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2026_Article_RESILINK_ML
Commit 863ccbb0 by Joaquín Irazábal González
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Add Batch 3 and Batch 4 changes for R1 manuscript

Batch 3: - Addressed reviewer comments on RBF training cost, motivation chain, novelty statement, component-level scope, seismic loading scope, resilience discussion, conclusions restructuring, consistency audit, and references. - Clarified scope and deferred certain analyses while ensuring all changes are documented and tracked. Batch 4: - Reconstructed optimization formulation and clarified damage constraints. - Introduced a hierarchical optimization approach with a focus on feasibility and shear-distortion performance. - Updated manuscript sections, figures, and addressed reviewer comments regarding methodology and terminology.
parent b99e6e24
Showing with 2345 additions and 119 deletions
Code/models/width_optimization/2W/ml_models/per_output_models_B34_H30/it1/local_oof_errors_2W_B34_H30_it1.csv
......@@ -44,3 +44,21 @@ tfmmax_frame,5,18.64,13.8,111.9409,114.5976509595643,2.656750959564306,0.0
tfmmax_frame,6,13.46,20.99,85.7607,93.48245442502184,7.721754425021842,0.0
tfmmax_frame,7,8.58,8.88,83.381,91.12531013954366,7.744310139543657,0.0
tfmmax_frame,8,15.17,17.61,94.4971,95.74313202279355,1.2460320227935426,0.0
min_window_tfd,0,15.22,18.46,79.5103,68.79487133175842,10.715428668241586,10.715428668241586
distortion_measure,0,15.22,18.46,2.2174451037299636e-06,2.525326484919471e-06,3.0788138118950727e-07,0.0
min_window_tfd,1,12.74,12.61,93.0272,97.73845131819569,4.711251318195693,0.0
distortion_measure,1,12.74,12.61,2.981550687625611e-06,3.6077026353863344e-06,6.261519477607236e-07,0.0
min_window_tfd,2,18.03,11.3,34.7023,37.2014881865335,2.4991881865335017,0.0
distortion_measure,2,18.03,11.3,3.9300911637151415e-06,3.533669953341776e-06,3.9642121037336555e-07,3.9642121037336555e-07
min_window_tfd,3,21.19,19.38,49.5951,71.22662844990502,21.63152844990502,0.0
distortion_measure,3,21.19,19.38,2.2790486312360745e-06,2.3069405841352184e-06,2.7891952899143904e-08,0.0
min_window_tfd,4,9.78,15.96,79.6009,95.8017268447797,16.200826844779698,0.0
distortion_measure,4,9.78,15.96,2.9513319664591657e-06,2.474073625458256e-06,4.772583410009098e-07,4.772583410009098e-07
min_window_tfd,5,18.64,13.8,37.1978,35.300995927654284,1.8968040723457165,1.8968040723457165
distortion_measure,5,18.64,13.8,3.190293870445974e-06,3.200119248881606e-06,9.825378435631842e-09,0.0
min_window_tfd,6,13.46,20.99,59.7863,82.66812785858757,22.88182785858757,0.0
distortion_measure,6,13.46,20.99,2.20994174682602e-06,2.5426629710617117e-06,3.327212242356917e-07,0.0
min_window_tfd,7,8.58,8.88,144.4093,109.54438649997257,34.86491350002743,34.86491350002743
distortion_measure,7,8.58,8.88,3.7422193481538176e-06,2.7666416698819123e-06,9.755776782719054e-07,9.755776782719054e-07
min_window_tfd,8,15.17,17.61,85.6117,82.19519087448936,3.4165091255106432,3.4165091255106432
distortion_measure,8,15.17,17.61,2.319332804682279e-06,2.2188032040150535e-06,1.0052960066722558e-07,1.0052960066722558e-07
Code/reports/width_optimization/2W/ml_optimization/it1/optimization_results_2W_B34_H30_TFD_W100.csv
......@@ -4,11 +4,22 @@ Configuration_B,34.0
Configuration_H,30.0
Configuration_TFD_W,100.0
Iteration,1.0
tw1_optimal,15.756209076636766
tw2_optimal,19.999999467023997
Objective_score,3.682647845498934
Exy_tw1,0.049785817649385625
Exy_tw2,0.04521727380072006
TFM_tw1,100.70532607837181
TFM_tw2,59.680000841580224
TFM_frame,93.6826477616043
tw1_optimal,14.389564101162314
tw2_optimal,16.689202205393375
candidate_tw1_raw,14.389564101162314
candidate_tw1,14.39
rounding_delta_tw1,0.00043589883768646587
candidate_tw2_raw,16.689202205393375
candidate_tw2,16.69
rounding_delta_tw2,0.000797794606626212
Objective_score,3.664084434454253
Exy_tw1,0.05236470345214329
Exy_tw2,0.05924118181423681
TFM_tw1,100.68676266732713
TFM_tw2,92.25103337960783
TFM_frame,93.66408432820377
manuf_Exy_tw1,0.052364206945286006
manuf_Exy_tw2,0.05923753693226129
manuf_TFM_tw1,100.68592131258363
manuf_TFM_tw2,92.24298645448413
manuf_TFM_frame,93.66445935040625
Code/reports/width_optimization/2W/ml_optimization/ml_2W_B34_H30_TFD_W100_it1_feasibility_de_convergence.csv 0 → 100644
run,seed,generation,best_objective,scipy_convergence_measure
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Code/reports/width_optimization/2W/ml_optimization/ml_2W_B34_H30_TFD_W100_it1_feasibility_de_summary.csv 0 → 100644
n_runs,min_objective,median_objective,mean_objective,std_objective,iqr_objective,median_nit,median_nfev,tw1_median,tw1_mean,tw1_std,tw1_iqr,tw2_median,tw2_mean,tw2_std,tw2_iqr,n_basins,reproducibility_rate,geometry_n_basins,geometry_reproducibility_rate,dominant_basin_count,geometry_dominant_basin_count,dominant_basin_indices,geometry_dominant_basin_indices,best_basin_run_idx,best_basin_objective,best_basin_x,tw1_dominant_basin_std,tw1_dominant_basin_iqr,tw1_dominant_basin_range,tw2_dominant_basin_std,tw2_dominant_basin_iqr,tw2_dominant_basin_range,performance_reproducibility_rate,performance_equivalent_count,performance_equivalent_indices,performance_best_objective,performance_best_run_idx,performance_relative_tolerance,performance_max_relative_error_equivalent_group,performance_objective_spread,performance_objective_iqr,n_total_runs,n_valid_runs,n_successful_runs,representative_run_idx,geometry_basin_rep_run_idx
30,3.664084434454253,3.6826480353498283,3.6795542324384294,0.007036442592627075,3.965913606407412e-07,63.0,3200.0,15.75620630861528,15.528432032447759,0.5180238811309796,8.011772978377962e-06,19.9999925106928,19.44819262770685,1.2549506563879291,1.992621255197946e-05,2,0.8333333333333334,2,0.8333333333333334,25,25,12;8;28;14;7;26;5;18;22;4;23;13;25;10;27;3;16;0;2;1;29;15;6;21;19,12;8;28;14;7;26;5;18;22;4;23;13;25;10;27;3;16;0;2;1;29;15;6;21;19,12,3.682647845498934,"[15.756209076636766, 19.999999467023997]",3.5130721387288365e-06,2.814688865981907e-06,1.1960812896560924e-05,8.74595445351237e-06,7.066766436025773e-06,2.9809761791455003e-05,0.16666666666666666,5,9;11;17;20;24,3.664084434454253,11,0.001,1.9422532869228138e-07,0.01856414989806865,3.965913606407412e-07,30,30,30,11,12
Code/reports/width_optimization/2W/ml_optimization/ml_2W_B34_H30_TFD_W100_it1_minimum_violation_solution.csv 0 → 100644
w_val,b_val,h_val,tfd_w_val,it,feasible,minimum_identified_violation,tw1,tw2,TFD_tw1,TFD_tw2,TFD_tw1_margin,TFD_tw2_margin,TFD_tw1_conservative,TFD_tw2_conservative,TFD_frame,TFD_frame_margin,TFD_frame_conservative,safety_margin_mode,safety_margin_factor,max_violation,viol_window_1,viol_window_2,viol_frame
2,34,30,100,1,False,3.664084434454253,14.389564101162314,16.689202205393375,100.68676266732713,92.25103337960783,2.9773217671271226,10.715428668241586,103.66408443445425,102.96646204784942,93.66408432820377,0.0,93.66408432820377,local_oof,1.0,3.664084434454253,3.664084434454253,2.9664620478494186,3.6640843282037707
Code/reports/width_optimization/3W/ml_optimization/it4/optimization_results_3W_B29_H45_TFD_W100.csv
......@@ -7,6 +7,15 @@ Iteration,4.0
tw1_optimal,5.107783743688125
tw2_optimal,7.449034156017617
tw3_optimal,8.371536009042268
candidate_tw1_raw,5.107783743688125
candidate_tw1,5.11
rounding_delta_tw1,0.0022162563118754974
candidate_tw2_raw,7.449034156017617
candidate_tw2,7.45
rounding_delta_tw2,0.0009658439823834186
candidate_tw3_raw,8.371536009042268
candidate_tw3,8.37
rounding_delta_tw3,-0.0015360090422689154
Objective_score,-1.306799128950891e-06
Exy_tw1,0.0531167629314931
Exy_tw2,0.05536691245735284
......@@ -15,3 +24,10 @@ TFM_tw1,99.21442333488952
TFM_tw2,94.93646818772172
TFM_tw3,90.29039221944643
TFM_frame,57.501208891048265
manuf_Exy_tw1,0.05308684408937943
manuf_Exy_tw2,0.05535816537536256
manuf_Exy_tw3,0.056690156921532175
manuf_TFM_tw1,99.14917150028359
manuf_TFM_tw2,94.90523864610947
manuf_TFM_tw3,90.32942764409073
manuf_TFM_frame,57.51345572087089
Code/reports/width_optimization/3W/ml_optimization/ml_3W_B29_H45_TFD_W100_it4_lexicographic_optimization_summary.csv
w_val,b_val,h_val,tfd_w_val,it,safety_margin_mode,safety_margin_factor,local_margin_support_pass,local_margin_k_requested,local_support_percentile,local_margin_fallback,TFD_tw1_margin_source,TFD_tw1_margin_in_support,TFD_tw1_margin_k,TFD_tw1_margin_nearest_distance,TFD_tw1_margin_support_radius,TFD_tw1_margin_neighbor_samples,TFD_tw1_margin_neighbor_distances,TFD_tw1_margin_neighbor_underprediction,TFD_tw2_margin_source,TFD_tw2_margin_in_support,TFD_tw2_margin_k,TFD_tw2_margin_nearest_distance,TFD_tw2_margin_support_radius,TFD_tw2_margin_neighbor_samples,TFD_tw2_margin_neighbor_distances,TFD_tw2_margin_neighbor_underprediction,TFD_tw3_margin_source,TFD_tw3_margin_in_support,TFD_tw3_margin_k,TFD_tw3_margin_nearest_distance,TFD_tw3_margin_support_radius,TFD_tw3_margin_neighbor_samples,TFD_tw3_margin_neighbor_distances,TFD_tw3_margin_neighbor_underprediction,TFD_frame_margin_source,TFD_frame_margin_in_support,TFD_frame_margin_k,TFD_frame_margin_nearest_distance,TFD_frame_margin_support_radius,TFD_frame_margin_neighbor_samples,TFD_frame_margin_neighbor_distances,TFD_frame_margin_neighbor_underprediction,min_de_reproducibility_required,stage1_performance_reproducibility_rate,stage2_performance_reproducibility_rate,stage1_performance_reproducibility_pass,stage2_performance_reproducibility_pass,de_performance_reproducibility_pass,stage1_geometry_reproducibility_rate,stage2_geometry_reproducibility_rate,stage1_geometry_n_basins,stage2_geometry_n_basins,geometry_unique_or_reproducible,stage1_performance_spread_min_window_tfd,stage2_best_distortion_measure,stage2_performance_spread_distortion,feasible_domain_found,minimum_domain_violation,feasible,tw1,tw2,tw3,TFD_tw1,TFD_tw2,TFD_tw3,TFD_tw1_margin,TFD_tw2_margin,TFD_tw3_margin,TFD_tw1_conservative,TFD_tw2_conservative,TFD_tw3_conservative,TFD_frame,TFD_frame_margin,TFD_frame_conservative,min_window_tfd,max_window_tfd,mean_window_tfd,window_tfd_range_diagnostic,distortion_measure,stage1_Tmin_star,stage1_preservation_threshold,max_constraint_violation,stage1_preservation_tolerance,stage1_preservation_tolerance_source,stage1_performance_group_count,stage1_rep_run,stage1_rep_seed,stage2_rep_run,stage2_rep_seed,stage2_tw1_dominant_basin_iqr,stage2_tw2_dominant_basin_iqr,stage2_tw3_dominant_basin_iqr,stage2_tw1_dominant_basin_range,stage2_tw2_dominant_basin_range,stage2_tw3_dominant_basin_range
3,29,45,100,4,local_oof,1.0,True,-1,95.0,global,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;0;0;0;0,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;5.0635262;0.4480527;0;0,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;9.7096075;0;0;3.0397123,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;0.8826844;0.27213773;0;0.094025442,0.8,1.0,1.0,True,True,True,0.3,1.0,6,1,False,3.28507100277875e-06,1.306799128950891e-06,2.3937105624913084e-13,True,0.0,True,5.107783743688125,7.449034156017617,8.371536009042268,99.21442333488952,94.93646818772172,90.29039221944643,0.0,5.063526170536292,9.709607450118042,99.21442333488952,99.99999435825801,99.99999966956447,57.501208891048265,0.8826844009775954,58.38389329202586,90.29039221944643,99.21442333488952,94.81376124735255,8.924031115443086,1.306799128950891e-06,90.2903925476171,90.2893925476171,0.0,0.001,performance_group_spread,30,5,47,27,69,0.0014176657899946932,9.24724214712569e-05,0.0005873247940790804,0.005601095003969547,0.0003629683766019909,0.002319526779546166
