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Numerical Descriptions of Hot Flow Behaviors across β Transus for as-Forged Ti–10V–2Fe–3Al Alloy by LHS-SVR and GA-SVR and Improvement in Forming Simulation Accuracy

Guo-Zheng Quan, Zhi-Hua Zhang, Le Zhang, Qing Liu

2016
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Applied Sciences
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Hot compression tests of as-forged Ti-10V-2Fe-3Al alloy in a wide temperature range of 948-1123 K and a strain rate range of 0.001-10 s´1 were conducted by a servo-hydraulic and computer-controlled Gleeble-3500 machine. In order to accurately and effectively model the non-linear flow behaviors, support vector regression (SVR), as a machine learning method, was combined with Latin hypercube sampling (LHS) and genetic algorithm (GA) to respectively characterize the flow behaviors, namely LHS-SVR
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... rs, namely LHS-SVR and GA-SVR. The significant characters of LHS-SVR and GA-SVR are that they, with identical training parameters, can maintain training accuracy and prediction accuracy at stable levels in different attempts. The study abilities, generalization abilities and modelling efficiencies of the mathematical regression model, artificial neural network (ANN), LHS-SVR and GA-SVR were compared in detail by using standard statistical parameters. After comparisons, the study abilities and generalization abilities of these models were shown as follows in ascending order: the mathematical regression model < ANN < GA-SVR < LHS-SVR. The modeling efficiencies of these models were shown as follows in ascending order: mathematical regression model < ANN < LHS-SVR < GA-SVR. The flow behaviors outside experimental conditions were predicted by the well-trained LHS-SVR, which improves the simulation precision of the load-stroke curve. Nowadays, there exist three representative models in characterizing flow behaviors of metals, i.e., the empirical/semiempirical model, the analytical model and the phenomenological model [4] [5] [6] . Hollomon presented a typical empirical model (σ " K H ε n ) for plastic deformation with a lower level of accuracy, in which K H and n are constants [7] . Afterwards, Guan et al. [8] proposed an improved empirical model with a higher level of accuracy (correlation coefficient R = 0.998) due to considering the effects of strain rate and strain on stress. However, the empirical model cannot accurately track the highly non-linear flow behaviors at different strain rates and temperatures. Additionally, many parameters of the empirical model need to be recalculated when some new experimental data are involved. The analytical model requires explicit and thorough study of microscopic deformation mechanisms, such as dislocation theory, DRV, DRX, etc. [9]. Vanini et al. established an analytical model (R « 0.988), which just considered the phase mixture law and boundary layer characteristics of the hot deformation behaviors of functionally-graded steels [10]. George Z. Voyiadjis et al. established the analytical constitutive models involving dislocations interaction mechanisms and thermal activation energy for the flow behaviors under different temperatures and strain rates of face centered cubic (FCC) and body centered cubic (BCC) metals [11] . In the investigation of George Z. Voyiadjis et al., the mobile dislocation density has different influences on flow behaviors at different temperatures and strain rates, so it needs to establish several constitutive models at different deformation conditions; otherwise, the physical-based analytical model cannot accurately characterize the highly non-linear deformation behaviors [11] . Besides, analytical models need many precise experiment data to construct a mathematical model of complicated microscopic deformation mechanisms. Thereby, analytical models have not been widely utilized in characterizing intricate flow behaviors of metals. The phenomenological model involves the mathematical regression equation and intelligence algorithm, and it does not need to deeply investigate complicated microscopic deformation mechanisms. The mathematical regression equations just need to calculate some essential material constants and further be fitted based on limited experimental data. At present, the typical Arrhenius-type equation of the phenomenological model and its modified forms were used to characterize the hot flow behaviors of many materials, such as Ti60 (R « 0.99) [12], Ti-6Al-4V [13], pure titanium [14], etc. Other phenomenological models involve the representative Johnson-Cook model (R = 0.9772), the Fields-Backofen model (R = 0.87025), the Khan-Huang-Liang model (R = 0.96559), the mechanical threshold stress model (R = 0.9614), etc.; nevertheless, they have large fluctuant accuracies at different strain rates and temperatures [15]. Additionally, Akbari et al. further pointed out that the original Johnson-Cook model was not able to predict the softening part of the flow stress curves [16]. The mathematical regression equations of the phenomenological model cannot accurately track the highly non-linear flow behaviors at different strain rates and temperatures [15, 17] . Because they are mathematically fitted based on limited experimental data. lately, the artificial neural network (ANN) of intelligence algorithm, which imitates biological neural systems, was adopted to characterize the flow behaviors of AZ80 magnesium alloy [18], A356 aluminum alloy [19], Ti-10V-2Fe-3Al alloy [20], etc. ANN needs to try many network topologies and training parameters to achieve a higher accuracy, which will consume much time. For a certain dataset, the identical network topology and training parameters of an ANN will obtain