Carlos Serrano
| Nestor Ulloa* | Christian Flores | Fausto Caicedo
© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
The novel class of composite materials known as polyethylene-carbon nanotube composites (PECNTs) has attracted significant interest from scientists. In this study, authors investigated how artificial intelligence (AI) is employed to calculate the elastic modulus of PECNTs. For the first time, an AI-based modeling methodology replaces nanoindentation techniques like depth sensing indentation (DSI). This study highlights the complexities inherent in traditional methods, where the proposed methodology utilizes a gene expression programming (GEP) model, addressing challenges associated with accuracy in PECNT simulation. The proposed AI model test uses 135 input/output data pairs taken from the literature and randomly split into 82 training and 53 testing sets. The elastic modulus (EM) whichever dynamic E′ or quasi-static E) employs as an output factor in the models created, with the method of analysis, matrix type, processing technique, nanofiller type, and its content serving as inputs. Though the modeling progression is complete with results from the training and testing sets, the nanometer sensitivity of the prominent designs of the AI model displayed significant promise for the effective application of artificial intelligence methods in measuring the elastic modulus of PECNTs through non-destructive testing.
artificial intelligence, carbon nanotubes, depth sending indentation, elastic modulus, mechanical performance, polyethylene
Depth sensing indentation (DSI) stands out as an essential mechanical method for material characterization [1]. The dynamic surface indentation (DSI) technique meticulously measures and records the depths of indentation and retraction of a penetrator while applying and removing a load on a standard material [2, 3]. The approach known as "nanoindentation" leverages DSI at a penetration depth of just a few microns [4, 5]. The DSI testing process involves distinct phases: loading, holding, and unloading, as illustrated (Figure 1). The analysis of the load-depth curve is carried out using elastic contact principles [6]. Estimating the quasi-static elastic modulus E is feasible due to the linear elastic behavior at the beginning of the unloading phase [7-9]. Instrumented indentation, by applying minimal focused deformation, facilitates the study of the material's mechanical properties [10]. Although limited research systems exist, particularly those with highly restricted parameters, can be beneficial, as in the case of thin films and coatings [11, 12]. However, DSI goes beyond by offering the capability to spatially resolve mechanical characteristics in heterogeneous materials, providing additional benefits, as seen in the case of polyethylene nanocomposites.
The incorporation of nanofillers into polyethylene materials yields a significant advantage in optical, electrical, mechanical, thermal, and fire-retardant qualities [13, 14]. These properties make such materials particularly useful in sectors such as electronics, automobiles, and aviation. Specifically, carbon nanotubes (CNTs) have attracted considerable interest as a potentially optimal nanofiller for polyethylene due to their exceptional electrical and mechanical characteristics [15, 16]. In this context, depth-sensing indentation emerges as the method of choice for evaluating the mechanical factors of polyethylene materials, thanks to its higher signal accuracy and examination of reduced material volumes [17-19]. However, to further optimize its utility, it is essential to emphasize the need for a smoother transition in the narrative, especially when transitioning from the discussion of DSI to exploring the potential of carbon nanotubes in polyethylene composites [20]. This improvement in narrative coherence and flow will help effectively contextualize the importance of artificial intelligence (AI) in enhancing the current limitations of DSI, providing a clearer and more comprehensive view of the presented research [21, 22].
Mechanical and physical characteristics of material are evaluated with the help of several AI-based methods, such as ANN and GEP [23, 24]. For many GEP models, projecting forward from current conditions is the ultimate goal [25, 26]. It's essential to consider the impact of input and output variables and product quality early in the process planning phase [27-29].
The GEP has several benefits but infrequently engages in the pitch of science materials [30, 31]. Applications of GEP in materials engineering range from predicting the synthesis time of materials to modeling binding strength in polyethylene to simulating the hardness of nanocomposites. The current work showed polyethylene-carbon nanotube composites' elastic module properties using GEP (PECNTs) [32]. The term "nanoindentation" has never before applying to a simulation [33, 34]. Across a suitable scope, this study introduced a safe approach to ascertain the elastic modulus (quasi-static) of polyethylene-carbon nanotube composites. The model outcomes suggest that the proposed method could serve as a viable substitute for the depth sensing indentation technique.
