Improve manuscript clarity and formatting in the section on buckling-delayed…

Manuscript/ComparisonSurrogatesOptimizationBDSL.tex

@@ -136,7 +136,7 @@ Figure \ref{fig:MethodologyFlowChart} summarizes the proposed workflow. The diff
 
 \section{Buckling-delayed shear-link damper}\label{sec:device}
 
-The BDSL dampers analysed in this work, with one representative configuration shown in Figure \ref{fig:Device}, are designed to concentrate energy dissipation in localized reduced-thickness zones, hereafter referred to as windows, while preserving the structural integrity of the surrounding frame. The optimization variables correspond to the window thicknesses, whereas the frame dimensions are kept fixed. The dissipative element is connected to a load-transfer system through a mechanism that allows imposed in-plane displacement while preventing axial force transmission, thereby promoting a shear-dominated response. Under cyclic loading, plastic deformation is intended to concentrate in the windows, whereas the frame provides load transfer and stability.
+The BDSL dampers analysed in this work, with one representative configuration shown in Figure~\ref{fig:Device}, are designed to concentrate energy dissipation in localized reduced-thickness zones, hereafter referred to as windows, while preserving the structural integrity of the surrounding frame. The dissipative element is connected to a load-transfer system through a mechanism that allows imposed in-plane displacement while preventing axial force transmission, thereby promoting a shear-dominated response. Under cyclic loading, plastic deformation is intended to concentrate in the windows, whereas the frame provides load transfer and stability.
 
 \begin{figure}[htbp]
   \centering
@@ -147,7 +147,7 @@ The BDSL dampers analysed in this work, with one representative configuration sh
 
 This separation of functions leads to a non-trivial design problem. Thin windows may enhance ductility and dissipative activation, but they may also promote excessive damage localization. Conversely, thicker windows may increase strength while transferring inelastic demand to the frame. Since severe frame damage may compromise the structural integrity of the device, frame damage must be penalized more strongly than window damage. At the same time, the dissipative demand should be distributed as uniformly as possible among the windows, avoiding configurations in which a single window absorbs most of the deformation while the remaining windows stay underused. Consequently, the design problem cannot be reduced to maximizing force or total dissipated energy alone, but must also control where damage develops and how the windows participate in the dissipative process.
 
-The design variables considered in this work are the window thicknesses,
+Accordingly, the design variables considered in this work are the window thicknesses, while the frame dimensions are kept fixed:
 \begin{equation}
 \mathbf{x}=\left[t_{w,1},t_{w,2},\ldots,t_{w,N_w}\right],
 \label{eq:design_vector}
@@ -182,7 +182,7 @@ $F_5$ & 2.00 & 1.17 & 5 & $t_{w,1},\ldots,t_{w,5}$ & 5--12 mm \\
 
 \section{Validation of the FEM numerical model}\label{sec:fem}
 
-The surrogate models developed in this work were trained using data generated from three-dimensional FEM simulations. The numerical model is based on a previously calibrated and validated representation of the BDSL device, described in detail in Ramirez et al. \cite{RamirezMachado2025}. An example of the numerical setup is shown in Figure \ref{fig:FEMSetup}. The simulations were carried out using the COMPACK code, an explicit dynamic FEM solver for linear and nonlinear problems \cite{Martinez2011}. The model accounts for large displacements, material and geometric nonlinearities, contact interactions and the boundary conditions associated with the experimental configuration.
+The surrogate models developed in this work were trained using data generated from three-dimensional FEM simulations. The numerical model is based on a previously calibrated and validated representation of the BDSL device, described in detail in Ramírez et al. \cite{RamirezMachado2025}. An example of the numerical setup is shown in Figure \ref{fig:FEMSetup}. The simulations were carried out using the COMPACK code, an explicit dynamic FEM solver for linear and nonlinear problems \cite{Martinez2011}. The model accounts for large displacements, material and geometric nonlinearities, contact interactions and the boundary conditions associated with the experimental configuration.
 
