Reduced-Order Computational Strategy Accelerates Nonlinear Coupled Thermo-Poroelasticity Simulations

Computational engineering researchers have established a non-incremental model order reduction strategy capable of resolving strongly coupled, nonlinear thermo-poroelasticity problems across parametric variations with significant reductions in computational runtime. Simulating porous media subjected to thermal, mechanical, and fluid loads traditionally demands the simultaneous or staggered solution of large, ill-conditioned algebraic systems. By deploying the Large Time Increment method coupled natively with Proper Generalized Decomposition, known as LATIN-PGD, the investigative team decoupled the global linear balance laws from the local nonlinear constitutive relationships. This configuration facilitates the progressive construction of independent reduced-order bases for solid displacement, fluid pore pressure, and thermal increments over the entire simulation time domain rather than proceeding through standard time-marching increments.

Thermo-poroelastic formulations govern interactions across three distinct physical behaviors: solid mechanics, fluid flow, and thermal conduction. The equations characterize applications spanning geotechnical reservoirs, radioactive waste repositories, and biomechanical systems where porous linear elastic skeletons interact with internal fluid pressure and temperature gradients. Standard monolithic approaches resolve all three fields simultaneously at each discrete time step, which guarantees formal coupling accuracy but generates substantial numerical expense and lacks modular adaptability when local nonlinearities are introduced. Conversely, conventional staggered approaches alternate the resolution of individual fields at each step, yielding reduced algebraic system sizes at the expense of potential numerical stability and accuracy deficits unless enhanced through multiple interfield iterations.

Within the presented formulation, the coupled continuous continuum is described by mechanical momentum balance, fluid mass balance, and thermal energy balance over a bounded domain. The constitutive equations link solid stress to strain, pore pressure, and thermal increments through modified Hooke’s law, while fluid velocity follows Darcy’s law and heat conduction follows Fourier’s law. In the generalized thermal source expression, two prominent nonlinear effects arise: the convective heat transport governed by the scalar product of Darcy’s velocity and the spatial temperature gradient, and the fluid viscous dissipation term proportional to the square of Darcy’s velocity divided by permeability.

The LATIN-PGD computational architecture partitions the problem into two distinct functional spaces to circumvent the repeated direct inversion of large nonlinear coupled matrices. One subspace encapsulates the linear admissibility and equilibrium equations across the domain, while the second subspace collects the local, time-dependent, and nonlinear constitutive laws along with initial conditions. The iterative algorithm alternates between a local coupled stage and a global decoupled stage using defined search direction operators. At the decoupled stage, Proper Generalized Decomposition expresses each primal unknown—displacement, pore pressure, and temperature—as a finite sum of separated spatial modes multiplied by time functions, generating independent bases tailored specifically to the disparate time scales and dynamics inherent to each physics.

A critical determinant of algorithm stability and convergence speed rests on identifying suitable search directions, which incorporate two characteristic time parameters corresponding to fluid and thermal transport. By conducting dimensional analyses on simplified one-dimensional governing equations, the authors derived analytical expressions for the fluid characteristic time and the thermal characteristic time. These derivations account directly for mechanical-fluid-thermal coupling parameters, including the drained bulk modulus, Biot coefficient, Biot modulus, and fluid permeability, alongside the discrete temporal increment size. Validation on a monodimensional poroelastic beam confirmed that the analytically predicted characteristic time closely matched the empirical optimum that minimizes iteration counts down to a preset error threshold.

The solver was initially evaluated against the Aboustit benchmark, a classic one-dimensional consolidation column subjected to constant mechanical and thermal boundary loadings on its upper surface. Spatial discretization incorporated composite finite elements, utilizing quadratic approximations for mechanical displacements and linear approximations for pore pressure and temperature fields. Under linear conditions, the LATIN-PGD solver matched the reference profiles produced by standard monolithic schemes, exhibiting identical field trends across the simulation duration. The presence of initial pressure oscillations in early time steps was linked directly to the rapid boundary application and temporal step sizing established in historical benchmark literature.

