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Optimization of Turbine Blade Aerodynamic Designs Using CFD and Neural Network Models

Optimization of Turbine Blade Aerodynamic Designs Using CFD and Neural Network Models International Journal of Turbomachinery Propulsion and Power Article Optimization of Turbine Blade Aerodynamic Designs Using CFD and Neural Network Models Chao Zhang * and Matthew Janeway Siemens Digital Industries Software, Cypress, CA 90630, USA; matt.janeway@siemens.com * Correspondence: chao.zhang1@siemens.com Abstract: Optimization methods have been widely applied to the aerodynamic design of gas turbine blades. While applying optimization to high-fidelity computational fluid dynamics (CFD) simulations has proven capable of improving engineering design performance, a challenge has been overcoming the prolonged run-time due to the computationally expensive CFD runs. Reduced-order models and, more recently, machine learning methods have been increasingly used in gas turbine studies to predict performance metrics and operational characteristics, model turbulence, and optimize designs. The application of machine learning methods allows for utilizing existing knowledge and datasets from different sources, such as previous experiments, CFD, low-fidelity simulations, 1D or system-level studies. The present study investigates inserting a machine learning model that utilizes such data into a high-fidelity CFD driven optimization process, and hence effectively reduces the number of required evaluations of the CFD model. Artificial Neural Network (ANN) models were trained on data from over three thousand two-dimensional (2D) CFD analyses of turbine blade cross-sections. The trained ANN models were then used as surrogates in a nested optimization process alongside a full three-dimensional Navier–Stokes CFD simulation. The much lower evaluation cost of the ANN model allows for tens of thousands of design evaluations to guide the search of the best blade profiles to be used in the more expensive, high-fidelity CFD runs, improving the progress of the optimization while reducing the required computation time. It is estimated that the current workflow achieves a five-fold reduction in computational time in comparison to an optimization process that Citation: Zhang, C.; Janeway, M. is based on three-dimensional (3D) CFD simulations alone. The methodology is demonstrated on Optimization of Turbine Blade Aerodynamic Designs Using CFD the NASA/General Electric Energy Efficient Engine (E3) high pressure turbine blade and found and Neural Network Models. Int. J. Pareto front designs with improved blade efficiency and power over the baseline. Quantitative Turbomach. Propuls. Power 2022, 7, 20. analysis of the optimization data reveals that some design parameters in the present study are more https://doi.org/10.3390/ijtpp7030020 influential than others, such as the lean angle and tip scaling factor. Examining the optimized designs also provides insight into the physics, showing that the optimized designs have a lower amount Academic Editor: Tom Verstraete of pressure drop near the trailing edge, but have an earlier onset of pressure drop on the suction Received: 25 April 2022 side surface when compared to the baseline design, contributing to the observed improvements in Accepted: 28 June 2022 efficiency and power. Published: 30 June 2022 Publisher’s Note: MDPI stays neutral Keywords: CFD; optimization; aerodynamics; gas turbines; machine learning; neural networks with regard to jurisdictional claims in published maps and institutional affil- iations. 1. Introduction The present study applies machine learning methods in a CFD-based optimization for turbine blade aerodynamics. Literature on optimization and machine learning methods Copyright: © 2022 by the authors. used in gas turbine studies will be reviewed, followed by a summary of the motivation of Licensee MDPI, Basel, Switzerland. the present work. This article is an open access article Numerical optimization has been widely used in the design and analysis of gas tur- distributed under the terms and conditions of the Creative Commons bines. In some earlier studies, specific optimization algorithms have been investigated to Attribution (CC BY-NC-ND) license leverage lower fidelity models to achieve fast optimization time. One study on aerody- (https://creativecommons.org/ namic wing optimization has used an approximation and model management optimization licenses/by-nc-nd/4.0/). Int. J. Turbomach. Propuls. Power 2022, 7, 20. https://doi.org/10.3390/ijtpp7030020 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 20 2 of 19 method to incorporate low fidelity, computationally cheaper models with occasional re- course to higher fidelity, more expensive models, resulting in threefold saving in optimiza- tion time [1]. Another study adopted a similar concept, employing a trust-region approach to interleave the exact models with cheaper surrogate models during optimization iter- ations [2]. These methods demonstrate the possibility of obtaining optimized solutions on a limited computational budget by incorporating lower-fidelity surrogate models. In more recent years, an increasing number of optimization studies have relied on using parametric CFD models. In optimization of the coolant flow passage of the NASA C3X vane, different designs were evaluated repeatedly through CFD runs [3]. In the study of a marine high-pressure turbine [4], ten design parameters controlling multi-row film cooling designs were built into the CFD model and optimized based on a non-dominated sorting genetic algorithm. Multiple studies were also conducted on ultra-super-critical steam turbines [5,6], in which the blade aerodynamic efficiency was optimized using 2D and 3D CFD simulations driven by Siemens’ Simcenter HEEDS commercial Sherpa opti- mization algorithm. A CFD-based co-optimization strategy was presented in [7], which demonstrates a workflow for coupling different disciplines into a nested optimization loop to conduct parallel blade aerodynamic and thermal optimizations. In addition to improving turbine blade designs, optimization has also been applied to improving the operations of gas turbine engines, for instance, to find the best valve setup parameters that reduce fuel consumption [8]. With the advancements in computer science and data storage, an increase in interest in application of machine learning methods to gas turbine designs has been observed. One area of such applications seeing increased interest has been the prediction of key performance metrics using models trained on input data gained through past simulations or experiments. In an earlier study [9], the outlet temperature and fuel mass flow rate at different operating conditions for a 255 MW single-shaft gas turbine were predicted by building a three-layer neural network model. Another application of the Artificial Neural Network (ANN) model has been seen in a turbine film cooling study to predict the instantaneous temperature distributions along the blade surface as well as the cooling effec- tiveness [10]. In another study of a jet engine power plant [11], a machine learning method combining physics-based and measurement-driven modeling was developed and used to conduct preventive maintenance and diagnose faults. Machine learning methods were also applied to a Viper 632-43 military turbojet engine to predict the exhaust temperature using models trained on data collected from a gas turbine simulation program [12]. Extending from predicting individual engineering metrics, machine learning has also been used to predict field quantities representing more complicated underlying physics. A method using gradient boosted trees was used to develop models of aerodynamic loads on vibrating turbine blades and demonstrated to have good agreement with detailed CFD results [13]. In another study, the turbine surface pressure distribution was predicted using transfer learning models, which transfer knowledge from a large-scale but low-fidelity dataset to a small-scale but high-fidelity dataset, shown to have a low prediction error with reduced cost [14]. Machine learning has also been applied to the prediction of operating charac- teristics of gas turbine engines, using real-time data of power plants to develop neural networks [15,16]. In addition to the above applications, machine learning has also been studied to develop turbulence closure models. In one study for wake mixing, a machine learning model was demonstrated to be robust across several different operating conditions when integrated into a RANS CFD model of a low-pressure turbine [17]. A review article on machine learning methods for science and engineering particularly highlighted the need for interpretable, generalizable, expandable, and certifiable machine learning techniques for safety-critical applications [18]. Recent works have also focused on using machine learning embedded into design opti- mization procedures. In the optimization of a centrifugal compressor impeller [19], an ANN model was first developed using CFD and FEA data from a Design of Experiment (DOE) study. Then, the ANN model was applied in an optimization procedure, which resulted in a Int. J. Turbomach. Propuls. Power 2022, 7, 20 3 of 19 1% increase in isentropic efficiency and 10% reduction in the blade stress. In another study investigating a carved blade tip [20], 55 CFD runs were conducted to generate ANN meta models, which were then used in a genetic algorithm routine to optimize the blade tip shape. In a missile control surface optimization study [21], machine learning, reinforcement learn- ing, and transfer learning were integrated into the optimization procedure and leveraged CFD in the evaluation iterations. In another study of 2D airfoil optimization [22], a deep convolutional generative adversarial network was trained and embedded as a surrogate model in an optimization framework. In still another study of a compact turbine rotor [23], machine learning models were trained and used to optimize the efficiency and torque based on a gradient-based multi-objective optimization algorithm. In addition to using machine learning models in optimization, several CFD and optimization studies have compared machine learning models with response surface models (RSM). In a study of aircyclone optimization [24], it is concluded that ANN offers an alternative and powerful approach to response surface methods for modeling the cyclone pressure drop, benchmarked against experimental data. In another study of modeling and optimizing a perforated baffle used for turbine blade passage cooling [25], both ANN and RSM methods were found capable of predicting friction factor and Nusselt number values, although the RSM method performed slightly better than ANN in that study. A more recent study on cyclone optimization also tested a RSM and several machine learning models. A GMDH-neural network model was found to be superior and chosen for the optimization process [26]. As discussed in the above literature, optimization that leverages high-fidelity CFD simulations can provide accurate and realistic optimal designs in general. While leveraging a neural network as a surrogate to replace the CFD evaluations in the optimization loop can significantly improve the computational time, the fundamental challenge is that a neural network model is a lower fidelity model compared to a CFD simulation, and therefore, it may not be as accurate as CFD in predicting certain design variants required by the optimization process. Further, neural network models may also be trained based on previously available datasets that come from different studies, such as 1D and 2D simulation data, and experimental data, all of which will result in the neural network being a reduced order model compared to 3D CFD. As such, relying solely on the neural network when the predictive accuracy is necessary can lead to errant results. The present study presents a nested optimization workflow that can leverage both neural networks and high-fidelity, 3D CFD simulations at the same time to ensure every “best” design is studied in detail by the CFD tool. The introduction of a neural network into the optimization allows for over 70% reduction in the number of CFD evaluations and thus makes significant reductions in computational cost compared to a process relying exclusively on CFD simulations. The methodology is demonstrated through an aerodynamic optimization using a rebuilt model of the NASA/General Electric E3 high pressure turbine blade [27–29]. 2. Methodology Artificial Neural Network models are used alongside 3D CFD simulations in a nu- merical optimization procedure to improve the aerodynamic performance of a turbine blade. ANN models are typically trained using large datasets obtained from previous experimental or numerical studies. In companies/organizations that conduct R&D on engi- neering designs, these datasets are usually available from previous studies. The present study proposes a methodology of using such existing knowledge from previous studies to train ANN models and then use the ANN models in an CFD-based design optimization process. A dataset from a previously published work is obtained to conduct the ANN modeling training in the present study. The overall framework of the research methodology is shown in Figure 1, which illustrates how different analysis models/tools required by the optimization process are created. First, data were obtained from a previous study [7] containing 3204 CFD design evaluations of different 2D turbine blade profiles. Performance metrics including efficiency and power were extracted as targets, with the blade geometric parameters serving as input features, to train ANN models. The ANN hyperparameters Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 4 of 23 [7] containing 3204 CFD design evaluations of different 2D turbine blade profiles. Perfor- mance metrics including efficiency and power were extracted as targets, with the blade Int. J. Turbomach. Propuls. Power 2022, 7, 20 4 of 19 geometric parameters serving as input features, to train ANN models. The ANN hyperpa- rameters were subsequently optimized. The blade cross-sectional profiles were parame- terized using a Class-Shape Transformation (CST) method [30]. Using the 2D blade cross- were subsequently optimized. The blade cross-sectional profiles were parameterized using sectional profile, the 3D blade CAD geometry was created with additional parameteriza- a Class-Shape Transformation (CST) method [30]. Using the 2D blade cross-sectional profile, tion allowing for the variation of further design transformation, such as the thickness, the 3D blade CAD geometry was created with additional parameterization allowing for twist, and lean. The blade CAD geometry was then used to build a 3D Reynolds Averag- the variation of further design transformation, such as the thickness, twist, and lean. The ing Navier–Stokes (RANS) CFD model, which solves for the flow field in a single blade blade CAD geometry was then used to build a 3D Reynolds Averaging Navier–Stokes pa (RANS) ssage. Onc CFD e the model, ANN which models solves and for the the 3D C flow FD field simu in laa tion single areblade indeppassage. endentlyOnce devethe loped, they ANN are inte models grand ated into a the 3D CFD neste simulation d optimizati are independently on procedure developed, for repeated ev they ar aluation e integrated s of dif- into a nested optimization procedure for repeated evaluations of different blade design ferent blade design variants to optimize the design parameters. In the nested optimization variants to optimize the design parameters. In the nested optimization process, an inner process, an inner optimization loop is executed using the ANN models to improve the optimization loop is executed using the ANN models to improve the blade cross-sectional blade cross-sectional profile. In each iteration of the global optimization, the best blade profile. In each iteration of the global optimization, the best blade profile is selected from profile is selected from the inner optimization step to create a 3D CFD blade model with the inner optimization step to create a 3D CFD blade model with additional parameters additional parameters such as the scaling factors, twist, and lean angles. The optimization such as the scaling factors, twist, and lean angles. The optimization targets improving the targets improving the efficiency and power of the blade. Details of the optimization pro- efficiency and power of the blade. Details of the optimization process will be discussed in a cess will be discussed in a later session in this chapter. later session in this chapter. Figure 1. The overall framework of the research methodology. (2D CFD data was collected from a Figure 1. The overall framework of the research methodology. (2D CFD data was collected from a previous study [7].) previous study [7].) 