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2026_Article_RESILINK_ML
Commit 737c0a63 by Joaquín Irazábal González
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Add loading pattern plotting script and update manuscript

- Introduced a new Python script `plot_load_patterns.py` to visualize loading pattern displacement histories from `LoadPatterns.csv`, generating figures in PNG, SVG, and PDF formats. - Removed the outdated `PatronesCarga.csv` file as it is no longer needed. - Revised the abstract and various sections of the manuscript to enhance clarity and detail, including the introduction of radial basis function surrogates and adjustments to the optimization methodology. - Updated references in the bibliography to include new citations for SciPy and Differential Evolution.
parent 7c430ea6
Showing with 1916 additions and 88 deletions
Manuscript/Figures/LoadPatterns/LoadPatterns.csv 0 → 100644
Time (s),Displacement H30 (mm),Displacement H45 (mm),Displacement H60 (mm)
0,0,0,0
0.0625,12,16,20
0.125,0,0,0
0.1875,-12,-16,-16
0.25,0,0,0
0.3125,12,16,16
0.375,0,0,0
0.4375,-12,-16,-16
0.5,0,0,0
0.5625,25,32,32
0.625,0,0,0
0.6875,-25,-32,-32
0.75,0,0,0
0.8125,25,32,32
0.875,0,0,0
0.9375,-25,-32,-32
1,0,0,0
1.0625,38,48,48
1.125,0,0,0
1.1875,-38,-48,-48
1.25,0,0,0
1.3125,38,48,48
1.375,0,0,0
1.4375,-38,-48,-48
1.5,0,0,0
1.5625,50,64,64
1.625,0,0,0
1.6875,-50,-64,-64
1.75,0,0,0
1.8125,50,64,64
1.875,0,0,0
1.9375,-50,-64,-64
2,0,0,0
2.0625,60,75,75
2.125,0,0,0
2.1875,-60,-75,-75
2.25,0,0,0
2.3125,60,75,75
2.375,0,0,0
2.4375,-60,-75,-75
2.5,0,0,0
2.5625,70,85,85
2.625,0,0,0
2.6875,-70,-85,-85
2.75,0,0,0
2.8125,70,85,85
2.875,0,0,0
2.9375,-70,-85,-85
3,0,0,0
Manuscript/Figures/LoadPatterns/LoadPatterns.svg 0 → 100644
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Manuscript/Figures/LoadPatterns/plot_load_patterns.py 0 → 100644
"""
Plot the loading pattern displacement histories.
The script reads LoadPatterns.csv and saves a single-panel figure as PNG, SVG,
and PDF using the same visual format as the FEM validation plots.
"""
from pathlib import Path
import matplotlib.pyplot as plt
import numpy as np
plt.rcParams.update(
{
"font.family": "serif",
"font.serif": ["Times New Roman", "DejaVu Serif"],
"mathtext.fontset": "stix",
"font.size": 11,
"axes.labelsize": 11,
"axes.titlesize": 11,
"xtick.labelsize": 11,
"ytick.labelsize": 11,
"axes.linewidth": 0.7,
"axes.spines.top": False,
"axes.spines.right": False,
"figure.dpi": 300,
"savefig.dpi": 300,
}
)
COLORS = ["#1a6faf", "#E07060", "#4D9A57"]
COLOR_GRID = "#d8d8d8"
X_LABEL = "Time (s)"
Y_LABEL = "Displacement (mm)"
SERIES_LABELS = ["H30", "H45", "H60"]
def load_loading_patterns(path):
data = np.genfromtxt(path, delimiter=",", skip_header=1)
if data.ndim != 2 or data.shape[1] != 4:
raise ValueError(f"{path.name} must contain four numeric columns.")
