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2026_Article_RESILINK_ML
Commit 397ec680 by Joaquín Irazábal González
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Add adaptive FEM-validated surrogate optimization manuscript for BDSL dampers

- Introduced a comprehensive manuscript detailing the optimization framework for buckling-delayed shear-link dampers. - Included sections on introduction, device description, FEM calibration, surrogate modeling, optimization algorithm, and adaptive validation. - Implemented a damage-aware objective function and outlined the methodology for generating high-fidelity FEM datasets. - Discussed the predictive performance of supervised ML models and RBF surrogates, along with planned numerical assessments. - Added author contributions, acknowledgments, financial disclosures, and conflict of interest statements.
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\documentclass[AMA,Times1COL]{WileyNJDv5} %STIX1COL,STIX2COL,STIXSMALL
% Own definitions
\newcommand{\tb}[1]{\textbf{#1}}
\newcommand{\ti}[1]{\textit{#1}}
\newcommand{\mcol}[3]{\multicolumn{#1}{#2}{#3}}
\newcommand{\mrow}[3]{\multirow{#1}{#2}{#3}}
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
\newcommand{\TFD}{\mathrm{TFDMap}}
\newcommand{\Exy}{\varepsilon_{xy}}
\newcommand{\tw}{t_{w}}
\articletype{Research Article}%
\received{Date Month Year}
\revised{Date Month Year}
\accepted{Date Month Year}
\journal{Journal}
\volume{00}
\copyyear{2026}
\startpage{1}
\raggedbottom
\begin{document}
\title{Adaptive FEM-validated surrogate optimization of buckling-delayed shear-link dampers for seismic damage mitigation}
\author[1]{J. Irazabal}
\author[1,2]{J. Ramirez}
\author[1,2]{J. Gonzalez}
\author[1]{L. Lazaro}
\author[1,2]{F. Rastellini}
\author[2,3]{G. Bozzo}
\author[4]{L. Bozzo}
\authormark{RAMIREZ \textsc{et al.}}
\titlemark{ADAPTIVE FEM-VALIDATED SURROGATE OPTIMIZATION OF BDSL DAMPERS}
\address[1]{\orgname{Centre Internacional de Metodes Numerics en Enginyeria (CIMNE)}, \orgaddress{\city{Barcelona}, \country{Spain}}}
\address[2]{\orgname{Universitat Politecnica de Catalunya (UPC)}, \orgaddress{\city{Barcelona}, \country{Spain}}}
\address[3]{\orgname{SLB Devices}, \orgaddress{\city{Barcelona}, \country{Spain}}}
\address[4]{\orgname{Luis Bozzo Estructuras y Proyectos S.L.}, \orgaddress{\city{Barcelona}, \country{Spain}}}
\corres{J. Irazabal. \email{jirazabal@cimne.upc.edu}}
\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{Rastellini F.},
\author{Bozzo G.}, and
\author{Bozzo L.}}.
\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 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.
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 impractical. This limitation motivates the use of surrogate models trained on FEM-generated data, which can approximate the structural response with significantly reduced computational effort, enabling the efficient evaluation of a large number of design configurations, facilitating systematic optimization.
Over the past decades, FEM has been widely used to study BDSL dampers and other passive 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 and parametric analysis of these devices by enabling systematic exploration of geometric configurations and performance criteria under prescribed loading \cite{Deng2014,Deng2015a}. In this context, FEM-based studies have been extensively applied 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, while 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. Simplified analytical and semi-empirical models have also been proposed to reduce computational cost \cite{Deng2014b} and more recent simulation-based studies have further explored the role of geometric and material variables in damper performance \cite{Kim2022}.
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}.
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 the 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 the probabilistic prediction of buckling stress. Physics-informed approaches have also been explored, such as the PINN framework proposed by Hu et al. \cite{Hu2022}.
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 framework in which the objective is not only to maximize distortion or energy dissipation, but to balance dissipative performance with damage indicators derived from FEM simulations. The proposed methodology combines: (i) experimentally calibrated nonlinear FEM models used as 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. The framework remains consistent with the underlying physics, as all surrogate models are trained on FEM-generated 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.
% \end{itemize}
\section{Buckling-delayed shear-link damper}\label{sec:device}
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.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.20\textwidth]{./Figures/Device.png}
\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.}}
\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.
