@@ -140,56 +140,38 @@ H60\_B34 & 60 cm & 34 cm & 5 & 30 mm & $t_{w,1},\ldots,t_{w,5}$ & 5--12 mm \\
\section{Validation of the FEM numerical model}\label{sec:fem}
\subsection{Nonlinear FEM model}\label{subsec:fem_model}
The surrogate models developed in this work are trained using data generated from three-dimensional FEM simulations. The numerical model is based on a previously calibrated and validated representation of the BDSL device, described in detail in Ramirez et al. \cite{RamirezMachado2025}. The model was developed using the COMPACK code, an explicit dynamic FEM solver based on an incremental formulation \cite{Martinez2011}. It accounts for large displacements, material and geometric nonlinearities, contact interactions, and the boundary conditions associated with the experimental setup.
\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. 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.
The dissipative steel component is modelled as ASTM A36 steel, whose cyclic plastic behaviour is represented by the Yoshida--Uemori model \cite{Yoshida2002,Jia2014}. This constitutive law allows the model to reproduce cyclic hardening, softening, and Bauschinger-type effects under large plastic deformation. The steel component is discretized using linear eight-node hexahedral solid elements, providing a structured three-dimensional mesh suitable for extracting local stress, strain, and damage-related fields.
\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.}}
\caption{FEM validation model of the BDSL device: mesh discretization, main components, boundary conditions and local/global buckling control.}
\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. 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}
\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.
The imposed displacement is applied through an actuator-like connector that transfers horizontal displacement while preventing axial load transmission, thereby reproducing the kinematic condition required to promote a shear-dominated response. Three experimental tests were performed following the loading protocols defined in the ANSI/AISC 341-16 \cite{American2002} and ASCE 7-22 \cite{American2017} qualification standards for seismic energy dissipation devices. Those protocols involve cyclic loading with progressively increasing amplitudes. Additional contact and confinement conditions are included to represent the experimental anti-buckling system and to control local and global instability.
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.
The model was calibrated and validated against cyclic experimental tests performed on representative BDSL specimens. The calibration involved the material parameters, assembled geometry, contact definitions, support flexibility, and boundary conditions. The validated model accurately reproduces the main global experimental responses, including hysteretic force--displacement loops, cumulative dissipated energy, stiffness degradation, and the skeleton curve. Figure \ref{fig:CalibrationCurves} shows the comparison between experimental and numerical results, confirming the suitability of the FEM model as a numerical reference for configurations beyond those experimentally tested.
\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?}}
\caption{Experimental--numerical validation of the BDSL model: (a) hysteretic response, (b) cumulative dissipated energy, and (c) skeleton curve.}
\label{fig:CalibrationCurves}
\end{figure}
\subsection{Damage and deformation indicators}\label{subsec:indicators}
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 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.
Once validated, the FEM model is used to generate the datasets required for surrogate training and optimization. This is essential because the proposed optimization strategy relies on internal response quantities that cannot be directly measured experimentally with sufficient spatial resolution. These quantities include local damage indicators in the dissipative windows and in the surrounding frame, as well as local distortion measures associated with the activation of the dissipative mechanism.
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.
The main damage indicator adopted in this work is the Triaxial Failure Damage Map (TFDMap) \cite{Rastellini2016}. This stress-triaxiality-based indicator evaluates the proximity of each material point to ductile failure by comparing its stress triaxiality and accumulated equivalent plastic strain with a reference failure envelope \cite{Rice1969,Bao2004,Wierzbicki2005,Bai2008}. In this study, the TFDMap is used as a post-processing damage-screening indicator, not as a constitutive fracture criterion. Its purpose is therefore not to explicitly predict crack initiation, but to compare geometrical configurations and ensure that optimized designs remain within acceptable damage levels.
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.
For optimization purposes, damage is evaluated separately in the windows and in the frame. The window damage indicator $\TFD_i$ is defined as the average TFDMap value of the 12 nodes with the highest values in window $i$, while the corresponding frame indicator is denoted by $\TFD_f$. This aggregation captures the most critical damage levels while reducing sensitivity to isolated numerical peaks. In addition, the maximum local shear distortion in each window is denoted by $\varepsilon_{xy,i}$ and is used as a proxy for dissipative activation. The contribution of each window is weighted according to its effective geometric volume, so that the optimization accounts not only for point-wise strain values but also for the 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.}
\caption{Example of TFDMap and local distortion fields used to define the damage and deformation indicators.}
\label{fig:TFDMapDistortion}
\end{figure}
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@@ -224,6 +206,8 @@ For each geometry family, the optimization cycle starts from an initial set of F
The number of samples in all cases is defined as a power of two. This choice facilitates the potential use of Progressive Latin Hypercube Sampling (PLHS) \cite{Sheikholeslami2017} in future iterations, allowing the DoE to be expanded while preserving its space-filling properties and avoiding the need to repeat previously computed simulations.
