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Abstract

This study proposes a backpropagation (BP) neural network model based on the multi-strategy improved sand cat swarm optimization (MISCSO) algorithm to predict the tensile properties of glass fiber-reinforced recycled polypropylene (GF/RPP) composites under different fused deposition modeling (FDM) parameter combinations. First, the MISCSO algorithm is built upon the SCSO algorithm by introducing a cubic chaotic reverse learning strategy to enhance population diversity, incorporating the dynamic nonlinear sensitivity range and Weibull flight strategy to strengthen global search capability, and further employing the Gaussian-Cauchy mutation strategy to avoid local optima and accelerate convergence. Subsequently, using printing temperature, layer thickness, infill density, and raster angle as input variables, 243 experimental samples were employed for model training and validation. Model performance was evaluated based on the mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), and coefficient of determination (R²). Finally, the results show that the proposed MISCSO-BP model achieved R² values of 0.93, 0.91, and 0.87 for tensile strength, elastic modulus, and elongation at break, with average prediction errors below 5%, outperforming existing methods in prediction accuracy, convergence speed, and stability. These findings demonstrate the effectiveness and robustness of the MISCSO-BP model in optimizing FDM process parameters and predicting the mechanical properties of sustainable polymer composites.

Keywords

FDM tensile property MISCSO backpropagation neural network printing parameter

Introduction

Additive manufacturing (AM) is an advanced manufacturing technology that, compared to traditional subtractive manufacturing, builds parts directly from a digital model by adding materials layer by layer.¹ AM enables the manufacture of complex geometries, lightweight structures, and customized designs that are difficult or even impossible to achieve using conventional manufacturing techniques.² A variety of AM processes have been developed, including stereolithography (SLA), selective laser sintering (SLS), and fused deposition modeling (FDM).³ By providing precise control over part geometry and internal structure, AM offers exceptional design flexibility and has been widely applied in aerospace, automotive, biomedical, and other high-value engineering fields.⁴

With the rapid advancement of AM technology, FDM has become one of the most widely adopted 3D printing techniques. In this process, thermoplastic materials are melted and extruded layer by layer and subsequently deposited onto a build platform, enabling the rapid fabrication of complex parts.^5,6 Compared with traditional manufacturing processes, FDM offers several advantages, including high material utilization, short production cycles, and relatively low costs.^7,8 Moreover, this process works with a wide range of thermoplastics, including polylactic acid (PLA), acrylonitrile-butadiene-styrene (ABS), polypropylene (PP), and polyamide (PA).

Glass fiber-reinforced recycled polypropylene (GF/RPP) composites have become one of the most promising sustainable materials in the field of 3D printing due to their excellent reuse value.^9,10 PP is a thermoplastic polymer characterized by low density, good chemical stability, and excellent processability. When reinforced with glass fibers, its stiffness, strength, and heat resistance are markedly improved, thereby providing the material with superior structural load-bearing capacity. Furthermore, RPP derived from the reuse of discarded polypropylene products helps alleviate plastic pollution and reduce raw material consumption, aligning with the development of green manufacturing and a circular economy. However, the mechanical properties of GF/RPP composites are significantly affected by numerous process parameters during FDM molding. Inappropriate parameter combinations can lead to insufficient interlayer bonding or internal defects. Therefore, thoroughly studying the influence of FDM parameters on GF/RPP properties and establishing reliable prediction models are essential for improving material utilization efficiency and the structural performance of printed parts.¹¹

In recent years, scholars have conducted a large number of experimental studies on the influence of FDM process parameters on the mechanical properties of thermoplastic composites. Studies have shown that key process parameters such as printing temperature, layer thickness, infill density, printing speed, and raster angle have a significant effect on the mechanical properties of printed parts.^12,13 Bharat et al. systematically explored how layer thickness, infill density, infill pattern, and raster angle affect the compressive strength of PLA/carbon fiber composites.¹⁴ Milovanović et al. explored the influence of process parameters on the tensile properties of PP materials.¹⁵ The results showed that layer thickness, infill density, and raster angle are critical to the elastic modulus. Zhang et al. studied the influence of infill percentage, infill pattern, layer thickness, and extrusion temperature on the dimensional accuracy of 3D-printed PP parts.¹⁶ Burnett et al. evaluated the influence of infill pattern, infill density, raster angle, and layer thickness on the tensile properties of short carbon fiber reinforced nylon and unreinforced nylon materials.¹⁷ Kumar et al. showed that barrel temperature, raster angle, and platform temperature have a marked influence on the tensile behavior of ethylene vinyl acetate (EVA) flexible materials.¹⁸ Pulipaka studied the impact of various FDM settings on the mechanical and tribological performance of polyether ether ketone (PEEK) and concluded that nozzle temperature and layer thickness play major roles.¹⁹ Recent studies have also explored hybrid artificial intelligence (AI) models for FDM parameter optimization. Raju et al. optimized the FDM parameters of ABS from the standpoints of mechanical characteristics and surface quality using a hybrid particle swarm optimization-bacterial foraging optimization (PSO-BFO) technique.²⁰ Lui et al. employed the Taguchi method integrated with grey relational analysis to identify the optimal process parameters for improving the overall mechanical properties of FDM parts.²¹ Deshwal et al. used hybrid optimization techniques such as genetic algorithm-artificial neural network (GA-ANN), genetic algorithm-response surface method (GA-RSM) and genetic algorithm-adaptive neuro-fuzzy interface system (GA-ANFIS) to optimize FDM process parameters to improve the tensile strength of PLA.²² However, most of these hybrid AI models still face challenges such as high computational cost and limited generalization capability.

Although previous studies have provided valuable insights into the relationship between FDM process parameters and material properties, they often suffer from limited parameter coverage and insufficient generalization. Therefore, it is essential to develop a predictive modeling method capable of accurately capturing the nonlinear relationships between processing parameters and mechanical properties of composite materials. With the continuous advancement of AI technologies, data-driven methods have been increasingly applied to performance prediction in AM.^23,24 Among them, machine learning and deep learning technologies have shown strong modeling capabilities.^25–29 Ulkir et al. employed machine learning (ML) algorithms, Bayesian linear regression (BLR) and Gaussian process regression (GPR) to predict the mechanical properties of 3D-printed polymer. Their results demonstrated that GPR achieved the highest overall prediction accuracy.³⁰ Hamouti et al. developed an intelligent model based on an artificial neural network (ANN), which is capable of predicting the optimal combination of 3D printing parameters to achieve maximum tensile strength.³¹ Mishra et al. proposed the Levenberg-Marquardt (LM) and scaled conjugate gradient (SCG) algorithms to predict the flexural and compressive strength of 3D-printed PLA/carbon parts, demonstrating robust predictive performance across all datasets.³² Fetecau et al. developed an ANN model to predict the Young’s modulus and tensile strength of 3D-printed PA and fiber-reinforced PA composites, achieving relative errors well below 5%.³³ Ulkir proposed an ANN model based on genetic algorithm (GA) to predict the coefficient of thermal expansion (CTE) of fiber-reinforced PLA composites, achieving a maximum prediction error below 1.17%.³⁴ Wei et al. used the Takagi-Sugeno (T-S) fuzzy neural network to construct a prediction model between different FDM parameter combinations and the tensile strength of PLA materials.³⁵ Nikzad et al. combined Taguchi design and the TabNet algorithm to predict the elastic modulus of PLA components.³⁶ Omer et al. applied a general recurrent neural network (GRNN) to predict the tensile properties of carbon fiber-PLA composites, while Tayyab et al. established a multi-parameter regression model based on an artificial neural network and optimized the network structure with the help of a genetic algorithm to achieve the prediction of the flexural strength of acrylonitrile-butadiene-styrene parts.^37,38 In addition, Schmidt et al. used a neural network method to predict the tensile strength of 3D-printed polyethylene terephthalate (PETG) parts.³⁹

