Enhance manuscript content by refining design of experiments section and adding…

Manuscript/RESILINK_surrogate_optimization_manuscript.tex

@@ -120,7 +120,7 @@ The admissible thickness ranges are defined according to the geometry family and
 
 \begin{table}[ht!]
 \centering
-\caption{Geometry families and surrogate input variables considered in the current implementation.}
+\caption{Geometry families in the current implementation and bounds of the window thicknesses.}
 \label{tab:families}
 \begin{tabular}{llllll}
 \toprule
@@ -186,38 +186,66 @@ In addition, the maximum local shear distortion in each window is denoted by $\v
 
 \subsection{Design of experiments}\label{subsec:doe}
 
-The FEM campaign is designed to cover the admissible design domain of each device family. Latin Hypercube Sampling (LHS) is used to generate a homogeneous set of thickness combinations within the prescribed bounds. This sampling strategy is well suited to the current problem because each window thickness has a bounded interval and the number of design variables increases with the number of windows. The FEM results stored: the input variables $t_{w,i}$ and the target outputs, including $\varepsilon_{xy,i}$, $\TFD_i$ and $\TFD_f$.
+The FEM campaign is designed to cover the admissible design domain of each device family while ensuring a sufficiently homogeneous exploration of the multidimensional parameter space. The design variables correspond to the window thicknesses $t_{w,i}$, whose combinations are generated using a Design of Experiments (DoE) strategy based on Latin Hypercube Sampling (LHS) optimized with the maximin criterion \cite{Joseph2008}. This approach provides a near-random yet space-filling distribution of samples, reducing clustering effects and improving the representation of the admissible domain.
 
-The training process is iterative. For each configuration family, the first optimization iteration starts from an initial number of FEM rows. In the current implementation, the starting number of rows is 8 for two-window devices, 16 for three-window devices and 64 for five-window devices. Additional FEM simulations can be appended in subsequent iterations when the optimized design does not satisfy the validation criteria.
+To improve the robustness of the surrogate models near the admissible limits, the sampling domain is extended slightly beyond the actual optimization ranges (see Table~\ref{tab:families}). This strategy reduces the risk of extrapolation when evaluating candidate solutions close to the true design limits. The ranges employed during the DoE are summarized in Table~\ref{tab:families_doe}.
+
+\begin{table}[ht!]
+\centering
+\caption{Geometry families and window thickness ranges considered during the DoE generation.}
+\label{tab:families_doe}
+\begin{tabular}{llllll}
+\toprule
+Family & Height $H$ & Width $B$ & Frame thickness & Design variables & Thickness bounds (DoE) \\
+\midrule
+2 windows & 30 cm & 29/34 cm & 30 mm & $t_{w,1},t_{w,2}$ & 8--22 mm \\
+3 windows & 45 cm & 29/34 cm & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm \\
+5 windows & 60 cm & 34 cm & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 4--14 mm \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+For every sampled configuration, a FEM simulation is performed, and the resulting dataset stores both the input variables and the corresponding structural response quantities. The target outputs include the maximum local distortions in each window, denoted by $\varepsilon_{xy,i}$, together with the window damage indicators $\TFD_i$ and the frame damage indicator $\TFD_f$.
+
+For each geometry family, the optimization cycle starts from an initial set of FEM simulations selected to provide a reasonable coverage of the design space while keeping the computational cost as low as possible, taking into account the dimensionality of each family. In the current implementation, the initial datasets contain 8 samples for two-window devices, 16 samples for three-window devices and 64 samples for five-window devices.
+
+The number of samples in all cases is defined as a power of two. This choice facilitates the potential use of Progressive Latin Hypercube Sampling (PLHS) \cite{Sheikholeslami2017} in future iterations, allowing the DoE to be expanded while preserving its space-filling properties and avoiding the need to repeat previously computed simulations.
 
 \subsection{Supervised ML surrogate models}\label{subsec:ml_models}
 
-In contrast to previous works focused on a single model or performance metric, this study provides a systematic comparison of surrogate techniques in terms of predictive accuracy and computational cost within the context of geometry optimization. The supervised surrogate set includes Random Forest (RF), Gradient Boosting Regression (GBR), XGBoost, Support Vector Regression (SVR), Multilayer Perceptron (MLP) and Gaussian Process Regression (GPR). Since preliminary calculations indicated that SVR and GPR often dominate the high-accuracy regime, RBF interpolants are also assessed as a computationally efficient alternative for fast optimization. The novelty is not the development of a new constitutive model, but the integration of FEM-calibrated damage indicators into an optimization workflow that explicitly distinguishes between damage in the windows and damage in the frame, encourages balanced window activation and verifies the optimized geometry with a high-fidelity FEM simulation before accepting it.
+This work provides a systematic comparison of surrogate modelling techniques in terms of predictive accuracy, robustness, and computational efficiency within the context of geometry optimization of BDSL dampers. The considered supervised surrogate models include Random Forest (RF), Gradient Boosting Regression (GBR), XGBoost, Support Vector Regression (SVR), Multilayer Perceptron (MLP), and Gaussian Process Regression (GPR).
 
