Unified numerical analysis for thermoelastic diffusion and thermo-poroelasticity of thin plates

TL;DR

This paper develops and analyzes a coupled numerical scheme for modeling thermoelastic diffusion and thermo-poroelasticity in thin plates, demonstrating convergence and effectiveness through numerical experiments.

Contribution

It introduces a unified numerical approach combining Galerkin, Newmark, Crank--Nicolson, and finite element methods for coupled hyperbolic-parabolic systems in plate models, with proven convergence.

Findings

Numerical schemes achieve quasi-optimal convergence rates.

2D models closely approximate 3D behavior as plate thickness decreases.

Numerical experiments validate theoretical convergence rates.

Abstract

We investigate a coupled hyperbolic-parabolic system modeling thermoelastic diffusion (resp. thermo-poroelasticity) in plates, consisting of a fourth-order hyperbolic partial differential equation for plate deflection and two second-order parabolic partial differential equations for the first moments of temperature and chemical potential (resp. pore pressure). The unique solvability of the system is established via Galerkin approach, and the additional regularity of the solution is obtained under appropriately strengthened data. For numerical approximation, we employ the Newmark method for time discretization of the hyperbolic term and a continuous interior penalty scheme for the spatial discretization of displacement. For the parabolic equations that represent the first moments of temperature and chemical potential (resp. pore pressure), we use the Crank--Nicolson method for time discretization and conforming finite elements for spatial discretization. The convergence of the fully discrete scheme with quasi-optimal rates in space and time is established. The numerical experiments demonstrate the effectiveness of the 2D Kirchhoff--Love plate model in capturing thermoelastic diffusion and thermo-poroelastic behavior in specific materials. We illustrate that as plate thickness decreases, the two-dimensional simulations closely approximate the results of three-dimensional problem. Finally, the numerical experiments also validate the theoretical rates of convergence.