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Meshless methods: A review and computer implementation aspects

Author

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  • Nguyen, Vinh Phu
  • Rabczuk, Timon
  • Bordas, Stéphane
  • Duflot, Marc

Abstract

The aim of this manuscript is to give a practical overview of meshless methods (for solid mechanics) based on global weak forms through a simple and well-structured MATLAB code, to illustrate our discourse. The source code is available for download on our website and should help students and researchers get started with some of the basic meshless methods; it includes intrinsic and extrinsic enrichment, point collocation methods, several boundary condition enforcement schemes and corresponding test cases. Several one and two-dimensional examples in elastostatics are given including weak and strong discontinuities and testing different ways of enforcing essential boundary conditions.

Suggested Citation

  • Nguyen, Vinh Phu & Rabczuk, Timon & Bordas, Stéphane & Duflot, Marc, 2008. "Meshless methods: A review and computer implementation aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 763-813.
    • Handle: RePEc:eee:matcom:v:79:y:2008:i:3:p:763-813
    DOI: 10.1016/j.matcom.2008.01.003
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    Cited by:

    1. Wesonga, Ronald, 2015. "Airport utility stochastic optimization models for air traffic flow management," European Journal of Operational Research, Elsevier, vol. 242(3), pages 999-1007.
    2. Ding, Rui & Shen, Quan & Yao, Yuan, 2022. "The element-free Galerkin method for the dynamic Signorini contact problems with friction in elastic materials," Applied Mathematics and Computation, Elsevier, vol. 415(C).
    3. Oanh, Dang Thi & Davydov, Oleg & Phu, Hoang Xuan, 2017. "Adaptive RBF-FD method for elliptic problems with point singularities in 2D," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 474-497.
    4. Hammou, Amina & Hilali, Youssef & Mesmoudi, Said & Boujmal, Radouane & Bourihane, Oussama, 2024. "A mesh-free Hermite-type approach for buckling analysis of functionally graded polygonal thin plates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 218(C), pages 112-132.
    5. Juan José Benito & Ángel García & Mihaela Negreanu & Francisco Ureña & Antonio M. Vargas, 2022. "A Novel Spatio-Temporal Fully Meshless Method for Parabolic PDEs," Mathematics, MDPI, vol. 10(11), pages 1-12, May.
    6. Bo He & Brahmanandam Javvaji & Xiaoying Zhuang, 2019. "Characterizing Flexoelectricity in Composite Material Using the Element-Free Galerkin Method," Energies, MDPI, vol. 12(2), pages 1-18, January.
    7. Piaopiao Peng & Ning Wang & Yumin Cheng, 2024. "A Hybrid Reproducing Kernel Particle Method for Three-Dimensional Helmholtz Equation," Mathematics, MDPI, vol. 12(12), pages 1-24, June.
    8. Li, Yang & Liu, Dejun & Yin, Zhexu & Chen, Yun & Meng, Jin, 2023. "Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    9. Nguyen, Vinh Phu & Anitescu, Cosmin & Bordas, Stéphane P.A. & Rabczuk, Timon, 2015. "Isogeometric analysis: An overview and computer implementation aspects," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 117(C), pages 89-116.
    10. Ai, Weilong & Augarde, Charles E., 2018. "A multi-cracked particle method for complex fracture problems in 2D," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 150(C), pages 1-24.
    11. Belinha, J. & Dinis, L.M.J.S. & Natal Jorge, R.M., 2016. "The analysis of the bone remodelling around femoral stems: A meshless approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 64-94.
    12. Ouagued, Sofiane & Amara, Yacine & Barakat, Georges, 2016. "Comparison of hybrid analytical modelling and reluctance network modelling for pre-design purposes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 130(C), pages 3-21.
    13. Li, Xiaolin, 2011. "Development of a meshless Galerkin boundary node method for viscous fluid flows," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(2), pages 258-280.
    14. Fengxin Sun & Jufeng Wang & Xiang Kong & Rongjun Cheng, 2021. "A Dimension Splitting Generalized Interpolating Element-Free Galerkin Method for the Singularly Perturbed Steady Convection–Diffusion–Reaction Problems," Mathematics, MDPI, vol. 9(19), pages 1-15, October.

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