return data[:, 0], data[:, 1:]
def style_axis(ax):
ax.grid(True, color=COLOR_GRID, linewidth=0.55, alpha=0.75)
ax.tick_params(axis="both", which="major", direction="in", length=4, width=0.7)
ax.tick_params(axis="both", which="minor", direction="in", length=2.5, width=0.6)
ax.minorticks_on()
def save_figure_all_formats(fig, out_base):
fig.savefig(f"{out_base}.png", format="png", bbox_inches="tight", dpi=300)
fig.savefig(f"{out_base}.svg", format="svg", bbox_inches="tight")
fig.savefig(f"{out_base}.pdf", format="pdf", bbox_inches="tight")
def main():
base_dir = Path(__file__).resolve().parent
time, displacements = load_loading_patterns(base_dir / "LoadPatterns.csv")
fig, ax = plt.subplots(figsize=(5.2, 3.8))
fig.subplots_adjust(left=0.14, right=0.985, bottom=0.28, top=0.94)
for index, label in enumerate(SERIES_LABELS):
ax.plot(
time,
displacements[:, index],
color=COLORS[index],
linewidth=1.6,
label=label,
)
ax.axhline(0, color="black", linewidth=0.55, alpha=0.65)
ax.set_xlabel(X_LABEL)
ax.set_ylabel(Y_LABEL)
ax.set_xlim(time.min(), time.max())
style_axis(ax)
handles, labels = ax.get_legend_handles_labels()
fig.legend(
handles,
labels,
loc="lower center",
ncol=3,
frameon=False,
framealpha=0.95,
edgecolor="#cccccc",
bbox_to_anchor=(0.5, 0.055),
)
output_base = base_dir / "LoadPatterns"
save_figure_all_formats(fig, output_base)
plt.close(fig)
print(f"Saved: {output_base}.png/.svg/.pdf")
if __name__ == "__main__":
main()
Manuscript/Figures/PatronesCarga/PatronesCarga.csv deleted 100644 → 0
Tiempo [s],Desplazamiento H30 [mm],Desplazamiento H45 [mm]
0,0,0
0.0625,12,16
0.125,0,0
0.1875,-12,-16
0.25,0,0
0.3125,12,16
0.375,0,0
0.4375,-12,-16
0.5,0,0
0.5625,25,32
0.625,0,0
0.6875,-25,-32
0.75,0,0
0.8125,25,32
0.875,0,0
0.9375,-25,-32
1,0,0
1.0625,38,48
1.125,0,0
1.1875,-38,-48
1.25,0,0
1.3125,38,48
1.375,0,0
1.4375,-38,-48
1.5,0,0
1.5625,50,64
1.625,0,0
1.6875,-50,-64
1.75,0,0
1.8125,50,64
1.875,0,0
1.9375,-50,-64
2,0,0
2.0625,60,75
2.125,0,0
2.1875,-60,-75
2.25,0,0
2.3125,60,75
2.375,0,0
2.4375,-60,-75
2.5,0,0
2.5625,70,85
2.625,0,0
2.6875,-70,-85
2.75,0,0
2.8125,70,85
2.875,0,0
2.9375,-70,-85
3,0,0
Manuscript/RESILINK_surrogate_optimization_manuscript.tex
...@@ -45,7 +45,7 @@ ...@@ -45,7 +45,7 @@
\corres{J. Irazábal. \email{jirazabal@cimne.upc.edu}} \corres{J. Irazábal. \email{jirazabal@cimne.upc.edu}}
\abstract[Abstract]{Buckling-delayed shear-link dampers are used in seismic-resistant structures as passive devices that concentrate energy dissipation while limiting damage to the primary system. Their geometric optimization requires balancing high energy dissipation with strict control of local damage. Finite element models can accurately reproduce the nonlinear cyclic response of these devices and provide internal quantities such as damage indicators and local distortion, but their computational cost prevents their direct use within iterative optimization loops. This work proposes an adaptive surrogate-assisted optimization framework for buckling-delayed shear-link dampers. First, experimentally calibrated nonlinear finite element models are used to generate reference datasets for dampers with different geometric and mechanical configurations. Supervised learning models are initially evaluated, with support vector regression and Gaussian process regression, both based on radial kernel functions, consistently providing high predictive accuracy. Motivated by this observation, radial basis function surrogates are introduced as a computationally efficient alternative. The surrogate predictions are coupled with a differential evolution algorithm through a damage-aware objective function that controls local damage and uses dissipated energy as a performance criterion. Optimized geometries are finally re-evaluated with finite element simulations. When the surrogate error exceeds the adopted tolerances, the new simulation result is added to the dataset and the surrogate models are retrained. In addition, SHapley Additive exPlanations are employed to quantify the influence of window thickness on damage distribution, with particular emphasis on the response of the surrounding frame. The proposed framework provides an efficient damage-aware optimization of seismic energy dissipation devices.