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A reproducing kernel particle method for solving generalized probability density evolution equation in stochastic dynamic analysis

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COMPUTATIONAL MECHANICS 2020年第3期65卷 597-607页

作者： Wang, Dan Li, Jie Tongji Univ Sch Civil Engn 1239 Siping Rd Shanghai 200092 Peoples R China Tongji Univ State Key Lab Disaster Reduct Civil Engn 1239 Siping Rd Shanghai 200092 Peoples R China

Analysis of stochastic dynamic system is still an open research issue. Recently a family of generalized probability density evolution equation, which provides an available way for general nonlinear systems, is put forward. In this paper, a numerical method based on reproducing kernel particle method (RKPM) for the solution of generalized probability density evolution equation, named the refined algorithm based on RKPM, is developed. Besides, the corresponding implementation procedure is elaborated. In this method, the time dependent probability distributions of the responses of interest can be obtained with less computational efforts. In addition, the mesh sensitivity problem in traditional probability density evolution method is settled well. Some details of parameter analysis are also discussed. To verify both the efficiency and accuracy of the method, a single-degree-of-freedom example and a 10-story frame structure are investigated. The refined algorithm based on RKPM can be applied to uni-variable and multi-variable, one-dimensional and multi-dimensional systems.

关键词： Stochastic dynamic system Probability density evolution method Refined algorithm reproducing kernel particle method

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The elastoplastic large deformation analysis based on meshless radial basis reproducing kernel particle method

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2023年第1期155卷 541-552页

作者： Qin, Shaopeng Yin, Deshun Wei, Gaofeng Han, Baozhi Tian, Mingyuan Ma, Liangzhu Hohai Univ Coll Mech & Mat Nanjing 211100 Peoples R China Qilu Univ Technol Shandong Acad Sci Sch Mech & Automot Engn Jinan 250353 Peoples R China Shandong Inst Mech Design & Res Jinan 250031 Peoples R China

In this paper, the radial basis reproducing kernel particle method (RB-RKPM) is extended to solve the elastoplastic large deformation problem. The Galerkin weak form of the elastoplastic large deformation problem based on the full Lagrange scheme is used to establish the solution equation, which is then solved using the Newton-Raphson iterative method. The RB-RKPM is established for the elastoplastic large deformation problem. The effect of the penalty factor, the scale parameter, and the shape parameter on the calculation accuracy are introduced, and the optimal parameters are obtained. Several numerical examples were used in the simulation. The RB-RKPM solution is compared with the reproducing kernel particle method (RKPM) solution and the finite element method (FEM) solution, the effectiveness of the RB-RKPM in solving elastoplastic large deformation problem is verified, and the RB-RKPM has high computational accuracy.

关键词： Elastoplastic large deformation problem Meshfree approach Radial basis reproducing kernel particle method reproducing kernel particle method

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The weighted reconstruction of reproducing kernel particle method for one-dimensional shock wave problems

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OCEAN ENGINEERING 2018年 149卷 325-340页

作者： Sun, C. T. Guan, P. C. Jiang, J. H. Kwok, O. L. A. Natl Taiwan Ocean Univ Dept Syst Engn & Naval Architecture 2 Beining Rd Keelung 202 Taiwan Natl Taiwan Univ Dept Civil Engn 1Sec 4Roosevelt Rd Taipei Taiwan

When high-order numerical approximation method is applied to model the propagation of shock wave or discontinuity, it Usually creates unstable unreal numerical oscillations around the discontinuous regions. In this research, we propose a non-oscillation meshfree scheme based on reproducing kernel particle method (RKPM) which can maintain the accuracy and minimize the oscillation in the modeling of shock wave propagation. In the proposed method, the original influence domain of high-order RK approximation is divided into several sub domains. Then we apply low-order RK approximation within each subdomain. Instead of directly using the discrete particles to build the numerical approximation, we consider that the high-order approximation is constructed by the summation of those low-order approximations multiplied by a local weight function. By adjusting these local weights with the "smoothness indicator, we can determine the "effect" of the corresponding sub domain and the discrete particles inside this subdomain. Therefore, the subdomain containing discontinuity would not participate in the high-order approximation, and the numerical oscillation is automatically suppressed. The proposed method does not need artificial viscosity or numerical damping to stabilize the solution. Several benchmark problems with shock wave propagation are tested. The results show that the proposed method can maintain high-order accuracy without numerical oscillation.

