Revise abstract and methodology sections for clarity; update design variables…

Manuscript/RESILINK_surrogate_optimization_manuscript.tex

@@ -45,7 +45,7 @@
 
 \corres{J. Irazábal. \email{jirazabal@cimne.upc.edu}}
 
-\abstract[Abstract]{Buckling-delayed shear-link (BDSL) dampers are used in seismic-resistant structures as passive devices that concentrate energy dissipation while limiting damage to the primary system. Their geometric optimization requires 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.}
+\abstract[Abstract]{Buckling-delayed shear-link (BDSL) dampers are used in seismic-resistant structures as passive devices that concentrate energy dissipation while limiting damage to the primary system. Their geometric optimization requires 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 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 dampers with different geometric and mechanical configurations. 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}
 
@@ -69,7 +69,7 @@
 
 Shear-link beam (SLB) dampers are passive energy dissipation devices widely used in seismic-resistant structures, designed to undergo stable inelastic deformations while limiting damage to the primary system. Their configuration concentrates inelastic demand in replaceable components, enabling high energy dissipation, ductile response and stable hysteretic behavior under severe cyclic loading \cite{Malley1984,Okazaki2007}. Within this family, buckling-delayed shear-link (BDSL) dampers incorporate a mechanical configuration that promotes shear-dominated behaviour while delaying local and global buckling.
 
-The global hysteretic response of BDSL dampers can be characterized through experimental testing. However, laboratory campaigns present important limitations when addressing design optimization problems: internal state variables such as local plastic strains, stress triaxiality or damage indicators cannot be directly measured and the high cost and logistical complexity of experimental programs restrict the number of geometric configurations that can be explored. These limitations become even more critical when the design objective is not only to increase global energy dissipation, but also to control where and how damage develops within the device, making systematic optimization impractical.
+The global hysteretic response of BDSL dampers can be characterized through experimental testing. However, laboratory campaigns present important limitations when addressing design optimization problems: internal state variables such as local plastic strains, stress triaxiality or damage indicators cannot be directly measured and the high cost and logistical complexity of experimental programs restrict the number of geometric configurations that can be explored. These limitations become even more critical when the design objective is not only to increase global energy dissipation and maximum displacement supported by the device, but also to control where and how damage develops within the device, making systematic optimization impractical.
 
 In this context, finite element method (FEM) simulations provide a powerful framework for generating high-fidelity datasets across a wide range of geometric configurations, making them an ideal foundation for surrogate-based optimization strategies. Advanced nonlinear FEM models can accurately reproduce experimental cyclic behaviour, including plasticity, geometric nonlinearity, contact interactions, local instability and damage evolution, while providing access to both global response quantities and local indicators governing failure mechanisms. However, their high computational cost makes their direct use within optimization loops inefficient. This limitation motivates the use of surrogate models trained on FEM-generated data, which can approximate the structural response with reduced computational effort. The use of surrogate models enables the possibility of evaluating a large number of design configurations, facilitating systematic optimization.
 
@@ -94,66 +94,68 @@ Figure \ref{fig:MethodologyFlowChart} summarizes the proposed workflow. The diff
 
 \section{Buckling-delayed shear-link damper}\label{sec:device}
 
-The BDSL dampers analysed in this work, with one representative configuration shown in Figure \ref{fig:Device}, are designed to concentrate energy dissipation in localized reduced-thickness zones, hereafter referred to as \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.
+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 preserving the structural integrity of the surrounding frame. The dissipative element is connected to a load-transfer system through a mechanism that allows imposed in-plane displacement while preventing axial force transmission, thereby promoting a shear-dominated response. Under cyclic loading, plastic deformation is intended to concentrate in the windows, whereas the frame provides load transfer and stability.
 
 \begin{figure}[!ht]
   \centering
   \includegraphics[width=0.20\textwidth]{./Figures/Device.png}
-  \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.}}
+  \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. Reduce caption.}}
   \label{fig:Device}
 \end{figure}
 
-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. 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.
+This separation of functions leads to a non-trivial design problem. Thin windows may enhance ductility and dissipative activation, but they may also promote excessive damage localization. Conversely, thicker windows may increase strength while transferring inelastic demand to the frame. Since severe frame damage may compromise the structural integrity of the device, frame damage must be penalized more strongly than window damage. At the same time, the dissipative demand should be distributed as uniformly as possible among the windows, avoiding configurations in which a single window absorbs most of the deformation while the remaining windows remain underused. Consequently, the design problem cannot be reduced to maximizing force or total dissipated energy alone, but must also control where damage develops and how the windows participate in the dissipative process.
 