 The dissipative steel component is modelled as ASTM A36 steel, whose cyclic plastic behaviour is represented by the Yoshida--Uemori model \cite{Yoshida2002,Jia2014}. This constitutive law allows the model to reproduce cyclic hardening, softening and Bauschinger-type effects under large plastic deformation. The steel component is discretized using linear eight-node hexahedral solid elements, providing a structured three-dimensional mesh suitable for extracting local stress, strain and damage-related fields.
 
@@ -195,7 +195,7 @@ The imposed displacement is applied through an actuator-like connector that tran
   \label{fig:FEMSetup}
 \end{figure}
 
-The model was calibrated and validated against cyclic experimental tests performed on representative BDSL specimens. The calibration involved the material parameters, assembled geometry, contact definitions, support flexibility and boundary conditions. The validated model accurately reproduces the main global experimental responses. Figure \ref{fig:FEM_validation_comparison} shows the comparison between experimental and numerical results, confirming the suitability of the FEM model as a numerical reference for configurations beyond those experimentally tested.
+The calibration involved the material parameters, assembled geometry, contact definitions, support flexibility and boundary conditions. The validated model accurately reproduces the main global experimental responses. Figure \ref{fig:FEM_validation_comparison} shows the comparison between experimental and numerical results, confirming the suitability of the FEM model as a numerical reference for configurations beyond those experimentally tested.
 
 \begin{figure*}[htbp]
   \centering
@@ -206,7 +206,7 @@ The model was calibrated and validated against cyclic experimental tests perform
 
 \section{Dataset generation and surrogate modelling}\label{sec:surrogates}
 
-Once validated, the FEM model is used to generate the datasets required for surrogate training and optimization. The optimization strategy relies on local damage indicators in the dissipative windows and surrounding frame, together with local distortion measures associated with the activation of the dissipative mechanism. Since these quantities are difficult to measure experimentally, the use of a high-fidelity FEM model provides valuable access to the internal state variables and local fields governing damage evolution and energy dissipation.
+Once validated, the FEM model is used to generate the datasets required for surrogate training and optimization. The optimization strategy relies on local damage indicators in the dissipative windows and surrounding frame, together with local distortion measures associated with the activation of the dissipative mechanism.
 
 The damage indicator adopted in this work is the Triaxial Failure Damage Map (TFDMap) \cite{Rastellini2016}. This stress-triaxiality-based indicator evaluates the proximity of each material point to ductile failure by comparing, for a given triaxiality, the accumulated equivalent plastic strain with a reference failure envelope \cite{Rice1969,Bao2004,Wierzbicki2005,Bai2008}. In this study, the TFDMap is used as a post-processing damage-screening indicator, not as a constitutive fracture criterion. Its purpose is therefore not to explicitly predict crack initiation, but to compare geometrical configurations and ensure that optimized designs remain within acceptable damage levels.
 
@@ -267,7 +267,7 @@ Model selection is performed in two stages. First, for each candidate algorithm,
   \label{fig:BayesianSearchCV}
 \end{figure*}
 
-Preliminary executions of the proposed workflow show that SVR and GPR always provide the highest, or second-highest, predictive accuracy for the considered datasets. These results suggest that kernel-based models, and in particular distance-based similarity measures, are well suited to approximate the FEM response surfaces involved in this problem. This observation motivates the assessment of Radial Basis Function (RBF) interpolation \cite{Gutmann2001} as a simpler and computationally efficient surrogate alternative. Although surrogate evaluation is negligible compared with FEM simulations, the training and hyperparameter optimization of complex supervised models can still become relevant when several outputs, geometry families and adaptive iterations are considered. RBF interpolation provides a non-parametric and fast-to-train alternative that can capture nonlinear response surfaces, making it attractive for low-dimensional and moderately sampled design spaces.
+Preliminary executions of the proposed workflow show that SVR or GPR always provide the highest, or second-highest, predictive accuracy for the considered datasets. These results suggest that kernel-based models, and in particular distance-based similarity measures, are well suited to approximate the FEM response surfaces involved in this problem. This observation motivates the assessment of Radial Basis Function (RBF) interpolation \cite{Gutmann2001} as a simpler and computationally efficient surrogate alternative. Although surrogate evaluation is negligible compared with FEM simulations, the training and hyperparameter optimization of complex supervised models can still become relevant when several outputs, geometry families and adaptive iterations are considered. RBF interpolation provides a non-parametric and fast-to-train alternative that can capture nonlinear response surfaces, making it attractive for low-dimensional and moderately sampled design spaces.
 
