# Multiple Quadratic Polynomial Regression Models and Quality Maps for Tensile Mechanical Properties and Quality Indices of Cast Aluminum Alloys according to Artificial Aging Heat Treatment Condition

Authors: Won-Chol Yang; Ji-Yon Yang; Ryong-Chol Kim
Journal: Advances in Materials Science and Engineering
 (2023)

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## Abstract

To evaluate the quality of cast aluminum alloys quantitatively and intuitively, quality index and quality map have been used. Quality index and quality map are to quantitatively evaluate the quality of cast aluminum alloys according to yield strength (YS), ultimate tensile strength (UTS), elongation to fracture (Ef), and strain energy density (W). There are some quality indices such as Q, QR, QC, and Q0. The quality maps are generated to intuitively evaluate the quality level based on the quality indices. These quality indices and quality maps show the quality levels according to the pairs of tensile mechanical properties such as UTS and Ef, or YS and Ef, or YS and W. By using these quality maps, it is impossible to directly evaluate the quality levels according to the artificial aging heat treatment condition. We develop multiple quadratic polynomial regression models and quality maps for tensile mechanical properties and quality indices of the cast aluminum alloys according to artificial aging heat treatment condition. The performances of the regression models are evaluated using the mean absolute errors, mean relative errors, and coefficients of determination. The regression models and quality maps could be widely used to evaluate the quality of the cast aluminum alloys according to the aging heat treatment conditions and determine the rational aging heat treatment condition.


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## Body

## 1. Introduction

Cast aluminum alloys are widely used in modern industry because of good corrosion resistance, high level of mechanical properties such as ultimate tensile strength, yield strength, and elongation, and a good castability [1]. Quality level of cast aluminum alloys is quantified using quality index. The quality index is a measure to evaluate the quality of cast aluminum alloys based on tensile mechanical properties such as yield strength (YS), ultimate tensile strength (UTS), elongation to fracture (Ef), and strain energy density (W). It enables the material designers and engineers to select the reasonable materials with high quality, and it is considered to be a key factor for selecting an alloy for a particular engineering application [2, 3]. The quality index allows to compare different castings, which may have received different heat treatment conditions, or whose chemical compositions are different [4].There are some popular quality indices such asQ, QR, QC, and QD. The calculation formulas are as follows [2, 5–9]:(1)Q=UTS+d⋅log10Ef,(2)QR=YS+m⋅Ef,(3)QC=UTS+0.4⋅YS,Ea⋅YSn⋅log10Ef,(4)QD=KD,Q0Q0=YS+10⋅W.In the above equations, UTS is the ultimate tensile strength,Ef is the elongation at fracture, YS is the yield strength, and W is the strain energy density. d is an empirical coefficient chosen such as to make Q practically independent of the aging condition and it takes the value of 150 MPa in the original work. m is a material constant expressed in MPa. The value of m varies according to considered alloys: 7.5–13 MPa for the Al-Cu alloys, 50 MPa for the Al-Si-Mg alloys, 7.5 MPa for A206 alloys, 13 MPa for A201 alloys, 50 MPa for A356 and A357 alloys, and 40 MPa for Al-7%Si-Mg Alloys. E is Young’s modulus and n is the strain hardening exponent and a is a scale factor of order [5]. In equation (4), (Q0) takes into account the average tensile mechanical properties of the materials and KD takes into account the scatter in the mechanical properties. Using the quality index QD and by neglecting the scatter in mechanical properties, i.e., by assuming KD = 1, the quality level of an alloy can be characterized by Q0 = YS+10W. [5, 9].The quality indexQ is more sensitive to tensile strength than to tensile ductility variation and the quality indices QC and Q follow absolutely the almost same alloy quality evaluation. The quality index QR is strongly governed by the ductility property of the material and more sensitive to ductility, and it favors the ductile materials more than QD [10]. The quality index QD is well balanced over strength and ductility of the materials [11]. It evaluates the material’s potential to offer combinations of tensile strength, ductility, and toughness. The quality index QD evaluates the material quality on the basis of a balance between the material properties’ yield strength (YS) and strain energy density W [12].The concept of quality map for supporting material selection was introduced since several decades and exploited in a series of applications [6, 8]. Ammar et al. [3] considered the quality of Al-Si casting alloys to be a key factor in selecting an alloy for a particular engineering application. They reviewed the development of the quality index concept and quality chart in relation to their application to Al-Si casting alloys. They reviewed different theories pertaining to the concept of the quality index Q, its calculation, and its use in the construction of quality charts. The