Figure 1. The load exerted on a semi-crystalline PEEK specimen versus indentation depth (degree of crystallinity of 40%)
2.1 AI in properties of composite materials
Recent progress in the use of artificial intelligence (AI) techniques has highlighted the significance of traditional machine learning (ML) and deep learning (DL) approaches for the prediction of composite materials' mechanical properties. These approaches have been effectively used to predict critical mechanical properties accurately. However, their success largely hinges on data availability and the efficiency of the learning models [35]. The performance of composite materials can be greatly affected by their microstructure and composition. While most existing techniques focus on predicting properties based on either microstructure or material components, there is often a lack of comprehensive data utilization. This challenge has prompted the creation of a new multimodal material performance prediction model that incorporates a network for extracting features from material microstructures [36].
2.2 Analysis of method
The elastic modulus derived from the methods often used to interpret load-depth data from DSI is highly accurate. In this work, authors have included the most crucial of these approaches as input parameters in the model—the model developed by Oliver and Pharr. Following the above inspections and measurements, a plot of applying load vs. penetration depth attain from a rough force curve. The following equation describes the relationship between the methodology, reduced modulus (Er), contact stiffness (S) as well as the predicted contact area (A):
${{E}_{r}}=\frac{\sqrt{\pi }}{2\beta }\frac{S}{\sqrt{A}}$ (1)
where,
$\frac{1}{{{E}_{r}}}=\frac{\left( 1-{{v}^{2}} \right)}{E}+\frac{\left( 1-v_{i}^{2} \right)}{{{E}_{i}}}$ (2)
$E$: elastic modulus f the material
$v$: Poisson's ratio of the material
${{E}_{i}}$: indenter of elastic modulus
${{v}_{i}}$: indenter of Poisson's ratio
The Oliver and Pharr method overestimates the contact area and underestimates the modulus of elasticity when used to pileup materials (where the pile-up is constrained) [37].
2.3 Dynamic strategy
As an alternate method for investigating the time-dependent behavior of polyethylene, dynamic indentation testing (or continuous stiffness measurements, CSM) offers exact results. During a dynamic indentation test, the quasi-static loading profile encloses an additional cycle (sinusoidal oscillation). Throughout the tests, the contact stiffness determines by measuring the amplitude of load, displacement, and phase lag among them. During a dynamic test, stiffness is a constant. The Hertz contact model is employed to illustrate the storage modulus (E') and loss modulus (E"), determined by the ratio of the applied force to the adhesion force [38]. This technique called nano dynamic mechanical analysis (nano DMA), has recently been brought to the marketplace. Specifically, the complex modulus of a viscoelastic material is:
${{E}^{\text{*}}}={E}'+i{E}''$ (3)
whereas,
${E}'=E\text{cos}~\delta $ (4)
and,
${E}''=E\text{sin}~\delta $ (5)
$\delta $: phase shifts among the complex
$E$: standard quasi-static modulus
There is no distinction between quasi-static and dynamic indentation moduli in the DSI literature.
2.4 Polyethylene-carbon nanotube composites
Carbon nanotubes are one of the most valuable nanofillers (CNT) to enhance polyethylene matrix. CNTs split into two distinct varieties. The initial type, known as single-walled carbon nanotubes (SWCNTs), consists of a single graphene sheet wrapped into a cylindrical tube with a diameter ranging from 0.7 to 3 nm. Figure 2 depicts a multi-walled carbon nanotube (MWCNT), which consists of more than two coaxial cylinders made from single sheets and has a diameter ranging from 2 to 40 nm. CNTs produced through diverse techniques such as arc discharge, laser ablation, and chemical vapor deposition, hydrothermal synthesis, electrolysis, and the solar process [39, 40]. The results of these multiple approaches vary significantly in terms of their purity and quality. Therefore, the "nanofiller type" can be employed as a second model parameter.