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 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 $W$ 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.
\section{FEM-calibrated numerical ground truth}\label{sec:fem}
\subsection{Nonlinear FEM model}\label{subsec:fem_model}
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]
\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?}}
\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.}
\subsection{Calibration and validation}\label{subsec:calibration}
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.
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?}}
\label{fig:CalibrationCurves}
\end{figure}
\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.
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.
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.
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.
\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 $\varepsilon_{xy,i}$, $\TFD_i$ 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 $N\leq 20$, repeated five-fold cross-validation with five repetitions is used for $21\leq N\leq 80$ 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.
\subsection{RBF surrogate models}\label{subsec:rbf_models}
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
\begin{equation}
\hat{y}(\mathbf{x}) = \sum_{j=1}^{N} \lambda_j \, \phi\left(\|\mathbf{x}-\mathbf{x}_j\|\right),
\label{eq:rbf}
\end{equation}
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.
\section{Damage-aware surrogate-assisted optimization}\label{sec:optimization}
\subsection{Optimization algorithm}\label{subsec:de}
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.
\subsection{Objective function}\label{subsec:objective}
The objective function is designed to encode four mechanical rules:
\begin{enumerate}
\item the damage indicator must remain limited in all structural components, including both windows and frame;
\item reaching complete or near-complete damage in the frame is much more critical than reaching a similar damage level in a window, because frame failure would imply loss of the structural integrity of the damper;
\item the windows should develop comparable damage levels so that the dissipation mechanism is distributed rather than concentrated in a single window;
\item if several geometries show similar damage performance, the preferred one is the geometry that dissipates more energy, approximated through stable window distortion and the corresponding volume contribution.
\end{enumerate}
The implemented scalar objective is
\begin{equation}
J(\mathbf{x}) = - \sum_{i=1}^{W} \hat{\Exy}_i^2\, t_{w,i}\, V_i +
\sum_{i=1}^{W} P_w\left(\widehat{\TFD}_i;\TFD_w^{\star}\right) +
P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right),
\label{eq:objective}
\end{equation}
where $V_i$ is the volume factor associated with window $i$, $\TFD_w^{\star}$ is the target TFDMap level for the windows and $\TFD_f^{\max}$ is the maximum admissible frame threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger distortion contributions reduce the objective value.
The window penalty is defined as
\begin{equation}
P_w\left(\widehat{\TFD}_i;\TFD_w^{\star}\right)=
\begin{cases}
\left(\widehat{\TFD}_i-\TFD_w^{\star}\right)^2, & \widehat{\TFD}_i>\TFD_w^{\star},\\
\left|\widehat{\TFD}_i-\TFD_w^{\star}\right|, & \widehat{\TFD}_i\leq\TFD_w^{\star}.
\end{cases}
\label{eq:window_penalty}
\end{equation}
This expression penalizes exceeding the target value quadratically, while also discouraging excessively under-utilized windows through a linear distance to the target. As a result, the optimizer tends to equalize the TFDMap levels among windows rather than forcing all windows to remain far below the admissible damage level.
The frame penalty is more severe:
\begin{equation}
P_f\left(\widehat{\TFD}_f;\TFD_f^{\max}\right)=
\begin{cases}
\left(\widehat{\TFD}_f-\TFD_f^{\max}\right)^3, & \widehat{\TFD}_f>\TFD_f^{\max},\\
0, & \widehat{\TFD}_f\leq\TFD_f^{\max}.
\end{cases}
\label{eq:frame_penalty}
\end{equation}
The cubic frame penalty reflects the fact that complete frame failure is a much more severe event than localized damage in a replaceable window. In the current scripts, $\TFD_f^{\max}=90$ is used, while the target window threshold $\TFD_w^{\star}$ is provided as an input argument to the optimizer.
\subsection{Volume factors}\label{subsec:volume_factors}
The distortion contribution is weighted by a geometric factor that accounts for the effective volume associated with each window. These factors differ between families because the windows have different dimensions and positions. The current values used in the scripts are summarized in Table~\ref{tab:volume_factors}. When a geometry family is not explicitly covered, the implementation falls back to an unweighted contribution.
