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Abstract

Hydrogen has a great potential for application as a green energy source. Composite hydrogen pipeline is an important way of hydrogen transportation, and its engineering demand prospects are also very broad. This paper uses the finite element simulation method in combination with the Tsai-Wu failure criterion to derive the burst pressure of hydrogen transport pipelines under internal pressure and bending conditions as 28.5 MPa and the minimum bending radius as 1591.6 mm. Through hydrostatic pressure testing, the internal pressure of the pipeline was determined to be 28.8 MPa, with an error of 1.04%, verifying the validity of the simulation. The reinforcement layer of the pipeline was optimized using a Kriging surrogate model combined with the NSGA-II genetic algorithm based on simulation results. Through normalized result processing, the optimized parameters of the reinforcement layer were obtained (layer angle 55°, number of layers 8, thickness of single layer 0.3 mm), and finite element simulation verification was conducted, yielding an internal pressure of 22.3 MPa and a minimum bending radius of 1439.6 mm for the pipeline. The optimization approach achieves improved lightweight and economic performance under engineering constraints, demonstrating its feasibility and application potential.

Keywords

Hydrogen pipeline finite element simulation kriging model NSGA-II genetic algorithm optimization

Introduction

Hydrogen, as a clean and efficient energy carrier, is the most promising secondary energy source for promoting a green and low-carbon economy.¹ Hydrogen energy also has the advantages of abundant sources, high energy utilization efficiency, high calorific value and high energy density.² However, in the hydrogen energy industry, production sites and end-use locations are often geographically separated, necessitating long-haul transportation to balance supply and demand. Among various transportation methods, hydrogen pipeline transmission enables large-scale, routine, and low-cost long-distance transportation, making it one of the most economical approaches.^3,4

Traditional hydrogen transportation usually uses steel pipelines with technical challenges as hydrogen embrittlement,⁵ which is the phenomenon of brittle fracture due to the penetration of hydrogen atoms into the metallic material, resulting in ruptures or leakage of pipelines.^6,7 In contrast, non-metallic composite pipelines can directly transport pure hydrogen, eliminating the hydrogen embrittlement with higher safety and economy.⁸ Reinforced thermoplastic pipe (RTP) is a newly developed non-metallic composite pipe, with the advantages of corrosion resistance, strong adaptability and low maintenance costs. So the comprehensive cost can be reduced by about 20%.⁹ The inner and outer functional layer materials of RTP composite pipe are mostly polyethylene that is mainly used for corrosion prevention and isolation of media, with anti-scratch and anti-static functions. High-strength fibers are used as the reinforcement layer to strengthen the pipe.¹⁰ In addition, an aluminum layer is inserted between the liner and the fiber layer to effectively block the permeation of hydrogen, and the special structure is shown in Figure 1.

Figure 1.

Structure of hydrogen transport composite pipeline.

Nonlinear properties of composite, as well as usage of multiple material in the pipeline, increases the difficulty of estimate the pipe’s strength. Finite Element Analysis (FEA) has become an efficient method for performance evaluation and optimization of composite hydrogen pipelines under complex loads and environments. By numerical simulation, the mechanical response of the pipeline under internal pressure and bending conditions can be systematically analyzed to guide the selection of materials, optimization of structural parameters and service life prediction.

Yu¹¹ used FEA to study the mechanical response of the pipe model under the combined loads of bending moment and internal pressure to predict its service life, and analyzed the effect of the changing internal pressure on the fatigue life of the pipe with different structural layers. Shao¹² established a theoretical model for calculating stress and deformations of the composite pipe by the principle of virtual work, which was verified by the predicted burst pressure from ABAQUS software. Yao¹³ used the ABAQUS UMAT and user development technique to define nonlinear materials for pipes, aiming at investigating the nonlinear mechanical behavior of RTP under the combined loads of bending and pressure. Gao¹⁴ built a finite element model of floating pipe including the nonlinear material properties of super-elastic rubber materials and fibers as well as the complex contact between them, to derive the failure pressure for different material parameters. Yu¹⁵ analyzed the mechanical behavior of RTP pipes under bending and external compressive loads, and examined the influence of the diameter-thickness ratio and the layup angle on the failure of the pipes. Their results showed that the pipeline with the optimal layer angle had the strongest load-bearing capacity. With FEA, Lu^16,17 simulated bending radius applied by combined axisymmetric and bending loads, introducing the equivalent thermal loads and orthotropic thermal expansion models. Bai^18,19 studied the crushing behavior of RTP under the combined loads of external pressure, tension and torsion, obtaining a simplified formula to compute the ultimate load capability of the pipe for the given parameters such as the initial ellipticity of the pipe, yield strength, and diameter-thickness ratio.

