\abstract[Abstract]{The geometric optimization of buckling-delayed shear-link (BDSL) dampers requires a compromise between high energy dissipation and strict control of local damage in the dissipative windows and in the surrounding frame. High-fidelity finite element method (FEM) models can reproduce the nonlinear cyclic response of these devices and provide internal quantities such as the Triaxial Failure Damage Map (TFDMap) and local distortion, but their computational cost prevents their direct use inside iterative optimization loops. This work proposes an adaptive surrogate-assisted optimization framework for BDSL dampers. First, experimentally calibrated nonlinear FEM models are used to generate ground-truth datasets for damper configurations with different numbers of windows and geometric proportions. Supervised machine learning (ML) models--including Random Forests, Gradient Boosting, XGBoost, Support Vector Regression, Multilayer Perceptrons, and Gaussian Process Regression--are trained to predict damage and distortion indicators from the window thicknesses. Since preliminary tests indicate that Support Vector Regression and Gaussian Processes are often the most accurate alternatives, Radial Basis Function (RBF) interpolants are also introduced as an ultra-fast surrogate strategy. The surrogate predictions are coupled with a Differential Evolution algorithm through a damage-aware objective function that limits the TFDMap in all components, strongly penalizes complete frame damage, promotes a balanced participation of all windows, and uses dissipated energy as a tie-breaking performance criterion. Optimized geometries are finally re-evaluated with FEM. When the surrogate error exceeds the adopted tolerances, the new FEM result is added to the dataset and the surrogate models are retrained. The proposed framework provides a scalable route for FEM-consistent, damage-aware, and computationally efficient optimization of seismic energy dissipation devices.}
\keywords{Buckling-delayed shear link, seismic energy dissipation, surrogate modelling, machine learning, radial basis functions, Differential Evolution, FEM validation, TFDMap}
\jnlcitation{\cname{%
\author{Ramirez J.},
\author{Gonzalez J.},
\author{Lazaro L.},
\author{Rastellini F.},
\author{Bozzo G.},
\author{Bozzo L.}, and
\author{Irazabal J.}}.
\ctitle{Adaptive FEM-validated surrogate optimization of buckling-delayed shear-link dampers for seismic damage mitigation.}\cjournal{\it Journal.}\cvol{2026;00(00):1--18}.}
\maketitle
\renewcommand\thefootnote{\fnsymbol{footnote}}
\setcounter{footnote}{1}
\section{Introduction}\label{sec:introduction}
Shear-link beam (SLB) dampers are widely used as passive energy dissipation devices in seismic-resistant structures due to their ability to undergo stable inelastic deformations while limiting damage to the primary structural system. Their design aims at maximizing energy dissipation capacity and ductility under cyclic loading, while simultaneously controlling local damage levels to ensure reliability, durability, and predictable failure mechanisms. Among metallic devices, shear-link dampers are especially attractive because they concentrate inelastic deformation in replaceable components and can develop stable hysteretic loops under severe cyclic loading \cite{Malley1984,Okazaki2007}.
Experimental testing plays a fundamental role in characterizing the global hysteretic response of SLB dampers. However, laboratory campaigns present inherent limitations when addressing design optimization problems. Internal state variables such as local plastic strains, stress triaxiality, or damage indicators cannot be directly measured with sufficient spatial resolution. In addition, the high cost and logistical complexity of experimental programs severely restrict the number of geometric configurations that can be explored, making systematic optimization impractical. These limitations become even more relevant when the design objective is not only to increase global energy dissipation, but also to control where and how damage develops within the device.
Buckling-delayed shear-link (BDSL) dampers extend the conventional shear-link concept by incorporating a mechanical configuration that promotes shear-dominated behaviour while delaying local and global buckling. In these devices, energy dissipation is mainly concentrated in reduced-thickness zones, hereafter referred to as \ti{windows}, while the surrounding frame provides load transfer, stability, and confinement. This separation of roles creates a non-trivial design problem: thin windows may increase ductility and dissipative activation, but they may also localize damage; thick windows may increase strength, but they can transfer inelastic demand to the frame. Since complete damage of the frame would compromise the structural integrity of the device, frame damage must be penalized more severely than window damage. At the same time, the dissipative windows should work in a balanced manner, avoiding configurations in which a single window absorbs most of the deformation demand while the remaining windows remain underused.
In this context, high-fidelity finite element method (FEM) simulations constitute a powerful alternative. Advanced nonlinear FEM models enable detailed representation of cyclic plasticity, geometric nonlinearity, contact interactions, local instability, and damage evolution, providing access to both global response quantities and local indicators governing failure mechanisms. Moreover, FEM simulations allow the systematic generation of data across a wide range of geometric parameters, making them particularly suitable as a foundation for data-driven optimization strategies. In particular, stress-triaxiality-based indicators, such as the Triaxial Failure Damage Map (TFDMap), provide a useful post-processing measure of proximity to ductile failure under multiaxial stress states \cite{Rice1969,Bao2004,Wierzbicki2005,Bai2008,Rastellini2016}. In the present work, TFDMap is used as a damage-screening indicator rather than as a constitutive fracture model.
Despite these advantages, the direct use of FEM models within optimization loops remains computationally prohibitive. The cyclic response of SLB and BDSL dampers is characterized by pronounced hysteresis and strong memory effects, where the force--displacement relationship depends on the full loading history. Accurately capturing such behaviour requires fine temporal discretization and sophisticated constitutive models, resulting in simulation times that can span several hours per configuration. At the same time, purely data-driven approaches face intrinsic limitations, particularly poor extrapolation capabilities outside the training domain. This creates the need for surrogate models that are both computationally efficient and sufficiently reliable, while remaining consistent with the underlying physics captured by FEM simulations. Furthermore, understanding the relative influence of geometric variables on performance indicators remains challenging, especially when surrogate models behave as black-box predictors.
Over the past decades, FEM has been widely used to study seismic energy dissipation devices, providing detailed insight into nonlinear cyclic response, stiffness degradation, and local inelastic mechanisms \cite{Deng2014a,Deng2015}. It has also supported the optimization of these devices by enabling systematic exploration of geometric configurations and performance criteria under prescribed loading \cite{Deng2014,Deng2015a}. Simplified analytical and semi-empirical models have been proposed to reduce computational cost in practice \cite{Deng2014b}, while more recent parametric and simulation-based studies have examined the influence of geometric and material variables on damper performance \cite{Kim2022}.
FEM-based parametric analyses have been widely used to characterize the mechanical response of metallic dampers. Motamedi et al. \cite{Motamedi2018} investigated accordion metallic dampers through combined experimental and numerical analyses, assessing the influence of key geometric variables on stiffness, strength, and energy dissipation. Ghamari et al. \cite{Ghamari2021} studied I-shaped shear links in concentrically braced frames, and Xiong et al. \cite{Xiong2024} examined replaceable steel shear links with different short-length ratios, highlighting the strong influence of geometry on cyclic performance and failure modes.
Geometric optimization has also been extensively explored. Zhang et al. \cite{Zhang2017} proposed a Kriging-assisted framework to maximize hysteretic energy in coupling beam dampers. Farzampour et al. \cite{Farzampour2019} optimized butterfly-shaped shear links by maximizing the ratio between dissipated energy and plastic strain, while Khatibinia et al. \cite{Khatibinia2019,Khatibinia2021} developed efficient strategies for U-shaped dampers using FEM and surrogate models. Shi et al. \cite{Shi2019} introduced a non-parametric shape optimization framework for shear panel dampers, and Saleh et al. \cite{Saleh2024,Saleh2026} extended this line through topology optimization of shear-link configurations. More recent contributions include the hybrid cellular automata approach by Mendoza-Cuy et al. \cite{MendozaCuy2025} and the statistical optimization framework by Rios et al. \cite{Rios2025}. While these approaches expand the design space, they remain strongly dependent on high-fidelity FEM simulations, which limits their efficiency in large-scale design exploration.
Data-driven approaches have mainly focused on response or property prediction. Chan et al. \cite{Chan2015} used nonlinear autoregressive exogenous (NARX) models to reproduce hysteretic behaviour. Bae et al. \cite{Bae2020} developed models for low-cycle fatigue estimation, and Almasabha et al. \cite{Almasabha2022} predicted shear strength of short steel links using ML. Elgammal et al. \cite{Elgammal2024} modelled hysteretic restoring forces using data-driven approaches, while Hu et al. \cite{Hu2023} proposed explainable ML models for probabilistic prediction of buckling stress. Physics-informed approaches have also been explored, such as the PINN framework proposed by Hu et al. \cite{Hu2022}. Despite their potential, these methods remain primarily focused on prediction rather than on integration into geometry optimization frameworks.
