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NUMERICAL SOLUTION OF BIDIRECTIONAL FUNCTIONALLY GRADED MATERIALS USING A NOVEL MESHLESS GLOBAL RADIAL BASIS reproducing kernel particle method

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JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES 2025年第1期20卷 33-53页

作者： Qin, Shaopeng Yin, Deshun Han, Baozhi Tian, Mingyuan Chen, Xuan Ma, Liangzhu Li, Lirui Hohai Univ Coll Mech & Engn Sci Nanjing 211100 Peoples R China

A novel meshless global radial basis reproducing kernel particle method (GRB-RKPM) is established based on the global radial basis function (GRBF) and the reproducing kernel particle method (RKPM). Subsequently, GRB-RKPM is used to solve the elastic mechanical of bidirectional functionally graded materials (BDFGMs). The influence of the shape parameter q, the shape parameter ac, the scale parameter dmax, and the penalty factor alpha on the GRB-RKPM solving the elastic mechanical of BDFGMs is analyzed, and the optimal value is given. The effectiveness of the GRB-RKPM in solving the elastic mechanical of BDFGMs is verified by comparing solutions calculated by the GRB-RKPM with reference solutions calculated by the finite element method (FEM). Compared with the RKPM, the GRB-RKPM converges faster. The GRB-RKPM solutions are compared with the local radial basis reproducing kernel particle method (LRB-RKPM) solutions and the RKPM solutions. It is proved that the GRB-RKPM has higher computational accuracy. The deformation behavior of the elastic mechanical of BDFGMs under different gradient indexes is studied.

关键词： meshless method global radial basis function reproducing kernel particle method bidirectional functionally graded materials elastic mechanical problem

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reproducing kernel particle method for two-dimensional time-space fractional diffusion equations in irregular domains

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2018年 97卷 131-143页

作者： Lin, Zeng Liu, Fawang Wang, Dongdong Gu, Yuantong Xiamen Univ Sch Math Sci Dept Civil Engn Xiamen 361005 Peoples R China Queensland Univ Technol Sch Math Sci GPO Box 2434 Brisbane Qld 4001 Australia Queensland Univ Technol Sch Chem Phys & Mech Engn GPO Box 2434 Brisbane Qld 4001 Australia

In recent years, the fractional differential equations have attracted a lot of attention due to their interested characteristics. Meshfree methods are highly accurate and have been extensively explored in engineering and mechanics fields. However, there is few research to develop the reproducing kernel particle method (RKPM), one of the widely used meshfree approach, for fractional partial differential equations. In this work, we solve time-space fractional diffusion equations in 2D regular and irregular domains. The temporal Caputo fractional derivatives are discretized by the L1 finite difference scheme and the spatial Laplacian fractional derivatives are discretized by RKPM based on the matrix transfer method. Especially, the corrected weighted shifted Griinwald-Letnikov scheme is utilized for temporally non-smooth solutions. Numerical examples in rectangular, circular, sector and human brain-like irregular domains are given to assess the efficiency and accuracy of the proposed numerical scheme. The spatial Laplacian fractional derivatives discretized by conventional finite difference method in the rectangular domain are also presented for comparison. The results indicate that RKPM is very effective for analyzing the considered fractional equations in various domains, which lays a concrete foundation for our further research of real application of human brain modeling.

关键词： Time-space fractional diffusion equations reproducing kernel particle method Finite difference method Corrected weighted shifted Griinwald-Letnikov scheme Temporally non-smooth solutions Irregular domain

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reproducing kernel particle method for coupled conduction-radiation phase-change heat transfer

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INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER 2018年 120卷 387-398页

作者： Tang, Jia-Dong He, Zhi-Hong Liu, Han Dong, Shi-Kui Tan, He-Ping Harbin Inst Technol Sch Energy Sci & Engn 92 West Dazhi St Harbin 150001 Heilongjiang Peoples R China

Interaction between conduction and radiation during the phase-change process within multidimensional participating media was numerically investigated. The solutions were based on the reproducing kernel particle method (RKPM). Test cases were analyzed and compared to results from analytical solutions and other studies to examine the performance of RKPM. Comparisons indicate that the RKPM is stable and accurate. The effect of thermo-optics effect on the phase-change process was carried out with the refractive index increases/decreases linearly with the temperature. The solidification process involving radiation was studied in a 2D enclosure. The effects of different parameters, i.e., extinction coefficients, scattering albedos, conduction-radiation parameters, and latent heat, on the temperature profiles were investigated, and the results show the parameters have significant effects on the solidification process. In addition, the solidification process during convection-radiation cooling was analyzed in a cuboid enclosure with the effect of various phase transition temperature ranges. (C) 2017 Elsevier Ltd. All rights reserved.