2.1. Artificial Neural Network 2.1. Artificial Neural Network A dataset representing different design variants of 2D aerodynamic CFD results was A dataset representing different design variants of 2D aerodynamic CFD results was obtained from a previous study [7] and used to train ANN models. The dataset features obtained from a previous study [7] and used to train ANN models. The dataset features blade design and performance parameters for 3204 designs. The input parameters for the blade design and performance parameters for 3204 designs. The input parameters for the ANN models include 20 blade design parameters. These parameters are the weighting ANN models include 20 blade design parameters. These parameters are the weighting factors used by the CST method [30] to control variations in the blade profile. Under the factors used by the CST method [30] to control variations in the blade profile. Under the CST method, these weighting factors are multiplied by Bernstein polynomials to define a CST method, these weighting factors are multiplied by Bernstein polynomials to define a shape function, and then subsequently multiplied by a class function to define the aero- shape function, and then subsequently multiplied by a class function to define the aero- dynamic blade profile. An order-9 Bernstein polynomial was deemed sufficiently flexible for the purposes of the study [7] and that choice is repeated here, resulting in 10 design dynamic blade profile. An order-9 Bernstein polynomial was deemed sufficiently flexible parameters for each of the suction and pressure sides of the blade: fw , w , . . . , w g for the purposes of the study [7] and that choice is repeated here, resulting in 10 design u,1 u,2 u,10 and w , w , . . . , w . These 20 parameters are used as input features in the ANN parameter l,1s for each of l,2 l,10 the suction and pressure sides of the blade: {𝑤 ,𝑤 ,…, 𝑤 } , , , models and optimization parameters in the subsequent blade optimization process. Two and {𝑤 ,𝑤 ,…, 𝑤 }. These 20 parameters are used as input features in the ANN mod- , , , performance metrics in the dataset were used as output parameters, isentropic efficiency, els and optimization parameters in the subsequent blade optimization process. Two per- formance metrics in the dataset were used as output parameters, isentropic efficiency, and Int. J. Turbomach. Propuls. Power 2022, 7, 20 5 of 19 and power output, for which two ANN models were developed, respectively. The efficiency and power are defined based on enthalpy quantities of the flow, as follows: h h i o h = , (1) h h i o,isen. Pow = Dh = h h (2) The Keras API for TensorFlow [31] was used to construct the ANN models. In consid- eration of the different distributions of the efficiency and power data and the robustness of the model, a separate ANN model was constructed for each of these two quantities. After some preliminary tests on ANN models of 5, 6, and 7 layers, 7 and 6 layers were constructed, respectively, into the ANN models representing efficiency and power. The choice of the number of layers in ANN usually varies in different problems. It will be shown that the prediction errors are within acceptable range for the chosen number of layers, and that the errors will be further reduced by optimizing the hyperparameters of the models. The first layer of each ANN model has 20 inputs representing the blade input features (CST weights) and the last layer has one output, which is either efficiency or power. The number of neurons in the hidden layers of each ANN, along with other topological and training parameters, were optimized in a separate hyperparameter optimization process. As a starting point for that process, the hyperparameters for each ANN model were set using the values provided in the first two columns of Table 1. Table 1. Hyperparameters of the initial and optimized ANN models. Initial ANN Initial ANN Optimized ANN Optimized ANN Hyperparameters for Efficiency for Power for Efficiency for Power Number of Neurons-Layer 2 20 20 5 23 Number of Neurons-Layer 3 40 40 35 15 Number of Neurons-Layer 4 20 20 53 36 Number of Neurons-Layer 5 10 5 50 42 Number of Neurons-Layer 6 5 – 62 – Activation Function-Layer 2 relu relu relu selu Activation Function-Layer 3 relu relu sigmoid softmax Activation Function-Layer 4 relu relu elu elu Activation Function-Layer 5 relu relu elu softmax Activation Function-Layer 6 relu – softmax – 5 3 Learning Rate 0.001 0.001 6.4895  10 1.47285  10 Number of Epochs 200 200 6123 4000 To improve the predictive performance of the ANN, the hyperparameters were opti- mized using Siemens commercial optimization tool HEEDS and its Sherpa algorithm [32]. A diagram of the optimization process is provided in Figure 2. The optimizer repeatedly tunes all the hyperparameters for the ANN models listed in the index column of Table 1 with an objective to minimize the model test error on a held-out testing dataset. In each optimization cycle, the dataset is randomly split into a training and testing set with a ratio of 95/5. The training set is used the train the ANN model. The training process involves a typical backpropagation procedure with an 80/20 split of the training set, over multiple epochs with TensorFlow’s ADAM optimizer. After the training step is completed, the model’s performance is then evaluated against the held-out test set. The evaluation error is passed back to the optimization solver to tune the hyperparameters for the next cycle. A total number of 240 iterations were executed during the optimization of each of the two ANNs. The evaluation number, 240, was based on the Sherpa algorithm best practice [32], which recommends the number of evaluations being about 10 times the number of input variables for single objective optimization problems, due to the hybrid and adaptive meth- ods employed by the algorithm. In the ANN optimization, 12 hyperparameters must be Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 6 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 6 of 19 and adaptive methods employed by the algorithm. In the ANN optimization, 12 hyperpa- rameters must be optimized, and thus it requires at least 120 evaluations according to the best practice. In consideration of the relatively fast run time for the ANN training process optimized, and thus it requires at least 120 evaluations according to the best practice. In consideration of the relatively fast run time for the ANN training process in comparison in comparison to the 3D CFD run time, that evaluation number is further doubled to 240 to the 3D CFD run time, that evaluation number is further doubled to 240 to provide to provide better accuracy. better accuracy. Figure 2. Figure ANN mod 2. ANN model el opti optimization mization d diagram. iagram. The optimized hyperparameters are obtained and provided in the last two columns of The optimized hyperparameters are obtained and provided in the last two columns Table 1. After optimization, the evaluation errors for efficiency and power were reduced, of Table 1. After optimization, the evaluation errors for efficiency and power were re- respectively, from 0.33% to 0.10%, and from 0.43% to 0.10%. The performance of the initial duced, r and optimized espective ANN ly, from models 0.3 ar 3e % to also examined 0.10%, and by comparing from 0.43% the pr to 0 edicted .10%values . The p vs. erfor truemance of values from the test set, shown in Figure 3. It is observed that the optimized ANN models the initial and optimized ANN models are also examined by comparing the predicted result in better alignment between the predicted values and the true values, as well as values vs. true values from the test set, shown in Figure 3. It is observed that the optimized narrower bandwidths of the prediction errors on the histogram plots. From Figure 3, it ANN models result in better alignment between the predicted values and the true values, is also observed that the bias of the optimized ANN for efficiency was greatly reduced, as well as narrower bandwidths of the prediction errors on the histogram plots. From as the data points are evenly distributed on both sides of the line of unity slope, shown Figuin re 3, it is al the second so subplot. observ This ed tha is consistent t the bias with of the theoptim histograms ized A of N the N for prediction efficienc errors y was gre in atly Figure 4, which shows that both the bias and bandwidth of the errors were improved reduced, as the data points are evenly distributed on both sides of the line of unity slope, in the optimized ANN models. The optimized ANN models are used in the subsequent shown in the second subplot. This is consistent with the histograms of the prediction er- blade optimization process in this study. The Python scripts for training the ANN models rors in Figure 4, which shows that both the bias and bandwidth of the errors were im- with the optimized hyperparameters are provided in [33]. The CPU time for training an proved in the optimized ANN models. The optimized ANN models are used in the sub- individual ANN largely depends on the number of Epochs. In the present study, the CPU times for training the ANN models representing efficiency and power with their respective sequent blade optimization process in this study. The Python scripts for training the ANN optimized hyperparameters were 14.3 min and 8.5 min, respectively. The total compute models with the optimized hyperparameters are provided in [33]. The CPU time for train- times for optimizing the hyperparameters of these ANN models were, respectively, 13.07 h ing an individual ANN largely depends on the number of Epochs. In the present study, and 11.65 h. This optimization was carried out based on parallel execution of 10 training the CPU times for training the ANN models representing efficiency and power with their models at the same time using a 6-core CPU. respective optimized hyperparameters were 14.3 min and 8.5 min, respectively. The total compute times for optimizing the hyperparameters of these ANN models were, respec- tively, 13.07 h and 11.65 h. This optimization was carried out based on parallel execution of 10 training models at the same time using a 6-core CPU. Int. J. Turbomach. Propuls. Power 2022, 7, 20 7 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 7 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 8 of 23 Figure 3. Predicted vs. true values of the initial and optimized ANN models. Figure 3. Predicted vs. true values of the initial and optimized ANN models. Figure 4. Histogram plots of the prediction errors of the initial and optimized ANN models. Figure 4. Histogram plots of the prediction errors of the initial and optimized ANN models. 2.2. Parametric Blade CAD and CFD Model 2.2. Parametric Blade CAD and CFD Model An initial turbine blade CAD model was built by adapting the E3 high pressure An initial turbine blade CAD model was built by adapting the E3 high pressure tur- turbine blade profiles. Using the first stage rotor blade coordinates provided at three bine blade profiles. Using the first stage rotor blade coordinates provided at three spanwise locations [27]—hub (12.693in), pitch (13.571in), and tip (14.41in)—and accounting spanwise locations [27]—hub (12.693in), pitch (13.571in), and tip (14.41in)—and account- for the incoming flow angle [28,29], a 3D CAD model was constructed by lofting these ing for the incoming flow angle [28,29], a 3D CAD model was constructed by lofting these cross-sectional profiles, as shown in Figure 5a, to provide a baseline case. Applying the cross-sectional profiles, as shown in Figure 5a, to provide a baseline case. Applying the same philosophy of lofting cross sections and at the same time utilizing the CST method same philosophy of lofting cross sections and at the same time utilizing the CST method for the cross-sectional profile definition [27], a parameterized blade CAD model was for the cross-sectional profile definition [27], a parameterized blade CAD model was cre- created for the design optimization study. First, the CST method creates a base cross- ated for the design optimization study. First, the CST method creates a base cross-sectional sectional profile, which is used as the pitch profile. The CST method allows varying the profile, which is used as the pitch profile. The CST method allows varying the base profile base profile in design optimization through manipulating the 20 weighting parameters, in design optimization through manipulating the 20 weighting parameters, fw , w , . . . , w g and w , w , . . . , w . Then, the base profile is scaled to create u,1 u,2 u,10 l,1 l,2 l,10 {𝑤 ,𝑤 ,…, 𝑤 } and {𝑤 ,𝑤 ,…,𝑤 }. Then, the base profile is scaled to create the , , , , , , the hub and tip profiles. In the scaling process, the twist angles and chord lengths of the hub and tip profiles. In the scaling process, the twist angles and chord lengths of the hub hub and tip profiles are matched to their counterparts of the original E3 blade; 4 additional n o and tip profiles are matched to their counterparts of the original E3 blade; 4 additional scaling factors, x , x , x , x , were applied to the lower and upper profiles hub,l hub,u ti p,l ti p,u scaling factors, {𝜉 , 𝜉 ,𝜉 ,𝜉 }, were applied to the lower and upper profiles of , , , , the hub and tip sections, respectively, to allow for small variations of the thicknesses of the hub and tip sections so that they can be optimized. The ranges of these variations are defined conservatively out of practical considerations of the original E3 engine blade shape. In general, the scaling allows the hub profile to be thicker and the tip profile to be thinner. Finally, a tilt angle, 𝜃 , and a lean angle, 𝛽 , were also built into the CAD model to be optimized later. Based on a reverse engineering study analyzing different engine blade geometries [34], the tilt angle can vary from 0 to 0.164°, and the lean angle can vary from −0.086° to 0.086° in the optimization study. A schematic plot of these angles is shown in Figure 5b. Int. J. Turbomach. Propuls. Power 2022, 7, 20 8 of 19 of the hub and tip sections, respectively, to allow for small variations of the thicknesses of the hub and tip sections so that they can be optimized. The ranges of these variations are defined conservatively out of practical considerations of the original E3 engine blade shape. In general, the scaling allows the hub profile to be thicker and the tip profile to be thinner. Finally, a tilt angle, q, and a lean angle, b, were also built into the CAD model to be optimized later. Based on a reverse engineering study analyzing different engine blade Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 9 of 23 geometries [34], the tilt angle can vary from 0 to 0.164 , and the lean angle can vary from 0.086 to 0.086 in the optimization study. A schematic plot of these angles is shown in Figure 5b. (a) (b) Figure 5. Blade geometry schematics: (a) blade geometry adapted from E3 profile coordinates; (b) Figure 5. Blade geometry schematics: (a) blade geometry adapted from E3 profile coordinates; blade angles schematic. (b) blade angles schematic. Using the 3D blade, a single-blade passage CFD model was developed using Siemens Using the 3D blade, a single-blade passage CFD model was developed using Siemens multi-physics package Simcenter STAR-CCM+. The continuity, momentum, and fluid en- multi-physics package Simcenter STAR-CCM+. The continuity, momentum, and fluid ergy transport equations are solved using a coupled solver (density-based) following a energy transport equations are solved using a coupled solver (density-based) following a finite volume approach with a 2nd order upwind discretization scheme on a polyhedral finite volume approach with a 2nd order upwind discretization scheme on a polyhedral grid. Menter’s SST K-Omega model [35] with all y+ treatment is used as a closure to the grid. Menter ’s SST K-Omega model [35] with all y+ treatment is used as a closure to the turbulence model. The all y+ wall treatment adjusts the application of a turbulence wall turbulence model. The all y+ wall treatment adjusts the application of a turbulence wall function based on the local y+ value of the near wall mesh cell. The CFD simulations were function based on the local y+ value of the near wall mesh cell. The CFD simulations were performed using the test conditions. Following E3 engine rotor testing conditions and 2D performed using the test conditions. Following E3 engine rotor testing conditions and 2D CFD practices from NASA studies [28,29], a total pressure of 344,777 Pa and a total tem- CFD practices from NASA studies [28,29], a total pressure of 344,777 Pa and a total temper- perature of 709 K were used for the inlet boundary condition, while an atmospheric pres- ature of 709 K were used for the inlet boundary condition, while an atmospheric pressure of sure of 101,325 Pa was defined at the outlet. The same boundary conditions representing 101,325 Pa was defined at the outlet. The same boundary conditions representing the engine the engine testing conditions were also applied in the previous 2D CFD simulations in [7]. testing conditions were also applied in the previous 2D CFD simulations in [7]. (These 2D (These 2D CFD datasets were used to develop ANN models in the present work.) A grid CFD datasets were used to develop ANN models in the present work.) A grid sequencing sequencing method was used to provide initial conditions, by solving inviscid flow equa- method was used to provide initial conditions, by solving inviscid flow equations repeat- tions repeatedly on a set of gradually refined mesh grids. Automatic CFL number control edly on a set of gradually refined mesh grids. Automatic CFL number control was also was also applied applied to adjustto the adj CFL ust the CF number L n inu rm esponse ber in respon to linear se solver to line conver ar solver conver gence behavior gence be during - havior durin the algebraic g the mul algebraic mu tigrid procedur ltigrid p e. Torocedure ensure overall . To ensure ove solutionrconver all solution con gence is r v eached, ergence in is re addition ached, in to add monitoring ition to m the onitor residuals ing the of resid the u governing als of the gov fluid ernin equations, g fluid estopping quations, s criteria top- ping criter were also ia set were also se based on the t based on asymptotic the asym values ptotic v of engineering alues of engin quantities, eering qu such anas titie isentr s, suopic ch efficiency, power output, maximum surface temperature, and total pressure difference. as isentropic efficiency, power output, maximum surface temperature, and total pressure The CFD methodology has been applied in a previous study based on a 2D version difference. of the E3 blade and validated against other experimental and numerical results of the E3 The CFD methodology has been applied in a previous study based on a 2D version blade [7]. In the present study, a grid independence investigation has also been performed of the E3 blade and validated against other experimental and numerical results of the E3 for the 3D CFD. Three sets of CFD grids with different resolutions were tested. The chosen blade [7]. In the present study, a grid independence investigation has also been performed grid features 12 million polyhedral cells with near-wall y+ values all below 1.4, based on for the 3D CFD. Three sets of CFD grids with different resolutions were tested. The chosen which the solution calculates a baseline efficiency of 96.0% and a power of 5.638  10 W. grid features 12 million polyhedral cells with near-wall y+ values all below 1.4, based on (Note power is calculated based on the single blade CFD model in the current study, which 4 which the solution calculates a baseline efficiency of 96.0% and a power of 5.638 × 10 W. has a pie sector angle of 4.737 ). The relative difference between the chosen grid and its (Note power is calculated based on the single blade CFD model in the current study, next-level refined grid is 0.1% for efficiency and 0.02% for power. which has a pie sector angle of 4.737°). The relative difference between the chosen grid and its next-level refined grid is 0.1% for efficiency and 0.02% for power. 