# Deep learning-driven predictive tools for damage prediction and optimization in composite hydrogen storage tanks

M. Nachtane, Mohammed El Fallaki Idrissi, M. Tarfaoui, Y. Qarssis, A. Abichou, A. Faik

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M. Nachtane, Mohammed El Fallaki Idrissi, M. Tarfaoui, Y. Qarssis, A. Abichou, et al., Deep learning-driven predictive tools for damage prediction and optimization in composite hydrogen storage tanks. Composites Communications, 2024, 51, pp.102079. 10.1016/j.coco.2024.102079. hal-05344644

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fDeep Learning-Driven Predictive Tools for Mechanical and
Damage Analysis in Composite Hydrogen Storage Tanks

M. Nachtane ¹*, M. El Fallaki Idrissi ², M. Tarfaoui ³,⁴, Y. Qarssis ¹,³, A. Abichou ⁵, A. Faik ¹

¹ UM6P, LIMSET, Benguerir 43150, Morocco.

² Arts et Métiers Institute of Technology, CNRS, PIMM-UMR 8006, Paris, France

³ ENSTA Bretagne, IRDL-UMR CNRS 6027, F-29200 Brest, France.

⁴ UM6P, Green Energy Park, GSMI, Benguerir 43150, Morocco

⁵ S Vertical Company, 92290 Paris, France.

Corresponding author:

* mourad.nachtane@um6p.ma

**Abstract**

This research presents a novel framework for optimizing and predicting damage in lightweight composite high-pressure tanks used for hydrogen storage. The framework combines advanced analytical and numerical methods with deep learning techniques to enhance design optimization and damage prediction. The 3D elasticity anisotropy theory is employed to predict the mechanical performances of the hydrogen storage vessel, and failure criteria are used to analyze damage. A parametric study is conducted to investigate the effects of wind angle, thickness, pressure, and diameter on the tank's performance. The analytical approach and finite element method using WoundSim Software are compared, highlighting the importance of modeling and analysis assumptions in determining the fidelity of results. A new deep learning framework is developed for stress and damage prediction, as well as a graphical user interface (GUI) is designed to automate calculations and predictions, facilitating user decision-making. This integrated approach offers a powerful tool to optimize the design and operation of lightweight composite high-pressure tanks, ensuring reliable and efficient hydrogen storage.

**Key words**

Composite materials, Hydrogen storage, High-pressure tanks, Analytical approach, WoundSim, Damage prediction.

**I. Introduction**

In recent years, hydrogen has emerged as a pivotal clean energy carrier, gaining significant traction due to its potential to decarbonize various sectors and contribute to a sustainable energy landscape [1,2]. Hydrogen storage, a critical step in the hydrogen value chain, ensures the reliability and efficiency of hydrogen as an energy carrier. The development and optimization of hydrogen storage technologies are essential to unlocking the full potential
of hydrogen as a clean energy source [3,4]. Leveraging their exceptional strength-to-weight ratio, composite hydrogen storage tanks are gaining prominence as a viable option for efficient hydrogen storage. This key advantage, coupled with improved thermal and mechanical properties, makes them highly versatile due to the ability to combine different constituent materials. Typically, these vessels are manufactured using filament winding processes, where the choice of materials plays a critical role in the final vessel characteristics. Despite their promising properties, the widespread adoption of composite materials in advanced hydrogen storage tanks is currently hindered by several factors, including high lifecycle costs and recycling challenges. Studies have shown that carbon fiber, a crucial component of these vessels, can account for a substantial portion of the total cost – ranging from 45% to 80% – depending on factors like tank size and production volume [5]. Furthermore, the design of hydrogen storage tanks requires careful consideration of application-specific requirements. These considerations include test pressure, external environmental factors, cycle life, and safety factors, all of which are crucial for stationary and portable applications [6-7]. Furthermore, material selection is strongly influenced by the anticipated failure modes and operating conditions experienced by the vessel during its lifetime [8].

Several key studies have significantly contributed to advancements in composite hydrogen storage vessel design. Xia et al. [9] shed light on the anisotropic nature of composite pipes. Their work presented an exact elastic solution for stresses and deformations based on 3D anisotropic elasticity. This research challenged conventional assumptions by highlighting the variations in hoop-to-axial stress ratios observed in cylindrical pressure vessels with different angle-ply designs. This understanding is crucial to optimize the design of composite vessels for specific pressure-load scenarios. Furthermore, Park et al. [10] focused on developing lightweight high pressure storage vessels suitable for hydrogen-fueled vehicles. Their study employed Classical Laminate Theory (CLT) to design composite layers for Type 4 hydrogen vessels. By optimizing the thickness and pattern of these composite layers, they were able to achieve lightweight construction while maintaining the required strength for hydrogen storage. This approach emphasizes the importance of tailoring the composite structure to meet specific weight and pressure requirements. Furthermore, Chapelle et al. [11] conducted a structural analysis of Type 3 high-pressure hydrogen storage vessels. Their analysis employed a CLT-based model to predict stresses and strains under static thermomechanical loading. This study provided valuable information to optimize the stacking sequences of composite layers within the walls of the vessel. Furthermore, it highlighted the growing use of filament-wound composite pipes within the oil and gas industry, where similar design considerations are applicable. Juszkiewicz et al. [12] conducted a comparative study on thin and thick cylinder models under thermomechanical loading conditions. When the responses of these two types of cylinders are compared, the research aims to provide information on structural performance and failure mechanisms under such loading conditions. Li et al. [13] investigated progressive damage in filament wound composite tubes with metal joints using FEA and specialized simulation software. Their findings revealed that metal joints can cause extrusion of outer composite layers, leading to further damage. They also demonstrated that strengthening the composite layer outside the metal joint can improve the overall tensile load
capacity. Importantly, their FEA results aligned well with experimental data, validating the effectiveness of FEA with the Puck criterion for predicting damage in these composite tubes. Almeida et al. [14] delved deeper into damage and failure mechanisms in carbon fiber reinforced epoxy filament wound composite tubes. They developed a damage model and a non-linear finite element model using the arc length method. Their research suggested that lower diameter-to-thickness (d/t) ratios tend to lead to buckling failures, while higher d/t ratios promote in-plane shear failures with delamination. Experimental validation confirmed the accuracy of their model, paving the way for its use in academic and industrial settings. Recent studies further emphasize the importance of FEA in the design of composite pressure vessels. Nebe et al. [15] demonstrated that the use of representative material properties in FE analysis allows an accurate reproduction of the mechanical response and final strength of composite pressure vessels. This approach is crucial to predict vessel behavior under various pressure conditions, ultimately leading to optimized designs with enhanced performance. Similarly, Rafiee et al. [16] employed multiscale analyses through FEA to estimate the burst pressure of composite pressure vessels under various internal pressure conditions. Kang et al. [17] performed a multiparametric sensitivity analysis on multilayered filament wound cylinders subjected to internal pressure. Their objective was to identify key parameters influencing the mechanical behavior of these vessels. This analysis serves as a valuable tool to develop a robust framework that can predict the behavior of composite pressure vessels under various operating conditions.