The backpropagation (BP) neural network is widely used due to its strong capability in modeling nonlinear relationships.⁴⁰ However, its training process is highly sensitive to initial parameter settings and tends to become trapped in local optima, which can adversely affect prediction performance.⁴¹ To overcome these limitations, swarm intelligence optimization algorithms are often combined with BP neural networks to enhance parameter optimization. Commonly applied methods include particle swarm optimization (PSO), whale optimization algorithm (WOA), sparrow search algorithm (SSA), slime mould algorithm (SMA), etc.^42,43 Tian et al. proposed a method that integrates response surface method (RSM), BP neural network and GA to evaluate the effects of key process parameters such as infill density, material flow rate, and layer thickness on the vertebral density of 3D-printed components.⁴⁴ Jiang et al. developed a PSO-BP algorithm that combines the PSO algorithm and BP neural network to predict the splitting tensile strength of track structures.⁴⁵ Although existing optimization algorithms combined with BP neural networks have demonstrated promising predictive performance, several critical issues remain. (1) Limited global optimization capability: many algorithms exhibit premature convergence in complex high-dimensional problems and are prone to becoming trapped in local optima. (2) High sensitivity to parameter settings: the performance of these models is often strongly affected by the selection and adjustment of parameters, thereby reducing robustness and generalization ability. (3) Inadequate balance between accuracy and efficiency: some methods achieve acceptable prediction accuracy but at the cost of high computational complexity or slow convergence, limiting their practical application in real-time optimization.

Compared with other swarm intelligence algorithms, the sand cat swarm optimization (SCSO) algorithm is a novel bio-inspired method that is inspired by the predatory behavior of sand cats. It effectively maintains a balance between global exploration and local exploitation, thereby avoiding local optima and exhibiting superior performance across diverse optimization tasks.⁴⁶ Moreover, SCSO has fewer tunable parameters, which simplifies the optimization process, reduces parameter adjustment complexity, and enhances computational efficiency, making it more suitable for nonlinear and high-dimensional complex datasets. Hameed et al. introduced a new method combining BP with variable adaptive momentum (BPVAM) and SCSO for classification of hyperspectral images.⁴⁷ Based on these advantages, this paper selects the SCSO algorithm as the improved basic algorithm. However, when dealing with high-dimensional and complex optimization problems, the original SCSO algorithm still exhibits limitations, such as restricted optimization accuracy, slow convergence, and susceptibility to being trapped in local optima. To overcome these limitations, this paper proposes a multi-strategy improved sand cat swarm optimization (MISCSO) algorithm to construct a MISCSO-BP prediction model. The MISCSO algorithm first incorporates a cubic chaotic reverse learning strategy to enhance the diversity of the initial population. Second, the dynamic nonlinear sensitivity range and Weibull flight strategy are used to improve individual position updates. Finally, a Gaussian-Cauchy mutation strategy is introduced to strengthen the algorithm’s ability to escape from local optima. These enhancements collectively enhance the performance and robustness of the SCSO algorithm, substantially increasing its global search capability, convergence speed, and optimization accuracy, while effectively preventing entrapment in local optima. By integrating MISCSO with the BP neural network, the proposed MISCSO-BP model enables more precise optimization of weights and biases, thereby enhancing prediction accuracy and generalization performance.

The main contributions of this study are as follows:

• The MISCSO algorithm incorporates multiple strategies, including cubic chaotic reverse learning, dynamic sensitivity range adjustment, Weibull flight, and Gaussian-Cauchy mutation, ensuring a better balance between global search and local exploitation, thereby avoiding premature convergence and local optima.

• The MISCSO algorithm effectively enhances global optimization capabilities, and when combined with the BP neural network, it significantly improves the accuracy and stability of mechanical property prediction.

• The adaptive parameter adjustment mechanism reduces the reliance on empirical settings, allowing the model to flexibly adapt to different datasets and parameter spaces.

• This model exhibits stronger robustness and generalization ability, making it applicable to a wide range of FDM materials and process parameters while providing more reliable guidance for practical optimization.

The rest of this paper is organized as follows: The experimental section details the materials used in the experiment, the sample preparation process, the design of the FDM printing parameters, and the tensile performance testing method; the methodology section explains the basic principles of the BP neural network and the SCSO algorithm, and constructs the MISCSO-BP prediction model; the discussion on correlations section analyzes the influence of 3D printing parameters on the tensile properties of GF/RPP parts; the results and discussion of the MISCSO-BP model section evaluates the effectiveness and stability of the MISCSO-BP model and compares it with the existing prediction models; the conclusion section summarizes the main research work of this paper.

Experimental

Material preparation

The RPP used in this study was derived from discarded meltblown nonwovens. After undergoing a recycling process involving hot pressing, melting, impurity removal, and pelletization, it was converted into a pelletized material suitable for FDM printing. It was purchased from Sikan 3D New Material Technology Co., Ltd in Tianjin, China. To enhance mechanical performance, short glass fibers (GF) with an average length of 4.5 mm and a diameter of 13 μm were selected as reinforcement and uniformly mixed with RPP at a weight ratio of 30 wt%. The GF/RPP composite wire was processed into a filament with a diameter of 1.75 mm using a twin-screw extruder (model 996, SUZHOU AIFUER MACHINERY) operating at a screw speed of 60 r/min, with barrel temperatures set to 175°C, 195°C, and 180°C for the feed, mixing, and extrusion zones, respectively. After extrusion, the filament was dried in a vacuum oven at 60°C for 6 hours to remove residual moisture. The complete material preparation process is illustrated in Figure 1.

Figure 1.

Preparation process of GF/RPP composite filament for FDM.

FDM printing and parameter setting

The sample was printed using a Bambu Lab A1 model FDM 3D printing device, as shown in Figure 2. In the FDM process, different process parameters have a significant impact on molding quality and mechanical properties. Among them, printing temperature (T), layer thickness (L), infill density (D), and raster angle (R) were selected as the key process parameters because they are the most influential factors affecting the mechanical properties and print quality of the GF/RPP composites.¹⁵ Printing temperature influences the material’s melting behavior and the interlayer fusion degree, layer thickness determines the interlayer contact area, infill density is directly related to the density and overall strength of the internal structure, and the raster angle determines the directionality of the material deposition path. To systematically evaluate the influence of these parameters on the mechanical properties of GF/RPP parts, each parameter was set at three levels, as shown in Table 1, resulting in a total of 81 different parameter combinations. To maintain consistency across all experiments, the printing parameters were set as listed in Table 2.