quality charts are used to evaluate the mechanical properties and the quality of the castings under the influence of various metallurgical parameters, and the quality chart concept has thus already been used as a selection tool for engineering materials. The quality charts are generated for use as a simple method of evaluating, selecting, and also predicting the most appropriate metallurgical conditions, which may be applied to the castings so as to obtain the best possible compromise between tensile properties and casting quality. Alexopoulos and Pantelakis [9] plotted a diagram of the ultimate tensile strength vs the logarithm of the elongation to fracture. In the diagram, Q=UTS+d⋅log10Ef and YS=a⋅UTS−b⋅log10Ef+c represent the sets of parallel lines called isoquality index and iso-yield strength lines, respectively; they fit the experimentally obtained Q and YS values resulting from variations in chemical composition, solidification conditions, and heat treatment of Al-Si-Mg aluminum alloys with a good approximation. Hence, the preceding diagram called quality index chart provides a very useful tool to reduce the experimental effort for developing or optimizing Al-Si-Mg cast alloys essentially. Tiryakioglu et al. [10] investigated the use of quality indices of cast aluminum alloys to support material selection by means of generation of quality maps. They generated the quality maps of cast aluminum alloys from different series, the same series and minor variations in chemical composition, and the same series and variations in heat treatment. The quality maps enable to support the material selection based on all proposed quality indices, devised for the evaluation of cast aluminum alloys to be used in aircraft applications. Alyaldin et al. [13] performed the study on the 354 (Al-9 wt% Si-1.8 wt% Cu-0.5 wt% Mg)-based alloy to which measured amounts of Zr, Ni, Mn, and Sc were added. They used the quality index Q to calculate the quality index values of the aluminum alloy castings and plot the iso-Q lines and the iso-YS lines, respectively. They carried out the analysis of the tensile data using quality charts and color contour maps. Pantelakis et al. [14] generated the quality map of the investigated AM cast magnesium alloys with variations in the Al content, the quality map of QE22A cast magnesium alloy for different solid solution heat treatment temperatures, and the quality map of QE22A cast magnesium alloy for the different artificial aging heat treatment conditions on the basis of the quality index Q0. They compared the mechanical performances of cast magnesium and aluminum alloys by means of the quality maps. The quality maps support the design engineer to easily select the process conditions of QE22A that optimize the material’s mechanical performance.The previous quality indices and quality maps show the quality levels of the cast aluminum alloys according to only the pairs of tensile mechanical properties such as UTS andEf, or YS and Ef, or YS and W. The research gaps of the previous studies are that it is impossible to show the tensile mechanical properties and quality levels according to artificial aging heat treatment conditions, directly.If we develop the relation models and quality maps to evaluate the tensile mechanical properties and quality indices of the cast aluminum alloys according to artificial aging heat treatment condition, it is possible to grasp the influences of the artificial aging heat treatment conditions to their quality levels and decide the reasonable artificial aging heat treatment conditions, directly. It is greatly convenient and effective to evaluate the quality level of the cast aluminum alloy according to the artificial aging heat treatment conditions in practice.For this purpose, we develop the regression models and quality maps for tensile mechanical properties and quality indices according to artificial aging heat treatment condition in this paper.Commonly, to model the relationship between the response variabley and several independent variables (factors), multiple linear regression model, multiple polynomial regression model, and artificial neural network (ANN) model are used. The multiple linear regression model is simple and easy to use in practice. When the performance is not satisfactory, the multiple polynomial regression model such as multiple quadratic or cubic polynomial regression model is used. When the performance is also not satisfactory, the ANN model such as the BPNN model is used. The ANN model has an excellent learning capacity. However, it needs large amount of data for developing the model and the performance is highly dependent on having sufficient data collection.To develop the relationship model with high performance, various research works were conducted by using artificial intelligent-based methodologies such as novel metaheuristic-based machine learning method, interpretable data-intelligence model, data driven models, and soft computing techniques [15–18].In this paper, the regression models for the tensile mechanical properties and quality indices of the cast aluminum alloy A357 according to artificial aging heat-treatment condition are developed using multiple quadratic polynomial regression models and the quality maps are plotted based on the regression models.