2.3. Optimization Strategy The present study adopts a nested optimization process. The nested optimization is a slight variation of a previous co-optimization strategy introduced in [7]. The nested op- timization accomplishes two essential tasks. First, because an ANN model calculation con- sumes significantly less computing resources and time than a 3D CFD run, nesting an inner optimization based on the ANN model can effectively utilize the time while waiting Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 10 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 9 of 19 for a 3D CFD run to complete. For each 3D CFD simulation run, 250 evaluations of differ- ent blade cross-sectional profiles were completed in the inner optimization loop using the 2.3. Optimization Strategy ANN model. Second, the inner, ANN-driven optimization passes the best blade profile of The present study adopts a nested optimization process. The nested optimization the present optimization cycle to the 3D CFD run. This architecture effectively allows us- is a slight variation of a previous co-optimization strategy introduced in [7]. The nested ing the ANN-driven optimization to guide the 3D CFD search, and thus reduces the num- optimization accomplishes two essential tasks. First, because an ANN model calculation consumes significantly less computing resources and time than a 3D CFD run, nesting an ber of 3D CFD runs compared to an optimization process that is solely based on evaluat- inner optimization based on the ANN model can effectively utilize the time while waiting ing 3D CFD models. for a 3D CFD run to complete. For each 3D CFD simulation run, 250 evaluations of different The goal of the optimization is to maximize blade efficiency, η, and power output, blade cross-sectional profiles were completed in the inner optimization loop using the ANN model. Second, the inner, ANN-driven optimization passes the best blade profile of the Pow. The design variables that are being explored in optimization are the ones discussed present optimization cycle to the 3D CFD run. This architecture effectively allows using the in the previous session, 𝑤 ,𝑤 ,…, 𝑤 , 𝑤 ,𝑤 ,…, 𝑤 , 𝜉 , 𝜉 ,𝜉 ,𝜉 , 𝜃 , , , , , , , , , , ANN-driven optimization to guide the 3D CFD search, and thus reduces the number of and 𝛽 . The optimization procedure repeatedly executes the ANN and CFD models to 3D CFD runs compared to an optimization process that is solely based on evaluating 3D CFD models. evaluate different blade design variants, while extracting performance metrics to search The goal of the optimization is to maximize blade efficiency, h, and power output, Pow. for better designs. As the ANN model is much less computationally expensive compared The design variables that are being explored in optimization are the ones discussed in the to CFD, it is leveraged in an optimization loop by itself to yield optimized cross-sectional previous session, w , w , . . . , w , w , w , . . . , w , x , x , x , x , q and u,2 u,1 u,10 l,1 l,2 l,10 hub,l hub,u ti p,l ti p,u b. The optimization procedure repeatedly executes the ANN and CFD models to evaluate blade profiles, which are subsequently used to construct 3D shapes for CFD simulations. different blade design variants, while extracting performance metrics to search for better Using the ANN embedded optimization to guide the search for the best blade profiles designs. As the ANN model is much less computationally expensive compared to CFD, it is allows for reducing the number of expensive CFD evaluations. As such, a nested optimi- leveraged in an optimization loop by itself to yield optimized cross-sectional blade profiles, which are subsequently used to construct 3D shapes for CFD simulations. Using the ANN zation workflow is constructed, as illustrated in Figure 6. The workflow is realized using embedded optimization to guide the search for the best blade profiles allows for reducing commercial optimization software Simcenter HEEDS and its Sherpa optimization algo- the number of expensive CFD evaluations. As such, a nested optimization workflow rithm [30] is used in both Optimizer 1 and Optimizer 2. The Sherpa algorithm constantly is constructed, as illustrated in Figure 6. The workflow is realized using commercial optimization software Simcenter HEEDS and its Sherpa optimization algorithm [30] is evaluates the characteristics of the problem by using a hybrid combination of search strat- used in both Optimizer 1 and Optimizer 2. The Sherpa algorithm constantly evaluates egies at each stage of the optimization process to best traverse the design landscape. In the characteristics of the problem by using a hybrid combination of search strategies at addition, it also adapts the tuning parameters of the search strategies to the specific region each stage of the optimization process to best traverse the design landscape. In addition, it also adapts the tuning parameters of the search strategies to the specific region of the of the search. The HEEDS’ Sherpa algorithm has been effectively applied to optimize gas search. The HEEDS’ Sherpa algorithm has been effectively applied to optimize gas turbine turbine applications, as shown in studies [5–7]. applications, as shown in studies [5–7]. Figure 6. Optimization iteration hierarchy and data Figure 6. Optimization iteration hierar flow. chy and data flow. In the global optimization loop (Optimizer 1 as shown in Figure 6), a multi-objective In the global optimization loop (Optimizer 1 as shown in Figure 6), a multi-objective trade-off problem is solved targeting maximizing both efficiency and power. Two con- straints are set: efficiency must be greater than 95% and power must also be greater than trade-off problem is solved targeting maximizing both efficiency and power. Two con- a baseline power value. These constraints are included in the optimization performance straints are set: efficiency must be greater than 95% and power must also be greater than function, based on considerations of practical gas turbine design metrics and general a baseline power value. These constraints are included in the optimization performance function, based on considerations of practical gas turbine design metrics and general per- formance characters of the E engine. The global optimization solver, Optimizer 1, affects the blade shape change by directly optimizing the section scaling factors and blade angles in the CAD. In addition, Optimizer 1 also affects an inner optimization process, which is represented by Optimizer 2. Optimization 1 controls the selection of the CST weights used as input parameters for Optimizer 2, through manipulating a random seed, ℛ, which sets the initial state for the global search strategies employed by the Sherpa algorithm. Alt- hough Optimizer 1 does not directly manipulate the CST weights, because ℛ controls their selection, it acts as a 1D encoding of the entire 20-dimensional design space for Op- Int. J. Turbomach. Propuls. Power 2022, 7, 20 10 of 19 performance characters of the E engine. The global optimization solver, Optimizer 1, affects the blade shape change by directly optimizing the section scaling factors and blade angles in the CAD. In addition, Optimizer 1 also affects an inner optimization process, which is represented by Optimizer 2. Optimization 1 controls the selection of the CST weights used as input parameters for Optimizer 2, through manipulating a random seed, R, which sets the initial state for the global search strategies employed by the Sherpa algorithm. Although Optimizer 1 does not directly manipulate the CST weights, becauseR controls their selection, it acts as a 1D encoding of the entire 20-dimensional design space for Optimizer 1 to explore. Optimizer 1 repeatedly evaluates and optimizes different design variables during the global optimization cycles; in each evaluation cycle, Optimizer 2 optimizes the CST weights using the ANN models, with an objective to maximize the efficiency and a constraint on minimum power value. Overall, 250 blade cross-sectional profiles were evaluated in each execution of Optimizer 2. It directly explores the shape profiles through manipulating the weighting parameters used in the CST method. In each evaluation cycle of Optimizer 1, the best blade profile found by Optimizer 2 is applied in the more expensive and high-fidelity CFD simulation. The performance function in each optimization is defined as the sum of normalized objectives minus the sum of normalized constraints. In a generic form, the performance function is formulated as follows: 4 2 i=N Obj 10  ConsViol j=N obj i cons P = , (3) å å i=1 j=1 2 Norm Norm i j In the above formula, Obj represents the objective quantity, such as h and Pow; ConsViol represents the amount of violation resulted from each constraint definition (h > 0.95, Pow > Pow ); the normalization factors Norm are selected using the respec- baseline tive h and Pow values of the baseline design. 3. Results 3.1. Optimization Results In the present optimization study, a total of 163 blade designs have been evalu- ated in 3D CFD, with 40,750 2D blade profiles being evaluated by the ANNs. An effi- ciency vs. power plot showing all 3D blade design points is provided in Figure 7. A Pareto front is formed in the upper right region of Figure 7, which contains optimal designs with the best trade-off relations between efficiency and power. Among the optimal designs, the maximum increase in efficiency is over 3%, and the maximum increase in power is around 8%. To obtain insight into the design variable distributions of the optimal designs, two groups of optimal designs are selected—a group of top 10 high-efficiency designs, and another group of top 10 high power designs. The responses and selected design input variables (which include the two blade angles and four scaling factors) for these two groups of designs are highlighted on a parallel plot in Figure 8. The parallel plot reveals some qual- itative trends in the design parameters: (1) both the high-efficiency and high power designs favor large lean angles, b; (2) the high-efficiency designs also favor a large scaling factor for the tip upper surface, x , while the high power designs favor a small-to-medium value ti p,u for x ; (3) a weaker influence of the scaling factor for the hub lower surface, x , is ti p,u hub,u also observed. Note that the lean angle was constrained conservatively within a smaller range in consideration of commercial turbine geometric characteristics provided in [27]. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 11 of 23 timizer 1 to explore. Optimizer 1 repeatedly evaluates and optimizes different design var- iables during the global optimization cycles; in each evaluation cycle, Optimizer 2 opti- mizes the CST weights using the ANN models, with an objective to maximize the effi- ciency and a constraint on minimum power value. Overall, 250 blade cross-sectional pro- files were evaluated in each execution of Optimizer 2. It directly explores the shape pro- files through manipulating the weighting parameters used in the CST method. In each evaluation cycle of Optimizer 1, the best blade profile found by Optimizer 2 is applied in the more expensive and high-fidelity CFD simulation. The performance function in each optimization is defined as the sum of normalized objectives minus the sum of normalized constraints. In a generic form, the performance function is formulated as follows: ∑ ∑ 𝒫= − , (3) In the above formula, 𝑂𝑏𝑗 represents the objective quantity, such as 𝜂 and 𝑃𝑜𝑤 ; 𝐶𝑜𝑛𝑠𝑉𝑖𝑜𝑙 represents the amount of violation resulted from each constraint definition (𝜂> 0.95, 𝑃𝑜𝑤 > 𝑃𝑜𝑤 ); the normalization factors Norm are selected using the respective 𝜂 and 𝑃𝑜𝑤 values of the baseline design. 3. Results 3.1. Optimization Results In the present optimization study, a total of 163 blade designs have been evaluated in 3D CFD, with 40,750 2D blade profiles being evaluated by the ANNs. An efficiency vs. power plot showing all 3D blade design points is provided in Figure 7. A Pareto front is formed in the upper right region of Figure 7, which contains optimal designs with the best trade-off relations between efficiency and power. Among the optimal designs, the maxi- mum increase in efficiency is over 3%, and the maximum increase in power is around 8%. To obtain insight into the design variable distributions of the optimal designs, two groups of optimal designs are selected—a group of top 10 high-efficiency designs, and another group of top 10 high power designs. The responses and selected design input variables (which include the two blade angles and four scaling factors) for these two groups of de- signs are highlighted on a parallel plot in Figure 8. The parallel plot reveals some qualita- tive trends in the design parameters: (1) both the high-efficiency and high power designs favor large lean angles, 𝛽 ; (2) the high-efficiency designs also favor a large scaling factor for the tip upper surface, 𝜉 , while the high power designs favor a small-to-medium value for 𝜉 ; (3) a weaker influence of the scaling factor for the hub lower surface, 𝜉 , is also observed. Note that the lean angle was constrained conservatively within a Int. J. Turbomach. Propuls. Power 2022, 7, 20 11 of 19 smaller range in consideration of commercial turbine geometric characteristics provided in [27]. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 12 of 23 Figure 7. Figure 7. Effi Efci ficiency ency vs vs. . power power for al for alll designs designpoints s points in in the the o optimization ptimization stu study. dy. Figure 8. Parallel plot highlighting selected design parameters of different groups of designs. Figure 8. Parallel plot highlighting selected design parameters of different groups of designs. A quantitative analysis has also been conducted. The Pearson correlation coefficients A quantitative analysis has also been conducted. The Pearson correlation coefficients of the data are calculated, based on the covariance and standard deviations. The definition of the data are calculated, based on the covariance and standard deviations. The definition is given as follows: is given as follows: (x x) Y Y i i i=1 r = q , (4) ∑ (𝓍 ̅)(𝓎 ) n 𝔯 = n , (x x) Y Y (4) å å i i i=1 i=1 ∑ (𝓍 ̅ ) ∑ (𝓎 ) n o In the above formula, x, Y 2 h, Pow, b, q, x , x , x , x , ( x 6= Y). hub,l hub,u ti p,l ti p,u In the above formula, 𝓍, 𝓎 ∈ 𝜂,𝑃𝑜𝑤,𝛽,𝜃,𝜉 , 𝜉 ,𝜉 ,𝜉 , ( 𝓍 ≠ 𝓎) . 𝓍 and 𝓎 , , , , x and Y represent sample values corresponding to each blade design, and n is the total i i represent sample values corresponding to each blade design, and n is the total number of number of designs, which is 163 in the present study. The Pearson correlation coefficients, designs, which is 163 in the present study. The Pearson correlation coefficients, the histo- the histogram distributions of selected parameters, and the data sample plots are shown on gram distributions of selected parameters, and the data sample plots are shown on a cor- a correlation plot in Figure 9, respectively, on the upper right region, along the diagonal relation plot in Figure 9, respectively, on the upper right region, along the diagonal line line joining the upper left and lower right corners, and on the lower left region. Large joining the upper left and lower right corners, and on the lower left region. Large magni- magnitudes of the Pearson correlation coefficients for x , h (1st row, last column) ti p,u tud and es ofbf , the PowPear g (2nd son cor row, 3rre dla column) tion coe indicate fficients str for ong 𝜉 relationships ,𝜂 (1stin rthese ow, ltwo ast c pairs olum of n) and variables. The large coefficient values are confirmed by their respective data sample plots {𝛽, 𝑃𝑜𝑤 } (2nd row, 3rd column) indicate strong relationships in these two pairs of varia- of x vs. h (last row, 1st column) and b vs. Pow (3rd row, 2nd column), in which most ti p,u bles. The large coefficient values are confirmed by their respective data sample plots of of the design points follow the linear regression lines closely. In addition, the correlation 𝜉 vs. 𝜂 (last row, 1st column) and 𝛽 vs. 𝑃𝑜𝑤 (3rd row, 2nd column), in which most plot also shows that the effect of the scaling factor for the hub lower surface, x , is hub,u of the design points follow the linear regression lines closely. In addition, the correlation rather mild. plot also shows that the effect of the scaling factor for the hub lower surface, 𝜉 , is rather mild. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 13 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 12 of 19 Figure 9. Correlation plot of selected design parameters of all designs. (The two most influential Figure 9. Correlation plot of selected design parameters of all designs. (The two most influential numbers and their corresponding plots are highlighted using the dashed arrows.) numbers and their corresponding plots are highlighted using the dashed arrows.) The efficiency and power are plotted versus the two parameters with the strongest The efficiency and power are plotted versus the two parameters with the strongest correlations, lean angle (b), and tip upper surface scaling factor (x ), shown in Figure 10. correlations, lean angle (𝛽 ), and tip upper surface scaling factor (𝜉 ), shown in Figure ti p,u It is observed that (1) most of the improved designs favor larger lean angles, and (2) among 10. It is observed that (1) most of the improved designs favor larger lean angles, and (2) the improved designs, a smaller x leads to greater power improvement, while a larger Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 14 of 23 ti p,u among the improved designs, a smaller 𝜉 leads to greater power improvement, while x leads to high-efficiency gain. To analyze the effects of the blade shape and the fluid ti p,u a larger 𝜉 leads to high-efficiency gain. To analyze the effects of the blade shape and dynamics, two designs are selected for further investigation—a high-efficiency design and the fluid dynamics, two designs are selected for further investigation—a high-efficiency a high power design, which will be referred to as “Design A” and “Design B,”, respectively. design and a high power design, which will be referred to as “Design A” and “Design B,”, These two designs are also marked in Figure 10 and will be discussed in the next session. respectively. These two designs are also marked in Figure 10 and will be discussed in the next session. (a) (b) Figure 10. Two views of a 3D plot showing efficiency, power, lean angle, and tip upper surface Figure 10. Two views of a 3D plot showing efficiency, power, lean angle, and tip upper surface scaling scaling factor of all designs: (a) a normal view; and (b) a view obtained by rotating the left-side factor of all designs: (a) a normal view; and (b) a view obtained by rotating the left-side plot around plot around the 𝜂 axis by 180°. the h axis by 180 . The total run time for the optimization process was around 30 h (with six CFD eval- uations running in parallel, each consuming 160 compute cores). There are 27 input vari- ables in the global optimization problem. As illustrated in Figure 6, the integration of the inner optimization loop allows for optimizing the 20 blade profile parameters using the inexpensive ANN models. Throughout the optimization process, the ANN models were evaluated 40,750 times; in each optimization cycle, the best blade cross-sectional profile obtained from the inner ANN optimization was passed on to the 3D CFD for evaluation. As a result, the number of design variables being directly exposed in the 3D CFD runs is reduced to six. Since the required number of CFD evaluations largely depends on the number of input variables in an optimization problem, the nested optimization strategy only requires 163 evaluations of 3D CFD simulations. In comparison, if an optimization study of the same problem does not adopt the nested optimization strategy and is solely based on CFD, then all 27 design variables (rather than 6) will be evaluated based on 3D CFD runs. Based on a quick estimation, this will result in 4.5 times (=27/6) the number of CFD runs, which is 733 CFD runs, and will cost roughly 135 h of optimization run time if the same compute resource settings were used. 3.2. Fluid Dynamics Analysis Two optimized designs are selected for detailed investigation of the aerodynamic results: a high-efficiency design, and a high-power design (a.k.a. Designs A and B as shown in Figure 9). They are compared with the baseline design. The CFD results show the optimized designs feature less pressure drop on the suction side near the trailing edge. A pressure coefficient is defined based on the average inlet velocity and density: 𝑐 = , (5) The pressure coefficient distributions on the hub, mid, and tip sections are compared in the upper row of plots in Figure 11. The baseline design shows significant low pressure regions on the suction surface near the trailing edge on all three streamwise sections. In Int. J. Turbomach. Propuls. Power 2022, 7, 20 13 of 19 The total run time for the optimization process was around 30 h (with six CFD evalua- tions running in parallel, each consuming 160 compute cores). There are 27 input variables in the global optimization problem. As illustrated in Figure 