w_val,b_val,h_val,tfd_w_val,it,safety_margin_mode,safety_margin_factor,local_margin_support_pass,local_margin_k_requested,local_support_percentile,local_margin_fallback,TFD_tw1_margin_source,TFD_tw1_margin_in_support,TFD_tw1_margin_k,TFD_tw1_margin_nearest_distance,TFD_tw1_margin_support_radius,TFD_tw1_margin_neighbor_samples,TFD_tw1_margin_neighbor_distances,TFD_tw1_margin_neighbor_underprediction,TFD_tw2_margin_source,TFD_tw2_margin_in_support,TFD_tw2_margin_k,TFD_tw2_margin_nearest_distance,TFD_tw2_margin_support_radius,TFD_tw2_margin_neighbor_samples,TFD_tw2_margin_neighbor_distances,TFD_tw2_margin_neighbor_underprediction,TFD_tw3_margin_source,TFD_tw3_margin_in_support,TFD_tw3_margin_k,TFD_tw3_margin_nearest_distance,TFD_tw3_margin_support_radius,TFD_tw3_margin_neighbor_samples,TFD_tw3_margin_neighbor_distances,TFD_tw3_margin_neighbor_underprediction,TFD_frame_margin_source,TFD_frame_margin_in_support,TFD_frame_margin_k,TFD_frame_margin_nearest_distance,TFD_frame_margin_support_radius,TFD_frame_margin_neighbor_samples,TFD_frame_margin_neighbor_distances,TFD_frame_margin_neighbor_underprediction,min_de_reproducibility_required,stage1_performance_reproducibility_rate,stage2_performance_reproducibility_rate,stage1_performance_reproducibility_pass,stage2_performance_reproducibility_pass,de_performance_reproducibility_pass,stage1_geometry_reproducibility_rate,stage2_geometry_reproducibility_rate,stage1_geometry_n_basins,stage2_geometry_n_basins,geometry_unique_or_reproducible,stage1_performance_spread_min_window_tfd,stage2_best_distortion_measure,stage2_performance_spread_distortion,feasible_domain_found,minimum_domain_violation,feasible,tw1,tw2,tw3,geometry_projection_applied,manufacturing_thickness_resolution_mm,candidate_tw1_raw,candidate_tw2_raw,candidate_tw3_raw,candidate_tw1,candidate_tw2,candidate_tw3,rounding_delta_tw1,rounding_delta_tw2,rounding_delta_tw3,manuf_min_window_tfd,manuf_max_window_tfd,manuf_mean_window_tfd,manuf_distortion_measure,manuf_frame_tfd,manuf_frame_tfd_conservative,manuf_max_constraint_violation,manuf_feasible,manuf_local_margin_support_pass,manuf_TFD_tw1,manuf_TFD_tw2,manuf_TFD_tw3,manuf_TFD_tw1_conservative,manuf_TFD_tw2_conservative,manuf_TFD_tw3_conservative,TFD_tw1,TFD_tw2,TFD_tw3,TFD_tw1_margin,TFD_tw2_margin,TFD_tw3_margin,TFD_tw1_conservative,TFD_tw2_conservative,TFD_tw3_conservative,TFD_frame,TFD_frame_margin,TFD_frame_conservative,min_window_tfd,max_window_tfd,mean_window_tfd,window_tfd_range_diagnostic,distortion_measure,stage1_Tmin_star,stage1_preservation_threshold,max_constraint_violation,stage1_preservation_tolerance,stage1_preservation_tolerance_source,stage1_performance_group_count,stage1_rep_run,stage1_rep_seed,stage2_rep_run,stage2_rep_seed,stage2_tw1_dominant_basin_iqr,stage2_tw2_dominant_basin_iqr,stage2_tw3_dominant_basin_iqr,stage2_tw1_dominant_basin_range,stage2_tw2_dominant_basin_range,stage2_tw3_dominant_basin_range
3,29,45,100,4,local_oof,1.0,True,-1,95.0,global,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;0;0;0;0,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;5.0635262;0.4480527;0;0,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;9.7096075;0;0;3.0397123,local_knn_max,True,5,0.19796330804369972,0.5722090560728312,17;2;19;16;18,0.19796331;0.21295065;0.31280287;0.38363723;0.4405638,0;0.8826844;0.27213773;0;0.094025442,0.8,1.0,1.0,True,True,True,0.3,1.0,6,1,False,3.28507100277875e-06,1.306799128950891e-06,2.3937105624913084e-13,True,0.0,True,5.107783743688125,7.449034156017617,8.371536009042268,True,0.01,5.107783743688125,7.449034156017617,8.371536009042268,5.11,7.45,8.37,0.0022162563118754974,0.0009658439823834186,-0.0015360090422689154,90.32942764409073,99.14917150028359,94.79461259682792,1.3068664539339266e-06,57.51345572087089,58.39614012184848,0.039035094208770715,False,True,99.14917150028359,94.90523864610947,90.32942764409073,99.14917150028359,99.96876481664576,100.03903509420877,99.21442333488952,94.93646818772172,90.29039221944643,0.0,5.063526170536292,9.709607450118042,99.21442333488952,99.99999435825801,99.99999966956447,57.501208891048265,0.8826844009775954,58.38389329202586,90.29039221944643,99.21442333488952,94.81376124735255,8.924031115443086,1.306799128950891e-06,90.2903925476171,90.2893925476171,0.0,0.001,performance_group_spread,30,5,47,27,69,0.0014176657899946932,9.24724214712569e-05,0.0005873247940790804,0.005601095003969547,0.0003629683766019909,0.002319526779546166
Code/reports/width_optimization/5W/rbf_optimization/rbf_5W_B34_H60_TFD_W100_it3_adaptive_acceptance_summary.csv
surrogate,W,B,H,TFD_W_limit,TFD_frame_limit,iteration,candidate_row_zero_based,geometry_match,geometry_match_tolerance_mm,geometry_projection_applied,manufacturing_thickness_resolution_mm,surrogate_validation_geometry_matches_fem_candidate,tfd_physical_acceptance_resolution,tfd_physical_acceptance_decimals,tfd_physical_acceptance_rounding,fem_tfmmax_tw1_raw,fem_tfmmax_tw1_acceptance,fem_tfmmax_tw1_limit,fem_tfmmax_tw1_physical_pass,fem_tfmmax_tw2_raw,fem_tfmmax_tw2_acceptance,fem_tfmmax_tw2_limit,fem_tfmmax_tw2_physical_pass,fem_tfmmax_tw3_raw,fem_tfmmax_tw3_acceptance,fem_tfmmax_tw3_limit,fem_tfmmax_tw3_physical_pass,fem_tfmmax_tw4_raw,fem_tfmmax_tw4_acceptance,fem_tfmmax_tw4_limit,fem_tfmmax_tw4_physical_pass,fem_tfmmax_tw5_raw,fem_tfmmax_tw5_acceptance,fem_tfmmax_tw5_limit,fem_tfmmax_tw5_physical_pass,fem_tfmmax_frame_raw,fem_tfmmax_frame_acceptance,fem_tfmmax_frame_limit,fem_tfmmax_frame_physical_pass,fem_feasible,damage_local_accuracy_pass,min_window_tfd_local_accuracy_pass,distortion_measure_local_accuracy_pass,surrogate_local_accuracy_pass,local_acceptance_support_pass,local_margin_support_pass,C_FEM_fem_feasibility,C_FEM_surrogate_local_accuracy,C_FEM_local_support,C_FEM,C_FEM_definition,local_oof_support_pass,surrogate_local_validity_pass,safety_margin_mode,de_reproducibility_warning_threshold,de_final_reproducibility_requirement,de_reproducibility_source,stage1_performance_reproducibility_rate,stage2_performance_reproducibility_rate,stage1_R_perf,stage2_R_perf,stage1_n_runs,stage2_n_runs,stage1_n_equivalent,stage2_n_equivalent,stage1_reproducibility_warning_pass,stage2_reproducibility_warning_pass,stage1_final_reproducibility_pass,stage2_final_reproducibility_pass,de_performance_reproducibility_pass,stage1_geometry_reproducibility_rate,stage2_geometry_reproducibility_rate,geometry_reproducibility_diagnostic,accepted,final_converged,recommended_action,diagnostic_recommendation,continuation_reason,failed_criteria,failed_subcriteria,diagnostic_warnings,C_DE,C_STABILITY,current_stage1_Tmin_star,previous_stage1_Tmin_star,delta_stage1_optimum,delta_stage1_optimum_percent,stage1_optimum_stable,stage1_optimum_stability_status,current_stage2_distortion,previous_stage2_distortion,delta_stage2_optimum,delta_stage2_optimum_percent,stage2_optimum_stable,stage2_optimum_stability_status,adaptive_stability_limit,adaptive_stability_limit_percent,report_schema_version,convergence_logic_version,fem_min_window_tfd,pred_min_window_tfd,fem_distortion_measure,pred_distortion_measure,surrogate_manuf_window_tfd_pass,surrogate_manuf_frame_tfd_pass,surrogate_manuf_conservative_pass,surrogate_manuf_stage1_preservation_pass,surrogate_manuf_feasibility_pass,candidate_tw1_raw,candidate_tw2_raw,candidate_tw3_raw,candidate_tw4_raw,candidate_tw5_raw,candidate_tw1,candidate_tw2,candidate_tw3,candidate_tw4,candidate_tw5,rounding_delta_tw1,rounding_delta_tw2,rounding_delta_tw3,rounding_delta_tw4,rounding_delta_tw5,delta_tw1,de_repeatability_range_tw1,delta_within_de_repeatability_tw1,delta_tw2,de_repeatability_range_tw2,delta_within_de_repeatability_tw2,delta_tw3,de_repeatability_range_tw3,delta_within_de_repeatability_tw3,delta_tw4,de_repeatability_range_tw4,delta_within_de_repeatability_tw4,delta_tw5,de_repeatability_range_tw5,delta_within_de_repeatability_tw5
rbf,5,34,60,100.0,90.0,3,67,True,0.01,True,0.01,True,0.1,1,ROUND_HALF_UP,79.1451,79.1,100.0,True,80.1063,80.1,100.0,True,80.1643,80.2,100.0,True,80.1658,80.2,100.0,True,79.489,79.5,100.0,True,69.417,69.4,90.0,True,True,True,True,True,True,True,True,True,True,True,True,physical_fem_feasibility_only,True,True,local_oof,0.8,1.0,optimizer_summary_performance_fields,1.0,1.0,1.0,1.0,30,30,30,30,True,True,True,True,True,1.0,1.0,PERFORMANCE_AND_GEOMETRY_REPRODUCIBLE,True,True,ACCEPT_FEM_VALIDATED_NUMERICAL_OPTIMUM,,All convergence criteria satisfied,,,,True,True,80.1796837078873,80.92166891936199,0.009254030162776698,0.9254030162776699,True,STABLE,1.88218321432158e-06,1.912078346172097e-06,0.015883220944190944,1.5883220944190943,True,STABLE,0.02,2.0,4,three_criteria_fem_feasibility_v2,79.1451,80.05590646560654,1.847173915819322e-06,1.8810684055988922e-06,True,True,True,False,False,6.207050750368829,7.855436182445988,8.903277365568599,7.027149222369977,5.000000134079064,6.21,7.86,8.9,7.03,5.0,0.002949249631170936,0.0045638175540121395,-0.003277365568598256,0.0028507776300230603,-1.3407906429563354e-07,0.011742264244341882,5.155261388267718e-05,False,0.008228910860080596,8.751570462983693e-05,False,0.033197042276370325,1.1715007552481891e-05,False,0.04141376545347786,6.298532078119479e-06,False,8.545370100421223e-08,7.24250991623876e-07,True
surrogate,W,B,H,TFD_W_limit,TFD_frame_limit,iteration,candidate_row_zero_based,geometry_match,geometry_match_tolerance_mm,geometry_projection_applied,manufacturing_thickness_resolution_mm,surrogate_validation_geometry_matches_fem_candidate,tfd_physical_acceptance_resolution,tfd_physical_acceptance_decimals,tfd_physical_acceptance_rounding,fem_tfmmax_tw1_raw,fem_tfmmax_tw1_acceptance,fem_tfmmax_tw1_limit,fem_tfmmax_tw1_physical_pass,fem_tfmmax_tw2_raw,fem_tfmmax_tw2_acceptance,fem_tfmmax_tw2_limit,fem_tfmmax_tw2_physical_pass,fem_tfmmax_tw3_raw,fem_tfmmax_tw3_acceptance,fem_tfmmax_tw3_limit,fem_tfmmax_tw3_physical_pass,fem_tfmmax_tw4_raw,fem_tfmmax_tw4_acceptance,fem_tfmmax_tw4_limit,fem_tfmmax_tw4_physical_pass,fem_tfmmax_tw5_raw,fem_tfmmax_tw5_acceptance,fem_tfmmax_tw5_limit,fem_tfmmax_tw5_physical_pass,fem_tfmmax_frame_raw,fem_tfmmax_frame_acceptance,fem_tfmmax_frame_limit,fem_tfmmax_frame_physical_pass,fem_feasible,damage_local_accuracy_pass,min_window_tfd_local_accuracy_pass,distortion_measure_local_accuracy_pass,surrogate_local_accuracy_pass,local_acceptance_support_pass,local_margin_support_pass,C_FEM_fem_feasibility,C_FEM_surrogate_local_accuracy,C_FEM_local_support,C_FEM,C_FEM_definition,local_oof_support_pass,surrogate_local_validity_pass,safety_margin_mode,de_reproducibility_warning_threshold,de_final_reproducibility_requirement,de_reproducibility_source,stage1_performance_reproducibility_rate,stage2_performance_reproducibility_rate,stage1_R_perf,stage2_R_perf,stage1_n_runs,stage2_n_runs,stage1_n_equivalent,stage2_n_equivalent,stage1_reproducibility_warning_pass,stage2_reproducibility_warning_pass,stage1_final_reproducibility_pass,stage2_final_reproducibility_pass,de_performance_reproducibility_pass,stage1_geometry_reproducibility_rate,stage2_geometry_reproducibility_rate,geometry_reproducibility_diagnostic,accepted,final_converged,recommended_action,diagnostic_recommendation,continuation_reason,failed_criteria,failed_subcriteria,diagnostic_warnings,C_DE,C_STABILITY,C_CONS,conservatism_gap_rtol,conservatism_gap_eps,stage1_Tmin_star_conservative,stage1_Tmin_star_shadow,stage1_conservatism_gap_absolute,stage1_conservatism_gap_relative,stage1_conservative_margin_materially_limiting,stage1_shadow_status,shadow_stage1_assessable,shadow_stage1_n_runs,shadow_stage1_seed_base,shadow_stage1_equivalent_runs,shadow_stage1_R_perf,shadow_stage1_geometry_reproducibility,shadow_stage1_max_constraint_violation,stage1_shadow_message,shadow_stage1_tw1,shadow_stage1_tw2,shadow_stage1_tw3,shadow_stage1_tw4,shadow_stage1_tw5,current_stage1_Tmin_star,previous_stage1_Tmin_star,delta_stage1_optimum,delta_stage1_optimum_percent,stage1_optimum_stable,stage1_optimum_stability_status,current_stage2_distortion,previous_stage2_distortion,delta_stage2_optimum,delta_stage2_optimum_percent,stage2_optimum_stable,stage2_optimum_stability_status,adaptive_stability_limit,adaptive_stability_limit_percent,report_schema_version,convergence_logic_version,fem_min_window_tfd,pred_min_window_tfd,fem_distortion_measure,pred_distortion_measure,surrogate_manuf_window_tfd_pass,surrogate_manuf_frame_tfd_pass,surrogate_manuf_conservative_pass,surrogate_manuf_stage1_preservation_pass,surrogate_manuf_feasibility_pass,candidate_tw1_raw,candidate_tw2_raw,candidate_tw3_raw,candidate_tw4_raw,candidate_tw5_raw,candidate_tw1,candidate_tw2,candidate_tw3,candidate_tw4,candidate_tw5,rounding_delta_tw1,rounding_delta_tw2,rounding_delta_tw3,rounding_delta_tw4,rounding_delta_tw5,delta_tw1,de_repeatability_range_tw1,delta_within_de_repeatability_tw1,delta_tw2,de_repeatability_range_tw2,delta_within_de_repeatability_tw2,delta_tw3,de_repeatability_range_tw3,delta_within_de_repeatability_tw3,delta_tw4,de_repeatability_range_tw4,delta_within_de_repeatability_tw4,delta_tw5,de_repeatability_range_tw5,delta_within_de_repeatability_tw5
rbf,5,34,60,100.0,90.0,3,67,True,0.01,True,0.01,True,0.1,1,ROUND_HALF_UP,79.1451,79.1,100.0,True,80.1063,80.1,100.0,True,80.1643,80.2,100.0,True,80.1658,80.2,100.0,True,79.489,79.5,100.0,True,69.417,69.4,90.0,True,True,True,True,True,True,True,True,True,True,True,True,physical_fem_feasibility_only,True,True,local_oof,0.8,1.0,optimizer_summary_performance_fields,1.0,1.0,1.0,1.0,30,30,30,30,True,True,True,True,True,1.0,1.0,PERFORMANCE_AND_GEOMETRY_REPRODUCIBLE,True,True,ACCEPT_FEM_VALIDATED_NUMERICAL_OPTIMUM,,All convergence criteria satisfied,,,,True,True,True,0.02,1e-12,80.1796837078873,80.17968698439654,3.2765092328190804e-06,4.086457999588735e-08,False,OK,True,30,42,30,1.0,1.0,0.0,,6.2070114742967935,7.8559031662161924,8.903243340424957,7.027129892110018,5.0000009198196365,80.1796837078873,80.92166891936199,0.009254030162776698,0.9254030162776699,True,STABLE,1.88218321432158e-06,1.912078346172097e-06,0.015883220944190944,1.5883220944190943,True,STABLE,0.02,2.0,5,four_criteria_fem_de_stability_conservatism_v3,79.1451,80.05590646560654,1.847173915819322e-06,1.8810684055988922e-06,True,True,True,False,False,6.207050750368829,7.855436182445988,8.903277365568599,7.027149222369977,5.000000134079064,6.21,7.86,8.9,7.03,5.0,0.002949249631170936,0.0045638175540121395,-0.003277365568598256,0.0028507776300230603,-1.3407906429563354e-07,0.011742264244341882,5.155261388267718e-05,False,0.008228910860080596,8.751570462983693e-05,False,0.033197042276370325,1.1715007552481891e-05,False,0.04141376545347786,6.298532078119479e-06,False,8.545370100421223e-08,7.24250991623876e-07,True
ManuscriptR1/ComparisonSurrogatesOptimizationBDSL_R1.tex
......@@ -66,12 +66,12 @@
country={Spain}}
\begin{abstract}
Buckling-delayed shear-link dampers are passive devices used in seismic-resistant structures to concentrate energy dissipation while limiting damage to the primary system. Optimizing their window thicknesses requires balancing high dissipative capacity with strict control of local damage. Nonlinear finite element models can reproduce their cyclic response and provide internal quantities for optimization, such as damage indicators and local distortion, but their computational cost prevents direct use within iterative optimization loops. This work proposes an adaptive surrogate-assisted optimization framework for damage-aware window-thickness optimization of buckling-delayed shear-link dampers within five predefined geometry families. Experimentally calibrated nonlinear finite element models are first used to generate reference datasets for different device configurations. Supervised learning models are evaluated, with support vector regression and Gaussian process regression providing the most accurate predictions among the candidate models, motivating the use of radial basis function surrogates as a more efficient distance-based alternative. The surrogate predictions are coupled with a differential evolution algorithm through a damage-aware objective function that controls local damage while using dissipative performance as an optimization criterion. The optimized geometries are then re-evaluated with finite element simulations. If the acceptance criteria are not met, the new simulation is added to the dataset and the surrogate models are retrained. The framework enables efficient damage-aware optimization of seismic energy dissipation devices.
Buckling-delayed shear-link dampers are passive devices used in seismic-resistant structures to concentrate energy dissipation while limiting damage to the primary system. Optimizing their window thicknesses requires balancing high dissipative capacity with strict control of local damage. Nonlinear finite element models can reproduce their cyclic response and provide internal quantities for optimization, such as damage indicators and local distortion, but their computational cost prevents direct use within iterative optimization loops. This work proposes an adaptive surrogate-assisted optimization framework for damage-aware window-thickness optimization of buckling-delayed shear-link dampers within five predefined geometry families. Experimentally calibrated nonlinear finite element models are first used to generate reference datasets for different device configurations. Supervised learning models are evaluated, with support vector regression and Gaussian process regression providing the most accurate predictions among the candidate models, motivating the use of radial basis function surrogates, whose training requires a simpler procedure, as a complementary distance-based alternative. The surrogate predictions are coupled with a differential evolution algorithm in a feasibility-first, hierarchical optimization that enforces prescribed damage-screening limits on the windows and on the surrounding frame and then uses a shear-distortion performance indicator as the secondary criterion. The optimized geometries are then re-evaluated with finite element simulations. If the acceptance criteria are not met, the new simulation is added to the dataset and the surrogate models are retrained. The added value of the framework is the integration of local, FEM-derived damage indicators into a window-thickness optimization that protects the surrounding frame and verifies each candidate design with FEM.
\end{abstract}
\begin{highlights}
\item Adaptive surrogates optimize damage-aware shear-link damper geometries
\item RBF interpolation offers comparable accuracy to supervised ML at lower training cost
\item RBF interpolation offers comparable accuracy with a simpler training procedure
\item FEM validation filters surrogate optima before accepting final designs
\item Optimized devices balance window activation and frame damage control
\end{highlights}
......@@ -99,7 +99,15 @@ Data-driven approaches have mainly focused on response or property prediction. C
All these works demonstrate the increasing interest in applying FEM-based and data-driven approaches, as well as in combining both, to analyse, understand and optimize seismic energy dissipation devices. However, most of these studies focus either on the prediction of the hysteretic response or on maximizing energy dissipation, leaving a critical aspect insufficiently explored: the need to control local damage while maintaining adequate dissipative capacity. In practice, excessive local damage may compromise structural integrity, reduce durability and lead to premature failure, even when global energy dissipation is improved.
The present work addresses this gap through a damage-aware surrogate-assisted optimization methodology in which the objective is not only to maximize distortion or energy dissipation, but also to balance dissipative performance with damage indicators derived from FEM simulations. The proposed approach combines: (i) experimentally calibrated nonlinear FEM models used as numerical ground truth; (ii) supervised surrogate models trained to predict local damage and distortion indicators; (iii) a Differential Evolution (DE) optimizer; and (iv) an adaptive FEM validation and retraining loop.
BDSL device performance therefore depends not only on global force or total dissipated energy, but on how local deformation and damage are distributed among the individual windows and the surrounding frame. The window thicknesses exert a direct control on this distribution, because they set the relative stiffness of each dissipative region and thus the balance between window activation and the inelastic demand transferred to the frame. Nonlinear FEM can resolve these local quantities, but its cost makes repeated direct optimization impractical. A surrogate-assisted approach therefore offers a practical route to search the window-thickness design space while explicitly accounting for damage-related responses.
The present work addresses this gap through a damage-aware surrogate-assisted optimization methodology for window-thickness optimization within five predefined BDSL geometry families. The proposed approach combines: (i) experimentally calibrated nonlinear FEM models used as numerical ground truth; (ii) supervised and radial basis function (RBF) 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 contribution of this study is not the use of surrogate optimization itself, which is well established for metallic dampers, but the combination of the following elements, all of which are implemented and assessed in the present framework: (i) damage-aware window-thickness optimization driven by local FEM-derived indicators in the windows and in the frame, rather than by global force or total energy alone; (ii) explicit protection of the surrounding frame and balanced activation of the dissipative windows; (iii) a common nested cross-validation framework that allows a like-for-like comparison of six supervised learning algorithms and RBF interpolation on identical data splits; and (iv) an adaptive FEM enrichment loop in which a candidate is accepted on the basis of FEM-confirmed feasibility and optimizer robustness, rather than on surrogate predictions alone.
The study is deliberately component-level. It optimizes the device and evaluates its cyclic response under prescribed, displacement-controlled loading, which isolates the nonlinear response of the device and provides controlled deformation histories for comparing geometry variants. It does not analyse the response of a complete building or structural system equipped with the optimized dampers.