Optimization design, namely topology optimization,²⁰ multi-objective genetic algorithm,²¹ response surface method,²² was employed to maximize the material utilization and minimize the cost under different working conditions (e.g., high pressure, bending, torsion, etc.) while guaranteeing the safety and strength of the structure. Gao²³ optimized the design parameters of steel wire winding angle, number of wires, diameter and thickness of the pipe, according to the strength analysis of steel-plastic winding reinforced plastic pipe. The optimized results showed effectively reduction of the cost with better load capacity of the pipe. For a fiber-wound structure with variable winding angle, Liu²⁴ established an optimization model which was solved by complex method, setting the winding angle of each layer as the design variable and the maximum load capacity as the objective function and determining the optimal winding angles. Yang²⁵ used NSGA-II genetic algorithm to optimize the corrugation height, pitch and wall thickness of LNG cryogenic flexible pipe, and obtained the Pareto solution set for stress minimization and stiffness minimization, in which the optimized solution reduced the internal compressive stress by about 11% and the bending stiffness by about 18%. Aiming at composite risers, Amaechi²⁶ optimized the layup angle configuration by genetic algorithm, and derived that the structure with a layer angle of ±55° had the best load capacity and fatigue life, i.e. the load capacity was increased by about 12%–15%, and the fatigue life improved by more than 20%.

However, local optimization with design variables and objective, is difficult to meet design requirements. Employing large scale sampling for finite element analysis, excessive computational costs is inevitable and unacceptable. By Kriging surrogate model, a preliminary model can be built with only a small number of initial samples, making it suitable for data-scarce scenarios in engineering. It can also be combined with Pareto-based genetic algorithms (such as NSGA-II) to address multi-objective optimization problems. Therefore, by integrating Kriging’s surrogate modeling with the global search characteristics of genetic algorithms (GA), an effective balance can be achieved between computational efficiency and optimization quality.

This paper employs the FEA to systematically analyze the mechanical response and failure behavior of composite hydrogen pipelines under internal pressure and bending conditions, with a special focus on the anisotropic characteristics of the fiber-wound structure and the influence of winding angles on non-axisymmetric stress distribution. By the results from FEA, the study effectively addresses stress prediction and failure assessment under complex loading conditions. Furthermore, by integrating the Kriging surrogate model and the NSGA-II multi-objective genetic algorithm, the reinforcement layer structure is optimized to achieve lightweight and cost-effective goals while meeting structural performance requirements. Through the optimal design of key parameters such as winding angle, single-layer thickness, and ply number, as well as structural verification under critical operating conditions, this study provides efficient and reliable analytical methods and theoretical support for the structural design and engineering application of composite pipelines.

Finite element modeling and experimental validation of pipeline

Finite element modeling

In this paper, ANSYS Workbench is used to perform multiple finite element analyses for glass fiber-reinforced composite pipes (DN150). Before creating the cylindrical geometry with diameter and length, the engineering material properties of high-density polyethylene (HDPE) and glass fiber must first be entered in the Engineering Data section of ANSYS. Subsequently, the thickness of the cylinder is defined by inputting the thickness of single layer, number of layers, and layer angle in the ANSYS Composite PrepPost (ACP) module. After finishing the modeling in ACP, it is transferred to the Static Structural analysis module. Before running the analysis and generating the expected results, the boundary conditions and internal pressure of the cylinder are set. The dimensions of the DN150 pipe are shown in Table 1, with same structure in Figure 1. The materials of liner and outer layer are both high-density polyethylene (HDPE), the aluminum layer is 6061-T6 type aluminum alloy, and the reinforcing layer is glass fiber (E-GLASS), whose parameters are listed in Tables 2 and 3 and 4.^27,28

Table 1.

Geometric parameters of composite pipes.

Parameter	Value (mm)	Parameter	Value (mm)
Inner diameter	150	Thickness of outer cover	4
Outer diameter	192	Thickness of impermeable layer	1
Thickness of a single reinforcing layer	0.5	Number of reinforcing layers	12
Winding angle (°)	+55/-55	Thickness of inner layer	10

Table 2.

High-density polyethylene (HDPE) material parameters.

Parameter	Value	Parameter	Value
E_HDPE (Mpa)	1080	ν _HDPE	0.42
σ(MPa)	28.39	$ρ$ (kg/m3)	958

Table 3.

Aluminum alloy layer 6061-T6 material parameters.

Materials	E (GPa)	ν	σ(MPa)
6061-T6	69	0.33	313

According to Saint-Venant’s Principle,²⁹ the distribution of stresses at a distance is determined primarily by the overall boundary conditions, independent of the way in which the loads are applied. It is generally considered that the length of the pipe is at least 5 times the inner diameter, which ensures that in the middle region of the pipe, the stress and strain distribution tends to be stable and is not affected by end loads or boundary conditions. So the length of the composite pipe for the finite element model in this paper is 1000 mm (Table 4).

