⚡ electrical engineering

Open-Source Hardening Soil Material-Point Integrator and Neural-Network Surrogate for Ho Chi Minh City Soils

This study presents an open-source Python implementation of a Hardening Soil material-point integrator validated against Ho Chi Minh City soil data, which successfully serves as a high-fidelity physics-based data generator for training a highly accurate neural-network surrogate model.

Original authors: Thao Hoang The, Phan Vo, Truong Xuan Dang

Published 2026-08-10

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CC BY 4.0

Original authors: Thao Hoang The, Phan Vo, Truong Xuan Dang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the ground beneath our feet not as a solid, unchanging rock, but as a living, breathing sponge that squishes, stretches, and fights back depending on how hard you squeeze it. This is the world of geotechnical engineering, the science of figuring out how soil behaves when we build skyscrapers, dig deep subway tunnels, or carve out massive construction sites. To predict if the ground will hold up or collapse, engineers use "constitutive models"—essentially complex mathematical recipes that tell a computer how soil should react to stress. One of the most famous recipes is called the "Hardening Soil" model. Think of it like a sophisticated video game physics engine for dirt: it knows that soil gets stiffer the more you squeeze it (like a stress ball) and that it behaves differently when you push it versus when you let it go. However, for years, this powerful engine has been locked inside expensive, closed-source software, making it hard for independent scientists to peek under the hood, check the math, or mix it with new technologies like artificial intelligence. This paper steps in to unlock that door, creating a free, open version of this soil physics engine and using it to train a super-smart AI assistant that can predict soil behavior in a flash.

The researchers, working with soils from Ho Chi Minh City, started by building their own open-source version of the Hardening Soil model using Python, a popular programming language. They focused on a specific type of soil behavior called "drained primary loading," which is like watching how soil settles and stiffens when you slowly add weight to it while letting water escape. They didn't just write code; they tested it rigorously. First, they checked if their code made sense internally, ensuring the math didn't contradict itself. Then, they compared their computer-generated soil curves against real-world data from 274 measurements taken from three different types of local soil: soft clay, medium clay, and silty sand. The results were incredibly precise. When they tried to predict how stiff the soil would be at different pressures, their open-source model matched the real-world data with a correlation of 0.999 (where 1.0 is a perfect match) and an average error of less than 2%. In fact, the internal math was so consistent that the error was less than 0.1%, proving their "digital soil" was behaving exactly as the physics equations demanded.

Once they proved their open-source model was trustworthy, they used it as a "data generator" to solve a major problem in geotechnical engineering: there isn't enough real-world soil data to train advanced AI models. Real soil tests are expensive and time-consuming. So, the team used their verified model to simulate 20,000 different stress-strain curves, creating a massive library of "what-if" scenarios covering a wide range of soil conditions. They then fed this library into a neural network—a type of AI designed to recognize patterns. This AI, acting as a "surrogate" or a fast-track substitute for the complex physics model, learned to predict the entire stress-strain curve of the soil just by looking at a few input numbers. The AI was astonishingly accurate, achieving a correlation of 0.9999 and an error of only 1.47 kPa (a tiny unit of pressure).

The paper makes it clear that this is a simulation-based success story, not a replacement for all future testing. The authors explicitly state that their model is currently limited to specific loading conditions (drained primary loading) and does not yet handle the full complexity of unloading, reloading, or massive real-world construction projects like entire tunnel networks. They also note that while the AI is perfect at interpolating within the data it was trained on, it hasn't been tested against new, unseen real-world field data yet. However, the core achievement is undeniable: they have created a reproducible, open-source foundation that bridges the gap between traditional soil mechanics and modern machine learning. By proving that a verified physics model can generate high-quality data for AI, they have provided a new toolkit for engineers to perform faster, more transparent, and more reliable analyses of how our cities interact with the ground beneath them.

