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The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme

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Abstract

An extended Maxwell viscoelastic model with a relaxation parameter is studied from mathematical and numerical points of view. It is shown that the model has a gradient flow property with respect to a viscoelastic energy. Based on the gradient flow structure, a structure-preserving time-discrete model is proposed and existence of a unique solution is proved. Moreover, a structure-preserving P1/P0 finite element scheme is presented and its stability in the sense of energy is shown by using its discrete gradient flow structure. As typical viscoelastic phenomena, two-dimensional numerical examples by the proposed scheme for a creep deformation and a stress relaxation are shown and the effects of the relaxation parameter are investigated.

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Acknowledgements

This work is partially supported by JSPS KAKENHI Grant Numbers JP16H02155, JP17H02857, JP26800091, JP16K13779, JP18H01135, and JP17K05609, JSPS A3 Foresight Program, and JST PRESTO Grant Number JPMJPR16EA.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Kanazawa University, Kanazawa, 920-1192, Japan

    Masato Kimura & Hirofumi Notsu

  2. Japan Science and Technology Agency, PRESTO, Kawaguchi, 332-0012, Japan

    Hirofumi Notsu

  3. Department of Environment and System Sciences, Graduate School of Environment and Information Sciences, Yokohama National University, Yokohama, 240-8501, Japan

    Yoshimi Tanaka

  4. Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, 920-1192, Japan

    Hiroki Yamamoto

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  1. Masato Kimura
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  2. Hirofumi Notsu
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  3. Yoshimi Tanaka
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  4. Hiroki Yamamoto
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Correspondence to Hirofumi Notsu.

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Kimura, M., Notsu, H., Tanaka, Y. et al. The Gradient Flow Structure of an Extended Maxwell Viscoelastic Model and a Structure-Preserving Finite Element Scheme. J Sci Comput 78, 1111–1131 (2019). https://doi.org/10.1007/s10915-018-0799-2

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  • DOI: https://doi.org/10.1007/s10915-018-0799-2

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