# Machine Learning and Statistical Methods for Studying Voids and Photothermal Effects of a Semiconductor Rotational Medium with Thermal Relaxation Time

Authors: Taghreed M. Jawa; Azhari A. Elhag; Tahani A. Aloafi; Neveen Sayed-Ahmed; F. S. Bayones; Jamel Bouslim
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Machine learning is the process of creating algorithms that extract useful facts from data automatically. The goal of this paper is to use an artificial neural network and a cubic spline model to predict various physical quantities displacement components in a thermoplastic solid, such as elastic waves, vector form, volume fraction field, thermal waves, stress components, and carrier density concentration (plasma waves). The mean absolute scaled error (MASE), the mean absolute percentage error (MAPE), and the symmetric mean absolute percentage errors (SMAPE) are used to compare the accuracy of two models. The true displacements are given their maximum expected values. These factors have also been described using various descriptive statistics and diagrams. Statistical significance was found in the examination of the correlation between the variables, and a comparison was conducted between the findings and prior results acquired by others. The findings show that voids, rotation, optical temperature, and thermal relaxation all have a significant impact on the phenomena, and they are in line with earlier physical findings. Furthermore, it is demonstrated that certain physical variables describing such systems may display this property, allowing for the development of an analytical criterion for the advent of dynamical chaos.


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## Body

## 1. Introduction

The interaction between elastic bodies and the heat field is known as thermoelasticity, and it has been examined in several studies. When the thermal relaxation period is fixed, void and photothermal properties on a semiconductor rotational surface have recently been examined [1]. Bayones investigated the influence of rotation on the composite of an infinite cylinder in viscoelastic media with nonhomogeneity [2]. El-Haina et al. studied the thermal buckling of thick functionally graded (FG) Sandwich plates using a simple analytical approach [3]. Menasria et al. also established a new and simple higher shear deformation theory (HSDT) for the thermal stability analysis of functionally graded (FG) Sandwich plates [4]. Attia et al. investigated the thermoelastic analysis of functionally graded material (FGM) plates sitting on variable elastic foundations using an improved four variables plate theory. Furthermore, the effect of rotation and beginning load on a generalized thermoelastic issue in an infinite circular cylinder was also studied [5]. In the context of external forces, some innovative contributions investigated the photothermal and semiconducting influences on void media [6–9]. More information on media with microstructures and voids can be found in [10–12].In the theory of thermoelasticity, statistical techniques have lately been applied. The prediction objectives employ the machine learning approach and the cubic spline model. The artificial neural network (ANN) as a machine learning method has recently been employed in several studies to create a prediction model for dependent variables. An artificial neural network technique was used to improve the prediction model for concrete chloride diffusion coefficient and evaluate chloride resistance in various environments [13]. Furthermore, in the research of nonionic and anionic individual surfactant adsorption behavior, an artificial neural network prediction model reduces the cost of operation and experiment time. In addition, when compared to the binary model [14], it provided accurate predictions. Furthermore, the ANN technique predicted carbonyl groups with good accuracy during photothermal and thermal ageing of polymers [15]. Cubic spline models, on the other hand, are utilized for prediction. The cubic spline model [16] can be used to assess nonlinear relationships between predictors and dependent variables. To overcome the problem of data quality for prediction, cubic spline interpolation and cubic B-spline interpolation were used. The cubic spline models considerably improved prediction accuracy [17, 18]. Various studies used different mathematical methods to investigate the influence of thermal relaxation durations on various modes. Lord–Shulman (L-S) and the dual-phase-lag (DPL) theories were considered to study the effect of photothermal, voids, rotation, initial stress, the process of semiconductor, and the thermic relaxation durations [19, 20]. However, statistical approaches have been utilized in a limited amount of research to investigate the effects of thermal relaxation durations.The goal of this paper is to predict the displacement components of elastic waves (u), vector form (v), volume fraction field (V), temperature (thermal waves) (T), stress components σxxandσxy, and carrier density concentration in a thermoelastic solid (plasma waves) using an artificial neural network as a machine learning model and the cubic spline model (N). The rotating, semiconducting, and photothermal effects discussed in [1, 5] have been generalized in this study. The symmetric mean absolute percentage errors (SMAPE), mean absolute scaled error (MASE), and mean absolute percentage error (MAPE) are also generated to compare the accuracy of two models. Furthermore, the real displacement components’ highest predicted values are obtained. Moreover, descriptive statistics and figures are examined to describe these variables. Because of their applications in physics, biology, and artificial intelligence, the behavior of artificial neural networks and cubic spline models has been explored to resolve differential equations.Finally, the simplest linear physical models were used to characterize the dynamics of harmonic oscillator behavior as a semichaotic that decreases to zero as the axial approaches’ infinity. Some physical factors that characterize such systems allow for the development of an analytical criterion for the commencement of chaotic behavior [21, 22]. Abouelregal discussed the magnetophotothermal interaction in a rotating solid cylinder of semiconductor silicone material under heat flow [23]. Also, Abouelregal investigated a rotating semiconductor considering a modified fractional photothermoelastic model for half-space under a magnetic field [24]. Wang et al. pointed out harnessing deep neural networks to solve inverse problems in quantum dynamics: machine-learned predictions of time-dependent optimal control fields [25]. Zhang et al. discussed the problem of accurate prediction and further dissection of neonicotinoid elimination in the water treatment, considering the neural network combined with the time-dependent Cox regression model [26].