 The design variables considered in this work are the window thicknesses
 \begin{equation}
 \mathbf{x}=\left[t_{w,1},t_{w,2},\ldots,t_{w,W}\right],
 \label{eq:design_vector}
 \end{equation}
-where $W$ denotes the number of windows. The width and height identifiers of the device are represented by $B$ and $H$, respectively. 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.
+where $W$ denotes the number of windows. The width and height identifiers of the device are represented by $B$ and $H$, respectively. Five geometry families are considered, as shown in Figure \ref{fig:GeometryFamilies}: H30\_B29, H30\_B34, H45\_B29, H45\_B34, and H60\_B34. Devices with $H=30$ cm have two windows, those with $H=45$ cm have three windows, and those with $H=60$ cm have five windows.
 
-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.
+\begin{figure}[ht!]
+\centering
+\fbox{\parbox[c][0.25\textheight][c]{0.85\textwidth}{\centering Placeholder for geometry family representations.}}
+\caption{Geometry families considered in the current implementation.}
+\label{fig:GeometryFamilies}
+\end{figure}
 
-\red{Revisar las medidas no me haya equivocado.}
+The admissible thickness ranges are defined according to the geometry family and manufacturing constraints. The main characteristics of each family are summarized in Table~\ref{tab:families}. In all cases, the frame thickness is kept constant at 30 mm.
 
 \begin{table}[ht!]
 \centering
-\caption{Geometry families in the current implementation and bounds of the window thicknesses.}
+\caption{Geometry families considered in the current implementation and admissible window thickness ranges.}
 \label{tab:families}
-\begin{tabular}{llllll}
+\begin{tabular}{lllllll}
 \toprule
-Family & Height $H$ & Width $B$ & Frame thickness & Design variables & Thickness bounds \\
+Family & Height $H$ & Width $B$ & Windows & Frame thickness & Design variables & Thickness bounds \\
 \midrule
-2 windows B29 & 30 cm & 29 cm & 30 mm & $t_{w,1},t_{w,2}$ & 10--20 mm \\
-2 windows B34 & 30 cm & 34 cm & 30 mm & $t_{w,1},t_{w,2}$ & 10--20 mm \\
-3 windows B29 & 45 cm & 29 cm & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
-3 windows B34 & 45 cm & 34 cm & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
-5 windows B34 & 60 cm & 34 cm & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 5--12 mm \\
+H30\_B29 & 30 cm & 29 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 10--20 mm \\
+H30\_B34 & 30 cm & 34 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 10--20 mm \\
+H45\_B29 & 45 cm & 29 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
+H45\_B34 & 45 cm & 34 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 5--14 mm \\
+H60\_B34 & 60 cm & 34 cm & 5 & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 5--12 mm \\
 \bottomrule
 \end{tabular}
 \end{table}
 
-
-\section{FEM-calibrated numerical ground truth}\label{sec:fem}
+\section{Validation of the FEM numerical model}\label{sec:fem}
 
 \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.
+The surrogate models are trained using data generated from three-dimensional FEM simulations. 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. Figure \ref{fig:FEMsetup} shows the FEM numerical model built for the simulations, including all the items here mentioned.
 
 \begin{figure}[!ht]
   \centering
   \includegraphics[width=0.90\textwidth]{./Figures/FEMsetup.png}
-  \caption{Finite element model used to generate the ground-truth dataset. The calibrated model reproduces the experimental cyclic response and provides internal quantities not directly accessible from laboratory tests. The image shows the mesh discretization, main components and boundary conditions, including local and global buckling control. \red{¿Tenemos una imagen de uno de los disipadores que se usan en este estudio?}}
+  \caption{Finite element model used to generate the ground-truth dataset. The calibrated model reproduces the experimental cyclic response and provides internal quantities not directly accessible from laboratory tests. The image shows the mesh discretization, main components and boundary conditions, including local and global buckling control. \red{¿Tenemos una imagen de uno de los disipadores que se usan en este estudio? Reducir caption.}}
   \label{fig:FEMsetup}
 \end{figure}
 
 The imposed displacement is applied through an actuator-like connector that transfers horizontal displacement while preventing 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 standard experimental qualification procedures for seismic energy dissipation devices.
 