-A set of supervised regression models is trained independently for each output variable. The considered algorithms are RF, GBR, XGBoost, SVR, MLP and GPR. The input vector is composed only of the window thicknesses of the corresponding device family. For each output, the model-selection routine evaluates all candidate algorithms and stores the best model as a serialized \texttt{joblib} file.
+The objective is not the development of a new constitutive formulation, but the integration of FEM-calibrated damage and distortion indicators into a surrogate-assisted optimization framework capable of distinguishing between damage in the dissipative windows and in the surrounding frame. The proposed methodology also promotes balanced window activation and incorporates an adaptive FEM validation loop to verify the optimized geometries before acceptance.
 
-Hyperparameters are optimized using Bayesian search with 40 iterations. The scoring metrics are root mean squared error (RMSE), mean absolute error (MAE) and $R^2$. Since the available dataset size changes between geometry families and between adaptive iterations, the cross-validation strategy is selected automatically. Leave-One-Out validation is used for datasets with $N\leq 20$, repeated five-fold cross-validation with five repetitions is used for $21\leq N\leq 80$ and standard shuffled five-fold cross-validation is used for larger datasets. For small datasets, the hyperparameter search spaces of tree-based models are reduced to mitigate overfitting.
+For each geometry family, a separate supervised regression model is trained for every target output variable. The input vector contains only the window thicknesses associated with the corresponding device configuration. The predicted outputs include the local distortion indicators $\varepsilon_{xy,i}$, the window damage indicators $\TFD_i$, and the frame damage indicator $\TFD_f$. For every output variable, all candidate algorithms are evaluated independently and the selected surrogate model is stored as a serialized \texttt{joblib} file.
 
-The best model is not selected exclusively from the lowest mean RMSE. Models whose RMSE lies within a 5\% band from the best RMSE are considered competitive; when the dispersion of fold-level RMSE is available for all competitive candidates, the model with the lowest coefficient of variation of RMSE is selected. This criterion favours not only accurate but also stable surrogates, which is relevant in optimization because a small local error near an active damage constraint can change the accepted geometry.
+Hyperparameter optimization is performed through Bayesian search using 40 iterations. Model performance is evaluated using the root mean squared error (RMSE), mean absolute error (MAE), and coefficient of determination ($R^2$). Since the available dataset size varies between geometry families and adaptive iterations, the cross-validation strategy is selected automatically. Leave-One-Out validation is used for datasets with $N\leq20$, repeated five-fold cross-validation with five repetitions is adopted for $21\leq N\leq80$, and standard shuffled five-fold cross-validation is employed for larger datasets. For small datasets, the hyperparameter search spaces of tree-based models are reduced to mitigate overfitting.
+
+The surrogate selection process is not based exclusively on the minimum average RMSE. Models whose RMSE falls within a 5\% tolerance band of the best-performing candidate are considered competitive. When fold-level dispersion metrics are available for all competitive candidates, the model with the lowest RMSE coefficient of variation is selected. This criterion favours not only accurate but also stable surrogates, which is particularly important in optimization problems where small local prediction errors near active damage constraints may alter the final accepted geometry.
 
 \begin{figure}[t]
 \centering
-\fbox{\parbox[c][0.24\textheight][c]{0.85\textwidth}{\centering Placeholder for supervised surrogate workflow: FEM dataset, adaptive CV, Bayesian hyperparameter search, model selection and model persistence.}}
-\caption{Supervised surrogate training workflow. A separate model is trained for each damage or distortion output. The selected model can differ from one output to another.}\label{fig:ml_workflow}
+\fbox{\parbox[c][0.24\textheight][c]{0.85\textwidth}{\centering Placeholder for supervised surrogate workflow: FEM dataset generation, adaptive cross-validation, Bayesian hyperparameter optimization, model selection, and surrogate persistence.}}
+\caption{Workflow of the supervised surrogate training strategy. A separate surrogate model is trained for each output variable, and the selected algorithm may differ depending on the predicted quantity.}
+\label{fig:ml_workflow}
 \end{figure}
 
-Preliminary executions of this workflow indicate that SVR and GPR frequently provide the best accuracy for the considered datasets. This observation motivated the additional assessment of RBF interpolation as a simpler and faster surrogate strategy, especially for low-dimensional or well-sampled design spaces.
+Preliminary executions of the proposed workflow indicate that SVR and GPR consistently provide the highest predictive accuracy for the considered datasets. Since both approaches rely on radial kernel functions, this observation motivated the additional assessment of Radial Basis Function (RBF) interpolation as a simpler and computationally efficient surrogate strategy, particularly suitable for low-dimensional and moderately sampled design spaces.
 