 \subsection{RBF surrogate models}\label{subsec:rbf_models}
 
@@ -280,7 +280,7 @@ 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 and $R^2$ in order to assess the interpolation accuracy.
+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 accuracy metrics in order to assess the interpolation precision.
 
 \section{Damage-aware surrogate-assisted optimization}\label{sec:optimization}
 
@@ -295,7 +295,7 @@ J(\mathbf{x}) = - \sum_{i=1}^{N_w} \hat{\varepsilon}_{xy,i}^2\, t_{w,i}\, A_i +
 P_f\left(\hat{\mathcal{D}}_f;\mathcal{D}_f^{\max}\right),
 \label{eq:objective}
 \end{equation}
-where $N_w$ is the number of windows, $t_{w,i}$ is the thickness of window $i$, $A_i$ is the corresponding area factor, $\mathcal{D}_w^{\star}$ is the target damage level for the windows and $\mathcal{D}_f^{\max}$ is the maximum admissible frame damage threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger energy dissipation contributions reduce the objective value and are favoured once the damage criteria are satisfied.
+where $N_w$ is the number of windows, $t_{w,i}$ is the thickness of window $i$, $A_i$ is the corresponding area factor, $\mathcal{D}_w^{\star}$ is the target damage level for the windows and $\mathcal{D}_f^{\max}$ is the maximum admissible frame damage threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger energy dissipation contributions reduce the objective value.
 
 The window penalty is defined as
 \begin{equation}
@@ -433,6 +433,8 @@ A comprehensive comparison of surrogate strategies was performed in terms of pre
 
 The proposed adaptive validation loop proved to be necessary and effective. Several initially optimized candidates did not satisfy the prescribed error tolerances. After incorporating the new FEM results into the training dataset and retraining the surrogates, the prediction errors decreased and all final optimized geometries satisfied the acceptance criteria after only two or three iterations. Therefore, the final designs are not accepted solely on the basis of surrogate predictions, but are explicitly verified through FEM in the region of the design space where the optimum is located.
 
+It is also worth noting that, although the framework allows the DoE to be expanded with additional FEM simulations when the surrogate accuracy is insufficient, this was not required in the present application. The initial DoE datasets were already adequate to obtain accurate optimized designs after adaptive validation and retraining. In particular, the final geometries were obtained from only 8 initial FEM simulations for the two-window devices, 16 for the three-window devices, and 64 for the five-window devices, with a maximum of three adaptive retraining iterations in all cases. This demonstrates that the proposed strategy can achieve FEM-consistent optimized designs with a limited number of high-fidelity simulations, making it highly competitive from a computational point of view.
+
 The proposed methodology also has some limitations that should be acknowledged. First, its reliability depends on the quality of the calibrated FEM model used to generate the training data and validate the optimized designs. Second, the TFDMap is used here as a post-processing damage indicator rather than as a constitutive fracture model; therefore, the optimized configurations should be interpreted in terms of relative damage control and proximity to critical states, not as direct predictions of crack initiation. Third, only the window thicknesses are considered as design variables. Although this leads to a controlled and interpretable optimization problem, it does not exploit the full geometric flexibility of BDSL dampers. Finally, the optimized geometries should ultimately be validated experimentally before being used to establish general design recommendations.
 