Table 4.

Reinforcement layer glass fiber (E-GLASS) material parameters.

Parameter	Value (MPa)	Parameter	Value (MPa)
E ₁	28,000	σ _1t	1000
E ₂	7500	σ _1c	−700
E ₃	7500	σ _2t	80
G ₁₂	4000	σ _2c	−150
G ₂₃	3500	σ _3t	80
G ₁₃	4000	σ _3c	−150
ν ₁₂	0.3	τ ₁₂	150
ν ₂₃	0.38	τ ₂₃	100
ν ₁₃	0.3	τ ₁₃	150
$ρ$	1900 kg/m³	c	−1

In FEA, Solid Elements are used to simulate each structural layer, and the interaction between the layers is defined by contact. The mesh is built by C3D8R elements, i.e., eight-node linear reduced integration element, which can effectively avoid the shear locking phenomenon and improve the solution accuracy. In this study, the final finite element model consists of 850,288 elements and 2,353,988 nodes. To increase the reliability of the calculation, Mesh Independence Verification (MIV) should be done by gradually refining the mesh to ensure a final reasonable mesh density for balancing the accuracy of the calculation and computational effort. The pipe is usually defined by the Cylindrical Coordinate System (CCS) to more accurately describe the laying direction of the reinforcement layer, where the layer angle φ is the angle between the axial direction (z-direction) of the pipe and the laying direction of the fibers. Several methods are available for simulating the reinforcement layer such as series-parallel models and loop models. However, the Halpin-Tsai model was used to simulate the reinforcement layer, in order to obtain high accuracy with less computational complexity in this work. The FE model is shown in Figure 2.

Figure 2.

Finite element modeling (a) Overall model of the pipe; (b) cross-section meshing of the pipe; (c) reinforcement layer of fiber pavement.

Two operation conditions, namely internal pressure and pure bending, are investigated by FEA and ANSYS workbench. Firstly, two remote points are generated at both endpoints of the pipeline, on which external loads or boundary constrains are applied according to mechanical behavior. As for the case of the internal pressure condition, one end of the pipe is fixed and another is limited to five degrees of freedom while keeping axial displacement free to allow the pipe’s contraction along the axia direction of the pipe. When exposed to internal pressure, the pipe is subjected to an axial pull-out force F, which acts at the open endpoint as a concentrated force.

F = π {r_{0}}^{2} P

(1)

where r₀ is the inner diameter of the pipe and P is the internal pressure. The boundary conditions and loads are shown in Figure 3.

Figure 3.

Boundary conditions for the internal pressure condition of the pipe.

For the bending condition, one end of the pipe is fixed, and another keeps free to rotate clockwise about the X-axis. The boundary conditions and applied rotation are shown in Figure 4.

Figure 4.

Boundary conditions for the bending condition of the pipe.

Failure criteria and performance indicators

In the FEA of the fiber-reinforced composite hydrogen pipeline, the failure was analyzed and evaluated by using the Tsai-Wu failure criterion,³⁰ which is essentially a quadratic function of stress and strength, reflecting damage mechanism of composite materials under complex stress states and multiaxial loading. So it has been widely used in the prediction of the failure of anisotropic materials. Tsai-Wu failure criterion is expressed as equation (2).

\begin{array}{l} I_{F} = \frac{1}{R} = {[- \frac{B}{2 A} + \sqrt{{(\frac{B}{2 A})}^{2} + \frac{1}{A}}]}^{- 1} \\ A = \frac{σ_{1}^{2}}{F_{1 t} F_{1 c}} + \frac{σ_{2}^{2}}{F_{2 t} F_{2 c}} + \frac{σ_{3}^{2}}{F_{3 t} F_{3 c}} + \frac{σ_{4}^{2}}{F_{4}^{2}} + \frac{σ_{5}^{2}}{F_{5}^{2}} + \frac{σ_{6}^{2}}{F_{6}^{2}} \\ + c_{4} \frac{σ_{2} σ_{3}}{\sqrt{F_{2 t} F_{2 c} F_{3 t} F_{3 c}}} + c_{5} \frac{σ_{1} σ_{3}}{\sqrt{F_{1 t} F_{1 c} F_{3 t} F_{3 c}}} + c_{6} \frac{σ_{1} σ_{2}}{\sqrt{F_{1 t} F_{1 c} F_{2 t} F_{2 c}}} \\ B = (F_{1 t}^{- 1} - F_{1 c}^{- 1}) σ_{1} + (F_{2 t}^{- 1} - F_{2 c}^{- 1}) σ_{2} + (F_{3 t}^{- 1} - F_{3 c}^{- 1}) σ_{3} \end{array}

(2)

where c₄,c₅,c₆ are the coupling coefficients, usually as −1. σ₁,σ₂,σ₃,σ₄,σ₅,σ₆ are the stress components in different directions, respectively. F_1t,F_1c,F_2t,F_2c,F_3t,F_3c, are the strength parameters of the material, i.e. the tensile and compressive strengths of the material in different directions.