\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 $V_i$ \\
\midrule
2 windows & 29 & 0.0208, 0.0185 \\
2 windows & 34 & 0.0263, 0.0240 \\
3 windows & 29 & 0.0229, 0.0210, 0.0185 \\
3 windows & 34 & 0.0262, 0.0262, 0.0240 \\
5 windows & -- & 0.0410, 0.0265, 0.0240, 0.0098, 0.0098 \\
\bottomrule
\end{tabular}
\end{table}
\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}
\end{figure}
\section{Planned numerical assessment}\label{sec:planned_results}
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: $\varepsilon_{xy,i}$, $\TFD_i$ 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] \\
\midrule
2W--B29--H30 & $\varepsilon_{xy,1}$ & [SVR/GPR/etc.] & [--] & [--] & [--] & [--] \\
2W--B29--H30 & $\TFD_1$ & [SVR/GPR/etc.] & [--] & [--] & [--] & [--] \\
2W--B29--H30 & $\TFD_f$ & [SVR/GPR/etc.] & [--] & [--] & [--] & [--] \\
3W--B34--H45 & $\varepsilon_{xy,i}$ & [SVR/GPR/etc.] & [--] & [--] & [--] & [--] \\
5W--H60 & $\TFD_i$ & [SVR/GPR/etc.] & [--] & [--] & [--] & [--] \\
\bottomrule
\end{tabular}
\end{table*}
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 \\
\midrule
2W--B29--H30 & $\varepsilon_{xy,1}$ & [--] & [--] & [--] & [--] \\
2W--B29--H30 & $\TFD_1$ & [--] & [--] & [--] & [--] \\
3W--B34--H45 & $\TFD_f$ & [--] & [--] & [--] & [--] \\
5W--H60 & $\varepsilon_{xy,i}$ & [--] & [--] & [--] & [--] \\
\bottomrule
\end{tabular}
\end{table*}
\subsection{Optimization results}\label{subsec:planned_opt}
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? \\
\midrule
2W--B29--H30 & ML & 80 & [--] & [--] & -- & [--] & [yes/no] \\
2W--B29--H30 & RBF & 80 & [--] & [--] & -- & [--] & [yes/no] \\
3W--B34--H45 & ML & 90 & [--] & [--] & [--] & [--] & [yes/no] \\
3W--B34--H45 & RBF & 90 & [--] & [--] & [--] & [--] & [yes/no] \\
\bottomrule
\end{tabular}
\end{table*}
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 $\varepsilon_{xy,i}$ 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 $\TFD_i$, $\TFD_f$, $\varepsilon_{xy,i}$ 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.
\bibliography{wileyNJD-AMA}
\bmsection*{Supporting information}
Additional supporting information may include the FEM database, trained surrogate models, optimization scripts and final FEM-validation simulations.
\end{document}
Manuscript/wileyNJD-AMA.bib
......@@ -379,4 +379,182 @@
urldate = {2026-04-22},
}
@Article{Malley1984,
author = {Malley, James O. and Popov, Egor P.},
journal = {Journal of Structural Engineering},
title = {Shear {Links} in {Eccentrically} {Braced} {Frames}},
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number = {9},
pages = {2275--2295},
volume = {110},
abstract = {Eccentrically braced steel framing in seismic applications can provide high elastic
stiffness and large inelastic energy dissipation capacity. The performance of this
framing system depends to a great extent on the behavior of short active link sections
...},
doi = {10.1061/(ASCE)0733-9445(1984)110:9(2275)},
keywords = {ASCE Awards \& Prizes, Connections, Energy efficiency, Seismic design, Steel frames, Stiffening},
language = {en},
publisher = {American Society of Civil Engineers},
urldate = {2026-05-05},
}
@Article{Okazaki2007,
author = {Okazaki, Taichiro and Engelhardt, Michael D.},
journal = {Journal of Constructional Steel Research},
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volume = {63},
abstract = {Cyclic loading tests were conducted to study the behavior of link beams in steel eccentrically braced frames. A total of thirty-seven link specimens were constructed from five different wide-flange sections, all of ASTM A992 steel, with link length varying from short shear yielding links to long flexure yielding links. The occurrence of web fracture in shear yielding link specimens led to further study on the cause of these fractures. Since the link web fracture appeared to be a phenomenon unique to modern rolled shapes, the potential role of material properties on these fractures is discussed. Based on the test data, a change in the flange slenderness limit is proposed. The link overstrength factor of 1.5, as assumed in the current U.S. code provisions, appears to be reasonable. The cyclic loading history used for testing was found to significantly affect link performance. Test observations also suggest new techniques for link stiffener design and detailing for link-to-column connections.},
doi = {10.1016/j.jcsr.2006.08.004},
keywords = {Cyclic tests, Eccentrically braced frame, Flange slenderness ratio, Fracture, k-area, Loading history, Seismic design, Steel structures},