The field of hydrogen storage systems is embracing the transformative power of artificial intelligence (AI). AI models are revolutionizing design optimization by leveraging training data from various sources. Experimental data, collected by testing vessels under real-world conditions, provide crucial insight into material behavior. Simulation data, generated by software tools that model vessel response based on material properties and operating conditions, offers a complementary perspective. These data sources enable AI models to tackle critical design challenges. One such challenge is crack detection. Islam et al. [18] demonstrated the effectiveness of AI in identifying potential vessel failures by achieving a remarkable 94.67% accuracy in crack classification using acoustic emission signals and deep learning techniques. AI can also optimize weight, a key factor for portable applications. Studies by Liu et al. and Xu et al. [19, 20] explored weight reduction strategies using the artificial immune system and adaptive genetic algorithms, respectively. Furthermore, AI offers valuable tools for analyzing probabilistic behavior and reliability. Azizian and Almeida Jr. [21] combined simulations with artificial neural networks to predict the performance of composite tubes under uncertain loading conditions. AI can even create "digital twins" of vessels, as demonstrated by Hopmann et al. [22] work on predicting future behavior based on manufacturing processes. These advancements highlight the vast potential of AI in creating the next generation of safe, lightweight, and efficient hydrogen storage solutions.

Recently, Li et al. [23] developed a framework integrating machine learning (ML) and finite element (FE) analysis to optimize type IV hydrogen storage vessel designs. Their method reduced computational costs while maintaining precision, improving the mechanical response under designed burst pressure from 145 MPa to 157.74 MP. In addition, Wang et al. [24] combined simulation of the finite element method (FEM) with machine learning to
predict failure and optimize composite pressure vessels (CPV). Using ABAQUS, they analyzed CPVs with various design parameters under different loadings and trained a deep neural network (DNN) to map the design parameters to failure factors. This approach, enhanced by transfer learning, was coupled with a genetic algorithm to efficiently optimize CPVs and predict failure factors for new designs.

To accelerate the development of hydrogen storage technology, this research introduces a groundbreaking framework that integrates advanced analytical methods, numerical simulations, and deep learning. By meticulously incorporating 3D elasticity anisotropy theory and conducting a comprehensive parametric study, we have significantly enhanced the computational efficiency and accuracy of predicting damage in lightweight composite high-pressure hydrogen storage tanks. Our novel deep neural network (DNN) model, coupled with a user-friendly graphical user interface (GUI), provides a powerful tool for precise damage prediction and design optimization. This integrated approach represents a substantial leap forward in ensuring the reliability and efficiency of hydrogen storage solutions.

## II. Analytical and Numerical Approaches

Composite hydrogen storage vessels endure complex loading scenarios during operation, subjected to mechanical stresses such as pressure, tension, and shear, as well as thermal stresses induced by temperature variations. Additionally, vessels experience cyclical loading from repeated filling and emptying processes, which significantly affect their structural and thermal performance. Key design considerations for composite hydrogen tanks include tank capacity, working pressure, material characteristics, and safety margins. The design process, as depicted in the provided figure 1, encompasses several key stages [3]. Initially, the conceptual design phase defines the overall requirements and constraints, including capacity, working pressure, and safety factors. This stage involves considering initial geometries and material types. Following this, the laminate design phase focuses on the design of the specific composite structure and layout. This includes selecting the types of composite materials, their orientations, and the thickness of each layer to achieve the desired mechanical properties.
<figure><img src="image_2.png" /><figcaption>Figure 1: Design methodology</figcaption></figure>

The next stage is process simulation, where the manufacturing process of the composite vessel is modeled to ensure that the design is feasible and can be accurately produced. This step helps in predicting any potential manufacturing issues and optimizing the process parameters. After the process simulation, The material characterization phase involves determining key mechanical properties such as Young's modulus, Poisson's ratio, and thermal expansion coefficient, which are essential for accurately predicting the vessel's behavior under various loading conditions. Finally, the structural design phase involves performing a detailed numerical analysis to predict performance under operational conditions. This is where the *3D Elasticity Anisotropy Theory comes into play*. The 3D elasticity anisotropy theory, as detailed in the section "3D Elasticity Anisotropy Theory: Theoretical background and computational implementation,» is utilized to predict the mechanical performance of the hydrogen storage vessel. This theory helps to accurately model the stress and strain distribution within the composite material, taking into account the anisotropic nature of the composites used. The computational implementation of this theory in Python allows for precise predictions of the vessel's response to various loading conditions, ensuring that the design is robust and reliable. Integrating the 3D elasticity anisotropy theory into the design process provides a robust framework for assessing the structural integrity and performance of composite hydrogen storage vessels. This integration ensures that all critical factors, such as material behavior under different loads and environmental conditions, are thoroughly analyzed, leading to the development of safe and efficient hydrogen storage solutions.