## 2. Methods

### 2.1. Method to Develop Multiple Quadratic Polynomial Regression Model

The general form of the multiple linear regression model of a dependent variable (response)y according to p independent variables (factors) x1, x2, …, xp can be expressed as follows [19, 20]:(5)y1=β0+β1x11+⋯+βpx1p+e1,y2=β0+β1x21+⋯+βpx2p+e2,⋯,yn=β0+β1xn1+⋯+βpxnp+en,i.e.,(6)yi=β0+β1xi1+⋯+βpxip+eifori=1,2,…,n,where yi is the value of the response variable y for the ith case, xij is the value of the jth independent variable xj for the ith case, β0 is the y-intercept of the regression surface (think multidimensionality), each βj is the slope of the regression surface with respect to variable xj, and ei is the random error component for the ith case. In equation (5), we have n observations and p factors (n > p+1).Equation (5) can be rewritten in matrix notation as follows:(7)y=X⋅β+e,where response vector y and error vector e are the column vectors of length n, vector of parameters β = (β0, β1, β2, …, βp)’ is a column vector of length p+1, and design matrix X is n by p+1 matrix (with its first column having all elements equal to 1, the second column being filled by the observed values of x1, the third column being filled by the observed values of x2, and so on).(8)y=y1y2⋯yn,X=1x11x12⋯x1p1x21x22⋯x2p⋮⋮⋮⋮1xn1xn2⋯xnp,β=β0β1⋯βp,e=e1e2⋯en.The vector of regression parametersβ could be estimated using the least square method as follows:(9)β^=XTX−1XTY,where β^ is an unbiased estimator of β.In practical applied work, the quadratic terms such asx21 and x22 are often included to model a curved relationship between y and several independent variables (factors).The general form of the multiple quadratic polynomial regression model is as follows:(10)y=β0+∑j=1pβjxj+∑j=1p∑k=jpβj,kxjxk,=β0+β1x1+β2x2+⋯+βpxp+β1,1x12+β1,2x1x2+⋯+β1,px1xp+β2,2x22+β2,3x2x3+⋯+β2,px2xp+⋯+βp−1,p−1xp−12+βp−1,pxp−1xp+βp,pxp2,where β0 is a constant, and βj, βj,j, and βj,k are the linear, pure quadratic and interaction coefficients, respectively.LetX1=x1,X1=x1,⋯,Xp=xp,Xp+1=x12,Xp+2=x1x2,Xp+3=x1x3,⋯,Xm−1=xp−1xp,Xm=xp2.The multiple quadratic regression model can be represented as the following multiple linear regression model:(11)y=β0+β1X1+β2X2+⋯+βpXp+β1,1Xp+1+β1,2Xp+2+⋯+βp,pXm.Therefore, the multiple quadratic regression model could be considered as a particular case of the multiple linear regression model.The least squared estimator for the multiple quadratic regression model is as follows:(12)β^=β^0,β^1,β^2,⋯,β^p,pT=XTX−1XTy,where(13)y=y1y2⋯yn,X=1X11X12⋯X1m1X21X22⋯X2m⋮⋮⋮⋮1Xn1Xn2⋯Xnm,=1x11x12⋯x1px112x11x12⋯x11x1px122x12x13⋯x12x1p⋯x1,p−1x1px1p2⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1xi1xi2⋯xipxi12xi1xi2⋯xi1xipxi22xi2xi3⋯xi2xip⋯xi,p−1xipxip2⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1xn1xn2⋯xnpxn12xn1xn2⋯xn1xnpxn22xn2xn3⋯xn2xnp⋯xn,p−1xnpxnp2.As a result, the multiple quadratic regression model is represented as follows:(14)y=β^0+β^1x1+β^2x2+⋯+β^pxp+β^1,1x12+β^1,2x1x2+⋯+β^1,px1xp+β^2,2x22+β^2,3x2x3+⋯+β^2,px2xp+⋯+β^p−1,p−1xp−12+β^p−1,pxp−1xp+β^p,pxp2.To evaluate the performance of the multiple regression model, the mean absolute error, mean relative error, andR-squared R2 (coefficient of determination) are calculated.The mean absolute error (MAE) and mean relative error (MRE) are calculated as follows:(15)MAE=1n∑i=1nyi−y^i,MRE=1n∑i=1nyi−y^iyi×100%,where yi is the observed value and y^i is the fitted value of the response variable y for the ith case.The coefficient of determination (R2) of the multiple regression model is calculated as follows:(16)R2=1−∑i=1nyi−y^i2∑i=1nyi−y¯2,where y¯ is the mean value of response variable y. R2 measures the percentage of variation in the response variable y explained by the explanatory variable x1, x2, …, xp. Thus, it is an important measure of how well the regression model fits the data.For external validation of the multiple regression model, the cross-validation (CV) method is used in this paper. The CV method was proposed to evaluate the predictive validity of a linear regression model [21]. One data sample from among the training data is set aside as validation data, and the remaining N − 1 data samples are used as a construction dataset to build the model. The quality of the model is assessed with the validation data. This procedure is repeated N times to get an average cross-validated indices such as MAE, MRE, and R2. The average cross-validated indices are used to validate the model.If MRE is less than 10%, it is interpreted as excellent accurate forecasting, between 10 and 20% as good forecasting, between 20 and 50% as acceptable forecasting, and over 50% as inaccurate forecasting [20]. An R2 value of 0.9 or above is very good, a value above 0.8 is good, and a value of 0.6 or above may be satisfactory in some applications although we must be aware of the fact that, in such cases, errors in prediction may be relatively high. When the R2 value is 0.5 or below, the regression explains only 50% or less of the variation in the data; therefore, prediction may be poor [22].


### 2.2. Method to Develop Multiple Quadratic Polynomial Regression Models and Quality Maps for Tensile Mechanical Properties and Quality Indices of Cast Aluminum Alloy according to Artificial Aging Heat Treatment Condition

In this subsection, a method to develop the multiple quadratic polynomial regression models and quality maps for the tensile mechanical properties and quality indices of the cast aluminum alloys according to artificial aging heat treatment condition is proposed.The main steps are as follows:Step 1. We develop the multiple quadratic regression model for the tensile mechanical property (YS, UTS, Ef, and W) and the quality index (Q, QR, QC, and Q0) according to the artificial aging heat-treatment conditions (aging temperature T and aging time t) in the following form:(17)y=β0+β1⋅T+β2⋅t+β3⋅T2+β4⋅T⋅t+β5⋅t2.This model is developed with data which consist of different artificial aging heat-treatment conditions (aging temperatureT and aging time t), tensile mechanical properties (YS, UTS, Ef, and W), and quality indices (Q, QR, QC, and Q0) of the investigated cast aluminum alloys.Step 2. We calculate and draw the lines of constant mechanical property (isomechanical property lines) or the lines of constant quality index (isoquality index lines) using the corresponding regression model on a 2D plane.The isoproperty lines or isoquality index lines are calculated from the following equation:(18)β0+β1⋅T+β2⋅t+β3⋅T2+β4⋅T⋅t+β5⋅t2=C, where C is a fixed constant.Step 3. Plot the pairs of aging temperature and time values of the investigated alloys in the above graph.This graph is called the “quality map” for the tensile mechanical property or “quality index” of the cast aluminum alloys. This map intuitively shows the quality levels of the cast aluminum alloys according to different heat treatment conditions (aging temperature and time).


