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2026_Article_RESILINK_ML
Commit 99f20325 by Joaquín Irazábal González
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Refactor manuscript to update geometry families and optimize design variables

- Replaced specific geometry family names (H30_B29, H30_B34, etc.) with generic identifiers (F_1, F_2, etc.) throughout the manuscript for confidentiality. - Adjusted descriptions and captions in figures and tables to reflect the new identifiers. - Updated design variable definitions and thickness ranges in the manuscript to align with the new naming convention. - Enhanced clarity in the discussion of optimization results and surrogate model comparisons. - Ensured consistency in the presentation of data across all sections of the manuscript.
parent 34b4262c
Showing with 1067 additions and 317 deletions
Manuscript/Figures/DeviceGeom.svg 0 → 100644
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Manuscript/Optimal-Design-layout/RESILINK_surrogate_optimization_OptimalDesignLayout.tex
......@@ -147,39 +147,35 @@ 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
The design variables considered in this work are the window thicknesses,
\begin{equation}
\mathbf{x}=\left[t_{w,1},t_{w,2},\ldots,t_{w,W}\right],
\label{eq:design_vector}
\end{equation}
where $W$ denotes the number of windows. The width and height identifiers of the device are represented by $B$ and $H$, respectively. Five geometry families are considered, as shown in Figure \ref{fig:GeometryFamilies}: H30\_B29, H30\_B34, H45\_B29, H45\_B34 and H60\_B34. Devices with $H=30$ cm have two windows, those with $H=45$ cm have three windows and those with $H=60$ cm have five windows.
where $W$ denotes the number of windows. The device width and height are identified by $B$ and $H$, respectively. Five geometry families are analysed, defined by different combinations of height, width and number of windows. Due to confidentiality constraints, the main dimensions are reported in normalized form with respect to reference values, $H_{\text{ref}}$ and $B_{\text{ref}}$. Figure~\ref{fig:DeviceGeom} shows the geometric scheme of one representative BDSL configuration. The example corresponds to a two-window device; throughout the paper, windows are numbered from top to bottom.
\begin{figure*}[htbp]
\begin{figure}[htbp]
\centering
\includegraphics[width=0.15\textwidth]{../Figures/H30_B29.png}\label{fig:H30_B29}
\includegraphics[width=0.15\textwidth]{../Figures/H30_B34.png}\label{fig:H30_B34}
\includegraphics[width=0.15\textwidth]{../Figures/H45_B29.png}\label{fig:H45_B29}
\includegraphics[width=0.15\textwidth]{../Figures/H45_B34.png}\label{fig:H45_B34}
\includegraphics[width=0.15\textwidth]{../Figures/H60_B34.png}\label{fig:H60_B34}
\caption{BDSL families considered for optimization in the current study. From left to right: H30\_B29, H30\_B34, H45\_B29, H45\_B34 and H60\_B34.}
\label{fig:GeometryFamilies}
\end{figure*}
\includegraphics[width=0.40\textwidth]{../Figures/DeviceGeom.pdf}
\caption{Geometric scheme of a representative BDSL device considered in this study.}
\label{fig:DeviceGeom}
\end{figure}
The admissible thickness ranges are defined according to the geometry family and manufacturing constraints. The main characteristics of each family are summarized in Table~\ref{tab:families}. In all cases, the frame thickness is kept constant at 30 mm.
The admissible thickness ranges are defined for each geometry family according to manufacturing constraints. The main characteristics of the analysed families are summarized in Table~\ref{tab:families}. In all cases, the frame thickness is kept constant.
