In this paper, we deal with the numerical analysis of the Lord–Shulman thermoelastic problem with porosity and microtemperatures. The thermomechanical problem leads to a coupled system composed of linear hyperbolic partial differential equations written in terms of transformations of the displacement field and the volume fraction, the temperature and the microtemperatures. An existence and uniqueness result is stated. Then, a fully discrete approximation is introduced using the finite element method and the implicit Euler scheme. A discrete stability property is shown, and an a priori error analysis is provided, from which the linear convergence is derived under suitable regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation, the comparison with the classical Fourier theory and the behavior of the solution in two-dimensional examples.
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This assumption is not relevant in the analysis proposed here. It only allows to simplify the calculations.
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J. Baldonedo acknowledges the funding by Xunta de Galicia (Spain) under the program Axudas á etapa predoutoral with Ref. ED481A-2019/230. The work of J.R. Fernández has been partially supported by Ministerio de Ciencia, Innovación y Universidades under the research Project PGC2018-096696-B-I00 (FEDER, UE). The work of R. Quintanilla has been supported by Ministerio de Economía y Competitividad under the research project “Análisis Matemático de Problemas de la Termomecánica” (MTM2016-74934-P), (AEI/FEDER, UE), and Ministerio de Ciencia, Innovación y Universidades under the research project “Análisis matemático aplicado a la termomecánica” (currently under evaluation).
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Baldonedo, J., Bazarra, N., Fernández, J.R. et al. An a priori error analysis of a Lord–Shulman poro-thermoelastic problem with microtemperatures. Acta Mech 231, 4055–4076 (2020). https://doi.org/10.1007/s00707-020-02738-z