Figure 2. Transmission electron microscopic of carbon nanotubes
Nanocomposites created by combining polyethylene and carbon nanotubes. To complete the process, the model needs to consider the production process for polyethylene-carbon nanotube composites [41, 42]. One of the biggest challenges in developing and using filled polyethylene is getting a uniform distribution of Carbon Nanotubes in the matrix by eliminating the Vander Waals interface among the individual tubes [43, 44]. Due to their high aspect ratio and non-Brownian nature, CNTs are not distributed uniformly in a polyethylene matrix. The reinforcing CNTs' inability to transfer loads efficiently across the CNT/matrix interface is mixed because they are passive and can only contact the matrix around them through van der Waals forces. Various methods, such as mechanical dispersion techniques like ultrasonication, ball-milling, plasma treatment, and chemical modification, are employed to address these two challenges. Pulverization, densification, and coagulant spinning are projects for preparing polyethylene/CNT nanocomposites. Milling; Melt blending; in situ polyethyleneization; and latex technology.
2.5 Gene expression programming theory
Genetic programming (GP), which draws inspiration from Darwin's idea of natural selection, is a rapidly expanding subfield of evolutionary algorithms [45]. It is a subfield of supervised machine learning that examines existing code instead of looking at data.
GP used the following three methods for developing software that solves problems:
(1). Repeat the following stage still the final condition attains.
(2). Create a random sample of computer programs with the problem's primitive functions and terminals.
(3). The best database from the population created throughout the run is identified as the product of genetic programming.
Figure 3. Expression tree and accompanying mathematical equation for a chromosome containing two genes
Evolutionary algorithms based on populations are called gene expression programming (GEP). People in GEP view them as either non-linear entities of varying sizes and forms or linear strings of a constant length (genome). Authors refer to these things as "expression trees" (ETs) [46, 47]. Each person has a single chromosome, and each gene on any given chromosome divides between the chromosome's head and tail. Genetic expression transcripts (ETs) are selected based on their fitness value to produce offspring.
Consider the algebraic statement $\left[ \left( b\times a \right)-c \right)\left] +\sqrt{\left( d-e \right)} \right]$. Figure 3 shows that a language with a linguistic basis in ETs classified as having either a chromosome with only two genes or an ET.
2.6 AI model
Gene expression programming (GEP) theory represents a powerful methodology in the field of artificial intelligence. This theory is based on principles inspired by genetics and evolution to create robust predictive models. In the design of a model using GEP, genetic program structures are employed that evolve and adapt over time. These programs, expressed in the form of expression trees, capture the complexity of genetic interactions and allow for modeling complex phenomena. GEP theory has been successfully used in predicting the mechanical properties of composite materials, highlighting its ability to address complex problems and extract meaningful patterns from data.
2.7 Data selection