## 2.1. Method to Develop Multiple Quadratic Polynomial Regression Model

The general form of the multiple linear regression model of a dependent variable (response)y according to p independent variables (factors) x1, x2, …, xp can be expressed as follows [19, 20]:(5)y1=β0+β1x11+⋯+βpx1p+e1,y2=β0+β1x21+⋯+βpx2p+e2,⋯,yn=β0+β1xn1+⋯+βpxnp+en,i.e.,(6)yi=β0+β1xi1+⋯+βpxip+eifori=1,2,…,n,where yi is the value of the response variable y for the ith case, xij is the value of the jth independent variable xj for the ith case, β0 is the y-intercept of the regression surface (think multidimensionality), each βj is the slope of the regression surface with respect to variable xj, and ei is the random error component for the ith case. In equation (5), we have n observations and p factors (n > p+1).Equation (5) can be rewritten in matrix notation as follows:(7)y=X⋅β+e,where response vector y and error vector e are the column vectors of length n, vector of parameters β = (β0, β1, β2, …, βp)’ is a column vector of length p+1, and design matrix X is n by p+1 matrix (with its first column having all elements equal to 1, the second column being filled by the observed values of x1, the third column being filled by the observed values of x2, and so on).(8)y=y1y2⋯yn,X=1x11x12⋯x1p1x21x22⋯x2p⋮⋮⋮⋮1xn1xn2⋯xnp,β=β0β1⋯βp,e=e1e2⋯en.The vector of regression parametersβ could be estimated using the least square method as follows:(9)β^=XTX−1XTY,where β^ is an unbiased estimator of β.In practical applied work, the quadratic terms such asx21 and x22 are often included to model a curved relationship between y and several independent variables (factors).The general form of the multiple quadratic polynomial regression model is as follows:(10)y=β0+∑j=1pβjxj+∑j=1p∑k=jpβj,kxjxk,=β0+β1x1+β2x2+⋯+βpxp+β1,1x12+β1,2x1x2+⋯+β1,px1xp+β2,2x22+β2,3x2x3+⋯+β2,px2xp+⋯+βp−1,p−1xp−12+βp−1,pxp−1xp+βp,pxp2,where β0 is a constant, and βj, βj,j, and βj,k are the linear, pure quadratic and interaction coefficients, respectively.LetX1=x1,X1=x1,⋯,Xp=xp,Xp+1=x12,Xp+2=x1x2,Xp+3=x1x3,⋯,Xm−1=xp−1xp,Xm=xp2.The multiple quadratic regression model can be represented as the following multiple linear regression model:(11)y=β0+β1X1+β2X2+⋯+βpXp+β1,1Xp+1+β1,2Xp+2+⋯+βp,pXm.Therefore, the multiple quadratic regression model could be considered as a particular case of the multiple linear regression model.The least squared estimator for the multiple quadratic regression model is as follows:(12)β^=β^0,β^1,β^2,⋯,β^p,pT=XTX−1XTy,where(13)y=y1y2⋯yn,X=1X11X12⋯X1m1X21X22⋯X2m⋮⋮⋮⋮1Xn1Xn2⋯Xnm,=1x11x12⋯x1px112x11x12⋯x11x1px122x12x13⋯x12x1p⋯x1,p−1x1px1p2⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1xi1xi2⋯xipxi12xi1xi2⋯xi1xipxi22xi2xi3⋯xi2xip⋯xi,p−1xipxip2⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1xn1xn2⋯xnpxn12xn1xn2⋯xn1xnpxn22xn2xn3⋯xn2xnp⋯xn,p−1xnpxnp2.As a result, the multiple quadratic regression model is represented as follows:(14)y=β^0+β^1x1+β^2x2+⋯+β^pxp+β^1,1x12+β^1,2x1x2+⋯+β^1,px1xp+β^2,2x22+β^2,3x2x3+⋯+β^2,px2xp+⋯+β^p−1,p−1xp−12+β^p−1,pxp−1xp+β^p,pxp2.To evaluate the performance of the multiple regression model, the mean absolute error, mean relative error, andR-squared R2 (coefficient of determination) are calculated.The mean absolute error (MAE) and mean relative error (MRE) are calculated as follows:(15)MAE=1n∑i=1nyi−y^i,MRE=1n∑i=1nyi−y^iyi×100%,where yi is the observed value and y^i is the fitted value of the response variable y for the ith case.The coefficient of determination (R2) of the multiple regression model is calculated as follows:(16)R2=1−∑i=1nyi−y^i2∑i=1nyi−y¯2,where y¯ is the mean value of response variable y. R2 measures the percentage of variation in the response variable y explained by the explanatory variable x1, x2, …, xp. Thus, it is an important measure of how well the regression model fits the data.For external validation of the multiple regression model, the cross-validation (CV) method is used in this paper. The CV method was proposed to evaluate the predictive validity of a linear regression model [21]. One data sample from among the training data is set aside as validation data, and the remaining N − 1 data samples are used as a construction dataset to build the model. The quality of the model is assessed with the validation data. This procedure is repeated N times to get an average cross-validated indices such as MAE, MRE, and R2. The average cross-validated indices are used to validate the model.If MRE is less than 10%, it is interpreted as excellent accurate forecasting, between 10 and 20% as good forecasting, between 20 and 50% as acceptable forecasting, and over 50% as inaccurate forecasting [20]. An R2 value of 0.9 or above is very good, a value above 0.8 is good, and a value of 0.6 or above may be satisfactory in some applications although we must be aware of the fact that, in such cases, errors in prediction may be relatively high. When the R2 value is 0.5 or below, the regression explains only 50% or less of the variation in the data; therefore, prediction may be poor [22].


## 2.2. Method to Develop Multiple Quadratic Polynomial Regression Models and Quality Maps for Tensile Mechanical Properties and Quality Indices of Cast Aluminum Alloy according to Artificial Aging Heat Treatment Condition