Overall, most existing studies address either response prediction or the maximization of energy dissipation. This leaves a critical aspect insufficiently explored: the need to control local damage while maintaining adequate dissipative capacity. In practice, excessive local damage may compromise structural integrity, reduce durability, and lead to premature failure even when global energy dissipation is improved. For BDSL dampers, this issue is particularly important because the same global dissipated energy may correspond to very different local damage distributions: one geometry may distribute deformation among all windows, whereas another may concentrate damage in a single window or transfer inelastic demand to the frame.
The present work addresses this gap through a damage-aware surrogate-assisted optimization framework in which the objective is not only to maximize distortion or energy dissipation, but to balance dissipative performance and damage indicators derived from FEM simulations. The proposed methodology combines: (i) experimentally calibrated nonlinear FEM models used as ground truth; (ii) supervised ML and Radial Basis Function (RBF) surrogate models trained to predict local damage and distortion indicators; (iii) a Differential Evolution (DE) optimizer; and (iv) an adaptive FEM validation and retraining loop. The framework is physically grounded, as all models are trained on simulation data that capture both global response and local damage mechanisms.
In contrast to previous works focused on a single model or performance metric, this study provides a systematic comparison of surrogate techniques in terms of predictive accuracy and computational cost within the context of geometry optimization. The supervised surrogate set includes Random Forest (RF), Gradient Boosting Regression (GBR), XGBoost, Support Vector Regression (SVR), Multilayer Perceptron (MLP), and Gaussian Process Regression (GPR). Since preliminary calculations indicated that SVR and GPR often dominate the high-accuracy regime, RBF interpolants are also assessed as a computationally efficient alternative for fast optimization. The novelty is not the development of a new constitutive model, but the integration of FEM-calibrated damage indicators into an optimization workflow that explicitly distinguishes between damage in the windows and damage in the frame, encourages balanced window activation, and verifies the optimized geometry with a high-fidelity FEM simulation before accepting it.
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.
The BDSL device consists of a steel dissipative element connected to a surrounding load-transfer system through a mechanism that allows imposed in-plane displacement while avoiding the transmission of axial force. This kinematic condition is essential because it promotes a shear-dominated response in the dissipative element. The central element contains a set of reduced-thickness regions, or windows, where plastic deformation is intended to concentrate. The remaining material forms the surrounding frame, which stabilizes the device and transfers load but should not become the dominant dissipative component.
\begin{figure}[t]
\centering
\fbox{\parbox[c][0.25\textheight][c]{0.85\textwidth}{\centering Placeholder for BDSL device: structural implementation, specimen detail, and definition of geometric parameters.}}
\caption{Buckling-delayed shear-link damper and main geometric parameters. The optimized variables are the window thicknesses $t_{w,i}$, ordered from top to bottom. The surrounding frame is monitored separately because excessive frame damage can trigger global structural failure.}\label{fig:device}
\end{figure}
The dissipation mechanism is governed by controlled yielding of the windows under cyclic shear deformation. The surrounding frame must remain sufficiently robust to prevent the transfer of damage away from the windows. In practice, however, the interaction between shear deformation in the windows and bending or longitudinal deformation in the frame leads to non-uniform damage patterns. Therefore, the optimization problem cannot be reduced to maximizing either force or total energy. It must also control where the damage is produced and whether all windows participate in a comparable way.
The design variables considered in this work are the window thicknesses
where $W$ is the number of windows. The width and height identifiers of the device are denoted by $B$ and $H$, respectively. In the current implementation, three families are considered: two-window devices with $H=30$ cm, three-window devices with $H=45$ cm, and five-window devices with $H=60$ cm. The corresponding admissible thickness ranges are defined according to the geometry family and manufacturing constraints.
\subsection{Nonlinear FEM model}\label{subsec:fem_model}
The surrogate models are trained using data generated from high-fidelity three-dimensional FEM simulations. The FEM model is based on a previously calibrated numerical representation of the BDSL device. The steel dissipator is modelled using ASTM A36 steel, or its European equivalent S235JR, with cyclic plasticity represented through the Yoshida--Uemori model \cite{Yoshida2002,Jia2014}. The formulation accounts for material and geometric nonlinearities, contact interactions, and the boundary conditions imposed by the experimental setup. Linear eight-node hexahedral elements are used to discretize the steel component, providing a structured three-dimensional representation suitable for extracting local stress and strain fields.
\begin{figure}[t]
\centering
\fbox{\parbox[c][0.23\textheight][c]{0.85\textwidth}{\centering Placeholder for FEM model: mesh, actuator/connector system, contact conditions, and buckling-control planes.}}
\caption{Finite element model used to generate the ground-truth dataset. The calibrated model reproduces the experimental cyclic response and provides internal quantities that are not directly available from tests.}\label{fig:fem_model}
\end{figure}
The imposed displacement is applied through an actuator-like connector that transfers horizontal displacement while avoiding axial load transmission. Additional contact and confinement conditions are included to reproduce the experimental anti-buckling configuration. The cyclic loading protocol is displacement controlled and follows progressively increasing amplitudes, consistent with the experimental qualification of seismic energy dissipation devices.
\subsection{Calibration and validation}\label{subsec:calibration}
The FEM model is calibrated against experimental cyclic tests. The calibration process involves the material model, the assembled geometry, contact conditions, and boundary conditions. The validated model reproduces the main global experimental quantities, including hysteretic force--displacement loops, cumulative dissipated energy, and the skeleton curve. After this validation, the FEM model is used as a reliable numerical ground truth for configurations that have not been experimentally tested.
\begin{figure}[t]
\centering
\fbox{\parbox[c][0.23\textheight][c]{0.85\textwidth}{\centering Placeholder for FEM validation: hysteretic response, cumulative energy, and skeleton curve.}}
\caption{Experimental--numerical validation of the BDSL model. Once calibrated, the FEM model is used to generate the datasets for surrogate training and optimization.}\label{fig:fem_validation}
\end{figure}
The use of FEM data is essential because the optimization requires internal response variables that are difficult or impossible to measure experimentally at the necessary resolution. These variables include local TFDMap values in each window and in the frame, local plastic distortion, and the spatial distribution of deformation among dissipative regions.
\subsection{Damage and deformation indicators}\label{subsec:indicators}
The main damage indicator used in this work is the TFDMap, a triaxiality-based post-processing quantity derived from the comparison between the local stress--strain state and a triaxial failure curve. For each integration or monitoring point, the stress triaxiality and equivalent plastic strain are tracked during the cyclic simulation. The TFDMap value provides an indicator of proximity to the adopted failure envelope. In this paper, TFDMap is used as a damage-screening indicator rather than as a constitutive fracture model.
For optimization, TFDMap values are aggregated separately in the dissipative windows and in the frame. The maximum value in each window is denoted by $\TFD_i$, and the frame value by $\TFD_f$. The local shear distortion in each window is denoted by $\Exy_i$. This variable is used as a proxy for dissipative activation because larger stable distortion is associated with a higher capacity to absorb seismic energy. In addition, the volume associated with each window can be included through geometric volume factors, so that the contribution of each window is not evaluated solely from a point-wise strain value.
\section{Dataset generation and surrogate modelling}\label{sec:surrogates}
\subsection{Design of experiments}\label{subsec:doe}
The FEM campaign is designed to cover the admissible design domain of each device family. Latin Hypercube Sampling (LHS) is used to generate a homogeneous set of thickness combinations within the prescribed bounds. This sampling strategy is well suited to the current problem because each window thickness has a bounded interval and the number of design variables increases with the number of windows. The FEM simulations generated from the LHS design are stored in CSV files containing the input variables $t_{w,i}$ and the target outputs, including $\Exy_i$, $\TFD_i$, and $\TFD_f$.
The training process is iterative. For each configuration family, the first optimization iteration starts from an initial number of FEM rows. In the current implementation, the starting number of rows is 8 for two-window devices, 16 for three-window devices, and 64 for five-window devices. Additional FEM simulations can be appended in subsequent iterations when the optimized design does not satisfy the validation criteria.
\begin{table}[t]
\centering
\caption{Geometry families and surrogate input variables considered in the current implementation.}\label{tab:families}
\begin{tabular}{llll}
\toprule
Family & Height identifier $H$& Design variables & Bounds in current scripts \\
\midrule
2 windows & 30 cm &$t_{w,1},t_{w,2}$& 10--20 mm \\
3 windows & 45 cm &$t_{w,1},t_{w,2},t_{w,3}$& 5--14 mm \\
5 windows & 60 cm &$t_{w,1},\ldots,t_{w,5}$& 5--12 mm \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Supervised ML surrogate models}\label{subsec:ml_models}
A set of supervised regression models is trained independently for each output variable. The considered algorithms are RF, GBR, XGBoost, SVR, MLP, and GPR. The input vector is composed only of the window thicknesses of the corresponding device family. For each output, the model-selection routine evaluates all candidate algorithms and stores the best model as a serialized \texttt{joblib} file.
