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Vibration and buckling analysis of functionally graded beams using reproducing kernel particle method

SCIENTIA IRANICA 2014年第6期21卷 1896-1906页

作者： Saljooghi, R. Ahmadiana, M. T. Farrahi, G. H. Sharif Univ Technol Sch Mech Engn Tehran Iran Sharif Univ Technol Ctr Excellence Design Robot & Automat Tehran Iran

This paper presents the vibration and buckling analysis of functionally graded beams with different boundary conditions, using the reproducing kernel particle method (RKPM). Vibration of simple Euler-Bernoulli beams using RKPM is already developed and reported in the literature. So far, the modeling of FGM beams using the theoretical method or the finite element technique has not evolved with accurate results for the power law form of FGM with a large power of "n" value. The accuracy of RKPM results is very good and is not sensitive to the n value. The system of equations of motion is derived using Lagrange's method under the assumption of the Euler-Bernoulli beam theory. Boundary conditions of the beam are taken into account using Lagrange multipliers. It is assumed that material properties of the beam vary continuously in the thickness direction, according to the power-law form. RKPM is implemented to obtain the equation of motion and, consequently, natural frequencies and buckling loads of the FGM beam are evaluated. Results are verified for special cases reported in the literature. Considering the displacement of the neutral axis, buckling loads, with respect to length and material distribution, are evaluated. For special cases of homogenous beams, RKPM matches theoretical evaluation with less than one percent error. (C) 2014 Sharif University of Technology. All rights reserved.

关键词： Functionally graded beam reproducing kernel particle method Natural frequencies Critical buckling loads

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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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A semi-Lagrangian reproducing kernel particle method with particle-based shock algorithm for explosive welding simulation

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COMPUTATIONAL MECHANICS 2021年第6期67卷 1601-1627页

作者： Baek, Jonghyuk Chen, Jiun-Shyan Zhou, Guohua Arnett, Kevin P. Hillman, Michael C. Hegemier, Gilbert Hardesty, Scott Univ Calif San Diego Dept Struct Engn La Jolla CA 92093 USA Optimal Inc Plymouth MI 48170 USA US Mil Acad Dept Civil & Mech Engn West Point NY 10996 USA Penn State Univ Dept Civil & Environm Engn University Pk PA 16802 USA Appl Res Associates Inc Rocky Mt Div Littleton CO 80127 USA

The explosive welding process is an extreme-deformation problem that involves shock waves, large plastic deformation, and fragmentation around the collision point, which are extremely challenging features to model for the traditional mesh-based methods. In this work, a particle-based Godunov shock algorithm under a semi-Lagrangian reproducing kernel particle method (SL-RKPM) is introduced into the volumetric strain energy to accurately embed the key shock physics in the absence of a mesh or grid, which is shown to also ensure the conservation of linear momentum. For kernel stability, a deformation-dependent anisotropic kernel support update algorithm is proposed, which is shown to capture excessive plastic flow and material separation. A quasi-conforming nodal integration is adopted to avoid the need of updating conforming cells which is tedious in extreme deformations. It is shown that the proposed formulation effectively captures shocks, jet formation, and smooth-to-wavy interface morphology transition with good agreement with experimental results.

关键词： reproducing kernel particle method kernel stability Shocks Nodal integration Explosive welding

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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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On background cells during the analysis of bulk forming processes by the reproducing kernel particle method

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COMPUTATIONAL MECHANICS 2007年第2期40卷 233-247页

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

The reproducing kernel particle method based on the irreducible flow formulation is utilised to perform the numerical simulation of bulk metal forming processes. Emphasis is given on analysing the influence of employing triangular or quadrilateral background cells on the predictions of material flow, forming load and distribution of strain. A new proposal to smooth the distribution of average stresses during stress computations in the background cells is also included. The effectiveness of the proposed method is discussed by comparing its numerical predictions with a benchmark test case, finite element calculations and experimental data. The benchmark test case is included with the objective of illustrating the influence of several theoretical and numerical subjects such as;order of the basis correction functions, dimension of the compact support and computation of the volume associated to each nodal point. Experimental data was acquired from metal forming controlled laboratory-based tests that were designed so that the proposed method could be tested on its ability to efficiently handle large plastic deformations. It is shown that adaptive arbitrary triangular background cells are capable of efficiently handling large plastic deformations without remeshing.