From a broader perspective, BDSL dampers are conceived as replaceable, sacrificial components that concentrate damage away from primary structural members, which can facilitate post-event inspection, repair or replacement \cite{Xiong2024}. The present study contributes to this objective at the component level by controlling where damage develops and by protecting the surrounding frame. It does not, however, quantify resilience: metrics such as downtime, repair cost or functional recovery are not computed, and system-level resilience assessment is outside the scope of this work.
Figure \ref{fig:MethodologyFlowChart} summarizes the proposed workflow. The different stages of the methodology, together with the surrogate modelling, optimization strategy, validation procedure and corresponding results and conclusions, are described in the following sections.
......@@ -121,7 +129,7 @@ The BDSL dampers analysed in this work, with one representative configuration sh
\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. 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 stay 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.
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 controlled more strictly 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 stay 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.
Accordingly, the design variables considered in this work are the window thicknesses, while the frame dimensions are kept fixed:
\begin{equation}
......@@ -237,7 +245,7 @@ To improve surrogate robustness near the admissible limits, the sampling domain
\end{tabular}
\end{table*}
For every sampled configuration, a FEM simulation is performed under a displacement-controlled cyclic loading protocol. Since the admissible deformation demand depends on the size of the device, different loading patterns are adopted according to the device height. As shown in Figure~\ref{fig:LoadPatterns}, the maximum displacement amplitude increase with the device height, consistently with the expected performance range of each geometry family.
For every sampled configuration, a FEM simulation is performed under a symmetric, displacement-controlled cyclic loading protocol, consistent with the qualification-oriented characterization of seismic energy dissipation devices. Since the admissible deformation demand depends on the size of the device, different loading patterns are adopted according to the device height. As shown in Figure~\ref{fig:LoadPatterns}, the maximum displacement amplitude increase with the device height, consistently with the expected performance range of each geometry family.
\begin{figure}[htbp]
\centering
......@@ -248,7 +256,7 @@ For every sampled configuration, a FEM simulation is performed under a displacem
The resulting dataset stores the input variables and the structural response quantities extracted at the final time of each simulation. The target outputs include damage indicators in the windows and in the frame, together with local distortion measures associated with dissipative activation. For compactness in the optimization formulation, the aggregated TFDMap indicator in window $i$ is denoted by $\TFD_i$, whereas the corresponding frame indicator is denoted by $\TFD_f$. Both quantities are obtained through a regional percentile-based aggregation, computed as the 98th percentile (P98) of the TFDMap values within the corresponding region. The same P98 regional statistic is used for the window distortion measure. A percentile-based regional statistic avoids defining the response through an arbitrary fixed number of nodes, while retaining the most critical damage levels and reducing sensitivity to isolated numerical peaks.
The local shear distortion in each window, aggregated with the same P98 regional statistic, is denoted by $\varepsilon_{xy,i}$ and is used as an indicator of the energy dissipation capacity. When included in the objective function, each window contribution is weighted according to its effective geometric volume, so that the optimization accounts for both local strain intensity and the material volume involved in the dissipation process.
The local shear distortion in each window, aggregated with the same P98 regional statistic, is denoted by $\varepsilon_{xy,i}$ and is used to build a shear-distortion performance indicator. When used in the optimization, each window contribution is weighted according to its effective volume, so that the indicator accounts for both the local distortion intensity and the amount of material involved in the shear mechanism.
\subsection{Supervised ML surrogate models}\label{subsec:ml_models}
......@@ -269,7 +277,7 @@ Model selection is performed in two stages using the inner cross-validation resu
\label{fig:BayesianSearchCV}
\end{figure*}
Preliminary executions of the proposed workflow show that SVR or GPR always provide the highest, or second-highest, predictive accuracy for the considered datasets. These results suggest that kernel-based models, and in particular distance-based similarity measures, are well suited to approximate the FEM response surfaces involved in this problem. This observation motivates the assessment of Radial Basis Function (RBF) interpolation \cite{Gutmann2001} as a simpler and computationally efficient surrogate alternative. Although surrogate evaluation is negligible compared with FEM simulations, the training and hyperparameter optimization of complex supervised models can still become relevant when several outputs, geometry families and adaptive iterations are considered. RBF interpolation provides a non-parametric and fast-to-train alternative that can capture nonlinear response surfaces, making it attractive for low-dimensional and moderately sampled design spaces.
Preliminary executions of the proposed workflow show that SVR or GPR always provide the highest, or second-highest, predictive accuracy for the considered datasets. These results suggest that kernel-based models, and in particular distance-based similarity measures, are well suited to approximate the FEM response surfaces involved in this problem. This observation motivates the assessment of Radial Basis Function (RBF) interpolation \cite{Gutmann2001} as a simpler surrogate alternative. Although surrogate evaluation is negligible compared with FEM simulations, the Bayesian hyperparameter optimization and cross-validation of several supervised models can still become relevant when many outputs, geometry families and adaptive iterations are considered. RBF interpolation provides a non-parametric alternative whose training requires only a small grid search over the kernel and smoothing parameters, and it can capture nonlinear response surfaces, making it attractive for low-dimensional and moderately sampled design spaces.
\subsection{RBF surrogate models}\label{subsec:rbf_models}
......@@ -286,47 +294,50 @@ For each output variable, a final RBF surrogate is trained using all available F
\section{Damage-aware surrogate-assisted optimization}\label{sec:optimization}
The proposed methodology seeks to balance damage among the dissipative windows while keeping it below a prescribed threshold. At the same time, it limits damage in the surrounding frame and promotes the highest possible energy dissipation through the activation of the windows. The window-thickness optimization is carried out using DE \cite{Storn1997}, a population-based global optimizer that does not require gradient information and is therefore suitable for nonlinear and non-convex surrogate response surfaces. In the current implementation, DE uses a best/1/bin strategy with a mutation factor sampled in $[0.5,1.0]$, a crossover probability of 0.7, a population size factor of 25, a maximum of 500 iterations and a convergence tolerance of $10^{-6}$; the initial population is generated by Latin hypercube sampling and no local polishing is applied. Each optimization stage is repeated with 30 independent runs using deterministic seeds derived from a fixed base seed. Once an optimal candidate is obtained, an adaptive FEM validation loop is applied to verify the predicted geometry before acceptance.
The proposed formulation treats damage admissibility through explicit constraints rather than through weighted penalty terms, and it promotes shear deformation in the intended dissipative windows. The window-thickness optimization is carried out using DE \cite{Storn1997}, a population-based global optimizer that does not require gradient information and is therefore suitable for nonlinear and non-convex surrogate response surfaces. In the current implementation, DE uses a best/1/bin strategy with a mutation factor sampled in $[0.5,1.0]$, a crossover probability of 0.7, a population size factor of 25, a maximum of 500 iterations and a convergence tolerance of $10^{-6}$; the initial population is generated by Latin hypercube sampling and no local polishing is applied. Each optimization stage is repeated with 30 independent runs using deterministic seeds derived from a fixed base seed. Once a candidate is obtained, an adaptive FEM validation loop is applied to verify the predicted geometry before acceptance.
For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window distortions $\hat{\varepsilon}_{xy,i}$, the window damage indicators $\hat{\mathcal{D}}_i$ and the frame damage indicator $\hat{\mathcal{D}}_f$. Damage is therefore controlled in all regions of the device, but with different mechanical relevance: frame damage is penalized more severely because it may compromise the structural integrity of the damper, whereas the window damage penalties are formulated to promote comparable damage levels among windows and avoid concentrating the dissipative demand in a single region. The dissipative contribution is estimated from $\hat{\varepsilon}_{xy,i}^2$, the window thickness and the corresponding area of the window, since the energy dissipated by each window depends not only on the distortion level but also on the amount of material involved. This term is several orders of magnitude smaller than the damage penalties and is intentionally left unscaled. As a result, damage control remains the dominant criterion, while the dissipative term acts as a tie-breaker among geometries with similar damage performance, favouring those with higher distortion and, consequently, greater energy dissipation capacity.
For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window distortion $\hat{\varepsilon}_{xy,i}$, the window damage-screening indicators $\hat{\mathcal{D}}_i$ and the frame damage-screening indicator $\hat{\mathcal{D}}_f$. A candidate is feasible at the surrogate level only if
\begin{equation}
\hat{\mathcal{D}}_i(\mathbf{x}) + m_i(\mathbf{x}) \le \mathcal{D}_W, \quad i=1,\ldots,N_w,
\qquad
\hat{\mathcal{D}}_f(\mathbf{x}) + m_f(\mathbf{x}) \le \mathcal{D}_F,
\label{eq:feasibility}
\end{equation}
where $\mathcal{D}_W=100$ and $\mathcal{D}_F=90$ are the prescribed damage-screening thresholds for the windows and for the surrounding frame, and $m_i$ and $m_f$ are conservative margins that account for local out-of-fold underprediction of the surrogate models. The thresholds are not soft optimization targets: they define the admissible damage region, and candidates that violate them are not feasible designs. Because the TFDMap is a post-processing damage-screening indicator rather than a constitutive fracture model, the thresholds are adopted to control the relative proximity to critical damage states within the optimization framework, and they are not interpreted as a mathematically exact point of complete physical failure.
The implemented objective function to be minimized is
Feasibility is assessed before the optimization stages. A dedicated pre-solve minimizes the maximum constraint violation
\begin{equation}
J(\mathbf{x}) = - \sum_{i=1}^{N_w} \hat{\varepsilon}_{xy,i}^2\, t_{w,i}\, A_i +
\sum_{i=1}^{N_w} P_w\left(\hat{\mathcal{D}}_i;\mathcal{D}_w^{\star}\right) +
P_f\left(\hat{\mathcal{D}}_f;\mathcal{D}_f^{\max}\right),
\label{eq:objective}
v(\mathbf{x}) = \max\Bigl(0,\; \max_{i}\bigl[\hat{\mathcal{D}}_i(\mathbf{x})+m_i(\mathbf{x})-\mathcal{D}_W\bigr],\; \hat{\mathcal{D}}_f(\mathbf{x})+m_f(\mathbf{x})-\mathcal{D}_F\Bigr).
\label{eq:violation}
\end{equation}
where $N_w$ is the number of windows, $t_{w,i}$ is the thickness of window $i$, $A_i$ is the corresponding area factor, $\mathcal{D}_w^{\star}$ is the target damage level for the windows and $\mathcal{D}_f^{\max}$ is the maximum admissible frame damage threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger energy dissipation contributions reduce the objective value.
If the minimum identified violation exceeds a small numerical tolerance, no surrogate-feasible region is identified. In that case the performance stages are not executed; the least-infeasible candidate is retained as an enrichment point and re-evaluated with FEM. This situation indicates that the surrogate-defined admissible region is empty or not yet resolved, and it is not, by itself, evidence that the physical/FEM design space contains no feasible solution.
The window penalty is defined as
When a surrogate-feasible region exists, the optimization proceeds hierarchically within that region. The first stage promotes the activation of every dissipative window by maximizing the smallest window damage-screening indicator,
\begin{equation}
P_w\left(\hat{\mathcal{D}}_i;\mathcal{D}_w^{\star}\right)=
\begin{cases}
\left(\hat{\mathcal{D}}_i-\mathcal{D}_w^{\star}\right)^2, & \hat{\mathcal{D}}_i>\mathcal{D}_w^{\star},\\
\left|\hat{\mathcal{D}}_i-\mathcal{D}_w^{\star}\right|, & \hat{\mathcal{D}}_i\leq\mathcal{D}_w^{\star}.
\end{cases}
\label{eq:window_penalty}
\max_{\mathbf{x}\in\mathcal{F}}\; \mathcal{J}_1(\mathbf{x}) = \min_{i} \hat{\mathcal{D}}_i(\mathbf{x}),
\label{eq:stage1}
\end{equation}
This formulation penalizes values above the target quadratically, while also discouraging excessively underused windows through a linear distance to the target. As a result, the optimizer tends to balance the damage levels among windows rather than forcing all windows to remain far below the admissible level.
where $\mathcal{F}$ denotes the feasible set defined by Eq.~\eqref{eq:feasibility}. Mechanically, this stage prevents configurations in which one or more windows remain essentially inactive while damage concentrates elsewhere, and it promotes a balanced participation of the dissipative regions. Let $\mathbf{x}_1^{\star}$ be the resulting optimum and $\mathcal{J}_1^{\star}$ the corresponding value.
The frame penalty is defined as
The second stage maximizes a shear-distortion performance indicator while preserving the first-stage optimum within a numerical tolerance,
\begin{equation}
P_f\left(\hat{\mathcal{D}}_f;\mathcal{D}_f^{\max}\right)=
\begin{cases}
\left(\hat{\mathcal{D}}_f-\mathcal{D}_f^{\max}\right)^3, & \hat{\mathcal{D}}_f>\mathcal{D}_f^{\max},\\
0, & \hat{\mathcal{D}}_f\leq\mathcal{D}_f^{\max}.
