Kai-Qi Li

Artificial ground freezing (AGF) is widely applied in civil engineering to construct temporary frozen curtains for stability and groundwater control. Accurate prediction of the temperature field induced by multiple freezing pipes is essential for the safe and efficient design of AGF systems. To overcome the convergence degradation of standard physics-informed neural networks (PINNs), we propose a novel hard-constrained PINN framework (AGF-PINN-HC), which explicitly embeds Dirichlet boundary conditions into the network architecture using distance-based weighting functions. This formulation automatically guarantees boundary satisfaction and eliminates the loss competition between partial differential equation residuals and boundary conditions, thereby enhancing training stability and solution accuracy. The proposed framework is validated through numerical experiments on both canonical (triple-pipe) and complex freezing configurations (circular, square, and horseshoe-shaped). AGF-PINN-HC achieves high-fidelity temperature predictions and accurately reconstructs isotherms under varying pipe layouts. Notably, the framework demonstrates strong generalization under nonuniform boundary conditions, with no structural modifications. In addition, by integrating transfer learning, the training time is reduced by up to 90% for new boundary scenarios, while maintaining predictive accuracy. These results highlight the first demonstration of a robust and boundary-aware PINN solution for practical AGF design, offering a promising tool for thermal analysis and rapid optimization in complex geotechnical applications.

Artificial Ground Freezing (AGF) has evolved into a cornerstone technique for geotechnical engineering, offering robust solutions for ground stabilization, seepage control, and infrastructure development in challenging environments

Artificial ground freezing (AGF) is a widely used technique for soil stabilization and waterproofing. Numerous studies have been devoted to solving the heat transfer problems in AGF while encountering limitations in handling complex geometries and boundary conditions and being computationally intensive. Recently, using machine learning methods to predict temperature fields has gained attention, demonstrating the potential to achieve higher accuracy than conventional models. However, these methods are typically limited by the need for large, labeled datasets, which are time-consuming and difficult to obtain. In this study, we address these challenges by applying physics-informed neural networks (PINNs) to solve the steady-state heat transfer problem in AGF, focusing on the temperature distribution around a single freezing pipe. By embedding the heat conduction equation into the loss function, PINNs reduce the need for extensive labeled data. To enhance accuracy and efficiency, transfer learning is employed, and results are compared against the finite element method. Results show that PINNs achieve high accuracy, particularly in larger domains with moderate temperature gradients, while providing competitive performance in more complex configurations involving steeper gradients. This approach offers a promising alternative for modeling temperature fields in geotechnical applications, with implications for reducing computational costs in AGF design.

Frozen wall thickness ( E) is an important index that reflects the freezing performance of artificial ground freezing (AGF). In previous design and numerical studies, the inherent spatial variability of soil properties is often neglected. Moreover, groundwater seepage can remarkably affect the freezing performance in the AGF system, while the specific relations between seepage and E remain unclear. Accordingly, this study aims to explore the evolution of frozen wall thickness under seepage conditions via a numerical model that considers the coupled thermo-hydraulic process and variations in hydrothermal properties. As two vital soil properties, spatial variability of thermal conductivity and intrinsic permeability is simulated by random field combined with Monte Carlo simulations. Based on the coupled model, the effects of seepage velocity, direction, and pipe spacing are examined by sensitivity analysis. Two indicators are introduced to quantify the influences of uncertainty in hydrothermal properties and seepage. The unfavourable scenarios (i.e., lower bound) of E from random models are tabulated and formulated via evolutionary polynomial regression for practical references. These findings contribute to an enhanced understanding of the freezing behaviours of AGF and provide a rule of thumb for predicting frozen wall thickness under complex seepage conditions and design parameters.

Numerous studies have investigated the coupled multi-field processes in frozen soils, focusing on the variation in frozen soils and addressing the influences of climate change, hydrological processes, and ecosystems in cold regions. The investigation of coupled multi-physics field processes in frozen soils has emerged as a prominent research area, leading to significant advancements in coupling models and simulation solvers. However, substantial differences remain among various coupled models due to the insufficient observations and in-depth understanding of multi-field coupling processes. Therefore, this study comprehensively reviews the latest research process on multi-field models and numerical simulation methods, including thermo-hydraulic (TH) coupling, thermo-mechanical (TM) coupling, hydro-mechanical (HM) coupling, thermo–hydro-mechanical (THM) coupling, thermo–hydro-chemical (THC) coupling and thermo–hydro-mechanical–chemical (THMC) coupling. Furthermore, the primary simulation methods are summarised, including the continuum mechanics method, discrete or discontinuous mechanics method, and simulators specifically designed for heat and mass transfer modelling. Finally, this study outlines critical findings and proposes future research directions on multi-physical field modelling of frozen soils. This study provides the theoretical basis for in-depth mechanism analyses and practical engineering applications, contributing to the advancement of understanding and management of frozen soils.

In recent decades, the constitutive modelling for frozen soils has attracted remarkable attention from scholars and engineers due to the continuously growing constructions in cold regions. Frozen soils exhibit substantial differences in mechanical behaviours compared to unfrozen soils, due to the presence of ice and the complexity of phase changes. Accordingly, it is more difficult to establish constitutive models to reasonably capture the mechanical behaviours of frozen soils than unfrozen soils. This study attempts to present a comprehensive review of the state of the art of constitutive models for frozen soils, which is a focal topic in geotechnical engineering. Various constitutive models of frozen soils under static and dynamic loads are summarised based on their underlying theories. The advantages and limitations of the models are thoroughly discussed. On this basis, the challenges and potential future research possibilities in frozen soil modelling are outlined, including the development of open databases and unified constitutive models with the aid of advanced techniques. It is hoped that the review could facilitate research on describing the mechanical behaviours of frozen soils, and promote a deeper understanding of the thermo-hydro-mechanical (THM) coupled process occurring in cold regions.

The precise determination of subsurface thermal properties is critical for ground-source heating systems. The geomaterials are inherently heterogeneous, and their thermal conductivity measured in laboratory and field tests often exhibits anisotropic behaviours. However, the accurate measurement of thermal responses in geomaterials presents a challenging task due to the anisotropy’s variation with the observed scale. Hence, a numerical method is developed in this work and illustrated by taking a typical anisotropic structure of geomaterials with the porosity of 0.5 as an example. The differences in data from laboratory measurements and field tests are discussed to explore the scale effect on anisotropic thermal properties. A series of simulation tests are conducted on specimens with varying dimensions using the finite element method. Results indicate that the thermal properties show a substantial sensitivity to the observation scale, the variation of which decreases with the sample dimensions. By comparing in situ data and laboratory results, the values of average thermal conductivity and corresponding anisotropy ratio are lower than those at small scales, indicating that careful consideration should be given to the thermal properties to account for heterogeneity and anisotropy. In addition, four upscaling schemes based on the averaging method are discussed. This study sheds light on the gap between the laboratory results and the field’s inherent properties and provides guidelines for upscaling small-scale results to field-scale applications.