Table 1 presents the compiled information from the preceding works. In this investigation, a Gene Expression Programming (GEP) model was employed to predict the elastic modulus of polyethylene-CNT composites, serving as the primary objective or cost function [48]. The possible values for the input factors are listed in Table 2. Further details regarding the results shown in Table 1 can be found in Table 3. For instance, "Raw MWCNTs" would be listed as the first option under "nanofiller type" (1).
Table 1. The collected data serves as input and target for training and testing sets obtained from previous studies
|
Matrix Type |
Kinds of Nanofi |
Processing Techni |
Content of Nanofiller (wt%) |
Analysis Method |
DSI Modul US (GPa) |
Ref. |
|
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 4 4 5 5 5 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 11 11 11 11 11 11 11 12 12 12 12 13 13 14 14 14 14 14 15 15 16 16 16 16 16 16 17 17 17 17 17 17 |
1 1 1 1 1 1 9 9 9 9 9 9 2 2 2 2 1 1 1 1 2 2 2 2 2 1 1 4 4 4 5 5 6 6 7 8 8 8 8 8 8 10 10 10 10 10 10 10 11 11 11 12 12 12 12 13 14 14 14 15 15 15 15 14 14 14 1 1 1 1 1 1 1 16 16 16 16 17 17 17 17 17 18 18 18 18 18 18 1 1 19 19 19 19 17 17 17 17 17 17 17 1 1 20 20 20 21 21 21 21 22 22 23 23 1 1 2 2 2 2 2 24 24 20 20 20 20 20 20 25 25 25 25 25 25 |
1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 4 4 4 5 5 3 3 3 5 5 5 5 3 6 6 6 6 6 6 7 7 8 9 9 9 9 10 10 10 2 2 2 2 11 12 12 12 13 13 13 13 12 12 12 14 14 14 14 14 14 14 15 15 15 15 16 16 16 16 16 17 17 17 17 17 17 18 18 13 13 13 13 19 19 19 19 19 19 19 20 20 21 21 21 12 12 12 12 22 22 22 22 23 23 24 24 24 24 24 25 25 21 21 21 21 21 21 26 26 26 26 26 26 |
0 0.25 0.75 0 0.25 0.75 0 0.1 0.5 1 3 5 0 0.1 0 3 0 0.1 0.5 1 0 5 0 1 2 0 1 0 5 5 0 1 0 1 0 0 1 3 0 1 3 0 0.5 0 0.75 5.5 14.2 14.2 0 0.75 3 0 1 3 5 0 0 1 10 0 0.1 0.2 0.5 0 1 10 0 0.4 1 1.3 1.6 2.2 2.4 0 0.2 0.5 1 0 1 2.5 5 7.5 0 1.25 2.5 3.75 5 6.25 0 0.5 0 0.05 0.1 0.2 0 0.5 1 1.5 2 2.5 5 0 0.3 0 0.25 0.5 0 0.2 0.4 0.6 0 1 0 1 0 4 0 2 4 6 15 0 0.5 0 0.5 1 2 4 8 0 1 3 5 2 4 |
1 1 1 1 1 1 3 3 3 3 3 3 2 2 2 2 2 2 2 2 3 3 2 2 2 2 2 3 3 3 2 2 2 2 2 2 2 2 3 3 3 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 2 2 2 2 2 2 2 2 2 2 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 2 2 2 2 2 2 2 1 1 1 1 1 2 2 2 2 3 3 3 3 2 2 2 2 2 2 2 3 3 1 1 1 1 1 1 2 2 2 2 2 2 |
1.81 2.2 2.6 2.2 2.5 2.05 3.9 3.78 3.98 4.06 4.25 4.56 4.22 4.7 4.53 4.75 3.16 3.19 3.35 3.44 3.7 7.4 2.26 3.1 3.64 3 3 3.7 7.55 4.25 3 3 3 3 4.22 4 4.4 5.6 4 5 5.9 3.6 4 3.6 4.8 4.9 6 8.8 4.8 4.6 5.1 4.9 5.6 6 7 2.23 5.66 7.62 11.74 2.3 2.6 3.15 3.5 0.12 0.15 0.53 1.45 2.22 2.23 1.58 2.64 2.02 2 1.18 1.33 1.6 2.02 0.95 1.56 2.14 2.38 2.66 3.6 4.4 5.8 6.6 7.5 7.8 2.02 2.23 2.75 3.1 4.26 4.35 0.617 0.728 0.808 0.925 1.67 1.75 1.89 0.26 0.31 7 9.4 10.4 0.66 6.93 7.3 7.8 3.9 4.25 3.9 4.6 0.0067 0.0089 4 4.2 4.4 4.5 5.2 5.23 8.41 2 2.24 3.1 2.56 2.96 6 3.6 3.38 3.67 3.53 3.48 3.73 |
[1] [1] [1] [1] [1] [2, 7] [1] [7] [1] [7] [1] [7] [1] [7] [1] [7] [13, 48] [13, 48] [36] [36] [32] [32] [32] [16] [16] [27, 28] [27, 28] [27, 28] [27, 28] [27, 28] [27, 28] [27, 28] [27, 28] [24] [24] [24] [24] [11] [12] [18] [18] [18] [18] [18] [18] [26] [26, 31] [43] [43] [43] [43] [43] [43] [43] [33] [33] [33] [33] [34] [46] [46] [46] [41] [46] [46] [46] [46] [46] [46] [14] [14] [14] [14] [14] [14] [14] [20] [20] [20] [20] [23] [23] [23] [23] [23] [39] [39] [39] [39] [39] [41, 42] [38] [46] [46] [46] [46] [41] [41] [41] [41] [41] [41] [41] [24] [29] [29] [29] [29] [33] [33] [33] [33] [23] [23] [23] [23] [23] [23] [18] [18] [18] [18] [18] [44] [44] [29] [29] [29] [29] [29] [29] [42, 44] [42, 44] [42, 44] [42] |