## 2. Statistical Methods

### 2.1. Multilayer Perceptron (MLP)

In machine learning approach, a multilayer perceptron (MLP) is an artificial neural network (ANN) that produces a set of outputs from a set of inputs. The multilayer perceptron [27, 28] is a method for generating prediction models for one or more dependent variables that are linked to one or more predictors. Multilayer perceptron networks are layered networks that are often trained using a specific process called constant backpropagation. The input layer, hidden layer, and output layer [29, 30] are three layers of nodes in the MLP network. The procedure of an artificial neural network (ANN) is to change the weights for a specific training set in order to accurately distribute the provided input patterns. During the network's learning stage, the weights are calculated. Throughout the learning phase, the feedforward of input training, the computation of error, and finally the calculation of weights are required for network training. After training, the network may create outputs incredibly quickly [31]. Using the Statistical Package for Social Sciences, artificial neural networks are used to build more accurate and effective predictive models in this study (SPSS). An ANN model’s architecture is described by one input layer and eight output layers.


### 2.2. Cubic Spline

A cubic spline is a piecewise function of three-degree polynomials. Cubic splines are chosen over traditional polynomial interpolation methods in interpolation situations because they provide a compromise between the smoothness of the curve and the degree of the polynomials. A cubic spline is a piecewise function defined on the interval [a, b] and divided into k intervals [xi−1,xi], with the property that(1)a=x0<x1<⋯<xk−1<xk=b.It is defined by the polynomial of degree 3 on the interval [a, b]. The spline S is given by(2)St=P1tift∈x0,x1,..St=Pktift∈xk−1,xk.P i (t), i = 1, 2, ..., k are three-degree polynomials in the interval [xi−1, xi]. The derivatives of polynomials are used to calculate the coefficients of the cubic spline [32]. The anticipated values of the real and imaginary parts of the displacement components are determined using the SPSS software after cubic spline models are fitted to the data.


### 2.3. The Accuracy Measurements

The proposed models’ performance is evaluated using standard accuracy measurements. In (3), (4), and (5), the symmetric mean absolute percentage error (SMAPE), mean absolute scaled error (MASE), and mean absolute percentage error (MAPE) will be calculated [33, 34].(3)SMAPE=1n∑i=1neiyi+y^i,(4)MASE=1/n∑i=1nei1/n−1∑i=2nyi−yi−1,(5)MAPE=1n∑i=1neiyi.Here, ei is a residual value, yi is an observed value, and y^i is a predicted value.


## 2.1. Multilayer Perceptron (MLP)

In machine learning approach, a multilayer perceptron (MLP) is an artificial neural network (ANN) that produces a set of outputs from a set of inputs. The multilayer perceptron [27, 28] is a method for generating prediction models for one or more dependent variables that are linked to one or more predictors. Multilayer perceptron networks are layered networks that are often trained using a specific process called constant backpropagation. The input layer, hidden layer, and output layer [29, 30] are three layers of nodes in the MLP network. The procedure of an artificial neural network (ANN) is to change the weights for a specific training set in order to accurately distribute the provided input patterns. During the network's learning stage, the weights are calculated. Throughout the learning phase, the feedforward of input training, the computation of error, and finally the calculation of weights are required for network training. After training, the network may create outputs incredibly quickly [31]. Using the Statistical Package for Social Sciences, artificial neural networks are used to build more accurate and effective predictive models in this study (SPSS). An ANN model’s architecture is described by one input layer and eight output layers.


## 2.2. Cubic Spline

A cubic spline is a piecewise function of three-degree polynomials. Cubic splines are chosen over traditional polynomial interpolation methods in interpolation situations because they provide a compromise between the smoothness of the curve and the degree of the polynomials. A cubic spline is a piecewise function defined on the interval [a, b] and divided into k intervals [xi−1,xi], with the property that(1)a=x0<x1<⋯<xk−1<xk=b.It is defined by the polynomial of degree 3 on the interval [a, b]. The spline S is given by(2)St=P1tift∈x0,x1,..St=Pktift∈xk−1,xk.P i (t), i = 1, 2, ..., k are three-degree polynomials in the interval [xi−1, xi]. The derivatives of polynomials are used to calculate the coefficients of the cubic spline [32]. The anticipated values of the real and imaginary parts of the displacement components are determined using the SPSS software after cubic spline models are fitted to the data.


## 2.3. The Accuracy Measurements

The proposed models’ performance is evaluated using standard accuracy measurements. In (3), (4), and (5), the symmetric mean absolute percentage error (SMAPE), mean absolute scaled error (MASE), and mean absolute percentage error (MAPE) will be calculated [33, 34].(3)SMAPE=1n∑i=1neiyi+y^i,(4)MASE=1/n∑i=1nei1/n−1∑i=2nyi−yi−1,(5)MAPE=1n∑i=1neiyi.Here, ei is a residual value, yi is an observed value, and y^i is a predicted value.


