Refine text for clarity and precision in the optimization manuscript

Manuscript/RESILINK_surrogate_optimization_manuscript.tex

@@ -71,7 +71,7 @@ Shear-link beam (SLB) dampers are passive energy dissipation devices widely used
 
 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.
+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.
 
 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}.
 
@@ -79,7 +79,7 @@ Geometric optimization has also been extensively explored. Zhang et al. \cite{Zh
 
 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.
+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.
 
@@ -156,7 +156,7 @@ The calibrated FEM model is subsequently employed to generate the datasets used
 
 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 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.
 
@@ -166,7 +166,7 @@ For the purpose of optimization, TFDMap values are aggregated separately in the
 
 \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 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 results stored: 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.