610065 Peoples R China

Using the point interpolation method, this study proposes a distinctive node selection scheme, T33-scheme, to construct the approximation over all triangles in the mesh. By the T33-scheme, we always selects out six nodes to take part in the interpolation to any element, creating a T33 element with a displacement approximation of completely quadratic polynomials. The six nodes may be real nodes in the mesh or assistant nodes temporarily generated during the calculation of the element matrices. The degrees of freedom of assistant nodes can be easily condensed off once the matrices of the element, which has one boundary edge, are formed. The T33 element can pass the patch test exactly and reproduce the displacement mode of quadratic polynomials. Numerical experiments confirmed that the T33 elements have an outstanding performance in its accuracy, convergence rates and stability.

关键词： point interpolation method T33 element Meshless Quadratic polynomials

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A GENERALIZED GRADIENT SMOOTHING TECHNIQUE AND THE SMOOTHED BILINEAR FORM FOR GALERKIN FORMULATION OF A WIDE CLASS OF COMPUTATIONAL methodS

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2008年第2期5卷 199-236页

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

This paper presents a generalized gradient smoothing technique, the corresponding smoothed bilinear forms, and the smoothed Galerkin weakform that is applicable to create a wide class of efficient numerical methods with special properties including the upper bound properties. A generalized gradient smoothing technique is first presented for computing the smoothed strain fields of displacement functions with discontinuous line segments, by "rudely" enforcing the Green's theorem over the smoothing domain containing these discontinuous segments. A smoothed bilinear form is then introduced for Galerkin formulation using the generalized gradient smoothing technique and smoothing domains constructed in various ways. The numerical methods developed based on this smoothed bilinear form will be spatially stable and convergent and possess three major important properties: (1) it is variationally consistent, if the solution is sought in a Hilbert space;(2) the stiffness of the discretized model will be reduced compared to the model of the finite element method (FEM) and often the exact model, which allows us to obtain upper bound solutions with respect to both the FEM solution and the exact solution;(3) the solution of the numerical method developed using the smoothed bilinear form is less insensitive to the quality of the mesh, and triangular meshes can be used perfectly without any problems. These properties have been proved, examined, and confirmed by the numerical examples. The smoothed bilinear form establishes a unified theoretical foundation for a class of smoothed Galerkin methods to analyze solid mechanics problems for solutions of special and unique properties: the node-based smoothed point interpolation method (NS-PIM), smoothed finite element method (SFEM), node-based smoothed finite element method (N-SFEM), edge-based smoothed finite element method (E-SFEM), cell-based smoothed point interpolation method (CS-PIM), etc.

关键词： Elasticity point interpolation method finite element method meshfree method solution bound smoothing operation bilinear form variational principle Galerkin weakform numerical method

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

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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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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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An efficient adaptive analysis procedure for certified solutions with exact bounds of strain energy for elasticity problems

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FINITE ELEMENTS IN ANALYSIS AND DESIGN 2008年第14期44卷 831-841页

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

We present an efficient adaptive analysis procedure to obtain certified solutions of desired accuracy with bounds to the exact solution in energy norm for elasticity problems. The procedure makes the use of the recent finding that the upper bound to the exact strain energy can be obtained using the linearly conforming point interpolation method (LC-PIM), and the well-known fact that the lower bound can be obtained using the standard displacement-based fully compatible finite element method (FEM). To perform the adaptive analysis, a residual error-based error indicator and a simple h-type refinement scheme are employed and the relative error of the computed strain energy is used as the global stopping criteria. A number of numerical examples, including problems with singularity, have been studied to demonstrate the effectiveness and efficiency of the present procedure. The numerical results have been found converging very fast to the exact solution, and the bounds to the exact strain energy can be obtained efficiently at any stage in the adaptive process whenever required. (C) 2008 Elsevier B.V. All rights reserved.

关键词： meshfree methods point interpolation method strain smoothing adaptive analysis solution bound elasticity problems

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Computation of stresses in non-homogeneous elastic solids by local integral equation method: a comparative study

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COMPUTATIONAL MECHANICS 2008年第6期41卷 827-845页

作者： Sladek, Vladimir Sladek, Jan Zhang, Chuanzeng Slovak Acad Sci Inst Construct & Architecture Bratislava Slovakia Univ Siegen Dept Civil Engn Siegen Germany

The paper deals with the numerical implementation of local integral equations for solution of boundary value problems and interior computations of displacements and their gradients in functionally graded elastic solids. Two kinds of meshless approximations and one element based approximation are employed in various formulations. The numerical stability, accuracy, convergence of accuracy and cost efficiency are investigated in numerous test examples with exact benchmark solutions.

关键词： elasticity local integral equations point interpolation method MLS-approximation accuracy numerical stability convergence CPU-time

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Computation of stresses in non-homogeneous elastic solids by local integral equation method: a comparative study

Computation of stresses in non-homogeneous elastic solids by...

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Symposium of the International-Association-for-Boundary-Element-methods

作者： Sladek, Vladimir Sladek, Jan Zhang, Chuanzeng Slovak Acad Sci Inst Construct & Architecture Bratislava Slovakia Univ Siegen Dept Civil Engn Siegen Germany

关键词： elasticity local integral equations point interpolation method MLS-approximation accuracy numerical stability convergence CPU-time

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