\end{cases}
\label{eq:frame_penalty}
\max_{\mathbf{x}\in\mathcal{F}}\; \mathcal{J}_2(\mathbf{x}) = \sum_{i=1}^{N_w} \hat{\varepsilon}_{xy,i}^{2}(\mathbf{x})\, t_{w,i}\, V_i
\qquad\text{subject to}\qquad
\min_{i} \hat{\mathcal{D}}_i(\mathbf{x}) \ge \mathcal{J}_1^{\star} - \delta_1 .
\label{eq:stage2}
\end{equation}
The cubic penalty reflects the greater severity of frame damage compared with localized damage in a window. Since a damage value $\mathcal{D}=100$ represents complete failure, $\mathcal{D}_f^{\max}=90$ is adopted in the current implementation. The target window threshold is also set to $\mathcal{D}_w^{\star}=90$.
Here $t_{w,i}$ is the thickness of window $i$, $V_i$ is the effective volume weighting factor of window $i$ (implemented as a thickness-scaled area/volume factor), and $\delta_1$ is a small preservation tolerance derived from the spread of the first-stage objective across the independent DE runs. The two stages are solved sequentially, so the formulation is hierarchical: damage admissibility is enforced through the constraints, the preferred damage distribution is secured first, and the shear-distortion performance indicator is optimized second within the tolerance that preserves the first-stage result.
The quantity $\mathcal{J}_2$ is a shear-distortion performance indicator. The BDSL windows are intended to accommodate their inelastic response predominantly through shear deformation, so a larger local shear distortion represents a stronger activation of the intended deformation mechanism. Weighting the squared distortion by the thickness and an effective volume factor favours both deformation intensity and the participation of material in the shear windows. The indicator is used only to rank feasible candidates; it does not integrate the stress--strain hysteresis loop, does not contain the complete stress history, and does not represent cumulative plastic work or the total hysteretic energy dissipated by the device. It is therefore not interpreted as an energy quantity, and establishing a quantitative relationship between this indicator and the cumulative hysteretic energy computed by the FEM model is left for future work.
The surrogate-optimized geometry is not accepted directly. Instead, 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. Acceptance is based on FEM-confirmed physical feasibility and on the robustness of the optimizer rather than on surrogate accuracy alone. Specifically, the candidate geometry is accepted only if: (i) the FEM damage indicators satisfy the prescribed physical limits in every window and in the frame; (ii) the Differential Evolution result is reproducible, requiring that the performance of all independent runs is numerically indistinguishable from the best run according to a scale-aware relative criterion, with a tolerance of $10^{-3}$ in each optimization stage; (iii) the optimum remains stable between consecutive adaptive iterations, with relative changes in the stage objectives below 2\%; and (iv) a conservatism check confirms that the surrogate safety margin does not materially limit the optimum. Surrogate--FEM prediction errors for the damage and distortion variables are monitored against a local out-of-fold error threshold as a diagnostic measure of surrogate fidelity, but do not by themselves determine acceptance. If the acceptance 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, that reduces the risk of accepting a geometry that is optimal only because of surrogate extrapolation error, is summarized in Figure~\ref{fig:OptimizationFlowChart}.
Compared with a scalar objective that combines damage penalties and a performance term, the constrained hierarchical formulation removes the need for arbitrary relative weights between dimensionally different quantities. The numerical values of the preservation tolerance, the conservative margin and the feasibility tolerance remain calibration parameters, and their sensitivity is not assessed in the present study.
The surrogate-optimized geometry is not accepted directly. The conservative feasibility screen of Eq.~\eqref{eq:feasibility} is only a surrogate-level admissibility check; engineering acceptance requires a FEM re-evaluation of the selected candidate. Once the hierarchical optimizer proposes a candidate, it is re-evaluated with FEM. Acceptance is based on FEM-confirmed physical feasibility and on the robustness of the optimizer rather than on surrogate accuracy alone. Specifically, the candidate geometry is accepted only if: (i) the FEM damage indicators satisfy the prescribed physical limits in every window and in the frame; (ii) the Differential Evolution result is reproducible, requiring that the performance of all independent runs is numerically indistinguishable from the best run according to a scale-aware relative criterion, with a tolerance of $10^{-3}$ in each optimization stage; (iii) the optimum remains stable between consecutive adaptive iterations, with relative changes in the stage objectives below 2\%; and (iv) a conservatism check confirms that the surrogate safety margin does not materially limit the optimum. Surrogate--FEM prediction errors for the damage and shear-distortion variables are monitored against a local out-of-fold error threshold as a diagnostic measure of surrogate fidelity, but do not by themselves determine acceptance. If the acceptance criteria are satisfied, the FEM-validated geometry is accepted as a feasible numerical candidate. If at least one criterion is not satisfied, the new FEM result is added to the dataset, the surrogate models are retrained and the optimization is repeated. This loop, that reduces the risk of accepting a geometry that is optimal only because of surrogate extrapolation error, is summarized in Figure~\ref{fig:OptimizationFlowChart}. If no feasible candidate is identified within the considered design space, the corresponding family is reported as such rather than treated as a feasible optimum.
\begin{figure*}[htbp]
\centering
\includegraphics[width=1.0\textwidth]{./images/OptimizationFlowChart/OptimizationFlowChart.pdf}
\caption{Surrogate-assisted optimization and FEM validation retraining loop.}
\caption{Feasibility-first hierarchical surrogate-assisted optimization and FEM validation loop. Damage-screening limits are enforced as constraints, and the shear-distortion performance indicator is optimized in the second stage.}
\label{fig:OptimizationFlowChart}
\end{figure*}
......@@ -341,7 +352,7 @@ The supervised-learning comparison shows a predominance of kernel-based models a
\label{fig:surrogate_selection_summary_barplot}
\end{figure*}
These results indicate that kernel-based models performed well under the adopted nested validation procedure for the datasets considered here. SVR provided the best compromise between accuracy and computational cost, while GPR was the second most frequently selected supervised strategy, especially in some higher-dimensional cases. Tree-based models were less frequently selected and MLP models were not competitive in terms of computational efficiency for the dataset sizes considered. As mentioned in previous sections, this behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. Because the RBF and the supervised models are assessed on the same outer cross-validation splits and with the same error metrics, the two strategies can be compared on a like-for-like basis at the level of each output variable. Across the analysed outputs neither strategy dominates: the selected supervised model is more accurate for some targets and the RBF surrogate for others, and both remain within the accuracy range required by the adaptive validation loop. The RBF surrogate, however, is obtained with a much smaller hyperparameter search and therefore requires substantially lower training effort, which makes it attractive for repeated surrogate updates within the adaptive optimization loop.
These results indicate that kernel-based models performed well under the adopted nested validation procedure for the datasets considered here. SVR provided the best compromise between accuracy and computational cost, while GPR was the second most frequently selected supervised strategy, especially in some higher-dimensional cases. Tree-based models were less frequently selected and MLP models were not competitive in terms of computational efficiency for the dataset sizes considered. As mentioned in previous sections, this behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. Because the RBF and the supervised models are assessed on the same outer cross-validation splits and with the same error metrics, the two strategies can be compared on a like-for-like basis at the level of each output variable. Across the analysed outputs neither strategy dominates: the selected supervised model is more accurate for some targets and the RBF surrogate for others, and both remain within the accuracy range required by the adaptive validation loop. The RBF surrogate, however, is obtained with a simpler training procedure, based on a small grid search over the kernel and smoothing parameters, that does not require Bayesian hyperparameter optimization, which makes it attractive for repeated surrogate updates within the adaptive optimization loop.
The predictive assessment is constrained by the small datasets available for the two-window families ($N=8$) and the three-window families ($N=16$). Nested cross-validation separates hyperparameter selection from generalization assessment, and the outer folds provide out-of-sample predictions together with bootstrap confidence intervals for RMSE and MAE. For datasets with $N\leq20$ the outer loop is Leave-One-Out, so the reported errors are obtained from a single deterministic partition and their dispersion is estimated by resampling the out-of-sample predictions rather than by repeating the outer split. These results should therefore be interpreted as an assessment of predictive accuracy within the sampled design domain and for the analysed geometry families, under the adopted validation procedure, rather than as a general guarantee. No independent FEM test set was retained, and the uncertainty of the accuracy estimates for the smallest datasets remains a limitation of the present comparison.
......@@ -414,7 +425,7 @@ Figure~\ref{fig:optimized_window_thickness_evolution} shows the evolution of the
\label{fig:optimized_window_thickness_evolution}
\end{figure*}
From a methodological point of view, these results highlight the trade-off between surrogate complexity, accuracy and computational efficiency. Supervised models, particularly SVR and GPR, provide competitive predictive accuracy under the adopted validation procedure, but require hyperparameter optimization and cross-validation for every output variable and adaptive iteration. RBF interpolation, in contrast, has a much lower training cost and provides very competitive final predictions once the relevant regions of the design domain have been adaptively sampled. Therefore, the comparison does not identify a universally superior surrogate strategy. Instead, it suggests that RBF interpolation is especially suitable for low- to moderate-dimensional design spaces with well-distributed FEM samples and relatively smooth input--output relationships, as occurs for the response variables analysed in this work. Supervised ML surrogates remain valuable when greater robustness is required or when the response surface is expected to involve stronger nonlinear interactions, local irregularities or higher-dimensional dependencies.
From a methodological point of view, these results highlight the trade-off between surrogate complexity, accuracy and computational efficiency. Supervised models, particularly SVR and GPR, provide competitive predictive accuracy under the adopted validation procedure, but require hyperparameter optimization and cross-validation for every output variable and adaptive iteration. RBF interpolation, in contrast, requires a simpler training procedure and provides competitive final predictions once the relevant regions of the design domain have been adaptively sampled. Therefore, the comparison does not identify a universally superior surrogate strategy. Instead, it suggests that RBF interpolation is especially suitable for low- to moderate-dimensional design spaces with well-distributed FEM samples and relatively smooth input--output relationships, as occurs for the response variables analysed in this work. Supervised ML surrogates remain valuable when greater robustness is required or when the response surface is expected to involve stronger nonlinear interactions, local irregularities or higher-dimensional dependencies.
The objective-function values should also be interpreted carefully. The objective-function error measures the consistency between surrogate predictions and FEM validation, not whether the final objective value is necessarily close to zero. Some geometry families, such as $F_2$, retain non-negligible penalty contributions because the prescribed damage targets cannot be fully achieved within the admissible design bounds. Nevertheless, the close agreement between surrogate and FEM objective values indicates that the accepted designs are not artifacts of surrogate extrapolation, but FEM-consistent optimized candidates within the explored design space.
......@@ -429,23 +440,18 @@ This behaviour is illustrated in Figure~\ref{fig:rbf_surface_evolution}, which s
\section{Conclusions and future work}\label{sec:conclusions}
This work presents an adaptive surrogate-assisted optimization framework for the seismic window-thickness optimization of buckling-delayed shear-link dampers within five predefined geometry families. The proposed methodology addresses the main limitation of direct FEM-based optimization, the high computational cost associated with repeatedly evaluating nonlinear cyclic simulations. By training surrogate models on FEM-generated datasets and validating the optimized candidates through additional FEM analyses, the optimizer can efficiently explore the design domain while remaining consistent with the mechanical response captured by the calibrated numerical model.
One of the key features of the proposed optimization framework, compared to approaches focused primarily on energy maximization, is the formulation of an objective function that takes damage into account. The aim is to prioritise local damage control in both the dissipative windows and the surrounding frame, whilst promoting a balanced contribution from all windows to the energy dissipation process. Thus, damage to the windows is permitted and expected, as they are intended to act as dissipative regions, provided that it remains controlled and reasonably distributed; conversely, damage to the frame is penalised more severely because it can compromise the structural integrity of the damper. Therefore, rather than merely maximizing the energy dissipated, the proposed formulation favours geometries that concentrate dissipative activation in the windows, prevent excessive damage localization in a single region and protect the frame from critical damage.
A comparison of surrogate strategies was performed in terms of predictive accuracy and computational efficiency. For the analysed datasets, the supervised ML results showed that kernel-based models, particularly SVR and GPR, were the most frequently selected, with SVR being the most frequently selected across the outputs. When evaluated on the same outer cross-validation splits, RBF interpolation provided predictive performance comparable to the supervised models for the considered low-dimensional response surfaces, while requiring a much smaller hyperparameter search and substantially lower training effort. For the optimized candidates, the RBF surrogate also achieved surrogate--FEM agreement comparable to, and in several cases better than, that of the supervised ML surrogates. These observations are limited to the analysed geometry families and to the sampled design domain, and their reliability is lower for the smallest datasets, for which the accuracy estimates carry larger uncertainty.