In this subsection, a method to develop the multiple quadratic polynomial regression models and quality maps for the tensile mechanical properties and quality indices of the cast aluminum alloys according to artificial aging heat treatment condition is proposed.The main steps are as follows:Step 1. We develop the multiple quadratic regression model for the tensile mechanical property (YS, UTS, Ef, and W) and the quality index (Q, QR, QC, and Q0) according to the artificial aging heat-treatment conditions (aging temperature T and aging time t) in the following form:(17)y=β0+β1⋅T+β2⋅t+β3⋅T2+β4⋅T⋅t+β5⋅t2.This model is developed with data which consist of different artificial aging heat-treatment conditions (aging temperatureT and aging time t), tensile mechanical properties (YS, UTS, Ef, and W), and quality indices (Q, QR, QC, and Q0) of the investigated cast aluminum alloys.Step 2. We calculate and draw the lines of constant mechanical property (isomechanical property lines) or the lines of constant quality index (isoquality index lines) using the corresponding regression model on a 2D plane.The isoproperty lines or isoquality index lines are calculated from the following equation:(18)β0+β1⋅T+β2⋅t+β3⋅T2+β4⋅T⋅t+β5⋅t2=C, where C is a fixed constant.Step 3. Plot the pairs of aging temperature and time values of the investigated alloys in the above graph.This graph is called the “quality map” for the tensile mechanical property or “quality index” of the cast aluminum alloys. This map intuitively shows the quality levels of the cast aluminum alloys according to different heat treatment conditions (aging temperature and time).


## 3. Results and Discussion

Cast aluminum alloy A357 is artificially aged under different artificial aging heat treatment conditions. The evaluated tensile mechanical properties and quality indices of A357 with 27 different artificial aging heat-treatment conditions are presented in Table1 [2, 11].Table 1
Mean values of the tensile mechanical properties and quality indices of cast aluminum alloy A357 according to 27 different artificial aging heat treatment conditions [2, 11].
Aging temperatureAging timeYield strengthTensile strengthElongation at fractureStrain energy densityQuality indicesT(°C)t(h)YS (MPa)UTS (MPa)Ef(%)W(MJ/m3)Q(MPa)QR(MPa)QC(MPa)Q0(MPa)1155616027719.5249.38470.51136.0356.0653.821551222531315.4344.85491.2996.5411.3673.531551622731515.3945.36493.1996.5413.9680.641552023231713.5840.31486.9911.0414.2635.151552425532712.6938.49492.5889.5430.3639.961553026533311.7937.43493.7854.5437.4639.371553628133810.6034.66491.8811.0444.2627.68155483113447.4725.71475.0684.5445.7568.19175115427621.1253.60474.71210.0354.4690.010175319029217.6848.36479.11074.0380.5673.611175625932312.7540.28488.8896.5427.9661.812175928733810.5235.45491.3813.0445.9641.513175122963409.7333.07488.2782.5448.0626.714175203043447.4225.81474.5675.0443.3562.115175362953337.5325.45464.5671.5430.2549.516175483113446.3722.45464.6629.5438.5535.517205126732511.2736.10482.8830.5428.6628.01820522933388.6329.17478.4724.5440.2584.71920533013368.3328.39474.1717.5439.3584.92020542863259.2430.27469.8748.0427.7588.72120552813279.1929.96471.5740.5427.8580.62220562913218.1326.68457.5697.5420.1557.823205102923258.4127.80463.7712.5425.8570.024205123043348.6029.43474.1734.0439.6598.325205163003329.2430.99476.8762.0439.2609.926205242572855.0514.58390.5509.5354.7402.827205362162507.4818.42381.1590.0322.6400.2In this section, the multiple quadratic polynomial regression models and quality maps for the tensile mechanical properties and quality indices of the cast aluminum alloy A357 according to artificial aging heat treatment condition are developed with the data from Table1.


### 3.1. Multiple Quadratic Polynomial Regression Model and Quality Map for Tensile Mechanical Properties according to Artificial Aging Heat Treatment Condition

The multiple quadratic polynomial regression model and quality map for YS of A357 are shown in equation (19) and Figure 1.(19)YS=−2538.186432+26.970192⋅T+32.097822⋅t−0.064578⋅T2−0.143513⋅T⋅t−0.111892⋅t2.Figure 1
3D surface graph of the regression model (a) and quality map (b) forYS of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(20)MAE=15.976MPa,MRE=6.819%,R2=0.786.By using equation (19) and Figure 1, it is possible to quantitatively and intuitively evaluate the YS value of the cast aluminum alloy A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat-treatment conditions that satisfy the design prerequisites for YS. In Figure 1, the number of alloys refers to Table 1.The multiple quadratic polynomial regression model and quality map for UTS of A357 are shown in equation (21) and Figure 2.(21)UTS=−1387.530241+17.131269⋅T+20.193788⋅t−0.042752⋅T2−0.096759⋅T⋅t−0.060772⋅t2.Figure 2
3D surface graph of the regression model (a) and quality map (b) forUTS of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(22)MAE=9.529MPa,MRE=3.052%,R2=0.748.By using equation (21) and Figure 2, it is possible to quantitatively and intuitively evaluate the UTS value of the cast aluminum alloy A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for UTS.The multiple quadratic polynomial regression model and quality map forEf of A357 are shown in equation (23) and Figure 3.(23)Ef=163.972973−1.373957⋅T−1.703747⋅t+0.003059⋅T2+0.006477⋅T⋅t+0.007572⋅t2.Figure 3
3D surface graph of the regression model (a) and quality map (b) for (E)f of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(24)MAE=1.224%MPa,MRE=11.795%,R2=0.853.By using equation (23) and Figure 3, it is possible to quantitatively and intuitively evaluate the Ef value of the cast aluminum alloy A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for Ef.The multiple quadratic polynomial regression model and quality map forW of A357 are shown in equation (25) and Figure 4.(25)W=248.534506−1.758050⋅T−2.157846⋅t+0.003477⋅T2+0.006329⋅T⋅t+0.011086⋅t2.Figure 4
3D surface graph of the regression model (a) and quality map (b) forW of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(26)MAE=2.79MJM3MPa,MRE=9.070%,R2=0.867.By using equation (25) and Figure 4, it is possible to quantitatively and intuitively evaluate the W value of the cast aluminum alloy A357 according to given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for W.