For the working condition of the internal pressure, the Tsai-Wu failure criterion is adopted as basis to determine the burst pressure. As for the bending condition, Tsai-Wu criterion is used as a auxiliary way to calculate the minimum bending radius R. The smaller R of pipeline indicates it is more convenient for storage and transportation. As shown in Figure 5, one end of the pipe is fixed, another is bending with the rotating angle θ, and R is computated as equation (3).

L = R \cdot θ

(3)

where L is the length of the pipe.

Figure 5.

Schematic diagram of composite tube bending.

Finite element results for internal pressure and bending

According to practical experience, the fiber layer serves as the main load-bearing component. So the stress and deformation of fiber layer are calculated for the cases with increasing internal pressure, simultaneously extracting the Tsai-Wu failure index. In other words, when the fiber layer of the pipe reaches the critical value of failure index in the FEA, the internal pressure at this time is the maximum value (i.e., burst pressure). As shown in Figure 6, the Tsai-Wu failure index is about 1 in the innerest reinforced layer of the pipe when the internal pressure is 28.5 MPa, which is defined as the burst pressure of the pipe.

Figure 6.

Critical value of the Tsai-Wu failure.

Subsequently, the stress and deformation are solved under the burst pressure 28.5 MPa. The von-mises stresses of each reinforced layer for the composite pipe are obtained as shown in Figure 7 and Table 5. Neglecting the influence of the boundary effects at both ends of the pipe, the results of mises stresses in the middle part of the pipe are only extracted and displayed. Morever, the stresses in each direction of each layer are calculated simultaneously and given in Figure 8. It is observed that the first layer of the reinforcement (The innermost layer of the pipeline is defined as the first layer, with subsequent layers increasing sequentially outward) is subjected to the highest stress, and therefore has the highest risk of failure.

Figure 7.

Distribution of mises stress in each layer of pipe reinforcement layer.

Table 5.

Mises stresses in each layer of the reinforced layer.

Layer	Von-mises stress (Mpa)	Layer	Von-mises stress (Mpa)
No.1	175.02	No.7	164.53
No.2	173.14	No.8	162.78
No.3	171.49	No.9	161.28
No.4	169.54	No.10	159.59
No.5	167.91	No.11	158.12
No.6	166.1	No.12	156.48

Figure 8.

Stresses in each direction in the pipe reinforcement layer.

For another working condition in FEA, a changing rotating angle is applied on the kinematic endpoint of pipe, and the stresses of all fiber layers are monitored as well as Tsai-Wu failure index. The rotating angle is recorded when Tsai-Wu failure index is up to 1. Then the minimum bending radius is calculated by equation (3).

From the results of several simulations, Tsai-Wu failure index for inner side of the outermost layer is firstly up to 1, shown in Figure 9. It means that the pipe’s bending angle and radius reaches the critical value at this time. By equation (3), the minimum bending radius is derived as 1591.6 mm, which is slightly below 1600 mm³¹ stipulated in industry standard for easy storage and transportation. It means that the design of composite pipe meets the corresponding specifications.

Figure 9.

Critical values for tsai-wu failure in bending.

As seen in Figures 10 and 11 and Table 6, the stress level rises sequentially from the innermost to the outermost layer. The outermost fiber layer on the inner side of the pipe is firstly failure under the bending load. It is essentially compression failure.

Figure 10.

Equivalent stress cloud of each layer of the pipe reinforcement layer.

Figure 11.

Stresses in each direction in the pipe reinforcement layer.

Table 6.

Mises stresses in each layer of the reinforced layer.

Layer	Von-mises stress (Mpa)		Layer	Von-mises stress (Mpa)
Layer	Outside	Inside	Layer	Outside	Inside
No.1	410.48	410.48	No.7	424.66	424.9
No.2	412.31	412.32	No.8	426.39	426.71
No.3	414.99	415.02	No.9	429.5	429.71
No.4	417.41	417.52	No.10	431.1	431.53
No.5	419.02	420.08	No.11	434.46	434.54
No.6	421.91	421.91	No.12	436.32	436.33