Figure 2.

(a) FDM process diagram (b) 3D printing equipment.

Table 1.

Parameters selected and their levels.

Table 2.

Printing parameter setting.

Parameter	Unit	Value
Printing speed	mm/s	50
Nozzle diameter	mm	0.4
Infill pattern		Linear
Line width	mm	0.45
Support structures		No
Build plate temperature	°C	60

The levels of FDM printing parameters were selected based on multiple test experiments to ensure that the GF/RPP composite material has optimal mechanical properties. The printing temperature range of 220–260°C was chosen because temperatures below 220°C resulted in poor filament flow and insufficient layer adhesion, while temperatures above 260°C led to thermal degradation of the polymer matrix and reduced mechanical properties. Layer thickness levels were set to 0.1–0.3 mm to balance dimensional accuracy and build efficiency. Infill densities ranging from 60 to 100% were selected to investigate the effect of internal structure on tensile performance. Raster angles of 0°, 45°, and 90° were adopted to study the anisotropic behavior of printed parts under different load orientations. These parameter levels provide a comprehensive basis for exploring the influence of FDM process conditions on the mechanical performance of GF/RPP composites.

Tensile property test

The GF/RPP specimens printed by the FDM process were subjected to tensile tests, and the key mechanical performance indicators such as tensile strength (TS), tensile modulus (E), and elongation at break (EL) were obtained, which respectively reflect the strength, stiffness, and ductility of the material. Tensile tests were carried out with a universal testing machine AGS-X (Shimadzu Corporation of Japan) at a loading speed of 5 mm/min, as seen in Figure 3(a). The tensile specimen was constructed using SolidWorks software and printed according to ASTM D638, with dimensions of 115 mm × 19 mm × 3.2 mm and a gauge length of 33 mm, as seen in Figure 3(b). After printing out each group of samples under the same conditions, the mechanical tests were carried out in sequence. To minimize experimental errors, three repeated tensile tests were conducted for each of the 81 parameter sets, as shown in Figure 3(c)–(d), resulting in a total of 243 data samples. This experimental design not only ensures repeatability but also provides comprehensive coverage of the parameter space, sufficiently capturing the nonlinear effects and interactions among printing temperature, layer thickness, infill density, and raster angle on the tensile properties of GF/RPP composites. The experimental results of different parameter combinations are shown in Table 3.

Figure 3.

(a) Specimens on a tensile testing machine (b) Tensile testing ASTM D638 standard (c) Samples with parameter combinations of 23, 47, and 66 (d) Stress-strain curves of samples with parameter combinations of 23, 47, and 66.

Table 3.

Experimental results of different parameter combinations.

Experiment number	Parameter				Experimental results
Experiment number	T (°C)	L (mm)	D (%)	R (°)	TS (MPa)	E (MPa)	EL (%)
1	220	0.1	60	0	12.38 ± 0.42	231.24 ± 5.2	17.94 ± 2.2
2	220	0.1	60	45	11.68 ± 0.39	259.53 ± 5.8	13.33 ± 1.8
3	220	0.1	60	90	12.87 ± 0.37	315.01 ± 4.7	9.92 ± 1.7
4	220	0.1	80	0	14.01 ± 0.25	246.92 ± 4.9	16.54 ± 2.5
5	220	0.1	80	45	14.67 ± 0.44	289.89 ± 5.3	10.87 ± 2.1
6	220	0.1	80	90	13.89 ± 0.36	317.94 ± 4.7	9.09 ± 1.5
7	220	0.1	100	0	15.61 ± 0.27	261.41 ± 5.6	18.29 ± 2.5
8	220	0.1	100	45	15.94 ± 0.40	293.45 ± 4.0	10.26 ± 2.3
9	220	0.1	100	90	16.23 ± 0.36	327.49 ± 5.1	7.46 ± 1.9
10	220	0.2	60	0	13.80 ± 0.33	250.86 ± 5.4	20.58 ± 2.0
…	…	…	…	…	…	…	…
81	260	0.3	80	90	15.33 ± 0.39	302.20 ± 5.6	19.76 ± 2.1

Methodology

BP neural network

BP neural network is a typical feedforward neural network, and its basic structure consists of three parts: input layer, hidden layer, and output layer. It iteratively adjusts the weights and biases using the backpropagation algorithm to minimize prediction error, and it is widely applied in the nonlinear modeling of complex systems.⁴⁸ This study employs a BP neural network to predict the tensile performance of GF/RPP parts under various FDM parameter settings. Printing temperature, layer thickness, infill density, and raster angle were used as input variables, whereas tensile strength, elastic modulus, and elongation at break served as the output parameters, as shown in Figure 4.

Figure 4.

BP neural network.

To balance prediction accuracy and computational efficiency, the number of hidden layer neurons is set using an empirical formula, as shown in formula (1).⁴⁹

α = \sqrt{(x + y)} + φ

(1)

where x is the number of input layers, y is the number of output layers, and φ is an adjustment factor within the range of 1 to 10.

MISCSO

To address the shortcomings of the BP neural network in global optimization and convergence speed, this paper introduces the MISCSO method, which significantly improves the initial quality of weights and biases while reducing oscillation and non-convergence during training.

SCSO

SCSO is an emerging swarm intelligence optimization algorithm, which is inspired by the high sensitivity and adaptive search behavior of sand cats in the process of hunting in the desert. The algorithm simulates two key behaviors of sand cats in the foraging process: prey search and attack, while enhancing the global search ability and effectively reducing the risk of falling into the local optimum.^50,51

Following initialization, the sand cat group enters the exploration and exploitation stage, which involves hunting for or attacking prey. The conversion coefficient R is the crucial parameter that controls the conversion between these two stages, and the formula is as follows:

R = 2 \times r_{G} \times r a n d (0, 1) - r_{G}

(2)

where r_G is the sensitivity coefficient of the sand cat group and rand (0,1) represents a random number between 0 and 1. The r_G calculation formula is:

r_{G} = S_{M} - (\frac{S_{M} \times t}{T_{\max}})

(3)

where T_max is the maximum number of iterations, t is the current iteration, and S_M is hearing coefficient of the sand cat group. With their sensitivity range between 0 and 2 kHz, this paper sets S_M is 2.

When the conversion coefficient R meets condition $| R | > 1$ , the sand cat enters the search stage, and when the conversion coefficient $| R | \leq 1$ , the sand cat enters the attack stage, as seen in Figure 5. The formula is given below:

P o s (t + 1) = {\begin{cases} P o s_{b e s t} (t) - r \times P o s_{r n d} \times \cos (θ), | R | \leq 1 \\ r \times (P o s_{b c} (t) - r a n d (0, 1) \times P o s_{c} (t)), | R | > 1 \end{cases}

(4)

where Pos_best is the best location, Pos_rnd is a random location, θ the random movement angle of the sand cat (0° < θ < 360°). r is the sensitivity coefficient and r = r_G × rand (0,1). Pos_bc is the best candidate position, Pos_c is the current position of the sand cat.

Figure 5.

SCSO exploration.

