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Title: | A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model |
Authors: | Jang, Y Shaw, S |
Keywords: | viscoelasticity;generalised Maxwell solid;symmetric interior penalty;discontinuous Galerkin finite element method;a priori analysis;internal variables |
Issue Date: | 22-Feb-2023 |
Publisher: | Walter de Gruyter |
Citation: | Jang, Y. and Shaw, S. (2023) 'A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model', Computational Methods in Applied Mathematics, 23 (3), pp. 1 - 21. doi: 10.1515/cmam-2022-0201. |
Abstract: | The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a fading memory kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a Dirichlet–Prony series of decaying exponentials to provide the fading memory in the Volterra kernel. We introduce two types of internal variable to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and – in time – a Crank–Nicolson method to formulate the fully discrete problems: one for each type of internal variable. We present a priori stability and error analyses without using Grönwall’s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially. In this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems. We also carry out a number of numerical experiments using the FEniCS environment (https://fenicsproject.org), describe a simulation using “real” material data, and explain how the codes can be obtained and all of the results reproduced. |
URI: | https://bura.brunel.ac.uk/handle/2438/25989 |
DOI: | https://doi.org/10.1515/cmam-2022-0201 |
ISSN: | 1609-4840 |
Other Identifiers: | ORCID iD: Yongseok Jang https://orcid.org/0000-0002-2036-558X ORCID iD: Simon Shaw https://orcid.org/0000-0003-1406-7225 |
Appears in Collections: | Dept of Mathematics Embargoed Research Papers |
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FullText.pdf | Embargoed until 22 February 2024 | 2.43 MB | Adobe PDF | View/Open |
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