## 3. Formulation of the Problem

It is considered a homogeneous isotropic generalized magnetothermoelastic half-space with initial tension. The key variables with which theorists assess the transport process in a semiconductor are coupled plasma waves, thermal waves, and elastic waves all at the same time.The transport mechanism is demonstrated during the interplay of coupled plasma waves, thermal waves, and elastic waves in a semiconductor medium. The problem was investigated when two-dimensional Cartesian coordinates ([x, z], []) were employed to represent the position vector and the time variablet, respectively. The circular plate is thought to be exceedingly thin, linear, uniform, and isotropic. When a thermal activation coupling parameter is present, the governing equations are used in the general case. The transport process of linked plasma, heat, magnetic field, starting tension, and elastic features of the medium can all be used to explain this.The basic governing equations of linear generalized thermoelasticity were obtained in [1] as follows:(6)τij,i+Fj=ρu¨+Ω⟶×Ω⟶×u⟶+2Ω×u˙⟶j,i,j=1,2,3,(7)kT,ii+k∗T˙,ii=ρCen1∂∂t+τ0∂2∂t2T+γT0n1∂∂t+n0τ0∂2∂t2∇.u,(8)σij=λuk,k−1+θo∂∂tTδij+2μeij,i,j,k=1,2,3,(9)eij=12ui,j+ui,j,i,j=1,2,3,(10)u=∂ψ1∂x−∂ψ2∂z,w=∂ψ1∂z−∂ψ2∂x,where ρ is the density, ∇2 is the Laplacian operator, δij is the Kronecker delta, γ is the material constant, μ,λ are Lame’s constant, eij is the strain tensor components, σij is the stress tensor components, k is thermal conductivity, r is the beam radius, T is the absolute temperature, u is the displacement vector, Ce is the specific heat at constant strain, k∗ is the material constant characteristic of the (GN) theory, T0 is the reference temperature chosen so that T−T0/T0≤1, Ω is the angular rotation, τ is relaxation time parameter, ψ1 and ψ2 are the scalar and vector potentials, ω is the angular frequency, and x,y,z are the coordinates of the system. The components in equations (6)–(10) which are related to the mentioned factors, voids, rotation, optical temperature, and thermal relaxation, were mentioned in detail in [1].In normal mode analysis, the solutions of the physical variables are decomposed in terms of normal modes in the following form:(11)ψ1x,z,t,ψ2x,z,t,Tx,z,t=ψ1∗x,ψ2∗x,T∗xexpωt+iaz,where ψ1∗x,ψ2∗x,T∗x are the amplitudes of the physical quantities, ω is the angular frequency, and x,y,z are the coordinates of the system. Considering the boundary conditions in [1], the variables are displayed and analysed.