-\red{Faltan por meter los ciclos de carga aplicados para cada geometría.}
+\red{Faltan por meter los ciclos de carga aplicados para cada geometría. En este apartado habría que explayarse un poco más: habría que hablar del procedimiento de calibración/validación a partir de tres ensayos, con tres patrones de carga según las correspondientes normativas y resultados de referencia.}
 
 \subsection{Calibration and validation}\label{subsec:calibration}
 
@@ -161,22 +163,22 @@ The imposed displacement is applied through an actuator-like connector that tran
 
 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.
+Figure \ref{fig:CalibrationCurves} presents the comparison between experimental and numerical results, showing good agreement in terms of global response and energy dissipation. Once validated, the numerical model is used as a reliable tool to provide a database to evaluate configurations beyond those experimentally tested.
 
 The calibrated FEM model is subsequently employed to generate the datasets used for surrogate training and optimization, providing access to local damage and distortion indicators that cannot be directly measured in experiments.
 
 \begin{figure}[!ht]
   \centering
   \includegraphics[width=1.0\textwidth]{./Figures/CalibrationCurves.png}
-  \caption{Experimental--numerical validation of the BDSL model. Comparison of hysteretic response, cumulative dissipated energy and skeleton curve, showing good agreement between FEM predictions and experimental results. \red{¿Tenemos una imagen de uno de los disipadores que se usan en este estudio?}}
+  \caption{Experimental--numerical validation of the BDSL model. a) Comparison of hysteretic response, b) cumulative dissipated energy and c) skeleton curve. \red{¿Tenemos una imagen de uno de los disipadores que se usan en este estudio?}}
   \label{fig:CalibrationCurves}
 \end{figure}
 
 \subsection{Damage and deformation indicators}\label{subsec:indicators}
 
-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.
+The optimization strategy proposed in this work is based in a set of output variables from the numerical simulations 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 observed at the different regions of the device.
 
-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.
+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 characteristic of the material. 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 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.
 
@@ -184,6 +186,13 @@ For optimization purposes, damage is evaluated separately in the dissipative win
 
 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.
 
+\begin{figure}[ht!]
+\centering
+\fbox{\parbox[c][0.25\textheight][c]{0.85\textwidth}{\centering Placeholder for an example of the TFDMap and distortion distribution in a device.}}
+\caption{Example of the TFDMap and distortion distribution.}
+\label{fig:TFDMapDistortion}
+\end{figure}
+
 \section{Dataset generation and surrogate modelling}\label{sec:surrogates}
 
 \subsection{Design of experiments}\label{subsec:doe}
@@ -196,15 +205,15 @@ To improve the robustness of the surrogate models near the admissible limits, th
 \centering
 \caption{Geometry families and window thickness ranges considered during the DoE generation.}
 \label{tab:families_doe}
-\begin{tabular}{llllll}
+\begin{tabular}{lllllll}
 \toprule
-Family & Height $H$ & Width $B$ & Frame thickness & Design variables & Thickness bounds (DoE) \\
+Family & Height $H$ & Width $B$ & Windows & Frame thickness & Design variables & Thickness bounds (DoE) \\
 \midrule
-2 windows B29 & 30 cm & 29 cm & 30 mm & $t_{w,1},t_{w,2}$ & 8--22 mm \\
-2 windows B34 & 30 cm & 34 cm & 30 mm & $t_{w,1},t_{w,2}$ & 8--22 mm \\
-3 windows B29 & 45 cm & 29 cm & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm \\
-3 windows B34 & 45 cm & 34 cm & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm \\
-5 windows B34 & 60 cm & 34 cm & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 4--14 mm \\
+H30\_B29 & 30 cm & 29 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 8--22 mm \\
+H30\_B34 & 30 cm & 34 cm & 2 & 30 mm & $t_{w,1},t_{w,2}$ & 8--22 mm \\
+H45\_B29 & 45 cm & 29 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm \\
+H45\_B34 & 45 cm & 34 cm & 3 & 30 mm & $t_{w,1},t_{w,2},t_{w,3}$ & 4--16 mm \\
+H60\_B34 & 60 cm & 34 cm & 5 & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 4--14 mm \\
 \bottomrule
 \end{tabular}
 \end{table}