Two-field mixed hp-finite elements for time-dependent problems in the refined theories of thermodynamics

TL;DR

This paper develops an $hp$-type finite element method for complex, time-dependent heat problems modeled by the Guyer--Krumhansl equation, demonstrating high efficiency and convergence in challenging scenarios.

Contribution

It introduces a novel $hp$-finite element technique tailored for complex geometries and advanced heat equations, with proven stability and superior computational speed.

Findings

Method achieves four times faster solutions than commercial algorithms.

Convergence properties are validated for various challenging scenarios.

Algorithm effectively handles complex geometries and fast propagation speeds.

Abstract

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.