A dynamic thermoviscoelastic contact problem with the second sound effect
A dynamic thermoviscoelastic contact problem with the second sound effect
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DOI：
10.1016/j.jmaa.2014.07.049
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发表时间：
2015-01
影响因子：
1.3
通讯作者：
A. Berti;M. Copetti;José R. Fernández;M. G. Naso
中科院分区：
文献类型：
--
作者：
A. Berti;M. Copetti;José R. Fernández;M. G. Naso
本文讨论了在Cattaneo定律下，可伸长的热粘弹性阻尼梁的接触问题，其中热流与温度梯度有关。梁在其左端被刚性地夹紧，而梁的右端像非线性弹簧一样在反作用止动件之间垂直移动。证明了解的存在性和唯一性，以及相关能量的指数衰减。然后，分别采用经典有限元法和隐式Euler格式对空间变量和时间导数进行全离散逼近。证明了一个先验误差估计，并由此推导出算法的线性收敛性。从解的存在性和长期性态的角度研究了两个停止点为刚性的情形。最后，给出了一些数值模拟，以证明近似的精度和解决方案的行为。
This paper deals with a contact problem describing the mechanical and thermal evolution of a damped extensible thermoviscoelastic beam under the Cattaneo law, relating the heat flux to the gradient of the temperature. The beam is rigidly clamped at its left end whereas the right end of the beam moves vertically between reactive stops like a nonlinear spring. Existence and uniqueness of the solution is proved, as well as the exponential decay of the related energy. Then, fully discrete approximations are introduced by using the classical finite element method and the implicit Euler scheme to approximate the spatial variable and to discretize the time derivatives, respectively. An a priori error estimates result is proved, from which the linear convergence of the algorithm is deduced. The case where the two stops are rigid is also studied from the point of view of the existence and longtime behavior of the solutions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation and the behavior of the solution.