Update manuscript to enhance clarity in RBF surrogate validation and streamline…

Manuscript/RESILINK_surrogate_optimization_manuscript.tex

@@ -242,9 +242,9 @@ The prediction model can be expressed as
 \end{equation}
 where $\mathbf{x}_j$ denotes the FEM-sampled geometries, $\lambda_j$ represents the interpolation weights, and $\phi$ corresponds to the selected radial basis function.
 
-For each output variable, a final RBF surrogate is trained using all available FEM samples in the current iteration and stored for subsequent use in the optimization process. Its predictive performance is assessed through Leave-One-Out validation: each FEM sample is iteratively removed from the dataset, a temporary RBF interpolant is trained with the remaining samples, and the excluded sample is predicted. The resulting out-of-sample predictions are then used to compute RMSE, MAE, $R^2$, and error-dispersion metrics.
+For each output variable, a final RBF surrogate is trained using all available FEM samples in the current iteration and stored for subsequent use in the optimization process. Its predictive performance is assessed through Leave-One-Out validation: each FEM sample is iteratively removed from the dataset, a temporary RBF interpolant is trained with the remaining samples, and the excluded sample is predicted. The resulting out-of-sample predictions are then used to compute RMSE, MAE and $R^2$ in order to assess the interpolation accuracy.
 
-This validation strategy is particularly suitable for the present application because the number of FEM simulations is limited and the interpolation accuracy may be sensitive to local variations in the density of training samples. In addition, the low training and evaluation cost of RBF interpolation makes it especially attractive for iterative optimization workflows involving repeated surrogate updates and evaluations.
+This validation strategy is suitable for the present application because the number of FEM simulations is limited and the interpolation accuracy may be sensitive to local variations in the density of training samples. In addition, the low training and evaluation cost of RBF interpolation makes it especially attractive for iterative optimization workflows involving repeated surrogate updates and evaluations.
 
 \section{Damage-aware surrogate-assisted optimization}\label{sec:optimization}
 
@@ -302,22 +302,7 @@ H30\_B34 & 0.0410, 0.0265, 0.0240, 0.0098, 0.0098 \\
 \end{tabular}
 \end{table}
 
-\section{Adaptive FEM validation and retraining}\label{sec:adaptive}
-
-The adaptive validation stage is a central component of the proposed framework. Once an optimal geometry is identified by the surrogate-assisted optimizer, it is re-evaluated with FEM to verify that the surrogate remains accurate in the region of the design space where the optimum lies. The candidate geometry is accepted only if: (i) the prediction error of all damage and distortion variables remains below the prescribed tolerance; (ii) the absolute error of the objective function remains within the admissible limit; and (iii) the optimized window thicknesses remain stable between consecutive optimization iterations, with variations smaller than a prescribed percentage of the full design range. If any of these criteria is not satisfied, the new FEM result is incorporated into the training dataset and the surrogate models are retrained.
-
-As a result, the main contribution of this work lies in the development of a robust, scalable and physically informed design methodology that explicitly accounts for the trade-off between energy dissipation and damage. To summarize, the main contributions of this work are:
-\begin{itemize}
-    \item generation of high-fidelity FEM datasets for BDSL dampers with increasing geometric complexity and different numbers of dissipative windows;
-    \item geometric optimization using surrogate models, including supervised ML techniques and RBF interpolants;
-    \item systematic comparison of surrogate strategies in terms of predictive accuracy, computational cost and practical suitability for optimization;
-    \item a damage-aware objective function that combines window damage control, severe frame-damage penalization, window-to-window damage balancing and dissipated-energy maximization;
-    \item an adaptive FEM validation and retraining strategy based on explicit tolerances for surrogate error, objective-function error and stability of the optimized geometry between successive iterations.
-\end{itemize}
-
-The surrogate-optimized geometry is not accepted directly. Instead, the optimal candidate proposed by the surrogate-assisted DE process is evaluated with the high-fidelity FEM model. This validation step checks whether the surrogate has remained reliable in the region of the design space selected by the optimizer.
-
-Three acceptance criteria are used. First, the prediction error of all variables entering the optimization process, including damage and distortion indicators, must be lower than 5\%. Second, the absolute error in the objective function must be lower than 10. Third, the optimized window thicknesses must be stable between consecutive optimization iterations: no thickness is allowed to change by more than 2\% of the total admissible range. For example, if a thickness is optimized within the interval 10--20 mm, the total range is 10 mm and the maximum admissible variation between iterations is 0.2 mm.
+The surrogate-optimized geometry is not accepted directly. Instead, once an optimal geometry is identified by the surrogate-assisted optimizer, it is re-evaluated with FEM to verify that the surrogate remains accurate in the region of the design space where the optimum lies. This validation step checks whether the surrogate has remained reliable in the region of the design space selected by the optimizer. The candidate geometry is accepted only if: (i) the prediction error of all damage and distortion variables remains below the prescribed tolerance, equal to 5\%; (ii) the absolute error of the objective function remains within the admissible limit, set to 10; and (iii) the optimized window thicknesses remain stable between consecutive optimization iterations, with variations smaller than the 5\% of the full design range. If any of these criteria is not satisfied, the new FEM result is incorporated into the training dataset and the surrogate models are retrained.
 
 If all criteria are satisfied, the FEM-validated geometry is accepted as the optimized design. If at least one criterion is not satisfied, the new FEM result is added to the dataset, the surrogate models are retrained and the DE optimization is repeated. This loop is summarized in Figure~\ref{fig:adaptive_loop}. The process reduces the risk of accepting a geometry that is optimal only because of surrogate extrapolation error.
 
@@ -411,6 +396,17 @@ Some limitations should be highlighted. First, the methodology is only as reliab
 
 \section{Conclusions}\label{sec:conclusions}
 
+
+As a result, the main contribution of this work lies in the development of a robust, scalable and physically informed design methodology that explicitly accounts for the trade-off between energy dissipation and damage. To summarize, the main contributions of this work are:
+\begin{itemize}
+    \item generation of high-fidelity FEM datasets for BDSL dampers with increasing geometric complexity and different numbers of dissipative windows;
+    \item geometric optimization using surrogate models, including supervised ML techniques and RBF interpolants;
+    \item systematic comparison of surrogate strategies in terms of predictive accuracy, computational cost and practical suitability for optimization;
+    \item a damage-aware objective function that combines window damage control, severe frame-damage penalization, window-to-window damage balancing and dissipated-energy maximization;
+    \item an adaptive FEM validation and retraining strategy based on explicit tolerances for surrogate error, objective-function error and stability of the optimized geometry between successive iterations.
+\end{itemize}
+
+
 This work presents an adaptive surrogate-assisted optimization framework for BDSL dampers under cyclic seismic loading. The methodology combines FEM-calibrated numerical simulations, supervised ML models, RBF interpolation, DE and FEM-based validation of the optimized geometries. The following conclusions can be drawn from the proposed formulation:
 
 \begin{enumerate}