\abstract[Abstract]{Buckling-delayed shear-link (BDSL) dampers are extensively used in seismic-resistant structures as passive devices that concentrate energy dissipation while limiting damage to the primary system. Their geometric optimization requires a compromise between high energy dissipation and control of local damage. Finite element method (FEM) models can reproduce with high accuracy the nonlinear cyclic response of these devices and provide internal quantities such as damage indicators and local distortion but their computational cost prevents their direct use 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 learning models are first evaluated, where Support Vector Regression (SVR) and Gaussian Process Regression (GPR)—both based on radial kernel functions—consistently provide the highest predictive accuracy. Motivated by this observation, Radial Basis Function (RBF) surrogates are subsequently introduced as a computationally efficient alternative. The surrogate predictions are coupled with a Differential Evolution algorithm through a damage-aware objective function that limits the damage and uses dissipated energy as a tie-breaking performance criterion. In addition, SHapley Additive exPlanations (SHAP) are employed to quantify the influence of window thickness on damage distribution, with particular emphasis on the response of the surrounding frame. 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 an efficient damage-aware 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{Irazabal J.},
\author{Ramirez J.},
\author{Gonzalez J.},
\author{Lazaro L.},
\author{Irazábal J.},
\author{Ramírez J.},
\author{González J. M.},
\author{Lázaro L.},
\author{Rastellini F.},
\author{Bozzo G.}, and
\author{Bozzo L.}}.
...
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@@ -81,7 +81,9 @@ Data-driven approaches have mainly focused on response or property prediction. C
All these works demonstrate the increasing interest in applying FEM-based and data-driven approaches, as well as in combining both, to analyse, understand and optimize seismic energy dissipation devices. However, most of these studies focus either on the prediction of the hysteretic response or on maximizing energy dissipation, leaving 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 surrogate-assisted optimization methodology in which the objective is not only to maximize distortion or energy dissipation, but also to balance dissipative performance with damage indicators derived from FEM simulations. The proposed approach combines: (i) experimentally calibrated nonlinear FEM models used as numerical ground truth; (ii) supervised 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. Figure \ref{fig:MethodologyFlowChart} summarizes the proposed workflow. The different stages of the methodology, together with the surrogate modelling, optimization strategy, validation procedure, and corresponding results, are described in the following sections.
The present work addresses this gap through a damage-aware surrogate-assisted optimization methodology in which the objective is not only to maximize distortion or energy dissipation, but also to balance dissipative performance with damage indicators derived from FEM simulations. The proposed approach combines: (i) experimentally calibrated nonlinear FEM models used as numerical ground truth; (ii) supervised 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.
Figure \ref{fig:MethodologyFlowChart} summarizes the proposed workflow. The different stages of the methodology, together with the surrogate modelling, optimization strategy, validation procedure, and corresponding results, are described in the following sections.
\begin{figure}[!ht]
\centering
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@@ -92,32 +94,52 @@ The present work addresses this gap through a damage-aware surrogate-assisted op
The BDSL dampers analysed in this work, shown in Figure \ref{fig:Device}, are designed to concentrate energy dissipation in localized regions while maintaining the overall structural integrity of the device. Energy dissipation is primarily achieved in reduced-thickness zones, hereafter referred to as windows, whereas the surrounding frame provides load transfer, stability and confinement.
The 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 promotes a shear-dominated response in the dissipative element. The windows are therefore intended to concentrate plastic deformation, while the remaining material forms a frame that stabilizes the system and transfers loads without becoming the main source of dissipation.
The BDSL dampers analysed in this work, with one representative configuration shown in Figure \ref{fig:Device}, are designed to concentrate energy dissipation in localized reduced-thickness zones, hereafter referred to as \ti{windows}, while maintaining the overall structural integrity of the device. The dissipative element is connected to a surrounding load-transfer system through a mechanism that allows imposed in-plane displacement while preventing axial force transmission, thereby promoting a shear-dominated response. Under cyclic loading, the windows are intended to concentrate plastic deformation and dissipative demand, whereas the surrounding frame provides load transfer, stability, and confinement without becoming the primary source of dissipation.