Table 2. The variety of input and output factors in the GEP model
|
Input |
Output |
|
Nanofiller type Processing method Nanofiller content (wt %) Method of analysis Type of Matrix |
1-25 1-26 0-15 1-3 1-17 |
Table 3. Methods and raw materials used in previous studies
|
Symbol |
Matrix Type |
Nanofiller Type |
Processing Method |
Method of Analysis |
|
1 |
Epoxyacrylate |
Raw MWCNTs |
Sonication/mixing photocuring |
Non-available |
|
2 |
Epoxy |
CVD raw MWCNTs |
Sonication/mixing curing in the oven |
From unloading (OP) |
|
3 |
Poly(3-hydroxybutyrate) |
Raw CVD MWCNTs |
Stirring/ mixing curing in the oven |
Dynamic |
|
4 |
Chitosan |
Acid-functionalized CVD MWCNTs |
Stirring/ mixing curing in a hot press |
- |
|
5 |
Poly(3-hydroxyoctanoate) |
Fluorinated-SWCNTs |
Stirring/ mixing curing in the oven |
- |
|
6 |
Polypropylene |
Silane functionalized SWCNTs |
Solution mixing curing in the oven |
|
|
7 |
Polyamide-6 |
Plasma functionalized CVD MWCNTs |
Mixing/ Curing |
|
|
8 |
Poly (L-lactic acid) |
MBZ functionalized Arc-discharge SWCNTs |
In situ wetting/Realign/curing |
|
|
9 |
Ultra-high molecular weight polyethylene |
Purified laser-grown SWCNTs |
Mechanical densification/capillary-induced wet-ting/curing |
|
|
10 |
Polyvinylidene fluoride |
Aligned CVDMWCNT forests |
Shearmixing/curing |
|
|
11 |
Polyvinyl alcohol |
NT forests CVDMWCNT forests |
Curing |
|
|
12 |
Poly (ether ketone)/GF |
Coiled CVDCNTs |
Solution-casting |
|
|
13 |
Polydimenthylsiloxane |
CVDCNTs grafted silica fiber |
Solution mixing/casting |
|
|
14 |
PC |
RawSWCNTs |
In-situpolyethyleneization compression-molding |
|
|
15 |
Polyimide; PLLA: poly (L-lactic acid) |
PEDOT-PSS functionalized |
Melt-bending/compression molding |
|
|
16 |
Poly (9-vinyl carbazole) |
Acid purifiedCVDMWCNTs |
Solution mixing/electro spinnig |
|
|
17 |
Poly (methyl methacrylate) |
Acid-functionalized MWCNTs |
Sonication/solution-casting |
|
|
18 |
|
Acid-purified Arc-discharge MWCNTs |
Milling/electrostatic spraying |
|
|
19 |
|
Plasma functionalized Arc discharge SWCNTs |
Ball milling/compression molding |
|
|
20 |
|
Raw arc-discharge MWCNTs |
Near-field electrospinning Solution mixing |
|
|
21 |
|
Acid functionalized Arc-discharge SWCNTs |
|
|
|
22 |
|
Arc purified SWCNTs |
Melt-blending/ hot compression |
|
|
23 |
|
PEES-wrapped Arc-purified SWCNTs |
Ultrasonication/curing |
|
|
24 |
|
|
Melt-extrusion |
|
|
25 |
|
Raw Arc-discharge SWCNTs |
Sonication/spin-coating |
|
|
26 |
|
Amide-functionalized CVD MWCNTs |
Solution mixing/spin coating |
|
3.1 Process parameters and GEP
The initial step involves selecting the fitness function for the formulations based on gene expression programming (GEP). First, in Eq. (6), authors may determine the "fi" equation as follows:
${{f}_{i}}=\sum _{j=1}^{{{C}_{i}}}\left( M-\left| {{C}_{\left( i,j \right)}}-{{T}_{j}} \right| \right)$ (6)
whereas,
i is fitness case.
C(i,j) is return value.
j is out of CT. fitness cases.
Tj is the target value to fitness case j.
M is selection range.
One benefit of these fitness functions is that allow the structure to discover the best course of action on its own. The GEP model comprises six input layers, namely the matrix type (d0), nanofiller type (d1), processing technique (d2), nanofiller content (d3), and method of analysis (d4). The output layer is characterized by the application of the elastic modulus (either quasi-static E or dynamic, E′). The models listed in Table 1 incorporate a range of input parameters.