# 1. A Theoretical Framework Based on 3D Elasticity and Literature Validation

Accurate prediction of stress and strain distribution within filament-wound composite pressure vessels (COPVs) is critical to ensuring their structural integrity and optimal performance. Traditional design approaches often rely on simplified material models that treat
the composite as isotropic, neglecting the directional dependence of its mechanical properties.
This section presents a theoretical framework that takes advantage of the established
principles of 3D elasticity for anisotropic materials to achieve a more accurate representation
of the stress and strain distribution within the COPV walls. Furthermore, the framework will
undergo a rigorous literature validation process to ensure its robustness and reliability.
Filament-wound composites exhibit anisotropic behavior due to the presence of reinforcing
fibers embedded within a matrix material. In this study, we focus on composites utilizing
T700 carbon fibers, a high-strength material known for its excellent stiffness and lightweight
properties. A detailed flowchart illustrating the steps involved in our analytical model can be
found in Figure 2. This flow chart provides a visual representation of the overall process,
starting from defining the geometry and material properties to solving the governing
equations and obtaining the stress and strain distribution within the COPV wall.

<figure><img src="image_3.png" /><figcaption>Figure 2: Flow chart for analytical model</figcaption></figure>

Figure 3 presents a comparison of the stress distribution within the wall of the composite overwrapped pressure vessel (COPV) predicted by our model (present work) with established results from the literature. These comparisons are made across various composite layups and through the thickness of the composite hydrogen storage vessel. The validation against existing literature helps to demonstrate the accuracy and reliability of our model. For this specific validation exercise, we analyze a multi-angle filament-wound (FW) pipe subjected to internal pressure, as detailed in reference [25,26]. The vessel has two winding layers, each containing two layers with a thickness of 0.5 mm and winding angles of ±55° and ±55°/±30°, respectively. The outer radius is 52 mm, the inner radius is 50 mm, and the internal pressure is 10 MPa. The figure presents the distributions of the distributions of the radial stress (σ<sub>r</sub>), hoop stress (σ<sub>θ</sub>), and axial stress (σ<sub>z</sub>) distributions within the wall of the COPV predicted by our model alongside the results of the reference. Any observed
discrepancies can be further investigated to potentially identify their source, such as limitations in analytical solutions or slight variations in material properties used. Overall, the observed close agreement between the stress distributions predicted by our model and the established result from the literature provides strong validation for the theoretical framework. This validation underpins the accuracy and reliability of the GUI, giving users confidence in its ability to predict the stress and strain response of filament-wound COPVs under various loading conditions. The GUI effectively leverages the validated framework to provide a user-friendly and efficient tool for COPV design and analysis.

<figure><img src="image_4.png" /><figcaption>Figure 3: Comparison with literature results</figcaption></figure>

Having validated our analytical model through comparisons with established analytical solutions, we have further enhanced its robustness by employing finite element analysis (FEA). FEA, a powerful numerical technique, discretizes the COPV geometry into small elements and solves the governing equations of elasticity at each element. This FEA approach enabled a highly detailed analysis of the stress and strain distribution within the vessel, taking into account complex geometries and material properties. By analyzing the behavior under various material selections and configurations, FEA provided valuable insights that complemented the existing validation process.
## 2. Finite element method: Model and Analysis

Building on the success of our validated analytical model, we employ a coupled Abaqus-WoundSim simulation to thoroughly examine the mechanical response of filament-wound composite pressure vessels (COPVs). Abaqus, a robust finite element analysis (FEA) software, allows us to explore the stress and strain distributions within the vessel walls in greater detail. Complementarily, WoundSim provides precise generation of composite lay-ups based on fiber properties and winding parameters. This integrated approach leverages the strengths of both software packages to enhance the insights gained from the analytical model.

In finite element modeling of filament-wound composite pressure vessels, it is essential to account for the complex architecture of composite materials [27]. Structural composite laminates are characterized by layers with varying thicknesses, materials, and orientations (layup definitions) designed to meet specific load-carrying requirements. Each composite layer comprises reinforcing fibers embedded in a supporting matrix, with the constitutive behavior of the layer defined by the properties of these constituents and their microstructural arrangement. Given the complexity of this architecture, complete incorporation into a structural finite element model is often impractical. Consequently, models are based on homogenizing assumptions to simplify the analysis. For accurate analysis of pressure vessel designs using the finite element method, adequate mesh discretization (nodes and elements) is critical. This ensures that the characteristics of the loading and boundary conditions are accurately captured, leading to reliable predictions. While symmetry in geometry, loading, response, and material properties can reduce modeling dimensions, it is not always applicable. In particular, in cases where the response is asymmetric, three-dimensional models are necessary. However, increasing the number of degrees of freedom in both the in-plane and thickness directions with three-dimensional modeling significantly raises the computational cost of the analysis.