Technical Summary: Open-Source Hardening Soil Material-Point Integrator and Neural-Network Surrogate for Ho Chi Minh City Soils

Problem Statement
The Hardening Soil (HS) model is a standard constitutive framework for analyzing deep excavations and metro tunnels due to its ability to capture nonlinear primary loading, unloading–reloading behavior, and stress-dependent stiffness. However, its widespread adoption is hindered by two primary limitations:

Proprietary Constraints: The model is predominantly implemented in closed-source commercial software (e.g., PLAXIS), limiting independent verification, modification, and integration with modern machine-learning workflows.
Data Scarcity: High-quality geotechnical test data are expensive and scarce, constraining the direct training of data-intensive deep-learning models for complex soil behaviors.

Methodology
This study addresses these limitations through a three-stage workflow:

Open-Source Implementation: A Python-based, open-source material-point integrator was developed for the drained triaxial primary-loading response. The implementation incorporates:
- The stress-dependent stiffness law (power law).
- The Mohr–Coulomb failure criterion.
- The hyperbolic stress–strain relation (Kondner/Duncan–Chang formulation).
- Scope Note: The current implementation focuses on primary loading and does not include the full multisurface HS formulation (e.g., cap mechanism, dilatancy, or separate unloading–reloading branches).
Experimental Validation: The integrator was validated against consolidated-drained (CD) triaxial test data from three representative Ho Chi Minh City soil types: flow-state soft clay, medium clay, and silty sand. The dataset comprised 274 measured points from 48 specimens tested at confining stresses of 100, 200, and 300 kPa. Validation utilized a leave-one-out protocol to assess predictive performance without overfitting.
Neural-Network Surrogate Training: The validated material-point model served as a physics-based data generator. It produced 20,000 synthetic stress–strain curves by sampling five input parameters ( $E_{50ref}$ , $c'$ , $\phi'$ , $\sigma_3$ , and $m$ ) from uniform distributions. These data trained a Multilayer Perceptron (MLP) surrogate to map input parameters directly to stress–strain curves.

Key Results

Stress-Dependency Exponent ( $m$ ): The fitted exponent $m$ ranged from 0.61 to 0.94 across the soil layers, with group means between 0.66 and 0.85. These values align with regional data and the theoretical range for clay-like to sand-like stress dependency.
Model Validation:
- Internal Consistency: The numerical implementation showed a maximum relative error of 0.071% between the prescribed secant modulus and the value extracted from the simulated curve.
- External Validation: Leave-one-out validation against experimental data yielded a pooled $R^2$ of 0.998, a Pearson correlation of 0.999, and a Mean Absolute Percentage Error (MAPE) of 1.9%. Group-specific $R^2$ values ranged from 0.966 to 0.996.
Surrogate Performance: The trained neural network achieved an $R^2$ of 0.9999 and a Root Mean Square Error (RMSE) of 1.47 kPa on a held-out test set of 3,000 samples. The surrogate demonstrated smooth, differentiable mapping suitable for gradient-based optimization.

Significance and Claims
The paper claims to provide a reproducible, open-source framework that bridges constitutive modeling and machine learning in geotechnical engineering. Its principal contributions are:

Transparency: By moving the HS formulation into an open-source environment, the study enables independent verification and customization, addressing the "black box" nature of commercial software.
Data Generation: The study demonstrates a viable workflow for using a verified physics-based model to generate large-scale synthetic datasets, thereby overcoming the scarcity of high-quality experimental data required for training deep learning models.
Inverse Analysis Capability: The resulting differentiable surrogate offers a computationally efficient tool for inverse analysis and large-scale simulations, such as those involving excavation–metro-tunnel interactions.

Limitations
The authors explicitly note that the study is limited to drained primary loading in conventional triaxial compression. The implementation does not yet reproduce the full boundary-value behavior of excavation or tunnel problems, nor does it include triaxial extension or oedometer validation (the latter being illustrative only). The surrogate's high accuracy is interpreted as precise interpolation within the sampled parameter domain rather than proven out-of-distribution generalization.

Technical Summary: Open-Source Hardening Soil Material-Point Integrator and Neural-Network Surrogate for Ho Chi Minh City Soils

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