When nonlinear terms within the Aboustit problem’s thermal source were activated, the LATIN-PGD solver maintained robust computational efficiency. Comparing runtime performance between linear and nonlinear configurations yielded computation durations of 19.53 seconds and 19.58 seconds, respectively, reflecting an operational ratio of only 1.003. Spatial mode decompositions revealed that while the temperature field required additional modes to resolve localized convective gradients, the fluid and mechanical fields retained stable base sizes. This stability demonstrates that localized thermal nonlinearities can be integrated without inducing numerical divergence or triggering widespread mode inflation across nonlinearly uncoupled field components.

Parametric investigations on the Aboustit benchmark evaluated system response across 30 sequential evaluations where the thermal expansion coefficient of the solid phase varied uniformly from 80 percent to 120 percent of its baseline value. Rather than initializing each parametric computation from an arbitrary undeformed state, each run adopted the solution and reduced basis of the nearest preceding parameter point. The initial computation required 20.31 seconds to establish the foundational reduced-order basis, whereas subsequent calculations averaged 1.157 seconds by relying predominantly on lightweight updates of the temporal functions. Summed across all 30 parametric evaluations, total runtime reached 53.87 seconds, delivering a computational speedup factor of 11.4 compared to an estimated naive sequence of independent single-run simulations.

Ceramic filtration relies on pressurized fluid transport across microporous channel geometries to filter bacteria, creating complex mechanical stresses, localized pressure drops, and elevated thermal gradients. Exploiting geometric symmetry, the numerical mesh examined a quadrant comprising 52,630 tetrahedral elements, yielding 65,442 mechanical nodes alongside 10,019 fluid and thermal nodes. Fluid pore pressure and temperature were increased linearly across an initial 12.5-millisecond interval before remaining steady through a 25-millisecond terminal duration.

Application of the analytical characteristic time expressions to the ceramic membrane yielded a fluid value that deviated by a factor of 14 from empirical optima, attributable to complex channel curvature and omitted spatial derivative variations in the analytical simplification, while the thermal characteristic time matched closely. Despite this approximation discrepancy, the LATIN-PGD algorithm converged within 28 iterations for the fully nonlinear regime compared to 15 iterations in the linear counterpart under an error tolerance of 0.001. A comparison of stress distributions revealed that omitting convective and viscous nonlinearities led to a 30 percent local discrepancy in equivalent Von Mises stress, emphasizing that nonlinear terms remain structurally indispensable for assessing membrane integrity.

By traversing the parametric space along an ordered path that altered only one property per consecutive run, the cumulative simulation time totaled 18,600 seconds, compared against an estimated naive cumulative duration of 380,730 seconds based on an average single-run cost of 1,110 seconds. This sequential initialization and progressive enrichment yielded an aggregate time gain factor of 20.35. Fluid and thermal bases expanded only intermittently when permeability shifted, whereas the mechanical basis incorporated up to 7 new modes per run or converged exclusively via temporal function updates.

The study establishes that constructing uncoupled, on-the-fly reduced bases within an iterative LATIN framework offers a tractable pathway for exploring multi-parameter material spaces in nonlinear coupled engineering physics. The authors noted that future developments must address distinct spatial and temporal discretizations tailored individually to the distinct dynamic rates of mechanical, fluid, and thermal fields. Further methodological extensions contemplate integrating multi-fidelity data-driven models to enrich reduced physical bases in scenarios where full mathematical descriptions of secondary multiphysics processes remain partially characterized.

Subject of Research: Model order reduction for parametrized nonlinear thermo-poroelasticity

Article Title: A strategy with reduced models dedicated to parametrized nonlinear strongly coupled thermo-poroelasticity problems

Article References: Foulatier, É., Néron, D., Louf, F., & Boucard, P.-A. (2025). A strategy with reduced models dedicated to parametrized nonlinear strongly coupled thermo-poroelasticity problems. Advanced Modeling and Simulation in Engineering Sciences, 12(1), Article 32. https://doi.org/10.1186/s40323-025-00314-5

Image Credits: AI Generated

DOI: 10.1186/s40323-025-00314-5

Keywords: Thermo-poroelasticity, LATIN method, Proper Generalized Decomposition, Model order reduction, Multiphysics simulation, Ceramic membranes, Nonlinear mechanics, Computational engineering, strategy, reduced, models, dedicated