### a) Material Properties

COPVs are manufactured by winding resin-impregnated fibers, typically carbon or glass, onto a mandrel (liner) in a specific pattern, as shown in Figure 4. Winding angle, which is the direction of the angle between the fiber and the axis of the vessel, plays a crucial role in mechanical performance. Different winding patterns can be used to achieve the desired strength and stiffness in various directions. The dome region of the COPV can have various shapes, such as spherical, elliptical, or geodesic, depending on the design needs. However, the cylindrical section is critical for the filament winding process. Here, straight or concave liners are preferred as they allow for tighter winding of the composite layers, ensuring good adhesion between the liner and the composite shell. This is essential to maintain the structural integrity of the vessel under pressure.
<figure><img src="image_5.png" /><figcaption>Figure 4. Schematic of a COPV</figcaption></figure>

**Table 1** summarizes the material properties of the T700 carbon fiber reinforced epoxy composite plies and the PA6 nylon liner material used in this study.

<table><caption>**Table 1:** Material properties.</caption><thead><tr><th></th><th></th><th>T700/epoxy</th><th></th><th>Liner (PA6)</th></tr></thead><tbody><tr><td rowspan="2">Elastic modulus (MPa)</td><td>E<sub>1</sub></td><td>156510</td><td></td><td></td></tr><tr><td>E<sub>2</sub> = E<sub>3</sub></td><td>1809</td><td>E</td><td>3000</td></tr><tr><td rowspan="2">Poisson ratio</td><td>ν<sub>12</sub> = ν<sub>13</sub></td><td>0.28</td><td></td><td></td></tr><tr><td>ν<sub>23</sub></td><td>0.305</td><td></td><td></td></tr><tr><td rowspan="2">Shear modulus (MPa)</td><td>G<sub>12</sub> = G<sub>13</sub></td><td>3930</td><td>ν</td><td>0.29</td></tr><tr><td>G<sub>23</sub></td><td>693</td><td></td><td></td></tr></tbody></table>

b) Geometry and Layup Definition

Our filament wound pressure vessel (COPV) for FEA analysis employs a meticulously defined geometry and layup configuration generated within WoundSim software [28]. The model geometry, as illustrated in the figure 5, represents a Type IV hydrogen storage vessel with a total length of 200 mm and a maximum diameter of 100 mm at the midsection of the cylindrical region. The dome regions feature highly customized curves designed to meet specific production requirements. The laminate comprises 14 distinct layers strategically designed to assess stress distribution and validate theoretical models. These layers fall into two categories: hoop layers and helical layers. Hoop layers, oriented entirely circumferentially, are confined to the cylindrical region and sandwiched between helical layers.
<figure><img src="image_6.png" /><figcaption>Figure 5: COPV Geometry</figcaption></figure>

Helical layers, consisting of fiber bands traversing the cylinder and partially extending onto the dome, are defined by their orientation relative to the dome profile. Layers solely defined by the dome profile are classified as "helical layers with no friction," while those with an additional turnaround radius specified (measured from the vessel's axis) are categorized as "helical layers with friction" to account for fiber path changes around the dome. WoundSim plays a key role: it assumes that a corresponding layer exists for each helical layer with the opposite wind angle, calculates equivalent homogenized properties for elements, and assigns an additional Y coordinate parameter to hoop layers to define their termination point within the cylinder. This comprehensive approach in WoundSim ensures an accurate representation of the COPV for stress and strain analysis within Abaqus.

c) Modeling using Axisymmetric Elements
The filament winding process creates a three-dimensional composite structure with multiple helical layers. Within each layer, the fibers are wound at opposing angles (θ and -θ) and wrap around the vessel multiple times. Accurately representing this architecture in a finite element model (FEM) would require capturing the continuously changing wind angle along both the axial and circumferential directions. However, modeling such a complex geometry can be computationally expensive or even impractical. Therefore, we employ simplifying assumptions to create a more manageable axisymmetric model, as shown in Figure 6. This model assumes symmetry around a central axis, allowing for efficient analysis. The figure depicts a finite element mesh with elements color-coded based on the assigned material properties. To model the interaction between the vessel liner and the composite layup, a contact interaction is defined between their respective surfaces. Although perfect bonding is assumed within the composite layers themselves, a coefficient of friction of 0.1 is assigned to the contact interaction between the liner and the lining, accounting for potential interfacial slip under loading conditions.

A pressure load of 70 MPa was applied internally to all models. The analysis was conducted using Abaqus, utilizing the incremental iterative Newton-Raphson solver within a static procedure. This solver is capable of handling static nonlinear simulations involving geometric, material, and boundary nonlinearities. However, for the current study, all materials are considered to exhibit linear elastic behavior, thus excluding material non-linearity from the analysis. The pressure vessel is the sole component in the assembly and does not interact with any other parts. Additionally, the boundary conditions remain constant throughout the simulation, thus eliminating contact or boundary non-linearity considerations. The bending loads cause rotation at the model's ends at specific reference points, necessitating the inclusion of geometric nonlinearity in the analysis.

To accurately model the geometric complexity of the dome region, where the fiber direction varies radially, the finite element (FE) model in Abaqus incorporates data from WoundSim. A fine meshing approach is adopted, with an angular increment of 0.5 degrees. Each composite layer is joined with two elements along its thickness to accurately capture variations in fiber orientation throughout the vessel wall. The mesh consists of 1440 linear quadrilateral elements of type CAX4R (4-node bilinear, reduced integration with hourglass control) and 25 linear triangular elements of type CAX3. This detailed meshing strategy ensures that the model can precisely represent the fiber architecture and its impact on the structural behavior of the composite hydrogen storage vessel.
<figure><img src="image_7.png" /><figcaption>Figure 6. Geometry characteristics of the composite layers of the vessel: (a) geometry model and stack sequences; (b) mesh (c) boundary conditions.</figcaption></figure>

d) Failure Analysis of COPVs

Building on the axisymmetric element model from the previous section, this work delves into the failure analysis of composite rolled pressure vessels (COPVs). We use finite element method (FEM) simulations to compute the failure factor and assess the structural integrity of the vessel under pressure loading. Three established failure criteria for composite materials are employed to assess potential failure modes: the Maximum Stress Criterion, the Tsai-Wu Criterion, the Tsai-Hill Criterion. By comparing the results of each criterion, we aim to comprehensively understand the behavior under pressure, identify potential weaknesses, and inform optimal design and safety considerations.