6, the integration of the inner optimization loop allows for optimizing the 20 blade profile parameters using the inexpen- sive ANN models. Throughout the optimization process, the ANN models were evaluated 40,750 times; in each optimization cycle, the best blade cross-sectional profile obtained from the inner ANN optimization was passed on to the 3D CFD for evaluation. As a result, the number of design variables being directly exposed in the 3D CFD runs is reduced to six. Since the required number of CFD evaluations largely depends on the number of input variables in an optimization problem, the nested optimization strategy only requires 163 evaluations of 3D CFD simulations. In comparison, if an optimization study of the same problem does not adopt the nested optimization strategy and is solely based on CFD, then all 27 design variables (rather than 6) will be evaluated based on 3D CFD runs. Based on a quick estimation, this will result in 4.5 times (=27/6) the number of CFD runs, which is 733 CFD runs, and will cost roughly 135 h of optimization run time if the same compute resource settings were used. 3.2. Fluid Dynamics Analysis Two optimized designs are selected for detailed investigation of the aerodynamic results: a high-efficiency design, and a high-power design (a.k.a. Designs A and B as shown in Figure 9). They are compared with the baseline design. The CFD results show the optimized designs feature less pressure drop on the suction side near the trailing edge. A pressure coefficient is defined based on the average inlet velocity and density: c = , (5) 1 2 r u in 2 in The pressure coefficient distributions on the hub, mid, and tip sections are compared in the upper row of plots in Figure 11. The baseline design shows significant low pressure regions on the suction surface near the trailing edge on all three streamwise sections. In comparison, the two optimized designs saw increased minimum pressure in those areas at the hub and mid sections. An enlarged view of the pressure coefficient distribution on the mid-section near the trailing edge is provided for each design in the lower row of plots in Figure 11. The baseline design features a shock in this region, highlighted by a relatively high pressure spot next to the low pressure zone, which is present inside the dotted circle marked on Figure 11a. This shock feature is absent in the two optimized designs due to the modifications in the blade shapes in those designs. This observation is consistent with the Mach number distributions that will be discussed in a later paragraph. To further investigate the pressure drop on the suction side near the trailing edge, volumetric renderings of the pressure coefficient scenes are shown in Figure 12, to highlight the flow in the low-pressure regime by focusing on c values between 0.35 and 0.1. It is observed that in the optimized designs, lower pressure coefficients are present near the tip, but the pressure coefficient has been increased in the mid-session and nearer the hub. The overall reduction of maximum negative pressure coefficients in the mid and hub regions offsets the effects of increased tip leakage flow in the optimized designs. Next, the pressure coefficient distributions on the blade surfaces are investigated. Scalar scenes of the pressure coefficients on the baseline, Design A, and Design B blades are shown in Figure 13. A large portion of the suction-side surface in the baseline design features a significantly negative c values between 0.35 and 0.23 starting around the three quarters chord. In the optimized designs, the onset of the negative pressure coefficient zone moves upstream, nearer the half chord, yet the magnitude of this zone is much less, with most areas featuring c values between 0.12 and 0. To achieve a more quantitative comparison, the pressure coefficient curves along the streamwise direction on the hub, mid, and tip sections of the three blades are plotted and compared, with the plots of the blade profiles, shown in Figure 14. The earlier onsets of the lower pressure coefficient region on Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 15 of 23 comparison, the two optimized designs saw increased minimum pressure in those areas Int. J. Turbomach. Propuls. Power 2022, 7, 20 at the hub and mid sections. An enlarged view of the pressure coefficient distribution on 14 of 19 the mid-section near the trailing edge is provided for each design in the lower row of plots in Figure 11. The baseline design features a shock in this region, highlighted by a relatively high pressure spot next to the low pressure zone, which is present inside the dotted circle the suction-side surfaces in the two optimized designs are demonstrated by the plots. The marked on Figure 11a. This shock feature is absent in the two optimized designs due to earlier onset allows for larger enclosed areas of negative c curves for the optimized design, the modifications in the blade shapes in those designs. This observation is consistent with contributing the Mach nu tom the ber dis power tribuimpr tions th ovements. at will be discussed in a later paragraph. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 16 of 23 (a) (b) (c) Figure 11. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. Figure 11. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. To further investigate the pressure drop on the suction side near the trailing edge, volumetric renderings of the pressure coefficient scenes are shown in Figure 12, to high- light the flow in the low-pressure regime by focusing on 𝑐 values between −0.35 and −0.1. It is observed that in the optimized designs, lower pressure coefficients are present near the tip, but the pressure coefficient has been increased in the mid-session and nearer the hub. The overall reduction of maximum negative pressure coefficients in the mid and hub regions offsets the effects of increased tip leakage flow in the optimized designs. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 17 of 23 (a) (b) (c) Figure 12. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. Figure 12. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. Next, the pressure coefficient distributions on the blade surfaces are investigated. Scalar scenes of the pressure coefficients on the baseline, Design A, and Design B blades are shown in Figure 13. A large portion of the suction-side surface in the baseline design features a significantly negative 𝑐 values between −0.35 and −0.23 starting around the three quarters chord. In the optimized designs, the onset of the negative pressure coeffi- cient zone moves upstream, nearer the half chord, yet the magnitude of this zone is much less, with most areas featuring 𝑐 values between −0.12 and 0. To achieve a more quanti- tative comparison, the pressure coefficient curves along the streamwise direction on the hub, mid, and tip sections of the three blades are plotted and compared, with the plots of the blade profiles, shown in Figure 14. The earlier onsets of the lower pressure coefficient region on the suction-side surfaces in the two optimized designs are demonstrated by the plots. The earlier onset allows for larger enclosed areas of negative 𝑐 curves for the op- (a) (b) (c) timized design, contributing to the power improvements. Figure 13. Pressure coefficient scenes on the surfaces of the blades: (a) Baseline design; (b) Design Figure 13. Pressure coefficient scenes on the surfaces of the blades: (a) Baseline design; (b) Design A; A; and (c) Design B. and (c) Design B. To further investigate the differences in solutions, a volume rendering scene is cre- ated by focusing on visualizing changes in Mach number within the range between 0.8 and 1.2, shown in Figure 15. The goal of this plot is to reveal where sudden changes in Mach number may occur in the transonic flow regime. A consistent shock structure near the trailing edge covering the full span length of the blade is observed in the baseline design, which is consistent with the high pressure spot in the same area highlighted by the dotted circle in Figure 11a. This shock feature is missing in the two optimized designs (Designs A and B), due to the weaker gradients in those designs. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 18 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 15 of 19 (a) (b) (c) Figure 14. Blade profiles (first row) and pressure coefficients in the streamwise direction (second Figure 14. Blade profiles (first row) and pressure coefficients in the streamwise direction (second row): (a) hub sections comparison; (b) mid sections comparison; and (c) tip sections comparison. row): (a) hub sections comparison; (b) mid sections comparison; and (c) tip sections comparison. To further investigate the differences in solutions, a volume rendering scene is created by focusing on visualizing changes in Mach number within the range between 0.8 and 1.2, shown in Figure 15. The goal of this plot is to reveal where sudden changes in Mach number may occur in the transonic flow regime. A consistent shock structure near the Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 19 of 23 trailing edge covering the full span length of the blade is observed in the baseline design, which is consistent with the high pressure spot in the same area highlighted by the dotted circle in Figure 11a. This shock feature is missing in the two optimized designs (Designs A and B), due to the weaker gradients in those designs. (a) (b) (c) Figure 15. Mach number rendering: (a) Baseline design; (b) Design A; and (c) Design B. Figure 15. Mach number rendering: (a) Baseline design; (b) Design A; and (c) Design B. A further comparison is made by plotting the streamlines in the tip leakage regions A further comparison is made by plotting the streamlines in the tip leakage regions and showing pressure coefficients on spanwise cross-sections in Figure 16. It appears the and showing pressure coefficients on spanwise cross-sections in Figure 16. It appears the two optimized designs feature slightly more chaotic Mach number distributions in the two optimized designs feature slightly more chaotic Mach number distributions in the local region near the trailing edge and tip leakage. The leakage flow’s influence on the local region near the trailing edge and tip leakage. The leakage flow’s influence on the overall performance is relatively small in comparison to the previously discussed suction- overall performance is relatively small in comparison to the previously discussed suction- side effects. It can be observed that the streamlines in the high-efficiency design (Design side effects. It can be observed that the streamlines in the high-efficiency design (Design A) are slightly more curved, yet the induced low-pressure spots by the secondary flow A) are slightly more curved, yet the induced low-pressure spots by the secondary flow effects of the tip leakage flows are weak and about the same magnitude. Overall, the dif- effects of the tip leakage flows are weak and about the same magnitude. Overall, the ference caused by the tip leakage flow is small in these three designs. The performance difference caused by the tip leakage flow is small in these three designs. The performance improvements in the optimized designs are more influenced by other flow effects dis- improvements in the optimized designs are more influenced by other flow effects discussed cussed earlier, such as the increased minimum pressure on the suction side near the trail- earlier, such as the increased minimum pressure on the suction side near the trailing edge ing edge (Figures 11–13), and the reduction of the shock structure (Figure 15). (Figures 11–13), and the reduction of the shock structure (Figure 15). Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 20 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 16 of 19 (a) (b) (c) Figure 16. Tip leakage flow streamlines and pressure coefficients on spanwise cross-sections: (a) Figure 16. Tip leakage flow streamlines and pressure coefficients on spanwise cross-sections: (a) Base- Baseline design; (b) Design A; and (c) Design B. line design; (b) Design A; and (c) Design B. 4. Conclusions 4. Conclusions The present study has demonstrated an optimization workflow combining the use of The present study has demonstrated an optimization workflow combining the use of neural networks and high-fidelity CFD. The neural network models were trained on over neural networks and high-fidelity CFD. The neural network models were trained on over three-thousand design data points from a previous publication. The practical implication three-thousand design data points from a previous publication. The practical implication of the overall strategy is that in engineering design analysis, existing data sets, which are of the overall strategy is that in engineering design analysis, existing data sets, which generally available from the previous simulation, experimental, or reduced-order studies, are generally available from the previous simulation, experimental, or reduced-order can be leveraged to build neural network models, which can then be used in combination studies, can be leveraged to build neural network models, which can then be used in with high-fidelity CFD simulations to guide optimization processes. This approach combination with high-fidelity CFD simulations to guide optimization processes. This achieves a reduction in the required number of high-fidelity CFD runs, and hence reduces approach achieves a reduction in the required number of high-fidelity CFD runs, and the computational cost while maintaining accuracy. The integration of computationally hence reduces the computational cost while maintaining accuracy. The integration of inexpensive ANN models, which were evaluated 40,750 times, allows for a relatively computationally inexpensive ANN models, which were evaluated 40,750 times, allows for small number (163) of CFD evaluations in the present optimization process, resulting in a a relatively small number (163) of CFD evaluations in the present optimization process, total run time of about 30 h (with 6 CFD evaluations running in parallel, each consuming resulting in a total run time of about 30 h (with 6 CFD evaluations running in parallel, 160 compute cores). It is estimated that if the nested optimization strategy based on ANN each consuming 160 compute cores). It is estimated that if the nested optimization strategy was not used, a total number of 733 CFD evaluations will be required due to the large based on ANN was not used, a total number of 733 CFD evaluations will be required due to number of design variables exposed in 3D CFD evaluations, resulting in roughly 135 h of the large number of design variables exposed in 3D CFD evaluations, resulting in roughly the optimization run time if the same compute resource settings were used. 135 h of The efficac the optimization y of the me runtho time dology if theis de same monstr compute ated o resour n a turbine b ce settings ladwer e aero e used. dynamic problem. ANN models The efficacy of the methodology with 7 layers ais nd 6 la demonstrated yers were b on uilt ato r turbine epresent blade two aer blad odynamic e per- formance metrics, efficiency, and power, respectively. The hyperparameters of the ANN problem. ANN models with 7 layers and 6 layers were built to represent two blade models were optimized, and the models were used as surrogate models along with high- performance metrics, efficiency, and power, respectively. The hyperparameters of the ANN fidelity CFD simulations in a nested optimization procedure to obtain optimized blade models were optimized, and the models were used as surrogate models along with high- designs. Pareto front designs representing improved efficiency and power were found by fidelity CFD simulations in a nested optimization procedure to obtain optimized blade the optimization procedure. A lean angle and a tip scaling factor were shown to be more designs. Pareto front designs representing improved efficiency and power were found by favored by the optimization procedure than other parameters in the context of the chosen the optimization procedure. A lean angle and a tip scaling factor were shown to be more blade and analysis methodology used in the current study. Examining the fluid dynamics favored by the optimization procedure than other parameters in the context of the chosen of the optimized designs vs. the baseline design reveals that the optimization (1) reduced blade and analysis methodology used in the current study. Examining the fluid dynamics the magnitude of the most negative pressure coefficients in the flow on the suction side of the optimized designs vs. the baseline design reveals that the optimization (1) reduced near the trailing edge, and (2) altered the blade geometry that reduced the shock near the the magnitude of the most negative pressure coefficients in the flow on the suction side trailing edge. Both of these aspects have effects on improved efficiency and power in the near the trailing edge, and (2) altered the blade geometry that reduced the shock near the optimized designs. trailing edge. Both of these aspects have effects on improved efficiency and power in the As a future extension of this study, the following may be considered. (1) Test the optimized designs. performance of other response surface methods and meta models against ANN and apply As a future extension of this study, the following may be considered. (1) Test the them in the nested optimization workflow. (2) The present study demonstrates a nested performance of other response surface methods and meta models against ANN and apply ANN-CFD optimization methodology, applied to an idealized turbine CFD problem as a them in the nested optimization workflow. (2) The present study demonstrates a nested proof-of-concept. In production-level gas turbine designs, more detailed constraints on ANN-CFD optimization methodology, applied to an idealized turbine CFD problem as a proof-of-concept. In production-level gas turbine designs, more detailed constraints on the geometry or loading curve must be incorporated to ensure more realistic optimiza- tion outcomes. Different power output conditions of the engine may also be explored in optimization. Int. J. Turbomach. Propuls. Power 2022, 7, 20 17 of 19 Author Contributions: Conceptualization, C.Z.; methodology, C.Z. and M.J.; software, C.Z. and M.J.; validation, C.Z. and M.J.; formal analysis, C.Z.; investigation, C.Z. and M.J.; resources, C.Z. and M.J.; data curation, C.Z.; writing—original draft preparation, C.Z.; writing—review and editing, C.Z. and M.J.; visualization, C.Z.; supervision, M.J.; project administration, C.Z. and M.J.; funding acquisition, M.J. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Not applicable. 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Optimization of Turbine Blade Aerodynamic Designs Using CFD and Neural Network Models