\begin{table*}[htbp]
\centering
\caption{Geometry families considered for optimization in the current study and admissible window thickness ranges.}
\caption{Geometry families considered in the optimization and admissible window thickness ranges.}
\label{tab:families}
\begin{tabular}{lllllll}
\begin{tabular}{llllll}
\toprule
Family & Height $H$ & Width $B$ & Windows & Frame thickness & Design variables & Thickness bounds \\
Family & $H/H_{\text{ref}}$ & $B/B_{\text{ref}}$ & $W$ & Design variables & Thickness bounds \\
\midrule
H30\_B29 & 30 cm & 29 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 10--20 mm \\
H30\_B34 & 30 cm & 34 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 10--20 mm \\
H45\_B29 & 45 cm & 29 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
H45\_B34 & 45 cm & 34 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
H60\_B34 & 60 cm & 34 cm & 5 & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 5--12 mm \\
$F_1$ & 1.00 & 1.00 & 2 & $t_{w,1},t_{w,2}$ & 10--20 mm \\
$F_2$ & 1.00 & 1.17 & 2 & $t_{w,1},t_{w,2}$ & 10--20 mm \\
$F_3$ & 1.50 & 1.00 & 3 & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
$F_4$ & 1.50 & 1.17 & 3 & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
$F_5$ & 2.00 & 1.17 & 5 & $t_{w,1},\ldots,t_{w,5}$ & 5--12 mm \\
\bottomrule
\end{tabular}
\end{table*}
......@@ -226,15 +222,15 @@ To improve surrogate robustness near the admissible limits, the sampling domain
\centering
\caption{Geometry families, window thickness ranges and number of simulations considered during the DoE generation.}
\label{tab:families_doe}
\begin{tabular}{llllllll}
\begin{tabular}{lllllll}
\toprule
Family & Height $H$ & Width $B$ & Windows & Frame thickness & Design variables & Thickness bounds (DoE) & Simulations (DoE) \\
Family & $H/H_{\text{ref}}$ & $B/B_{\text{ref}}$ & $W$ & Design variables & Thickness bounds & Simulations (DoE) \\
\midrule
H30\_B29 & 30 cm & 29 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 8--22 mm & 8 \\
H30\_B34 & 30 cm & 34 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 8--22 mm & 8 \\
H45\_B29 & 45 cm & 29 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm & 16 \\
H45\_B34 & 45 cm & 34 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm & 16 \\
H60\_B34 & 60 cm & 34 cm & 5 & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 4--14 mm & 64 \\
$F_1$ & 1.00 & 1.00 & 2 & $t_{w,1},t_{w,2}$ & 8--22 mm & 8 \\
$F_2$ & 1.00 & 1.17 & 2 & $t_{w,1},t_{w,2}$ & 8--22 mm & 8 \\
$F_3$ & 1.50 & 1.00 & 3 & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm & 16 \\
$F_4$ & 1.50 & 1.17 & 3 & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm & 16 \\
$F_5$ & 2.00 & 1.17 & 5 & $t_{w,1},\ldots,t_{w,5}$ & 4--14 mm & 64 \\
\bottomrule
\end{tabular}
\end{table*}
......@@ -290,24 +286,7 @@ For each output variable, a final RBF surrogate is trained using all available F
The proposed methodology seeks to minimize local damage in both the dissipative windows and the surrounding frame, while promoting a balanced contribution of all windows to the energy dissipation process. The geometric optimization is carried out using DE \cite{Storn1997}, a population-based global optimizer that does not require gradient information and is therefore suitable for nonlinear and non-convex surrogate response surfaces. In the current implementation, DE is run with a maximum of 500 iterations, a population size factor of 25 and a convergence tolerance of $10^{-6}$. Once an optimal candidate is obtained, an adaptive FEM validation loop is applied to verify the predicted geometry before acceptance.
For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window distortions $\hat{\varepsilon}_{xy,i}$, the window damage indicators $\hat{\mathcal{D}}_i$, and the frame damage indicator $\hat{\mathcal{D}}_f$. Damage is therefore controlled in all regions of the device, but with different mechanical relevance: frame damage is penalized more severely because it may compromise the structural integrity of the damper, whereas the window penalties are formulated to promote comparable damage levels among windows and avoid concentrating the dissipative demand in a single region. The dissipative contribution is estimated from $\hat{\varepsilon}_{xy,i}^2$, the window thickness, and the corresponding area factor (Table~\ref{tab:volume_factors}), since the energy dissipated by each window depends not only on the distortion level but also on the amount of material involved. This term is several orders of magnitude smaller than the damage penalties and is intentionally left unscaled. As a result, damage control remains the dominant criterion, while the dissipative term acts as a tie-breaker among geometries with similar damage performance, favouring those with higher distortion and, consequently, greater energy dissipation capacity.
