Source-linked AI summary
Parameters Robust Parallelizable Iterative Decoupled Algorithms for Thermo-Poroelasticity with Nonlinear Convective Transport
Ziliang Li, Jingzhi Li
TL;DR
Nonlinear thermo-poroelasticity couples fluid flow, heat transfer, and porous-material deformation, while extreme parameter regimes pose numerical challenges. The paper introduces a locking-free four-field formulation and stabilized parallel iterative algorithms; its analysis establishes convergence under stated conditions, and experiments report expected convergence rates in challenging regimes.
Problem
Method
Results
Across the nearly incompressible, convection-dominated, and strong-coupling cases, the algorithms retain expected convergence rates, with decoupled-method errors comparable to the coupled algorithm in the latter two cases.
Source: §5.1 · p. 18, §5.3 · p. 20Takeaways & Limitations
The analysis establishes convergence of the iterative algorithms to the discretized coupled solution under stated conditions, while the paper reports parameter-independent convergence to the nonlinear coupled solution.
Source: §1 · p. 3, §3.1.5 · p. 7
Abstract
from arXiv · showhide
This paper presents several parallel iterative algorithms for the nonlinear thermo-poroelasticity. We introduce an auxiliary variable to transform the original model into a four-field formulation to overcome Poisson locking. The iterative idea is extended to linearize and decouple the four-field model. Different coupling strategies are employed among these three subproblems, combined with appropriately designed iteration settings, yielding distinct iterative parallel algorithms. Stabilization terms are incorporated into these algorithms to ensure parameter robustness in extreme situations. We derive error estimates and prove the convergence of these parallel iterative decoupled algorithms. Finally, numerical experiments are presented to compare the performance of the different parallel algorithms in extreme parameter settings, thereby validating their efficiency and robustness.
1. Introduction
Nonlinear thermo-poroelasticity couples non-isothermal fluid flow with porous-material deformation, combining Biot’s equation with energy conservation. The paper develops parallel iterative algorithms for this nonlinear system, addressing parameter variation, Poisson locking, and convergence.
1. Introduction
- The model couples mechanical effects, fluid flow, and heat transfer in porous media through Biot’s equation [3, 20, 4] and energy conservation.It describes interactions between non-isothermal fluid flow and porous-material deformation.Source: §1 · p. 1
- The analysis assumes mixed boundary conditions for displacement, Dirichlet conditions for pressure and temperature, and homogeneous boundary data.It also specifies regularity assumptions for coefficients, initial conditions, and source terms.Source: §1 · p. 2
- Iterative algorithms for thermo-poroelastic models remain relatively limited, despite prior coupled, decoupled, and iterative approaches.Earlier work includes a five-field formulation, a four-field iterative decoupling method, and iterative solution with reduced-order modeling.Source: §1 · p. 2
- The paper introduces an auxiliary variable to form a four-field model, then splits the system into parallelizable mechanical, flow, and heat subproblems.Stabilization terms address ill-conditioning and pressure or temperature instability under extreme parameter conditions; the analysis derives an error estimate and proves iterative convergence.Source: §1 · p. 3
2. Poisson Locking-Free Four-Field Formulation
An auxiliary variable extends the three-field thermo-poroelastic model to four fields to address Poisson locking as the Lamé parameter tends to infinity. The formulation is discretized and analyzed under stated assumptions and boundary conditions.
2. Poisson Locking-Free Four-Field Formulation
- As the Lamé constant λ tends to infinity, displacement approaches a divergence-free state, called Poisson locking; an auxiliary variable extends the model to four fields to eliminate it.The formulation follows approaches in.Source: §2 · p. 3
- The formulation retains the initial conditions and Dirichlet boundary condition; the analysis assumes homogeneous boundary data, with an extension to nonhomogeneous conditions described as straightforward.The considered displacement boundary conditions are mixed, while pressure and temperature have Dirichlet conditions.Source: §2 · pp. 3–4
- The four-field variational formulation is discretized in time with backward Euler on a uniform time partition.Its finite-element approximation uses displacement, auxiliary-variable, pressure, and temperature spaces.Source: §2 · p. 4
- The paper states that its discrete nonlinear algorithm is free of Poisson locking as λ tends to infinity, under the theorem’s assumptions and discrete inf-sup condition.The error-estimate constant is described as depending on 1/λ and the listed model parameters.Source: §2 · pp. 4–5
3. Parallel Iterative Algorithms with Stabilization Terms
Extreme parameter regimes can make the pressure-temperature block ill-posed, motivating stabilization and distinct iterative coupling strategies. The algorithms use lagged convection and parallel subproblem solves, with convergence established under stated time-step and stabilization conditions.