Hyperparameters are optimized using Bayesian search with 40 iterations. The scoring metrics are root mean squared error (RMSE), mean absolute error (MAE), and $R^2$. Since the available dataset size changes between geometry families and between adaptive iterations, the cross-validation strategy is selected automatically. Leave-One-Out validation is used for datasets with $N\leq20$, repeated five-fold cross-validation with five repetitions is used for $21\leq N\leq80$, and standard shuffled five-fold cross-validation is used for larger datasets. For small datasets, the hyperparameter search spaces of tree-based models are reduced to mitigate overfitting.
The best model is not selected exclusively from the lowest mean RMSE. Models whose RMSE lies within a 5\% band from the best RMSE are considered competitive; when the dispersion of fold-level RMSE is available for all competitive candidates, the model with the lowest coefficient of variation of RMSE is selected. This criterion favours not only accurate but also stable surrogates, which is relevant in optimization because a small local error near an active damage constraint can change the accepted geometry.
\begin{figure}[t]
\centering
\fbox{\parbox[c][0.24\textheight][c]{0.85\textwidth}{\centering Placeholder for supervised surrogate workflow: FEM dataset, adaptive CV, Bayesian hyperparameter search, model selection, and model persistence.}}
\caption{Supervised surrogate training workflow. A separate model is trained for each damage or distortion output. The selected model can differ from one output to another.}\label{fig:ml_workflow}
\end{figure}
Preliminary executions of this workflow indicate that SVR and GPR frequently provide the best accuracy for the considered datasets. This observation motivated the additional assessment of RBF interpolation as a simpler and faster surrogate strategy, especially for low-dimensional or well-sampled design spaces.
The RBF surrogate is implemented as a wrapper around the \texttt{scipy.interpolate.Rbf} interpolator. For each output variable, the model is trained using the same input features as the supervised ML surrogates. The current implementation uses a multiquadric radial basis function with zero smoothing and automatic shape parameter selection. The prediction model can be written generically as
where $\mathbf{x}_j$ are the FEM-sampled geometries, $\lambda_j$ are interpolation weights, and $\phi$ is the selected radial basis function.
RBF models are evaluated through Leave-One-Out validation. For each output, the procedure repeatedly removes one FEM sample, trains the RBF interpolant with the remaining samples, predicts the left-out sample, and computes RMSE, MAE, $R^2$, and dispersion metrics. This validation is especially useful because the number of FEM simulations is limited and because interpolation accuracy may be sensitive to the local density of training samples.
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 predictions are then combined into a scalar objective function to be minimized.
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.
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.
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.
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 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.
\begin{table}[t]
\centering
\caption{Window volume factors used to weight the distortion contribution in the objective function.}\label{tab:volume_factors}
\begin{tabular}{lll}
\toprule
Family & Width identifier $B$& Volume factors $V_i$\\
\section{Adaptive FEM validation and retraining}\label{sec:adaptive}
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.
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.
\begin{figure}[t]
\centering
\fbox{\parbox[c][0.24\textheight][c]{0.85\textwidth}{\centering Placeholder for adaptive loop: FEM dataset $\rightarrow$ surrogate training $\rightarrow$ DE optimization $\rightarrow$ FEM validation $\rightarrow$ accept or retrain.}}
\caption{Adaptive FEM validation and retraining loop. The optimized geometry is accepted only when prediction errors, objective error, and geometry stability criteria are simultaneously satisfied.}\label{fig:adaptive_loop}
The full numerical results are currently being generated. This section is therefore structured as the target results section to be completed once the final training, optimization, and FEM-validation outputs are available. The tables and figures indicated below should be filled with the final values obtained from the scripts.
\subsection{Predictive performance of supervised ML models}\label{subsec:planned_ml}
For each device family and each adaptive iteration, the supervised training script produces two CSV files: a summary table containing the selected model for each output and a detailed table containing the performance of every candidate model. The final paper should report RMSE, MAE, $R^2$, training time, and selected hyperparameters for the relevant outputs. Special attention should be paid to the variables entering the objective function: $\Exy_i$, $\TFD_i$, and $\TFD_f$.
\begin{table*}[t]
\centering
\caption{Template for reporting the supervised surrogate performance. Replace placeholders with the final results obtained from the training scripts.}\label{tab:ml_results_template}
\begin{tabular}{lllllll}
\toprule
Family & Output & Best model & RMSE & MAE &$R^2$& Training time [s] \\
Based on preliminary tests, it is expected that SVR and GPR will be among the most accurate models for several target variables. This should be confirmed quantitatively using the final cross-validation summaries.
\subsection{RBF validation and computational efficiency}\label{subsec:planned_rbf}
The RBF training script performs Leave-One-Out validation for each output and stores RMSE, MAE, $R^2$, and error-dispersion indicators. The comparison with supervised ML models should be presented in terms of both predictive accuracy and computational efficiency. RBF models are expected to be particularly competitive when the design space is low-dimensional and the FEM samples cover the domain adequately.
\begin{table*}[t]
\centering
\caption{Template for comparing supervised ML and RBF surrogates.}\label{tab:rbf_results_template}
\begin{tabular}{llllll}
\toprule
Family & Output & Best supervised model & Supervised RMSE & RBF LOO RMSE & Relative speed-up \\
For each surrogate type, device family, width identifier, window threshold, and adaptive iteration, the optimization scripts export a CSV file containing the optimal thicknesses, objective value, predicted distortions, predicted window TFDMap values, and predicted frame TFDMap. These results should be compared with the corresponding FEM validation values.
\begin{table*}[t]
\centering
\caption{Template for reporting optimized geometries and FEM validation.}\label{tab:optimization_template}
\begin{tabular}{llllllll}
\toprule
Family & Surrogate &$\TFD_w^{\star}$&$t_{w,1}$&$t_{w,2}$&$t_{w,3}$&$\max(\TFD_f)$ FEM & Accepted? \\
The expected interpretation is that the optimized designs should approach the target window TFDMap level while keeping the frame below its admissible threshold. A successful optimization should also avoid concentrating almost all damage in a single window. Therefore, figures showing the window-to-window distribution of $\TFD_i$ and $\Exy_i$ are recommended.
\begin{figure}[t]
\centering
\fbox{\parbox[c][0.25\textheight][c]{0.85\textwidth}{\centering Placeholder for final optimization results: predicted vs FEM $\TFD_i$, $\TFD_f$, $\Exy_i$, and objective value for ML and RBF surrogates.}}
\caption{Recommended final validation plot comparing surrogate predictions and FEM results for the optimized geometries.}\label{fig:validation_results}
\end{figure}
\section{Discussion}\label{sec:discussion}
The proposed methodology addresses a central limitation of direct FEM-based damper optimization: the high cost of evaluating many nonlinear cyclic simulations. By training surrogate models on FEM-calibrated datasets, the optimizer can explore the design domain efficiently while retaining a connection to the underlying mechanics of the device. The adaptive validation loop is essential because it prevents the optimizer from relying blindly on surrogate predictions in regions where the training data may be sparse.
The objective function also reflects the mechanical hierarchy of the BDSL device. Damage in the windows is not intrinsically undesirable; it is the intended mechanism for energy dissipation, provided that it remains below the adopted threshold and is reasonably distributed among windows. By contrast, damage in the frame is structurally more critical. The cubic frame penalty therefore gives the optimizer a clear preference for geometries that keep the frame safe, even if another geometry could dissipate slightly more energy. This is consistent with the design philosophy of replaceable dissipative devices, where controlled damage should be localized in predefined sacrificial regions.
The inclusion of RBF models is motivated by practical computational considerations. Supervised models such as SVR and GPR may provide high predictive accuracy, but they require hyperparameter optimization and cross-validation. RBF models, in contrast, can be trained with minimal overhead and can provide very fast predictions. Their performance is expected to depend strongly on the dimensionality of the design space and on the density of FEM sampling. Therefore, the final comparison should not identify a universally best surrogate, but rather establish when each surrogate class is preferable.
Some limitations should be highlighted. First, the methodology is only as reliable as the calibrated FEM model used to generate the data. Second, TFDMap is used as a damage indicator and not as a direct fracture model. Third, the current optimization considers window thicknesses as design variables; additional variables such as window height, spacing, frame thickness, or filler properties could be incorporated in future work but would require a larger FEM dataset. Finally, the optimized geometries should ultimately be validated experimentally before being used for design recommendations.
\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:
\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 FEM simulations provide the necessary ground truth for training because they supply internal indicators such as TFDMap and local distortion, which cannot be obtained experimentally with the same spatial resolution.
\item Supervised ML models and RBF interpolants provide complementary surrogate strategies. SVR and GPR are expected to be highly accurate based on preliminary tests, while RBF models offer a faster alternative for well-sampled design domains.