### 3.2. Multiple Quadratic Polynomial Regression Model and Quality Map for Quality Indices according to Artificial Aging Heat-Treatment Condition

The multiple quadratic polynomial regression model and quality map forQ of A357 are shown in equation (27) and Figure 5.(27)Q=−438.958673+9.912151⋅T+13.043748⋅t−0.026478⋅T2−0.071426⋅T⋅t−0.029495⋅t2.Figure 5
3D surface graph of the regression model (a) and quality map (b) for (Q) of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(28)MAE=7.598MPa,MRE=1.663%,R2=0.806.By using equation (27) and Figure 5, it is possible to quantitatively and intuitively evaluate the Q value of A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat-treatment conditions that satisfy the design prerequisites for Q.The multiple quadratic polynomial regression model and quality map forQR of A357 are shown in equation (29) and Figure 6.(29)QR=5660.462202−41.727683⋅T−53.089517⋅t+0.088387⋅T2+0.180326⋅T⋅t+0.266725⋅t2.Figure 6
3D surface graph of the regression model (a) and quality map (b) forQR of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE andR2 of this model are as follows:(30)MAE=49.829MPa,MRE=6.222%,R2=0.856.By using equation (29) and Figure 6, it is possible to quantitatively and intuitively evaluate the QR value of the cast aluminum alloy A357 according to the given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for QR.The multiple quadratic polynomial regression model and quality map forQC of A357 are shown in equation (31) and Figure 7.(31)QC=−1846.121845+22.618754⋅T+28.431034⋅t−0.056177⋅T2−0.136343⋅T⋅t−0.089533⋅t2.Figure 7
3D surface graph of the regression model (a) and quality map (b) forQC of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(32)MAE=13.456MPa,MRE=3.313%,R2=0.729.By using equation (31) and Figure 7, it is possible to quantitatively and intuitively evaluate the QC value of the cast aluminum alloy A357 according to a given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for QC.The multiple quadratic polynomial regression model and quality map forQ0 of A357 are shown in equation (33) and Figure 8.(33)Q0=−52.841375+9.389694⋅T+10.519360⋅t−0.029807⋅T2−0.080221⋅T⋅t−0.001030⋅t2.Figure 8
3D surface graph of the regression model (a) and quality map (b) forQ0 of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(34)MAE=20.395MPa,MRE=3.626%,R2=0.828.By using equation (33) and Figure 8, it is possible to quantitatively and intuitively evaluate the Q0 value of the cast aluminum alloy A357 according to a given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for Q0.In overall, the multiple quadratic polynomial regression models for tensile mechanical properties and quality indices according to artificial aging heat-treatment condition are as follows:(35)YS=−2538.186432+26.970192⋅T+32.097822⋅t−0.064578⋅T2−0.143513⋅T⋅t−0.111892⋅t2,UTS=−1387.530241+17.131269⋅T+20.193788⋅t−0.042752⋅T2−0.096759⋅T⋅t−0.060772⋅t2,Ef=163.972973−1.373957⋅T−1.703747⋅t+0.003059⋅T2+0.006477⋅T⋅t+0.007572⋅t2,W=248.534506−1.758050⋅T−2.157846⋅t+0.003477⋅T2+0.006329⋅T⋅t+0.011086⋅t2,Q=−438.958673+9.912151⋅T+13.043748⋅t−0.026478⋅T2−0.071426⋅T⋅t−0.029495⋅t2,QR=5660.462202−41.727683⋅T−53.089517⋅t+0.088387⋅T2+0.180326⋅T⋅t+0.266725⋅t2,QC=−1846.121845+22.618754⋅T+28.431034⋅t−0.056177⋅T2−0.136343⋅T⋅t−0.089533⋅t2,Q0=−52.841375+9.389694⋅T+10.519360⋅t−0.029807⋅T2−0.080221⋅T⋅t−0.001030⋅t2.The corresponding quality maps are shown in Figures1–8.The MREs of the above regression models are less than or slightly higher than 10%; hence, the regression models and quality maps are excellently accurate. TheR2 values of the above regression models are higher than 0.8 or 0.7, indicating that the regression models and quality maps are very good or may be satisfactory.


### 3.3. Comparison of the Developed Models with Multiple Linear Regression Models and External Validation Results

Table2 shows the performance comparison result between the multiple quadratic and linear regression models for the tensile mechanical properties and quality indices of A357.Table 2
Comparison result of the multiple quadratic and linear regression models for tensile mechanical properties and quality indices of A357.
Multiple quadratic regression modelsMultiple linear regression modelsMAEMRE (%)R2MAEMRE (%)R2YS (MPa)15.9766.8190.78626.42311.5220.316UTS (MPa)9.5293.0520.74817.4615.8030.020Ef (%)1.22411.7950.8531.51514.7220.719W (MJ/m3)2.7939.0700.8672.9349.9880.831Q (MPa)7.5981.6630.80613.8543.0950.475QR (MPa)49.8296.2220.85656.8617.1890.779QC (MPa)13.4563.3130.72925.1306.4510.007Q0 (MPa)20.3953.6260.82824.2134.4990.752As can be seen in Table2, the performance of the multiple quadratic regression models is much superior than that of the multiple linear regression model. Therefore, it is reasonable to use multiple quadratic polynomial regression models rather than multiple linear regression models for the tensile mechanical properties and quality indices of the cast aluminum alloy A357 according to the artificial aging heat treatment condition.For external validation of the multiple quadratic polynomial regression models for tensile mechanical properties of A357, the CV method is used. Table3 shows the CV results of the regression models for tensile mechanical properties and quality indices of A357 using the CV method.Table 3
CV result of the multiple quadratic regression models for tensile mechanical properties and quality indices of A357.
MAEMRE (%)R2YS (MPa)35.04213.7390.702UTS (MPa)16.7885.4140.726Ef (%)3.46618.2600.999W (MJ/m3)8.02818.6540.995Q (MPa)11.2504.8650.993QR (MPa)89.24514.3350.898QC (MPa)21.5777.4350.976Q0 (MPa)21.9556.3610.982As it can be seen in Table3, the MREs are less than 20%; hence, the performance of the multiple quadratic regression models is excellent and good accurate. The R2 values are larger than 0.9 except for YS and UTS; hence, the regression models are very good. The R2 values for YS and UTS are larger than 0.7; hence, the regression models are satisfactory [20, 22].