Figure 4 shows the expression trees generated by the GEP method developed in this study for predicting the values of the elastic modulus (EM) of PECNTs. Within this diagram, d0, d1, d2, d3, and d4 correspond to the aforementioned input layers. Specifically, d0 represents the matrix type, d1 signifies the type of nanofiller, d2 denotes the processing method, d3 indicates the nanofiller content, and d4 represents the analysis method.
A total of 135 experimental sets became rid of the literature; 82 selects for the training phase, and the remaining 53 sites went through their paces in the testing phase. Genetic evolution used the training data as its input. Elastic modulus was determined by evaluating GEP's best model against data not used in the initial model generation. The model with the highest EM on the practicing and experimenting value sets is selected based on these evaluations.
Figure 4. Predicting the elastic modulus of PECNTs using an expression tree including five genes
Table 4. Factors of the GEP approach model
|
Factor Definition |
Values |
|
Gene transposition rate Chromosomes Weight of functions Head size Inversion rate Number of genes Linking function Gene recombination rate Mutation rate One-point recombination rate Two-point recombination rate Lower bound Upper bound Constants per gene |
0.1 30 7 10 0.1 5 Multiplication 0.1 0.044 0.3 0.3 10 10 5 |
All modeling methodologies strive to accurately predict the target parameter in the output layer, whether by maximizing the root mean square error (RMSE) or minimizing the absolute percentage error (MAPE). Only if four genes are employed, and the connecting function employs multiplication, will these conditions be satisfied (x). It reveals that 30 chromosomes were present in the most predictive cohort for the elastic module. The GEP values that determine for this study are in Table 4.
As the second stage in chromosome construction, the pivotal task is to choose the set of terminals (T) and the set of functions (F). In this instance, the terminal set comprises the independent variables: Elastic modulus value = d0, d1, d2, d3, d4.
Some elementary functions (ln, sin, cos, Arctan, Exp, √${{\text{x}}^{2}},{{\text{x}}^{3}}$,) and the four elementary operators (+), (-), (x), and (/) are measured because selecting the best function set can be challenging for modeling. For ${{f}_{s}}$, an explicit model based on the GEP technique was used to obtain a formulation.
$EMV=f\left\{ d0,d1,d2,d3,d4 \right\}$ (7)
It is crucial to describe the principles for evaluating the results of the model, which are the model's performance and the accuracy of its predictions. The model's efficacy calculates across a variety of statistical scales. Comparing experimental (target) and predicted values, we employed criteria such as root-mean-squared error (RMSE), R-squared (R2), and mean absolute error (MAE).
$MAE=\frac{1}{n}\left[ \frac{\mathop{\sum }_{i=1}^{n}\left| {{t}_{i}}-{{o}_{i}} \right|}{\mathop{\sum }_{i=1}^{n}{{t}_{i}}} \right]$ (8)
$RMSE=\sqrt{\frac{1}{n}\underset{i=1}{\overset{n}{\mathop \sum }}\,{{\left( {{t}_{i}}-{{o}_{i}} \right)}^{2}}}$ (9)
${{R}^{2}}=\frac{{{\left( n\sum {{t}_{i}}{{o}_{i}}-\sum {{t}_{i}}\sum {{o}_{i}} \right)}^{2}}}{\left( n\sum t_{i}^{2}-{{\left( \sum {{t}_{i}} \right)}^{2}} \right)\left( n\sum o_{i}^{2}-{{\left( \sum {{o}_{i}} \right)}^{2}} \right)}$ (10)
whereas, (t, o, n) denotes the experimented range, the anticipated range, and the total no of data points, respectively. The Root means square statistic indicates how close the model results are to the experimental (target) results; a smaller value indicates an adjacent fit. If the Root Means Square statistical range is extensive, however, the outcome provided by the models diverges significantly from the experimented (target) values.
Eq. (11) follows from Figure 4 and is related to the GEP model.
