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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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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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Geometrically nonlinear analysis of functionally graded materials based on reproducing kernel particle method

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INTERNATIONAL JOURNAL OF MECHANICS AND MATERIALS IN DESIGN 2020年第3期16卷 487-502页

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

Using the total Lagrange formulation, the reproducing kernel particle method (RKPM) for the geometrically nonlinear problem of the functionally graded materials (FGM) is proposed, and the corresponding formulae are derived. The displacement boundary condition is applied by the penalty method, and the numerical solution is solved by Newton-Raphson (N-R) iterative method. Furthermore, penalty factor, the control parameter of influence domain radius, loading step number and node distribution are discussed. Finally, the numerical examples illustrate that the RKPM for the geometrically nonlinear problem of the FGM is correct and effective.

关键词： Functionally graded materials Meshless method reproducing kernel particle method Geometrically nonlinear problem Penalty method

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Numerical solution of functionally graded materials based on radial basis reproducing kernel particle method

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2020年 111卷 32-43页

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

In this paper, the radial basis function (RBF) is used to construct the approximating function of the reproducing kernel particle method (RKPM), which can eliminate the negative effect of different kernel functions on the calculating accuracy. The radial basis reproducing kernel particle method (RRKPM) is proposed. Furthermore, the RRKPM is applied to solve the elastic mechanical problems of the functionally graded materials (FGM), and the RRKPM for FGM is established. The corresponding formulae of the RRKPM for FGM are derived. Compared with the traditional RKPM, the RRKPM has higher calculating accuracy and stability. Finally, the numerical examples illustrate that the RRKPM is correct and effective to solve the elastic mechanical problems of the FGM.

关键词： Meshless method Radial basis function reproducing kernel particle method Functionally graded materials Elastic mechanical problem

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Analyzing three-dimensional transient heat conduction problems with the dimension splitting reproducing kernel particle method

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ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS 2020年 121卷 180-191页

作者： Peng, P. P. Cheng, Y. M. Shanghai Univ Sch Mech & Engn Sci Shanghai Inst Appl Math & Mech Shanghai Key Lab Mech Energy Engn Shanghai 200072 Peoples R China

In this paper, to improve the computational speed of the reproducing kernel particle method (RKPM), the dimension splitting reproducing kernel particle method (DSRKPM) is presented for solving three-dimensional (3D) transient heat conduction problems. The idea of the proposed method is transforming the 3D problem into a series of two-dimensional (2D) ones by using the finite difference method in the splitting direction. Since the shape function of the RKPM for 2D problem is much simpler than the one of 3D problem, the DSRKPM has higher computational speed than the RKPM. To demonstrate the applicability of the proposed method, four example problems are presented, and each of them is solved by the DSRKPM and the RKPM, respectively. And the numerical solutions show that the DSRKPM not only greatly improves the computational efficiency, but also has a higher computational accuracy.

关键词： reproducing kernel particle method Dimension splitting method Dimension splitting reproducing kernel particle method Finite difference method Transient heat conduction problems

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The radial basis reproducing kernel particle method for geometrically nonlinear problem of functionally graded materials

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APPLIED MATHEMATICAL MODELLING 2020年 85卷 244-272页

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

In this paper, the radial basis function (RBF) is introduced into the reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is proposed for solving geometrically nonlinear problem of functionally graded materials (FGM). Compared with the RKPM, the advantages of the proposed method are that it can eliminate the negative effect of different kernel functions on the computational accuracy, and has higher computational accuracy and stability. Using the Total Lagrange (T.L.) formulation and the weak form of Galerkin integration, the corresponding formulae for geometrically nonlinear problem of FGM are derived. The penalty factor, shaped parameter of the RBF, the control parameter of influence domain radius, loading step number and node distribution are discussed. Furthermore, the effects of different gradient functions and exponents on displacement and stress are analyzed. Newton-Raphson (N-R) iterative method is utilized for numerical solution. The proposed method is correct and effective for solving geometrically nonlinear problem of FGM, which can be demonstrated by several numerical examples. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Geometrically nonlinear problem Meshless method reproducing kernel particle method Radial basis function Functionally graded materials

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The dimension splitting reproducing kernel particle method for three-dimensional potential problems

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INTERNATIONAL JOURNAL FOR NUMERICAL methodS IN ENGINEERING 2020年第1期121卷 146-164页

作者： Peng, P. P. Wu, Q. Cheng, Y. M. Shanghai Univ Sch Mech & Engn Sci Shanghai Inst Appl Math & Mech Shanghai Key Lab Mech Energy Engn Shanghai Peoples R China

In this paper, the dimension splitting reproducing kernel particle method (DSRKPM) for three-dimensional (3D) potential problems is presented. In the DSRKPM, a 3D potential problem can be transformed into a series of two-dimensional (2D) ones in the dimension splitting direction. The reproducing kernel particle method (RKPM) is used to solve each 2D problem, the essential boundary conditions are imposed by penalty method, and the discretized equation is obtained from Galerkin weak form of potential problems. Finite difference method is used in the dimension splitting direction. Then, by combining a series of the equations of the RKPM for solving 2D problems, the final equation of the DSRKPM for 3D potential problems is obtained. Five example problems on regular or irregular domains are selected to show that the DSRKPM has higher computational efficiency than the RKPM and the improved element-free Galerkin method for 3D potential problems.

关键词： dimension splitting method dimension splitting reproducing kernel particle method finite difference method potential problem reproducing kernel particle method

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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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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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