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International Journal of Turbomachinery Propulsion and Power Article Optimization of Turbine Blade Aerodynamic Designs Using CFD and Neural Network Models Chao Zhang * and Matthew Janeway Siemens Digital Industries Software, Cypress, CA 90630, USA; matt.janeway@siemens.com * Correspondence: chao.zhang1@siemens.com Abstract: Optimization methods have been widely applied to the aerodynamic design of gas turbine blades. While applying optimization to high-fidelity computational fluid dynamics (CFD) simulations has proven capable of improving engineering design performance, a challenge has been overcoming the prolonged run-time due to the computationally expensive CFD runs. Reduced-order models and, more recently, machine learning methods have been increasingly used in gas turbine studies to predict performance metrics and operational characteristics, model turbulence, and optimize designs. The application of machine learning methods allows for utilizing existing knowledge and datasets from different sources, such as previous experiments, CFD, low-fidelity simulations, 1D or system-level studies. The present study investigates inserting a machine learning model that utilizes such data into a high-fidelity CFD driven optimization process, and hence effectively reduces the number of required evaluations of the CFD model. Artificial Neural Network (ANN) models were trained on data from over three thousand two-dimensional (2D) CFD analyses of turbine blade cross-sections. The trained ANN models were then used as surrogates in a nested optimization process alongside a full three-dimensional Navier–Stokes CFD simulation. The much lower evaluation cost of the ANN model allows for tens of thousands of design evaluations to guide the search of the best blade profiles to be used in the more expensive, high-fidelity CFD runs, improving the progress of the optimization while reducing the required computation time. It is estimated that the current workflow achieves a five-fold reduction in computational time in comparison to an optimization process that Citation: Zhang, C.; Janeway, M. is based on three-dimensional (3D) CFD simulations alone. The methodology is demonstrated on Optimization of Turbine Blade Aerodynamic Designs Using CFD the NASA/General Electric Energy Efficient Engine (E3) high pressure turbine blade and found and Neural Network Models. Int. J. Pareto front designs with improved blade efficiency and power over the baseline. Quantitative Turbomach. Propuls. Power 2022, 7, 20. analysis of the optimization data reveals that some design parameters in the present study are more https://doi.org/10.3390/ijtpp7030020 influential than others, such as the lean angle and tip scaling factor. Examining the optimized designs also provides insight into the physics, showing that the optimized designs have a lower amount Academic Editor: Tom Verstraete of pressure drop near the trailing edge, but have an earlier onset of pressure drop on the suction Received: 25 April 2022 side surface when compared to the baseline design, contributing to the observed improvements in Accepted: 28 June 2022 efficiency and power. Published: 30 June 2022 Publisher’s Note: MDPI stays neutral Keywords: CFD; optimization; aerodynamics; gas turbines; machine learning; neural networks with regard to jurisdictional claims in published maps and institutional affil- iations. 1. Introduction The present study applies machine learning methods in a CFD-based optimization for turbine blade aerodynamics. Literature on optimization and machine learning methods Copyright: © 2022 by the authors. used in gas turbine studies will be reviewed, followed by a summary of the motivation of Licensee MDPI, Basel, Switzerland. the present work. This article is an open access article Numerical optimization has been widely used in the design and analysis of gas tur- distributed under the terms and conditions of the Creative Commons bines. In some earlier studies, specific optimization algorithms have been investigated to Attribution (CC BY-NC-ND) license leverage lower fidelity models to achieve fast optimization time. One study on aerody- (https://creativecommons.org/ namic wing optimization has used an approximation and model management optimization licenses/by-nc-nd/4.0/). Int. J. Turbomach. Propuls. Power 2022, 7, 20. https://doi.org/10.3390/ijtpp7030020 https://www.mdpi.com/journal/ijtpp Int. J. Turbomach. Propuls. Power 2022, 7, 20 2 of 19 method to incorporate low fidelity, computationally cheaper models with occasional re- course to higher fidelity, more expensive models, resulting in threefold saving in optimiza- tion time [1]. Another study adopted a similar concept, employing a trust-region approach to interleave the exact models with cheaper surrogate models during optimization iter- ations [2]. These methods demonstrate the possibility of obtaining optimized solutions on a limited computational budget by incorporating lower-fidelity surrogate models. In more recent years, an increasing number of optimization studies have relied on using parametric CFD models. In optimization of the coolant flow passage of the NASA C3X vane, different designs were evaluated repeatedly through CFD runs [3]. In the study of a marine high-pressure turbine [4], ten design parameters controlling multi-row film cooling designs were built into the CFD model and optimized based on a non-dominated sorting genetic algorithm. Multiple studies were also conducted on ultra-super-critical steam turbines [5,6], in which the blade aerodynamic efficiency was optimized using 2D and 3D CFD simulations driven by Siemens’ Simcenter HEEDS commercial Sherpa opti- mization algorithm. A CFD-based co-optimization strategy was presented in [7], which demonstrates a workflow for coupling different disciplines into a nested optimization loop to conduct parallel blade aerodynamic and thermal optimizations. In addition to improving turbine blade designs, optimization has also been applied to improving the operations of gas turbine engines, for instance, to find the best valve setup parameters that reduce fuel consumption [8]. With the advancements in computer science and data storage, an increase in interest in application of machine learning methods to gas turbine designs has been observed. One area of such applications seeing increased interest has been the prediction of key performance metrics using models trained on input data gained through past simulations or experiments. In an earlier study [9], the outlet temperature and fuel mass flow rate at different operating conditions for a 255 MW single-shaft gas turbine were predicted by building a three-layer neural network model. Another application of the Artificial Neural Network (ANN) model has been seen in a turbine film cooling study to predict the instantaneous temperature distributions along the blade surface as well as the cooling effec- tiveness [10]. In another study of a jet engine power plant [11], a machine learning method combining physics-based and measurement-driven modeling was developed and used to conduct preventive maintenance and diagnose faults. Machine learning methods were also applied to a Viper 632-43 military turbojet engine to predict the exhaust temperature using models trained on data collected from a gas turbine simulation program [12]. Extending from predicting individual engineering metrics, machine learning has also been used to predict field quantities representing more complicated underlying physics. A method using gradient boosted trees was used to develop models of aerodynamic loads on vibrating turbine blades and demonstrated to have good agreement with detailed CFD results [13]. In another study, the turbine surface pressure distribution was predicted using transfer learning models, which transfer knowledge from a large-scale but low-fidelity dataset to a small-scale but high-fidelity dataset, shown to have a low prediction error with reduced cost [14]. Machine learning has also been applied to the prediction of operating charac- teristics of gas turbine engines, using real-time data of power plants to develop neural networks [15,16]. In addition to the above applications, machine learning has also been studied to develop turbulence closure models. In one study for wake mixing, a machine learning model was demonstrated to be robust across several different operating conditions when integrated into a RANS CFD model of a low-pressure turbine [17]. A review article on machine learning methods for science and engineering particularly highlighted the need for interpretable, generalizable, expandable, and certifiable machine learning techniques for safety-critical applications [18]. Recent works have also focused on using machine learning embedded into design opti- mization procedures. In the optimization of a centrifugal compressor impeller [19], an ANN model was first developed using CFD and FEA data from a Design of Experiment (DOE) study. Then, the ANN model was applied in an optimization procedure, which resulted in a Int. J. Turbomach. Propuls. Power 2022, 7, 20 3 of 19 1% increase in isentropic efficiency and 10% reduction in the blade stress. In another study investigating a carved blade tip [20], 55 CFD runs were conducted to generate ANN meta models, which were then used in a genetic algorithm routine to optimize the blade tip shape. In a missile control surface optimization study [21], machine learning, reinforcement learn- ing, and transfer learning were integrated into the optimization procedure and leveraged CFD in the evaluation iterations. In another study of 2D airfoil optimization [22], a deep convolutional generative adversarial network was trained and embedded as a surrogate model in an optimization framework. In still another study of a compact turbine rotor [23], machine learning models were trained and used to optimize the efficiency and torque based on a gradient-based multi-objective optimization algorithm. In addition to using machine learning models in optimization, several CFD and optimization studies have compared machine learning models with response surface models (RSM). In a study of aircyclone optimization [24], it is concluded that ANN offers an alternative and powerful approach to response surface methods for modeling the cyclone pressure drop, benchmarked against experimental data. In another study of modeling and optimizing a perforated baffle used for turbine blade passage cooling [25], both ANN and RSM methods were found capable of predicting friction factor and Nusselt number values, although the RSM method performed slightly better than ANN in that study. A more recent study on cyclone optimization also tested a RSM and several machine learning models. A GMDH-neural network model was found to be superior and chosen for the optimization process [26]. As discussed in the above literature, optimization that leverages high-fidelity CFD simulations can provide accurate and realistic optimal designs in general. While leveraging a neural network as a surrogate to replace the CFD evaluations in the optimization loop can significantly improve the computational time, the fundamental challenge is that a neural network model is a lower fidelity model compared to a CFD simulation, and therefore, it may not be as accurate as CFD in predicting certain design variants required by the optimization process. Further, neural network models may also be trained based on previously available datasets that come from different studies, such as 1D and 2D simulation data, and experimental data, all of which will result in the neural network being a reduced order model compared to 3D CFD. As such, relying solely on the neural network when the predictive accuracy is necessary can lead to errant results. The present study presents a nested optimization workflow that can leverage both neural networks and high-fidelity, 3D CFD simulations at the same time to ensure every “best” design is studied in detail by the CFD tool. The introduction of a neural network into the optimization allows for over 70% reduction in the number of CFD evaluations and thus makes significant reductions in computational cost compared to a process relying exclusively on CFD simulations. The methodology is demonstrated through an aerodynamic optimization using a rebuilt model of the NASA/General Electric E3 high pressure turbine blade [27–29]. 2. Methodology Artificial Neural Network models are used alongside 3D CFD simulations in a nu- merical optimization procedure to improve the aerodynamic performance of a turbine blade. ANN models are typically trained using large datasets obtained from previous experimental or numerical studies. In companies/organizations that conduct R&D on engi- neering designs, these datasets are usually available from previous studies. The present study proposes a methodology of using such existing knowledge from previous studies to train ANN models and then use the ANN models in an CFD-based design optimization process. A dataset from a previously published work is obtained to conduct the ANN modeling training in the present study. The overall framework of the research methodology is shown in Figure 1, which illustrates how different analysis models/tools required by the optimization process are created. First, data were obtained from a previous study [7] containing 3204 CFD design evaluations of different 2D turbine blade profiles. Performance metrics including efficiency and power were extracted as targets, with the blade geometric parameters serving as input features, to train ANN models. The ANN hyperparameters Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 4 of 23 [7] containing 3204 CFD design evaluations of different 2D turbine blade profiles. Perfor- mance metrics including efficiency and power were extracted as targets, with the blade Int. J. Turbomach. Propuls. Power 2022, 7, 20 4 of 19 geometric parameters serving as input features, to train ANN models. The ANN hyperpa- rameters were subsequently optimized. The blade cross-sectional profiles were parame- terized using a Class-Shape Transformation (CST) method [30]. Using the 2D blade cross- were subsequently optimized. The blade cross-sectional profiles were parameterized using sectional profile, the 3D blade CAD geometry was created with additional parameteriza- a Class-Shape Transformation (CST) method [30]. Using the 2D blade cross-sectional profile, tion allowing for the variation of further design transformation, such as the thickness, the 3D blade CAD geometry was created with additional parameterization allowing for twist, and lean. The blade CAD geometry was then used to build a 3D Reynolds Averag- the variation of further design transformation, such as the thickness, twist, and lean. The ing Navier–Stokes (RANS) CFD model, which solves for the flow field in a single blade blade CAD geometry was then used to build a 3D Reynolds Averaging Navier–Stokes pa (RANS) ssage. Onc CFD e the model, ANN which models solves and for the the 3D C flow FD field simu in laa tion single areblade indeppassage. endentlyOnce devethe loped, they ANN are inte models grand ated into a the 3D CFD neste simulation d optimizati are independently on procedure developed, for repeated ev they ar aluation e integrated s of dif- into a nested optimization procedure for repeated evaluations of different blade design ferent blade design variants to optimize the design parameters. In the nested optimization variants to optimize the design parameters. In the nested optimization process, an inner process, an inner optimization loop is executed using the ANN models to improve the optimization loop is executed using the ANN models to improve the blade cross-sectional blade cross-sectional profile. In each iteration of the global optimization, the best blade profile. In each iteration of the global optimization, the best blade profile is selected from profile is selected from the inner optimization step to create a 3D CFD blade model with the inner optimization step to create a 3D CFD blade model with additional parameters additional parameters such as the scaling factors, twist, and lean angles. The optimization such as the scaling factors, twist, and lean angles. The optimization targets improving the targets improving the efficiency and power of the blade. Details of the optimization pro- efficiency and power of the blade. Details of the optimization process will be discussed in a cess will be discussed in a later session in this chapter. later session in this chapter. Figure 1. The overall framework of the research methodology. (2D CFD data was collected from a Figure 1. The overall framework of the research methodology. (2D CFD data was collected from a previous study [7].) previous study [7].) 2.1. Artificial Neural Network 2.1. Artificial Neural Network A dataset representing different design variants of 2D aerodynamic CFD results was A dataset representing different design variants of 2D aerodynamic CFD results was obtained from a previous study [7] and used to train ANN models. The dataset features obtained from a previous study [7] and used to train ANN models. The dataset features blade design and performance parameters for 3204 designs. The input parameters for the blade design and performance parameters for 3204 designs. The input parameters for the ANN models include 20 blade design parameters. These parameters are the weighting ANN models include 20 blade design parameters. These parameters are the weighting factors used by the CST method [30] to control variations in the blade profile. Under the factors used by the CST method [30] to control variations in the blade profile. Under the CST method, these weighting factors are multiplied by Bernstein polynomials to define a CST method, these weighting factors are multiplied by Bernstein polynomials to define a shape function, and then subsequently multiplied by a class function to define the aero- shape function, and then subsequently multiplied by a class function to define the aero- dynamic blade profile. An order-9 Bernstein polynomial was deemed sufficiently flexible for the purposes of the study [7] and that choice is repeated here, resulting in 10 design dynamic blade profile. An order-9 Bernstein polynomial was deemed sufficiently flexible parameters for each of the suction and pressure sides of the blade: fw , w , . . . , w g for the purposes of the study [7] and that choice is repeated here, resulting in 10 design u,1 u,2 u,10 and w , w , . . . , w . These 20 parameters are used as input features in the ANN parameter l,1s for each of l,2 l,10 the suction and pressure sides of the blade: {𝑤 ,𝑤 ,…, 𝑤 } , , , models and optimization parameters in the subsequent blade optimization process. Two and {𝑤 ,𝑤 ,…, 𝑤 }. These 20 parameters are used as input features in the ANN mod- , , , performance metrics in the dataset were used as output parameters, isentropic efficiency, els and optimization parameters in the subsequent blade optimization process. Two per- formance metrics in the dataset were used as output parameters, isentropic efficiency, and Int. J. Turbomach. Propuls. Power 2022, 7, 20 5 of 19 and power output, for which two ANN models were developed, respectively. The efficiency and power are defined based on enthalpy quantities of the flow, as follows: h h i o h = , (1) h h i o,isen. Pow = Dh = h h (2) The Keras API for TensorFlow [31] was used to construct the ANN models. In consid- eration of the different distributions of the efficiency and power data and the robustness of the model, a separate ANN model was constructed for each of these two quantities. After some preliminary tests on ANN models of 5, 6, and 7 layers, 7 and 6 layers were constructed, respectively, into the ANN models representing efficiency and power. The choice of the number of layers in ANN usually varies in different problems. It will be shown that the prediction errors are within acceptable