## 4. Data Analysis

The artificial neural network (ANN) and cubic spline model are used to predict the displacement components: elastic waves (u), vector form (v), volume fraction field (ϕv), temperature (thermal waves) (T), stress components (σxx and σxy), and carrier density concentration in a thermoelastic solid (plasma waves) (N) [1]. The data set includes the real (Re) and the imaginary (Im) parts of the displacement components.The projected values of real and imaginary portions for all displacement components with varying rotation and photothermal were extracted and saved using the ANN on SPSS application utilizing the multilayer perceptron approach. In addition, the data was fitted with a cubic spline model, and the anticipated values were recorded.In this investigation, a sample size of 100 was used. The training sample makes up 70% of the total, whereas the testing sample makes up 30%. The mean, variance, range, minimum, maximum, and correlation are used in Tables1–7 to describe the behavior (measures of central tendency and variability) of the real (Re) and imaginary (Im) parts of displacement components with a change of rotation under photothermal and nonphotothermal conditions. The boundary conditions (BC) and constants from [1] are considered. The methodology was developed using ANN computations and cubic interpolations, and the physical experiments were proven.Table 1
The descriptive analysis for the real (Re) and imaginary (Im) displacementu with a change of rotation (Ω) under photothermal and nonphotothermal effects.
Re(u)xΩ=0.1 PhotothermalΩ=0.4 PhotothermalΩ=0.1 NonphotothermalΩ=0.4 NonphotothermalMean2.250.00940790.017762440.014702920.02853617Variance1.7390.0020.0030.0020.005Range4.500.18314100.225001000.219758000.28189700Minimum0.00−0.015641−0.0141010−0.0146480−0.0115470Maximum4.500.16750000.210900000.205110000.27035000Correlation betweenx and real displacement1−0.556∗∗−0.636∗∗−0.603∗∗−0.689∗∗Im(u)Mean2.250.03681090.0437773550.0403181470.045819144Variance1.7390.0030.0030.0030.003Range4.500.16155360.1758198500.1620076500.157121090Minimum0.00−0.000764−0.00057985−0.00063765−0.00040109Maximum4.500.16079000.1752400000.1613700000.156720000Correlation betweenx and imaginary displacement1−0.868∗∗−0.888∗∗−0.882∗∗−0.910∗∗Table 2
The descriptive analysis for the real (Re) and imaginary (Im) displacementsv with a change of rotation (Ω) under photothermal and nonphotothermal effects.
Re(v)xΩ=0.1 PhotothermalΩ=0.4 PhotothermalΩ=0.1 NonphotothermalΩ=0.4 NonphotothermalMean2.250.0070650.01165850.00833920.0126411Variance1.7390.0000.0000.0000.000Range4.500.0745280.09072600.08164800.1001720Minimum0.00−0.04229−0.0441300−0.0455750−0.0506870Maximum4.500.0322410.04659600.03607300.0494850Correlation betweenx and real displacement1−0.392∗∗−0.486∗∗−0.384∗∗−0.412∗∗Im(v)Mean2.250.000488−0.0039748−0.0005080−0.0071368Variance1.7390.0000.0000.0000.000Range4.500.0341400.059473000.039825000.06887100Minimum0.00−0.02338−0.0474430−0.0283140−0.0581960Maximum4.500.0107570.012030000.011511000.01067500Correlation betweenx and imaginary displacement10.432∗∗0.615∗∗0.487∗∗0.673∗∗Table 3
The descriptive analysis for the real (Re) and imaginary (Im) displacementϕv with a change of rotation (Ω) under photothermal and nonphotothermal effects.
xΩ=0.1 PhotothermalΩ=0.4 PhotothermalΩ=0.1 NonphotothermalΩ=0.4 NonphotothermalRe (ϕv)Mean2.25−0.085877−0.10799−0.0746369−0.0446341Variance1.7390.0060.0050.0080.006Range4.500.27705700.20198300.357500000.33548000Minimum0.00−0.218040−0.215190−0.2290000−0.1433000Maximum4.500.0590170−0.0132070.128500000.19218000Correlation betweenx and real displacement10.540∗∗0.702∗∗0.419∗∗−0.023Mean2.250.03840650.02420730.200341390.22243681Im (ϕv)Variance1.7390.0250.0140.0960.093Range4.500.5602270.437222000.974992500.95549380Minimum0.00−0.07915−0.0646920−0.0055625−0.0039838Maximum4.500.4810800.372530000.969430000.95151000Correlation betweenx and imaginary displacement1−0.638∗∗−0.634∗∗−0.830∗∗−0.872∗∗Table 4
The descriptive analysis for the real (Re) and imaginary (Im) displacementσxx with a change of rotation (Ω) under photothermal and nonphotothermal effects.
xΩ=0.1 PhotothermalΩ=0.4 PhotothermalΩ=0.1 NonphotothermalΩ=0.4 NonphotothermalRe (σxx)Mean2.25−0.085877−0.10799−0.0746369−0.0446341Variance1.7390.0060.0050.0080.006Range4.500.27705700.20198300.357500000.33548000Minimum0.00−0.218040−0.215190−0.2290000−0.1433000Maximum4.500.0590170−0.0132070.128500000.19218000Correlation betweenx and real displacement10.540∗∗0.702∗∗0.419∗∗−0.023Mean2.250.03840650.02420730.200341390.22243681Im (σxx)Variance1.7390.0250.0140.0960.093Range4.500.5602270.437222000.974992500.95549380Minimum0.00−0.07915−0.0646920−0.0055625−0.0039838Maximum4.500.4810800.372530000.969430000.95151000Correlation betweenx and imaginary displacement1−0.638∗∗−0.634∗∗−0.830∗∗−0.872∗∗Table 5
The descriptive analysis for the real (Re) and imaginary (Im) displacementσxy with a change of rotation (Ω) under photothermal and without photothermal.
