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A Hermite interpolation element-free Galerkin method for functionally graded structures

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  • Ma, Xiao
  • Zhou, Bo
  • Xue, Shifeng

Abstract

In this paper, a Hermite interpolation element-free Galerkin method (HIEFGM) for functionally graded structures (FGSs) is proposed by combining the Hermite interpolation and interpolation element-free Galerkin method (IEFGM). The proposed method considers the normal derivative of the displacements at the boundary, which can improve the computational accuracy without adding the nodes. The material properties of the FGSs are assumed to vary exponentially along the thickness direction. Employing the constitutive equation, geometric equation and equilibrium equation, the Galerkin weak form for governing equation of the FGSs is obtained. The Hermite interpolation and moving least squares method are utilized for approximating the field variables in problem domain. The discrete governing equation is procured using the variational principle and the HIEFGM formulation for the FGSs is established. The shape function constructed in meshless approximation has Kronecker delta property, which is conducive to the application of the essential boundary conditions. Furthermore, an error analysis is performed to determine the effects of weight function, scale factor and node density in the performance of this method. Three benchmark numerical examples considering the different gradient indexes are conducted, and the results uniformly demonstrate the HIEFGM is accurate and reliable for the FGSs.

Suggested Citation

  • Ma, Xiao & Zhou, Bo & Xue, Shifeng, 2022. "A Hermite interpolation element-free Galerkin method for functionally graded structures," Applied Mathematics and Computation, Elsevier, vol. 419(C).
    • Handle: RePEc:eee:apmaco:v:419:y:2022:i:c:s0096300321009486
    DOI: 10.1016/j.amc.2021.126865
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    References listed on IDEAS

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    1. Ma, Xiao & Zhou, Bo & Xue, Shifeng, 2021. "A meshless Hermite weighted least-square method for piezoelectric structures," Applied Mathematics and Computation, Elsevier, vol. 400(C).
    2. Dai, Baodong & Wei, Dandan & Ren, Hongping & Zhang, Zhu, 2017. "The complex variable meshless local Petrov–Galerkin method for elastodynamic analysis of functionally graded materials," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 17-26.
    3. Ma, Jichao & Wei, Gaofeng & Liu, Dandan & Liu, Gongtian, 2017. "The numerical analysis of piezoelectric ceramics based on the Hermite-type RPIM," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 170-182.
    4. Lal, Roshan & Dangi, Chinika, 2021. "Dynamic analysis of bi-directional functionally graded Timoshenko nanobeam on the basis of Eringen's nonlocal theory incorporating the surface effect," Applied Mathematics and Computation, Elsevier, vol. 395(C).
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