} \abstract[Abstract]{Buckling-delayed shear-link dampers are used in seismic-resistant structures as passive devices that concentrate energy dissipation while limiting damage to the primary system. Their geometric optimization requires balancing high energy dissipation with strict control of local damage. Finite element models can accurately reproduce the nonlinear cyclic response of these devices and provide internal quantities such as damage indicators and local distortion, but their computational cost prevents their direct use within iterative optimization loops. This work proposes an adaptive surrogate-assisted optimization framework for buckling-delayed shear-link dampers. First, experimentally calibrated nonlinear finite element models are used to generate reference datasets for dampers with different geometric and mechanical configurations. Supervised learning models are initially evaluated, with support vector regression and Gaussian process regression consistently providing high predictive accuracy. This suggests that kernel-based, distance-dependent approximations are well suited to the problem, motivating the introduction of radial basis function surrogates as a computationally efficient alternative. The surrogate predictions are coupled with a differential evolution algorithm through a damage-aware objective function that controls local damage and uses dissipated energy as a performance criterion. Optimized geometries are finally re-evaluated with finite element simulations. When the surrogate error exceeds the adopted tolerances, the new simulation result is added to the dataset and the surrogate models are retrained. In addition, SHapley Additive exPlanations are employed to quantify the influence of window thickness on damage distribution, with particular emphasis on the response of the surrounding frame. The proposed framework provides an efficient damage-aware optimization of seismic energy dissipation devices.}
\keywords{Buckling-delayed shear link, seismic energy dissipation, FEM validation, surrogate modelling, machine learning, radial basis functions, Differential Evolution, SHAP.} \keywords{Buckling-delayed shear link, seismic energy dissipation, FEM validation, surrogate modelling, machine learning, radial basis functions, Differential Evolution, SHAP.}
...@@ -197,23 +197,30 @@ To improve surrogate robustness near the admissible limits, the sampling domain ...@@ -197,23 +197,30 @@ To improve surrogate robustness near the admissible limits, the sampling domain
\end{tabular} \end{tabular}
\end{table} \end{table}
For every sampled configuration, a FEM simulation is performed, and the resulting dataset stores the input variables and the corresponding structural response quantities. The target outputs include damage indicators in the windows and in the frame, together with local distortion measures associated with dissipative activation. The aggregated TFDMap indicator in window $i$ is denoted by $\TFD_i$, whereas the corresponding frame indicator is denoted by $\TFD_f$. Both quantities are computed as the average TFDMap value of the 12 nodes with the highest values within each region, capturing the most critical damage levels while reducing sensitivity to isolated numerical peaks. The maximum local shear distortion in each window is denoted by $\varepsilon_{xy,i}$ and is used as a proxy for dissipative activation. When included in the objective function, each window contribution is weighted according to its effective geometric volume, so that the optimization accounts for both local strain intensity and the material volume involved in the dissipation process. For every sampled configuration, a FEM simulation is performed under a displacement-controlled cyclic loading protocol. Since the admissible deformation demand depends on the size of the device, different loading patterns are adopted according to the device height. As shown in Figure~\ref{fig:LoadPatterns}, both the number of cycles and the maximum displacement amplitude increase with the device height, consistently with the expected performance range of each geometry family.
\red{Faltan por meter los ciclos de carga aplicados para cada geometría.} \begin{figure}[!ht]
\centering
\includegraphics[width=0.50\textwidth]{./Figures/LoadPatterns/LoadPatterns.png}
\caption{Displacement-controlled cyclic loading patterns adopted for the different device heights considered in the FEM campaign.}
\label{fig:LoadPatterns}
\end{figure}
The resulting dataset stores the input variables and the structural response quantities extracted at the final time of each simulation. The target outputs include damage indicators in the windows and in the frame, together with local distortion measures associated with dissipative activation. For compactness in the optimization formulation, the aggregated TFDMap indicator in window $i$ is denoted by $\TFD_i$, whereas the corresponding frame indicator is denoted by $\TFD_f$. Both quantities are computed as the average TFDMap value of the 12 nodes with the highest values within each region, capturing the most critical damage levels while reducing sensitivity to isolated numerical peaks.
The maximum local shear distortion in each window is denoted by $\varepsilon_{xy,i}$ and is used as a proxy for dissipative activation. When included in the objective function, each window contribution is weighted according to its effective geometric volume, so that the optimization accounts for both local strain intensity and the material volume involved in the dissipation process.