Experimental verification

Verification of the simulation was carried out by hydraulic pressure experiments, which is usually used to evaluate the ultimate load capacity and safety of fiber-reinforced composite hydrogen pipes. For the preparation of the test, the pipe specimen that meets the standard size was firstly selected and sealed on the both ends to ensure no liquid leakage. Then, the specimen was fixed in the base of test flat and connected with the pressure pump, sensor and data acquisition system. The pressurization process was conducted using an XGNB-Y pressure cycling machine (Chengde Precision Testing Machine Co., Ltd), and the pressure was increased at a constant rate of 0.5 MPa/s. After preparation, the sealing of the whole system should be checked. In the early pressurization stage, air was slowly removed to avoid errors or shocks caused by sudden compression of gas. During the subsequent pressurization process, the pressure pump gradually increases the pressure inside the pipe at a constant rate, and the sensor recorded the pressure change in real time. With the increase of pressure, the pipe would undergo elastic deformation, plastic deformation, and finally damage under the ultimate pressure, such as fiber fracture, matrix cracking or delamination. The ultimate pressure is the burst pressure of the pipe. The test record is shown in Figure 12.

Figure 12.

Pipe burst pressure test results.

The experimental results show that the pressure loaded in composite pipe increases from 0 and to the peak value 28.8 MPa, and then starts to unload, as shown in Figure 13. So the burst pressure of the composite pipe is 28.8 MPa, close to the simulated value of 28.5 MPa with the error of 1.04%. Therefore, it verifies the simulation results.

Figure 13.

Comparison of experimental and simulation results.

Kriging surrogate model and optimization design

Latin Hypercubic Sampling (LHS)

Traditional engineering design is often overly conservative, using high safety factors that result in material waste and avoidable cost escalation. So it is necessary to make a balance between safety and economy by optimization. In this section, a kriging surrogate model was built to approximate the system response for different combinations of design variables. The stress results of a composite hydrogen pipeline were utilized to analyze the influence of each design variable on the service performance (e.g., bursting pressure, minimum bending radius) by means of numerical modeling and optimization algorithm. During the process, the first step is to determine the layer angle, thickness of single layer, and the number of layers for the composite pipeline as input variables, so as to compute the corresponding bursting pressure and minimum bending radius through finite element simulation. The Genetic Algorithm is used to compute the optimal results for the design parameters. The optimization flow of the composite hydrogen pipeline based on Kriging model is shown in Figure 14.

Figure 14.

Composite pipe structure optimization process.

Latin Hypercube Sampling (LHS) has the capability of ensuring that each value of a variable is equally important in the sample. It can not only decrease the number of simulations, but keep proper orthogonality and proportionality of the sample. In this paper, LHS³² is then utilized to generate design samples with the values range as listed in Table 7. Considering the balance between computational accuracy and efforts, the minimum number of samples is determined as 50 for finite element simulation. Figure 15 shows that the uniform distribution of the sample points, which is conducive to improving the representativeness of the selected sample points, making it easier to guarantee the accuracy of the surrogate model.

Table 7.

Range of values of design variables.

Inputs	Value range	Outputs	Unit
Layer angle [φ]	30°–70°	Burst pressure (P)	Mpa
Number of layers [n]	8–14	Minimum bending radius (R)	mm
Thickness of single layer (d)	0.3–0.8 mm

Figure 15.

Distribution of sample points.

Kriging model

Kriging Model is a statistics-based modeling method, originally proposed by Danie Krige and later systematized by Georges Matheron. It is widely used in spatial interpolation, uncertainty quantification, and optimization design, particularly serving as a substitute model for computationally expensive engineering simulations and experiments. By nonparametric interpolation, the Kriging Model can integrate the given sample points to approximate the model parameters and forecasting the unknown response of a new design point. It balances the ability to model accuracy and uncertainty by predicting the response of unknown points through statistical modeling of existing data points.^33,34

Considering a performance function with k random inputs, Kriging Model can be constructed with n samples, in which any sample is (x_i,y_i), x_i =(x1 i…x2 i), (i = 1…n) are simple inputs and y_i is the system performance when the system is given the inputs x_i.

In the Kriging Model, the system performances are generated from:

y (x) = f (x) + Z (x)

(4)

where y(x) is the unknown function, f(x) is a polynomial function of x, and Z(x) is the realization of a Gaussian stochastic process with zero mean and variance σ². The polynomial term f(x) is simplified by a constant value μ. So The Kriging model can be rewritten as:

y (x) = μ + Z (x)

(5)

The entry correlation matrix Z(x) is expressed by

C o r r [G (x_{i}, x_{j})] = σ^{2} R (x_{i}, x_{j})

(6)

where the R (x_i,y_i) represents the (i, j) entry of an n × n matrix, and this correlation matrix is defined by the distance between two sample points, with ones along the diagonal, which can be expressed as:

R (x_{i}, x_{j}) = \exp (- \sum_{k = 1}^{d} θ_{k} {(x_{i k} - x_{j k})}^{2})