xΩ=0.1 (photothermal)Ω=0.4 (photothermal)Ω=0.1 (nonphotothermal)Ω=0.4 (nonphotothermal)Re (σxy)Mean2.25−0.085877−0.10799−0.0746369−0.0446341Variance1.7390.0060.0050.0080.006Range4.500.27705700.20198300.357500000.33548000Minimum0.00−0.218040−0.215190−0.2290000−0.1433000Maximum4.500.0590170−0.0132070.128500000.19218000Correlation betweenx and real displacement10.540∗∗0.702∗∗0.419∗∗−0.023Mean2.250.03840650.02420730.200341390.22243681Im (σxy)Variance1.7390.0250.0140.0960.093Range4.500.5602270.437222000.974992500.95549380Minimum0.00−0.07915−0.0646920−0.0055625−0.0039838Maximum4.500.4810800.372530000.969430000.95151000Correlation betweenx and imaginary displacement1−0.638∗∗−0.634∗∗−0.830∗∗−0.872∗∗Table 6
The descriptive analysis for the real (Re) and imaginary (Im) displacementsT with a change of rotation (Ω) under photothermal and without photothermal.
xΩ=0.1 (photothermal)Ω=0.4 (photothermal)Ω=0.1 (nonphotothermal)Ω=0.4 (nonphotothermal)Re (T)Mean2.25−0.085877−0.10799−0.0746369−0.0446341Variance1.7390.0060.0050.0080.006Range4.500.27705700.20198300.357500000.33548000Minimum0.00−0.218040−0.215190−0.2290000−0.1433000Maximum4.500.0590170−0.0132070.128500000.19218000Correlation betweenx and real displacement10.540∗∗0.702∗∗0.419∗∗−0.023Im (T)Mean2.250.03840650.02420730.200341390.22243681Variance1.7390.0250.0140.0960.093Range4.500.5602270.437222000.974992500.95549380Minimum0.00−0.07915−0.0646920−0.0055625−0.0039838Maximum4.500.4810800.372530000.969430000.95151000Correlation betweenx and imaginary displacement1−0.638∗∗−0.634∗∗−0.830∗∗−0.872∗∗Table 7
The descriptive analysis for the real (Re) and imaginary (Im) displacementsN with a change of rotation (Ω) under photothermal and nonphotothermal effects.
xΩ=0.1 (photothermal)Ω=0.4 (photothermal)Ω=0.1 (nonphotothermal)Ω=0.4 (nonphotothermal)Re (N)Mean2.250.010.0126−0.0241−0.2016Variance1.7390.000.00000.3750.230Range4.500.020.03002.7502.310Minimum0.000.000.0000−0.450−0.610Maximum4.500.020.03002.30001.700Correlation betweenx and real displacement1− 0.984− 0.983− 0.586− 0.390Im (N)Mean2.250.0010−0.0011−1.0760−1.1098Variance1.7390.00000.0001.4851.8110Range4.500.01000.0004.1404.6800Minimum0.000.0000.000−4.110−4.6300Maximum4.500.0000.0000.0300.05Correlation betweenx and imaginary displacement1−0.932− 0.7460.9220.907Figures1–7 demonstrate the change in observed values of the real and imaginary parts of the horizontal displacement components as a function of x for various values of rotational effect in the presence or absence of photothermal effects. Figures 1–7 demonstrate the difference in the predicted values of the real and imaginary parts of the horizontal displacement components as a function of x, with different values of rotational impact while considering the presence or absence of photothermal effects. Furthermore, the maximum anticipated value range for each displacement component is determined. The maximum predicted value is the highest point in the curve, and it is calculated to infer the behavior of the function.Figure 1
The displacementu as a function of (x) with a change of rotation and photothermal and its prediction. (a) The real part of the horizontal displacement component u as a function of distance x. (b) The imaginary part of the horizontal displacement component u as a function of distance x. (c) The prediction of the real part of the horizontal displacement component u using ANN. The maximum predicted value is a range of between 0.14741 and 0.26007. (d) The prediction of the imaginary part of the horizontal displacement component u using ANN. The maximum predicted value is a range of between 0.17847 and 0.17941. (e) The prediction of the real part of the horizontal displacement component u using cubic spline is made. The maximum predicted value is a range of between 0.14741 and 0.2600. This is a common misunderstanding. (f) The prediction of the imaginary part of the horizontal displacement component u using cubic spline is made. The maximum predicted value is a range of between 0.17847 and 0.17941.
(a)(b)(c)(d)(e)(f)Figure 2
The displacementv as a function of (x) with a change of rotation and photothermal. (a) The real part of the horizontal displacement component v as a function of distance x. (b) The imaginary part of the horizontal displacement component v as a function of distance x. (c) The prediction of the real part of the horizontal displacement component v using ANN. The maximum predicted value is a range of between 0.023833 and 0.0.040835. (d) The prediction of the imaginary part of the horizontal displacement component v using ANN. The maximum predicted value is a range of between 0.05831 and 0.006057. (e) The prediction of the real part of the horizontal displacement component v using cubic spline is made. The maximum predicted value is a range of between 0.02445 and 0.09983. (f) The prediction of the imaginary part of the horizontal displacement component v using cubic spline is made. The maximum predicted value is a range of between 0.00730 and 0.00949.