\subsection{Supervised ML surrogate models}\label{subsec:ml_models} \subsection{Supervised ML surrogate models}\label{subsec:ml_models}
This work compares several supervised surrogate models for predicting FEM-derived damage and distortion indicators. The considered models cover three families: tree-based methods, including Random Forest (RF) \cite{Breiman2001}, Gradient Boosting Regression (GBR) \cite{Friedman2001} and XGBoost \cite{Chen2016}; kernel-based methods, including Support Vector Regression (SVR) \cite{Drucker1996} and Gaussian Process Regression (GPR) \cite{Williams1995}; and neural-network models, represented by the Multilayer Perceptron (MLP) \cite{Rosenblatt1958,Rumelhart1986}. RF relies on bootstrap aggregation of decision trees, GBR and XGBoost are sequential boosting approaches, SVR and GPR exploit kernel functions to model nonlinear relationships and MLP approximates nonlinear input--output mappings through interconnected layers. This work compares several supervised surrogate models for predicting FEM-derived damage and distortion indicators. The considered models cover three families: tree-based methods, including Random Forest (RF) \cite{Breiman2001}, Gradient Boosting Regression (GBR) \cite{Friedman2001} and XGBoost \cite{Chen2016}; kernel-based methods, including Support Vector Regression (SVR) \cite{Drucker1996} and Gaussian Process Regression (GPR) \cite{Williams1995}; and neural-network models, represented by the Multilayer Perceptron (MLP) \cite{Rosenblatt1958,Rumelhart1986}. RF relies on bootstrap aggregation of decision trees, GBR and XGBoost are sequential boosting approaches, SVR and GPR exploit kernel functions to model nonlinear relationships and MLP approximates nonlinear input--output mappings through interconnected layers.
For each geometry family, an independent regression model is trained for every target output. The input vector contains the window thicknesses of the corresponding device, while the outputs are the local distortion indicators $\varepsilon_{xy,i}$, the window damage indicators $\TFD_i$ and the frame damage indicator $\TFD_f$. Therefore, for a device with $W$ windows, $2W+1$ surrogate models are trained: $W$ for distortion, $W$ for window damage and one for frame damage. For each output, all candidate algorithms are evaluated and the selected model is stored. For each geometry family, an independent regression model is trained for every target output. The input vector contains the window thicknesses of the corresponding device, while the outputs are the local distortion indicators in each window $\varepsilon_{xy,i}$, the window damage indicators $\TFD_i$ and the frame damage indicator $\TFD_f$. Therefore, for a device with $W$ windows, $2W+1$ surrogate models are trained: $W$ for windows distortion, $W$ for windows damage and one for frame damage. For each output, all candidate algorithms are evaluated and the selected model is stored.
Hyperparameters are optimized using Bayesian optimization \cite{Snoek2012}, with 40 evaluations per model. The first 10 evaluations are randomly sampled to explore the search space, while the remaining 30 are guided by the Bayesian surrogate model. This strategy provides a more efficient alternative to exhaustive grid search, particularly given the number of geometry families, target outputs and candidate algorithms considered. Hyperparameters are optimized using Bayesian optimization \cite{Snoek2012}, with 40 evaluations per model. The first 10 evaluations are randomly sampled to explore the search space, while the remaining 30 are guided by the Bayesian surrogate model. This strategy provides a more efficient alternative to exhaustive grid search, particularly given the number of geometry families, target outputs and candidate algorithms considered.
The cross-validation strategy is adapted to the dataset size. Leave-One-Out validation is used for $N\leq20$, repeated five-fold cross-validation with five repetitions for $21\leq N\leq80$, and shuffled five-fold cross-validation for larger datasets. For small datasets, the search spaces of tree-based models are restricted to reduce overfitting. The cross-validation strategy is adapted to the dataset size. Leave-One-Out validation is used for $N\leq20$, repeated five-fold cross-validation with five repetitions for $21\leq N\leq80$, and shuffled five-fold cross-validation for larger datasets. For small datasets, the search spaces of tree-based models are restricted to reduce overfitting.