\begin{table}[htbp]
\centering
\caption{Window area factors used to weight the distortion contribution in the objective function.}
\label{tab:volume_factors}
\begin{tabular}{ll}
\toprule
Family & Area factors $A_i$ \\
\midrule
H30\_B29 & 0.0208, 0.0185 \\
H30\_B34 & 0.0263, 0.0240 \\
H30\_B29 & 0.0229, 0.0210, 0.0185 \\
H30\_B34 & 0.0262, 0.0262, 0.0240 \\
H30\_B34 & 0.0410, 0.0265, 0.0240, 0.0098, 0.0098 \\
\bottomrule
\end{tabular}
\end{table}
For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window distortions $\hat{\varepsilon}_{xy,i}$, the window damage indicators $\hat{\mathcal{D}}_i$, and the frame damage indicator $\hat{\mathcal{D}}_f$. Damage is therefore controlled in all regions of the device, but with different mechanical relevance: frame damage is penalized more severely because it may compromise the structural integrity of the damper, whereas the window penalties are formulated to promote comparable damage levels among windows and avoid concentrating the dissipative demand in a single region. The dissipative contribution is estimated from $\hat{\varepsilon}_{xy,i}^2$, the window thickness and the corresponding area of the window, since the energy dissipated by each window depends not only on the distortion level but also on the amount of material involved. This term is several orders of magnitude smaller than the damage penalties and is intentionally left unscaled. As a result, damage control remains the dominant criterion, while the dissipative term acts as a tie-breaker among geometries with similar damage performance, favouring those with higher distortion and, consequently, greater energy dissipation capacity.
The implemented objective function to be minimized is
\begin{equation}
......@@ -351,7 +330,7 @@ The surrogate-optimized geometry is not accepted directly. Instead, once an opti
\section{Numerical results and discussion}\label{sec:results}
The supervised-learning comparison shows a clear hierarchy among the candidate surrogate models. A total of 100 output-specific training problems were considered, corresponding to the $2W+1$ target variables required for each geometry family and adaptive iteration. The H30\_B29 family required two optimization iterations, whereas the remaining families required three. Across all geometry families, iterations and output variables, SVR was the most frequently selected model and also the model that most often achieved the lowest cross-validated RMSE. As summarized in Figure~\ref{fig:surrogate_selection_summary_barplot}, SVR was selected in 71 cases, followed by GPR, GBR, XGBoost and MLP, while Random Forest was not selected in any case. This dominance was particularly clear for the two-window families, where only one output was assigned to a GPR model and for the damage-related outputs, for which SVR was selected in 47 out of 57 cases. For more complex devices, those with three and five windows, and for distortion-related outputs, the model selection became more heterogeneous, although SVR still provided the best overall performance. From the computational point of view, SVR also offered the lowest median training times among the supervised models, whereas MLP required substantially longer training times without providing gain in accuracy.
The supervised-learning comparison shows a clear hierarchy among the candidate surrogate models. A total of 100 output-specific training problems were considered, corresponding to the $2W+1$ target variables required for each geometry family and adaptive iteration. The $F_1$ family required two optimization iterations, whereas the remaining families required three. Across all geometry families, iterations and output variables, SVR was the most frequently selected model and also the model that most often achieved the lowest cross-validated RMSE. As summarized in Figure~\ref{fig:surrogate_selection_summary_barplot}, SVR was selected in 71 cases, followed by GPR, GBR, XGBoost and MLP, while Random Forest was not selected in any case. This dominance was particularly clear for the two-window families, where only one output was assigned to a GPR model and for the damage-related outputs, for which SVR was selected in 47 out of 57 cases. For more complex devices, those with three and five windows, and for distortion-related outputs, the model selection became more heterogeneous, although SVR still provided the best overall performance. From the computational point of view, SVR also offered the lowest median training times among the supervised models, whereas MLP required substantially longer training times without providing gain in accuracy.
