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A quasi-conforming embedded reproducing kernel particle method for heterogeneous materials

COMPUTER methodS IN APPLIED MECHANICS AND ENGINEERING 2023年第1期416卷

作者： Schlinkman, Ryan T. Baek, Jonghyuk Beckwith, Frank N. Nelson, Stacy M. Chen, J. S. Univ Calif San Diego La Jolla CA 92093 USA Sandia Natl Labs Albuquerque NM USA

We present a quasi-conforming embedded reproducing kernel particle method (QCE-RKPM) for modeling heterogeneous materials that makes use of techniques not available to mesh-based methods such as the finite element method (FEM) and avoids many of the drawbacks in current embedded and immersed formulations which are based on meshed methods. The different material domains are discretized independently thus avoiding time-consuming, conformal meshing. In this approach, the superposition of foreground (inclusion) and background (matrix) domain integration smoothing cells are corrected by a quasi-conforming quadtree subdivision on the background integration smoothing cells. Due to the non-conforming nature of the background integration smoothing cells near the material interfaces, a variationally consistent (VC) correction for domain integration is introduced to restore integration constraints and thus optimal convergence rates at a minor computational cost. Additional interface integration smoothing cells with area (volume) correction, while non-conforming, can be easily introduced to further enhance the accuracy and stability of the Galerkin solution using VC integration on non-conforming cells. To properly approximate the weak discontinuity across the material interface by a penalty-free Nitsche's method with enhanced coercivity, the interface nodes on the surface of the foreground discretization are also shared with the background discretization. As such, there are no tunable parameters, such as those involved in the penalty type method, to enforce interface compatibility in this approach. The advantage of this meshfree formulation is that it avoids many of the instabilities in mesh-based immersed and embedded methods. The effectiveness of QCE-RKPM is illustrated with several examples.(c) 2023 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://***/licenses/by/4.0/).

关键词： Meshfree modeling reproducing kernel particle method Heterogeneous materials Embedded discretization Quasi-conforming Quadtree subdivision

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A Hybrid reproducing kernel particle method for Three-Dimensional Elasticity Problems

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INTERNATIONAL JOURNAL OF APPLIED MECHANICS 2023年第9期15卷

作者： Peng, Piaopiao Cheng, Heng Cheng, Yumin Shanxi Univ Sch Elect Power Civil Engn & Architecture Taiyuan 030006 Peoples R China Taiyuan Univ Sci & Technol Sch Appl Sci Taiyuan 030024 Peoples R China Shanghai Univ Shanghai Key Lab Mech Energy Engn Shanghai Inst Appl Math & Mech Sch Mech & Engn Sci Shanghai 200072 Peoples R China

This study presents a fast meshless method called the hybrid reproducing kernel particle method (HRKPM) for the solution of three-dimensional (3D) elasticity problems. The equilibrium equations of 3D elasticity are divided into three groups of equations, and two equilibrium equations are contained in each group. By coupling the discrete equations for solving two arbitrary groups of equations, the complete solution of 3D elasticity can be obtained. For an arbitrary group of equations, the 3D elasticity problem is transformed into a series of associated two-dimensional (2D) ones, which is solved by the RKPM to derive the discrete formulae. The discrete equations of 2D problems are combined using the difference method in dimension splitting direction. Then, arbitrarily choosing another group of equilibrium equations, the discrete equation of another group of 2D problems can be obtained similarly. By combining the discrete equations for these two groups of 2D problems, the solution to an original 3D problem will be reached. The numerical results show that the HRKPM performs better than RKPM in solution efficiency.

关键词： Meshless method reproducing kernel particle method dimension splitting method hybrid reproducing kernel particle method elasticity

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The meshfree analysis of elasticity problem utilizing radial basis reproducing kernel particle method

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RESULTS IN PHYSICS 2020年 17卷

作者： Liu, Zheng Gao, Hongfen Wei, Gaofeng Wang, Zhiming Qilu Univ Technol Sch Mech & Automot Engn Shandong Acad Sci Jinan 250353 Shandong Peoples R China Shandong Management Univ Sch Intelligent Engn Jinan 250357 Shandong Peoples R China

The computational accuracy of the traditional reproducing kernel particle method (RKPM) is susceptible to different kernel functions. To eliminate the adverse effects of the different kernel functions on the computational accuracy of the RKPM, the radial basis function (RBF) is introduced and the radial basis reproducing kernel particle method (RRKPM) is proposed. Compared with the RKPM, the proposed method has the advantages of greater computational accuracy and faster convergence speed. Furthermore, the proposed method is applied to analyze the elastic deformation process of materials with various defects. Utilizing the Galerkin weak form of elastic mechanical problem, the RRKPM of elastic mechanics is established and the corresponding formulae are derived. Combined with the numerical examples, the shaped parameter of the RBF, the scaling parameter and the penalty factor are discussed. Finally, several different examples are adopted to demonstrate the applicability and correctness of the RRKPM.

