⚡ electrical engineering

Modeling method for axial symmetric liquid storage tank sloshing equivalent parameters based on random forests

This paper proposes a rapid, high-precision modeling method for the equivalent sloshing parameters of axisymmetric liquid storage tanks using a Random Forest proxy model trained on a large-scale dataset, which significantly improves design efficiency and reveals that fill ratio and length-to-slenderness ratio are the most critical geometric factors influencing sloshing behavior.

Original authors: Yulin Chen

Published 2026-07-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yulin Chen

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 you are designing a rocket to carry astronauts to the moon. Inside that rocket are huge tanks filled with liquid fuel. Now, picture a glass of water in your hand. If you walk quickly, the water sloshes back and forth, right? In a rocket, that sloshing isn't just a messy spill; it's a powerful force that can push the rocket off course, making it wobble or even spin out of control. Engineers call this "sloshing." To keep the rocket steady, they need to predict exactly how that liquid will move. They do this by creating a "virtual twin" of the liquid—a simplified mathematical model that acts like a swinging weight on a string (a pendulum) or a weight bouncing on a spring. These models tell the rocket's computer how to adjust its engines to fight the wobble.

The problem is, figuring out how these virtual weights behave is incredibly hard. Traditionally, engineers have to run massive, slow computer simulations that can take hours or even days for just one tank shape. If they want to try a new shape, they have to wait again. It's like trying to design a new car by building a full-size clay model, putting it in a wind tunnel, and waiting a week to see if it flies. They need a faster way to guess the answer without building the whole thing every time. This is where the new study comes in, looking for a shortcut that is both fast and accurate.

The Paper's Big Idea: Teaching a Computer to Guess the Wobble

This paper introduces a clever new method to predict how liquid sloshes in rocket tanks, using a type of artificial intelligence called a "Random Forest." Think of a Random Forest not as a bunch of trees, but as a massive team of detectives. Instead of one detective trying to solve a mystery alone, you have hundreds of them, each looking at the clues from a slightly different angle. They all vote on the answer, and the final result is usually spot-on.

The researchers, led by Yulin Chen from Shanghai Jiao Tong University, wanted to see if this "team of detectives" could learn to predict the behavior of liquid fuel in tanks shaped like cylinders, capsules, or spheres. First, they built a digital playground where they could generate thousands of different tank shapes. They used seven "knobs" to tweak the tanks: changing the length of the straight middle part, the shape of the top and bottom caps (making them pointy or round), and how full the tank was.

To teach their AI team, they didn't just guess. They used a trusted, high-precision physics simulator (called SLOSH-ML) to calculate the exact "sloshing rules" for 1,500 different tank scenarios. This took a while, but it created a perfect "answer key." Then, they fed this data into the Random Forest. The AI studied the patterns, learning how changing a tank's shape or fill level changed the way the liquid swung.

The Results: Speed and Surprise

The results were impressive. Once trained, the AI could predict the sloshing behavior in milliseconds. That is thousands of times faster than the traditional physics simulations, which can take hours. But speed wasn't the only win; the accuracy was there, too.

The AI's predictions were incredibly close to the "real" physics answers. For eight different parameters (like the weight of the swinging liquid, how long the virtual pendulum is, and how stiff the virtual spring is), the model got an average accuracy score (called $R^2$ ) of 0.946. In the world of predictions, a score of 1.0 is perfect, and 0.98 or higher is considered excellent. Six of the eight parameters the AI predicted hit that 0.98 mark or higher. Even the ones that were slightly lower were still accurate enough to be useful for real engineering, with errors so small they wouldn't cause a rocket to crash.

What Actually Matters? (The "Sobol" Secret)

The researchers didn't stop at just making a fast calculator. They also asked a deeper question: "Which parts of the tank shape actually matter the most?" To answer this, they used a method called "Sobol sensitivity analysis," which is like a spotlight that shines on the most important factors while dimming the unimportant ones.

They found two "superstars" that controlled almost everything:

The Fill Ratio: How full the tank is.
The Slenderness Ratio: How long and skinny the tank is compared to how wide it is.

These two factors were the heavy lifters, influencing the sloshing behavior more than all the other details combined. Things like the exact curve of the top or bottom of the tank mattered much less. It's like realizing that when you're swinging on a playground swing, the most important things are how high you are sitting and how long the chains are; the color of the seat or the exact shape of the handle doesn't change how you swing.

What This Means for the Future

This paper doesn't claim to have solved every problem in rocket science, but it offers a powerful new tool. By using this Random Forest method, engineers can now test hundreds of different tank designs in the time it used to take to test just one. They can iterate quickly, finding the best shape to keep the fuel from sloshing too wildly, all without spending months on slow computer runs.

The study confirms that machine learning can capture the complex, wiggly physics of liquid fuel with high precision. It suggests that the future of rocket design might involve AI assistants that instantly tell engineers, "If you make the tank 10% skinnier, the fuel will swing less," allowing them to build safer, more efficient spacecraft much faster than before.