Improvement strategy

While the SCSO algorithm performs well in global exploration and convergence efficiency, there are still some shortcomings in practical applications, which limit the further improvement of its optimization effect. For instance, it suffers from uneven initial population distribution, slow convergence, and a tendency to get trapped in local optima. To overcome the above defects, this paper proposes an MISCSO algorithm to improve its search efficiency and global optimization ability.

Strategy 1: Cubic chaos reverse learning

The SCSO algorithm faces reduced accuracy in subsequent searches due to insufficient initial population diversity. To address this issue, this paper uses a cubic chaotic map, characterized by strong ergodicity and homogeneity, to initialize the population. The cubic mapping method has greater diversity and sensitivity compared to other chaotic methods. The calculation formula is:

2 y_{i + 1} = 4 y_{i}^{3} - 3 y_{i}

(5)

δ (i) = l_{b} + \frac{(u_{b} - l_{b}) \times (y_{i} + 1)}{2}

(6)

where δ(i) represents the position after cubic chaotic mapping, −1< y_i <1, y_i ≠ 0, i = 0,1,…,N, and N is the population size. u_b and l_b represent the upper and lower bounds of the solution, respectively.

This paper introduces a learning strategy based on the refraction principle to further increase the quality of the initial population generated by the cubic chaotic map.⁵² This strategy is based on the notion of convex lens imaging, and by creating individual relative position solutions, it increases population diversity, expands the search space, and speeds up the algorithm’s convergence. The calculation formula is:

X^{'} = \frac{u_{b} + l_{b}}{2} + \frac{u_{b} + l_{b}}{2 β} - \frac{X}{β}

(7)

where

X^{'}

is the reverse solution of the initial solution obtained after refraction learning, X is current solution and β is the scaling factor, which is set to 0.75 in this paper.

Strategy 2: Dynamic nonlinear sensitivity range

The sensitivity range parameter r_G determines how the algorithm switches between searching and attacking behaviors. A higher value favors global search, while a lower value favors local attack. Traditional linear decreasing strategies are difficult to apply to complex multi-peak issues and may degrade optimization accuracy. This paper introduces a dynamic nonlinear sensitivity range and combines it with the control factor ψ ∈ (1, 3) to achieve adaptive adjustment of the search intensity, thereby enhancing the search ability and avoiding local optima at different stages. The formula is given below:

r_{G} = S_{M} \times {(\frac{T_{\max} - t}{T_{\max}})}^{ψ} \times r a n d (0, 1)

(8)

Strategy 3: Weibull flight strategy

Due to the predatory behavior of sand cats, the SCSO algorithm exhibits rapid convergence. However, this feature may also cause premature stagnation at local optima, thereby reducing its global search capability and overall search accuracy. To enhance population diversity and avoid getting trapped in local optima, the Weibull flight strategy introduces a stochastic hopping mechanism to expand the exploration range and improve search coverage. The position update model is given below:

P o s_{n e w} (t + 1) = r \times R_{w} \times (P o s_{b c} (t) - r a n d \times P o s_{c} (t))

(9)

where R_w represents a Weibull-distributed random variable, Pos_bc(t) and Pos_c(t) represent the best candidate and current positions, respectively.

The probability density function is expressed as:

f (x, λ, k) = (\frac{k}{λ}) \times {(\frac{x}{λ})}^{k - 1} \times \exp (- {(\frac{x}{λ})}^{k})

(10)

where x represents the random variable, while k and λ determine the shape and scale of the distribution. For this study, λ = 0.5 and k = 4.

Strategy 4: Gaussian-Cauchy mutation strategy

The Gaussian-Cauchy mutation strategy was employed to optimize the positions of the sand cat population.⁵³ The Gaussian mutation introduces small random perturbations that enhance local search accuracy and prevent premature convergence, while the Cauchy mutation applies larger stochastic variations that help individuals escape local optima and improve the algorithm’s global exploration ability. This mechanism ensures efficient local convergence while maintaining strong global optimization performance. X (b (i + 1)) is the perturbed position using Gaussian-Cauchy variation, the specifics are:

X (b (i + 1)) = P o s_{b e s t} (t) \times (1 + μ_{1} \times Gauss (σ) + μ_{2} \times Cauchy (σ))

(11)

μ_{1} = t / T_{\max}, μ_{2} = 1 - μ_{1}

(12)

where Gauss(σ) and Cauchy(σ) are random variables from Gaussian and Cauchy distributions, respectively.

After the Gauss-Cauchy mutation disturbance, the greedy strategy is used to update positions, continuously improving solutions in each iteration and enhancing search efficiency while avoiding local optima. This strategy evaluates the fitness scores of both the original and mutated individuals, retaining the superior one. When the mutated solution outperforms the original, it replaces it, thereby enhancing the performance of the algorithm and guiding the population steadily toward the global optimum. The formula is as follows:

P o s (b (i + 1)) = {\begin{cases} X (b (i + 1)), f (X (b (i + 1))) < f (H (b (i + 1))) \\ H (b (i + 1)), f (X (b (i + 1))) \geq f (H (b (i + 1))) \end{cases}

(13)

where Pos(b (i + 1)) is the current position, f is the fitness function and H (b (i + 1)) is the position after Gauss-Cauchy perturbation.

In this study, the MISCSO algorithm’s fitness function was defined as the mean squared error (MSE) between the experimental and predicted values. The objective of the optimization process was to minimize this error, thereby improving the prediction accuracy of the BP neural network. The fitness function f can be expressed as:

f = \frac{1}{N} \sum_{i = 1}^{N} {(y_{i} - {\hat{y}}_{i})}^{2}

(14)

where

y_{i}

and

{\hat{y}}_{i}

denote the experimental and predicted mechanical properties values, respectively, and N is the number of samples.

The pseudo-code of the MISCSO algorithm is as follows:

MISCSO-BP model

This section introduces the MISCSO-BP model to predict the tensile properties of GF/RPP 3D-printed parts. To improve prediction accuracy and accelerate convergence, the MISCSO algorithm is employed to globally optimize the weights and biases of the BP neural network. The overall structure of the model is shown in Figure 6, and the main steps are as follows:

Step 1: Initialize the neural network structure and optimization parameters. Before starting the optimization process, first determine the structural configuration of the BP neural network, including the number of neurons in the input layer, hidden layer, and output layer. Each individual in the MISCSO population represents a candidate set of weights and biases for the BP neural network. Then initialize the key parameters of the MISCSO algorithm, such as population size, position boundary, maximum number of iterations, etc. To enhance population diversity and solution quality, the cubic chaotic reverse learning strategy is applied, effectively expanding the search space and improving early-stage global exploration.

Step 2: Weight optimization and fitness evaluation. The MISCSO algorithm iteratively optimizes the weights and biases of the BP neural network. In each iteration, the current individual’s solution is applied to the network, and its performance on the training set is evaluated to calculate the fitness value. To enhance search efficiency across different stages, the dynamic nonlinear sensitivity range strategy is employed to modify the sensitivity coefficient adaptively, thereby increasing the number of iterations in the search stage and avoiding premature convergence. This mechanism improves convergence stability and enhances predictive performance on unseen data.