\item The proposed objective function explicitly encodes the desired mechanical behaviour: controlled window damage, severe penalization of frame damage, balanced participation of windows, and preference for higher dissipated energy when damage performance is comparable.
\item The adaptive FEM validation and retraining loop is a key component of the framework. A candidate geometry is accepted only if surrogate predictions match FEM results within the defined tolerances and if the optimized thicknesses remain stable between consecutive iterations.
\end{enumerate}
Once the final numerical campaign is completed, the placeholders in Section~\ref{sec:planned_results} should be replaced by the final model-selection tables, RBF validation metrics, optimized geometries, and FEM-validation comparisons.
%\backmatter
\bmsection*{Author contributions}
[To be completed according to the final author list.] Conceptualization: J. Ramirez, J. Gonzalez, F. Rastellini, G. Bozzo, L. Bozzo, J. Irazabal. Methodology: J. Ramirez, J. Gonzalez, F. Rastellini, J. Irazabal. Software and surrogate optimization: J. Ramirez, J. Gonzalez, J. Irazabal. FEM modelling and validation: J. Ramirez, F. Rastellini, G. Bozzo, L. Bozzo. Writing--original draft: J. Ramirez and J. Irazabal. Writing--review and editing: all authors.
\bmsection*{Acknowledgments}
The authors acknowledge the financial support of Project ACE100/23/000022, ``Edificacions resilients equipades amb dissipadores Shear Link'', funded by the Government of Catalonia through ACCIO and with the support of the Catalan Office for Climate Change, with the participation of Luis Bozzo Estructuras y Proyectos S.L. and the Centre Internacional de Metodes Numerics en Enginyeria (CIMNE).
\bmsection*{Financial disclosure}
None reported.
\bmsection*{Conflict of interest}
The authors declare no potential conflict of interests.
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\bmsection*{Supporting information}
Additional supporting information may include the FEM database, trained surrogate models, optimization scripts, and final FEM-validation simulations.
% \presentaddress{This is sample for present address text this is sample for present address text.}
%\fundingInfo{Text}
%\JELinfo{ejlje}
\abstract[Abstract]{The geometric optimization of \red{shear-link beam (SLB) dampers} under cyclic loading is challenged by strong nonlinear behavior and the need to control local damage mechanisms. High-fidelity finite element method (FEM) simulations provide detailed information on global response and internal damage, but their computational cost limits their use in design optimization. This work proposes a surrogate-assisted optimization framework based on FEM-generated datasets covering SLB configurations with varying geometric complexity. Supervised learning models and Radial Basis Function (RBF) approximations are used to evaluate the performance indicators and enable efficient exploration of the design space. The different surrogate approaches are \red{systematically compared in terms of predictive accuracy and computational cost}. An interpretability analysis based on SHapley Additive exPlanations (SHAP) is incorporated to quantify the influence of geometric variables in the response of the device. \red{Results demonstrate an effective trade-off between accuracy and efficiency, and provide insight into key design drivers, enabling fast and reliable optimization of SLB dampers.}}
\red{Shear-link beam (SLB) dampers} are widely used as passive energy dissipation devices in seismic-resistant structures due to their ability to undergo stable inelastic deformations while limiting damage to the primary structural system. Their design aims at maximizing energy dissipation capacity and ductility under cyclic loading, while simultaneously controlling local damage levels to ensure reliability, durability, and predictable failure mechanisms.
Experimental testing plays a fundamental role in characterizing the global hysteretic response of SLB dampers. However, laboratory campaigns present inherent limitations when addressing design optimization problems. Internal state variables such as local plastic strains, stress triaxiality, or damage indicators cannot be directly measured with sufficient spatial resolution. In addition, the high cost and logistical complexity of experimental programs severely restrict the number of geometric configurations that can be explored, making systematic optimization impractical.
In this context, high-fidelity finite element method (FEM) simulations constitute a powerful alternative. Advanced nonlinear FEM models enable detailed representation of cyclic plasticity, geometric nonlinearity, and damage evolution, providing access to both global response quantities and local indicators governing failure mechanisms. Moreover, FEM simulations allow the systematic generation of data across a wide range of geometric parameters, making them particularly suitable as a foundation for data-driven optimization strategies.
Despite these advantages, the direct use of FEM models within optimization loops remains computationally prohibitive. The cyclic response of SLB dampers is characterized by pronounced hysteresis and strong memory effects, where the force–displacement relationship depends on the full loading history. Accurately capturing such behavior requires fine temporal discretization and sophisticated constitutive models, resulting in simulation times that can span several hours per configuration. At the same time, purely data-driven approaches face intrinsic limitations, particularly poor extrapolation capabilities outside the training domain. This creates the need for surrogate models that are both computationally efficient and sufficiently reliable, while remaining consistent with the underlying physics captured by FEM simulations. Furthermore, understanding the relative influence of geometric variables on performance indicators remains challenging, especially when surrogate models behave as black-box predictors.
Over the past decades, FEM has been widely used to study seismic energy dissipation devices, providing detailed insight into nonlinear cyclic response, stiffness degradation, and local inelastic mechanisms \cite{Deng2014a, Deng2015}. It has also supported the optimization of these devices by enabling systematic exploration of geometric configurations and performance criteria under prescribed loading \cite{Deng2014, Deng2015a}. Simplified analytical and semi-empirical models have been proposed to reduce computational cost in practice \cite{Deng2014b}, while more recent parametric and simulation-based studies have examined the influence of geometric and material variables on damper performance \cite{Kim2022}.
FEM-based parametric analyses have been widely used to characterize the mechanical response of metallic dampers. Motamedi et al. \cite{Motamedi2018} investigated accordion metallic dampers through combined experimental and numerical analyses, assessing the influence of key geometric variables on stiffness, strength, and energy dissipation. Ghamari et al. \cite{Ghamari2021} studied I-shaped shear links in concentrically braced frames, and Xiong et al. \cite{Xiong2024} examined replaceable steel shear links with different short-length ratios, highlighting the strong influence of geometry on cyclic performance and failure modes.
Geometric optimization has also been extensively explored. Zhang et al. \cite{Zhang2017} proposed a Kriging-assisted framework to maximize hysteretic energy in coupling beam dampers. Farzampour et al. \cite{Farzampour2019} optimized butterfly-shaped shear links by maximizing the ratio between dissipated energy and plastic strain, while Khatibinia et al. \cite{Khatibinia2019, Khatibinia2021} developed efficient strategies for U-shaped dampers using FEM and surrogate models. Shi et al. \cite{Shi2019} introduced a non-parametric shape optimization framework for shear panel dampers, and Saleh et al. \cite{Saleh2024, Saleh2026} extended this line through topology optimization of shear-link configurations. More recent contributions include the hybrid cellular automata approach by Mendoza-Cuy et al. \cite{MendozaCuy2025} and the statistical optimization framework by Rios et al. \cite{Rios2025}. While these approaches expand the design space, they remain strongly dependent on high-fidelity FEM simulations, which limits their efficiency in large-scale design exploration.
Data-driven approaches have mainly focused on response or property prediction. Chan et al. \cite{Chan2015} used nonlinear autoregressive exogenous (NARX) models to reproduce hysteretic behavior. Bae et al. \cite{Bae2020} developed models for low-cycle fatigue estimation, and Almasabha et al. \cite{Almasabha2022} predicted shear strength of short steel links using ML. Elgammal et al. \cite{Elgammal2024} modeled hysteretic restoring forces using data-driven approaches, while Hu et al. \cite{Hu2023} proposed explainable ML models for probabilistic prediction of buckling stress. Physics-informed approaches have also been explored, such as the PINN framework proposed by Hu et al. \cite{Hu2022}. Despite their potential, these methods remain primarily focused on prediction rather than on integration into geometry optimization frameworks.
Overall, most existing studies address either response prediction or the maximization of energy dissipation. This leaves a critical aspect insufficiently explored: the need to control local damage while maintaining adequate dissipative capacity. In practice, excessive local damage may compromise structural integrity, reduce durability, and lead to premature failure even when global energy dissipation is improved.
The present work addresses this gap through a damage-aware optimization framework in which the objective is not only to maximize \red{distortion or energy} dissipation, but to balance dissipative performance and minimized damage indicators derived from FEM simulations. The proposed methodology combines high-fidelity FEM-generated datasets with multiple surrogate strategies, including supervised learning models and radial basis functions, to efficiently explore the design space of SLB dampers with different geometric complexities. The framework is physically grounded, as all models are trained on simulation data that capture both global response and local damage mechanisms.
\red{In contrast to previous works focused on a single model or performance metric, this study provides a systematic comparison of surrogate techniques in terms of predictive accuracy and computational cost within the context of geometry optimization.} An interpretability analysis based on SHapley Additive exPlanations (SHAP) is also incorporated to quantify the influence of geometric variables on the predicted performance metrics, providing insight into the governing design drivers. In addition, the framework includes an adaptive validation and retraining loop in which selected optimal candidates are re-evaluated with FEM and iteratively incorporated into the training dataset.