(7)

where θ_k denotes the weight parameter on the dimension of each input variable. After n sample points {x⁽ⁱ⁾,y⁽ⁱ⁾} are known, the response mean of the predicted point (x^*) is:

\hat{y} (x^{*}) = μ + r^{T} (x^{*}) R^{- 1} (y - A μ)

(8)

where A is a n × 1 vector filled with ones. r (x^*) is the correlation vector between the prediction point and the design points x₁∼x_n, which are given by

r^{T} (x^{*}) = [C o r r [Z (x^{*}, x_{1})], \dots C o r r [Z (x^{*}, x_{n})]]

(9)

In the following study, layer angle, number of layers, and thickness of single layer are the design variables, and the burst pressure and bending radius are investigated as the objective functions. Using the Kriging model with 50 design points, a surrogate model was built to predict objective function and assume the response surfaces. For the burst pressure as objective, the parameter estimation of the final model is the means μ = 35.4 MPa with the kernel function weight parameter θ = [0.02,0.01,0.005]. The sample point covariance matrix R can be computed from the θ and input samples. The final prediction function can be written as:

{\hat{y}}_{P} (x) = 35.4 + r_{P}^{T} R_{P}^{- 1} (y - 35.4 A)

(10)

where x = [x₁, x₂, x₃] denotes the layer angle, number of layers, and thickness of single layer, which can be used for subsequent optimization.

In similar way, the predictive function for the minimum bending radius is also constructed and expressed as:

{\hat{y}}_{\min R} (x) = 2100 + r_{R}^{T} R_{R}^{- 1} (y - 2100 A)

(11)

From Kriging surrogate model, analysis was carried out to investigate the influence of reinforced layer’s parameters on burst pressure and minimum bending radius by response surfaces shown in Figures 16 and 17. As for burst pressure, it increases nonlinearly with all design parameters. When increasing the layer angle(φ) from ±30° to ±70° and the number of layers (n) rises from eight to 14, the burst pressure(P) escalates from about 25 MPa to over 50 MPa, indicating that a high-angle, multi-ply configuration optimally enhances pressure resistance. In contrast, the influence of thickness of single layer(d) is relatively slight. It increases from 0.3 mm to 0.8 mm, but only resulting in enhancement of burst pressure(P) 15 MPa and small marginal effect.

Figure 16.

Analysis of the effect of layup parameters on blast pressure; (a) Burst pressure vs. layer angle vs. number of layers response surface map; (b) Burst pressure vs. layer angle vs. number of layers contour map; (c) Burst pressure vs. layer angle vs. thickness of single layer response surface map; (d) Burst pressure vs. layer angle vs. thickness of single layer contour map; (e) Burst pressure vs. number of layers vs. thickness of single layer response surface map; (f) Burst pressure vs. number of layers vs. thickness of single layer contour map.

Figure 17.

Analysis of the effect of layup parameters on the minimum bending radius; (a) Minimum bending radius vs. layer angle vs. number of layers response surface map; (b) Minimum bending radius vs. layer angle vs. number of layers contour map; (c) Minimum bending radius vs. layer angle vs. thickness of single layer response surface map; (d) Minimum bending radius vs. layer angle vs. thickness of single layer contour map; (e) Minimum bending radius vs. number of layers vs. thickness of single layer response surface map; (f) Minimum bending radius vs. number of layers vs. thickness of single layer contour map.

For the minimum bending radius(R), the thickness of single layer(d) becomes the dominant factor with very significant negative correlation. When d increases from 0.3 mm to 0.8 mm, R decreases from 1700 mm to 1580 mm with reduction of over 10%. Moreover, the effect of layer angle (φ) on the R is non-monotonic, i.e. R reaches its minimum near ±55° taken as optimal layup angle for minimizing bending radius. The number of layers (n) is secondary influence factor on R because it only has a moderate effect at smaller thicknesses.

Contour density variations in the figures can further reveal interactions between variables. For the case of high-thickness, burst pressure(P) is sensitive to the layer angle(φ), with the effect that small increasement of angle can lead to the significant enhancement of burst pressure(P). On the other hand, for a lower layer angle, thickness of single layer d is the most important factor to derive the minimum bending radius. It is inferred the number of layers(n) and layer angle(φ) are critical for maximizing burst pressure, whereas thickness is key to dominate bending performance. In summary, complex interaction exists among the design variables, indicating that trade-off is necessary in multi-objective optimization.