Step 3: Global exploration and local optimization balance mechanism. To mitigate the risk of premature convergence and local entrapment during the optimization process, the Weibull flight strategy is employed to enhance global exploration and increase the likelihood of the algorithm jumping out of the local optimal solution. This mechanism accelerates convergence and directs the weights and biases of the BP neural network towards optimal solutions.

Step 4: Population update and robustness improvement. After each iteration, the Gaussian-Cauchy mutation strategy is applied to perform perturbation updates on the current optimal solution, thereby enabling more robust optimization of the BP neural network’s weights and biases and preventing overfitting to the training data. Meanwhile, the greedy selection strategy compares the fitness of mutated and original individuals, retaining the superior ones to further enhance the overall population quality.

Step 5: BP fine-tuning and model output. After global optimization by MISCSO, the BP algorithm performs local fine-tuning to further refine the parameters. This combination leverages MISCSO algorithm’s strong global search capability and BP neural network’s nonlinear mapping strength, enabling the MISCSO-BP model to efficiently capture complex high-dimensional nonlinear relationships and achieve high prediction accuracy with rapid convergence. When the stopping condition is met, the final trained neural network will be verified and performance evaluated on the test set, and the prediction results will be output.

Figure 6.

MISCSO-BP model flowchart.

Discussion on correlations

The influence of 3D printing parameters on tensile properties

To analyze the influence of printing parameters on the tensile properties of 3D-printed GF/RPP parts, the Pearson correlation coefficient r_XY was employed to quantify the linear relationship between parameter variables, as defined in equation (15).⁵⁴ When r = +1, the variables are perfectly positively correlated, whereas r = −1 indicates a perfect negative correlation. The absolute value of r reflects the strength of the relationship, with values close to one representing strong correlation and those close to 0 indicating weak correlation.

r_{X Y} = \frac{\sum_{i = 1}^{n} (X_{i} - \bar{X}) (Y_{i} - \bar{Y})}{\sqrt{\sum_{i = 1}^{n} {(X_{i} - \bar{X})}^{2}} \sqrt{\sum_{i = 1}^{n} {(Y_{i} - \bar{Y})}^{2}}}

(15)

where X_i and Y_i being individual samples, and

\bar{X}

and

\bar{Y}

is the mean values X and Y, respectively.

This paper constructed a correlation matrix between four process parameters, including T, L, D, and R, as well as tensile performance indicators TS, E, and EL, and visualized it in the form of a heat map, as shown in Figure 7.

Figure 7.

Correlation heatmap between printing parameters and tensile properties.

Tensile strength

Figure 7 illustrates that infill density (0.50) has the most significant effect on the tensile strength of GF/RPP composites.²⁰ A higher infill density provides more effective load-bearing paths, enabling the material to exhibit higher ultimate strength under tensile loading. Layer thickness is negatively correlated with tensile strength, and an increase in layer thickness will reduce tensile strength. This trend is mainly due to the fact that the larger layer thickness leads to a reduction in contact area between layers, insufficient bonding, and a weak interface, which is easy to destroy first during the stretching process. Printing temperature has a relatively smaller influence compared to the first two parameters. An appropriate temperature improves the interfacial adhesion between the RPP matrix and the glass fibers, thereby enhancing the bonding strength. The influence of the raster angle on tensile strength is marginally adverse. When the raster arrangement aligns with the primary loading direction, the bearing capacity of the entire structure in a singular tensile direction will diminish.

Elasticity modulus

In terms of elastic modulus, the infill density (0.55) is still the most significant influencing parameter.⁵⁵ A higher infill density represents a more compact internal structure with fewer voids, thereby enhancing the mechanical synergy between the GF and the RPP matrix, which is manifested as an improvement in the overall stiffness and deformation resistance of the material. The raster angle also has a certain effect on the elastic modulus. A properly selected raster angle enhances the alignment between fiber orientation and the loading direction, thus boosting stress transfer performance. The influence of layer thickness and printing temperature is small. While an increase in layer thickness may lead to insufficient interlayer bonding, its impact on the elasticity modulus is less significant than that of the infill density. Additionally, temperature variations within the experimental range do not significantly affect the melting-solidification behavior, so the effect on the elasticity modulus can be disregarded.

Elongation at break

In terms of elongation at break, layer thickness is the most significant factor. When the layer thickness is too large, obvious interfaces form between layers, causing uneven strain and allowing microcracks to propagate, significantly reducing the material’s elongation capacity. The raster angle also has a considerable negative effect. An unreasonable raster arrangement limits the material’s uniform ductility in all directions, leading to premature failure in localized areas. The infill density also negatively impacts elongation at break, suggesting that while higher density improves stiffness and strength, it sacrifices ductility. Printing temperature has the least impact on elongation at break, indicating that within the experimental temperature range, its effect on molecular chain movement is minimal.

In summary, correlation analysis confirms that the infill density and layer thickness have a significant effect on the mechanical properties of GF/RPP composite materials, while the printing temperature has little effect within the experimental range, and the raster angle mainly affects the ductility of the material. Specifically, higher infill density contributes to improved tensile strength and elastic modulus by creating denser internal structures with more effective load-bearing paths and enhanced interfacial bonding between glass fibers and the RPP matrix.¹⁵ Smaller layer thickness ensures sufficient interlayer fusion and continuity, which benefits tensile strength and elongation at break, while excessive thickness weakens bonding and promotes microcrack propagation. The raster angle determines the degree of fiber arrangement and stress transfer directionality, thereby influencing the deformation behavior and ductility of the printed parts. In contrast, the printing temperature exhibits only minor effects on all three mechanical properties, suggesting that within the tested range, variations in temperature have limited impact on interfacial adhesion and molecular mobility. Therefore, to meet diverse mechanical performance requirements in industrial FDM applications, the infill density and layer thickness should be carefully optimized, and the raster angle should be strategically designed. By coordinating these parameters, manufacturers can achieve a balanced improvement in strength, stiffness, and ductility of 3D-printed GF/RPP parts, ensuring reliable performance in structural and functional applications. These findings provide practical guidance for process parameter selection in industrial settings, facilitating the production of high-performance FDM parts.

Results and discussion of the MISCSO-BP model

Testing and discussion on the MISCSO algorithm

Test function evaluation

To verify the search capacity and robustness of the proposed MISCSO algorithm, this study used six widely used test functions, which include unimodal, multimodal, and complex functions, as shown in Table 4. In performance comparisons, the unimodal function has only one unique global optimal solution and does not contain interfering local extreme points, which primarily reflects the algorithm’s convergence speed and local search accuracy. Multimodal and complex functions, on the other hand, contain a high number of local optimal solutions, have a complicated search space structure, and may effectively test the algorithm’s global search capabilities as well as its ability to escape local optima. The simulation experiments in this study were conducted using MATLAB R2023b on a 64-bit Windows 11 system with an AMD Ryzen 9 7945HX processor and 32 GB of memory.

Table 4.

Six types of test functions.