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, while enhancing model interpretability for engineering decision-making.
To summarize, the main contributions of this work are:
\begin{itemize}
\item Generation of high-fidelity FEM datasets for SLB dampers with increasing geometric complexity.
\item Geometric optimization using surrogate models, including supervised learning techniques and radial basis functions.
\item Systematic comparison of surrogate strategies in terms of accuracy and computational cost.
\item Interpretability analysis using SHAP to assess the influence of geometric variables.
\item Adaptive validation and retraining strategy based on additional FEM simulations.
\end{itemize}
\subsection{Contributions of this work}\label{sec1_4}
This paper proposes a surrogate-assisted optimization framework for SLB dampers subjected to cyclic loading, combining FEM-calibrated datasets with data-driven modeling techniques. The main contributions of the work can be summarized as follows:
\begin{itemize}
\item Generation of high-fidelity FEM datasets for SLB dampers with increasing geometric complexity, considering configurations with two, three, and five windows.
\item Geometric optimization based on surrogate models, including supervised learning algorithms—such as Random Forest (RF), Gradient Boosting Trees (GBT), XGBoost, Support Vector Regression (SVR), and Multilayer Perceptrons (MLP)—as well as Radial Basis Function (RBF) approximations.
\item Systematic comparison of the different surrogate strategies in terms of predictive accuracy and computational cost within the optimization process.
\item Interpretability analysis using SHapley Additive exPlanations (SHAP) to quantify the influence of geometric variables on performance indicators, providing insight into the governing design parameters.
\item Adaptive validation and feedback strategy, where selected optimized designs are verified through additional FEM simulations and incorporated into the dataset when necessary.
\end{itemize}
\section{Numerical simulation and ground truth generation}\label{sec2}
This section describes the finite element modeling strategy adopted to generate the reference datasets used in this study. The numerical simulations are designed to accurately capture the cyclic hysteretic behavior of shear-link beam (SLB) dampers, as well as relevant internal indicators associated with damage and energy dissipation. The resulting FEM outputs constitute the ground truth employed for training, validation, and comparison of the surrogate models considered in the subsequent sections.
\subsection{Geometry of SLB dampers}\label{sec2_1}
The study considers SLB dampers with increasing geometric complexity, focusing on configurations featuring two, three, and five shear windows. These layouts are representative of typical design solutions and allow the influence of geometric complexity on both structural response and surrogate modeling performance to be systematically assessed.
The geometry of each damper is parameterized through a reduced set of design variables. In particular, the thicknesses of the shear windows, denoted as tw, are treated as independent parameters, enabling local stiffness and plastic deformation patterns to be controlled. Additionally, the thickness of the surrounding frame, tf, is included as a global design variable, governing the overall stiffness and load transfer capacity of the device. This parameterization provides sufficient flexibility to explore a wide range of feasible geometries while maintaining a manageable dimensionality of the design space.
All numerical simulations are performed under displacement-controlled loading conditions, where the relative displacement is imposed by an idealized actuator. This choice ensures stable numerical convergence and allows the hysteretic response of the dampers to be directly characterized in terms of force-displacement relationships.
The applied loading histories follow cyclic protocols with progressively increasing displacement amplitudes, which are commonly adopted in the assessment of energy dissipation devices. Such protocols are particularly suitable for capturing stiffness degradation, strength evolution, and cumulative damage effects under repeated loading. The specific sequence of displacement amplitudes is adapted to the height of each damper configuration, ensuring comparable deformation demands across the different geometric layouts.
\subsection{Finite element outputs and quantities of interest}\label{sec2_3}
For each simulated configuration, the FEM model provides both global and local response quantities. At the global level, the primary quantity of interest is the hysteretic force-displacement curve, which characterizes the energy dissipation capacity and overall nonlinear behavior of the damper under cyclic loading.
In addition to the global response, internal indicators related to damage and deformation localization are extracted from the simulations. In particular, the triaxial failure damage map (TFDMap) is employed to quantify the proximity to failure at critical regions of the device. Furthermore, a local distortion measure associated with the deformation of the shear windows is evaluated, as it is directly related to the activation of plastic mechanisms responsible for energy dissipation. These quantities provide essential information for defining damage-based constraints in the subsequent optimization process.
\subsection{Design of the simulation dataset}\label{sec2_4}
The numerical campaign is designed to ensure a systematic and sufficiently uniform exploration of the geometric design space. The values of the geometric parameters are sampled using a quasi-random strategy, enabling efficient coverage of the admissible domain while avoiding clustering of samples in restricted regions.
The resulting dataset is explicitly divided into two subsets. A first subset is used for training the surrogate models, while a second subset is reserved for validation purposes and includes configurations not seen during the training phase. This separation allows the generalization capability of the surrogate models to be rigorously assessed and provides a consistent basis for evaluating their performance in both prediction and optimization tasks.
% \caption{This is the sample figure caption.\label{fig2}}
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% \section{Hysteresis prediction using LSTM models}\label{sec3}
% Accurate prediction of hysteretic force-displacement curves is a key requirement for surrogate-based optimization of SLB dampers. The nonlinear cyclic response of these devices is inherently history-dependent, as stiffness, strength, and energy dissipation mechanisms evolve with the accumulated deformation and loading direction. In this work, Long Short-Term Memory (LSTM) neural networks are adopted to model such behavior, as they are specifically designed to capture temporal dependencies and memory effects in sequential data.
% \subsection{Problem formulation with memory effects}\label{sec3_1}
% The objective of the hysteresis prediction task is to estimate the resisting force
% F(t)
% F(t) at a given time step as a function of the displacement history and the geometric characteristics of the damper. Unlike monotonic or memoryless systems, the force response of SLB dampers under cyclic loading cannot be uniquely determined from the instantaneous displacement alone.
% Two key aspects justify the explicit inclusion of memory effects. First, the response depends on the loading direction, as unloading and reloading paths differ from the initial loading branch. Second, the amplitude and sequence of previous cycles influence stiffness degradation, plastic deformation accumulation, and energy dissipation capacity. As a result, an appropriate formulation must account for both the temporal evolution of the displacement signal and the geometric parameters governing the mechanical behavior of the device.
% \subsection{Sequential representation of the input signal}\label{sec3_2}
% To enable sequence-based learning, the input to the LSTM model is formulated as a time series of feature vectors. At each discrete time step t, the input vector is defined as $x_t=[u_t, \dot{u}_t, c_t, A_t, \theta]$, where $u_t$ denotes the imposed displacement, $\dot{u}_t$ its associated velocity or loading direction indicator, $c_t$ the cycle index, $A_t$ the maximum displacement amplitude reached in the current cycle, and $\theta$ the set of geometric parameters defining the damper configuration. This representation combines instantaneous kinematic information with cycle-level descriptors and fixed geometric features.
% The output of the network is the corresponding resisting force $F_t$. Two prediction strategies are considered: an autoregressive formulation, in which the force is predicted sequentially at each time step, and a many-to-many formulation, where the full force-displacement curve is predicted as a sequence given the input displacement history.
% Different LSTM-based architectures are explored to assess their suitability for hysteresis prediction. A standard many-to-many LSTM architecture is employed as a baseline model, enabling direct mapping between input and output sequences of equal length. This architecture is particularly effective for reproducing complete hysteretic loops once the loading protocol is specified.
% To improve generalization across different loading histories, an encoder-decoder architecture is also considered. In this configuration, the encoder processes the input displacement sequence and compresses the relevant temporal information into a latent representation, which is then decoded into the corresponding force response. This approach facilitates the handling of variable-length sequences and differing loading protocols.
% Optionally, attention mechanisms can be incorporated into the decoder to enhance interpretability and to emphasize critical portions of the loading history, such as displacement reversals or peak amplitudes. Although not essential for all cases, attention-based models provide additional insight into the temporal regions that most strongly influence the predicted response.
% \subsection{Training and validation strategy}\label{sec3_4}
% Prior to training, all input and output variables are normalized to ensure numerical stability and to prevent scale imbalances from biasing the learning process. The networks are trained using loss functions based on mean absolute error (MAE) or mean squared error (MSE), optionally augmented with additional penalties to emphasize accurate prediction of force peaks, which are particularly relevant for energy dissipation assessment.
% To rigorously evaluate generalization performance, the dataset is split at the level of complete simulations rather than individual time steps. Entire hysteretic responses corresponding to specific geometric configurations are reserved for validation, ensuring that no temporal segments of a given simulation appear simultaneously in both training and validation sets. This strategy prevents data leakage and provides a realistic assessment of predictive capability for unseen configurations.