关键词： Meshless method reproducing kernel particle method Radiative heat transfer Phase change

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Improved reproducing kernel particle method for piezoelectric materials

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Chinese Physics B 2018年第1期27卷 215-222页

作者：马吉超魏高峰刘丹丹 School of Mechanical and Automotive Engineering Qilu University of Technology College of Foreign Languages Shandong Normal University

In this paper, the normal derivative of the radial basis function （RBF） is introduced into the reproducing kernel particle method （RKPM）, and the improved reproducing kernel particle method （IRKPM） is proposed. The method can decrease the errors on the boundary and improve the accuracy and stability of the algorithm. The proposed method is applied to the numerical simulation of piezoelectric materials and the corresponding governing equations are derived. The numerical results show that the IRKPM is more stable and accurate than the RKPM.

关键词： meshless methods piezoelectric materials reproducing kernel particle method radial basis func-tion method

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The complex variable reproducing kernel particle method for two-dimensional elastodynamics

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Chinese Physics B 2010年第9期19卷 59-70页

作者：陈丽程玉民 Department of Engineering Mechanics Chang'an University Shanghai Institute of Applied Mathematics and Mechanics Shanghai University

On the basis of the reproducing kernel particle method （RKPM）, a new meshless method, which is called the complex variable reproducing kernel particle method （CVRKPM）, for two-dimensional elastodynamics is presented in this paper. The advantages of the CVRKPM are that the correction function of a two-dimensional problem is formed with one-dimensional basis function when the shape function is obtained. The Galerkin weak form is employed to obtain the discretised system equations, and implicit time integration method, which is the Newmark method, is used for time history analysis. And the penalty method is employed to apply the essential boundary conditions. Then the corresponding formulae of the CVRKPM for two-dimensional elastodynamics are obtained. Three numerical examples of two-dimensional elastodynamics are presented, and the CVRKPM results are compared with the ones of the RKPM and analytical solutions. It is evident that the numerical results of the CVRKPM are in excellent agreement with the analytical solution, and that the CVRKPM has greater precision than the RKPM.

关键词： meshless method reproducing kernel particle method complex variable reproducing kernel particle method elastodvnamics

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Level set topology optimization for design-dependent pressure loads using the reproducing kernel particle method

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2020年第5期61卷 1805-1820页

作者： Neofytou, Andreas Picelli, Renato Huang, Tsung-Hui Chen, Jiun-Shyan Kim, H. Alicia Cardiff Univ Cardiff Sch Engn Queens Bldg14-17 Parade Cardiff CF24 3AA Wales Univ Sao Paulo Dept Min & Petr Engn Praca Narciso Andrade Vila Mathias Santos 11013560 SP Brazil Univ Calif San Diego Struct Engn Dept 9500 Gilman Dr San Diego CA 92093 USA

This paper presents a level set topology optimization method in combination with the reproducing kernel particle method (RKPM) for the design of structures subjected to design-dependent pressure loads. RKPM allows for arbitrary particle placement in discretization and approximation of unknowns. This attractive property in combination with the implicit boundary representation given by the level set method provides an effective framework to handle the design-dependent loads by moving the particles on the pressure boundary without the need of remeshing or special numerical treatments. Moreover, the reproducing kernel (RK) smooth approximation allows for the Young's modulus to be interpolated using the RK shape functions. This is another advantage of the proposed method as it leads to a smooth Young's modulus distribution for smooth boundary sensitivity calculation which yields a better convergence. Numerical results show good agreement with those in the literature.

