Zeitschrift fur Angewandte Mathematik und Physik最新文献

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A MGT thermoelastic problem with two relaxation parameters 具有两个松弛参数的MGT热弹性问题

3区数学 Q2 MATHEMATICS, APPLIED

Zeitschrift fur Angewandte Mathematik und Physik

Pub Date : 2023-09-15 DOI: 10.1007/s00033-023-02080-z

Noelia Bazarra, José R. Fernández, Ramón Quintanilla

Abstract In this paper, we consider, from both analytical and numerical viewpoints, a thermoelastic problem. The so-called MGT model, with two different relaxation parameters, is used for both the displacements and the thermal displacement, leading to a linear coupled system made by two third-order in time partial differential equations. Then, using the theory of linear semi-groups the existence and uniqueness to this problem is proved. If we restrict ourselves to the one-dimensional case, the exponential decay of the energy is obtained assuming some conditions on the constitutive parameters. Then, using the classical finite element method and the implicit Euler scheme, we introduce a fully discrete approximation of a variational formulation of the thermomechanical problem. A main a priori error estimates result is shown, from which we conclude the linear convergence under suitable additional regularity conditions. Finally, we present some one-dimensional numerical simulations to demonstrate the convergence of the fully discrete approximation, the behavior of the discrete energy decay and the dependence on a coupling parameter.

摘要本文从解析和数值两方面考虑了一个热弹性问题。对位移和热位移采用两种不同松弛参数的MGT模型，得到由两个三阶时间偏微分方程组成的线性耦合系统。然后利用线性半群理论证明了该问题的存在唯一性。如果我们限制在一维情况下，假设本构参数的某些条件，就可以得到能量的指数衰减。然后，利用经典的有限元方法和隐式欧拉格式，我们引入了热力学问题变分公式的完全离散近似。给出了一个主要的先验误差估计结果，并由此得出在适当的附加正则性条件下线性收敛的结论。最后，我们给出了一些一维数值模拟来证明完全离散近似的收敛性，离散能量衰减的行为以及对耦合参数的依赖。

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3区数学 Q2 MATHEMATICS, APPLIED

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