Model selection is performed in two stages. First, for each candidate algorithm, Bayesian hyperparameter optimization is carried out using cross-validated RMSE as the refit criterion. The best hyperparameter configuration is therefore the one with the lowest mean RMSE. Then, the best configurations from all candidate algorithms are compared. The model with the lowest mean RMSE defines a competitive threshold, and all models with an RMSE within 5\% of this value are retained. Among these competitive models, the final selection is based on the lowest relative RMSE dispersion, computed as the standard deviation of the fold-wise RMSE divided by the mean RMSE. If two models have the same dispersion, the one with the lower mean RMSE is preferred. Model selection is performed in two stages. First, for each candidate algorithm, Bayesian hyperparameter optimization is carried out using cross-validated Root Mean Squared Error (RMSE) as the refit criterion. The best hyperparameter configuration is therefore the one with the lowest mean RMSE. Then, the best configurations from all candidate algorithms are compared. The model with the lowest mean RMSE defines a competitive threshold, and all models with an RMSE within 5\% of this value are retained. Among these competitive models, the final selection is based on the lowest relative RMSE dispersion, computed as the standard deviation of the fold-wise RMSE divided by the mean RMSE. If two models have the same dispersion, the one with the lower mean RMSE is preferred. This procedure favours surrogates that are both accurate and stable, which is important because small prediction errors near active damage constraints may alter the optimized geometry.
This procedure favours surrogates that are both accurate and stable, which is important because small prediction errors near active damage constraints may alter the optimized geometry.
\begin{figure}[!ht] \begin{figure}[!ht]
\centering \centering
...@@ -222,11 +229,11 @@ This procedure favours surrogates that are both accurate and stable, which is im ...@@ -222,11 +229,11 @@ This procedure favours surrogates that are both accurate and stable, which is im
\label{fig:BayesianSearchCV} \label{fig:BayesianSearchCV}
\end{figure} \end{figure}
Preliminary executions of the proposed workflow show that SVR and GPR frequently provide the highest, or second-highest, predictive accuracy for the considered datasets. Since both methods rely on radial kernel functions, this result motivates the assessment of Radial Basis Function (RBF) interpolation \cite{Gutmann2001} as a simpler and computationally efficient surrogate alternative. Although the cost of surrogate evaluation is negligible compared with FEM simulations, training and repeatedly evaluating complex supervised models can still become relevant within optimization loops involving many candidate geometries. RBF interpolation offers a non-parametric and fast-to-train alternative that can capture nonlinear response surfaces with limited hyperparameter tuning, making it attractive for low-dimensional and moderately sampled design spaces. Preliminary executions of the proposed workflow show that SVR and GPR frequently 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 with limited hyperparameter tuning, making it attractive for low-dimensional and moderately sampled design spaces.
\subsection{RBF surrogate models}\label{subsec:rbf_models} \subsection{RBF surrogate models}\label{subsec:rbf_models}
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 RBF surrogate is implemented using the interpolator available in the \textit{SciPy} library \cite{Virtanen2025}. 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 selection. Unlike the Gaussian RBF kernel used in SVR, the multiquadric function leads here to an interpolation-based approximation of the FEM response surface.
The prediction model can be expressed as The prediction model can be expressed as
\begin{equation} \begin{equation}
...@@ -235,38 +242,24 @@ The prediction model can be expressed as ...@@ -235,38 +242,24 @@ The prediction model can be expressed as
\end{equation} \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. 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. 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.
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. 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.
\section{Damage-aware surrogate-assisted optimization}\label{sec:optimization} \section{Damage-aware surrogate-assisted optimization}\label{sec:optimization}
\subsection{Optimization algorithm}\label{subsec:de} 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.
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. Following optimization, an adaptive FEM validation loop is incorporated to verify the predicted optimal geometries before acceptance.
The geometric optimization is carried out with Differential Evolution. DE is a population-based global optimizer that does not require gradient information and is therefore suitable for nonlinear, non-convex surrogate response surfaces. In the current implementation, the DE algorithm is run with a maximum of 500 iterations, a population size factor of 25, a convergence tolerance of $10^{-6}$ and a fixed random seed equal to 42 for reproducibility. For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window distortions $\hat{\Exy}_i$, the window damage indicators $\widehat{\TFD}_i$ and the frame damage indicator $\widehat{\TFD}_f$. These quantities are combined into a scalar objective function designed to encode four mechanical criteria: damage must remain limited in all structural components; frame damage must be penalized more severely than window damage because frame failure would compromise the structural integrity of the damper; the windows should develop comparable damage levels to avoid concentration of the dissipative demand in a single region; and, among geometries with similar damage performance, the preferred solution should be the one with higher energy dissipation capacity.