<table><thead><tr><td>Failure criteria</td><td>Formulas</td></tr></thead></table>
<table><tr><td>Tsai-Wu Criterion</td><td>I<sub>F</sub> = F<sub>1</sub>σ<sub>11</sub> + F<sub>2</sub>σ<sub>22</sub> + F<sub>11</sub>σ<sub>11</sub><sup>2</sup> + F<sub>22</sub>σ<sub>22</sub><sup>2</sup> + F<sub>66</sub>σ<sub>12</sub><sup>2</sup> + 2F<sub>12</sub>σ<sub>11</sub>σ<sub>22</sub> &lt; 1.0<br>The Tsai-Wu coefficients are defined as follows:<br>F<sub>1</sub> = <sup>1</sup>&frasl;<sub>X<sub>t</sub></sub> + <sup>1</sup>&frasl;<sub>X<sub>c</sub></sub>, F<sub>2</sub> = <sup>1</sup>&frasl;<sub>Y<sub>t</sub></sub> + <sup>1</sup>&frasl;<sub>Y<sub>c</sub></sub>, F<sub>11</sub> = -<sup>1</sup>&frasl;<sub>X<sub>t</sub>X<sub>c</sub></sub>, F<sub>22</sub> = -<sup>1</sup>&frasl;<sub>Y<sub>t</sub>Y<sub>c</sub></sub>, F<sub>66</sub> = <sup>1</sup>&frasl;<sub>S<sup>2</sup></sub><br>σ<sub>biax</sub> is the equibiaxial stress at failure. If it is known, then<br>F<sub>12</sub> = <sup>1</sup>&frasl;<sub>2σ<sub>biax</sub><sup>2</sup></sub> [1 - (<sup>1</sup>&frasl;<sub>X<sub>t</sub></sub> + <sup>1</sup>&frasl;<sub>X<sub>c</sub></sub> + <sup>1</sup>&frasl;<sub>Y<sub>t</sub></sub> + <sup>1</sup>&frasl;<sub>Y<sub>c</sub></sub>)σ<sub>biax</sub> + (<sup>1</sup>&frasl;<sub>X<sub>t</sub>X<sub>c</sub></sub> + <sup>1</sup>&frasl;<sub>Y<sub>t</sub>Y<sub>c</sub></sub>)σ<sub>biax</sub><sup>2</sup>]</td></tr><tr><td>Tsai-Hill Criterion</td><td>I<sub>F</sub> = <sup>σ<sub>11</sub><sup>2</sup></sup>&frasl;<sub>X<sup>2</sup></sub> - <sup>σ<sub>11</sub>σ<sub>22</sub></sup>&frasl;<sub>X<sup>2</sup></sub> + <sup>σ<sub>22</sub><sup>2</sup></sup>&frasl;<sub>Y<sup>2</sup></sub> + <sup>σ<sub>12</sub><sup>2</sup></sup>&frasl;<sub>S<sup>2</sup></sub> &lt; 1.0<br>If σ<sub>11</sub> &gt; 0, X = X<sub>t</sub>; otherwise, X = X<sub>c</sub>. If σ<sub>22</sub> &gt; 0, Y = Y<sub>t</sub>; otherwise, Y = Y<sub>c</sub>.</td></tr><tr><td>Maximum Stress Criterion</td><td>I<sub>F</sub> = max (<sup>σ<sub>11</sub></sup>&frasl;<sub>X</sub>, <sup>σ<sub>22</sub></sup>&frasl;<sub>Y</sub>, |<sup>σ<sub>12</sub></sup>&frasl;<sub>S</sub>|) &lt; 1.0<br>If σ<sub>11</sub> &gt; 0, X = X<sub>t</sub>; otherwise, X = X<sub>c</sub>. If σ<sub>22</sub> &gt; 0, Y = Y<sub>t</sub>; otherwise, Y = Y<sub>c</sub></td></tr></table>

### III. Comparison and model validation

Prior to data generation and deep learning predictions, a critical step involved validating the consistency of the analytical and numerical models. Figure 7 presents the stress distributions obtained from both models along a predefined path for various directions of the material and failure criteria. The first graph depicts stress distributions for the fiber direction, transverse direction, and in-plane shear (Figure 7a). Results from both the analytical model and WoundSim software are included for comparison. As expected, fiber direction stress exhibits the most significant fluctuations, reaching a maximum of approximately 2500 MPa. On the contrary, the transverse and in-plane shear stresses show relatively stable patterns with lower peak values. This aligns with the inherent material properties of composites, where the fiber direction offers superior strength compared to the transverse and shear directions. Importantly, this comparison reveals close agreement between the analytical results and WoundSim, indicating the validity of the chosen numerical simulation approach. The second graph (Figure 7b) focuses on stress distributions based on three established failure criteria: Tsai-Wu, Tsai-Hill, and Maximum Stress. As expected, the Tsai-Wu criterion, known for its sensitivity to multiaxial stress states, exhibits the highest stress peaks, reaching approximately 35 MPa. The Tsai-Hill criterion follows with peaks around 10 MPa, while the maximum stress criterion presents the lowest values. This observation aligns with the theoretical underpinnings of each criterion, where Tsai-Wu prioritizes multi-axial loading conditions.
Crucially, this comparison again demonstrates a close match between the analytical and WoundSim results, further validating the accuracy of the numerical model.

