Thanks to modern manufacturing technologies, heterogeneous materials with complex inner structures (e.g., foams) can be easily produced. However, their utilization is not straightforward, as the classical constitutive laws are not necessarily valid. According to various experimental observations, the Guyer--Krumhansl equation stands as a promising candidate to model such complex structures. However, the practical applications need a reliable and efficient algorithm that is capable of handling both complex geometries and advanced heat equations. In the present paper, we present the development of a $hp$-type finite element technique, which can be reliably applied. We investigate its convergence properties for various situations, being challenging in relation to stability and the treatment of fast propagation speeds. That algorithm is also proved to be outstandingly efficient, providing solutions four magnitudes faster than commercial algorithms.
翻译:由于现代制造技术,可以很容易地生产具有复杂内部结构(如泡沫)的多种材料,但是,由于古典成份法不一定有效,它们的利用并非直截了当,根据各种实验观察,Guyer-Krumhansl等式是建模这种复杂结构的有希望的候选方,然而,实际应用需要一种可靠和高效的算法,能够处理复杂的地貌和先进的热方程。我们在本文件中介绍了一种可可靠应用的$hp$型有限元素技术的发展。我们调查了各种情况的趋同特性,在稳定性和快速传播速度的处理方面,这种算法也非常有效,提供了比商业算法更快的四级解决方案。