For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window distortions $\hat{\Exy}_i$, the window damage indicators $\widehat{\TFD}_i$ and the frame damage indicator $\widehat{\TFD}_f$. These predictions are then combined into a scalar objective function to be minimized. The implemented objective function to be minimized is
\subsection{Objective function}\label{subsec:objective}
The objective function is designed to encode four mechanical rules:
\begin{enumerate}
\item the damage indicator must remain limited in all structural components, including both windows and frame;
\item reaching complete or near-complete damage in the frame is much more critical than reaching a similar damage level in a window, because frame failure would imply loss of the structural integrity of the damper;
\item the windows should develop comparable damage levels so that the dissipation mechanism is distributed rather than concentrated in a single window;
\item if several geometries show similar damage performance, the preferred one is the geometry that dissipates more energy, approximated through stable window distortion and the corresponding volume contribution.
\end{enumerate}
The implemented scalar objective is
\begin{equation} \begin{equation}
J(\mathbf{x}) = - \sum_{i=1}^{W} \hat{\Exy}_i^2\, t_{w,i}\, V_i + J(\mathbf{x}) = - \sum_{i=1}^{W} \hat{\Exy}_i^2\, t_{w,i}\, V_i +
\sum_{i=1}^{W} P_w\left(\widehat{\TFD}_i;\TFD_w^{\star}\right) + \sum_{i=1}^{W} P_w\left(\widehat{\TFD}_i;\TFD_w^{\star}\right) +
P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right), P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right),
\label{eq:objective} \label{eq:objective}
\end{equation} \end{equation}
where $V_i$ is the volume factor associated with window $i$, $\TFD_w^{\star}$ is the target TFDMap level for the windows and $\TFD_f^{\max}$ is the maximum admissible frame threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger distortion contributions reduce the objective value. where $W$ is the number of windows, $t_{w,i}$ is the thickness of window $i$, $V_i$ is the corresponding volume factor, $\TFD_w^{\star}$ is the target damage level for the windows, and $\TFD_f^{\max}$ is the maximum admissible frame damage threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger stable distortion contributions reduce the objective value and are favoured once the damage criteria are satisfied.
The window penalty is defined as The window penalty is defined as
\begin{equation} \begin{equation}
...@@ -277,9 +270,9 @@ P_w\left(\widehat{\TFD}_i;\TFD_w^{\star}\right)= ...@@ -277,9 +270,9 @@ P_w\left(\widehat{\TFD}_i;\TFD_w^{\star}\right)=
\end{cases} \end{cases}
\label{eq:window_penalty} \label{eq:window_penalty}
\end{equation} \end{equation}
This expression penalizes exceeding the target value quadratically, while also discouraging excessively under-utilized windows through a linear distance to the target. As a result, the optimizer tends to equalize the TFDMap levels among windows rather than forcing all windows to remain far below the admissible damage level. This formulation penalizes values above the target quadratically, while also discouraging excessively underused windows through a linear distance to the target. As a result, the optimizer tends to balance the damage levels among windows rather than forcing all windows to remain far below the admissible level.
The frame penalty is more severe: The frame penalty is defined as
\begin{equation} \begin{equation}
P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right)= P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right)=
\begin{cases} \begin{cases}
...@@ -288,15 +281,14 @@ P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right)= ...@@ -288,15 +281,14 @@ P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right)=
\end{cases} \end{cases}
\label{eq:frame_penalty} \label{eq:frame_penalty}
\end{equation} \end{equation}
The cubic frame penalty reflects the fact that complete frame failure is a much more severe event than localized damage in a replaceable window. In the current scripts, $\TFD_f^{\max}=90$ is used, while the target window threshold $\TFD_w^{\star}$ is provided as an input argument to the optimizer. The cubic penalty reflects the greater severity of frame damage compared with localized damage in a replaceable window. In the current implementation, $\TFD_f^{\max}=90$ is adopted, while the target window threshold $\TFD_w^{\star}$ is provided as an input parameter to the optimizer.