Function type	Test function	Dim	Range	Objective
Unimodal	$F_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	30	[−100, 100]	0
Unimodal	$F_{2} (x) = \sum_{i = 1}^{n - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}]$	30	[−30, 30]	0
Multimodal	$F_{3} (x) = (\frac{1}{500} + \sum_{j = 1}^{25} \frac{1}{j + \sum_{i = 1}^{2} {(x_{i} - a_{i j})}^{6}})$	2	[−65, 65]	1
	$\begin{array}{l} F_{4} (x) = - 20 \exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - \exp (\frac{1}{n} \sum_{i = 1}^{n} \cos (2 π x_{i})) \\ + 20 + \exp (1) \end{array}$	30	[−32, 32]	0
	$F_{5} (x) = \sum_{i = 1}^{n} [x_{i}^{2} - 10 \cos (2 π x_{i}) + 10]$	30	[−5, 5]	0
	$F_{6} (x) = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} \cos (\frac{x_{i}}{\sqrt{i}}) + 1$	30	[−600, 600]	0

To verify the effectiveness of the proposed MISCSO algorithm, this study used six algorithms, including MISCSO, SCSO, PSO, SSA, WOA, and SMA for performance comparison. The specific parameter settings of each algorithm are shown in Table 5. To ensure the fairness of the comparison and the repeatability of the results, all algorithms adopt the same experimental parameters, with the maximum number of iterations set to 2000 and the population size at 100. Each algorithm is run independently 100 times on each test function to reduce randomness’s impact on the results. To more intuitively show the performance of each algorithm in the search process, Figure 8 shows the convergence curve of MISCSO and five comparison algorithms on the test function.

Table 5.

Different algorithm parameter settings.

Algorithms	Parameter settings
SCSO	S_M = 2, Roulette wheel selection ∈ [0, 360]
PSO	c₁ = c₂ = 1.5, ω = 0.9
SSA	Leader position update prob-ability = 0.5, v₀ = 0
SMA	z = 0.03
WOA	a = 2
MISCSO	S_M = 2, β = 0.75, ψ = 2, k = 4, λ = 0.5

Figure 8.

Comparison of convergence curves of different algorithms in six test functions (a) F1 (b) F2 (c) F3 (d) F4 (e) F5 (f) F6.

As shown in Figure 8, the MISCSO algorithm demonstrates strong convergence capacity on all six test functions, not only able to converge stably to the global optimal solution but also outperforming the comparison algorithm in terms of convergence speed and accuracy. Although the final results of some algorithms on individual functions are close, MISCSO has more advantages in overall performance, especially in the early iteration stages of showing a faster downward trend. Furthermore, in the F1 and F4 functions, MISCSO explores solution spaces that other algorithms fail to touch, highlighting its capabilities in global search, which is mainly due to the synergy of the introduced cubic chaos reverse learning and dynamic nonlinear sensitivity range strategy. In multimodal complex functions like F2, F3, F5, and F6, several comparison methods’ convergence curves tend to plateau after mid-term iteration and fall into local optimality. In contrast, MISCSO can continue to improve at this stage and eventually achieve a better solution. This indicates that MISCSO is has a greater advantage in breaking the local optimum, mainly due to the introduction of the Weibull flight and Gaussian-Cauchy mutation strategy. Specifically, the Weibull flight strategy expands the search space by generating new individual positions, enhancing the algorithm’s global exploration capability and effectively reducing the risk of getting trapped in local optima. Meanwhile, the Gaussian-Cauchy mutation strategy enhances the local exploitation ability and avoids premature convergence by perturbing the optimal position of the sand cat population. By combining these mechanisms, MISCSO achieves an effective balance between global exploration and local exploitation, ensuring faster convergence and higher convergence accuracy compared to SCSO, PSO, SSA, WOA, and SMA.

Ablation study on MISCSO

To evaluate the effectiveness of the different enhancement strategies in the MISCSO algorithm, ablation experiments were conducted by comparing MISCSO with its single-strategy algorithms, including the original SCSO, C-SCSO with cubic chaos reverse learning strategy, D-SCSO with dynamic nonlinear sensitivity range strategy, and W-SCSO incorporating the Weibull flight and Gaussian-Cauchy mutation strategies. The optimization performance of each algorithm was analyzed on six test functions based on convergence curves, and the contribution of each strategy to global search capability, convergence speed, and local exploitation ability was comprehensively evaluated, as illustrated in Figure 9.

Figure 9.

Comparison of convergence curves of different strategy algorithms in six test functions (a) F1 (b) F2 (c) F3 (d) F4 (e) F5 (f) F6.

As shown in Figure 9, the MISCSO algorithm achieved optimal solutions in all six test functions, indicating its good stability and reliability in optimization performance. In contrast, while the overall performance of the C-SCSO algorithm is slightly lower than that of MISCSO, it has improved significantly when compared to the original SCSO algorithm, owing primarily to algorithm optimization at the initial population stage, demonstrating the effectiveness of initial population optimization in improving algorithm performance. The D-SCSO algorithm introduces dynamic nonlinear sensitivity range strategies, which improve the search diversity and convergence accuracy, but its overall optimization effect on the original SCSO algorithm is limited. Moreover, the W-SCSO algorithm exhibits excellent performance on both the F2, F3 and F4 functions, all achieving optimal values. This demonstrates that the incorporation of the Weibull flight and Gaussian-Cauchy mutation strategies effectively enhances the algorithm’s global search capability while avoiding premature convergence. However, comparisons indicate that algorithms employing a single strategy are less effective than MISCSO in optimizing complex functions. Therefore, the multi-strategy design of MISCSO demonstrates superior convergence accuracy and a stronger ability to avoid local optima compared with single-strategy algorithms.

Analysis of prediction results

To verify the effectiveness of the proposed prediction model MISCSO-BP, this paper compares and analyzes the experimental values of the tensile properties of GF/RPP composite materials with the model prediction values. According to the universal approximation theorem proposed by Cybenko,⁵⁶ a single hidden layer with a sufficient number of neurons can approximate any continuous nonlinear function, which makes it suitable for capturing the complex nonlinear relationships between printing parameters and mechanical properties. The MSE is used to evaluate and select the optimal number of hidden layer nodes. Based on the empirical formula for determining the number of hidden layer nodes (formula (1)), this study evaluated networks with 4 to 14 hidden neurons. The average normalized MSE across all three mechanical properties was calculated for each network, as shown in Figure 10. Comparative experiments indicated that the network with 9 neurons achieved the lowest prediction error, and thus the optimal architecture was determined to be a single hidden layer containing 9 neurons. The hidden layer activation function uses the logsig function, and the output layer uses the purelin function. To avoid getting trapped in local optima and improve prediction accuracy, this study uses the MISCSO algorithm to globally optimize the weights and biases of the BP neural network. Through multi-strategy improvements, MISCSO can cover a wider candidate solution space, avoid premature convergence, and enhance the BP neural network’s stability when dealing with complex high-dimensional nonlinear problems. This enables the network to update weights more accurately, suppress overfitting, and maintain reliable prediction performance on unseen data, thereby significantly improving the generalization ability of the MISCSO-BP model. The key parameters of the MISCSO algorithm are as follows: population size is 100, and the maximum number of iterations is 200.

Figure 10.

Normalized MSE scores under different numbers of hidden layer neurons.