(a)(b)(c)(d)(e)(f)Figure 3
The concentrationϕv as a function of (x) with a change of rotation and photothermal effect. (a) The real part of the horizontal displacement component ϕv as a function distance x. (b) The imaginary part of the horizontal displacement component as a function of distance x. (c) The prediction of the real part of the horizontal displacement component ϕv using ANN is made. The maximum predicted value is a range of between 0.06683 and 0.18141. (d) The prediction of the imaginary part of the horizontal displacement component ϕv using ANN is made. The maximum predicted value is a range of between 0.498232 and 0.97550. (e) The prediction of the real part of the horizontal displacement component ϕv using cubic spline is made. The maximum predicted value is a range of between 0.00161 and 0.00905. (f) The prediction of the imaginary part of the horizontal displacement component ϕv using cubic spline is made. The maximum predicted value is a range of between 0.10853 and 0.44991.
(a)(b)(c)(d)(e)(f)Figure 4
The normal stressσxx as a function of (x) with a change of rotation and photothermal. (a) The real part of the horizontal displacement component σxx as a function of distance (x). (b) The imaginary part of the horizontal displacement component σxx as a function of distance (x). (c) The prediction of the real part of the horizontal displacement component σxx using ANN is made. The maximum predicted value is a range of between 0.00 and 0.1. (d) The prediction of the imaginary part of the horizontal displacement component σxx using ANN. The maximum predicted value is a range of between 0.1 and 0.00. (e) The prediction of the real part of the horizontal displacement component σxx using cubic spline is made. The maximum predicted value is a range of between 0.086678 and 0.12471. (f) The prediction of the imaginary part of the horizontal displacement component σxx using cubic spline is made. The maximum predicted value is a range of between 0.05675 and 0.03448.
(a)(b)(c)(d)(e)(f)Figure 5
The normal stressσxy as a function of (x) with a change of rotation and photothermal. (a) The real part of the horizontal displacement component σxy as a function of distance (x). (b) The imaginary part of the horizontal displacement component σxy as a function of distance (x). (c) The prediction of the real part of the horizontal displacement component σxy using ANN is made. The maximum predicted value is a range of between 0.11 and 0.13. (d) The prediction of the imaginary part of the horizontal displacement component σxy using ANN. The maximum predicted value is a range of between 0.02 and 0.02. (e) The prediction of the real part of the horizontal displacement component σxy using cubic spline is made. The maximum predicted value is a range of between 0.085 and 0.13. (f) The prediction of the imaginary part of the horizontal displacement component σxy using cubic spline is made. The maximum predicted value is a range of between 0.02 and 0.10546.
(a)(b)(c)(d)(e)(f)Figure 6
The temperatureT as a function of (x) with a change of rotation and photothermal. (a) The real part of the horizontal displacement component T as a function of distance (x). (b) The imaginary part of the horizontal displacement component T as a function of distance (x). (c) The prediction of the real part of the horizontal displacement component T using ANN. The maximum predicted value is a range of between −0.00003 and 0.00002. (d) The prediction of the imaginary part of the horizontal displacement component T using ANN. The maximum predicted value is a range of between 0.00014 and 0.00032. (e) The prediction of the real part of the horizontal displacement component T using cubic spline is made. The maximum predicted value is a range of between −0.00003 and 0.00002. (f) The prediction of the imaginary part of the horizontal displacement component T using cubic spline is made. The maximum predicted value is a range of between 0.00014 and 0.00032.
(a)(b)(c)(d)(e)(f)Figure 7
The componentN as a function of (x) with a change of rotation and photothermal. (a) The real part of the horizontal displacement component N as a function of distance (x). (b) The imaginary part of the horizontal displacement component N as a function of distance (x). (c) The prediction of the real part of the horizontal displacement component N using ANN. The maximum predicted value is predicted to range between −0.02 and 1.49. (d) The prediction of the imaginary part of the horizontal displacement component N using ANN is made. The maximum predicted value is a range of between 0.01 and 0.07. (e) The prediction of the real part of the horizontal displacement component N using cubic spline is made. The maximum predicted value is range of between 0.01818 and 0.09107. (f) The prediction of the imaginary part of the horizontal displacement component N using cubic spline is made. The maximum predicted value ranges between 0.00294 and 0.56346.
(a)(b)(c)(d)(e)(f)