NSGA-Ⅱ genetic algorithm

When solving optimization problems with conflicting objectives, such as maximizing the burst pressure and minimizing the bending radius, the traditional single-objective optimization methods is incompetent to trade-off the conflicts. However, the non-dominated sorting genetic algorithm (NSGA-II) is widely used for its efficient multi-objective processing capability and good retention of the Pareto-optimal solution set. The core idea of NSGA-II is to select the solutions based on the non-dominated sorting and the crowding distance between individuals in each generation of the population, thus ensuring the superiority and diversity of the solutions.^35,36

In this work, the NSGA-II is utilized to obtain the optimal combination of design variables, ensuring the maximum burst pressure and minimum bending radius, as well as keeping the number of layers and the thickness of a single layer as small as possible. The algorithm flowchart is described in Figure 18.

Figure 18.

Flowchart of NSGA-II algorithm.

In the optimization process, the design variables are first solved iteratively based on a genetic algorithm, and the optimal solution on the current Pareto front is extracted in each generation in order to analyze the evolution characteristics of the design variables and the objective function. Figure 19 shows that the design variables change with the number of generations. It can be seen that the layup angle fluctuates at the beginning of the iteration, and gradually tend to be static, which indicates that the algorithm has sufficiently searched the angle variable to explore the potential solution domain in the initial stage. The number of layers is kept at eight throughout the whole evolution process without any change, which reflects that this value is the optimal and stable solution in the process of convergence, under the control of the current design requirements and the prediction model. The thickness of single layer shows an obvious convergence trend, from the initial distribution gradually gathered in a specific interval, which verifies the effectiveness of the algorithm in continuous variable optimization. As for the objective function, the burst pressure value shows an overall upward trend during the iteration process, i.e., its amplitude decreases continuously, indicating that the load capacity of the pipe is gradually enhanced. Besides, the minimum bending radius tends to decrease, implying the optimization of the flexibility performance. These evolutionary trends reveal the synergistic relationship and optimization path between the design variables and performance indexes in the multi-objective optimization process.

Figure 19.

Iterative plot of variables; (a) Iterative plot of layer angle; (b) Iterative plot of number of layers; (c) Iterative plot of thickness of single layer; (d) Iterative plot of burst pressure; (e) Iterative plot of minimum bending radius.

In the Pareto frontier diagram, the undominated solutions are distributed in the two objective function dimensions of bursting pressure and minimum bending radius, forming a continuous and smooth boundary shape, which reveals the distribution characteristics of the solution set in the objective space. As seen in Figure 20, the frontier shape reflects the trade-off relationship between the two performance indexes, i.e., while enhancing the burst pressure, the bending performance needs to be sacrificed to a certain extent, and vice versa. It is obvious for the realization of the multi-objective balance in the optimization process, which provides theoretical support for the screening and decision-making of the subsequent solutions.

Figure 20.

Pareto frontier diagram.

Normalization process

On the basis of the Pareto solution set, the normalization method is used to normalize the objective function values and calculate the total normalized score of each solution, in order to further assess the comprehensive performance of each solution. The solution with the lowest weighted normalized objective sum, as shown in the total score plot, can be considered the one with the best overall performance.

Figure 21 systematically presents the multi-objective optimization results based on the Kriging surrogate model and NSGA-II. In Figure 21(a), the four normalized iteration results (the burst pressure, the bending radius, the number of layers, and thickness of single layer) of the 70 design solutions exhibit significant fluctuations, demonstrating highly nonlinear characteristics. Among them, the burst pressure and the number of plies show a positive correlation in most solutions, with 21.4% (15/70) of the solutions having both normalized values greater than 0.8. It indicates that increasing the number of plies enhances pressure-bearing performance. Conversely, when the normalized thickness exceeds 0.7, the normalized bending radius is notably lower (<0.4), suggesting that thickness is the dominant factor controlling bending performance.

Figure 21.

Normalization results; (a) normalized iteration results; (b) normalized objective space; (c) tolal normalised objective score; (d) genetic algorithm convergence.

Figure 21(b) illustrates the distribution of objective functions in the normalized multidimensional space. Most solutions are uniformly distributed, revealing no direct correlation between the burst pressure/the bending radius and thickness. Additionally, multiple non-dominated solutions (Pareto Front) form a distinct frontier, highlighting significant trade-offs among different objectives in multi-objective optimization, necessitating appropriate weighting or Pareto screening.

In Figure 21(c), the solutions are ranked by weighted total normalized scores, ranging from 0.96 to 2.75. Among them, eight solutions score are above 2.5, corresponding to the optimal combinations of burst pressure and bending radius in the original design. The lowest-scoring solutions generally exhibit normalized thickness values below 0.2 and burst pressures under 0.3, confirming their poor overall structural performance.

Figure 21(d) displays the convergence curve of the genetic algorithm. The initial optimal value is 1.21, which rapidly drops below 0.98 within the first 10 generations and stabilizes between 0.96 and 1.00 after the 15th generation. It demonstrates that the optimization algorithm achieves over 95% convergence within the first 15 generations, with fast and stable optimization capability.