<figure><img src="image_8.png" /><figcaption>Figure 7: Comparison of Stress Distributions from Analytical and Numerical Models. (a) Stress Distributions for Different Directions. (b) Failure criteria.</figcaption></figure>
**Figure 8** presents a series of contour plots visualizing the stress distribution within a Composite Overwrapped Pressure Vessel (COPV) under various loading conditions and failure criteria. These plots offer valuable insight into the stress state of the vessel, aiding in design optimization and ensuring structural integrity. The first three subplots (Figures 8a-c) depict the stress distributions in distinct directions: fiber direction (a), transverse direction (b), and in-plane shear (c). As expected, fiber direction stress (a) exhibits the highest concentration in the central region, visualized by the red zones. This aligns with the inherent strength of composite materials along the fiber direction compared to the transverse and shear directions (b, c). The lower and more uniform stress distribution observed in the transverse and shear directions is consistent with this material behavior. The following subplots (Figures 8d-f) focus on stress distributions based on failure criteria. Figure 8d shows the maximum stress distribution, highlighting areas experiencing the most significant overall stress. These regions are critical for assessing the structural integrity and potential failure locations. Figures 8e and 8f illustrate the stress distributions according to the Tsai-Hill and Tsai-Wu failure criteria, respectively. These criteria account for multi-axial stress states, providing a more comprehensive assessment of potential failure compared to the simple maximum stress approach. The red zones in these plots represent regions where the composite is more susceptible to failure under the applied loading conditions, considering the combined effects of different stress components.

<figure><img src="image_9.png" /><figcaption>**Figure 8:** Stress Distribution Analysis in a COPV and Failure Criteria. (a) Fiber direction stress, (b) transverse direction stress, (c) in-plane shear stress, (d) maximum stress distribution, (e) Tsai-Hill failure criterion stress, and (f) Tsai-Wu failure criterion stress.</figcaption></figure>

# IV. Artificial Intelligence-Driven Assessment of Hydrogen Storage Tanks
# 1. Construction of a deep learning model

Deep neural networks (DNNs) have emerged as powerful tools within the field of machine learning, demonstrating exceptional capabilities in pattern recognition and extraction from intricate data sets [29]. This has led to significant success in predicting various aspects of material performance, including elasticity, load-bearing capacity, fatigue failure, and damage evolution. One of the key strengths of DNNs lies in their ability to rapidly derive meaningful insights from complex data features. These networks are typically made up of interconnected artificial neurons that mimic the structure of the human brain [30]. This architecture commonly includes an input layer that receives data, hidden layers that process the information, and an output layer that delivers the predicted results. Each neuron within the network employs a mathematical model for information processing. The network learns through iterative cycles called epochs, where it continuously adjusts the connections between neurons (neural weights) to refine the final prediction. A critical component of the model training process is the loss function, which measures the discrepancy between predicted and observed values. The network utilizes an optimization algorithm, such as gradient descent, to adjust the neural weights based on the calculated loss, ultimately minimizing the prediction error. Figure 9 illustrates a comprehensive process for developing a deep learning model aimed at predicting the failure pressure of composite overwrapped pressure vessels (COPV). The process begins with the collection of essential input data, including material properties such as elastic moduli (E1, E2), thermal expansion coefficients (α1, α2), and strength properties (XT, XC). These parameters are crucial for defining the mechanical behavior of the composite materials used in COPVs. Next, the mandrel definition step involves specifying the shape and dimensions of the mandrel around which the composite material is wound. The mandrel serves as the core structure during the manufacture of the pressure vessel. Alongside this, tank layers are defined by detailing the winding parameters and the layup sequence, which includes the winding angle and layer thickness. These parameters are integral to the structural integrity and performance of the COPVs. All of these input data are then consolidated into laminate data, which serves as a comprehensive dataset for further analysis. To simulate the mechanical behavior of COPVs under different conditions, the dataset is analyzed using 3D elasticity anisotropy theory and Finite Element Analysis (FEA). Boundary conditions are applied during this analysis, including internal pressure, temperature variations, and moisture variations, to reflect real-world operating environments. The results from the FEA provide detailed stress and strain distributions within the COPV. These distributions are then used to calculate 3D failure measures, which help to understand the failure mechanisms and critical stress points in the vessel. The processed data is compiled into a data set of results. This data set is then used to train a deep learning model. The architecture of the neural network is designed to correlate specific input features with the corresponding failure pressures. By learning from the stress and strain distributions, the model can accurately predict the failure pressure of new COPVs based on their input characteristics. Once trained, the deep learning model can be used to predict the failure pressure of new designs, aiding in the optimization and safety evaluation of COPVs. This predictive capability is crucial for advancing the design and manufacturing processes of hydrogen storage tanks and ensuring their reliability and safety in practical applications.
<figure><img src="image_10.png" /><figcaption>Figure 9: Flow chart of the construction of deep learning model</figcaption></figure>