fluctuant accuracies in different attempts. ANN can meet network topology and training parameters well to achieve a higher accuracy level; however, these accurate results have poor reproducibility. Worse still, ANN easily falls into local extreme values and cannot attain a globally-optimal solution. Support vector regression (SVR), as a machine learning method according to the structural risk minimization principle and statistical learning theory, is mostly used in the regression analysis field [21] . SVR has the advantages of strong generalization ability, robustness and a systemic theoretical system. Compared to ANN, SVR can avoid falling into local extreme values and can attain a globally-optimal solution. For a certain dataset, an SVR with identical training parameters will maintain training accuracy and prediction accuracy at stable levels in different attempts. In this work, SVR was adopted to characterize the hot flow behaviors of as-forged Ti-10V-2Fe-3Al alloy on account of Appl. Sci. 2016, 6, 210 3 of 23 its excellent advantages. The learning ability and generalization ability of an SVR rely on three parameters (penalty factor C, kernel parameter γ and insensitive loss function ζ), especially the mutual impacts among them. Therefore, SVR needs to adjust the three parameters (C, γ and ζ) to attain a precise prediction model. An SVR with appropriate parameters C, γ and ζ will accurately study the stress-strain curves and appropriately ignore some singular points of stress-strain data to accord with the overall trend of the curves. The influence of the combination of the three parameters (C, γ and ζ) on the learning ability and generalization ability of an SVR should be synthetically considered. It is time consuming to independently adjust the three parameters one by one to establish an SVR that can accurately characterize the hot flow behaviors of as-forged Ti-10V-2Fe-3Al alloy. Thereby, it is important to find a stable and efficient method to realize the optimal selection of the three parameters in SVR. Lou et al. established an SVR combined with particle swarm optimization (PSO) to characterize the flow behaviors of AZ80 magnesium alloy in which PSO was adopted to select the parameters C, γ and ζ, and this study indicates that the model is more accurate than ANN and the constitutive equation; besides, the sample dependence of the SVR is lower [22] . Raghuram Karthik Desu et al. constructed an SVR to characterize the flow behaviors of Austenitic Stainless Steel 304, and they found that SVR is more precise, efficient and reliable than the mathematical regression equations, such as the revised-Arrhenius model, the Johnson-Cook model, the revised Zerrili-Armstrong model and the intelligence algorithm ANN model [23] . The best correlation coefficient (R) in the work of Raghuram Karthik Desu et al. is 0.9989 at a high accuracy level; nevertheless, they only attempted a few parameter combinations of the three parameters (C, γ and ζ), and there is still room for improvement in accuracy and efficiency [23] . The Latin hypercube sampling (LHS) method, as a uniform sampling method, was used to ensure that sampling areas can be uniformly covered by all sampling points [24] . In this work, the sampling points represent the combinations of the three parameters (C, γ and ζ) in three-dimensional space. There are two advantages of LHS. Firstly, the sampling points generated by LHS can effectively fill the sampling space. Secondly, less sampling points generated by LHS can represent more combinations of parameters. In this work, a novel SVR for the flow behaviors of as-forged Ti-10V-2Fe-3Al alloy combined with LHS was established, i.e., the LHS-SVR. Compared to the work of Raghuram Karthik Desu et al. [23] the sampling points generated by LHS-SVR can effectively fill the sampling space and represent more combinations of the three parameters (C, γ and ζ). LHS-SVR can show the influences of the three parameter combinations on the accuracy of the model, and this influence law can be used as a reference to researchers when they need to select the three parameters. However, LHS-SVR needs to calculate the many parameter combinations in the search space, and there is still room for improvement in efficiency. GA, as a global optimization algorithm, has been widely used in the multi-parameter optimization area on account of the advantages of parallel processing, high efficiency and strong robustness. GA searches the optimal parameters in the solution space by imitating the natural selection process and genetic mechanism. In order to use the advantages of GA, an SVR model of the hot flow behaviors of as-forged Ti-10V-2Fe-3Al alloy combined with GA was constructed, i.e., the GA-SVR. In GA-SVR, GA was utilized to efficiently seek the optimal parameter combination of the three parameters (C, γ and ζ). The GA-SVR only requires representative training samples from the research problem and then self-adaptively and dynamically adjusts the three parameters (C, γ and ζ) to attain the most accurate SVR. In this work, the comparisons of the learning abilities, generalization abilities and modelling efficiencies of the mathematical regression model, ANN, LHS-SVR and GA-SVR were investigated. A standard statistical parameter, average absolute relative error (AARE), was utilized to estimate the predicted performance of these four prediction models. In the comparisons of study abilities, the LHS-SVR and GA-SVR have larger R-values and lower AARE-values, which show that the LHS-SVR and GA-SVR can sufficiently and accurately learn the training samples. The study abilities of these models were shown as follows in ascending order: ANN < GA-SVR < LHS-SVR. In the comparisons of

doi:10.3390/app6080210
fatcat:kckbmupqgfctzh6qtvawz7nbii