By the multi-objective optimization genetic algorithm, the final optimized results for fiber-reinforced hydrogen pipe are listed in Table 8. In short, this optimizaton scheme takes into account the good bending performance while guaranteeing high bursting strength, which is a typical application of the multi-objective genetic algorithm (NSGA-Ⅱ) to approximate the optimal solution set under the given design space.

Table 8.

Structural parameters of optimized fiber reinforced layer.

Layer angle	Number of layers	Single layer thickness
55°	8	0.3 mm

Validation and interpretation of optimized model results

Using the optimized parameters as optimal layer angle, number of layers and thickness of single layer, the FEM was re-built to obtain the response results, which were compared with the Kriging prediction results shown in Figure 22 and Table 9, in order to verify whether its performances in terms of burst pressure, bending radius and other performances meet the expectations.

Figure 22.

Simulation results with optimized structural parameters.

Table 9.

Comparison of optimization and simulation results.

Outputs	Optimization results	Simulation results	Error
Burst pressure (MPa)	22.42	22.3	0.54%
Minimum bending radius (mm)	1435.18	1439.6	0.31%

As seen from Figure 22, the safety factors of operationally internal pressure and bending are 1.0528 and 1.0079 respectively, which are very close to 1 and agree with the objective values. Table 9 lists the comparison between the optimized value from Kriging-NSGA-Ⅱ and FEM results. In terms of the burst pressure, it decreases from 28.5 MPa to 22.3 MPa after optimization. Although the burst pressure is reduced, it still meets the pipeline’s rupture pressure rating standard, i.e. three times the nominal pressure of 19.2 MPa. On the other hand, the minimum bending radius of the pipeline is reduced from the initial value 1591.6 mm to the optimized 1439.6 mm. Also it meets the standard requirement that the minimum bending radius does not exceed 1600 mm. Therefore, the group of design parameters for the composite hydrogen pipeline are selected as the optimal values as follows: the layer angle 55°, the number of layers 8, the thickness of a single layer 0.3 mm. This design have several advantages of lightweight, material saving, and transportability.

Conclusion

(1) The internal pressure and bending conditions of the composite hydrogen transmission pipeline are analyzed by finite element, and the burst pressure of the pipeline is 28.5 MPa and the minimum bending radius is 1591.6 mm by using Tsai-wu failure criterion. The burst pressure of the pipeline is 28.8M Pa by hydrostatic bursting test, the error is 1.04%. Which proves the accuracy of the simulation.

(2) NSGA-II genetic optimization algorithm is used to find the optimal parameter combination on the basis of Kriging response surface model, and the Pareto optimal solution is obtained after several iterative calculations, and the optimal solution is obtained through normalization. The laying angle of the pipe is 55°, the thickness of the single layer for the pipe is 0.3 mm, and the number of laying layers is 8. The optimization results are systematically analyzed by finite element simulation, and the feasibility of the optimization results is verified. At the same time, the reduction in the number of pipe layers and the thickness of each layer has led to a decrease in weight and volume, as well as a significant reduction in economic investment.

(3) The burst pressure of the pipeline was reduced from 28.5 MPa to 22.3 MPa, with a decrease of 21.7%, but it still meets the engineering requirement of three times the nominal pressure 19.2 MPa. The minimum bending radius was reduced from 1591.6 mm to 1439.6 mm, with a decrease of 9.55%. The reduction in the minimum bending radius allows the pipeline to adapt to more complex environments while facilitating transportation.

(4) Although reasonable results obatined in this work, some limitations still exist such as idealized material model and perfect bonding between layers, without consiering local defects of voids, delamination, and manufacturing imperfections. Future work will aim to refine these assumptions by incorporating more accuarte material model, multi-scale finite element modeling, imperfect interfaces and residual stresses. Thus complicated problem can be addressed by reliability and probabilistic analysis methods.

Footnotes

Author contributions

Caixia Guo: Key Conceptual Idea,Writing – Review & Editing. Feng Gao:Methodology, Data curation. Yuewen Su:Methodology, Investigation, writing - Original Draft. Ke Zhang:Methodology, Data curation.Xiaoye Wang:Supervision, project administration and writing - review & editing. Zhifeng Yang:Data curation, Formal analysis. Xiaoyu Zhang:Methodology, Investigation. Yaozhou Wang:Supervision, Validation, Methodology.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the Youth Innovation Team of Shaanxi Universities: Shaanxi Provincial Natural Science Foundation (2024JC-YBMS-262), Shaanxi Provincial Department of Education Youth Innovation Team Scientific Research Program Project (24JP006), Special Plan Project for Serving the Local Area of the Shaanxi Provincial Education Department (22JC003).

ORCID iD

Feng Gao

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