## 3.1. Multiple Quadratic Polynomial Regression Model and Quality Map for Tensile Mechanical Properties according to Artificial Aging Heat Treatment Condition

The multiple quadratic polynomial regression model and quality map for YS of A357 are shown in equation (19) and Figure 1.(19)YS=−2538.186432+26.970192⋅T+32.097822⋅t−0.064578⋅T2−0.143513⋅T⋅t−0.111892⋅t2.Figure 1
3D surface graph of the regression model (a) and quality map (b) forYS of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(20)MAE=15.976MPa,MRE=6.819%,R2=0.786.By using equation (19) and Figure 1, it is possible to quantitatively and intuitively evaluate the YS value of the cast aluminum alloy A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat-treatment conditions that satisfy the design prerequisites for YS. In Figure 1, the number of alloys refers to Table 1.The multiple quadratic polynomial regression model and quality map for UTS of A357 are shown in equation (21) and Figure 2.(21)UTS=−1387.530241+17.131269⋅T+20.193788⋅t−0.042752⋅T2−0.096759⋅T⋅t−0.060772⋅t2.Figure 2
3D surface graph of the regression model (a) and quality map (b) forUTS of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(22)MAE=9.529MPa,MRE=3.052%,R2=0.748.By using equation (21) and Figure 2, it is possible to quantitatively and intuitively evaluate the UTS value of the cast aluminum alloy A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for UTS.The multiple quadratic polynomial regression model and quality map forEf of A357 are shown in equation (23) and Figure 3.(23)Ef=163.972973−1.373957⋅T−1.703747⋅t+0.003059⋅T2+0.006477⋅T⋅t+0.007572⋅t2.Figure 3
3D surface graph of the regression model (a) and quality map (b) for (E)f of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(24)MAE=1.224%MPa,MRE=11.795%,R2=0.853.By using equation (23) and Figure 3, it is possible to quantitatively and intuitively evaluate the Ef value of the cast aluminum alloy A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for Ef.The multiple quadratic polynomial regression model and quality map forW of A357 are shown in equation (25) and Figure 4.(25)W=248.534506−1.758050⋅T−2.157846⋅t+0.003477⋅T2+0.006329⋅T⋅t+0.011086⋅t2.Figure 4
3D surface graph of the regression model (a) and quality map (b) forW of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(26)MAE=2.79MJM3MPa,MRE=9.070%,R2=0.867.By using equation (25) and Figure 4, it is possible to quantitatively and intuitively evaluate the W value of the cast aluminum alloy A357 according to given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for W.


## 3.2. Multiple Quadratic Polynomial Regression Model and Quality Map for Quality Indices according to Artificial Aging Heat-Treatment Condition

The multiple quadratic polynomial regression model and quality map forQ of A357 are shown in equation (27) and Figure 5.(27)Q=−438.958673+9.912151⋅T+13.043748⋅t−0.026478⋅T2−0.071426⋅T⋅t−0.029495⋅t2.Figure 5
3D surface graph of the regression model (a) and quality map (b) for (Q) of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(28)MAE=7.598MPa,MRE=1.663%,R2=0.806.By using equation (27) and Figure 5, it is possible to quantitatively and intuitively evaluate the Q value of A357 according to the given artificial aging heat-treatment condition and select or determine the rational artificial aging heat-treatment conditions that satisfy the design prerequisites for Q.The multiple quadratic polynomial regression model and quality map forQR of A357 are shown in equation (29) and Figure 6.(29)QR=5660.462202−41.727683⋅T−53.089517⋅t+0.088387⋅T2+0.180326⋅T⋅t+0.266725⋅t2.Figure 6
3D surface graph of the regression model (a) and quality map (b) forQR of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE andR2 of this model are as follows:(30)MAE=49.829MPa,MRE=6.222%,R2=0.856.By using equation (29) and Figure 6, it is possible to quantitatively and intuitively evaluate the QR value of the cast aluminum alloy A357 according to the given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for QR.The multiple quadratic polynomial regression model and quality map forQC of A357 are shown in equation (31) and Figure 7.(31)QC=−1846.121845+22.618754⋅T+28.431034⋅t−0.056177⋅T2−0.136343⋅T⋅t−0.089533⋅t2.Figure 7
3D surface graph of the regression model (a) and quality map (b) forQC of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(32)MAE=13.456MPa,MRE=3.313%,R2=0.729.By using equation (31) and Figure 7, it is possible to quantitatively and intuitively evaluate the QC value of the cast aluminum alloy A357 according to a given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for QC.The multiple quadratic polynomial regression model and quality map forQ0 of A357 are shown in equation (33) and Figure 8.(33)Q0=−52.841375+9.389694⋅T+10.519360⋅t−0.029807⋅T2−0.080221⋅T⋅t−0.001030⋅t2.Figure 8
3D surface graph of the regression model (a) and quality map (b) forQ0 of cast aluminum alloy A357 according to artificial aging heat treatment condition.
(a)(b)The MAE, MRE, andR2 of this model are as follows:(34)MAE=20.395MPa,MRE=3.626%,R2=0.828.By using equation (33) and Figure 8, it is possible to quantitatively and intuitively evaluate the Q0 value of the cast aluminum alloy A357 according to a given artificial aging heat treatment condition and select or determine the rational artificial aging heat treatment conditions that satisfy the design prerequisites for Q0.In overall, the multiple quadratic polynomial regression models for tensile mechanical properties and quality indices according to artificial aging heat-treatment condition are as follows:(35)YS=−2538.186432+26.970192⋅T+32.097822⋅t−0.064578⋅T2−0.143513⋅T⋅t−0.111892⋅t2,UTS=−1387.530241+17.131269⋅T+20.193788⋅t−0.042752⋅T2−0.096759⋅T⋅t−0.060772⋅t2,Ef=163.972973−1.373957⋅T−1.703747⋅t+0.003059⋅T2+0.006477⋅T⋅t+0.007572⋅t2,W=248.534506−1.758050⋅T−2.157846⋅t+0.003477⋅T2+0.006329⋅T⋅t+0.011086⋅t2,Q=−438.958673+9.912151⋅T+13.043748⋅t−0.026478⋅T2−0.071426⋅T⋅t−0.029495⋅t2,QR=5660.462202−41.727683⋅T−53.089517⋅t+0.088387⋅T2+0.180326⋅T⋅t+0.266725⋅t2,QC=−1846.121845+22.618754⋅T+28.431034⋅t−0.056177⋅T2−0.136343⋅T⋅t−0.089533⋅t2,Q0=−52.841375+9.389694⋅T+10.519360⋅t−0.029807⋅T2−0.080221⋅T⋅t−0.001030⋅t2.The corresponding quality maps are shown in Figures1–8.The MREs of the above regression models are less than or slightly higher than 10%; hence, the regression models and quality maps are excellently accurate. TheR2 values of the above regression models are higher than 0.8 or 0.7, indicating that the regression models and quality maps are very good or may be satisfactory.