range for the chosen number of layers, and that the errors will be further reduced by optimizing the hyperparameters of the models. The first layer of each ANN model has 20 inputs representing the blade input features (CST weights) and the last layer has one output, which is either efficiency or power. The number of neurons in the hidden layers of each ANN, along with other topological and training parameters, were optimized in a separate hyperparameter optimization process. As a starting point for that process, the hyperparameters for each ANN model were set using the values provided in the first two columns of Table 1. Table 1. Hyperparameters of the initial and optimized ANN models. Initial ANN Initial ANN Optimized ANN Optimized ANN Hyperparameters for Efficiency for Power for Efficiency for Power Number of Neurons-Layer 2 20 20 5 23 Number of Neurons-Layer 3 40 40 35 15 Number of Neurons-Layer 4 20 20 53 36 Number of Neurons-Layer 5 10 5 50 42 Number of Neurons-Layer 6 5 – 62 – Activation Function-Layer 2 relu relu relu selu Activation Function-Layer 3 relu relu sigmoid softmax Activation Function-Layer 4 relu relu elu elu Activation Function-Layer 5 relu relu elu softmax Activation Function-Layer 6 relu – softmax – 5 3 Learning Rate 0.001 0.001 6.4895  10 1.47285  10 Number of Epochs 200 200 6123 4000 To improve the predictive performance of the ANN, the hyperparameters were opti- mized using Siemens commercial optimization tool HEEDS and its Sherpa algorithm [32]. A diagram of the optimization process is provided in Figure 2. The optimizer repeatedly tunes all the hyperparameters for the ANN models listed in the index column of Table 1 with an objective to minimize the model test error on a held-out testing dataset. In each optimization cycle, the dataset is randomly split into a training and testing set with a ratio of 95/5. The training set is used the train the ANN model. The training process involves a typical backpropagation procedure with an 80/20 split of the training set, over multiple epochs with TensorFlow’s ADAM optimizer. After the training step is completed, the model’s performance is then evaluated against the held-out test set. The evaluation error is passed back to the optimization solver to tune the hyperparameters for the next cycle. A total number of 240 iterations were executed during the optimization of each of the two ANNs. The evaluation number, 240, was based on the Sherpa algorithm best practice [32], which recommends the number of evaluations being about 10 times the number of input variables for single objective optimization problems, due to the hybrid and adaptive meth- ods employed by the algorithm. In the ANN optimization, 12 hyperparameters must be Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 6 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 6 of 19 and adaptive methods employed by the algorithm. In the ANN optimization, 12 hyperpa- rameters must be optimized, and thus it requires at least 120 evaluations according to the best practice. In consideration of the relatively fast run time for the ANN training process optimized, and thus it requires at least 120 evaluations according to the best practice. In consideration of the relatively fast run time for the ANN training process in comparison in comparison to the 3D CFD run time, that evaluation number is further doubled to 240 to the 3D CFD run time, that evaluation number is further doubled to 240 to provide to provide better accuracy. better accuracy. Figure 2. Figure ANN mod 2. ANN model el opti optimization mization d diagram. iagram. The optimized hyperparameters are obtained and provided in the last two columns of The optimized hyperparameters are obtained and provided in the last two columns Table 1. After optimization, the evaluation errors for efficiency and power were reduced, of Table 1. After optimization, the evaluation errors for efficiency and power were re- respectively, from 0.33% to 0.10%, and from 0.43% to 0.10%. The performance of the initial duced, r and optimized espective ANN ly, from models 0.3 ar 3e % to also examined 0.10%, and by comparing from 0.43% the pr to 0 edicted .10%values . The p vs. erfor truemance of values from the test set, shown in Figure 3. It is observed that the optimized ANN models the initial and optimized ANN models are also examined by comparing the predicted result in better alignment between the predicted values and the true values, as well as values vs. true values from the test set, shown in Figure 3. It is observed that the optimized narrower bandwidths of the prediction errors on the histogram plots. From Figure 3, it ANN models result in better alignment between the predicted values and the true values, is also observed that the bias of the optimized ANN for efficiency was greatly reduced, as well as narrower bandwidths of the prediction errors on the histogram plots. From as the data points are evenly distributed on both sides of the line of unity slope, shown Figuin re 3, it is al the second so subplot. observ This ed tha is consistent t the bias with of the theoptim histograms ized A of N the N for prediction efficienc errors y was gre in atly Figure 4, which shows that both the bias and bandwidth of the errors were improved reduced, as the data points are evenly distributed on both sides of the line of unity slope, in the optimized ANN models. The optimized ANN models are used in the subsequent shown in the second subplot. This is consistent with the histograms of the prediction er- blade optimization process in this study. The Python scripts for training the ANN models rors in Figure 4, which shows that both the bias and bandwidth of the errors were im- with the optimized hyperparameters are provided in [33]. The CPU time for training an proved in the optimized ANN models. The optimized ANN models are used in the sub- individual ANN largely depends on the number of Epochs. In the present study, the CPU times for training the ANN models representing efficiency and power with their respective sequent blade optimization process in this study. The Python scripts for training the ANN optimized hyperparameters were 14.3 min and 8.5 min, respectively. The total compute models with the optimized hyperparameters are provided in [33]. The CPU time for train- times for optimizing the hyperparameters of these ANN models were, respectively, 13.07 h ing an individual ANN largely depends on the number of Epochs. In the present study, and 11.65 h. This optimization was carried out based on parallel execution of 10 training the CPU times for training the ANN models representing efficiency and power with their models at the same time using a 6-core CPU. respective optimized hyperparameters were 14.3 min and 8.5 min, respectively. The total compute times for optimizing the hyperparameters of these ANN models were, respec- tively, 13.07 h and 11.65 h. This optimization was carried out based on parallel execution of 10 training models at the same time using a 6-core CPU. Int. J. Turbomach. Propuls. Power 2022, 7, 20 7 of 19 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 7 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 8 of 23 Figure 3. Predicted vs. true values of the initial and optimized ANN models. Figure 3. Predicted vs. true values of the initial and optimized ANN models. Figure 4. Histogram plots of the prediction errors of the initial and optimized ANN models. Figure 4. Histogram plots of the prediction errors of the initial and optimized ANN models. 2.2. Parametric Blade CAD and CFD Model 2.2. Parametric Blade CAD and CFD Model An initial turbine blade CAD model was built by adapting the E3 high pressure An initial turbine blade CAD model was built by adapting the E3 high pressure tur- turbine blade profiles. Using the first stage rotor blade coordinates provided at three bine blade profiles. Using the first stage rotor blade coordinates provided at three spanwise locations [27]—hub (12.693in), pitch (13.571in), and tip (14.41in)—and accounting spanwise locations [27]—hub (12.693in), pitch (13.571in), and tip (14.41in)—and account- for the incoming flow angle [28,29], a 3D CAD model was constructed by lofting these ing for the incoming flow angle [28,29], a 3D CAD model was constructed by lofting these cross-sectional profiles, as shown in Figure 5a, to provide a baseline case. Applying the cross-sectional profiles, as shown in Figure 5a, to provide a baseline case. Applying the same philosophy of lofting cross sections and at the same time utilizing the CST method same philosophy of lofting cross sections and at the same time utilizing the CST method for the cross-sectional profile definition [27], a parameterized blade CAD model was for the cross-sectional profile definition [27], a parameterized blade CAD model was cre- created for the design optimization study. First, the CST method creates a base cross- ated for the design optimization study. First, the CST method creates a base cross-sectional sectional profile, which is used as the pitch profile. The CST method allows varying the profile, which is used as the pitch profile. The CST method allows varying the base profile base profile in design optimization through manipulating the 20 weighting parameters, in design optimization through manipulating the 20 weighting parameters, fw , w , . . . , w g and w , w , . . . , w . Then, the base profile is scaled to create u,1 u,2 u,10 l,1 l,2 l,10 {𝑤 ,𝑤 ,…, 𝑤 } and {𝑤 ,𝑤 ,…,𝑤 }. Then, the base profile is scaled to create the , , , , , , the hub and tip profiles. In the scaling process, the twist angles and chord lengths of the hub and tip profiles. In the scaling process, the twist angles and chord lengths of the hub hub and tip profiles are matched to their counterparts of the original E3 blade; 4 additional n o and tip profiles are matched to their counterparts of the original E3 blade; 4 additional scaling factors, x , x , x , x , were applied to the lower and upper profiles hub,l hub,u ti p,l ti p,u scaling factors, {𝜉 , 𝜉 ,𝜉 ,𝜉 }, were applied to the lower and upper profiles of , , , , the hub and tip sections, respectively, to allow for small variations of the thicknesses of the hub and tip sections so that they can be optimized. The ranges of these variations are defined conservatively out of practical considerations of the original E3 engine blade shape. In general, the scaling allows the hub profile to be thicker and the tip profile to be thinner. Finally, a tilt angle, 𝜃 , and a lean angle, 𝛽 , were also built into the CAD model to be optimized later. Based on a reverse engineering study analyzing different engine blade geometries [34], the tilt angle can vary from 0 to 0.164°, and the lean angle can vary from −0.086° to 0.086° in the optimization study. A schematic plot of these angles is shown in Figure 5b. Int. J. Turbomach. Propuls. Power 2022, 7, 20 8 of 19 of the hub and tip sections, respectively, to allow for small variations of the thicknesses of the hub and tip sections so that they can be optimized. The ranges of these variations are defined conservatively out of practical considerations of the original E3 engine blade shape. In general, the scaling allows the hub profile to be thicker and the tip profile to be thinner. Finally, a tilt angle, q, and a lean angle, b, were also built into the CAD model to be optimized later. Based on a reverse engineering study analyzing different engine blade Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 9 of 23 geometries [34], the tilt angle can vary from 0 to 0.164 , and the lean angle can vary from 0.086 to 0.086 in the optimization study. A schematic plot of these angles is shown in Figure 5b. (a) (b) Figure 5. Blade geometry schematics: (a) blade geometry adapted from E3 profile coordinates; (b) Figure 5. Blade geometry schematics: (a) blade geometry adapted from E3 profile coordinates; blade angles schematic. (b) blade angles schematic. Using the 3D blade, a single-blade passage CFD model was developed using Siemens Using the 3D blade, a single-blade passage CFD model was developed using Siemens multi-physics package Simcenter STAR-CCM+. The continuity, momentum, and fluid en- multi-physics package Simcenter STAR-CCM+. The continuity, momentum, and fluid ergy transport equations are solved using a coupled solver (density-based) following a energy transport equations are solved using a coupled solver (density-based) following a finite volume approach with a 2nd order upwind discretization scheme on a polyhedral finite volume approach with a 2nd order upwind discretization scheme on a polyhedral grid. Menter’s SST K-Omega model [35] with all y+ treatment is used as a closure to the grid. Menter ’s SST K-Omega model [35] with all y+ treatment is used as a closure to the turbulence model. The all y+ wall treatment adjusts the application of a turbulence wall turbulence model. The all y+ wall treatment adjusts the application of a turbulence wall function based on the local y+ value of the near wall mesh cell. The CFD simulations were function based on the local y+ value of the near wall mesh cell. The CFD simulations were performed using the test conditions. Following E3 engine rotor testing conditions and 2D performed using the test conditions. Following E3 engine rotor testing conditions and 2D CFD practices from NASA studies [28,29], a total pressure of 344,777 Pa and a total tem- CFD practices from NASA studies [28,29], a total pressure of 344,777 Pa and a total temper- perature of 709 K were used for the inlet boundary condition, while an atmospheric pres- ature of 709 K were used for the inlet boundary condition, while an atmospheric pressure of sure of 101,325 Pa was defined at the outlet. The same boundary conditions representing 101,325 Pa was defined at the outlet. The same boundary conditions representing the engine the engine testing conditions were also applied in the previous 2D CFD simulations in [7]. testing conditions were also applied in the previous 2D CFD simulations in [7]. (These 2D (These 2D CFD datasets were used to develop ANN models in the present work.) A grid CFD datasets were used to develop ANN models in the present work.) A grid sequencing sequencing method was used to provide initial conditions, by solving inviscid flow equa- method was used to provide initial conditions, by solving inviscid flow equations repeat- tions repeatedly on a set of gradually refined mesh grids. Automatic CFL number control edly on a set of gradually refined mesh grids. Automatic CFL number control was also was also applied applied to adjustto the adj CFL ust the CF number L n inu rm esponse ber in respon to linear se solver to line conver ar solver conver gence behavior gence be during - havior durin the algebraic g the mul algebraic mu tigrid procedur ltigrid p e. Torocedure ensure overall . To ensure ove solutionrconver all solution con gence is r v eached, ergence in is re addition ached, in to add monitoring ition to m the onitor residuals ing the of resid the u governing als of the gov fluid ernin equations, g fluid estopping quations, s criteria top- ping criter were also ia set were also se based on the t based on asymptotic the asym values ptotic v of engineering alues of engin quantities, eering qu such anas titie isentr s, suopic ch efficiency, power output, maximum surface temperature, and total pressure difference. as isentropic efficiency, power output, maximum surface temperature, and total pressure The CFD methodology has been applied in a previous study based on a 2D version difference. of the E3 blade and validated against other experimental and numerical results of the E3 The CFD methodology has been applied in a previous study based on a 2D version blade [7]. In the present study, a grid independence investigation has also been performed of the E3 blade and validated against other experimental and numerical results of the E3 for the 3D CFD. Three sets of CFD grids with different resolutions were tested. The chosen blade [7]. In the present study, a grid independence investigation has also been performed grid features 12 million polyhedral cells with near-wall y+ values all below 1.4, based on for the 3D CFD. Three sets of CFD grids with different resolutions were tested. The chosen which the solution calculates a baseline efficiency of 96.0% and a power of 5.638  10 W. grid features 12 million polyhedral cells with near-wall y+ values all below 1.4, based on (Note power is calculated based on the single blade CFD model in the current study, which 4 which the solution calculates a baseline efficiency of 96.0% and a power of 5.638 × 10 W. has a pie sector angle of 4.737 ). The relative difference between the chosen grid and its (Note power is calculated based on the single blade CFD model in the current study, next-level refined grid is 0.1% for efficiency and 0.02% for power. which has a pie sector angle of 4.737°). The relative difference between the chosen grid and its next-level refined grid is 0.1% for efficiency and 0.02% for power. 