\caption{Example of a BDSL damper representative of the devices analysed in this work. The optimization variables correspond to the window thicknesses. These windows are surrounded by the main frame, whose dimensions are kept fixed. \red{Cambiar Figura por una que incluya las dimensiones.}}
\caption{Representative BDSL damper configuration analysed in this work. The optimization variables correspond to the window thicknesses, while the surrounding frame dimensions remain fixed. \red{Replace figure with a version including geometric dimensions.}}
\label{fig:Device}
\end{figure}
This separation of roles leads to a non-trivial design problem. Thin windows may enhance ductility and promote dissipative activation, but they may also localize damage excessively. Conversely, thicker windows may increase strength but transfer inelastic demand to the frame. Since severe damage in the frame can compromise the structural integrity of the device, frame damage must be penalized more strongly 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.
This separation of functions leads to a non-trivial design problem. Thin windows may enhance ductility and dissipative activation, but they may also promote excessive damage localization. Conversely, thicker windows may increase strength while transferring inelastic demand to the frame. Since severe frame damage may compromise the structural integrity of the device, frame damage must be penalized more strongly than window damage. In addition, the dissipative demand should be distributed as uniformly as possible among the windows, avoiding configurations in which a single window absorbs most of the deformation while the remaining windows remain underused.
The dissipation mechanism is governed by controlled yielding of the windows under cyclic shear deformation. However, the interaction between shear deformation in the windows and bending or longitudinal effects in the frame leads to non-uniform damage distributions. As a result, the design problem cannot be reduced to maximizing force or total dissipated energy alone. It must also explicitly control where damage occurs and ensure a balanced participation of all windows in the dissipative process.
The dissipation mechanism is governed by controlled yielding of the windows under cyclic shear deformation. However, the interaction between shear deformation in the windows and bending or longitudinal effects in the frame leads to non-uniform damage distributions. Consequently, the design problem cannot be reduced to maximizing force or total dissipated energy alone. It must also explicitly control where damage develops and ensure a balanced participation of all windows in the dissipative process.
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.
where $W$ denotes the number of windows. The width and height identifiers of the device are represented by $B$ and $H$, respectively. Three geometry families are considered in the current implementation: two-window devices with $H=30$ cm, three-window devices with $H=45$ cm, and five-window devices with $H=60$ cm. For the two-window and three-window families, two different widths are analysed, namely $B=29$ cm and $B=34$ cm, whereas the five-window family is currently implemented only for a single width configuration $B=34$ cm.
The admissible thickness ranges are defined according to the geometry family and manufacturing constraints. The corresponding bounds for each design variable are summarized in Table~\ref{tab:families}. In all cases, the frame thickness is kept constant at 30 mm.
\red{Revisar las medidas no me haya equivocado.}
\begin{table}[ht!]
\centering
\caption{Geometry families and surrogate input variables considered in the current implementation.}
\subsection{Nonlinear FEM model}\label{subsec:fem_model}
\red{El material es acero ASTM A36, ¿no? Estaría bien poner una referncia al software utilizado para los cálculos FEM.}
The surrogate models are trained using data generated from three-dimensional FEM simulations, as shown in Figure \ref{fig:FEMsetup}. The numerical model is based on a previously calibrated representation of the BDSL device. The steel dissipator is modelled using ASTM A36 steel, with cyclic plasticity described by the Yoshida--Uemori model \cite{Yoshida2002,Jia2014}. The formulation accounts for both material and geometric nonlinearities, as well as contact interactions and boundary conditions consistent with the experimental setup. The steel component is discretized using linear eight-node hexahedral elements, providing a structured three-dimensional representation suitable for extracting local stress and strain fields.
\begin{figure}[!ht]
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@@ -133,6 +155,8 @@ The imposed displacement is applied through an actuator-like connector that tran
\subsection{Calibration and validation}\label{subsec:calibration}
\red{¿Tenemos alguna publicación que podamos citar sobre la calibración de los modelos FEM?}
The FEM model is calibrated against experimental cyclic tests. The calibration procedure involves the definition of the material model, the assembled geometry, as well as the contact and boundary conditions. The validated model accurately reproduces the main global experimental responses, including hysteretic force--displacement loops, cumulative dissipated energy and the corresponding skeleton curve.
Once validated, the numerical model is used as a reliable ground truth to evaluate configurations beyond those experimentally tested. Figure \ref{fig:CalibrationCurves} presents the comparison between experimental and numerical results, showing good agreement in terms of global response and energy dissipation.
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@@ -148,13 +172,15 @@ The calibrated FEM model is subsequently employed to generate the datasets used
\subsection{Damage and deformation indicators}\label{subsec:indicators}
The use of FEM data is essential in this work because the optimization requires internal response variables that cannot be measured experimentally with sufficient spatial resolution. These variables include local damage indicators within each dissipative window and in the surrounding frame, as well as the distribution of deformation among the different regions of the device.