## 3.3. Comparison of the Developed Models with Multiple Linear Regression Models and External Validation Results

Table2 shows the performance comparison result between the multiple quadratic and linear regression models for the tensile mechanical properties and quality indices of A357.Table 2
Comparison result of the multiple quadratic and linear regression models for tensile mechanical properties and quality indices of A357.
Multiple quadratic regression modelsMultiple linear regression modelsMAEMRE (%)R2MAEMRE (%)R2YS (MPa)15.9766.8190.78626.42311.5220.316UTS (MPa)9.5293.0520.74817.4615.8030.020Ef (%)1.22411.7950.8531.51514.7220.719W (MJ/m3)2.7939.0700.8672.9349.9880.831Q (MPa)7.5981.6630.80613.8543.0950.475QR (MPa)49.8296.2220.85656.8617.1890.779QC (MPa)13.4563.3130.72925.1306.4510.007Q0 (MPa)20.3953.6260.82824.2134.4990.752As can be seen in Table2, the performance of the multiple quadratic regression models is much superior than that of the multiple linear regression model. Therefore, it is reasonable to use multiple quadratic polynomial regression models rather than multiple linear regression models for the tensile mechanical properties and quality indices of the cast aluminum alloy A357 according to the artificial aging heat treatment condition.For external validation of the multiple quadratic polynomial regression models for tensile mechanical properties of A357, the CV method is used. Table3 shows the CV results of the regression models for tensile mechanical properties and quality indices of A357 using the CV method.Table 3
CV result of the multiple quadratic regression models for tensile mechanical properties and quality indices of A357.
MAEMRE (%)R2YS (MPa)35.04213.7390.702UTS (MPa)16.7885.4140.726Ef (%)3.46618.2600.999W (MJ/m3)8.02818.6540.995Q (MPa)11.2504.8650.993QR (MPa)89.24514.3350.898QC (MPa)21.5777.4350.976Q0 (MPa)21.9556.3610.982As it can be seen in Table3, the MREs are less than 20%; hence, the performance of the multiple quadratic regression models is excellent and good accurate. The R2 values are larger than 0.9 except for YS and UTS; hence, the regression models are very good. The R2 values for YS and UTS are larger than 0.7; hence, the regression models are satisfactory [20, 22].


## 4. Conclusions

In this paper, the multiple quadratic polynomial regression models and quality maps for tensile mechanical properties and quality indices of the cast aluminum alloy A357 according to artificial aging heat-treatment condition have been developed.The developed multiple quadratic polynomial regression models and quality maps for tensile mechanical properties and quality indices of the cast aluminum alloy, A357, evaluate the tensile mechanical properties and quality indices of the cast aluminum alloys according to the artificial aging heat treatment condition with comparatively good accuracy. The multiple quadratic regression models may be widely used in practice because they have better performance than the multiple linear regression models and they are simpler than the artificial neural network models. By using the regression models and quality maps, materials designers and engineers can grasp the influences of the artificial aging heat-treatment conditions to the tensile mechanical properties and quality levels and decide the optimal or reasonable artificial aging heat-treatment conditions in practice.The proposed methodology may be applied to the other aluminum alloys.The limitation of this work is that the work did not consider the problems to improve the quality of the prediction models by developing the optimal regression models for tensile mechanical properties and quality indices according to artificial aging heat-treatment condition using artificial intelligent-based methodologies such as artificial neural network, metaheuristic-based machine learning, and soft computing methods owing to limited space. There is something yet to be studied. Future work is needed to study about the problems.



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*Source: 7069987-2023-02-06.xml*