2.3. Optimization Strategy The present study adopts a nested optimization process. The nested optimization is a slight variation of a previous co-optimization strategy introduced in [7]. The nested op- timization accomplishes two essential tasks. First, because an ANN model calculation con- sumes significantly less computing resources and time than a 3D CFD run, nesting an inner optimization based on the ANN model can effectively utilize the time while waiting Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 10 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 9 of 19 for a 3D CFD run to complete. For each 3D CFD simulation run, 250 evaluations of differ- ent blade cross-sectional profiles were completed in the inner optimization loop using the 2.3. Optimization Strategy ANN model. Second, the inner, ANN-driven optimization passes the best blade profile of The present study adopts a nested optimization process. The nested optimization the present optimization cycle to the 3D CFD run. This architecture effectively allows us- is a slight variation of a previous co-optimization strategy introduced in [7]. The nested ing the ANN-driven optimization to guide the 3D CFD search, and thus reduces the num- optimization accomplishes two essential tasks. First, because an ANN model calculation consumes significantly less computing resources and time than a 3D CFD run, nesting an ber of 3D CFD runs compared to an optimization process that is solely based on evaluat- inner optimization based on the ANN model can effectively utilize the time while waiting ing 3D CFD models. for a 3D CFD run to complete. For each 3D CFD simulation run, 250 evaluations of different The goal of the optimization is to maximize blade efficiency, η, and power output, blade cross-sectional profiles were completed in the inner optimization loop using the ANN model. Second, the inner, ANN-driven optimization passes the best blade profile of the Pow. The design variables that are being explored in optimization are the ones discussed present optimization cycle to the 3D CFD run. This architecture effectively allows using the in the previous session, 𝑤 ,𝑤 ,…, 𝑤 , 𝑤 ,𝑤 ,…, 𝑤 , 𝜉 , 𝜉 ,𝜉 ,𝜉 , 𝜃 , , , , , , , , , , ANN-driven optimization to guide the 3D CFD search, and thus reduces the number of and 𝛽 . The optimization procedure repeatedly executes the ANN and CFD models to 3D CFD runs compared to an optimization process that is solely based on evaluating 3D CFD models. evaluate different blade design variants, while extracting performance metrics to search The goal of the optimization is to maximize blade efficiency, h, and power output, Pow. for better designs. As the ANN model is much less computationally expensive compared The design variables that are being explored in optimization are the ones discussed in the to CFD, it is leveraged in an optimization loop by itself to yield optimized cross-sectional previous session, w , w , . . . , w , w , w , . . . , w , x , x , x , x , q and u,2 u,1 u,10 l,1 l,2 l,10 hub,l hub,u ti p,l ti p,u b. The optimization procedure repeatedly executes the ANN and CFD models to evaluate blade profiles, which are subsequently used to construct 3D shapes for CFD simulations. different blade design variants, while extracting performance metrics to search for better Using the ANN embedded optimization to guide the search for the best blade profiles designs. As the ANN model is much less computationally expensive compared to CFD, it is allows for reducing the number of expensive CFD evaluations. As such, a nested optimi- leveraged in an optimization loop by itself to yield optimized cross-sectional blade profiles, which are subsequently used to construct 3D shapes for CFD simulations. Using the ANN zation workflow is constructed, as illustrated in Figure 6. The workflow is realized using embedded optimization to guide the search for the best blade profiles allows for reducing commercial optimization software Simcenter HEEDS and its Sherpa optimization algo- the number of expensive CFD evaluations. As such, a nested optimization workflow rithm [30] is used in both Optimizer 1 and Optimizer 2. The Sherpa algorithm constantly is constructed, as illustrated in Figure 6. The workflow is realized using commercial optimization software Simcenter HEEDS and its Sherpa optimization algorithm [30] is evaluates the characteristics of the problem by using a hybrid combination of search strat- used in both Optimizer 1 and Optimizer 2. The Sherpa algorithm constantly evaluates egies at each stage of the optimization process to best traverse the design landscape. In the characteristics of the problem by using a hybrid combination of search strategies at addition, it also adapts the tuning parameters of the search strategies to the specific region each stage of the optimization process to best traverse the design landscape. In addition, it also adapts the tuning parameters of the search strategies to the specific region of the of the search. The HEEDS’ Sherpa algorithm has been effectively applied to optimize gas search. The HEEDS’ Sherpa algorithm has been effectively applied to optimize gas turbine turbine applications, as shown in studies [5–7]. applications, as shown in studies [5–7]. Figure 6. Optimization iteration hierarchy and data Figure 6. Optimization iteration hierar flow. chy and data flow. In the global optimization loop (Optimizer 1 as shown in Figure 6), a multi-objective In the global optimization loop (Optimizer 1 as shown in Figure 6), a multi-objective trade-off problem is solved targeting maximizing both efficiency and power. Two con- straints are set: efficiency must be greater than 95% and power must also be greater than trade-off problem is solved targeting maximizing both efficiency and power. Two con- a baseline power value. These constraints are included in the optimization performance straints are set: efficiency must be greater than 95% and power must also be greater than function, based on considerations of practical gas turbine design metrics and general a baseline power value. These constraints are included in the optimization performance function, based on considerations of practical gas turbine design metrics and general per- formance characters of the E engine. The global optimization solver, Optimizer 1, affects the blade shape change by directly optimizing the section scaling factors and blade angles in the CAD. In addition, Optimizer 1 also affects an inner optimization process, which is represented by Optimizer 2. Optimization 1 controls the selection of the CST weights used as input parameters for Optimizer 2, through manipulating a random seed, ℛ, which sets the initial state for the global search strategies employed by the Sherpa algorithm. Alt- hough Optimizer 1 does not directly manipulate the CST weights, because ℛ controls their selection, it acts as a 1D encoding of the entire 20-dimensional design space for Op- Int. J. Turbomach. Propuls. Power 2022, 7, 20 10 of 19 performance characters of the E engine. The global optimization solver, Optimizer 1, affects the blade shape change by directly optimizing the section scaling factors and blade angles in the CAD. In addition, Optimizer 1 also affects an inner optimization process, which is represented by Optimizer 2. Optimization 1 controls the selection of the CST weights used as input parameters for Optimizer 2, through manipulating a random seed, R, which sets the initial state for the global search strategies employed by the Sherpa algorithm. Although Optimizer 1 does not directly manipulate the CST weights, becauseR controls their selection, it acts as a 1D encoding of the entire 20-dimensional design space for Optimizer 1 to explore. Optimizer 1 repeatedly evaluates and optimizes different design variables during the global optimization cycles; in each evaluation cycle, Optimizer 2 optimizes the CST weights using the ANN models, with an objective to maximize the efficiency and a constraint on minimum power value. Overall, 250 blade cross-sectional profiles were evaluated in each execution of Optimizer 2. It directly explores the shape profiles through manipulating the weighting parameters used in the CST method. In each evaluation cycle of Optimizer 1, the best blade profile found by Optimizer 2 is applied in the more expensive and high-fidelity CFD simulation. The performance function in each optimization is defined as the sum of normalized objectives minus the sum of normalized constraints. In a generic form, the performance function is formulated as follows: 4 2 i=N Obj 10  ConsViol j=N obj i cons P = , (3) å å i=1 j=1 2 Norm Norm i j In the above formula, Obj represents the objective quantity, such as h and Pow; ConsViol represents the amount of violation resulted from each constraint definition (h > 0.95, Pow > Pow ); the normalization factors Norm are selected using the respec- baseline tive h and Pow values of the baseline design. 3. Results 3.1. Optimization Results In the present optimization study, a total of 163 blade designs have been evalu- ated in 3D CFD, with 40,750 2D blade profiles being evaluated by the ANNs. An effi- ciency vs. power plot showing all 3D blade design points is provided in Figure 7. A Pareto front is formed in the upper right region of Figure 7, which contains optimal designs with the best trade-off relations between efficiency and power. Among the optimal designs, the maximum increase in efficiency is over 3%, and the maximum increase in power is around 8%. To obtain insight into the design variable distributions of the optimal designs, two groups of optimal designs are selected—a group of top 10 high-efficiency designs, and another group of top 10 high power designs. The responses and selected design input variables (which include the two blade angles and four scaling factors) for these two groups of designs are highlighted on a parallel plot in Figure 8. The parallel plot reveals some qual- itative trends in the design parameters: (1) both the high-efficiency and high power designs favor large lean angles, b; (2) the high-efficiency designs also favor a large scaling factor for the tip upper surface, x , while the high power designs favor a small-to-medium value ti p,u for x ; (3) a weaker influence of the scaling factor for the hub lower surface, x , is ti p,u hub,u also observed. Note that the lean angle was constrained conservatively within a smaller range in consideration of commercial turbine geometric characteristics provided in [27]. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 11 of 23 timizer 1 to explore. Optimizer 1 repeatedly evaluates and optimizes different design var- iables during the global optimization cycles; in each evaluation cycle, Optimizer 2 opti- mizes the CST weights using the ANN models, with an objective to maximize the effi- ciency and a constraint on minimum power value. Overall, 250 blade cross-sectional pro- files were evaluated in each execution of Optimizer 2. It directly explores the shape pro- files through manipulating the weighting parameters used in the CST method. In each evaluation cycle of Optimizer 1, the best blade profile found by Optimizer 2 is applied in the more expensive and high-fidelity CFD simulation. The performance function in each optimization is defined as the sum of normalized objectives minus the sum of normalized constraints. In a generic form, the performance function is formulated as follows: ∑ ∑ 𝒫= − , (3) In the above formula, 𝑂𝑏𝑗 represents the objective quantity, such as 𝜂 and 𝑃𝑜𝑤 ; 𝐶𝑜𝑛𝑠𝑉𝑖𝑜𝑙 represents the amount of violation resulted from each constraint definition (𝜂> 0.95, 𝑃𝑜𝑤 > 𝑃𝑜𝑤 ); the normalization factors Norm are selected using the respective 𝜂 and 𝑃𝑜𝑤 values of the baseline design. 3. Results 3.1. Optimization Results In the present optimization study, a total of 163 blade designs have been evaluated in 3D CFD, with 40,750 2D blade profiles being evaluated by the ANNs. An efficiency vs. power plot showing all 3D blade design points is provided in Figure 7. A Pareto front is formed in the upper right region of Figure 7, which contains optimal designs with the best trade-off relations between efficiency and power. Among the optimal designs, the maxi- mum increase in efficiency is over 3%, and the maximum increase in power is around 8%. To obtain insight into the design variable distributions of the optimal designs, two groups of optimal designs are selected—a group of top 10 high-efficiency designs, and another group of top 10 high power designs. The responses and selected design input variables (which include the two blade angles and four scaling factors) for these two groups of de- signs are highlighted on a parallel plot in Figure 8. The parallel plot reveals some qualita- tive trends in the design parameters: (1) both the high-efficiency and high power designs favor large lean angles, 𝛽 ; (2) the high-efficiency designs also favor a large scaling factor for the tip upper surface, 𝜉 , while the high power designs favor a small-to-medium value for 𝜉 ; (3) a weaker influence of the scaling factor for the hub lower surface, 𝜉 , is also observed. Note that the lean angle was constrained conservatively within a Int. J. Turbomach. Propuls. Power 2022, 7, 20 11 of 19 smaller range in consideration of commercial turbine geometric characteristics provided in [27]. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 12 of 23 Figure 7. Figure 7. Effi Efci ficiency ency vs vs. . power power for al for alll designs designpoints s points in in the the o optimization ptimization stu study. dy. Figure 8. Parallel plot highlighting selected design parameters of different groups of designs. Figure 8. Parallel plot highlighting selected design parameters of different groups of designs. A quantitative analysis has also been conducted. The Pearson correlation coefficients A quantitative analysis has also been conducted. The Pearson correlation coefficients of the data are calculated, based on the covariance and standard deviations. The definition of the data are calculated, based on the covariance and standard deviations. The definition is given as follows: is given as follows: (x x) Y Y i i i=1 r = q , (4) ∑ (𝓍 ̅)(𝓎 ) n 𝔯 = n , (x x) Y Y (4) å å i i i=1 i=1 ∑ (𝓍 ̅ ) ∑ (𝓎 ) n o In the above formula, x, Y 2 h, Pow, b, q, x , x , x , x , ( x 6= Y). hub,l hub,u ti p,l ti p,u In the above formula, 𝓍, 𝓎 ∈ 𝜂,𝑃𝑜𝑤,𝛽,𝜃,𝜉 , 𝜉 ,𝜉 ,𝜉 , ( 𝓍 ≠ 𝓎) . 𝓍 and 𝓎 , , , , x and Y represent sample values corresponding to each blade design, and n is the total i i represent sample values corresponding to each blade design, and n is the total number of number of designs, which is 163 in the present study. The Pearson correlation coefficients, designs, which is 163 in the present study. The Pearson correlation coefficients, the histo- the histogram distributions of selected parameters, and the data sample plots are shown on gram distributions of selected parameters, and the data sample plots are shown on a cor- a correlation plot in Figure 9, respectively, on the upper right region, along the diagonal relation plot in Figure 9, respectively, on the upper right region, along the diagonal line line joining the upper left and lower right corners, and on the lower left region. Large joining the upper left and lower right corners, and on the lower left region. Large magni- magnitudes of the Pearson correlation coefficients for x , h (1st row, last column) ti p,u tud and es ofbf , the PowPear g (2nd son cor row, 3rre dla column) tion coe indicate fficients str for ong 𝜉 relationships ,𝜂 (1stin rthese ow, ltwo ast c pairs olum of n) and variables. The large coefficient values are confirmed by their respective data sample plots {𝛽, 𝑃𝑜𝑤 } (2nd row, 3rd column) indicate strong relationships in these two pairs of varia- of x vs. h (last row, 1st column) and b vs. Pow (3rd row, 2nd column), in which most ti p,u bles. The large coefficient values are confirmed by their respective data sample plots of of the design points follow the linear regression lines closely. In addition, the correlation 𝜉 vs. 𝜂 (last row, 1st column) and 𝛽 vs. 𝑃𝑜𝑤 (3rd row, 2nd column), in which most plot also shows that the effect of the scaling factor for the hub lower surface, x , is hub,u of the design points follow the linear regression lines closely. In addition, the correlation rather mild. plot also shows that the effect of the scaling factor for the hub lower surface, 𝜉 , is rather mild. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 13 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 12 of 19 Figure 9. Correlation plot of selected design parameters of all designs. (The two most influential Figure 9. Correlation plot of selected design parameters of all designs. (The two most influential numbers and their corresponding plots are highlighted using the dashed arrows.) numbers and their corresponding plots are highlighted using the dashed arrows.) The efficiency and power are plotted versus the two parameters with the strongest The efficiency and power are plotted versus the two parameters with the strongest correlations, lean angle (b), and tip upper surface scaling factor (x ), shown in Figure 10. correlations, lean angle (𝛽 ), and tip upper surface scaling factor (𝜉 ), shown in Figure ti p,u It is observed that (1) most of the improved designs favor larger lean angles, and (2) among 10. It is observed that (1) most of the improved designs favor larger lean angles, and (2) the improved designs, a smaller x leads to greater power improvement, while a larger Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 14 of 23 ti p,u among the improved designs, a smaller 𝜉 leads to greater power improvement, while x leads to high-efficiency gain. To analyze the effects of the blade shape and the fluid ti p,u a larger 𝜉 leads to high-efficiency gain. To analyze the effects of the blade shape and dynamics, two designs are selected for further investigation—a high-efficiency design and the fluid dynamics, two designs are selected for further investigation—a high-efficiency a high power design, which will be referred to as “Design A” and “Design B,”, respectively. design and a high power design, which will be referred to as “Design A” and “Design B,”, These two designs are also marked in Figure 10 and will be discussed in the next session. respectively. These two designs are also marked in Figure 10 and will be discussed in the next session. (a) (b) Figure 10. Two views of a 3D plot showing efficiency, power, lean angle, and tip upper surface Figure 10. Two views of a 3D plot showing efficiency, power, lean angle, and tip upper surface scaling scaling factor of all designs: (a) a normal view; and (b) a view obtained by rotating the left-side factor of all designs: (a) a normal view; and (b) a view obtained by rotating the left-side plot around plot around the 𝜂 axis by 180°. the h axis by 180 . The total run time for the optimization process was around 30 h (with six CFD eval- uations running in parallel, each consuming 160 compute cores). There are 27 input vari- ables in the global optimization problem. As illustrated in Figure 6, the integration of the inner optimization loop allows for optimizing the 20 blade profile parameters using the inexpensive ANN models. Throughout the optimization process, the ANN models were evaluated 40,750 times; in each optimization cycle, the best blade cross-sectional profile obtained from the inner ANN optimization was passed on to the 3D CFD for evaluation. As a result, the number of design variables being directly exposed in the 3D CFD runs is reduced to six. Since the required number of CFD evaluations largely depends on the number of input variables in an optimization problem, the nested optimization strategy only requires 163 evaluations of 3D CFD simulations. In comparison, if an optimization study of the same problem does not adopt the nested optimization strategy and is solely based on CFD, then all 27 design variables (rather than 6) will be evaluated based on 3D CFD runs. Based on a quick estimation, this will result in 4.5 times (=27/6) the number of CFD runs, which is 733 CFD runs, and will cost roughly 135 h of optimization run time if the same compute resource settings were used. 3.2. Fluid Dynamics Analysis Two optimized designs are selected for detailed investigation of the aerodynamic results: a high-efficiency design, and a high-power design (a.k.a. Designs A and B as shown in Figure 9). They are compared with the baseline design. The CFD results show the optimized designs feature less pressure drop on the suction side near the trailing edge. A pressure coefficient is defined based on the average inlet velocity and density: 𝑐 = , (5) The pressure coefficient distributions on the hub, mid, and tip sections are compared in the upper row of plots in Figure 11. The baseline design shows significant low pressure regions on the suction surface near the trailing edge on all three streamwise sections. In Int. J. Turbomach. Propuls. Power 2022, 7, 20 13 of 19 The total run time for the optimization process was around 30 h (with six CFD evalua- tions running in parallel, each consuming 160 compute cores). There are 27 input variables in the global optimization problem. As illustrated in Figure 6, the integration of the inner optimization loop allows for optimizing the 20 blade profile parameters using the inexpen- sive ANN models. Throughout the optimization process, the ANN models were evaluated 40,750 times; in each optimization cycle, the best blade cross-sectional profile obtained from the inner ANN optimization was passed on to the 3D CFD for evaluation. As a result, the number of design variables being directly exposed in the 3D CFD runs is reduced to six. Since the required number of CFD evaluations largely depends on the number of input variables in an optimization problem, the nested optimization strategy only requires 163 evaluations of 3D CFD