 \subsection{RBF surrogate models}\label{subsec:rbf_models}
 
-The RBF surrogate is implemented as a wrapper around the \texttt{scipy.interpolate.Rbf} interpolator. For each output variable, the model is trained using the same input features as the supervised ML surrogates. The current implementation uses a multiquadric radial basis function with zero smoothing and automatic shape parameter selection. The prediction model can be written generically as
+The RBF surrogate is implemented using the \texttt{scipy.interpolate.Rbf} interpolator. For each output variable, the model is trained using the same input features employed by the supervised ML surrogates. In the current implementation, a multiquadric radial basis function is adopted with zero smoothing and automatic shape-parameter estimation.
+
+The prediction model can be expressed as
 \begin{equation}
 \hat{y}(\mathbf{x}) = \sum_{j=1}^{N} \lambda_j \, \phi\left(\|\mathbf{x}-\mathbf{x}_j\|\right),
 \label{eq:rbf}
 \end{equation}
-where $\mathbf{x}_j$ are the FEM-sampled geometries, $\lambda_j$ are interpolation weights and $\phi$ is the selected radial basis function.
+where $\mathbf{x}_j$ denotes the FEM-sampled geometries, $\lambda_j$ represents the interpolation weights, and $\phi$ corresponds to the selected radial basis function.
+
+RBF models are evaluated through Leave-One-Out validation. For each output variable, one FEM sample is iteratively removed from the dataset, the interpolant is trained using the remaining samples, and the excluded sample is subsequently predicted. The resulting predictions are used to compute RMSE, MAE, $R^2$, and dispersion-related metrics.
 
-RBF models are evaluated through Leave-One-Out validation. For each output, the procedure repeatedly removes one FEM sample, trains the RBF interpolant with the remaining samples, predicts the left-out sample and computes RMSE, MAE, $R^2$ and dispersion metrics. This validation is especially useful because the number of FEM simulations is limited and because interpolation accuracy may be sensitive to the local density of training samples.
+This validation strategy is particularly suitable for the present application because the number of FEM simulations remains limited and the interpolation accuracy may be sensitive to local variations in the density of training samples. In addition, the low computational cost of RBF interpolation makes it especially attractive for iterative optimization workflows involving repeated surrogate evaluations.
 
 \section{Damage-aware surrogate-assisted optimization}\label{sec:optimization}
 

Manuscript/wileyNJD-AMA.bib

@@ -557,4 +557,35 @@ steel from coupon test results available. First, the theory of metal plasticity
   urldate   = {2026-05-05},
 }
 
+@Article{Joseph2008,
+  author    = {Joseph, V. Roshan and Hung, Ying},
+  journal   = {Statistica Sinica},
+  title     = {Orthogonal-{Maximin} {Latin} {Hypercube} {Designs}},
+  year      = {2008},
+  issn      = {1017-0405},
+  number    = {1},
+  pages     = {171--186},
+  volume    = {18},
+  abstract  = {A randomly generated Latin hypercube design (LHD) can be quite structured: the variables may be highly correlated or the design may not have good space-filling properties. There are procedures for finding good LHDs by minimizing the pairwise correlations or by maximizing the inter-site distances. In this article we show that these two criteria need not be in close agreement. We propose a multi-objective optimization approach to find good LHDs by combining correlation and distance performance measures. We also propose a new exchange algorithm for efficiently generating such designs. Several examples are presented to show that the new algorithm is fast, and that the optimal designs are good in terms of both the correlation and distance criteria.},
+  publisher = {Institute of Statistical Science, Academia Sinica},
+  urldate   = {2026-05-06},
+}
+
+@Article{Sheikholeslami2017,
+  author     = {Sheikholeslami, Razi and Razavi, Saman},
+  journal    = {Environmental Modelling \& Software},
+  title      = {Progressive {Latin} {Hypercube} {Sampling}: {An} efficient approach for robust sampling-based analysis of environmental models},
+  year       = {2017},
+  issn       = {1364-8152},
+  month      = jul,
+  pages      = {109--126},
+  volume     = {93},
+  abstract   = {Efficient sampling strategies that scale with the size of the problem, computational budget, and users’ needs are essential for various sampling-based analyses, such as sensitivity and uncertainty analysis. In this study, we propose a new strategy, called Progressive Latin Hypercube Sampling (PLHS), which sequentially generates sample points while progressively preserving the distributional properties of interest (Latin hypercube properties, space-filling, etc.), as the sample size grows. Unlike Latin hypercube sampling, PLHS generates a series of smaller sub-sets (slices) such that (1) the first slice is Latin hypercube, (2) the progressive union of slices remains Latin hypercube and achieves maximum stratification in any one-dimensional projection, and as such (3) the entire sample set is Latin hypercube. The performance of PLHS is compared with benchmark sampling strategies across multiple case studies for Monte Carlo simulation, sensitivity and uncertainty analysis. Our results indicate that PLHS leads to improved efficiency, convergence, and robustness of sampling-based analyses.},
+  doi        = {10.1016/j.envsoft.2017.03.010},
+  keywords   = {Design of computer experiments, Monte Carlo simulation, Optimal Latin hypercube sampling, Sensitivity analysis, Sequential sampling, Uncertainty analysis},
+  shorttitle = {Progressive {Latin} {Hypercube} {Sampling}},
+  url        = {https://www.sciencedirect.com/science/article/pii/S1364815216305096},
+  urldate    = {2026-05-06},
+}
+
 @Comment{jabref-meta: databaseType:bibtex;}