urldate = {2026-05-05},
}
@Article{Bao2004,
author = {Bao, Yingbin and Wierzbicki, Tomasz},
journal = {International Journal of Mechanical Sciences},
title = {On fracture locus in the equivalent strain and stress triaxiality space},
year = {2004},
issn = {0020-7403},
month = jan,
number = {1},
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volume = {46},
abstract = {The stress triaxiality is, besides the strain intensity, the most important factor that controls initiation of ductile fracture. In this study, a series of tests including upsetting tests, shear tests and tensile tests on 2024-T351 aluminum alloy providing clues to fracture ductility for a wide range of stress triaxiality was carried out. Numerical simulations of each test was performed using commercial finite element code ABAQUS. Good correlation of experiments and numerical simulations was achieved. Based on the experimental and numerical results, the relation between the equivalent strain to fracture versus the stress triaxiality was quantified and it was shown that there are three distinct branches of this function with possible slope discontinuities in the transition regime. For negative stress triaxialities, fracture is governed by shear mode. For large triaxialities void growth is the dominant failure mode, while at low stress triaxialities between above two regimes, fracture may develop as a combination of shear and void growth modes.},
doi = {10.1016/j.ijmecsci.2004.02.006},
keywords = {Experiment, Fracture, Numerical simulation, Stress triaxiality},
urldate = {2026-05-05},
}
@Article{Wierzbicki2005,
author = {Wierzbicki, Tomasz and Bao, Yingbin and Lee, Young-Woong and Bai, Yuanli},
journal = {International Journal of Mechanical Sciences},
title = {Calibration and evaluation of seven fracture models},
year = {2005},
issn = {0020-7403},
month = apr,
number = {4},
pages = {719--743},
volume = {47},
abstract = {Over the past 5 years, there has been increasing interest of the automotive, aerospace, aluminum, and steel industries in numerical simulation of the fracture process of typical structural materials. Accordingly, there is a pressure on the developers of leading commercial codes, such as ABAQUS, LS-DYNA, and PAM-CRASH to implement reliable fracture criteria into those codes. Even though there are several options to address fracture in these and other commercial codes, no guidelines are given for the users as to which fracture criterion is suitable for a particular application and how to calibrate a given material for fracture. The objective of the present paper is to address the above issues and present a thorough comparative study of seven fracture criteria that are included in libraries of material models of non-linear finite element codes. A set of 15 tests recently conducted by the authors on 2024-T351 aluminum alloy is taken as a reference for the present study. The plane stress prevails in all these tests. These experiments are compared with the constant equivalent strain criterion, the Xue–Wierzbicki (X–W) fracture criterion, the Wilkins (W), the Johnson–Cook (J–C) and the CrachFEM fracture models. Additionally, the maximum shear (MS) stress model, and the fracture forming limit diagram (FFLD) are included in the present evaluation. All criteria are formulated in the general 3-D case for the power law hardening materials and then are specified for the plane stress condition. The advantage of working with plane stress is that there is one-to-one mapping from the stress to the strain space. Therefore, the fracture criteria formulated in the stress space can be compared with those expressed in the strain space and vice versa. Fracture loci for all seven cases were constructed in the space of the equivalent fracture strain and the stress triaxiality. Interesting observations were made regarding the range of applicability and expected errors of some of the most common fracture criteria. Besides evaluating the applicability of several fracture criteria, a detailed calibration procedure for each criterion is presented in the present paper. It was found rather unexpectedly that the MS stress fracture model closely follows the trend of all tests except the round bar tensile tests. The X–W criterion and the CrachFEM models predict correctly fracture in all types of experiments. The W criterion is working well in certain ranges of the stress triaxiality.},
doi = {10.1016/j.ijmecsci.2005.03.003},
keywords = {Calibration, Fracture criterion, Plane stress},