$EMV=\left( \ln 2.1+\sin \left( \frac{\left( {{d}_{4}}\times \arctan 3.73 \right)}{\sqrt[3]{{{d}_{2}}}} \right) \right)x \\ \sqrt{{{d}_{4}}+\left( \left( \left( {{d}_{0}}+1.11 \right)\times \cos {{d}_{0}} \right)+{{d}_{0}}-1.11 \right)}\\ \sin \left( \ln \sqrt{{{d}_{3}}}+{{\sin }^{10}}{{d}_{0}} \right)\text{co}{{\text{s}}^{2}}\left( d_{3}^{3}-{{\left( \sqrt[3]{4.21-{{d}_{1}}} \right)}^{2}} \right)\\ \left( \sqrt[3]{{{d}_{2}}}-\text{cos}\left( \left( {{d}_{2}}+\text{arctan}3.29 \right)-{{d}_{0}} \right) \right)$ (11)
Table 5. Evaluation of training and testing data statistically
|
Statistics |
GEP Model |
|
|
Training |
Testing |
|
|
R2 Minimizing the absolute Percentage error Root mean square error |
0.99 0.43 0.31 |
0.95 0.76 0.69 |
Training and testing outcomes of the GEP model, along with the results of experimental studies, are depicted in Figure 5(a) and (b) respectively. These figures showcase the linear least square fit line, its computation, and the corresponding R2 values for both datasets. During both the training and testing phases, these findings effectively forecast the elastic modulus values for polyethylene-CNT composites—Table 5 illustrates the R2, MAPE, and RMSE values for both the training and testing sets. The EM values of polyethylene-Carbon Nanotube composites forecasts with the suggested GEP model. They are near the experimental values, as shown by the complete R2, MAPE, and RMSE values.
Results indicate that the modeling technique suggested in this project is a replacement for the Deep Sensing Indentation technique to calculate the elastic modulus of PECNTs with a high correlation to the experimental ones.
(a) During training
(b) During testing
Figure 5. Comparing the accuracy of the GEP model’s predicted and observed values for elastic modules during (a) training and (b) testing
3.2 Statistical analysis of the AI model's performance
The statistical analysis of the performance of the artificial intelligence (AI) model, based on the data provided in Table 5, reveals outstanding results in both the training and testing sets. In the training set, the gene expression programming (GEP) model demonstrates an exceptionally high coefficient of determination (R2) of 0.99, indicating outstanding ability to explain variability in the data used during training. Additionally, it minimizes the absolute percentage error to a low value of 0.43, signaling significant accuracy in predictions. The root mean square error (RMSE) in the training set is 0.31, demonstrating a well-fitted adjustment to the training data. These results underscore the GEP model's ability to adapt and fit well to the data used for training.
The R2 achieves a high value of 0.95, indicating that the model effectively generalizes to unseen data during training. The absolute percentage error in the testing set is 0.76, slightly higher than the training set but still acceptable, suggesting consistent accuracy in predicting previously unused data. The RMSE in the testing set remains low, with a value of 0.69, emphasizing the model's ability to effectively forecast in new observations.
3.3 Compare AI predictions with traditional methods
In comparison to traditional methods used to simulate the behavior of polyethylene-carbon nanotube composites (PECNT), this research highlights the inherent complexity in such conventional approaches. The challenges associated with accuracy in simulating PECNT through traditional methods are addressed by the proposed method, employing a gene expression programming (GEP) model. The promising results of the GEP model in determining the elastic modulus (quasi-static) of PECNT suggest that it could overcome limitations observed in traditional methods, providing a more precise and effective alternative to tackle the complexity of these compounds. The application of GEP, with its evolutionary approach, could represent a significant advancement in the simulation of composite materials, surpassing the inherent limitations of traditional methods.
Since polyethylene-carbon nanotube composites (PECNTs) are complicated, accurately simulating their behavior is challenging to model. This research establishes a reliable method for determining the elastic modulus (quasi-static) of PECNTs over the relevant range. After applying the proposed method, the model's accuracy demonstrates that it can be a viable alternative to the DSI approach. As the connecting function, multiplication employs the GEP model, five sub-expression trees, thirty chromosomes, and a head size of ten. Based on the obtained data, it appears that GEP can serve as a viable alternative method for determining the EM values of PECNTs. The optimal values for R2, minimizing absolute percentage error, and root mean square error in the training set are 0.99, 0.43, and 0.31, respectively. In the testing set, these values are 0.95, 0.76, and 0.69. The GEP model demonstrates its ability to provide accurate predictions for the EM values of PECNTs in the measured data, as evidenced by the favorable comparisons of R2, MAPE, and root mean square error values.
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