\begin{figure*}[htbp]
\centering
......@@ -373,54 +352,54 @@ The FEM validation of the optimized geometries is summarized in Table~\ref{tab:f
\toprule
Family & Surrogate & Iterations & $\mathbf{t}_w^{\star}$ [mm] & $J_{\mathrm{surr}}$ & $J_{\mathrm{FEM}}$ & $|e_J|$ & $e_{\max}$ [\%] \\
\midrule
H30\_B29 & RBF & 3
$F_1$ & RBF & 3
& $[12.56,\;14.79]$
& 0.00 & 0.41 & 0.41 & 0.29 \\
H30\_B29 & Supervised ML & 2
$F_1$ & Supervised ML & 2
& $[12.53,\;14.75]$
& 0.00 & 0.08 & 0.08 & 3.14 \\
\midrule
H30\_B34 & RBF & 3
$F_2$ & RBF & 3
& $[14.77,\;18.95]$
& 688.53 & 691.06 & 2.53 & 0.10 \\
H30\_B34 & Supervised ML & 3
$F_2$ & Supervised ML & 3
& $[15.20,\;20.00]$
& 796.29 & 802.64 & 6.35 & 0.64 \\
\midrule
H45\_B29 & RBF & 3
$F_3$ & RBF & 3
& $[5.81,\;7.88,\;8.98]$
& 0.00 & 0.93 & 0.93 & 1.70 \\
H45\_B29 & Supervised ML & 3
$F_3$ & Supervised ML & 3
& $[5.72,\;7.87,\;8.96]$
& 0.00 & 7.47 & 7.47 & 2.85 \\
\midrule
H45\_B34 & RBF & 2
$F_4$ & RBF & 2
& $[6.81,\;9.02,\;9.65]$
& 0.00 & 1.39 & 1.39 & 4.21 \\
H45\_B34 & Supervised ML & 3
$F_4$ & Supervised ML & 3
& $[6.84,\;9.05,\;9.73]$
& 0.00 & 1.35 & 1.35 & 2.91 \\
\midrule
H60\_B34 & RBF & 3
$F_5$ & RBF & 3
& $[5.77,\;7.38,\;8.37,\;6.18,\;5.00]$
& 17.74 & 19.26 & 1.51 & 2.66 \\
H60\_B34 & Supervised ML & 3
$F_5$ & Supervised ML & 3
& $[5.74,\;7.46,\;8.50,\;6.37,\;5.00]$
& 16.90 & 26.87 & 9.97 & 3.81 \\
\bottomrule
\end{tabular}
\end{table*}
The final validation results show that both surrogate strategies provide FEM-consistent optimized geometries after only a few adaptive iterations. RBF surrogates generally lead to lower objective-function discrepancies, with an average $|e_J|$ of 1.36, compared with 5.04 for the supervised ML surrogates. The average maximum variable error is also lower for RBF, with 1.79\% compared with 2.67\% for supervised ML. The RBF surrogate provides particularly accurate predictions for the two-window devices, with $e_{\max}$ below 0.3\%, while remaining below 4.3\% for the three- and five-window families. The supervised ML surrogates also satisfy all validation criteria, although larger discrepancies are observed in some cases, especially for the H60\_B34 family, where the objective-function error reaches 9.97.
The final validation results show that both surrogate strategies provide FEM-consistent optimized geometries after only a few adaptive iterations. RBF surrogates generally lead to lower objective-function discrepancies, with an average $|e_J|$ of 1.36, compared with 5.04 for the supervised ML surrogates. The average maximum variable error is also lower for RBF, with 1.79\% compared with 2.67\% for supervised ML. The RBF surrogate provides particularly accurate predictions for the two-window devices, with $e_{\max}$ below 0.3\%, while remaining below 4.3\% for the three- and five-window families. The supervised ML surrogates also satisfy all validation criteria, although larger discrepancies are observed in some cases, especially for the $F_5$ family, where the objective-function error reaches 9.97.
The adaptive validation loop is essential to reach these levels of agreement. In several cases, the first surrogate-optimized candidate did not satisfy the prescribed error tolerances, particularly for the three- and five-window devices. After incorporating the additional FEM results and retraining the surrogates, the prediction errors decreased significantly. For instance, the maximum variable error of the RBF surrogate decreased from 13.70\% to 1.70\% in the H45\_B29 family and from 11.44\% to 2.66\% in the H60\_B34 family. A similar behaviour was observed for the supervised ML surrogates, whose final candidates also satisfied all acceptance criteria. This confirms that the adaptive loop reduces the risk of accepting geometries that appear optimal only because of surrogate prediction errors in sparsely sampled regions of the design space.