关键词： Meshfree method Radial basis function reproducing kernel particle method Elasticity problem Numerical analysis

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Numerical solution of potential problems using radial basis reproducing kernel particle method

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RESULTS IN PHYSICS 2019年 13卷 102122-000页

作者： Gao, Hongfen Wei, Gaofeng Shandong Management Univ Coll Mech & Elect Engn Jinan 250357 Shandong Peoples R China Qilu Univ Technol Sch Mech & Automot Engn Shandong Acad Sci Jinan 250353 Shandong Peoples R China

The paper presents the radial basis reproducing kernel particle method (RRKPM) for potential problems. The proposed RRKPM can eliminate the negative effect of different reproducing kernel functions (RKF) on computational stability and accuracy. The integral weak form is used to derive a discretized system equation for potential problem, and the essential boundary condition is imposed by the Lagrange interpolation method, the corresponding governing equations of RRKPM are investigated and obtained. Compared with the conventional RKPM, the RRKPM has higher computational stability and accuracy. Finally, the RRKPM is applied to numerical simulation of potential problems, and the accuracy and stability of the RRKPM can be illustrated by the two numerical results.

关键词： Radial basis functions Meshless methods reproducing kernel particle method The Lagrange interpolation method Potential problems

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The Hermit-Type reproducing kernel particle method for Elasticity Problems

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INTERNATIONAL JOURNAL OF COMPUTATIONAL methodS 2019年第2期16卷

作者： Ma, Jichao Wei, Gaofeng Gao, Hongfen Qilu Univ Technol Shandong Acad Sci Sch Mech & Automot Engn Jinan 250353 Shandong Peoples R China Shandong Management Univ Coll Mech & Elect Engn Jinan 250357 Shandong Peoples R China

To reduce the error on the boundary and improve computational accuracy, the normal derivative of radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the Hermit-type reproducing kernel particle method (Hermit-type RKPM) is proposed. The Hermit-type RKPM approximation function is constructed and the governing equation of the elasticity problems is deduced. Then the Hermit-type RKPM is applied to the numerical simulation of the elasticity problems, and the results illustrate that the proposed method is more accurate than the RKPM.

关键词： Meshless methods elasticity problems Hermit-type radial basis functions reproducing kernel particle method numerical simulation

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Analysis of Kirchhoff plate on Winkler foundation by reproducing kernel particle method

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ROCK AND SOIL MECHANICS 2010年第SUPPL. 2期31卷 98-103页

作者： Zeng Xiang-yong Peng Chuan-hai Deng An-fu Chongqing Univ Coll Civil Engn Chongqing 400045 Peoples R China

reproducing kernel particle method (RKPM) is a typical meshless method based on kernel function (spline function) simulation. The first and second derivatives of RKPM shape function are deduced completely;with Kirchhoff thin plate theory, the meshless method RKPM governing equations for deflection solution of thin plate on Winkler foundation are established;numerical analysis results indicate that the above method and corresponding program are feasible and effective.

关键词： computational method reproducing kernel particle method meshless method Kirchhoff thin plate Winkler foundation

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On the use of reproducing kernel particle finite strip method in the static, stability and free vibration analysis of FG plates with different boundary conditions and diverse internal supports

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APPLIED MATHEMATICAL MODELLING 2021年 92卷 380-409页

作者： Esfahani, Sadegh Ghamsari Sarrami-Foroushani, Saeid Azhari, Mojtaba Isfahan Univ Technol Dept Civil Engn Esfahan *** Iran

This paper is focused on the application of a novel reproducing kernel particle finite strip method (RKP-FSM) for the static, mechanical and thermal buckling, and free vibration analyses of rectangular functionally graded (FG) plates with different boundary conditions and internal supports. Conventional reproducing kernel particle method (RKPM) is incorporated into the finite strip method (FSM), and the governing equations are derived based on the first-order shear deformation theory (FSDT). As a convenient approach, boundary conditions are imposed using a simple method based on the RKPM correction function and the essence of the FSDT displacement field. Several examples are studied using this method, and the results are compared to those obtained from other numerical methods. The proposed method is also extended to consider different support geometries and complex inner conditions, and the results are validated using COMSOL finite element software. The shear locking error, which occurs when considering C-0 plate theories, is noticeably reduced by considering the RKPM parameters in the finite strip formulation. (c) 2020 Elsevier Inc. All rights reserved.