关键词： Shock capturing Non-oscillating method Meshfree methods High-order approximation reproducing kernel particle method

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Radial Basis reproducing kernel particle method for Damped Elastic Dynamics Problems

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IRANIAN JOURNAL OF SCIENCE AND TECHNOLOGY-TRANSACTIONS OF MECHANICAL ENGINEERING 2024年第3期48卷 1161-1176页

作者： Liu, Tengda Wei, Gaofeng Zhang, Yijie Qilu Univ Technol Shandong Acad Sci Sch Mech Engn Jinan 250353 Shandong Peoples R China Shandong Inst Mech Design & Res Jinan 250353 Shandong Peoples R China

The reproducing kernel particle method (RKPM) is a widely used meshless method that has been extensively applied in numerical analysis. The drawback of RKPM is that when different kernel functions are chosen during the computation process, there are different calculation accuracy, and significant discreteness. To address this issue, the radial basis function is introduced to RKPM, and the radial basis reproducing kernel particle method (RB-RKPM) is proposed. The negative impacts of different kernel functions on calculation accuracy can be eliminated by RB-RKPM, which possesses some advantages, such as good convergence, high computational accuracy and efficiency. Furthermore, the RB-RKPM is applied to the damped elastic dynamics problems (DEDPs), the governing equations for the DEDPs are derived based on the weak integral formulation, and the time is integrated by using the Newmark-linear acceleration method. Finally, the correctness of the proposed method in analyzing the DEDPs is verified through numerical examples.

关键词： reproducing kernel particle method Meshless method Damped elastic dynamics problems Weak integral formulation Radial basis function

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Numerical solution of bulk metal forming processes by the reproducing kernel particle method

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JOURNAL OF MATERIALS PROCESSING TECHNOLOGY 2006年第1-3期177卷 49-52页

作者： Xiong, Shangwu Martins, P. A. F. Inst Super Tecn Dept Engn Mech P-1049001 Lisbon Portugal Northwestern Univ Dept Mech Engn Evanston IL 60208 USA

A new mesh free approach 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 incompressible and slightly compressible rigid-plastic materials. Special emphasis is placed on the utilization of adaptive cell procedures that are capable of generating a new set of background cells at the end of each increment of deformation. Results show that the proposed methodology is capable of efficiently handling large plastic deformations without remeshing and providing results that are in close agreement with both finite element predictions and experimental measurements. (c) 2006 Elsevier B.V All rights reserved.

关键词： metal forming reproducing kernel particle method

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Splitting Rolling Simulated by reproducing kernel particle method

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Journal of Iron and Steel Research International 2007年第3期14卷 42-46页

作者： CUI Qing-ling LIU Xiang-hua WANG Guo-dong State Key Laboratory of Rolling and Automation Northeastern University Shenyang 110004 Liaoning China

During splitting rolling simulation, re-meshing is necessary to prevent the effect of severe mesh distortion when the conventional finite element method is used. However, extreme deformation cannot be solved by the finite element method in splitting rolling. The reproducing kernel particle method can solve this problem because the continuum body is discretized by a set of nodes, and a finite element mesh is unnecessary, and there is no explicit limitation of mesh when the metal is split. To ensure stability in the large deformation elastoplastic analysis, the Lagrange material shape function was introduced. The transformation method was utilized to impose the essential boundary conditions. The splitting rolling method was simulated and the simulation results were in accordance with the experimental ones in the literature.

关键词： elasto-plasticity large deformation reproducing kernel particle method splitting rolling

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RKPM2D: an open-source implementation of nodally integrated reproducing kernel particle method for solving partial differential equations

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COMPUTATIONAL particle MECHANICS 2020年第2期7卷 393-433页

作者： Huang, Tsung-Hui Wei, Haoyan Chen, Jiun-Shyan Hillman, Michael C. Univ Calif San Diego Dept Struct Engn La Jolla CA 92093 USA Penn State Univ Dept Civil & Environm Engn State Coll PA USA

We present an open-source software RKPM2D for solving PDEs under the reproducing kernel particle method (RKPM)-based meshfree computational framework. Compared to conventional mesh-based methods, RKPM provides many attractive features, such as arbitrary order of continuity and discontinuity, relaxed tie between the quality of the discretization and the quality of approximation, simple h-adaptive refinement, and ability to embed physics-based enrichment functions, among others, which make RKPM promising for solving challenging engineering problems. The aim of the present software package is to support reproducible research and serve as an efficient test platform for further development of meshfree methods. The RKPM2D software consists of a set of data structures and subroutines for discretizing two-dimensional domains, nodal representative domain creation by Voronoi diagram partitioning, boundary condition specification, reproducing kernel shape function generation, domain integrations with stabilization, a complete meshfree solver, and visualization tools for post-processing. In this paper, a brief overview that covers the key theoretical aspects of RKPM is given, such as the reproducing kernel approximation, weak form using Nitsche's method for boundary condition enforcement, various domain integration schemes (Gauss quadrature and stabilized nodal integration methods), as well as the fully discrete equations. In addition, the computer implementation aspects employed in RKPM2D are discussed in detail. Benchmark problems solved by RKPM2D are presented to demonstrate the convergence, efficiency, and robustness of the RKPM implementation.