The adaptive validation loop is essential to reach these levels of agreement. In several cases, the first surrogate-optimized candidate did not satisfy the prescribed error tolerances, particularly for the three- and five-window devices. After incorporating the additional FEM results and retraining the surrogates, the prediction errors decreased significantly. For instance, the maximum variable error of the RBF surrogate decreased from 13.70\% to 1.70\% in the $F_3$ family and from 11.44\% to 2.66\% in the $F_5$ family. A similar behaviour was observed for the supervised ML surrogates, whose final candidates also satisfied all acceptance criteria. This confirms that the adaptive loop reduces the risk of accepting geometries that appear optimal only because of surrogate prediction errors in sparsely sampled regions of the design space.
The optimized geometries obtained with RBF and supervised ML surrogates are very similar in most geometry families. In general, the differences in the optimized window thicknesses are below 0.1 mm and always below 0.2 mm, except for the H30\_B34 family. In this case, the differences reach 0.43 mm and 1.05 mm for the first and second windows, respectively. This family also presents the largest objective-function values, indicating that the prescribed damage targets, $\TFD_i$ and $\TFD_f$, cannot be fully maintained below the desired threshold within the admissible thickness range. Consequently, the optimizer is forced to find the best compromise between damage control and dissipative activation, which may amplify the differences between surrogate predictions.
The optimized geometries obtained with RBF and supervised ML surrogates are very similar in most geometry families. In general, the differences in the optimized window thicknesses are below 0.1 mm and always below 0.2 mm, except for the $F_2$ family. In this case, the differences reach 0.43 mm and 1.05 mm for the first and second windows, respectively. This family also presents the largest objective-function values, indicating that the prescribed damage targets, $\TFD_i$ and $\TFD_f$, cannot be fully maintained below the desired threshold within the admissible thickness range. Consequently, the optimizer is forced to find the best compromise between damage control and dissipative activation, which may amplify the differences between surrogate predictions.
Figure~\ref{fig:optimized_window_thickness_evolution} shows the evolution of the window thicknesses during the adaptive optimization process. In most cases, the largest adjustment occurs between the first and second iterations, while the changes between the second and third iterations are considerably smaller, indicating convergence of the proposed geometry. According to the acceptance criteria, the maximum variation in any window thickness between the last two iterations must remain below 5\% of the corresponding design range. This condition is satisfied in all validated cases, with a maximum observed variation of 4.44\%. Therefore, the adaptive process converges not only in terms of surrogate prediction accuracy, but also in terms of the optimized geometry proposed by the optimizer.
......@@ -433,14 +412,14 @@ Figure~\ref{fig:optimized_window_thickness_evolution} shows the evolution of the
From a methodological point of view, these results highlight the trade-off between surrogate complexity, accuracy and computational efficiency. Supervised models, particularly SVR and GPR, provide high predictive accuracy and robustness, but require hyperparameter optimization and cross-validation for every output variable and adaptive iteration. RBF interpolation, in contrast, has a much lower training cost and provides very competitive final predictions once the relevant regions of the design domain have been adaptively sampled. Therefore, the comparison does not identify a universally superior surrogate strategy. Instead, it suggests that RBF interpolation is especially suitable for low- to moderate-dimensional design spaces with well-distributed FEM samples and relatively smooth input--output relationships, as occurs for the response variables analysed in this work. Supervised ML surrogates remain valuable when greater robustness is required, or when the response surface is expected to involve stronger nonlinear interactions, local irregularities or higher-dimensional dependencies.
The objective-function values should also be interpreted carefully. The objective-function error measures the consistency between surrogate predictions and FEM validation, not whether the final objective value is necessarily close to zero. Some geometry families, such as H30\_B34, retain non-negligible penalty contributions because the prescribed damage targets cannot be fully achieved within the admissible design bounds. Nevertheless, the close agreement between surrogate and FEM objective values indicates that the accepted designs are not artifacts of surrogate extrapolation, but FEM-consistent optimized candidates within the explored design space.