3. Parallel Iterative Algorithms with Stabilization Terms
- Small material, permeability, and conductivity parameters can make the pressure-temperature block ill-posed, and this issue persists in the four-field formulation.Large convection coefficients also make the convection term dominant and introduce a nonsymmetric perturbation that worsens ill-posedness.Source: §3 · p. 5
- Diagonal stabilization restores algebraic invertibility and guarantees convergence; additional stabilization is required for the MF-H and MH-F algorithms.The stated role of stabilization is to address ill-posedness and ensure convergence.Source: §3 · p. 5
3.1.1. MFH algorithm:
- The fully coupled iterative algorithm replaces the nonlinear convection term with its value from the previous iteration, preserving symmetry and simplicity.Other linearizations, including fixed-point and Newton-type approaches, are also possible.Source: §3.1.1 · p. 6
- Different splitting strategies divide the fully coupled system into two or three decoupled subproblems, yielding distinct iterative algorithms.The algorithms repeat parallel subproblem solves until the iterates meet a prescribed tolerance.Source: §3.1.1 · p. 6, §3.1.5 · p. 7
3.1.5. Parallel M-F-H algorithm:
- Theorem 3.1 establishes convergence for MFH and M-FH under its time-step condition, and gives stabilization bounds for MF-H and MH-F.The M-F-H, M-FH, MF-H, and MH-F methods compute subproblems in parallel at each iteration.Source: §3.1.5 · p. 7
4. Proofs of Error Estimates and Convergence Analysis
This section establishes an error estimate for the nonlinear model and proves convergence of the iteratively decoupled algorithms. The estimate remains uniform as the Lamé parameter tends to infinity.
4. Proofs of Error Estimates and Convergence Analysis
- The error estimate remains uniform with respect to the Lamé parameter as λ→∞, the nearly incompressible limit.The analysis partitions each field’s error into interpolation and discretization components.Source: §4 · pp. 8, 15
- The convergence analysis proves the most intractable case, MF-H, and states that the remaining results follow analogously.Source: §4 · p. 15
- When δ and B_i are positive, A_i tends to zero and the stated iteration errors tend to zero as i→∞.The convergence proof treats the MF-H algorithm; the remaining results follow analogously.Source: §4 · pp. 15, 17
5. Numerical experiments
The experiments assess accuracy, robustness, and computational performance across challenging parameter regimes. The algorithms retain expected convergence rates in the reported regimes, and two parallel variants have the shortest reported CPU times.
5. Numerical experiments
- The benchmark uses a unit-square domain with k = 2, s = l = 1 finite elements and reports terminal-time errors and convergence rates across mesh sizes.Unless otherwise stated, τ = 0.001, Δt = 10^-4, and each time step uses 10 iterations.Source: §5 · p. 18
5.1. Large Lamé parameters
- At ν = 0.4999999, all algorithms retain optimal energy-norm convergence rates as λ approaches infinity.Other parameters remain at baseline values, with ρ_p = ρ_T = ρ_ξ = 1.Source: §5.1 · p. 18, Table 2 · p. 19
5.2. Pressure and temperature numerical instability
- With a_0, b_0, and c_0 sufficiently small, the unstabilized three-field model produces spurious pressure instability, while stabilization eliminates these artifacts.The temperature counterpart exhibits the same behavior, though only pressure results are shown.Source: §5.2 · pp. 18–19
5.3. Convection dominate and strong coupling effects
- Across convection-dominated and strong-coupling cases, all iterative algorithms retain expected convergence rates, while decoupled methods produce errors comparable to the coupled algorithm.The cases use c_f = 100 and a_0 = c_0 = 1, b_0 = 0, α = β = 2.Source: §5.3 · p. 20, Table 3 · p. 20, Table 4 · p. 20
5.4. Computational efficiency
- MFH takes the longest CPU time, while parallel M-F-H and parallel M-FH take the shortest in the reported comparison.Subproblems for the parallel algorithms run concurrently on two or three CPU cores; MFH runs sequentially on one core.Source: §5.4 · p. 21, Table 5 · p. 21