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The Role of Boundary Conditions on Convergence Properties of Peridynamic Model for Transient Heat Transfer

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Abstract

Normally, peridynamic (PD) models for transient heat transfer converge, in the limit of the horizon size going to zero, to the solution of the corresponding PDE-based heat transfer. However, different ways of imposing local boundary conditions have been observed to lead to some interesting properties that may deliver the classical solution at a point in space and time from a series of PD solutions obtained with relatively large horizons. Here, we use analytical derivation of PD solutions to explain how approximations introduced by the one-point Gaussian discretization in space (the so-called “meshfree” PD method) and by specific implementations of boundary conditions lead to the intersection of m-convergence curves at the exact value of the corresponding classical model solution. This leads to a strategy of approximating the local solution better with a sequence of PD models that use relatively large horizons compared to a PD model that uses a small horizon. We analyze this property for transient heat conduction in homogeneous and heterogeneous bars. We find that material interfaces influence the intersection of m-convergence curves for transient heat conduction in a 1D heterogeneous bar.

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Acknowledgements

This work was supported by the Natural Science Foundation of China (No. 11802098) and the Natural Science Foundation of Hubei Province (No. 2018CFB111). The work of F.B. and J. Z. was supported by the US National Science Foundation CMMI CDS&E Grant No. 1953346 (program manager Joanne Culbertson). The authors would like to thank the anonymous reviewers for constructive comments and suggestions that have led to significant improvements.

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Authors and Affiliations

  1. Department of Engineering Mechanics, School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China

    Tie Mei, Zian Liu, Xuhao Peng & Ziguang Chen

  2. Hubei Key Laboratory of Engineering Structural Analysis and Safety Assessment, 1037 Luoyu Road, Wuhan, 430074, China

    Ziguang Chen

  3. Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE, 68588-0526, USA

    Jiangming Zhao & Florin Bobaru

Authors
  1. Tie Mei
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  2. Jiangming Zhao
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  3. Zian Liu
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  4. Xuhao Peng
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  5. Ziguang Chen
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  6. Florin Bobaru
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Correspondence to Ziguang Chen.

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Mei, T., Zhao, J., Liu, Z. et al. The Role of Boundary Conditions on Convergence Properties of Peridynamic Model for Transient Heat Transfer. J Sci Comput 87, 50 (2021). https://doi.org/10.1007/s10915-021-01469-0

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  • DOI: https://doi.org/10.1007/s10915-021-01469-0

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