The objective-function values should also be interpreted carefully. The objective-function error measures the consistency between surrogate predictions and FEM validation, not whether the final objective value is necessarily close to zero. Some geometry families, such as $F_2$, retain non-negligible penalty contributions because the prescribed damage targets cannot be fully achieved within the admissible design bounds. Nevertheless, the close agreement between surrogate and FEM objective values indicates that the accepted designs are not artifacts of surrogate extrapolation, but FEM-consistent optimized candidates within the explored design space.
This behaviour is illustrated in Figure~\ref{fig:rbf_surface_evolution}, which shows the evolution of the RBF objective surface during the adaptive optimization process for the two-window families. The response surfaces remain relatively smooth, supporting the suitability of RBF interpolation for this problem. The evolution between adaptive iterations also shows how the surrogate surface is progressively corrected as new FEM information is incorporated near the optimized region.
\begin{figure*}[htbp]
\centering
\includegraphics[width=1.0\textwidth]{../Figures/RBFOptimizationSurfaceEvolution/rbf_surface_evolution.png}
\caption{Evolution of the RBF objective surface during the adaptive optimization process for the two-window families. Left: H30\_B29. Right: H30\_B34.}
\caption{Evolution of the RBF objective surface during the adaptive optimization process for the two-window families. Left: $F_1$. Right: $F_2$.}
\label{fig:rbf_surface_evolution}
\end{figure*}
......@@ -569,8 +548,8 @@ This appendix summarizes the surrogate-predicted optimized configurations, the c
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H30\_B29 family using Supervised ML surrogates.}
\label{tab:app_ml_h30_b29}
\caption{Optimization results for the $F_1$ family using Supervised ML surrogates.}
\label{tab:app_ml_f1}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -591,8 +570,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H30\_B34 family using Supervised ML surrogates.}
\label{tab:app_ml_h30_b34}
\caption{Optimization results for the $F_2$ family using Supervised ML surrogates.}
\label{tab:app_ml_f2}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -617,8 +596,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H45\_B29 family using Supervised ML surrogates.}
\label{tab:app_ml_h45_b29}
\caption{Optimization results for the $F_3$ family using Supervised ML surrogates.}
\label{tab:app_ml_f3}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -643,8 +622,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H45\_B34 family using Supervised ML surrogates.}
\label{tab:app_ml_h45_b34}
\caption{Optimization results for the $F_4$ family using Supervised ML surrogates.}
\label{tab:app_ml_f4}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -669,8 +648,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H60\_B34 family using Supervised ML surrogates.}
\label{tab:app_ml_h60_b34}
\caption{Optimization results for the $F_5$ family using Supervised ML surrogates.}
\label{tab:app_ml_f5}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -695,8 +674,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H30\_B29 family using RBF surrogates.}
\label{tab:app_rbf_h30_b29}
\caption{Optimization results for the $F_1$ family using RBF surrogates.}
\label{tab:app_rbf_f1}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -721,8 +700,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H30\_B34 family using RBF surrogates.}
\label{tab:app_rbf_h30_b34}
\caption{Optimization results for the $F_2$ family using RBF surrogates.}
\label{tab:app_rbf_f2}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -747,8 +726,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H45\_B29 family using RBF surrogates.}
\label{tab:app_rbf_h45_b29}
\caption{Optimization results for the $F_3$ family using RBF surrogates.}
\label{tab:app_rbf_f3}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -773,8 +752,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H45\_B34 family using RBF surrogates.}
\label{tab:app_rbf_h45_b34}
\caption{Optimization results for the $F_4$ family using RBF surrogates.}
\label{tab:app_rbf_f4}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......@@ -795,8 +774,8 @@ Iter. & Source & $\mathbf{t}_w$ [mm] & $\boldsymbol{\varepsilon}_{xy}$ & $\bolds
\begin{table*}[htbp]
\centering
\caption{Optimization results for the H60\_B34 family using RBF surrogates.}
\label{tab:app_rbf_h60_b34}
\caption{Optimization results for the $F_5$ family using RBF surrogates.}
\label{tab:app_rbf_f5}
\scriptsize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.08}
......
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