## 5. The Accuracy Measurements Results

The displacement components are predicted using an ANN and a cubic spline model. SMAPE, MASE, and MAPE accuracy metrics are utilized using equations (3), (4), and (5), respectively, to investigate the performance of two proposed models: the ANN and the cubic spline model. The accuracy measurements for ANN models of the real and imaginary sections of the horizontal displacement components for cubic spline models are shown in Tables 8 –11. When the accuracy data in Tables 8–11 is compared, the artificial neural network outperforms the cubic spline model [1].Table 8
The accuracy measurements for ANN models of the real (Re) parts of the horizontal displacement components.
Ω=0.1 (photothermal)Ω=0.4 (photothermal)Ω=0.1 (nonphotothermal)Ω=0.4 (nonphotothermal)Re (u)SMAPE3.70E − 054.55E − 054.44E − 055.69E − 05MASE0.6354810.4664030.3501340.629209MAPE0.532990.4374490.3121060.383288Re (v)SMAPE0.120660.112640.114340.36563MASE0.687053230.417623990.407123030.944657MAPE0.3085498240.3771337150.483519350.878681207Re (ϕv)SMAPE0.0527130.21380.3160770.244785MASE0.7124494.0174152.9577287.352439MAPE0.0908210.5770890.554390.689852Re (σxx)SMAPE0.0001680.0002670.0001840.000207MASE0.0715662650.4650.2828571430.246292683MAPE2.34222.34222.34222.3422Re (σxy)SMAPE2.02E − 052.63E − 052.22E − 052.63E − 05MASE0.6930.3426923080.180.875769231MAPE0.5700356.3931110.4438250.627785Re (T)SMAPE5.14444E − 076.60626E − 072.69919E − 081.82626E − 08MASE0.194898130.2024803550.3910274010.666676842MAPE2.34222.34222.34222.3422Re (N)SMAPE4.46909E − 065.70869E − 060.0005551720.000466313MASE0.7434468590.8035138720.7919506410.849342995MAPE0.0474240.0421730.1717830.1717830.162719Table 9
The accuracy measurements for ANN models of the imaginary (Im) parts of the horizontal displacement components.
Ω=0.1 (photothermal)Ω=0.4 (photothermal)Ω=0.1 (nonphotothermal)Ω=0.4 (nonphotothermal)Im (u)SMAPE0.18140.16410.15090.2530MSE1.17280.87920.65751.2108MAPE0.53300.43740.31210.3833Im (v )SMAPE0.123120.1520.166720.10719MASE1.920282251.912509761.997004851.84391622MAPE0.3458785060.4202773450.5954571780.265812308Im (ϕv)SMAPE0.08510.14590.18660.2026MASE0.68581.40090.59090.7221MAPE0.22100.56100.63050.8968Im (σxx)SMAPE7.58E − 055.45E − 055.66E − 054.65E − 05MASE0.40920.3666666670.4242857140.365869565MAPE0.3617330.2213260.270261.197851Im (σxy)SMAPE2.22E − 053.43E − 052.63E − 054.04E − 05MASE1.4851.3394111.4469230771.485MAPE2.8392280.9889991.2238581.917065Im (T)SMAPE1.72414E − 071.19812E − 075.64606E − 085.96303E − 08MASE0.6914069750.9870602310.3612106110.446970393MAPE0.3458785060.4202773450.5954571780.265812308Im (N)SMAPE1.14684E − 066.28596E − 070.0008370080.00094504MASE0.7219116091.0033183640.4066172910.520677279MAPE0.0908210.065330.4230384.207863Table 10
The accuracy measurements for cubic spline models of the real (Re) parts of the horizontal displacement components.
Ω=0.1 (photothermal)Ω=0.4 (photothermal)Ω=0.1 (nonphotothermal)Ω=0.4 (nonphotothermal)Re (u)SMAPE3.70E − 054.55E − 054.44E − 055.69E − 05MSE1.6920831.2434571.5166141.003554MAPE2.9450432.3984022.6899892.078128Re (v )SMAPE1.24E − 051.55E − 051.42E − 051.89E − 05MASE0.0420520.0436780.0416790.042014MAPE0.0165680.017230.0197740.01527Re (ϕv)SMAPE5.60E − 054.35E − 057.22E − 056.78E − 05MSE4.2345442.5580474.0119012.010292MAPE0.617330.2168732.5207060.973477Re (σxx)SMAPE0.0001680.0001680.0001840.000184MASE7.8406747598.5797097592.34228.643227637MAPE0.5700356.3931110.4438250.627785Re (σxy)SMAPE2.02E − 052.63E − 052.22E − 052.62626E − 05MASE6.7924895.1630403856.0098854.721728846MAPE0.5700356.3931110.4438250.627785Re (T)SMAPE5.14444E − 076.60626E − 072.69919E − 081.82626E − 08MASE0.7680751620.7963852331.187699911.776765265MAPE2.34221.2207590.2959520.289676Re (N)SMAPE4.46909E − 065.70869E − 060.0005551720.000555172MASE0.6283705250.6481823381.0009203331.000920333MAPE0.6283705250.0745260.3087790.308779Table 11
The accuracy measurements for cubic spline models of the imaginary (Im) parts of the horizontal displacement components.
Ω=0.1 (photothermal)Ω=0.4 (photothermal)Ω=0.1 (nonphotothermal)Ω=0.4 (nonphotothermal)Im (u)SMAPE3.26E − 053.55E − 053.27E − 055.69E − 05MASE0.8604511.0758611.1429134.810742MAPE1.698813.2854662.8212024.276973Im (v )SMAPE6.59E − 061.30E − 057.28E − 061.48E − 05MASE0.0326460.0311880.0321570.030266MAPE0.0116940.0109570.0106790.01026Im (ϕv)SMAPE0.08510.14590.18660.2026MASE0.68581.40090.59090.7221MAPE0.22100.56100.63050.8968Im (σxx)SMAPE6.06E − 055.45E − 055.66E − 054.65E − 05MASE10.17040210.028736679.3346216077.56075913MAPE0.3617330.2213260.270261.197851Im (σxy)SMAPE2.22222E − 053.43434E − 052.62626E − 054.0404E − 05MASE10.3314610.0906332410.294362694.0404E-05MAPE0.5700356.3931110.4438250.627785Im (T)SMAPE1.72414E − 071.19812E − 075.64606E − 085.96303E − 08MASE1.1988169781.3763882850.8483614210.945056408MAPE0.2825990.2010040.2634070.342063Im (N)SMAPE1.14684E − 066.28596E − 070.0008370080.00094504MASE1.2196529971.8136914220.1508962130.076863319MAPE0.2950690.2423910.1167250.302983