## 2. Results and Discussion

In this study, four deep neural networks (DNN) were developed to tackle regression problems, specifically designed to approximate individual stress components ($\sigma_L, \sigma_T, \sigma_{LT}, \sigma_Z$) within composite laminate structures. These stress components are critical for understanding the mechanical behavior and integrity of the laminate under various loading conditions. Each DNN model was structured to accept three key input parameters: pressure, layer thickness, and tank radius. These inputs are crucial as they significantly influence the stress distribution across the laminate. The outputs of the DNN were designed to produce fourteen distinct stress values, each corresponding to the stresses present within the various layers of the laminate structure. This multi-output approach enables a comprehensive stress analysis across the entire laminate, providing detailed insights into the stress profile. The architecture of the developed DNNs consisted of fully connected feedforward networks. Each network featured two hidden layers, each comprising eight neurons. This configuration was chosen on a balance between model complexity and computational efficiency. The hidden layers employ activation functions to introduce nonlinearity, enabling the network to capture complex stress-strain relationships within the laminate. The architecture can be visualized as having an input layer with three neurons (corresponding to pressure, layer thickness, and tank radius), two hidden layers with eight neurons each, and an output layer with fourteen neurons (corresponding to the stress values for each layer).
The MATLAB Deep Learning Toolbox was chosen for developing and training these DNN models due to its extensive suite of functions and seamless integration with other MATLAB tools. The toolbox offers pre-built functions for designing, training, and validating neural networks, which significantly streamlined the development process. The training workflow involved several critical steps, starting with dataset preparation. The dataset was created by simulating various scenarios with parameter ranges, including pressure (P) from 5 MPa to 70 MPa, tank radius (R) from 10 mm to 200 mm, and layer thickness (t) from 0.01 mm to 0.14 mm. Stress values corresponding to these parameters were computed using a detailed finite element analysis (FEA) model, which served as a reference data for both training and evaluating the DNNs. To optimize training efficiency and enhance model performance, the input data were normalized before use in the network. Normalization ensures that all input features contribute equally to the learning process. The dataset was then divided into three subsets: 60% for training, 20% for validation, and 20% for testing. The training subset was used to train the DNNs with the Levenberg-Marquardt algorithm, known for its effectiveness with small to medium-sized networks and its balance between convergence speed and accuracy. The validation and testing subsets were reserved to evaluate the generalization capability of the DNNs. Although the MATLAB Deep Learning Toolbox was utilized in this study, alternative libraries such as PyTorch or TensorFlow could also be employed for similar purposes. These libraries offer extensive support for neural network development and have strong community support. They also provide advanced features such as automatic differentiation and GPU acceleration, which can further enhance the training efficiency and scalability of the models. The development of these DNNs for approximating stress components in composite laminates showcases the potential of neural networks in solving complex engineering problems. By leveraging key input parameters and a well-structured DNN architecture, it is possible to achieve accurate and reliable predictions of stress distributions, making it easier to design and analyze advanced composite structures.
<figure><img src="image_11.png" /><figcaption>(a) Loss vs. Epochs for $DNN\sigma_L$.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Prediction ($\sigma_L^P$) vs. Reference ($\sigma_L^R$).</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c) Error analysis of the $DNN\sigma_L$ model.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(d) Loss vs. Epochs for $DNN\sigma_T$.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(e) Prediction ($\sigma_T^P$) vs. Reference ($\sigma_T^R$).</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>(f) Error analysis of the $DNN\sigma_T$ model.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>(g) Loss vs. Epochs for $DNN\sigma_{LT}$.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>(h) Prediction ($\sigma_{LT}^P$) vs. Reference ($\sigma_{LT}^R$).</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>(i) Error analysis of the $DNN\sigma_{LT}$ model.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>(j) Loss vs. Epochs for $DNN\sigma_Z$.</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>(k) Prediction ($\sigma_Z^P$) vs. Reference ($\sigma_Z^R$).</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>(l) Error analysis of the $DNN\sigma_Z$ model.</figcaption></figure>

<caption>**Figure 10:** Performance and accuracy of DNNs ($DNN\sigma_L$, $DNN\sigma_T$, $DNN\sigma_{LT}$, and $DNN\sigma_Z$) used to approximate the stress components ($\sigma_L$, $\sigma_T$, $\sigma_{LT}$, $\sigma_Z$) as function of (t, R, and P) in 14 layers of the laminate composite.</caption>
Mean Squared Error (MSE) was employed as the loss function, and the model was trained for a maximum of 2000 epochs. An early stopping strategy was implemented to stop training once the model achieved a high level of accuracy, thereby preventing overfitting. The hidden layers of the network employed sigmoid activation functions to introduce nonlinearity, while the output layer used a linear activation function to produce continuous predictions. To further validate and interpret the performance of the developed DNN models, a detailed analysis was conducted using various evaluation metrics and visualization techniques. The following subfigures provide an in-depth examination of the results obtained from these analyses:

* The subfigures 10 (a), 10 (d), 10 (g), and 10(j) illustrate the variation of the loss function with respect to the number of epochs for the training, validation, and test datasets. As depicted in these subfigures, the loss function consistently decreases as training progresses, with the curves eventually flattening out, indicating that the model's performance is stabilizing. Furthermore, the loss function values for the training, validation and test datasets are closely aligned, suggesting that DNN models effectively avoid overfitting issues.

* The subfigures 10 (b), 10 (e), 10 (h) and 10 (k) show the DNN predictions compared to the reference values for the stress components in the training, validation, and test datasets. These figures demonstrate a strong correlation between the predicted and reference values, as all points are closely aligned with the 1-1 line. This alignment indicates that the DNN models achieve high accuracy in their predictions for seen and unseen values of parameters.

* Subfigures 10(c), 10(f), 10(i), and 10(l) provide an alternative evaluation of the DNN models through error analysis. Errors in the training, validation, and test data sets are computed and presented as histograms. This method clearly depicts the distribution and magnitude of the errors for each data set. The figures reveal a Gaussian distribution of errors, with a significant number of instances exhibiting errors close to zero.

To further evaluate the performance and precision of DNN, a specific point in the parametric space was selected for comparison, with parameters set to P = 20 MPa, R = 50 mm and t = 0.09 mm. Subfigures 11(a), 11(b), 11(c), and 11(d) present a comparative analysis of the stress components $\sigma_L$, $\sigma_T$, $\sigma_{LT}$, and $\sigma_Z$, respectively. These subfigures illustrate the DNN-predicted values compared to reference values across the thickness of the laminate. The figures demonstrate a strong agreement between the DNN predictions and reference curves for all stress components and throughout the laminate layers.
<figure><img src="image_23.png" /></figure>

(a) Longitudinal stress ($\sigma_L$) across the thickness of the laminate: $DNN\sigma_L$ Predictions vs. Reference values.