Technical Summary: Modeling Method for Axial Symmetric Liquid Storage Tank Sloshing Equivalent Parameters Based on Random Forests

Problem Statement
In aerospace engineering, the dynamic behavior of liquid propellant sloshing within storage tanks critically impacts the attitude stability and control accuracy of launch vehicles and spacecraft. Accurate determination of equivalent sloshing parameters—such as natural frequencies, equivalent masses, pendulum lengths, and damping ratios—is a prerequisite for Guidance, Navigation, and Control (GNC) system design. Traditional methods for obtaining these parameters face significant limitations:

Analytical methods are computationally fast but restricted to regular, axisymmetric tanks and cannot handle complex configurations or anti-slosh structures.
Numerical simulations (e.g., Boundary Element Method or CFD) offer higher accuracy but suffer from low computational efficiency, with single simulations taking hours to days, making them unsuitable for rapid multi-scheme iteration.
Model testing is reliable but time-consuming and costly, typically reserved for final verification.

Consequently, there is a need for a rapid, high-precision modeling method capable of handling diverse tank geometries to support efficient design cycles and reduce R&D costs.

Methodology
The paper proposes a rapid modeling framework that integrates parametric geometry definition, high-fidelity simulation, and machine learning surrogate modeling.

Parametric Tank Description: A fully parametric system was constructed to characterize axisymmetric liquid storage tanks using seven geometric parameters with clear physical significance. These parameters define the cylindrical section length, upper and lower head straight edge lengths, upper and lower head shape indices (controlling the transition from flat to spherical/ellipsoidal profiles), and the slenderness ratio. This framework unifies the representation of common engineering configurations, including cylindrical, capsule, and spherical tanks.
Dataset Generation: A large-scale, standardized dataset was generated using SLOSH-ML, a verified boundary element method (BEM) simulation tool based on linear potential flow theory. SLOSH-ML reduces the 3D sloshing problem to a 2D meridional plane, offering high engineering accuracy with significantly lower computational cost than CFD.
- Sampling: Latin Hypercube Sampling (LHS) was used to generate 1,500 sample sets across the feasible parameter space.
- Targets: For each sample, eight equivalent parameters were calculated: fixed mass, moving mass (sloshing mass), equivalent pendulum length, suspension point height, sloshing amplitude ratio (CMI), spring-mass sloshing mass, spring-mass height, and spring stiffness.
- Data Split: The dataset was divided into a training set (1,200 samples) and a test set (300 samples).
Random Forest Surrogate Model: A Random Forest (RF) regression model was trained to map the seven geometric input parameters to the eight output equivalent parameters.
- Optimization: Hyperparameters (number of trees, max depth, min samples split, etc.) were optimized using grid search combined with 5-fold cross-validation.
- Evaluation: Model performance was assessed using the coefficient of determination ( $R^2$ ), Root Mean Square Error (RMSE), and Mean Absolute Error (MAE).
Sensitivity Analysis: To interpret the model and identify key design drivers, Sobol global sensitivity analysis was applied. This method quantifies both the independent effects of individual geometric parameters and their interaction effects on the output variables.

Key Results

Prediction Accuracy: The Random Forest surrogate model demonstrated high predictive capability. The average coefficient of determination ( $R^2$ $R^{2}$ ) across the eight target variables was 0.946. Six of the eight parameters achieved an $R^2$ $R^{2}$ exceeding 0.98.
- The sloshing amplitude ratio (CMI) showed the highest accuracy ( $R^2 = 0.990$ ).
- Moving mass, equivalent pendulum length, and spring stiffness also achieved $R^2$ values above 0.98.
- Parameters with narrow numerical ranges, such as suspension height and spring installation height, showed lower $R^2$ scores (0.824 and 0.891, respectively) due to the amplification of relative errors, though their absolute errors remained extremely low and within engineering allowable limits.
Computational Efficiency: The trained model achieves millisecond-level predictions, representing an improvement in computational efficiency of 3 to 4 orders of magnitude compared to traditional BEM and CFD methods.
Sensitivity Findings: Sobol analysis revealed distinct patterns in parameter influence:
- Fill Ratio and Slenderness Ratio are the dominant factors, ranking first and second in total sensitivity indices for all eight equivalent parameters.
- Fixed mass, sloshing mass moment of inertia, and spring mounting height are predominantly driven by the fill ratio.
- Sloshing mass, equivalent pendulum length, and equivalent stiffness are jointly dominated by the slenderness ratio and fill ratio.
- Other structural parameters (e.g., head shape indices, specific section lengths) exhibit negligible first-order sensitivity, with their influence primarily arising through interactions with the core geometric variables.

Significance and Contributions
The paper claims that the proposed method provides a robust theoretical and practical foundation for the rapid design and optimization of spacecraft storage containers. By successfully capturing the strong nonlinear mapping relationships between complex tank geometries and equivalent sloshing parameters, the Random Forest surrogate model enables:

Rapid Design Iteration: The millisecond-level prediction speed allows for the evaluation of thousands of design schemes in the time previously required for a single simulation, significantly shortening design cycles.
Cost Reduction: The method reduces reliance on expensive and time-consuming physical model testing and high-fidelity CFD simulations during the early and mid-stages of development.
Design Guidance: The sensitivity analysis provides clear, quantitative evidence that the fill ratio and slenderness ratio are the most critical design parameters. This insight directs engineers to prioritize these variables when optimizing tank configurations to control sloshing characteristics, offering a clear basis for configuration optimization.

The study concludes that machine learning-based surrogate modeling, specifically using Random Forests, is a feasible and highly effective approach for the rapid prediction of tank sloshing equivalent parameters, bridging the gap between high-fidelity simulation accuracy and the speed required for modern aerospace engineering design.

Technical Summary: Modeling Method for Axial Symmetric Liquid Storage Tank Sloshing Equivalent Parameters Based on Random Forests

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