## 6. Discussion

Table1 shows the statistical significance of the real and imaginary displacements u with a change of rotation () under photothermal and nonphotothermal conditions. Figure 1 showed that as the rotation effect grows with the presence or absence of photothermal, the values of Im (u) and Re (u) increase. Furthermore, employing artificial neural network (ANN) and cubic spline models, the values of the observed data for Im (u) and Re (u) almost match the anticipated values. ANN was used to forecast the maximum Im (u) and Re (u) values, which were (0.17847, 0.17941) and (0.14741, 0.26007), respectively. For cubic spline, typical results for the highest projected value were also achieved.Table2 also reveals a statistically significant relationship between the real and imaginary displacements v with a change in rotation (Ω) under photothermal and nonphotothermal conditions. The variation in vertical displacement v versus x is seen in Figure 2. In the period 0 × 1.0, it was discovered that the real component Re (v) grew in lockstep with the expansion of the distance x. Then, it starts to go down until it reaches x = 3. There was also a considerable difference in the rotational and photothermal effects. The imaginary part Im (v), on the other hand, behaves in the opposite way to the real part Re (v), with an obvious circulation and photothermal effect. In other words, the imaginary portion behaves similarly to the real part Re (v) in the vertical axis’ negative segment. In addition, the presence of both rotation and photothermal had a clear impact on the imaginary component Im (v). The maximum Im (v) and Re (v) values predicted by ANN were (0.05831, 0.006057) and (0.023833, 0.0.040835), respectively, whereas the maximum Im (v) and Re (v) values predicted by cubic spline were (0.00730, 0.00949) and (0.02445, 0.09983), respectively.Moreover, a statistical significance correlation between the real and imaginary displacementϕv with a change of rotation (Ω) under photothermal and nonphotothermal conditions is observed in Table 3. Figure 3 clarifies the alteration in the volume fraction field ϕv against the distance x. The maximum predicted value of Im(ϕv) and Re(ϕv) was ranged between (0.498232, 0.97550) and (0.06683, 0.18141), respectively, using ANN and between (0.00730, 0.00949) and (0.06683, 0.18141), respectively, using cubic spline.In addition, Table4 also reveals a statistical significance correlation between the real and imaginary displacements σxx with a change of rotation (Ω) under photothermal and nonphotothermal conditions. Table 5 shows a statistically significant relationship between the real and imaginary displacements σxy with a change of rotation (Ω) under photothermal and nonphotothermal conditions. Figures 4 and 5 show how the change of normal and tangential stress components (σxx, σxy) fluctuate as a function of the distance x when the effects of both rotation and photothermal effects are present. With changes in rotation and photothermal on both actual and fiction portions, an effect can be noticed. It also has a slight oscillating pattern around the axis x. The maximum predicted value of Im (σxx) and Re (σxx) was ranged between (0.498232, 0.97550) and (0.06683, 0.18141), respectively, using ANN and between (0.00730, 0.00949) and (0.06683, 0.18141), respectively, using cubic spline. However, the maximum predicted value of Im (σxy) and Re (σxy) was ranged between (0.11, 0.9755013) and (0.02, 0.02), respectively, using ANN and between (0.085, 0.13) and (0.02, 0.11), respectively, using cubic spline.Finally, Table6 shows a statistical significance correlation between the real and imaginary displacement T with a change of rotation (Ω) under photothermal and nonphotothermal conditions. Similarly, Table 7 shows a statistical significance correlation between the real and imaginary displacement N with a change of rotation (Ω) under photothermal and nonphotothermal conditions. Figures 6 and 7 illustrated the difference between the temperature T and the carrier density N versus the distance x with the variation of the values of rotation and photothermal. It was shown in Figures 6 and 7 there is clear dependence on these variables on both the real and imaginary parts of the temperature and the carrier density variables. Therefore, their impact should be taken into consideration when using them in practical and technological applications. The maximum predicted value of Im (T) and Re (T) was ranged between (0.00014, 0.00032) and (0.00003, 0.00002), respectively, using ANN and between (0.00730, 0.00949) and (0.06683, 0.18141), respectively, using cubic spline. In addition, the maximum predicted value of Im (N) and Re (N) was ranged between (0.01, 0.07) and (0.02, 1.49), respectively, using ANN and between (0.00294, 0.56346) and (0.01818, 0.09107), respectively, using cubic spline.Lastly, Tables8–11 show that the artificial neural network improves the forecasting accuracy over the cubic spline interpolation model. The results obtained in this study showed that machine learning is very effective on the phenomenon and has a good and strong interpretation on the physical meaning for the phenomenon and it is applicable in diverse fields such as biology, bioengineering, and so on.


## 7. Conclusion

This paper aims to use the artificial neural network method and cubic spline model to predict both the real and the imaginary parts of different physical quantities displacement components, such as elastic waves (u), vector form (v), volume fraction field (ϕv), temperature (thermal waves) (T), stress components (σxx and σxy), and carrier density concentration in a thermoelastic solid (plasma waves) (N). Furthermore, the maximum predicted values of the real and the imaginary parts of those physical quantities’ displacement components are obtained.In conclusion, the following points are obtained:(i)
A strong effect of voids, rotation, optical temperature, and thermal relaxation on the phenomenon is consistent with the physical results that obtained by Kilany et al. [1] about photothermal and voids effect of a semiconductor rotational medium with thermal relaxation time(ii)
The finding in this study is very important, and it is interesting in the wave’s propagation phenomena(iii)
The effect of thermal field, rotation, and magnetism has various applications in medicine and bioinformatics such as the human skin and peristaltic flowFinally, the treatment has strong applications in cancer, stomach, diabetic, and skin diseases due to the high temperature that the human can endure, especially in the hot regions



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*Source: 7205380-2022-02-24.xml*