<figure><img src="image_24.png" /></figure>

(b) Transversal stress ($\sigma_T$) across the thickness of the laminate: $DNN\sigma_T$ Predictions vs. Reference values.

<figure><img src="image_25.png" /></figure>

(c) Shear stress ($\sigma_{LT}$) across the thickness of the laminate: $DNN\sigma_{LT}$ Predictions vs. Reference values.

<figure><img src="image_26.png" /></figure>

(d) Out-of-plane stress ($\sigma_Z$) across the thickness of the laminate: $DNN\sigma_Z$ Predictions vs. Reference values.

**Figure 11:** Comparison of stress components ($\sigma_L$, $\sigma_T$, $\sigma_{LT}$, $\sigma_Z$) predicted by DNN with reference values across the thickness of the laminate composite for a randomly selected configuration (P= 20 MPa, R=50 mm and t=0.09 mm).

The subfigures 12 (a), 12 (b), 12 (c), and 12(d) utilize 4D visualization to represent the variation of stress components in the parametric space for layer 10 of the laminate. These figures effectively illustrate how each parameter—pressure, tank radius, and layer thickness— affects the stress components, providing a comprehensive view of stress distribution variations. For example, both longitudinal and transversal stresses increase when high pressure is applied to a large tank radius with a thin layer thickness.
<figure><img src="image_27.png" /><figcaption>(a) Variation of the longitudinal stress $\sigma_L(R, P, t)$ in layer No. 10 (90°) across the parametric space (R, P, and t).</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>(b) Variation of the transversal stress $\sigma_T(R, P, t)$ in layer No. 10 (90°) across the parametric space (R, P and t).</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>(c) Variation of the shear stress $\sigma_{LT}(R, P, t)$ in layer No. 10 (90°) across the parametric space (R, P and t).</figcaption></figure>

<figure><img src="image_30.png" /><figcaption>(d) Variation of the out-of-plane stress $\sigma_Z(R, P, t)$ in layer No. 10 (90°) across the parametric space (R, P, and t).</figcaption></figure>

<caption>**Figure 12: 4D Visualization of stress components across the entire parametric space (R, P, t) for Layer No. 10 (90°) in the laminate composite.**</caption>

A Graphical User Interface (GUI) application has been developed, as illustrated in Fig. 13. This GUI provides an intuitive and efficient platform for designers and engineers to quickly determine the stress components and failure criteria in each layer of laminate composites used in hydrogen storage tanks. The user-friendly interface allows users to adjust key parameters such as tank radius, layer thickness, and pressure using interactive cursors. This enables the instant retrieval of results, eliminating the need for time-consuming finite element analysis
(FEA) or the generation of complex meshes and geometries for the tanks. The tool facilitates real-time assessment of how input parameters impact overall behavior of the tanks. Users can visualize the effects of parameter changes on the stress distribution and potential failure points within the laminate structure. This immediate feedback is crucial for optimizing design choices and ensuring the safety and reliability of hydrogen storage tanks. Additionally, the GUI serves as a valuable decision-making aid for selecting the optimal tank configuration for specific applications. By streamlining the design process, the tool enhances the efficiency and effectiveness of engineering evaluations. It allows engineers to quickly compare different design scenarios and make informed decisions based on the computed stress components and failure criteria. The development of this GUI application represents a significant advancement in the design and analysis of composite laminate structures for hydrogen storage. By integrating sophisticated computational models into an accessible interface, the tool democratizes advanced engineering analyses, making them available to a broader range of users. This not only accelerates the design process but also ensures that high-quality, data-driven decisions are made, ultimately leading to safer and more efficient hydrogen storage solutions.

<figure><img src="image_31.png" /><figcaption>Figure 13: Graphical User Interface (GUI) for Real-Time Evaluation of Hydrogen Storage Tank Performance: Instantaneous prediction of Stress Components ($\sigma_L$, $\sigma_T$, $\sigma_{LT}$, $\sigma_z$) and failure criteria (maximum stress, Tsai-Wu, Tsai-Hill, and Azzi-Tsai) based on Tank Radius, Layer Thickness, and pressure amplitude.</figcaption></figure>

## V. Conclusion

A novel integrated framework has been developed in this research to revolutionize the design and analysis of lightweight composite high-pressure hydrogen storage tanks. By synergistically combining advanced analytical techniques, sophisticated numerical simulations, and powerful deep learning, we have achieved unprecedented accuracy and
efficiency in predicting tank performance. The cornerstone of this framework is the
development of highly accurate DNN models capable of predicting stress components and
potential failure points with exceptional precision. These models, combined with the
enhanced computational efficiency afforded by our 3D elasticity anisotropy theory, provide a
robust foundation for optimizing tank design. To bridge the gap between complex
computational models and practical engineering applications, we have created a user-friendly
GUI. This tool empowers designers and engineers to rapidly assess tank performance under
various conditions, facilitating informed decision-making and accelerating the development of
safer, more efficient hydrogen storage solutions. Our comprehensive parametric study has
yielded valuable insights into the influence of key design parameters on tank behavior,
providing a solid basis for future optimization efforts. By democratizing access to advanced
engineering analysis, this research represents a significant step forward in the sustainable and
safe deployment of hydrogen energy technologies.

Acknowledgments

The authors appreciate the financial support of the Laboratory of Inorganic Materials for
Sustainable Energy Technologies (LIMSET) and Mohammed VI Polytechnic University
(UM6P), Morocco, for their support. We also like to thank the National Research Agency
(ANR) under the project "ANR-23-LERE-0005-01," which was crucial for this study.

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