The proposed adaptive validation loop proved to be necessary and effective. Several initially optimized candidates did not satisfy the prescribed error tolerances. After incorporating the new FEM results into the training dataset and retraining the surrogates, the prediction errors decreased and the optimization converged after only two or three iterations. Therefore, the final designs are not accepted solely on the basis of surrogate predictions, but are explicitly verified through FEM in the region of the design space where the optimum is located.
It is also worth noting that, although the framework allows the DoE to be expanded with additional FEM simulations when the surrogate accuracy is insufficient, this was not required in the present application. The initial DoE datasets were already adequate to obtain accurate optimized designs after adaptive validation and retraining. In particular, the final geometries were obtained from only 8 initial FEM simulations for the two-window devices, 16 for the three-window devices and 64 for the five-window devices, with a maximum of three adaptive retraining iterations in all cases. This demonstrates that the proposed strategy can achieve FEM-consistent optimized designs with a limited number of simulations, making it competitive from a computational point of view.
The proposed methodology also has some limitations that should be acknowledged. First, its reliability depends on the quality of the calibrated FEM model used to generate the training data and validate the optimized designs. Second, the TFDMap is used here as a post-processing damage indicator rather than as a constitutive fracture model; therefore, the optimized configurations should be interpreted in terms of relative damage control and proximity to critical states, not as direct predictions of crack initiation. Third, only the window thicknesses are considered as design variables; although this leads to a controlled and interpretable optimization problem, it does not exploit the full geometric flexibility of BDSL dampers. Finally, the optimized geometries should ultimately be validated experimentally before being used to establish general design recommendations.
This work presented an adaptive surrogate-assisted optimization framework for the seismic window-thickness optimization of buckling-delayed shear-link dampers within five predefined geometry families. Damage admissibility is enforced through explicit feasibility limits on the window and frame damage-screening indicators, and the preference among feasible designs is expressed hierarchically through a maximum window-activation stage followed by a shear-distortion performance indicator. The framework combines FEM-calibrated reference models, supervised and RBF surrogates, Differential Evolution and an adaptive FEM validation loop, and it addresses the main limitation of direct FEM-based optimization, namely the cost of repeatedly evaluating nonlinear cyclic simulations.
Future work should extend the design space by including additional geometric and mechanical variables, such as window height, window spacing, frame thickness or global device proportions. This extension would increase the dimensionality and complexity of the surrogate task. In those cases, the performance of RBF interpolation should therefore be reassessed. While RBF models performed very well in the present study, their efficiency and accuracy may decrease as the input space becomes larger or the response surfaces develop stronger local nonlinearities. In such cases, supervised ML models or hybrid surrogate strategies may become more advantageous.
The main findings of this study can be summarised as follows.
\begin{enumerate}
\item BDSL optimization is a damage-distribution problem rather than a pure energy-maximization problem. The same level of global dissipation can correspond to very different local damage patterns, so the design must control damage in both the dissipative windows and the surrounding frame.
\item FEM-calibrated surrogate models can predict the local damage and distortion indicators required by the optimizer. Under a common nested cross-validation framework, the kernel-based supervised models (SVR and GPR) were the most frequently selected, and RBF interpolation achieved comparable predictive performance with a simpler training procedure.
\item The adaptive FEM enrichment loop, in which acceptance is based on FEM-confirmed feasibility and optimizer robustness rather than on surrogate accuracy alone, reduces the risk of accepting candidates that are optimal only because of surrogate prediction error.
\end{enumerate}
A complementary line of future work is the consideration of non-symmetric cyclic loading histories or recorded seismic displacement demands. The present study focuses on symmetric cyclic protocols because they provide a standardized and industry-relevant basis for the qualification of seismic energy dissipation devices, which must satisfy prescribed cyclic testing requirements before being implemented in practice. Nevertheless, earthquake-induced demands may lead to non-symmetric deformation histories in structural components. Extending the proposed framework to asymmetric cyclic protocols or representative seismic displacement histories would therefore be an interesting step towards broader performance assessment conditions.
The scope of these findings is limited. Only the window thicknesses were optimized; window height, spacing, corner radius, frame thickness and global proportions remained fixed within each family. The study is component-level and considers a symmetric, prescribed displacement-controlled cyclic protocol. It does not reproduce irregular, asymmetric or pulse-like earthquake demands, nor record-to-record variability, and it does not quantify structural-system-level seismic performance or resilience. In addition, the results depend on the calibrated FEM model used as ground truth, TFDMap is used as a post-processing damage indicator rather than as a constitutive fracture model, and no independent experimental test set was retained. The optimized geometries are therefore FEM-validated numerical candidates rather than experimentally validated designs.
Another future direction is the incorporation of interpretability analyses, such as SHapley Additive exPlanations \cite{Lundberg2017}, to quantify the influence of each geometric variable on window damage, frame damage and dissipative activation. Although such analysis lies outside the main scope of the present study, it could provide valuable insight into the design drivers governing the behaviour of BDSL dampers and support more transparent engineering decision-making. Overall, the proposed methodology establishes a scalable basis for FEM-consistent, damage-aware optimization of seismic energy dissipation devices, while leaving room for broader design variables, richer surrogate strategies and experimental validation of the optimized configurations.
Future work should extend the design space to additional geometric and mechanical variables, such as window height, window spacing, frame thickness or global device proportions, and should reassess the surrogate strategies in the resulting higher-dimensional space. Assessing the optimized devices at the structural-system level through nonlinear time-history analyses, able to quantify inter-storey drift, floor acceleration and global energy dissipation, is a necessary next step before broader design recommendations can be established. Extending the framework to asymmetric cyclic protocols or recorded seismic displacement histories, and incorporating interpretability analyses such as SHapley Additive exPlanations \cite{Lundberg2017} to identify the main geometric drivers of damage and shear-distortion performance, are also natural developments.
\appendix
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ManuscriptR1/ComparisonSurrogatesOptimizationBDSL_R1_changes.tex
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......@@ -108,13 +108,14 @@
country={Spain}}
\begin{abstract}
Buckling-delayed shear-link dampers are passive devices used in seismic-resistant structures to concentrate energy dissipation while limiting damage to the primary system. \DIFdelbegin \DIFdel{Their geometric optimization }\DIFdelend \DIFaddbegin \DIFadd{Optimizing their window thicknesses }\DIFaddend requires balancing high dissipative capacity with strict control of local damage. Nonlinear finite element models can reproduce their cyclic response and provide internal quantities for optimization, such as damage indicators and local distortion, but their computational cost prevents direct use within iterative optimization loops. This work proposes an adaptive surrogate-assisted optimization framework for \DIFaddbegin \DIFadd{damage-aware window-thickness optimization of }\DIFaddend buckling-delayed shear-link dampers \DIFaddbegin \DIFadd{within five predefined geometry families}\DIFaddend . Experimentally calibrated nonlinear finite element models are first used to generate reference datasets for different device configurations. Supervised learning models are evaluated, with support vector regression and Gaussian process regression \DIFdelbegin \DIFdel{showing high predictive accuracy}\DIFdelend \DIFaddbegin \DIFadd{providing the most accurate predictions among the candidate models}\DIFaddend , motivating the use of radial basis function surrogates as a more efficient distance-based alternative. The surrogate predictions are coupled with a differential evolution algorithm through a damage-aware objective function that controls local damage while using dissipative performance as an optimization criterion. The optimized geometries are then re-evaluated with finite element simulations. If the \DIFdelbegin \DIFdel{surrogate error exceeds the adopted tolerances}\DIFdelend \DIFaddbegin \DIFadd{acceptance criteria are not met}\DIFaddend , the new simulation is added to the dataset and the surrogate models are retrained. The framework enables efficient damage-aware optimization of seismic energy dissipation devices.
Buckling-delayed shear-link dampers are passive devices used in seismic-resistant structures to concentrate energy dissipation while limiting damage to the primary system. \DIFdelbegin \DIFdel{Their geometric optimization }\DIFdelend \DIFaddbegin \DIFadd{Optimizing their window thicknesses }\DIFaddend requires balancing high dissipative capacity with strict control of local damage. Nonlinear finite element models can reproduce their cyclic response and provide internal quantities for optimization, such as damage indicators and local distortion, but their computational cost prevents direct use within iterative optimization loops. This work proposes an adaptive surrogate-assisted optimization framework for \DIFaddbegin \DIFadd{damage-aware window-thickness optimization of }\DIFaddend buckling-delayed shear-link dampers \DIFaddbegin \DIFadd{within five predefined geometry families}\DIFaddend . Experimentally calibrated nonlinear finite element models are first used to generate reference datasets for different device configurations. Supervised learning models are evaluated, with support vector regression and Gaussian process regression \DIFdelbegin \DIFdel{showing high predictive accuracy}\DIFdelend \DIFaddbegin \DIFadd{providing the most accurate predictions among the candidate models}\DIFaddend , motivating the use of radial basis function surrogates\DIFdelbegin \DIFdel{as a more efficient }\DIFdelend \DIFaddbegin \DIFadd{, whose training requires a simpler procedure, as a complementary }\DIFaddend distance-based alternative. The surrogate predictions are coupled with a differential evolution algorithm \DIFdelbegin \DIFdel{through a damage-aware objective function that controls local damage while using dissipative performance as an optimization }\DIFdelend \DIFaddbegin \DIFadd{in a feasibility-first, hierarchical optimization that enforces prescribed damage-screening limits on the windows and on the surrounding frame and then uses a shear-distortion performance indicator as the secondary }\DIFaddend criterion. The optimized geometries are then re-evaluated with finite element simulations. If the \DIFdelbegin \DIFdel{surrogate error exceeds the adopted tolerances}\DIFdelend \DIFaddbegin \DIFadd{acceptance criteria are not met}\DIFaddend , the new simulation is added to the dataset and the surrogate models are retrained. The \DIFdelbegin \DIFdel{framework enables efficient damage-aware optimization of seismic energy dissipation devices}\DIFdelend \DIFaddbegin \DIFadd{added value of the framework is the integration of local, FEM-derived damage indicators into a window-thickness optimization that protects the surrounding frame and verifies each candidate design with FEM}\DIFaddend .
\end{abstract}
\begin{highlights}
\item Adaptive surrogates optimize damage-aware shear-link damper geometries
\item RBF interpolation \DIFdelbegin \DIFdel{matches supervised ML with lower retraining }\DIFdelend \DIFaddbegin \DIFadd{offers comparable accuracy to supervised ML at lower training }\DIFaddend cost
\item FEM validation filters surrogate optima before accepting final designs
\item RBF interpolation \DIFdelbegin \DIFdel{matches supervised ML with lower retraining cost
}\DIFdelend \DIFaddbegin \DIFadd{offers comparable accuracy with a simpler training procedure
}\DIFaddend \item FEM validation filters surrogate optima before accepting final designs
\item Optimized devices balance window activation and frame damage control
\end{highlights}
......@@ -141,9 +142,23 @@ Data-driven approaches have mainly focused on response or property prediction. C
All these works demonstrate the increasing interest in applying FEM-based and data-driven approaches, as well as in combining both, to analyse, understand and optimize seismic energy dissipation devices. However, most of these studies focus either on the prediction of the hysteretic response or on maximizing energy dissipation, leaving a critical aspect insufficiently explored: the need to control local damage while maintaining adequate dissipative capacity. In practice, excessive local damage may compromise structural integrity, reduce durability and lead to premature failure, even when global energy dissipation is improved.
The present work addresses this gap through a damage-aware surrogate-assisted optimization methodology in which the objective is not only to maximize distortion or energy dissipation, but also to balance dissipative performance with damage indicators derived from FEM simulations. The proposed approach combines: (i) experimentally calibrated nonlinear FEM models used as numerical ground truth; (ii) supervised surrogate models trained to predict local damage and distortion indicators; (iii) a Differential Evolution (DE) optimizer; and (iv) an adaptive FEM validation and retraining loop.
\DIFaddbegin \DIFadd{BDSL device performance therefore depends not only on global force or total dissipated energy, but on how local deformation and damage are distributed among the individual windows and the surrounding frame. The window thicknesses exert a direct control on this distribution, because they set the relative stiffness of each dissipative region and thus the balance between window activation and the inelastic demand transferred to the frame. Nonlinear FEM can resolve these local quantities, but its cost makes repeated direct optimization impractical. A surrogate-assisted approach therefore offers a practical route to search the window-thickness design space while explicitly accounting for damage-related responses.
}
\DIFaddend The present work addresses this gap through a damage-aware surrogate-assisted optimization methodology \DIFdelbegin \DIFdel{in which the objective is not only to maximize distortion or energy dissipation, but also to balance dissipative performance with damage indicators derived from FEM simulations}\DIFdelend \DIFaddbegin \DIFadd{for window-thickness optimization within five predefined BDSL geometry families}\DIFaddend . The proposed approach combines: (i) experimentally calibrated nonlinear FEM models used as numerical ground truth; (ii) supervised \DIFaddbegin \DIFadd{and radial basis function (RBF) }\DIFaddend 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.