The optimization strategy proposed in this work requires internal response variables that cannot be directly measured experimentally, making FEM simulations an appropriate source of data. These variables include local damage indicators in both the dissipative windows and the surrounding frame, as well as the distribution of deformation among the different regions of the device.
In particular, stress-triaxiality-based indicators provide a meaningful measure of ductile damage under multiaxial loading conditions. Among them, the Triaxial Failure Damage Map (TFDMap) \cite{Rastellini2016} is used in this study as a post-processing quantity to assess the proximity to ductile failure \cite{Rice1969,Bao2004,Wierzbicki2005,Bai2008}. The TFDMap is constructed by comparing the local stress state, characterized by stress triaxiality and the accumulated equivalent plastic strain with a reference failure envelope. As the loading progresses, each material point follows a trajectory in the triaxiality--strain space and the corresponding TFDMap value quantifies how close that state is to the onset of ductile fracture.
Among the available damage measures, stress-triaxiality-based indicators provide a meaningful description of ductile damage under multiaxial loading conditions. In this study, the Triaxial Failure Damage Map (TFDMap) \cite{Rastellini2016} is adopted as a post-processing indicator to evaluate the proximity to ductile failure \cite{Rice1969,Bao2004,Wierzbicki2005,Bai2008}. The TFDMap is obtained by comparing the local stress state, characterized by the stress triaxiality, together with the accumulated equivalent plastic strain, against a reference failure envelope. During cyclic loading, each material point evolves through a trajectory in the triaxiality--strain space, and the associated TFDMap value provides a quantitative measure of proximity to ductile fracture.
In this work, the TFDMap is not used as a constitutive fracture model, but as a robust damage-screening indicator that allows comparing different geometrical configurations in terms of their proximity to failure. This distinction is important, as the objective is not to predict crack initiation, but to ensure that the optimized designs remain within acceptable damage levels.
In the present work, the TFDMap is not employed as a constitutive fracture criterion, but rather as a robust damage-screening indicator suitable for comparing different geometrical configurations in terms of their relative proximity to failure. The objective is therefore not to predict crack initiation explicitly, but to ensure that the optimized configurations remain within acceptable damage levels while maintaining adequate dissipative performance.
For the purpose of optimization, TFDMap values are aggregated separately in the dissipative windows and in the frame. The maximum value in each window is denoted by $\TFD_i$, while the corresponding value in the frame is denoted by $\TFD_f$. In addition, the local shear distortion in each window is denoted by $\varepsilon_{xy,i}$. This variable is used as a proxy for dissipative activation, since larger stable distortions are associated with higher energy dissipation capacity. When required, the contribution of each window can be weighted using geometric volume factors, ensuring that the evaluation is not based solely on point-wise strain values but also accounts for the effective volume involved in the dissipation process.
For optimization purposes, damage is evaluated separately in the dissipative windows and in the surrounding frame. The damage index is computed using the average TFDMap value of the 12 nodes with the highest values within each region. This approach captures the most critical damage levels while avoiding excessive sensitivity to isolated local peaks. The resulting damage indicator associated with each window is denoted by $\TFD_i$, whereas the corresponding value in the frame is represented by $\TFD_f$.
In addition, the maximum local shear distortion in each window is denoted by $\varepsilon_{xy,i}$. This quantity is used as a proxy for dissipative activation, since larger stable distortions are generally associated with higher energy dissipation capacity. The contribution of each window is weighted depending on its geometric volume, ensuring that the evaluation accounts not only for point-wise strain values but also for the effective material volume involved in the dissipation process.
\section{Dataset generation and surrogate modelling}\label{sec:surrogates}
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@@ -164,20 +190,6 @@ The FEM campaign is designed to cover the admissible design domain of each devic
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}
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.
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@@ -400,7 +412,7 @@ Once the final numerical campaign is completed, the placeholders in Section~\ref
%\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.
[To be completed according to the final author list.] Conceptualization: J. Irazábal, J. Ramírez, J. M. González, G. Bozzo, L. Bozzo. Methodology: J. Irazábal, J. Ramírez, J. M. González. Software and surrogate optimization: J. Irazábal. FEM modelling and validation: J. Ramírez, J. M. González, L. Lázaro, F. Rastellini. Writing--original draft: J. Irazábal and J. Ramírez. Writing--review and editing: all authors.