series = {A {Special} {Issue} in {Honour} of {Professor} {Stephen} {R}. {Reid}'s 60th {Birthday}},
urldate = {2026-05-05},
}
@Article{Bai2008,
author = {Bai, Yuanli and Wierzbicki, Tomasz},
journal = {International Journal of Plasticity},
title = {A new model of metal plasticity and fracture with pressure and {Lode} dependence},
year = {2008},
issn = {0749-6419},
month = jun,
number = {6},
pages = {1071--1096},
volume = {24},
abstract = {Classical metal plasticity theory assumes that the hydrostatic pressure has no or negligible effect on the material strain hardening, and that the flow stress is independent of the third deviatoric stress invariant (or Lode angle parameter). However, recent experiments on metals have shown that both the pressure effect and the effect of the third deviatoric stress invariant should be included in the constitutive description of the material. A general form of asymmetric metal plasticity, considering both the pressure sensitivity and the Lode dependence, is postulated. The calibration method for the new metal plasticity is discussed. Experimental results on aluminum 2024-T351 are shown to validate the new material model. From the similarity between yielding surface and fracture locus, a new 3D asymmetric fracture locus, in the space of equivalent fracture strain, stress triaxiality and the Lode angle parameter, is postulated. Two methods of calibration of the fracture locus are discussed. One is based on classical round specimens and flat specimens in uniaxial tests, and the other one uses the newly designed butterfly specimen under biaxial testing. Test results of Bao (2003) [Bao, Y., 2003. Prediction of ductile crack formation in uncracked bodies. PhD Thesis, Massachusetts Institute of Technology] on aluminum 2024-T351, and test data points of A710 steel from butterfly specimens under biaxial testing validated the postulated asymmetric 3D fracture locus.},
doi = {10.1016/j.ijplas.2007.09.004},
keywords = {Calibration method, Fracture locus, Lode dependence, Pressure effect, Yield surface},
urldate = {2026-05-05},
}
@Article{Yoshida2002,
author = {Yoshida, Fusahito and Uemori, Takeshi},
journal = {International Journal of Plasticity},
title = {A model of large-strain cyclic plasticity describing the {Bauschinger} effect and workhardening stagnation},
year = {2002},
number = {5--6},
pages = {661--686},
volume = {18},
doi = {10.1016/S0749-6419(01)00050-X},
}
@Article{Yoshida2002a,
author = {Yoshida, Fusahito and Uemori, Takeshi and Fujiwara, Kenji},
journal = {International Journal of Plasticity},
title = {Elastic–plastic behavior of steel sheets under in-plane cyclic tension–compression at large strain},
year = {2002},
issn = {0749-6419},
month = oct,
number = {5},
pages = {633--659},
volume = {18},
abstract = {Elastic–plastic behavior of two types of steel sheets for press-forming (an aluminum-killed mild steel and a dual-phase high strength steel of 590MPa ultimate tensile strength) under in-plane cyclic tension–compression at large strain (up to 25\% strain for mild steel and 13\% for high strength steel) have been investigated. From the experiments, it was found that the cyclic hardening is strongly influenced by cyclic strain range and mean strain. Transient softening and workhardening stagnation due to the Bauschinger effect, as well as the decrease in Young's moduli with increasing prestrain, were also observed during stress reversals. Some important points in constitutive modeling for such large-strain cyclic elasto-plasticity are discussed by comparing the stress–strain responses calculated by typical constitutive models of mixed isotropic–kinematic hardening with the corresponding experimental observations.},
doi = {10.1016/S0749-6419(01)00049-3},
keywords = {B. Constitutive behaviour, B. Cyclic loading, B. Finite strain, B. Mechanical testing, Bauschinger effect},
urldate = {2026-05-05},
}
@Article{Jia2014a,
author = {Jia, Liang-Jiu and Kuwamura, Hitoshi},
journal = {Journal of Structural Engineering},
title = {Prediction of {Cyclic} {Behaviors} of {Mild} {Steel} at {Large} {Plastic} {Strain} {Using} {Coupon} {Test} {Results}},
year = {2014},
month = feb,
number = {2},
pages = {04013056},
volume = {140},
abstract = {AbstractIn practice, engineers can usually only obtain material properties from monotonic
tensile coupon tests. The aim of this paper is to predict cyclic plasticity of mild
steel from coupon test results available. First, the theory of metal plasticity ...},
doi = {10.1061/(ASCE)ST.1943-541X.0000848},