\DIFaddbegin \DIFadd{The contribution of this study is not the use of surrogate optimization itself, which is well established for metallic dampers, but the combination of the following elements, all of which are implemented and assessed in the present framework: (i) damage-aware window-thickness optimization driven by local FEM-derived indicators in the windows and in the frame, rather than by global force or total energy alone; (ii) explicit protection of the surrounding frame and balanced activation of the dissipative windows; (iii) a common nested cross-validation framework that allows a like-for-like comparison of six supervised learning algorithms and RBF interpolation on identical data splits; and (iv) an adaptive FEM enrichment loop in which a candidate is accepted on the basis of FEM-confirmed feasibility and optimizer robustness, rather than on surrogate predictions alone.
}
\DIFadd{The study is deliberately component-level. It optimizes the device and evaluates its cyclic response under prescribed, displacement-controlled loading, which isolates the nonlinear response of the device and provides controlled deformation histories for comparing geometry variants. It does not analyse the response of a complete building or structural system equipped with the optimized dampers.
}
\DIFadd{From a broader perspective, BDSL dampers are conceived as replaceable, sacrificial components that concentrate damage away from primary structural members, which can facilitate post-event inspection, repair or replacement \mbox{%DIFAUXCMD
\cite{Xiong2024}}\hskip0pt%DIFAUXCMD
. The present study contributes to this objective at the component level by controlling where damage develops and by protecting the surrounding frame. It does not, however, quantify resilience: metrics such as downtime, repair cost or functional recovery are not computed, and system-level resilience assessment is outside the scope of this work.
}
Figure \ref{fig:MethodologyFlowChart} summarizes the proposed workflow. The different stages of the methodology, together with the surrogate modelling, optimization strategy, validation procedure and corresponding results and conclusions, are described in the following sections.
\DIFaddend Figure \ref{fig:MethodologyFlowChart} summarizes the proposed workflow. The different stages of the methodology, together with the surrogate modelling, optimization strategy, validation procedure and corresponding results and conclusions, are described in the following sections.
\begin{figure*}[htbp]
\centering
......@@ -163,7 +178,7 @@ The BDSL dampers analysed in this work, with one representative configuration sh
\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. 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 stay 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.
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 \DIFdelbegin \DIFdel{penalized more strongly }\DIFdelend \DIFaddbegin \DIFadd{controlled more strictly }\DIFaddend 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 stay 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.
Accordingly, the design variables considered in this work are the window thicknesses, while the frame dimensions are kept fixed:
\begin{equation}
......@@ -279,7 +294,7 @@ To improve surrogate robustness near the admissible limits, the sampling domain
\end{tabular}
\end{table*}
For every sampled configuration, a FEM simulation is performed under a displacement-controlled cyclic loading protocol. Since the admissible deformation demand depends on the size of the device, different loading patterns are adopted according to the device height. As shown in Figure~\ref{fig:LoadPatterns}, the maximum displacement amplitude increase with the device height, consistently with the expected performance range of each geometry family.
For every sampled configuration, a FEM simulation is performed under a \DIFaddbegin \DIFadd{symmetric, }\DIFaddend displacement-controlled cyclic loading protocol\DIFaddbegin \DIFadd{, consistent with the qualification-oriented characterization of seismic energy dissipation devices}\DIFaddend . Since the admissible deformation demand depends on the size of the device, different loading patterns are adopted according to the device height. As shown in Figure~\ref{fig:LoadPatterns}, the maximum displacement amplitude increase with the device height, consistently with the expected performance range of each geometry family.
\begin{figure}[htbp]
\centering
......@@ -290,7 +305,7 @@ For every sampled configuration, a FEM simulation is performed under a displacem
The resulting dataset stores the input variables and the structural response quantities extracted at the final time of each simulation. The target outputs include damage indicators in the windows and in the frame, together with local distortion measures associated with dissipative activation. For compactness in the optimization formulation, the aggregated TFDMap indicator in window $i$ is denoted by $\TFD_i$, whereas the corresponding frame indicator is denoted by $\TFD_f$. Both quantities are \DIFaddbegin \DIFadd{obtained through a regional percentile-based aggregation, }\DIFaddend computed as the \DIFdelbegin \DIFdel{average TFDMap value of the 12 nodes with the highest values within each region. This allows to capture }\DIFdelend \DIFaddbegin \DIFadd{98th percentile (P98) of the TFDMap values within the corresponding region. The same P98 regional statistic is used for the window distortion measure. A percentile-based regional statistic avoids defining the response through an arbitrary fixed number of nodes, while retaining }\DIFaddend the most critical damage levels \DIFdelbegin \DIFdel{while }\DIFdelend \DIFaddbegin \DIFadd{and }\DIFaddend reducing sensitivity to isolated numerical peaks.
The \DIFdelbegin \DIFdel{maximum }\DIFdelend local shear distortion in each window\DIFaddbegin \DIFadd{, aggregated with the same P98 regional statistic, }\DIFaddend is denoted by $\varepsilon_{xy,i}$ and is used as an indicator of the energy dissipation capacity. When included in the objective function, each window contribution is weighted according to its effective geometric volume, so that the optimization accounts for both local strain intensity and the material volume involved in the dissipation process.
The \DIFdelbegin \DIFdel{maximum }\DIFdelend local shear distortion in each window\DIFaddbegin \DIFadd{, aggregated with the same P98 regional statistic, }\DIFaddend is denoted by $\varepsilon_{xy,i}$ and is used \DIFdelbegin \DIFdel{as an indicatorof the energy dissipation capacity. When included in the objective function}\DIFdelend \DIFaddbegin \DIFadd{to build a shear-distortion performance indicator. When used in the optimization}\DIFaddend , each window contribution is weighted according to its effective \DIFdelbegin \DIFdel{geometric }\DIFdelend volume, so that the \DIFdelbegin \DIFdel{optimization }\DIFdelend \DIFaddbegin \DIFadd{indicator }\DIFaddend accounts for both \DIFdelbegin \DIFdel{local strain }\DIFdelend \DIFaddbegin \DIFadd{the local distortion }\DIFaddend intensity and the \DIFdelbegin \DIFdel{material volume }\DIFdelend \DIFaddbegin \DIFadd{amount of material }\DIFaddend involved in the \DIFdelbegin \DIFdel{dissipation process}\DIFdelend \DIFaddbegin \DIFadd{shear mechanism}\DIFaddend .
\subsection{Supervised ML surrogate models}\label{subsec:ml_models}
......@@ -311,7 +326,7 @@ Model selection is performed in two stages \DIFaddbegin \DIFadd{using the inner
\label{fig:BayesianSearchCV}
\end{figure*}
Preliminary executions of the proposed workflow show that SVR or GPR always provide the highest, or second-highest, predictive accuracy for the considered datasets. These results suggest that kernel-based models, and in particular distance-based similarity measures, are well suited to approximate the FEM response surfaces involved in this problem. This observation motivates the assessment of Radial Basis Function (RBF) interpolation \cite{Gutmann2001} as a simpler and computationally efficient surrogate alternative. Although surrogate evaluation is negligible compared with FEM simulations, the training and hyperparameter optimization of complex supervised models can still become relevant when several outputs, geometry families and adaptive iterations are considered. RBF interpolation provides a non-parametric and fast-to-train alternative that can capture nonlinear response surfaces, making it attractive for low-dimensional and moderately sampled design spaces.
Preliminary executions of the proposed workflow show that SVR or GPR always provide the highest, or second-highest, predictive accuracy for the considered datasets. These results suggest that kernel-based models, and in particular distance-based similarity measures, are well suited to approximate the FEM response surfaces involved in this problem. This observation motivates the assessment of Radial Basis Function (RBF) interpolation \cite{Gutmann2001} as a simpler \DIFdelbegin \DIFdel{and computationally efficient }\DIFdelend surrogate alternative. Although surrogate evaluation is negligible compared with FEM simulations, the \DIFdelbegin \DIFdel{training and hyperparameter optimization of complex }\DIFdelend \DIFaddbegin \DIFadd{Bayesian hyperparameter optimization and cross-validation of several }\DIFaddend supervised models can still become relevant when \DIFdelbegin \DIFdel{several }\DIFdelend \DIFaddbegin \DIFadd{many }\DIFaddend outputs, geometry families and adaptive iterations are considered. RBF interpolation provides a non-parametric \DIFdelbegin \DIFdel{and fast-to-train alternative that }\DIFdelend \DIFaddbegin \DIFadd{alternative whose training requires only a small grid search over the kernel and smoothing parameters, and it }\DIFaddend can capture nonlinear response surfaces, making it attractive for low-dimensional and moderately sampled design spaces.
\subsection{RBF surrogate models}\label{subsec:rbf_models}
......@@ -328,47 +343,83 @@ For each output variable, a final RBF surrogate is trained using all available F
\section{Damage-aware surrogate-assisted optimization}\label{sec:optimization}
The proposed methodology seeks to balance damage among the dissipative windows while keeping it below a prescribed threshold. At the same time, it limits damage in the surrounding frame and promotes the highest possible energy dissipation through the activation of the windows. The \DIFdelbegin \DIFdel{geometric }\DIFdelend \DIFaddbegin \DIFadd{window-thickness }\DIFaddend optimization is carried out using DE \cite{Storn1997}, a population-based global optimizer that does not require gradient information and is therefore suitable for nonlinear and non-convex surrogate response surfaces. In the current implementation, DE \DIFdelbegin \DIFdel{is run with a maximum of 500 iterations}\DIFdelend \DIFaddbegin \DIFadd{uses a best/1/bin strategy with a mutation factor sampled in $[0.5,1.0]$, a crossover probability of 0.7}\DIFaddend , a population size factor of 25\DIFaddbegin \DIFadd{, a maximum of 500 iterations }\DIFaddend and a convergence tolerance of $10^{-6}$\DIFaddbegin \DIFadd{; the initial population is generated by Latin hypercube sampling and no local polishing is applied. Each optimization stage is repeated with 30 independent runs using deterministic seeds derived from a fixed base seed}\DIFaddend . Once an optimal candidate is obtained, an adaptive FEM validation loop is applied to verify the predicted geometry before acceptance.
The proposed \DIFdelbegin \DIFdel{methodology seeks to balance damage among the dissipative windows while keeping it below a prescribed threshold. At the same time, it limits damage in the surrounding frame and promotes the highest possible energy dissipation through the activation of the }\DIFdelend \DIFaddbegin \DIFadd{formulation treats damage admissibility through explicit constraints rather than through weighted penalty terms, and it promotes shear deformation in the intended dissipative }\DIFaddend windows. The \DIFdelbegin \DIFdel{geometric }\DIFdelend \DIFaddbegin \DIFadd{window-thickness }\DIFaddend optimization is carried out using DE \cite{Storn1997}, a population-based global optimizer that does not require gradient information and is therefore suitable for nonlinear and non-convex surrogate response surfaces. In the current implementation, DE \DIFdelbegin \DIFdel{is run with a maximum of 500 iterations}\DIFdelend \DIFaddbegin \DIFadd{uses a best/1/bin strategy with a mutation factor sampled in $[0.5,1.0]$, a crossover probability of 0.7}\DIFaddend , a population size factor of 25\DIFaddbegin \DIFadd{, a maximum of 500 iterations }\DIFaddend and a convergence tolerance of $10^{-6}$\DIFdelbegin \DIFdel{. Once an optimal }\DIFdelend \DIFaddbegin \DIFadd{; the initial population is generated by Latin hypercube sampling and no local polishing is applied. Each optimization stage is repeated with 30 independent runs using deterministic seeds derived from a fixed base seed. Once a }\DIFaddend candidate is obtained, an adaptive FEM validation loop is applied to verify the predicted geometry before acceptance.
For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window \DIFdelbegin \DIFdel{distortions }\DIFdelend \DIFaddbegin \DIFadd{distortion }\DIFaddend $\hat{\varepsilon}_{xy,i}$, the window \DIFdelbegin \DIFdel{damage }\DIFdelend \DIFaddbegin \DIFadd{damage-screening }\DIFaddend indicators $\hat{\mathcal{D}}_i$ and the frame \DIFdelbegin \DIFdel{damage }\DIFdelend \DIFaddbegin \DIFadd{damage-screening }\DIFaddend indicator $\hat{\mathcal{D}}_f$. \DIFdelbegin \DIFdel{Damage is therefore controlled in all regions of the device, but with different mechanical relevance: frame damage is penalized more severely because it may compromise the structural integrity of the damper, whereas the window damage penalties are formulated to promote comparable damage levels among windows and avoid concentrating the dissipative demand in a single region.
The dissipative contribution is estimated from $\hat{\varepsilon}_{xy,i}^2$, the window thickness and the corresponding area of the window, since the energy dissipated by each window depends not only on the distortion level but also on the amount of material involved. This term is several orders of magnitude smaller than the damage penalties and is intentionally left unscaled.