In this study, a total of 243 sets of samples with combinations of process parameters were collected and inputted into the prediction model for training. The data set was split into a 7:3 ratio, of which 170 sets of samples were used for model training and 73 sets were used for performance verification. Random sampling was used to ensure the consistency of data distribution to improve the generalization ability of the model and the reliability of the results. The MSE was used as a loss function during training. To prevent overfitting and improve the network’s generalization performance, L2 weight regularization was applied to constrain excessive growth of network weights, and an early stopping mechanism was employed to terminate training when the validation loss failed to decrease for several consecutive epochs. The optimization algorithm used batch gradient descent with a learning rate of 0.1 and a maximum of 1000 training rounds. Figure 11 illustrates the comparison of actual and predicted values based on MISCSO-BP. It can be seen from the figure that the predicted values of TS, E, and EL have a satisfactory fitting effect with the experimental values, and the mean prediction error is maintained below 5%, showing the high prediction accuracy of the MISCSO-BP model.

Figure 11.

Comparison of actual and predicted values based on MISCSO-BP (a) TS (b) E (c) EL.

To further evaluate the model’s performance, this study employs the coefficient of determination (R²), mean absolute error (MAE), mean squared error (MSE), and root mean squared error (RMSE) as performance metrics. These indicators comprehensively reflect the model’s fitting accuracy and generalization capability. Among them, R² shows the model’s variance explanation capability, with values close to one indicating a better fit. The MAE reflects the average magnitude of prediction errors, whereas the MSE emphasizes larger deviations and is more sensitive to outliers. RMSE intuitively represents error magnitude, and the calculation formulas are as follows:

R^{2} = 1 - \frac{\sum_{i = 1}^{N} {(Y_{i} - {\overset{⌢}{Y}}_{i})}^{2}}{\sum_{i = 1}^{N} {(Y_{i} - \bar{Y})}^{2}}

(16)

M A E = \frac{1}{N} \sum_{i = 1}^{N} (Y_{i} - {\overset{⌢}{Y}}_{i})

(17)

M S E = \frac{1}{N} \sum_{i = 1}^{N} {(Y_{i} - {\overset{⌢}{Y}}_{i})}^{2}

(18)

R M S E = \sqrt{\frac{1}{N} \sum_{i = 1}^{N} {(Y_{i} - {\overset{⌢}{Y}}_{i})}^{2}}

(19)

where N is the total number of samples,

Y_{i}

{\overset{⌢}{Y}}_{i}

and

{\bar{Y}}_{i}

represent the actual values, predicted values and means of actual values.

Table 6 presents the prediction results of the MISCSO-BP model for the three mechanical performance indicators. The results show that the R² of TS, E, and EL are all above 0.85, and the MAE and RMSE values remain relatively low, indicating that the model has high predictive accuracy in the multiple performance parameters. Among these indicators, the prediction accuracy of tensile strength is the highest, with an R² reaching 0.931 and MAE only 0.158, indicating that the model effectively captures the influence of printing parameters on strength variations. The prediction results of elastic modulus are also relatively ideal, with an R² of 0.918, indicating that the model effectively identifies key influencing factors related to rigidity, such as infill density and raster angle. Although elongation at break exhibits large volatility and is influenced by nonlinear factors such as local defects, the R² for this indicator still reaches 0.875, and the prediction error remains within an acceptable range, confirming the robustness of the model in handling uncertain outputs.

Table 6.

Performance indices of MISCSO-BP model for TS, E, and EL prediction.

Performance index	R²	MAE	MSE	RMSE
TS	0.931	0.158	0.028	0.167
E	0.918	6.957	67.79	8.233
EL	0.875	1.902	5.110	2.260

Comparative analysis of methods

To verify the effectiveness of the proposed MISCSO-BP model in performance prediction, this study compares it with four methods: BP neural network,⁴⁸ PSO-BP,⁴⁵ SCSO-BP,⁴⁷ and WOA-BP.⁵⁷ All models use the same data set partitioning (the ratio of training set to test set is 7:3) and maximum training times of 1000 to ensure fairness of the comparison. The number of particles of PSO is set to 100, the population size of WOA and SCSO is 100, and the maximum number of iterations is 200. Figure 12 shows a comparative analysis of the prediction performance of different models on TS, E, and EL.

Figure 12.

Comparative analysis of the prediction performance of different models on TS, E and EL (a) R² (b) MAE (c) MSE (d) RMSE.

Figure 12 illustrates that the MISCSO-BP model predicts three mechanical properties with lower error values and higher R², which fully demonstrates its significant advantages in prediction accuracy. Compared to the BP neural network, the PSO-BP, WOA-BP, and SCSO-BP models all show considerable decreases in MAE and RMSE, demonstrating that the incorporation of the swarm intelligence optimization method can enhance the predictive ability of the BP neural network. Among them, SCSO-BP utilizes the global optimization ability of the SCSO algorithm and outperforms PSO-BP and WOA-BP in some data on the three performance indicators. Compared with the traditional BP neural network and other hybrid models (PSO-BP, WOA-BP, and SCSO-BP), the MISCSO-BP model achieves superior prediction performance, indicating that the integration of multiple optimization strategies effectively enhances the BP network’s capability to handle complex nonlinear relationships. Furthermore, compared to the WOS-BP model used by Ji et al.,⁴⁹ MISCSO-BP achieved a 20% higher R² value for tensile properties prediction, further underscoring its robustness in modeling nonlinear FDM parameter-property relationships. Specifically, the MISCSO algorithm performs global optimization of the BP neural network’s weights and biases, thereby ensuring more accurate and stable parameter updates. Among its strategies, the dynamic nonlinear sensitivity range strategy adaptively regulates the search step size, prevents premature convergence, and enhances predictive performance on unseen data. Meanwhile, the Gaussian-Cauchy mutation strategy introduces stochastic perturbations to candidate solutions, enhancing the robustness of weight optimization and preventing overfitting to the training data. Combined with the BP neural network’s inherent nonlinear mapping ability, these mechanisms enable the MISCSO-BP model to capture high-dimensional nonlinear correlations more effectively, thereby achieving higher predictive accuracy than all compared models.

Figure 13 illustrates the comparative prediction errors of five models: BP neural network, PSO-BP, WOA-BP, SCSO-BP, and the proposed MISCSO-BP, across three mechanical performance indicators: TS, E, and EL. As shown in Figure 13(a)–(c), compared with the traditional BP neural network, the prediction errors of the PSO-BP, WOA-BP, and SCSO-BP models that use swarm intelligence optimization algorithms are significantly reduced. This indicates that the integration of intelligent optimization algorithms effectively enhances the global search capability of the BP neural network. Moreover, the MISCSO-BP model exhibits prediction results that are closer to the experimental values, with a more concentrated error distribution, demonstrating superior stability and prediction accuracy compared with the other four models.

Figure 13.

Comparison of prediction errors (a) TS (b) E (c) EL.