% \subsection{Advantages and limitations}\label{sec3_5}
% The use of LSTM models for hysteresis prediction offers several advantages. By construction, LSTMs explicitly capture memory effects and temporal dependencies, enabling high-fidelity reproduction of complex hysteretic behavior under cyclic loading. This makes them particularly well suited for modeling systems where the response is governed by accumulated deformation and loading history.
% However, these benefits come at the cost of increased model complexity and training time. LSTM networks typically require larger datasets and careful tuning to achieve stable performance, and their extrapolation capabilities remain limited when applied outside the range of training data. Consequently, while LSTMs provide an accurate tool for hysteresis prediction, they are best employed as part of a broader surrogate-based framework rather than as standalone optimization models.
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% \end{quote}
\section{Geometric optimization using surrogate models}\label{sec4}
Within each optimization iteration, a large number of candidate geometries are evaluated using the surrogate models, allowing rapid estimation of objective and constraint functions. Based on these surrogate predictions, the most promising candidate designs are identified and retained for further exploration. This surrogate-assisted DE framework enables efficient identification of optimal or near-optimal geometries while drastically reducing the number of costly FEM simulations required.
\subsection{Problem formulation}\label{sec4_1}
The geometric optimization problem is formulated in terms of a set of design variables and performance objectives. The design variables are defined as $\theta={t_{wi}, t_f}$, where
$t_{wi}$ denotes the thickness of the i-th shear window and $t_f$ represents the thickness of the surrounding frame. This parameterization allows local deformation mechanisms and global stiffness characteristics to be simultaneously controlled.
The optimization objectives are twofold. On the one hand, the goal is to minimize damage indicators derived from the FEM simulations, thereby ensuring structural integrity and limiting the risk of premature failure. On the other hand, the objective is to maximize the deformation capacity of the device, quantified through distortion measures directly related to energy dissipation under cyclic loading. These competing objectives reflect the fundamental trade-off between performance and safety in the design of energy dissipation devices.
To ensure physically admissible solutions, constraints are imposed based on damage-related quantities. Candidate designs exceeding prescribed thresholds of damage indicators are considered infeasible and are penalized or excluded from the optimization process. This constraint-based formulation ensures that optimized geometries satisfy both performance and reliability requirements.
\subsection{Surrogate models}\label{sec4_2}
The computational cost of direct FEM evaluations motivates the use of surrogate models to approximate the relationship between geometric parameters and response quantities. Two classes of surrogate approaches are considered in this study.
The first class comprises supervised machine learning models, including Random Forest (RF), Gradient Boosting Trees (GBT), XGBoost, Support Vector Regression (SVR), and Multilayer Perceptrons (MLP). These models are trained on the FEM-generated datasets to predict damage indicators and distortion-related quantities as functions of the geometric design variables. Their flexibility and ability to capture nonlinear relationships make them suitable for modeling complex response surfaces.
As an alternative, Radial Basis Function (RBF) approximations are employed as a direct interpolation-based surrogate. RBF models provide smooth functional representations of the design space and are characterized by very low computational cost at inference time. Their performance, however, depends strongly on the density and coverage of the available data, particularly in higher-dimensional parameter spaces.
\subsection{Optimization algorithm}\label{sec4_3}
The exploration of the design space is performed using the Differential Evolution (DE) algorithm, a population-based evolutionary optimization method well suited for nonlinear and nonconvex problems. DE operates through iterative mutation, crossover, and selection steps, enabling robust search without requiring gradient information.
Within each optimization iteration, a large number of candidate geometries are evaluated using the surrogate models, allowing rapid estimation of objective and constraint functions. Based on these surrogate predictions, the most promising candidate designs are identified and retained for further exploration. This surrogate-assisted DE framework enables efficient identification of optimal or near-optimal geometries while drastically reducing the number of costly FEM simulations required.
\section{Numerical validation and adaptive feedback}\label{sec5}
To assess the reliability of the surrogate-assisted optimization framework, selected optimized designs are subjected to additional finite element simulations. This validation stage serves a dual purpose: verifying the accuracy of surrogate predictions for previously unseen configurations and providing a mechanism for adaptive refinement of the surrogate models when necessary.
\subsection{FEM validation of optimized designs}\label{sec5_1}
A subset of candidate geometries identified as optimal by the surrogate-based optimization process is re-evaluated using high-fidelity FEM simulations. For each selected design, the FEM results are directly compared with the corresponding surrogate predictions.
The comparison focuses on both global and local response quantities. At the global level, the predicted hysteretic force-displacement curves are evaluated against their FEM counterparts to assess the accuracy of the surrogate models in reproducing energy dissipation and cyclic response characteristics. At the local level, damage-related indicators extracted from the FEM simulations are compared with the surrogate-based estimates, enabling verification of the constraint satisfaction and safety margins associated with the optimized designs.
This validation step provides a quantitative measure of the surrogate model performance in the vicinity of the optimal solutions, which is critical for ensuring the practical applicability of the proposed optimization framework.
An adaptive feedback mechanism is introduced to enhance the robustness of the surrogate models. A predefined admissible error criterion is established for both global response quantities and damage indicators. When the discrepancy between surrogate predictions and FEM results for a validated design exceeds this threshold, the surrogate model is considered insufficiently accurate in that region of the design space.
In such cases, the newly generated FEM simulation is incorporated into the existing dataset, and the surrogate models are retrained to account for the additional information. This iterative enrichment of the training data progressively improves surrogate accuracy in regions of interest, particularly near optimal or constraint-active designs. The adaptive retraining strategy thus ensures that the optimization process remains reliable while minimizing the total number of high-fidelity FEM simulations required.
\section{Comparative assessment of supervised surrogate models and RBF approaches}\label{sec6}
This section presents a systematic comparison between supervised machine learning surrogate models and Radial Basis Function (RBF) approximations when employed within the proposed optimization framework. The comparison focuses on predictive accuracy, computational efficiency, and practical usability, with the objective of identifying the most suitable surrogate strategy for different design scenarios.
\subsection{Predictive accuracy}\label{sec6_1}
The predictive performance of the surrogate models is first assessed in terms of their ability to reproduce FEM-derived damage indicators and performance-related quantities. For each surrogate approach, the error in damage prediction is evaluated using validation cases not included in the training dataset. Particular attention is given to configurations near constraint boundaries, as accurate damage estimation in these regions is critical for reliable optimization.
In addition to damage-related quantities, errors in performance metrics associated with deformation capacity and energy dissipation are analyzed. The results show that supervised learning models generally provide higher predictive accuracy across the explored design space, especially in cases involving complex geometric configurations or strong nonlinear interactions between design variables. RBF models, while capable of accurately interpolating within densely sampled regions, exhibit increased sensitivity to data sparsity and dimensionality, which can lead to larger prediction errors in less populated areas of the design space.
\subsection{Computational cost}\label{sec6_2}
Computational efficiency is evaluated by analyzing both training and inference times. Supervised learning models typically require a non-negligible training phase, the cost of which depends on model complexity, hyperparameter tuning, and dataset size. However, once trained, their inference time remains relatively low and largely independent of the number of training samples.
In contrast, RBF models are characterized by minimal training effort and extremely fast inference, making them attractive for scenarios where rapid evaluation of candidate designs is required. Nevertheless, the computational cost of RBF approaches tends to increase with the dimensionality of the problem and the number of basis functions needed to adequately represent the response surface. As a result, scalability becomes a limiting factor for RBF models when applied to high-dimensional design spaces or when extensive datasets are employed.
\subsection{Practical implications and model selection}\label{sec6_3}
The comparative results highlight that the choice of surrogate model should be guided by the specific requirements of the design problem. For low-dimensional problems with limited but well-distributed datasets, RBF models offer an efficient and reliable solution, providing fast evaluations with minimal implementation complexity. Conversely, for higher-dimensional problems or scenarios involving complex nonlinear interactions, supervised learning models demonstrate superior robustness and accuracy, albeit at the cost of increased training time.
When rapid design iterations are required and computational resources are constrained, RBF approaches may be preferable, provided that the design space is adequately sampled. In contrast, when predictive fidelity and generalization are critical—particularly near damage thresholds—supervised surrogate models represent a more reliable choice. These insights provide practical guidance for selecting appropriate surrogate strategies in surrogate-assisted optimization of SLB dampers.
\section{Discussion}\label{sec7}
The results presented in this study highlight the inherent trade-offs involved in surrogate-assisted optimization of SLB dampers under cyclic loading. In particular, the balance between predictive accuracy, computational efficiency, and robustness emerges as a central aspect governing the selection and deployment of surrogate models within practical design workflows.