\red{Faltan por meter los ciclos de carga aplicados para cada geometría.}
\subsection{Supervised ML surrogate models}\label{subsec:ml_models}
This work compares several supervised surrogate models for predicting FEM-derived damage and distortion indicators. The considered models cover three families: tree-based methods, including Random Forest (RF) \cite{Breiman2001}, Gradient Boosting Regression (GBR) \cite{Friedman2001}, and XGBoost \cite{Chen2016}; kernel-based methods, including Support Vector Regression (SVR) \cite{Drucker1996} and Gaussian Process Regression (GPR) \cite{Williams1995}; and neural-network models, represented by the Multilayer Perceptron (MLP) \cite{Rosenblatt1958,Rumelhart1986}. RF relies on bootstrap aggregation of decision trees, GBR and XGBoost are sequential boosting approaches, SVR and GPR exploit kernel functions to model nonlinear relationships, and MLP approximates nonlinear input--output mappings through interconnected layers.
@@ -727,4 +727,42 @@ steel from coupon test results available. First, the theory of metal plasticity
urldate = {2026-05-07},
}
@InProceedings{RamirezMachado2025,
author = {Ramírez Machado, Brain Junior and Bozzo Fernández, Guillermo and González Lopez, Jose Manuel and Rastellini, Fernando G. and Irazábal González, Joaquín and Bozzo Rotondo, Luis Miguel},
title = {Advanced modeling of buckling delayed shear links under strong cyclic loads},
year = {2025},
organization = {National Technical University of Athens},
abstract = {This study investigates the advanced numerical modelling of a Buckling Delayed Shear Link (BDSL) system as a response control device aimed at improving the seismic resilience of structures. This system combines shear-based dissipative devices with an improved connection that transmits shear loads while preventing axial force, allowing for in-plane rotation. It currently accommodates displacements up to 60 mm. Experimental tests using cyclic loading were performed on specific dissipative element geometry to calibrate the mechanical properties and validate the numerical model. The tests, performed at the structures laboratory of the University of Lima, Peru, adhered to cyclic loading protocols following standard codes. The tests provided the structural response of the system in terms of hysteretic curves and failure modes. The methodology focuses on creating a three-dimensional numerical model of the BDSL using the Finite Element Method (FEM) to simulate its behaviour under cyclic loads. The FEM model of the dissipator features large displacements and incorporates both material and contact nonlinearities. ASTM A36 steel has been evaluated using a combined hardening model to represent its cyclic behaviour. A penalty-based algorithm handles contact in the special connection, ensuring accurate seismic response. The results show strong agreement regarding displacements, total shear forces, hysteretic curves, dissipated energy, and effective stiffness. Additionally, the experimental failure mode is analysed by considering key numerical variables: equivalent plastic strain and stress triaxiality. The analysis of key variables from calibrated models and numerical simulation data will be used to train a surrogate model using machine learning techniques. With this approach, the system can be optimised, and the structural response can be predicted at a low computational cost.},
language = {eng},
urldate = {2026-05-08},
}
@Article{Martinez2011,
author = {Martinez, Xavier and Rastellini, Fernando and Oller, Sergio and Flores, Fernando and Oñate, Eugenio},
journal = {Composites Part B: Engineering},
title = {Computationally optimized formulation for the simulation of composite materials and delamination failures},
year = {2011},
issn = {1359-8368},
month = mar,
number = {2},
pages = {134--144},
volume = {42},
abstract = {The numerical simulation of complex failure modes of composite materials, such as delamination, can be computationally very demanding, as it requires special elements and/or numerical strategies to characterize damage onset and propagation. This work presents several formulations developed to optimize the computational performance of an explicit finite element code designed specifically for the simulation of large scale composite structures. The composite mechanical performance is obtained with the matrix-reinforced mixing theory, a simplified version of the serial/parallel mixing theory that does not require an iterative procedure or the calculation of the tangent stiffness matrix. The number of elements required to perform the simulation is reduced by stacking several layers inside a single finite element. This work also proposes a modification of the isotropic damage law, capable of taking into account the residual strength provided by friction in type II fracture modes. The ability of these formulations to successfully predict the mechanical performance of composite materials is assessed with the ply drop-off test. In this test a laminate with a change of thickness in its mid-span is loaded until it breaks due to a delamination process. The formulation proposed obtains a very accurate prediction of the experimental response of the test, as it provides a very good characterization of the initial laminate stiffness, the delamination onset, and its propagation along the specimen.},
doi = {10.1016/j.compositesb.2010.09.013},
keywords = {A. Laminates, B. Delamination, C. Damage mechanics, C. Finite element analysis},
urldate = {2026-05-08},
}
@Misc{American2002,
title = {Seismic provisions for structural steel buildings},
year = {2002},
howpublished = {Standard, American Institute of Steel Construction},
}
@Misc{American2017,
title = {Minimum design loads and associated criteria for buildings and other structures},
year = {2017},
howpublished = {Standard, American Society of Civil Engineers},