simulations. In comparison, if an optimization study of the same problem does not adopt the nested optimization strategy and is solely based on CFD, then all 27 design variables (rather than 6) will be evaluated based on 3D CFD runs. Based on a quick estimation, this will result in 4.5 times (=27/6) the number of CFD runs, which is 733 CFD runs, and will cost roughly 135 h of optimization run time if the same compute resource settings were used. 3.2. Fluid Dynamics Analysis Two optimized designs are selected for detailed investigation of the aerodynamic results: a high-efficiency design, and a high-power design (a.k.a. Designs A and B as shown in Figure 9). They are compared with the baseline design. The CFD results show the optimized designs feature less pressure drop on the suction side near the trailing edge. A pressure coefficient is defined based on the average inlet velocity and density: c = , (5) 1 2 r u in 2 in The pressure coefficient distributions on the hub, mid, and tip sections are compared in the upper row of plots in Figure 11. The baseline design shows significant low pressure regions on the suction surface near the trailing edge on all three streamwise sections. In comparison, the two optimized designs saw increased minimum pressure in those areas at the hub and mid sections. An enlarged view of the pressure coefficient distribution on the mid-section near the trailing edge is provided for each design in the lower row of plots in Figure 11. The baseline design features a shock in this region, highlighted by a relatively high pressure spot next to the low pressure zone, which is present inside the dotted circle marked on Figure 11a. This shock feature is absent in the two optimized designs due to the modifications in the blade shapes in those designs. This observation is consistent with the Mach number distributions that will be discussed in a later paragraph. To further investigate the pressure drop on the suction side near the trailing edge, volumetric renderings of the pressure coefficient scenes are shown in Figure 12, to highlight the flow in the low-pressure regime by focusing on c values between 0.35 and 0.1. It is observed that in the optimized designs, lower pressure coefficients are present near the tip, but the pressure coefficient has been increased in the mid-session and nearer the hub. The overall reduction of maximum negative pressure coefficients in the mid and hub regions offsets the effects of increased tip leakage flow in the optimized designs. Next, the pressure coefficient distributions on the blade surfaces are investigated. Scalar scenes of the pressure coefficients on the baseline, Design A, and Design B blades are shown in Figure 13. A large portion of the suction-side surface in the baseline design features a significantly negative c values between 0.35 and 0.23 starting around the three quarters chord. In the optimized designs, the onset of the negative pressure coefficient zone moves upstream, nearer the half chord, yet the magnitude of this zone is much less, with most areas featuring c values between 0.12 and 0. To achieve a more quantitative comparison, the pressure coefficient curves along the streamwise direction on the hub, mid, and tip sections of the three blades are plotted and compared, with the plots of the blade profiles, shown in Figure 14. The earlier onsets of the lower pressure coefficient region on Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 15 of 23 comparison, the two optimized designs saw increased minimum pressure in those areas Int. J. Turbomach. Propuls. Power 2022, 7, 20 at the hub and mid sections. An enlarged view of the pressure coefficient distribution on 14 of 19 the mid-section near the trailing edge is provided for each design in the lower row of plots in Figure 11. The baseline design features a shock in this region, highlighted by a relatively high pressure spot next to the low pressure zone, which is present inside the dotted circle the suction-side surfaces in the two optimized designs are demonstrated by the plots. The marked on Figure 11a. This shock feature is absent in the two optimized designs due to earlier onset allows for larger enclosed areas of negative c curves for the optimized design, the modifications in the blade shapes in those designs. This observation is consistent with contributing the Mach nu tom the ber dis power tribuimpr tions th ovements. at will be discussed in a later paragraph. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 16 of 23 (a) (b) (c) Figure 11. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. Figure 11. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. To further investigate the pressure drop on the suction side near the trailing edge, volumetric renderings of the pressure coefficient scenes are shown in Figure 12, to high- light the flow in the low-pressure regime by focusing on 𝑐 values between −0.35 and −0.1. It is observed that in the optimized designs, lower pressure coefficients are present near the tip, but the pressure coefficient has been increased in the mid-session and nearer the hub. The overall reduction of maximum negative pressure coefficients in the mid and hub regions offsets the effects of increased tip leakage flow in the optimized designs. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 17 of 23 (a) (b) (c) Figure 12. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. Figure 12. Static pressure distributions: (a) Baseline design; (b) Design A; and (c) Design B. Next, the pressure coefficient distributions on the blade surfaces are investigated. Scalar scenes of the pressure coefficients on the baseline, Design A, and Design B blades are shown in Figure 13. A large portion of the suction-side surface in the baseline design features a significantly negative 𝑐 values between −0.35 and −0.23 starting around the three quarters chord. In the optimized designs, the onset of the negative pressure coeffi- cient zone moves upstream, nearer the half chord, yet the magnitude of this zone is much less, with most areas featuring 𝑐 values between −0.12 and 0. To achieve a more quanti- tative comparison, the pressure coefficient curves along the streamwise direction on the hub, mid, and tip sections of the three blades are plotted and compared, with the plots of the blade profiles, shown in Figure 14. The earlier onsets of the lower pressure coefficient region on the suction-side surfaces in the two optimized designs are demonstrated by the plots. The earlier onset allows for larger enclosed areas of negative 𝑐 curves for the op- (a) (b) (c) timized design, contributing to the power improvements. Figure 13. Pressure coefficient scenes on the surfaces of the blades: (a) Baseline design; (b) Design Figure 13. Pressure coefficient scenes on the surfaces of the blades: (a) Baseline design; (b) Design A; A; and (c) Design B. and (c) Design B. To further investigate the differences in solutions, a volume rendering scene is cre- ated by focusing on visualizing changes in Mach number within the range between 0.8 and 1.2, shown in Figure 15. The goal of this plot is to reveal where sudden changes in Mach number may occur in the transonic flow regime. A consistent shock structure near the trailing edge covering the full span length of the blade is observed in the baseline design, which is consistent with the high pressure spot in the same area highlighted by the dotted circle in Figure 11a. This shock feature is missing in the two optimized designs (Designs A and B), due to the weaker gradients in those designs. Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 18 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 15 of 19 (a) (b) (c) Figure 14. Blade profiles (first row) and pressure coefficients in the streamwise direction (second Figure 14. Blade profiles (first row) and pressure coefficients in the streamwise direction (second row): (a) hub sections comparison; (b) mid sections comparison; and (c) tip sections comparison. row): (a) hub sections comparison; (b) mid sections comparison; and (c) tip sections comparison. To further investigate the differences in solutions, a volume rendering scene is created by focusing on visualizing changes in Mach number within the range between 0.8 and 1.2, shown in Figure 15. The goal of this plot is to reveal where sudden changes in Mach number may occur in the transonic flow regime. A consistent shock structure near the Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 19 of 23 trailing edge covering the full span length of the blade is observed in the baseline design, which is consistent with the high pressure spot in the same area highlighted by the dotted circle in Figure 11a. This shock feature is missing in the two optimized designs (Designs A and B), due to the weaker gradients in those designs. (a) (b) (c) Figure 15. Mach number rendering: (a) Baseline design; (b) Design A; and (c) Design B. Figure 15. Mach number rendering: (a) Baseline design; (b) Design A; and (c) Design B. A further comparison is made by plotting the streamlines in the tip leakage regions A further comparison is made by plotting the streamlines in the tip leakage regions and showing pressure coefficients on spanwise cross-sections in Figure 16. It appears the and showing pressure coefficients on spanwise cross-sections in Figure 16. It appears the two optimized designs feature slightly more chaotic Mach number distributions in the two optimized designs feature slightly more chaotic Mach number distributions in the local region near the trailing edge and tip leakage. The leakage flow’s influence on the local region near the trailing edge and tip leakage. The leakage flow’s influence on the overall performance is relatively small in comparison to the previously discussed suction- overall performance is relatively small in comparison to the previously discussed suction- side effects. It can be observed that the streamlines in the high-efficiency design (Design side effects. It can be observed that the streamlines in the high-efficiency design (Design A) are slightly more curved, yet the induced low-pressure spots by the secondary flow A) are slightly more curved, yet the induced low-pressure spots by the secondary flow effects of the tip leakage flows are weak and about the same magnitude. Overall, the dif- effects of the tip leakage flows are weak and about the same magnitude. Overall, the ference caused by the tip leakage flow is small in these three designs. The performance difference caused by the tip leakage flow is small in these three designs. The performance improvements in the optimized designs are more influenced by other flow effects dis- improvements in the optimized designs are more influenced by other flow effects discussed cussed earlier, such as the increased minimum pressure on the suction side near the trail- earlier, such as the increased minimum pressure on the suction side near the trailing edge ing edge (Figures 11–13), and the reduction of the shock structure (Figure 15). (Figures 11–13), and the reduction of the shock structure (Figure 15). Int. J. Turbomach. Propuls. Power 2022, 7, x FOR PEER REVIEW 20 of 23 Int. J. Turbomach. Propuls. Power 2022, 7, 20 16 of 19 (a) (b) (c) Figure 16. Tip leakage flow streamlines and pressure coefficients on spanwise cross-sections: (a) Figure 16. Tip leakage flow streamlines and pressure coefficients on spanwise cross-sections: (a) Base- Baseline design; (b) Design A; and (c) Design B. line design; (b) Design A; and (c) Design B. 4. Conclusions 4. Conclusions The present study has demonstrated an optimization workflow combining the use of The present study has demonstrated an optimization workflow combining the use of neural networks and high-fidelity CFD. The neural network models were trained on over neural networks and high-fidelity CFD. The neural network models were trained on over three-thousand design data points from a previous publication. The practical implication three-thousand design data points from a previous publication. The practical implication of the overall strategy is that in engineering design analysis, existing data sets, which are of the overall strategy is that in engineering design analysis, existing data sets, which generally available from the previous simulation, experimental, or reduced-order studies, are generally available from the previous simulation, experimental, or reduced-order can be leveraged to build neural network models, which can then be used in combination studies, can be leveraged to build neural network models, which can then be used in with high-fidelity CFD simulations to guide optimization processes. This approach combination with high-fidelity CFD simulations to guide optimization processes. This achieves a reduction in the required number of high-fidelity CFD runs, and hence reduces approach achieves a reduction in the required number of high-fidelity CFD runs, and the computational cost while maintaining accuracy. The integration of computationally hence reduces the computational cost while maintaining accuracy. The integration of inexpensive ANN models, which were evaluated 40,750 times, allows for a relatively computationally inexpensive ANN models, which were evaluated 40,750 times, allows for small number (163) of CFD evaluations in the present optimization process, resulting in a a relatively small number (163) of CFD evaluations in the present optimization process, total run time of about 30 h (with 6 CFD evaluations running in parallel, each consuming resulting in a total run time of about 30 h (with 6 CFD evaluations running in parallel, 160 compute cores). It is estimated that if the nested optimization strategy based on ANN each consuming 160 compute cores). It is estimated that if the nested optimization strategy was not used, a total number of 733 CFD evaluations will be required due to the large based on ANN was not used, a total number of 733 CFD evaluations will be required due to number of design variables exposed in 3D CFD evaluations, resulting in roughly 135 h of the large number of design variables exposed in 3D CFD evaluations, resulting in roughly the optimization run time if the same compute resource settings were used. 135 h of The efficac the optimization y of the me runtho time dology if theis de same monstr compute ated o resour n a turbine b ce settings ladwer e aero e used. dynamic problem. ANN models The efficacy of the methodology with 7 layers ais nd 6 la demonstrated yers were b on uilt ato r turbine epresent blade two aer blad odynamic e per- formance metrics, efficiency, and power, respectively. The hyperparameters of the ANN problem. ANN models with 7 layers and 6 layers were built to represent two blade models were optimized, and the models were used as surrogate models along with high- performance metrics, efficiency, and power, respectively. The hyperparameters of the ANN fidelity CFD simulations in a nested optimization procedure to obtain optimized blade models were optimized, and the models were used as surrogate models along with high- designs. Pareto front designs representing improved efficiency and power were found by fidelity CFD simulations in a nested optimization procedure to obtain optimized blade the optimization procedure. A lean angle and a tip scaling factor were shown to be more designs. Pareto front designs representing improved efficiency and power were found by favored by the optimization procedure than other parameters in the context of the chosen the optimization procedure. A lean angle and a tip scaling factor were shown to be more blade and analysis methodology used in the current study. Examining the fluid dynamics favored by the optimization procedure than other parameters in the context of the chosen of the optimized designs vs. the baseline design reveals that the optimization (1) reduced blade and analysis methodology used in the current study. Examining the fluid dynamics the magnitude of the most negative pressure coefficients in the flow on the suction side of the optimized designs vs. the baseline design reveals that the optimization (1) reduced near the trailing edge, and (2) altered the blade geometry that reduced the shock near the the magnitude of the most negative pressure coefficients in the flow on the suction side trailing edge. Both of these aspects have effects on improved efficiency and power in the near the trailing edge, and (2) altered the blade geometry that reduced the shock near the optimized designs. trailing edge. Both of these aspects have effects on improved efficiency and power in the As a future extension of this study, the following may be considered. (1) Test the optimized designs. performance of other response surface methods and meta models against ANN and apply As a future extension of this study, the following may be considered. (1) Test the them in the nested optimization workflow. (2) The present study demonstrates a nested performance of other response surface methods and meta models against ANN and apply ANN-CFD optimization methodology, applied to an idealized turbine CFD problem as a them in the nested optimization workflow. (2) The present study demonstrates a nested proof-of-concept. In production-level gas turbine designs, more detailed constraints on ANN-CFD optimization methodology, applied to an idealized turbine CFD problem as a proof-of-concept. In production-level gas turbine designs, more detailed constraints on the geometry or loading curve must be incorporated to ensure more realistic optimiza- tion outcomes. Different power output conditions of the engine may also be explored in optimization. Int. J. Turbomach. Propuls. Power 2022, 7, 20 17 of 19 Author Contributions: Conceptualization, C.Z.; methodology, C.Z. and M.J.; software, C.Z. and M.J.; validation, C.Z. and M.J.; formal analysis, C.Z.; investigation, C.Z. and M.J.; resources, C.Z. and M.J.; data curation, C.Z.; writing—original draft preparation, C.Z.; writing—review and editing, C.Z. and M.J.; visualization, C.Z.; supervision, M.J.; project administration, C.Z. and M.J.; funding acquisition, M.J. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Data Availability Statement: Not applicable. Acknowledgments: This work is a part of the authors’ professional development initiatives spon- sored by the Simcenter Customer Support organization of Siemens Digital Industries Software. The authors wish to thank their colleagues at Siemens Digital Industries Software. Additional thanks go to Yutao He at Jet Propulsion Laboratory, Caltech, for discussions on the research methodology. Conflicts of Interest: The authors declare no conflict of interest. Nomenclature c pressure coefficient h fluid enthalpy J/kg p pressure Pa P performance function Pow power W r radius m r Pearson coefficient u velocity m/s Dh enthalpy change J/kg w blade profile parameter h efficiency r density kg/m s centrifugal stress Pa W rotating speed rad/s Subscripts h hub i domain inlet l lower surface max maximum quantity o domain outlet s solid t tip u upper surface References 1. Alexandrov, N.M.; Lewis, R.M.; Gumbert, G.R.; Green, L.L.; Newman, P.A. 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Journal

"International Journal of Turbomachinery, Propulsion and Power"Multidisciplinary Digital Publishing Institute

Published: Jun 30, 2022

Keywords: CFD; optimization; aerodynamics; gas turbines; machine learning; neural networks

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