CHAPTER III CASE STUDY AND METHODOLOGY III.1 Case Studies As previously discussed, this research deals with three kinds case studies that involve the flutter prediction case as a less complex problem, and also a wind turbine blade design and re-entry trajectory analysis as the more complex problems. This section provide explanations of the cases. III.1.1 Flutter Prediction Case The first case study used to benchmark the various machine learning models’ predictive performance and their solution interpretability is a flutter behavior prediction task in a two-degree-of-freedom NACA64A010 airfoil aeroelastic system researched by Palar et al. (Palar et al., 2020). This task was chosen as one of the benchmark case study in this research due to its non-linearity phenomena of the problem that makes it more challenging to model and its low-dimentionality that fits as a less-complex problem case study. The task is to predict theγof the aeroelastic system as the output variable based on the variations of the Mach numberM∈[0.6,0.9] and the flutter speed indexVµ∈[0.4,2.0] as input variables. An aeroelastic system’s stability could be determined to experience flutter (destructive unstable condition) or not, based onγvalue where positive value results in an unstable system and negative value as a stable system. The boundary that acts as the decisive stability condition whereγ= 0 is regarded as the flutter boundary. The dataset used to train and test the machine learning models in this research was obtained from the past work of Palar et al. (Palar et al., 2020) where they generated 300 sample points with a latin hypercube sampling (LHS) method (McKay et al., 1979) to ensure uniform sample across the parameter space. The scatter plot of the 300 sample points are as shown by Fig. III.1 . 32 Figure III.1 . Dataset of 300 samples from latin hypercube sampling This study case was selected to be the study case to benchmark the models’ performance in less training data environment where the dataset was divided into the training set withNtrainnumber of samples, where it contains sample points used to train the machine learning model and the test set withNtest number of samples used to test the model’s ability to predict unseen samples. The samples in the training and test set were shuffled and chosen randomly. The number ofNtestdepends onNtrainasNtest= 300−Ntrain. When comparing various machine learning performances throughout various numbers ofNtrain, for each number ofNtrain, this research used 21 different training and test sets with randomly chosen sample points and evaluate the models’ performance based on how it performs generally from those 21 different train and test sets. This aims to prevent training set sampling point bias that could favor certain machine learning models and to measure the machine learning model’s performance and robustness regardless of the training sample in the less training data condition. 33 III.1.2 Wind Turbine Blade Design Case The second case study is more complex when compared to the first case study as it is a higher-dimension non-linear problem that involves the design analysis of floating offshore wind turbines (FOWTs) based on the environmental condition at the East Coast of Mexico. The environment variables considered in the analysis are theVhub,θwwith respect to the wind,Hs,Tp, andθp. In this work, those five variables were considered as the input variables, where the task is to predict the side-side tower base bending momentMtbas the output prediction. The dataset for this problem was obtained from Graf et al.(Graf et al., 2016) with 10,000 instance samples as shown by Fig. III.2 . The training and test set ratio is 4 : 1, thus resulting inNtrain= 8000 andNtest= 2000. Figure III.2 . Wind turbine case 10,000 sample points 34 III.1.3 Re-entry Trajectory Analysis Case The last case study which is the most complex problem compared to the previous case studies as it has higher dimensions and seems to have discontinue- clustered variables. The problem is about a stochastic re-entry trajectory analysis of an Apollo-type capsule with seven uncertain initial conditions of the capsule’s variables defined as the input variables. Those seven variables are theAlt,Vo,γp,α, M,MoI, andq. The task is to predict theVgof the vehicle as the output prediction. The dataset for this problem was obtained from the work of Tokunaga et al. (Tokunaga et al., 2019) with 10,000 instance samples as shown by Fig. III.3 . Similarly to the wind turbine case, the training and test set ratio is 4 : 1, thus resulting inNtrain= 8000 andNtest= 2000. Figure III.3 . Re-entry trajectory case 10,000 sample points 35 III.2 Machine Learning Modeling As there are various machine learning models used in this research, this section provides explanations of the modeling procedure for each machine learning model algorithm, including the calculation procedure, parameters, hyperparameters, and implementation methodology used for each machine learning algorithm. III.2.1 Polynomial Chaos Expansion Modeling The method used in this research to construct the orthogonal polynomial expansion for the PCE model is by the Stieltjes procedure (Stieltjes, 1884) as it is one of the most popular method due to its well-known numerically stable property. The polynomial expansion was limited up to the third order and the distribution space where the polynomials are orthogonal used was the joint probability distribution of the input variables with uniform distribution. For the flutter prediction case, The orthogonal polynomials set constructed by the aforementioned setups yield the polynomials as shown by Table. III.1 where the polynomials are represented in terms of the input variables of the flutter prediction problem case study. Table III.1 . PCE orthogonal polynomial set for flutter prediction case Orthogonal PCE Polynomials M 3 −1.5M 2 + 0.6M −0.05 M 2 Vµ−MVµ−0.5M 2 + 0.17V µ+ 0.5M −0.08 MV 2 µ−0.5V 2 µ−MVµ+ 0.5V µ+ 0.17M −0.08 V 3 µ−1.5V 2 µ+ 0.6V µ−0.05 M 2 −M+ 0.17 MVµ−0.5Vµ−0.5M + 0.25 V 2 µ−Vµ+ 0.17 M−0.5 Vµ−0.5 1.0 After generating the orthogonal polynomials, the coefficients of the PCEshould be calculated. This research used a non-intrusive method which is 36 the point collocation method (Hosder et al., 2007) where the coefficients could be obtained by solving a linear system as previously shown by Eq. (II.3). The method used to solve the linear system in Eq. (II.3) in this research was by the least squares minimization method (Aldrich, 1998) where it is considered as a basic and common approach to fit a polynomial to its training sample points. The general simplified form of the polynomial equation with the coefficients is as shown by Eq. (III.1). γ= ˆ f(M, Vµ) =c9V 3 µ+c8MV 2 µ+c7M 2 Vµ+c6M 3 +c5V 2 µ+c4MVµ+c3M 2 +c2Vµ+c1M+c0 (III.1) As for the wind turbine case, the general simplified form of the polynomial equation of up to second order PCE with the coefficients are as shown by Eq. (III.2). Notice that by using PCE to model the wind turbine case even if its still on the second order polynomial, the model contains up to 21 terms where it starts to be considered as hardly interpretable, thus for the wind turbine case, model-specific interpretation by PCE is not feasible.