To further evaluate the optimization efficiency and convergence performance of the proposed MISCSO-BP model during training. Figure 14 shows the MSE convergence curves of four prediction models: MISCSO-BP, PSO-BP, SCSO-BP, and WOA-BP. As shown in the figure, the MISCSO-BP model exhibits a rapid convergence trend at the early stage of iteration, with the prediction error decreasing sharply. It reaches stable convergence within approximately 300 iterations, achieving the lowest final MSE value of approximately 0.0077, significantly outperforming the other models. This result demonstrates its significant advantages in global search capability and convergence speed. In contrast, while PSO-BP and SCSO-BP improve the convergence performance of traditional BP models to a certain extent, their rate of descent and final error levels still lag significantly behind MISCSO-BP, indicating that they are prone to falling into local optimal solutions during the parameter optimization process. Although the WOA-BP model eventually achieved a relatively small error level, it exhibited significant fluctuations in the early iterations, indicating a certain degree of instability during training. These superior convergence characteristics of the MISCSO-BP model can be attributed to the multi-strategy improvement mechanisms embedded within the MISCSO algorithm. Among these strategies, the cubic chaotic reverse learning strategy diversifies the initial population and expands the search space coverage, enabling the BP neural network’s weights and biases toward high-quality solutions more rapidly and reducing the likelihood of premature convergence. In addition, the Weibull flight strategy facilitates long-distance exploratory jumps, allowing the algorithm to more rapidly adjust the BP neural network parameters toward global optima. Through the combination of these mechanisms, the MISCSO-BP model achieves faster and more stable convergence while maintaining strong global optimization ability during BP neural network parameter optimization.

Figure 14.

Comparison of convergence curves of different prediction models during the training process.

Stability analysis of MISCSO-BP model

To assess the stability of the proposed MISCSO-BP model, Monte Carlo cross-validation was performed with 10 independent trainings on five models: BP neural network, PSO-BP, WOA-BP, SCSO-BP, and MISCSO-BP. In each training, the data set is randomly divided into a training set and a validation set in a ratio of 7:3 to obtain the model’s performance under different data partitions. Each training was run under the same parameter configuration, and the corresponding MAE was recorded. The MAE performance of each model was visualized using box plots, as shown in Figure 15.

Figure 15.

Comparison of prediction errors box plots (a) TS (b) E (c) EL.

As can be seen from the figure, the MISCSO-BP model exhibits a lower median error and a smaller box height across all three mechanical properties. This shows that the model not only achieves lower prediction errors but also demonstrates smaller variations in training outcomes, reflecting its superior stability. In contrast, the traditional BP neural network displays larger error fluctuations, wider boxes, and numerous outliers, indicating instability during training and limited reliability of their predictions. The PSO-BP, WOA-BP, and SCSO-BP models using swarm intelligence optimization algorithms show certain improvements in error concentration, but their error boxes remain significantly higher than that of MISCSO-BP, indicating residual fluctuations. Compared with other models, the MISCSO-BP model effectively avoids premature convergence to local optima, enhances generalization capability, and maintains consistently stable predictive performance across repeated training processes. These advantages indicate that the MISCSO-BP prediction model can serve as a reliable and practical tool for industrial FDM applications, enabling precise optimization of printing parameters, reducing experimental costs, and improving the mechanical properties of 3D-printed GF/RPP parts. In actual production, this model helps to achieve efficient control of process parameters and provides a scientific basis for performance prediction, thereby improving the overall production efficiency and part performance stability of additive manufacturing.

Conclusion

This paper presents an MISCSO-BP prediction model that uses four FDM process parameters, namely printing temperature, layer thickness, infill density, and raster angle, as inputs to predict the tensile strength, elastic modulus, and elongation at break of GF/RPP composites in the FDM 3D printing process. The following conclusions can be drawn:

(1) Correlation analysis between printing parameters and mechanical properties revealed that infill density and layer thickness are the primary factors influencing the tensile properties of 3D-printed GF/RPP composites, whereas raster angle predominantly affects ductility, and printing temperature has little effect. By optimizing infill density and layer thickness and selecting an appropriate raster angle, the strength, stiffness, and ductility of the printed components can be effectively balanced, providing practical guidance for industrial FDM applications and enabling the reliable fabrication of parts that meet specific engineering performance requirements.

(2) Building on the original SCSO algorithm, the MISCSO algorithm integrates multiple strategies, including cubic chaotic reverse learning, dynamic nonlinear sensitivity range, the Weibull flight, and the Gaussian-Cauchy mutation strategy. Comparative experiments on six test functions and ablation studies with various single-strategy improvement algorithms demonstrate that MISCSO markedly enhances population diversity and global search capability, achieves faster convergence, and attains higher optimization accuracy in complex and multimodal optimization problems, thereby verifying the effectiveness of the multi-strategy collaborative improvement.

(3) The performance of the MISCSO-BP model was evaluated using metrics including R², MAE, MSE, and RMSE, and compared with several prediction models, including the traditional BP, PSO-BP, WOA-BP, and SCSO-BP. The results demonstrate the superior prediction accuracy, faster convergence, and greater stability of the proposed model. Specifically, the R² values for tensile strength, elastic modulus, and elongation at break are 0.93, 0.91, and 0.87, with average prediction errors below 5%, indicating that the MISCSO-BP model provides reliable predictions and is suitable for FDM printing parameter optimization and performance forecasting tasks.

Although the proposed MISCSO algorithm exhibits superior convergence speed and prediction accuracy, this study still has certain limitations. The experimental validation was performed within a restricted range of FDM parameters and focused on a single GF/RPP composite material, which may limit the model’s generalizability to other materials or printing conditions. Moreover, the current optimization model primarily focuses on mechanical properties, without considering other critical aspects such as printing efficiency, surface quality, and dimensional accuracy. Future research will aim to improve the scalability and adaptability of the MISCSO algorithm by incorporating adaptive learning strategies, hybrid metaheuristic techniques, and multi-objective optimization. Expanding the experimental dataset to cover diverse materials and a wider range of process parameters will further enhance the robustness and practical applicability of the proposed method for FDM process optimization.

Footnotes

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 research was funded by National Natural Science Foundation of China (Nos. 51205288, 62401391) and the Natural Science Foundation of Tianjin (Nos. 17JCYBJC19400, 24JCZDJC00910).

ORCID iDs

Shuaijie Zhao

Yuhong Du

Jiaqi Li

Xinlong Li

Xiangyu Nie

Weijia Ren

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Prediction of tensile properties in 3D printed GF/RPP composites using MISCSO-BP neural network

Abstract

Keywords

Introduction

Experimental

Material preparation

FDM printing and parameter setting

Tensile property test

Methodology

BP neural network

MISCSO

SCSO

Improvement strategy

Strategy 1: Cubic chaos reverse learning

Strategy 2: Dynamic nonlinear sensitivity range

Strategy 3: Weibull flight strategy

Strategy 4: Gaussian-Cauchy mutation strategy

MISCSO-BP model

Discussion on correlations

The influence of 3D printing parameters on tensile properties

Tensile strength

Elasticity modulus

Elongation at break

Results and discussion of the MISCSO-BP model

Testing and discussion on the MISCSO algorithm

Test function evaluation

Ablation study on MISCSO

Analysis of prediction results

Comparative analysis of methods

Stability analysis of MISCSO-BP model

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References