A clear trade-off between accuracy and computational speed is observed across the different surrogate strategies. Supervised learning models generally provide higher predictive fidelity, especially in regions of the design space characterized by strong nonlinearities and near active damage constraints. This improved accuracy comes at the expense of increased training effort and model complexity. In contrast, Radial Basis Function approaches offer extremely fast evaluation times and minimal training overhead, but their performance is strongly dependent on the density and distribution of the available data, as well as on the dimensionality of the design space. As a consequence, robustness with respect to extrapolation and sparse sampling becomes a limiting factor for purely interpolative methods.
The role of feature engineering is particularly relevant in this context. By incorporating physically meaningful descriptors—such as geometric parameters directly linked to deformation mechanisms and damage evolution—supervised surrogate models are able to capture complex response patterns with relatively compact representations. This contrasts with purely interpolative approaches, where predictive capability relies primarily on local proximity in the input space and rapidly degrades as dimensionality increases. The results indicate that informed feature selection, grounded in the underlying mechanics of the problem, is a key enabler for achieving reliable surrogate performance in high-dimensional optimization tasks.
The use of LSTM models for hysteresis prediction further complements the surrogate-based optimization framework. LSTMs are shown to be highly effective in reproducing history-dependent force-displacement relationships, explicitly accounting for memory effects that cannot be captured by static surrogate models. However, their computational cost and data requirements make them less suitable as direct optimization surrogates. Instead, LSTMs are best regarded as a complementary tool, providing accurate hysteresis predictions and insight into cyclic response behavior, while supervised or interpolative surrogates are employed for fast evaluation within optimization loops.
From a broader perspective, the proposed framework has important implications for computer-aided design in structural and seismic engineering. By combining high-fidelity FEM simulations, data-driven surrogates, and evolutionary optimization, the methodology enables systematic exploration of complex design spaces that would otherwise be inaccessible using conventional approaches. This paradigm supports more informed design decisions, facilitates performance-based optimization under realistic loading conditions, and contributes to the development of efficient and reliable workflows for the design of energy dissipation devices in seismic applications.
\section{Conclusions}\label{sec8}
This study has presented a surrogate-assisted framework for the geometric optimization of shear-link beam (SLB) dampers subjected to cyclic loading, combining high-fidelity finite element simulations with data-driven modeling and evolutionary optimization techniques. Based on the results obtained, the following conclusions can be drawn:
High-fidelity FEM simulations provide a reliable and information-rich ground truth for characterizing both global hysteretic response and internal damage-related quantities, enabling the systematic generation of datasets suitable for surrogate modeling and optimization.
Long Short-Term Memory (LSTM) neural networks are shown to be effective in predicting hysteretic force-displacement curves, successfully capturing the memory effects and history dependence inherent to cyclic structural response.
Surrogate-based optimization strategies significantly reduce computational cost compared to direct FEM-driven approaches, making accelerated geometric optimization of SLB dampers feasible without compromising predictive fidelity.
Radial Basis Function (RBF) models offer an ultra-fast surrogate alternative when the design space is well sampled and of limited dimensionality, whereas supervised learning models provide greater robustness and accuracy for higher-dimensional or more complex optimization problems.
Clear practical recommendations can be established regarding surrogate selection, depending on dataset size, design space complexity, and computational constraints, supporting informed decision-making in engineering design workflows.
Overall, the proposed methodology demonstrates the potential of integrating FEM-calibrated surrogates, sequence-based learning, and evolutionary optimization to enable efficient and reliable design of energy dissipation devices for seismic applications.
% \paragraph{Fourth level head}
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% \begin{boxwithhead}
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% \begin{table*}[!ht]%
% \caption{This is sample table caption.\label{tab1}}
% col1 text & col2 text & col3 text & 12.34 & col5 text\tnote{1} \\
% col1 text & col2 text & col3 text & \hphantom{0}1.62 & col5 text\tnote{2} \\
% col1 text & col2 text & col3 text & 51.809 & col5 text \\
% \bottomrule
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% \begin{tablenotes}%%[341pt]
% \item[$^{\rm a}$] Example for a first table footnote.
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% \end{table*}
% \end{center}
% Curabitur tellus magna, porttitor a, commodo a, commodo in, tortor. Donec interdum (Table~\ref{tab1}). Praesent scelerisque. Maecenas posuere sodales odio. Vivamus metus lacus, varius quis, imperdiet quis, rhoncus a, turpis. Etiam ligula arcu,
% elementum a, venenatis quis, sollicitudin sed, metus. Donec nunc pede, tincidunt in, venenatis vitae, faucibus vel,
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% Maecenas viverra nulla in massa (Table~\ref{tab2}).
% Nulla in ipsum. Praesent eros nulla, congue vitae, euismod ut, commodo a, wisi. Pellentesque habitant morbi
% tristique senectus et netus et malesuada fames ac turpis egestas. Aenean nonummy magna non leo. Sed felis erat,
% ullamcorper in, dictum non, ultricies ut, lectus. Proin vel arcu a odio lobortis euismod. Vestibulum ante ipsum primis
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% venenatis arcu wisi vel nisl. Vestibulum diam. Aliquam pellentesque, augue quis sagittis posuere, turpis lacus congue
% quam, in hendrerit risus eros eget felis. Maecenas eget erat in sapien mattis porttitor. Vestibulum porttitor. Nulla
% facilisi. Sed a turpis eu lacus commodo facilisis. Morbi fringilla, wisi in dignissim interdum, justo lectus sagittis dui, et
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% Nulla egestas. Curabitur a leo. Quisque egestas wisi eget nunc. Nam feugiat lacus vel est. Curabitur consectetuer.
% \begin{table*}[!t]%
% \centering %
% \caption{This is sample table caption.\label{tab2}}%
% col1 text & col2 text & col3 text & col4 text & col5 text\tnote{$^\dagger$} \\
% col1 text & col2 text & col3 text & col4 text & col5 text \\
% col1 text & col2 text & col3 text & col4 text & col5 text\tnote{$^\ddagger$} \\
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% \begin{tablenotes}
% \item[$^\dagger$] Example for a first table footnote.
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% Below is the example for bulleted list. Below is the example for bulleted list. Below is the example for bulleted list. Below is the example for bulleted list. Below is the example for bulleted list. Below is the example for bulleted list\footnote{This is an example for footnote.}:
% \begin{itemize}
% \item bulleted list entry sample bulleted list entry , sample list entry text.
% \item bulleted list entry sample bulleted list entry. bulleted list entry sample bulleted list entry. bulleted list entry sample bulleted list entry.
% \item bulleted list entry sample bulleted list entry , bulleted list entry sample bulleted list entry , sample list entry text. bulleted list entry sample bulleted list entry.
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% \end{itemize}
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% eu, sodales cursus, magna. Donec eu purus. Quisque vehicula, urna sed ultricies auctor, pede lorem egestas dui, et
% convallis elit erat sed nulla. Donec luctus. Curabitur et nunc. Aliquam dolor odio, commodo pretium, ultricies non,
% pharetra in, velit. Integer arcu est, nonummy in, fermentum faucibus, egestas vel, odio.
% Sed commodo posuere pede. Mauris ut est. Ut quis purus. Sed ac odio. Sed vehicula hendrerit sem. Duis non
% odio. Morbi ut dui. Sed accumsan risus eget odio. In hac habitasse platea dictumst. Pellentesque non elit. Fusce
% sed justo eu urna porta tincidunt. Mauris felis odio, sollicitudin sed, volutpat a, ornare ac, erat. Morbi quis dolor. Donec pellentesque, erat ac sagittis semper, nunc dui lobortis purus, quis congue purus metus ultricies tellus. Proin
% et quam. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos hymenaeos. Praesent sapien
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% et quam. Below is the example for description list. Below is the example for description list. Below is the example for description list. Below is the example for description list. Below is the example for description list. Below is the sample for description list. Below is the example for description list. Below is the example for description list. Below is the example for description list. Below is the example for description list. Below is the example for description list:\vskip12pt
% \item First level numbered list entry, sample numbered list entry.
% \item First numbered list entry, sample numbered list entry. Numbered list entry, sample numbered list entry. Numbered list entry, sample numbered list entry.
% \begin{enumerate}[a.]
% \item Second level alpabetical list entry. Second level alpabetical list entry. Second level alpabetical list entry. Second level alpabetical list entry.
% \item Second level alpabetical list entry. Second level alpabetical list entry.
% \begin{enumerate}[i.]
% \item Third level lowercase roman numeral list entry. Third level lowercase roman numeral list entry. Third level lowercase roman numeral list entry.
% \item Third level lowercase roman numeral list entry. Third level lowercase roman numeral list entry.
% \end{enumerate}
% \item Second level alpabetical list entry. Second level alpabetical list entry.
% \end{enumerate}
% \item First level numbered list entry, sample numbered list entry. Numbered list entry, sample numbered list entry. Numbered list entry.
% \item Another first level numbered list entry, sample numbered list entry. Numbered list entry, sample numbered list entry. Numbered list entry.