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

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Finite volume reproducing kernel particle method

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 167卷 150-159页

作者： Lin, Zeng Chinese Acad Sci Shenzhen Inst Adv Technol Shenzhen 518055 Peoples R China

In the realm of computational mechanics, the reproducing kernel particle method (RKPM) shines in addressing challenges associated with intricate geometries and substantial deformations. However, in the Galerkin framework, RKPM has a substantial computational workload due to the complexities of numerical integration. To mitigate this limitation, we present a pioneering approach: the finite volume reproducing kernel particle method (FV-RKPM). FV-RKPM amalgamates the strengths of meshfree methods and finite volume methods, offering benefits such as grid independence, conservation properties, and numerical stability. In this paper, we elaborate on the construction methodology for FV-RKPM, alongside its associated integration strategies. We consistently validate the efficacy of this innovative methodology through a series of numerical results.

关键词： Finite volume method reproducing kernel particle method Computational efficiency

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Research on elasticity behavior of Ni-Ti shape memory alloys using the reproducing kernel particle method

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JOURNAL OF STRAIN ANALYSIS FOR ENGINEERING DESIGN 2024年第4期59卷 291-302页

作者： Zhang, Yijie Wei, Gaofeng Liu, Tengda Qilu Univ Technol Shandong Acad Sci Sch Mech Engn Jinan Shandong Peoples R China Shandong Inst Mech Design & Res Jinan Shandong Peoples R China Qilu Univ Technol Shandong Acad Sci Sch Mech Engn 3501 Daxue Rd Jinan 250353 Shandong Peoples R China

The unique superelastic properties of the Ni-Ti shape memory alloys (SMAs) have been found extensive practical application in many engineering fields, but the Ni-Ti SMAs exhibits elastic behavior in small load, so the elastic behavior of the Ni-Ti SMAs is studied by using meshless method of reproducing kernel particle method (RKPM) in this paper. The displacement boundary conditions are applied by utilizing the penalty function method, and the impact of penalty factor and parameter of influence domain on errors are discussed, and computational stability is analyzed. At last, the correctness of the RKPM in studying the elastic behavior of the Ni-Ti SMAs is evidenced by using three numerical examples.

关键词： Meshless method shape memory alloys reproducing kernel particle method elasticity behavior penalty method

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Image-based modeling of coupled electro-chemo-mechanical behavior of Li-ion battery cathode using an interface-modified reproducing kernel particle method

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ENGINEERING WITH COMPUTERS 2024年 1-32页

作者： Susuki, Kristen Allen, Jeffery Chen, Jiun-Shyan Univ Calif San Diego Dept Struct Engn La Jolla CA 92093 USA Natl Renewable Energy Lab Computat Sci Ctr Golden CO USA

An interface-modified reproducing kernel particle method (IM-RKPM) is introduced in this work to allow for a direct model construction from image pixels of heterogeneous polycrystalline Li-ion battery microstructures. The interface-modified reproducing kernel (IM-RK) approximation is constructed through scaling of a kernel function by a regularized distance function in conjunction with strategic placement of interface node locations. This leads to RK shape functions with either weak or strong discontinuities across material interfaces, suitable for modeling various interface mechanics. With the placement of a triple junction node and distance-based scaling of kernel functions, the resulting IM-RK shape function also possesses proper discontinuities at the triple junctions. This IM-RK approximation effectively remedies the well-known Gibb's oscillation in the smooth approximation of discontinuities. Different from the conventional meshfree approaches for interface discontinuities, this IM-RK approach is done without additional degrees of freedom associated with the enrichment functions, and it is formulated with the standard procedures in the RK shape function construction. This work focuses on identifying the accuracy and convergence properties of IM-RKPM for modeling the coupled electro-chemo-mechanical system. A linear patch test is formulated and numerically tested for the electro-chemo-mechanical coupled problem with a Butler-Volmer boundary condition representing the physical conditions in Li-ion battery microstructures. This is followed by verification of the optimal rates of convergence of IM-RKPM for solving the coupled problem with higher order solutions. The image-based modeling of Li-ion battery microstructures in the numerical examples demonstrates the applicability of the proposed method to realistic Li-ion battery materials modeling.

关键词： reproducing kernel particle method Image based modeling Electro-chemo-mechanical coupling Interface-modified reproducing kernel approximation Li-ion battery

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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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