keywords = {Chaboche model, Cyclic plasticity, Cyclic tests, Metal and composite structures, Mild steel, Plasticity, Predictions, Structural steel, Yield plateau, Yoshida-Uemori model},
language = {en},
publisher = {American Society of Civil Engineers},
urldate = {2026-05-05},
}
@Article{Jia2014,
author = {Jia, Liang-Jiu},
journal = {Computers \& Structures},
title = {Integration algorithm for a modified {Yoshida}–{Uemori} model to simulate cyclic plasticity in extremely large plastic strain ranges up to fracture},
year = {2014},
issn = {0045-7949},
month = dec,
pages = {36--46},
volume = {145},
abstract = {Rate-independent metal plasticity models are widely applied to many fields, where most of the models are only effective for small strain ranges. A modified Yoshida–Uemori model was previously proposed and validated by the author, which can well evaluate metal plasticity at strain ranges until fracture. The model is difficult to implement into finite element software due to its complicated formation, which greatly limits its application. This paper aims to implement the modified model using a robust integration algorithm called adaptive substepping method. The implemented model is further validated by cyclic tests on mild steels at both material and member levels.},
doi = {10.1016/j.compstruc.2014.08.010},
keywords = {Cyclic loading, Large plasticity, Substepping, Yoshida–Uemori model},
urldate = {2026-05-05},
}
@Article{Rice1969,
author = {Rice, J. R. and Tracey, D. M.},
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year = {1969},
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number = {3},
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volume = {17},
abstract = {The fracture of ductile solids has frequently been observed to result from the large growth and coalescence of microscopic voids, a process enhanced by the superposition of hydrostatic tensile stresses on a plastic deformation field. The ductile growth of voids is treated here as a problem in continuum plasticity. First, a variational principle is established to characterize the flow field in an elastically rigid and incompressible plastic material containing an internal void or voids, and subjected to a remotely uniform stress and strain rate field. Then an approximate Rayleigh-Ritz procedure is developed and applied to the enlargement of an isolated spherical void in a nonhardening material. Growth is studied in some detail for the case of a remote tensile extension field with superposed hydrostatic stresses. The volume changing contribution to void growth is found to overwhelm the shape changing part when the mean remote normal stress is large, so that growth is essentially spherical. Further, it is found that for any remote strain rate field, the void enlargement rate is amplified over the remote strain rate by a factor rising exponentially with the ratio of mean normal stress to yield stress. Some related results are discussed, including the long cylindrical void considered by F.A. McClintock (1968, J. appl. Mech. 35, 363), and an approximate relation is given to describe growth of a spherical void in a general remote field. The results suggest a rapidly decreasing fracture ductility with increasing hydrostatic tension.},
doi = {10.1016/0022-5096(69)90033-7},
urldate = {2026-05-05},
}
@Article{Rastellini2016,
author = {Rastellini, F. and Socorro, G. and Forgas, A. and Oñate, E.},
journal = {J. Phys.: Conf. Ser.},
title = {A {Triaxial} {Failure} {Diagram} to predict the forming limit of {3D} sheet metal parts subjected to multiaxial stresses},
year = {2016},
issn = {1742-6596},
month = aug,
number = {3},
pages = {032020},
volume = {734},
abstract = {Accurate prediction of failure and forming limits is essential when modelling sheet metal forming processes. Since traditional Forming Limit Curves (FLCs) are not valid for materials subjected to triaxial loading, a new failure criterion is proposed in this paper based on the stress triaxility and the effective plastic strain accumulated during the history of material loading. Formability zones are identified inside the proposed Triaxial Failure Diagram (TFD). FLCs may be mapped into the TFD defining a new Triaxial Failure Curve, or it can be defined by triaxial failure experiments. Several TFD examples are validated and constrasted showing acceptable accuracy in the numerical prediction of forming failure/limit of 3D thick sheet parts.},
doi = {10.1088/1742-6596/734/3/032020},
language = {en},
publisher = {IOP Publishing},
urldate = {2026-05-05},
}
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