 Future work should extend the design space by including additional geometric and mechanical variables, such as window height, window spacing, frame thickness or global device proportions. This extension would increase the dimensionality and complexity of the surrogate task. In those cases, the performance of RBF interpolation should therefore be reassessed. While RBF models performed very well in the present study, their efficiency and accuracy may decrease as the input space becomes larger or the response surfaces develop stronger local nonlinearities. In such cases, supervised ML models or hybrid surrogate strategies may become more advantageous.
@@ -484,30 +486,46 @@ Model & Preprocessing / kernel & Hyperparameter & Search space \\
 
 \multirow{5}{*}{RF}
 & \multirow{5}{*}{--}
-& $n_{\mathrm{estimators}}$ & $[200,2000]$ \\
-& & max depth & $[5,40]$ \\
-& & min samples split & $[2,10]$ \\
-& & min samples leaf & $[1,6]$ \\
-& & max features & $[0.5,1.0]$ \\
+& $n_{\mathrm{estimators}}$
+& S/M/L: $[200,800]/[300,1200]/[200,2000]$ \\
+& & max depth
+& S/M/L: $[1,4]/[2,10]/[5,40]$ \\
+& & min samples split
+& S/M/L: $[2,8]/[2,10]/[2,10]$ \\
+& & min samples leaf
+& S/M/L: $[2,4]/[1,6]/[1,6]$ \\
+& & max features
+& S/M/L: $[0.5,1.0]/[0.5,1.0]/[0.5,1.0]$ \\
 
 \midrule
 \multirow{5}{*}{GBR}
 & \multirow{5}{*}{--}
-& $n_{\mathrm{estimators}}$ & $[200,3000]$ \\
-& & learning rate & $[10^{-3},10^{-1}]$ \\
-& & max depth & $[2,6]$ \\
-& & subsample & $[0.6,1.0]$ \\
-& & max features & $[0.5,1.0]$ \\
+& $n_{\mathrm{estimators}}$
+& S/M/L: $[200,1500]/[300,2500]/[200,3000]$ \\
+& & learning rate
+& S/M/L: $[10^{-2},8{\times}10^{-2}]/[10^{-3},10^{-1}]/[10^{-3},10^{-1}]$ \\
+& & max depth
+& S/M/L: $[1,3]/[1,4]/[2,6]$ \\
+& & subsample
+& S/M/L: $[0.7,0.9]/[0.6,1.0]/[0.6,1.0]$ \\
+& & max features
+& S/M/L: $[0.6,1.0]/[0.5,1.0]/[0.5,1.0]$ \\
 
 \midrule
 \multirow{6}{*}{XGBoost}
 & \multirow{6}{*}{--}
-& $n_{\mathrm{estimators}}$ & $[200,2000]$ \\
-& & max depth & $[3,8]$ \\
-& & learning rate & $[10^{-3},0.3]$ \\
-& & subsample & $[0.6,1.0]$ \\
-& & colsample by tree & $[0.6,1.0]$ \\
-& & min child weight & $[1,10]$ \\
+& $n_{\mathrm{estimators}}$
+& S/M/L: $[200,1200]/[300,1600]/[200,2000]$ \\
+& & max depth
+& S/M/L: $[1,4]/[2,6]/[3,8]$ \\
+& & learning rate
+& S/M/L: $[10^{-2},0.2]/[10^{-3},0.3]/[10^{-3},0.3]$ \\
+& & subsample
+& S/M/L: $[0.7,0.95]/[0.6,1.0]/[0.6,1.0]$ \\
+& & colsample by tree
+& S/M/L: $[0.7,1.0]/[0.6,1.0]/[0.6,1.0]$ \\
+& & min child weight
+& S/M/L: $[3,20]/[1,15]/[1,10]$ \\
 
 \midrule
 \multirow{3}{*}{SVR}
@@ -538,6 +556,9 @@ Model & Preprocessing / kernel & Hyperparameter & Search space \\
 \bottomrule
 \end{tabular}
 
+\vspace{2mm}
+\parbox{0.95\textwidth}{\footnotesize \textit{Note:} For tree-based models, the hyperparameter ranges are adapted to the dataset size: S denotes small sample $N\leq20$, M denotes medium sample $21\leq N\leq80$, and L denotes large sample $N>80$. For the remaining models, the same search spaces are used for all dataset sizes.}
+
 \end{table*}
 
 \bmsection{Summary of optimization results}