关键词： reproducing kernel particle method Static analysis Free vibration Buckling First-order shear deformation Internal support

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The analysis of geometrically nonlinear behavior of SMAs using RKPM

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2025年 142卷

作者： Zhang, Yijie Wei, Gaofeng Liu, Tengda Hua, Fengfeng Zhou, Shasha 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 SDEE Hitachi High Voltage Switchgear Co Ltd Jinan Peoples R China

As the temperature surpasses the threshold for the completion of austenitic transformation, shape memory alloys (SMAs) necessitate a substantial external force to trigger internal phase transformation. Given the substantial deformation induced by the external force on SMAs, the application of geometrically nonlinear analysis becomes essential. In this paper, reproducing kernel particle method (RKPM) is employed to investigate the geometrically nonlinear behavior of SMAs. The penalty function method is applied to impose the displacement boundary conditions. The study utilizes the Galerkin weak form with total Lagrangian (TL) framework to develop geometrically nonlinear SMAs equations, solved via Newton-Raphson (N-R) iteration. The effects of varying penalty factor and radius control parameter of the influence domain on error and computational stability are investigated. Ultimately, the suitability of applying RKPM for exploring the geometrically nonlinearity behavior of SMAs is demonstrated via numerical examples.

关键词： Phase transformation Shape memory alloys Geometrically nonlinear behavior reproducing kernel particle method Numerical analysis

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An efficient numerical method for the distributed-order time-fractional diffusion equation with the error analysis and stability properties

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MATHEMATICAL methodS IN THE APPLIED SCIENCES 2025年第3期48卷 2743-2765页

作者： Irandoust-Pakchin, Safar Hossein Derakhshan, Mohammad Rezapour, Shahram Adel, Mohamed Univ Tabriz Fac Math Stat & Comp Sci Dept Appl Math Tabriz Iran Apadana Inst Higher Educ Dept Ind Engn Shiraz Iran Azarbaijan Shahid Madani Univ Dept Math Tabriz Iran Insurance Res Ctr IRC Tehran Iran China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan Islamic Univ Madinah Fac Sci Dept Math Medina Saudi Arabia

In this manuscript, we study and examine the time-fractional modified anomalous sub-diffusion model of distributed-order. Two numerical approaches are used to study the approximate solutions of the presented model. For the first approach, we use a second-order difference method based on the L1 formula for the temporal variable. In this case, stability and convergence analysis for time discretization using the L1 formula is displayed. For the second approach, we display and demonstrate the numerical method for the full-discrete based on the Galerkin weak form based on various kernels and shape functions of reproducing kernel particle method which do not have the delta$$ \delta $$-Kronecker property. Moreover, in this manuscript, in order to be able to use the necessary boundary conditions, the two straight strategies are used: one is the Lagrange multiplier method, and the other one is the penalty method. By using penalty method, we can the main boundary value model is changed to a new BVP with Robin boundary condition. At the end of the article, some numerical examples are presented and analyzed for the efficiency of the proposed numerical method. Also, the presented numerical method is compared with other numerical approaches, and the results of this comparison are reported in the form of a table.

关键词： distributed-order meshless Galerkin method reproducing kernel particle method stability and convergence time-fractional sub-diffusion model

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Meshless analysis of three-dimensional steady-state heat conduction problems

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Chinese Physics B 2010年第9期19卷 36-41页

作者：程荣军葛红霞 Ningbo Institute of Technology Zhejiang University Faculty of Science Ningbo University

Steady-state heat conduction problems arisen in connection with various physical and engineering problems where the functions satisfy a given partial differential equation and particular boundary conditions, have attracted much attention and research recently. These problems are independent of time and involve only space coordinates, as in Poisson＇s equation or the Laplace equation with Dirichlet, Neuman, or mixed conditions. When the problems are too complex, it is difficult to find an analytical solution, the only choice left is an approximate numerical solution. This paper deals with the numerical solution of three-dimensional steady-state heat conduction problems using the meshless reproducing kernel particle method （RKPM）. A variational method is used to obtain the discrete equations. The essential boundary conditions are enforced by the penalty method. The effectiveness of RKPM for three-dimensional steady-state heat conduction problems is investigated by two numerical examples.

关键词： reproducing kernel particle method meshless method steady-state heat conduction problem

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