% \end{enumerate}
% \noindent\textbf{un-numbered list items sample:}
% \section{Examples for enunciations}\label{sec15}
% \begin{theorem}\label{thm1}
% Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text. Example theorem text.
% \end{theorem}
% Quisque ullamcorper placerat ipsum. Cras nibh. Morbi vel justo vitae lacus tincidunt ultrices. Lorem ipsum dolor sit
% amet, consectetuer adipiscing elit. In hac habitasse platea dictumst. Integer tempus convallis augue. Etiam facilisis.
% Nunc elementum fermentum wisi. Aenean placerat. Ut imperdiet, enim sed gravida sollicitudin, felis odio placerat
% quam, ac pulvinar elit purus eget enim. Nunc vitae tortor. Proin tempus nibh sit amet nisl. Vivamus quis tortor
% vitae risus porta vehicula.
% Fusce mauris. Vestibulum luctus nibh at lectus. Sed bibendum, nulla a faucibus semper, leo velit ultricies tellus, ac
% venenatis arcu wisi vel nisl. Vestibulum diam. Aliquam pellentesque, augue quis sagittis posuere, turpis lacus congue
% quam, in hendrerit risus eros eget felis. Maecenas eget erat in sapien mattis porttitor. Vestibulum porttitor. Nulla
% facilisi. Sed a turpis eu lacus commodo facilisis. Morbi fringilla, wisi in dignissim interdum, justo lectus sagittis dui, et
% vehicula libero dui cursus dui. Mauris tempor ligula sed lacus. Duis cursus enim ut augue. Cras ac magna. Cras nulla.
% Nulla egestas. Curabitur a leo. Quisque egestas wisi eget nunc. Nam feugiat lacus vel est. Curabitur consectetuer.
% \begin{proposition}
% Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text. Example proposition text.
% \end{proposition}
% Nulla malesuada porttitor diam. Donec felis erat, congue non, volutpat at, tincidunt tristique, libero. Vivamus
% viverra fermentum felis. Donec nonummy pellentesque ante. Phasellus adipiscing semper elit. Proin fermentum massa
% ac quam. Sed diam turpis, molestie vitae, placerat a, molestie nec, leo. Maecenas lacinia. Nam ipsum ligula, eleifend
% at, accumsan nec, suscipit a, ipsum. Morbi blandit ligula feugiat magna. Nunc eleifend consequat lorem. Sed lacinia
% nulla vitae enim. Pellentesque tincidunt purus vel magna. Integer non enim. Praesent euismod nunc eu purus. Donec
% bibendum quam in tellus. Nullam cursus pulvinar lectus. Donec et mi. Nam vulputate metus eu enim. Vestibulum
% pellentesque felis eu massa.
% Quisque ullamcorper placerat ipsum. Cras nibh. Morbi vel justo vitae lacus tincidunt ultrices. Lorem ipsum dolor sit
% amet, consectetuer adipiscing elit. In hac habitasse platea dictumst. Integer tempus convallis augue. Etiam facilisis.
% Nunc elementum fermentum wisi. Aenean placerat. Ut imperdiet, enim sed gravida sollicitudin, felis odio placerat
% quam, ac pulvinar elit purus eget enim. Nunc vitae tortor. Proin tempus nibh sit amet nisl. Vivamus quis tortor
% vitae risus porta vehicula.
% \begin{definition}
% Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text. Example definition text.
% \end{definition}
% Sed commodo posuere pede. Mauris ut est. Ut quis purus. Sed ac odio. Sed vehicula hendrerit sem. Duis non
% odio. Morbi ut dui. Sed accumsan risus eget odio. In hac habitasse platea dictumst. Pellentesque non elit. Fusce
% sed justo eu urna porta tincidunt. Mauris felis odio, sollicitudin sed, volutpat a, ornare ac, erat. Morbi quis dolor.
% Donec pellentesque, erat ac sagittis semper, nunc dui lobortis purus, quis congue purus metus ultricies tellus. Proin
% et quam. Class aptent taciti sociosqu ad litora torquent per conubia nostra, per inceptos hymenaeos. Praesent sapien
% Pellentesque habitant morbi tristique senectus et netus et malesuada fames ac turpis egestas. Donec odio elit,
% dictum in, hendrerit sit amet, egestas sed, leo. Praesent feugiat sapien aliquet odio. Integer vitae justo. Aliquam
% vestibulum fringilla lorem. Sed neque lectus, consectetuer at, consectetuer sed, eleifend ac, lectus. Nulla facilisi.
% Pellentesque eget lectus. Proin eu metus. Sed porttitor. In hac habitasse platea dictumst. Suspendisse eu lectus. Ut
% mi mi, lacinia sit amet, placerat et, mollis vitae, dui. Sed ante tellus, tristique ut, iaculis eu, malesuada ac, dui.
% Mauris nibh leo, facilisis non, adipiscing quis, ultrices a, dui.
% \begin{proof}
% Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text.
% \end{proof}
% Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi. Morbi auctor lorem non justo. Nam lacus libero,
% pretium at, lobortis vitae, ultricies et, tellus. Donec aliquet, tortor sed accumsan bibendum, erat ligula aliquet magna,
% vitae ornare odio metus a mi. Morbi ac orci et nisl hendrerit mollis. Suspendisse ut massa. Cras nec ante. Pellentesque
% a nulla. Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Aliquam tincidunt
% urna. Nulla ullamcorper vestibulum turpis. Pellentesque cursus luctus mauris.
% Nulla malesuada porttitor diam. Donec felis erat, congue non, volutpat at, tincidunt tristique, libero. Vivamus
% viverra fermentum felis. Donec nonummy pellentesque ante. Phasellus adipiscing semper elit. Proin fermentum massa
% ac quam. Sed diam turpis, molestie vitae, placerat a, molestie nec, leo. Maecenas lacinia. Nam ipsum ligula, eleifend
% at, accumsan nec, suscipit a, ipsum. Morbi blandit ligula feugiat magna. Nunc eleifend consequat lorem. Sed lacinia
% nulla vitae enim. Pellentesque tincidunt purus vel magna. Integer non enim. Praesent euismod nunc eu purus. Donec
% bibendum quam in tellus. Nullam cursus pulvinar lectus. Donec et mi. Nam vulputate metus eu enim. Vestibulum
% pellentesque felis eu massa.
% \begin{proof}[Proof of Theorem~{\rm\ref{thm1}}]
% Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text. Example for proof text.
% \end{proof}
% \begin{sidewaystable}%[h]
% \def\d{\hphantom{0}}
% \caption{Sideways table caption. For decimal alignment refer column 4 to 9 in tabular* preamble.\label{tab3}}%
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%\backmatter
\bmsection*{Author contributions}
This is an author contribution text. This is an author contribution text. This is an author contribution text. This is an author contribution text. This is an author contribution text.
\bmsection*{Acknowledgments}
This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here. This is acknowledgment text. Provide text here.
\bmsection*{Financial disclosure}
None reported.
\bmsection*{Conflict of interest}
The authors declare no potential conflict of interests.
\bibliography{wileyNJD-AMA}
\bmsection*{Supporting information}
Additional supporting information may be found in the
online version of the article at the publisher’s website.
% \appendix
% \bmsection{Program codes appear in Appendix\label{app1}}
% \vspace*{12pt}
% Using the package {\tt listings} you can add non-formatted text as you would do with \verb|\begin{verbatim}| but its main aim is to include the source code of any programming language within your document.\newline Use \verb|\begin{lstlisting}...\end{lstlisting}| for program codes without mathematics.
% The {\tt listings} package supports all the most common languages and it is highly customizable. If you just want to write code within your document, the package provides the {\tt lstlisting} environment; the output will be in Computer Modern typewriter font. Refer to the below example:
% col1 text & col2 text & col3 text & col4 text & col5 text & col6 text\\
% col1 text & col2 text & col3 text & col4 text & col5 text & col6 text\\
% col1 text & col2 text & col3 text& col4 text & col5 text & col6 text\\
% \bottomrule
% \end{tabular*}
% \end{table*}
% Example for an equation inside appendix
% \begin{equation}
% {\mathcal{L}} = i \bar{\psi} \gamma^\mu D_\mu \psi - \frac{1}{4} F_{\mu\nu}^a F^{a\mu\nu} - m \bar{\psi} \psi\label{eq25}
% \end{equation}
% \bmsection{Example of another appendix section\label{app3}}%
% \vspace*{12pt}
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% \begin{equation}
% \mathcal{L} = i \bar{\psi} \gamma^\mu D_\mu \psi
% - \frac{1}{4} F_{\mu\nu}^a F^{a\mu\nu} - m \bar{\psi} \psi
% \label{eq26}
% \end{equation}
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% \nocite{*}% Show all bib entries - both cited and uncited; comment this line to view only cited bib entries;