关键词： Meshfree method reproducing kernel particle method Nodal integration Open-source software RKPM2D

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Application of reproducing kernel particle method in an Analysis of Elasto-plastic Deformation Under Taylor Impact

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Journal of China University of Mining and Technology 2006年第4期16卷 485-489页

作者： ZHAO Guang-ming SONG Shun-cheng MENG Xiang-rui Department of Resource Exploration and Management Engineering Anhui University of Science and Technology Huainan Anhui 232001 China Department of Applied Mechanics and Engineering Southwest Jiaotong University Chengdu Sichuan 610031 China

The reproducing kernel particle method (RKPM) is one of several new meshless numerical methods de- veloped internationally in recent years. The ideal elasto-plastic constitutive model of material under a Taylor impact is characterized by the Jaumann stress- and strain-rates. An updated Lagrangian format is used for the calculation in a nu- merical analysis. With the RKPM, this paper deals with the calculation model for the Taylor impact and deduces the control equation for the impact process. A program was developed to simulate numerically the Taylor impact of projec- tiles composed of several kinds of material. The simulation result is in good accordance with both the test results and the Taylor analysis outcome. Since the meshless method is not limited by meshes, it is believed to be widely applicable to such complicated processes as the Taylor impact, including large deformation and strain and to the study of the dy- namic qualities of materials.

关键词： reproducing kernel particle method Taylor impact large deformation meshless method

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USING reproducing kernel particle method FOR SHALLOW WATER PROBLEMS

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JOURNAL OF MARINE SCIENCE AND TECHNOLOGY-TAIWAN 2018年第3期26卷 431-440页

作者： Sun, Chien-Ting Kwok, On-Lei Annie Guan, Pai-Chen Shih, Wen-Kai CR Classificat Soc Taipei Taiwan Natl Taiwan Ocean Univ Dept Syst Engn & Naval Architecture Keelung Taiwan Natl Taiwan Univ Dept Civil Engn Taipei Taiwan Ctr Excellence Ocean Engn Taipei Taiwan

In this paper, a meshfree numerical scheme, which is based on the reproducing kernel particle method (RKPM), is proposed to solve the shallow water equations (SWEs). By applying the split coefficient matrix method on SWEs, RK approximation with upstream scheme can be employed for spatial discretization. Temporal discretization of SWEs is handled by the second order total-variation diminishing Runge-Kutta method. The merits of the present method are verified by performing three numerical experiments, which are problems related to open channel flow, oblique hydraulic jump and two-dimensional dam break. It is found that the proposed meshfree numerical scheme is able to efficiently model the shallow water problems with high convergence rate and accuracy. When non-uniform discretization is used in conjunction with the proposed method, the order of approximation and accuracy can still be maintained.

关键词： shallow water equations meshfree method reproducing kernel particle method high order approximation

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ON SOME ENRICHMENTS OF reproducing kernel particle method

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2004年第3期1卷 519-533页

作者： Zhou, J. X. Wang, X. M. Zhang, Z. Q. Zhang, L. Xi An Jiao Tong Univ Sch Civil Engn & Mech Xian 710049 Peoples R China

In this paper efforts are made to enrich reproducing kernel particle method (RKPM). Firstly, the RKPM shape functions are expressed explicitly in terms of kernel function moments. This avoids numerical matrix inversions and solutions of linear algebraic equations which are involved in classical RKPM, and thus makes RKPM more accurate, faster and more efficient. Then, a recently developed truly meshless body integration technique is introduced into RKPM. It is based on a partition of unity by a set of overlapping patches covering the domain and eliminates background cells completely. We borrow an idea from computational geometry and propose a sweeping-line method to determine the quadrature points inside the domain. The method is robust and effective even for domain with complicated shapes. The truly meshless integration technique in combination with this sweeping-line method make implementation of RKPM quite simple, smart and especially very advantageous when nodes are irregularly scattered. Numerical results presented herein demonstrate that these enrichments make RKPM more efficient, versatile and particularly truly meshless.

关键词： Meshless method reproducing kernel particle method partition of unity

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