关键词： Topology optimization Level set method Design-dependent pressure load reproducing kernel particle method

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The Isoparametric reproducing kernel particle method for nonlinear deformation of plates

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2014年 42卷 67-76页

作者： Guan, Pai-Chen Sun, Chien-Ting Natl Taiwan Ocean Univ Dept Syst Engn & Naval Architecture Keelung Taiwan

This paper proposed the new Isoparametric reproducing kernel particle method (IsoRKPM) for modeling nonlinear plate and shell deformation problems. The reproducing kernel shape functions are constructed on a two-dimensional parent domain. Following the concept of isoparametric mapping, the RK shape functions are directly used to approximate the plate geometry. A High Order Nodal Integration (HONI) is developed to integrate the Galerkin weak form of the Mindlin plate equilibrium equations. The proposed IsoRKPM with HONI is applied to solve several benchmark problems. Both modal and convergence analyses show that HONI provides more stable and accurate solutions than the nodal integration method. (C) 2013 Elsevier Ltd. All rights reserved.

关键词： Meshfree reproducing kernel particle method Nonlinear Plates and shells Numerical integration

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Simulation of bulk metal forming processes using the reproducing kernel particle method

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COMPUTERS & STRUCTURES 2005年第8-9期83卷 574-587页

作者： Xiong, SW Liu, WK Cao, H Li, CS Rodrigues, JMC Martins, PAF Inst Super Tecn Dept Engn Mecan P-1049001 Lisbon Portugal Northwestern Univ Dept Mech Engn Evanston IL 60208 USA Northeastern Univ State Key Lab Rolling & Automat Shenyang 110004 Peoples R China

A new approach based on the utilization of the reproducing kernel particle method is put forwards for the numerical simulation of bulk metal forming processes. The approach is based on the utilization of the reproducing kernel particle method in conjunction with the flow formulation for slightly compressible rigid-plastic materials. The effectiveness of the proposed methodology is discussed by comparing theoretical predictions with finite element calculations and experimental data. The later were obtained by means of simple metal forming experiments that were performed under laboratory-controlled conditions using aluminium, A16060 (natural aged). The overall comparisons comprise a wide range of topics such as material flow, geometry, forming load, strain and stress distributions. As results show, the proposed approach is capable of efficiently handling large plastic deformations without the need of remeshing procedures and providing results that are in close agreement with both finite element predictions and experimental measurements. (c) 2004 Elsevier Ltd. All rights reserved.

关键词： bulk metal forming reproducing kernel particle method finite element method remeshing

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On coupling of reproducing kernel particle method and boundary element method

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COMPUTATIONAL MECHANICS 2004年第4期34卷 282-297页

作者： Lee, CK Lie, ST Shuai, YY Nanyang Technol Univ Sch Civil & Environm Engn Singapore 639798 Singapore

In this paper a new coupling method to merge the reproducing kernel particle method (RKPM) and the boundary element method (BEM) is proposed. In this formulation, the whole problem domain will first be divided into two subdomains in which the RKPM and the BEM are applied respectively. A simple and direct coupling procedure is then applied to preserve the compatibility of the solution along the interface of the two subdomains. Unlike other coupling procedures suggested previously, the present coupling procedure neither requires modification of the RKPM and the BEM formulations nor the use of finite elements along the interface boundary between the two subdomains. The validity and efficiency of the coupling procedure are demonstrated by using it to solve five benchmark elastostatic stress analysis problems. In addition, a simple analysis of the a priori convergence rate of the coupling procedure is given. Preliminary assessment of the convergence characteristics of the coupling procedure is done by carrying out uniform refinements on the benchmark problems.

关键词： meshless method reproducing kernel particle method boundary element method a priori error estimation

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Investigations on reproducing kernel particle method enriched by partition of unity and visibility criterion

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COMPUTATIONAL MECHANICS 2004年第4期34卷 310-329页

作者： Zhang, ZQ Zhou, JX Wang, XM Zhang, YF Zhang, L Xi An Jiao Tong Univ Sch Civil Engn & Mech Xian 710049 Peoples R China

This work introduces a numerical integration technique based on partition of unity (PU) to reproducing kernel particle method (RKPM) and presents an implementation of the visibility criterion for meshfree methods. According to the theory of PU and the inherent features of Gaussian quadrature, the convergence property of the PU integration is studied in the paper. Moreover, the practical approaches to implement the PU integration are presented in different strategies. And a method to carry out visibility criterion is presented to handle the problems with a complex domain. Furthermore, numerical examples have been performed on the h-version and p-like version convergence studies of the PU integration and the validity of visibility criterion. The results demonstrate that PU integration is a feasible and effective numerical integration technique, and RKPM enriched by PU integration and visibility criterion is of more efficiency, versatility and high performance.

关键词： meshfree method reproducing kernel particle method partition of unity visibility criterion

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