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Projection-based reduced order modeling of an iterative scheme for linear thermo-poroelasticity

Francesco Ballarin, Sanghyun Lee, Son-Young Yi

TL;DR

We address the computational challenge of linear thermo-poroelasticity by coupling a fixed-stress iterative scheme with projection-based ROMs trained from high-fidelity data. The offline stage uses a fixed-stress high-fidelity solver and POD to build reduced bases, while the online stage performs Galerkin projection for both monolithic (M-ROM) and fixed-stress (FS-ROM) formulations. The paper proves convergence of FS-HF to the monolithic HF solution and demonstrates FS-ROM convergence to FS-HF, achieving over two orders of magnitude speedups and effectively handling parametric heterogeneities. This approach enables fast, repeated simulations and multi-query analyses for THM systems with potential applications in geothermal and environmental settings.

Abstract

This paper explores an iterative coupling approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative coupling technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot's poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.

Projection-based reduced order modeling of an iterative scheme for linear thermo-poroelasticity

TL;DR

We address the computational challenge of linear thermo-poroelasticity by coupling a fixed-stress iterative scheme with projection-based ROMs trained from high-fidelity data. The offline stage uses a fixed-stress high-fidelity solver and POD to build reduced bases, while the online stage performs Galerkin projection for both monolithic (M-ROM) and fixed-stress (FS-ROM) formulations. The paper proves convergence of FS-HF to the monolithic HF solution and demonstrates FS-ROM convergence to FS-HF, achieving over two orders of magnitude speedups and effectively handling parametric heterogeneities. This approach enables fast, repeated simulations and multi-query analyses for THM systems with potential applications in geothermal and environmental settings.

Abstract

This paper explores an iterative coupling approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative coupling technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot's poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.
Paper Structure (19 sections, 4 theorems, 61 equations, 17 figures, 1 table)

This paper contains 19 sections, 4 theorems, 61 equations, 17 figures, 1 table.

Table of Contents

  1. Introduction
  2. Governing Equations
  3. High-fidelity Discretization
  4. Monolithic high-fidelity (M-HF) method
  5. Fixed-stress high-fidelity (FS-HF) iterative scheme
  6. Convergence analysis of the FS-HF scheme
  7. Projection-Based Reduced Order Models
  8. Proper orthogonal decomposition
  9. Monolithic reduced order model (M-ROM)
  10. Fixed-stress reduced order model (FS-ROM)
  11. Convergence analysis of the FS-ROM
  12. Numerical Experiments
  13. Example 1. Validation of ROM on test cases with smooth analytical solution
  14. Example 1A. Validation of HF and ROM solvers by a mesh refinement analysis
  15. Example 1B. Comparison of HF and ROM solvers in time
  16. ...and 4 more sections

Key Result

Lemma 3.1

The following coercivity condition is satisfied for any ${\bf v} \in \boldsymbol{V}_h$.

Figures (17)

  • Figure 1: Example 1A: convergence of the errors for the monolithic schemes (M-HF and M-ROM).
  • Figure 2: Example 1A: convergence of the errors for the fixed-stress iterative schemes (FS-HF and FS-ROM).
  • Figure 3: Example 1A: average CPU time per time step.
  • Figure 4: Example 1A: average number of fixed-stress iterations.
  • Figure 5: Example 1B: relative errors ($\delta=h,r$) with respect to the analytical solution for the fixed-stress iterative schemes (FS-HF and FS-ROM).
  • ...and 12 more figures

Theorems & Definitions (7)

  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.1
  • Remark 3.1
  • Remark 4.1
  • Theorem 4.1
  • Remark 4.2