\subsection{Volume factors}\label{subsec:volume_factors}
The distortion contribution is weighted by a geometric factor that accounts for the effective volume associated with each window. These factors differ between families because the windows have different dimensions and positions. The current values used in the scripts are summarized in Table~\ref{tab:volume_factors}. When a geometry family is not explicitly covered, the implementation falls back to an unweighted contribution. The distortion contribution in Eq.~\eqref{eq:objective} is weighted by a geometric factor $V_i$ that accounts for the effective volume associated with each window. These factors depend on the geometry family because the windows differ in size and position. The values currently used in the optimization scripts are summarized in Table~\ref{tab:volume_factors}. If a geometry family is not explicitly covered, the implementation falls back to an unweighted contribution.
\begin{table}[t] \begin{table}[ht!]
\centering \centering
\caption{Window volume factors used to weight the distortion contribution in the objective function.}\label{tab:volume_factors} \caption{Window volume factors used to weight the distortion contribution in the objective function.}
\label{tab:volume_factors}
\begin{tabular}{lll} \begin{tabular}{lll}
\toprule \toprule
Family & Width identifier $B$ & Volume factors $V_i$ \\ Family & Width identifier $B$ & Volume factors $V_i$ \\
...@@ -419,7 +411,7 @@ Some limitations should be highlighted. First, the methodology is only as reliab ...@@ -419,7 +411,7 @@ Some limitations should be highlighted. First, the methodology is only as reliab
\section{Conclusions}\label{sec:conclusions} \section{Conclusions}\label{sec:conclusions}
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, Differential Evolution and FEM-based validation of the optimized geometries. The following conclusions can be drawn from the proposed formulation: 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} \begin{enumerate}
\item The BDSL optimization problem must be formulated as a damage-aware design problem rather than as a pure energy-maximization problem. The same energy level may correspond to different local damage distributions and different safety margins in the frame. \item The BDSL optimization problem must be formulated as a damage-aware design problem rather than as a pure energy-maximization problem. The same energy level may correspond to different local damage distributions and different safety margins in the frame.
......
Manuscript/wileyNJD-AMA.bib
...@@ -765,4 +765,31 @@ steel from coupon test results available. First, the theory of metal plasticity ...@@ -765,4 +765,31 @@ steel from coupon test results available. First, the theory of metal plasticity
howpublished = {Standard, American Society of Civil Engineers}, howpublished = {Standard, American Society of Civil Engineers},
} }
@Article{Virtanen2025,
author = {Virtanen, Pauli and Gommers, Ralf and Oliphant, Travis E. and Haberland, Matt and Reddy, Tyler and Cournapeau, David and Burovski, Evgeni and Peterson, Pearu and Weckesser, Warren and Bright, Jonathan and {van der Walt}, St{\'e}fan J. and Brett, Matthew and Wilson, Joshua and Millman, K. Jarrod and Mayorov, Nikolay and Nelson, Andrew R. J. and Jones, Eric and Kern, Robert and Larson, Eric and Carey, C J and Polat, {\.I}lhan and Feng, Yu and Moore, Eric W. and {VanderPlas}, Jake and Laxalde, Denis and Perktold, Josef and Cimrman, Robert and Henriksen, Ian and Quintero, E. A. and Harris, Charles R. and Archibald, Anne M. and Ribeiro, Ant{\^o}nio H. and Pedregosa, Fabian and {van Mulbregt}, Paul and {SciPy 1.0 Contributors}},
title = {{{SciPy} 1.0: Fundamental Algorithms for Scientific Computing in Python}},
journal = {Nature Methods},
year = {2020},
volume = {17},
pages = {261--272},
doi = {10.1038/s41592-019-0686-2},
}
@Article{Storn1997,
author = {Storn, Rainer and Price, Kenneth},
journal = {Journal of Global Optimization},
title = {Differential {Evolution} – {A} {Simple} and {Efficient} {Heuristic} for global {Optimization} over {Continuous} {Spaces}},
year = {1997},
issn = {1573-2916},
month = dec,
number = {4},
pages = {341--359},
volume = {11},
abstract = {A new heuristic approach for minimizing possiblynonlinear and non-differentiable continuous spacefunctions is presented. By means of an extensivetestbed it is demonstrated that the new methodconverges faster and with more certainty than manyother acclaimed global optimization methods. The newmethod requires few control variables, is robust, easyto use, and lends itself very well to parallelcomputation.},
doi = {10.1023/A:1008202821328},
keywords = {evolution strategy, genetic algorithm, global optimization, nonlinear optimization, Stochastic optimization},
language = {en},
urldate = {2026-05-13},
}
@Comment{jabref-meta: databaseType:bibtex;} @Comment{jabref-meta: databaseType:bibtex;}
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