As a result, damage control remains the dominant criterion, while the dissipative term acts as a tie-breaker among geometries with similar damage performance,
favouring those with higher distortion and, consequently, greater energy dissipation capacity}\DIFdelend \DIFaddbegin \DIFadd{A candidate is feasible at the surrogate level only if
}\begin{equation}
\DIFadd{\hat{\mathcal{D}}_i(\mathbf{x}) + m_i(\mathbf{x}) \le \mathcal{D}_W, \quad i=1,\ldots,N_w,
\qquad
\hat{\mathcal{D}}_f(\mathbf{x}) + m_f(\mathbf{x}) \le \mathcal{D}_F,
\label{eq:feasibility}
}\end{equation}
\DIFadd{where $\mathcal{D}_W=100$ and $\mathcal{D}_F=90$ are the prescribed damage-screening thresholds for the windows and for the surrounding frame, and $m_i$ and $m_f$ are conservative margins that account for local out-of-fold underprediction of the surrogate models. The thresholds are not soft optimization targets: they define the admissible damage region, and candidates that violate them are not feasible designs. Because the TFDMap is a post-processing damage-screening indicator rather than a constitutive fracture model, the thresholds are adopted to control the relative proximity to critical damage states within the optimization framework, and they are not interpreted as a mathematically exact point of complete physical failure.
}
\DIFadd{Feasibility is assessed before the optimization stages. A dedicated pre-solve minimizes the maximum constraint violation
}\begin{equation}
\DIFadd{v(\mathbf{x}) = \max\Bigl(0,\; \max_{i}\bigl[\hat{\mathcal{D}}_i(\mathbf{x})+m_i(\mathbf{x})-\mathcal{D}_W\bigr],\; \hat{\mathcal{D}}_f(\mathbf{x})+m_f(\mathbf{x})-\mathcal{D}_F\Bigr).
\label{eq:violation}
}\end{equation}
\DIFadd{If the minimum identified violation exceeds a small numerical tolerance, no surrogate-feasible region is identified. In that case the performance stages are not executed; the least-infeasible candidate is retained as an enrichment point and re-evaluated with FEM. This situation indicates that the surrogate-defined admissible region is empty or not yet resolved, and it is not, by itself, evidence that the physical/FEM design space contains no feasible solution.
}
For each candidate geometry $\mathbf{x}$, the trained surrogate models predict the window distortions $\hat{\varepsilon}_{xy,i}$, the window damage indicators $\hat{\mathcal{D}}_i$ and the frame damage indicator $\hat{\mathcal{D}}_f$. Damage is therefore controlled in all regions of the device, but with different mechanical relevance: frame damage is penalized more severely because it may compromise the structural integrity of the damper, whereas the window damage penalties are formulated to promote comparable damage levels among windows and avoid concentrating the dissipative demand in a single region. The dissipative contribution is estimated from $\hat{\varepsilon}_{xy,i}^2$, the window thickness and the corresponding area of the window, since the energy dissipated by each window depends not only on the distortion level but also on the amount of material involved. This term is several orders of magnitude smaller than the damage penalties and is intentionally left unscaled. As a result, damage control remains the dominant criterion, while the dissipative term acts as a tie-breaker among geometries with similar damage performance, favouring those with higher distortion and, consequently, greater energy dissipation capacity.
\DIFadd{When a surrogate-feasible region exists, the optimization proceeds hierarchically within that region. The first stage promotes the activation of every dissipative window by maximizing the smallest window damage-screening indicator,
}\begin{equation}
\DIFadd{\max_{\mathbf{x}\in\mathcal{F}}\; \mathcal{J}_1(\mathbf{x}) = \min_{i} \hat{\mathcal{D}}_i(\mathbf{x}),
\label{eq:stage1}
}\end{equation}
\DIFadd{where $\mathcal{F}$ denotes the feasible set defined by Eq.~\eqref{eq:feasibility}. Mechanically, this stage prevents configurations in which one or more windows remain essentially inactive while damage concentrates elsewhere, and it promotes a balanced participation of the dissipative regions. Let $\mathbf{x}_1^{\star}$ be the resulting optimum and $\mathcal{J}_1^{\star}$ the corresponding value}\DIFaddend .
The implemented objective function to be minimized is
\begin{equation}
J(\mathbf{x}) = - \sum_{i=1}^{N_w} \hat{\varepsilon}_{xy,i}^2\, t_{w,i}\, A_i +
The \DIFdelbegin \DIFdel{implemented objective function to be minimized is
}\begin{displaymath}
\DIFdel{J(\mathbf{x}) = - \sum_{i=1}^{N_w} \hat{\varepsilon}_{xy,i}^2\, t_{w,i}\, A_i +
\sum_{i=1}^{N_w} P_w\left(\hat{\mathcal{D}}_i;\mathcal{D}_w^{\star}\right) +
P_f\left(\hat{\mathcal{D}}_f;\mathcal{D}_f^{\max}\right),
\label{eq:objective}
\end{equation}
where $N_w$ is the number of windows, $t_{w,i}$ is the thickness of window $i$, $A_i$ is the corresponding area factor, $\mathcal{D}_w^{\star}$ is the target damage level for the windows and $\mathcal{D}_f^{\max}$ is the maximum admissible frame damage threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger energy dissipation contributions reduce the objective value.
The window penalty is defined as
\begin{equation}
P_w\left(\hat{\mathcal{D}}_i;\mathcal{D}_w^{\star}\right)=
%DIFDELCMD < \label{eq:objective}%%%
}\end{displaymath}%DIFAUXCMD
\DIFdelend \DIFaddbegin \DIFadd{second stage maximizes a shear-distortion performance indicator while preserving the first-stage optimum within a numerical tolerance,
}\begin{equation}
\DIFadd{\max_{\mathbf{x}\in\mathcal{F}}\; \mathcal{J}_2(\mathbf{x}) = \sum_{i=1}^{N_w} \hat{\varepsilon}_{xy,i}^{2}(\mathbf{x})\, t_{w,i}\, V_i
\qquad\text{subject to}\qquad
\min_{i} \hat{\mathcal{D}}_i(\mathbf{x}) \ge \mathcal{J}_1^{\star} - \delta_1 .
\label{eq:stage2}
}\end{equation}\DIFaddend
\DIFdelbegin \DIFdel{where $N_w$ is the number of windows,
}\DIFdelend \DIFaddbegin \DIFadd{Here }\DIFaddend $t_{w,i}$ is the thickness of window $i$, \DIFdelbegin \DIFdel{$A_i$ is the corresponding areafactor, $\mathcal{D}_w^{\star}$ is }\DIFdelend \DIFaddbegin \DIFadd{$V_i$ is the effective volume weighting factor of window $i$ (implemented as a thickness-scaled area/volume factor), and $\delta_1$ is a small preservation tolerance derived from the spread of the first-stage objective across the independent DE runs. The two stages are solved sequentially, so the formulation is hierarchical: damage admissibility is enforced through the constraints, }\DIFaddend the \DIFdelbegin \DIFdel{target damage level for the windows and $\mathcal{D}_f^{\max}$ is the maximum admissible frame damage threshold. The first term is negative because the optimizer minimizes $J$; therefore, larger energy dissipation contributions reduce the objective value}\DIFdelend \DIFaddbegin \DIFadd{preferred damage distribution is secured first, and the shear-distortion performance indicator is optimized second within the tolerance that preserves the first-stage result}\DIFaddend .
The \DIFdelbegin \DIFdel{window penalty is defined as
}\begin{displaymath}
\DIFdel{P_w\left(\hat{\mathcal{D}}_i;\mathcal{D}_w^{\star}\right)=
\begin{cases}
\left(\hat{\mathcal{D}}_i-\mathcal{D}_w^{\star}\right)^2, & \hat{\mathcal{D}}_i>\mathcal{D}_w^{\star},\\
\left|\hat{\mathcal{D}}_i-\mathcal{D}_w^{\star}\right|, & \hat{\mathcal{D}}_i\leq\mathcal{D}_w^{\star}.
\end{cases}
\label{eq:window_penalty}
\end{equation}
This formulation penalizes values above the target quadratically, while also discouraging excessively underused windows through a linear distance to the target. As a result, the optimizer tends to balance the damage levels among windows rather than forcing all windows to remain far below the admissible level.
}\end{displaymath}%DIFAUXCMD
\DIFdel{This formulation penalizes values above the target quadratically, while also discouraging excessively underused windows through }\DIFdelend \DIFaddbegin \DIFadd{quantity $\mathcal{J}_2$ is }\DIFaddend a \DIFdelbegin \DIFdel{linear distance to the target. As a result, the optimizer tends to balance the damage levels among windows rather than forcing all windowsto remain far below the admissible level. }\DIFdelend \DIFaddbegin \DIFadd{shear-distortion performance indicator. The BDSL windows are intended to accommodate their inelastic response predominantly through shear deformation, so a larger local shear distortion represents a stronger activation of the intended deformation mechanism. Weighting the squared distortion by the thickness and an effective volume factor favours both deformation intensity and the participation of material in the shear windows. The indicator is used only to rank feasible candidates; it does not integrate the stress--strain hysteresis loop, does not contain the complete stress history, and does not represent cumulative plastic work or the total hysteretic energy dissipated by the device. It is therefore not interpreted as an energy quantity, and establishing a quantitative relationship between this indicator and the cumulative hysteretic energy computed by the FEM model is left for future work.
}\DIFaddend
The frame penalty is defined as
\begin{equation}
P_f\left(\hat{\mathcal{D}}_f;\mathcal{D}_f^{\max}\right)=
\DIFdelbegin \DIFdel{The frame penalty is defined as }\begin{displaymath}
\DIFdel{P_f\left(\hat{\mathcal{D}}_f;\mathcal{D}_f^{\max}\right)=
\begin{cases}
\left(\hat{\mathcal{D}}_f-\mathcal{D}_f^{\max}\right)^3, & \hat{\mathcal{D}}_f>\mathcal{D}_f^{\max},\\
0, & \hat{\mathcal{D}}_f\leq\mathcal{D}_f^{\max}.
\end{cases}
\label{eq:frame_penalty}
\end{equation}
The cubic penalty reflects the greater severity of frame damage compared with localized damage in a window. Since a damage value $\mathcal{D}=100$ represents complete failure, $\mathcal{D}_f^{\max}=90$ is adopted in the current implementation. The target window threshold is also set to $\mathcal{D}_w^{\star}=90$.
}\end{displaymath}%DIFAUXCMD
\DIFdel{The cubic penalty reflects the greater severity of frame damage compared with localized damage in a window.
Since a damage value $\mathcal{D}=100$ represents complete failure, $\mathcal{D}_f^{\max}=90$ is adopted in the current implementation. The target window threshold is also set to $\mathcal{D}_w^{\star}=90$}\DIFdelend \DIFaddbegin \DIFadd{Compared with a scalar objective that combines damage penalties and a performance term, the constrained hierarchical formulation removes the need for arbitrary relative weights between dimensionally different quantities. The numerical values of the preservation tolerance, the conservative margin and the feasibility tolerance remain calibration parameters, and their sensitivity is not assessed in the present study}\DIFaddend .
The surrogate-optimized geometry is not accepted directly. Instead, 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. \DIFdelbegin \DIFdel{This validation step checks whether the surrogate has remained reliable in the region of the design space selected by the optimizer . The }\DIFdelend \DIFaddbegin \DIFadd{Acceptance is based on FEM-confirmed physical feasibility and on the robustness of the optimizer rather than on surrogate accuracy alone. Specifically, the }\DIFaddend candidate geometry is accepted only if: (i) the \DIFdelbegin \DIFdel{prediction error of all damage and distortion variables remains below the prescribed tolerance, equal to 5\%}\DIFdelend \DIFaddbegin \DIFadd{FEM damage indicators satisfy the prescribed physical limits in every window and in the frame}\DIFaddend ; (ii) the \DIFdelbegin \DIFdel{absolute error of the objective function remains within the admissible limit, set to 10; and }\DIFdelend \DIFaddbegin \DIFadd{Differential Evolution result is reproducible, requiring that the performance of all independent runs is numerically indistinguishable from the best run according to a scale-aware relative criterion, with a tolerance of $10^{-3}$ in each optimization stage; }\DIFaddend (iii) the \DIFdelbegin \DIFdel{optimized window thicknesses remain }\DIFdelend \DIFaddbegin \DIFadd{optimum remains }\DIFaddend stable between consecutive \DIFdelbegin \DIFdel{optimization }\DIFdelend \DIFaddbegin \DIFadd{adaptive }\DIFaddend iterations, with \DIFdelbegin \DIFdel{variations smaller than the 5\%of the full design range. If all }\DIFdelend \DIFaddbegin \DIFadd{relative changes in the stage objectives below 2\%; and (iv) a conservatism check confirms that the surrogate safety margin does not materially limit the optimum. Surrogate--FEM prediction errors for the damage and distortion variables are monitored against a local out-of-fold error threshold as a diagnostic measure of surrogate fidelity, but do not by themselves determine acceptance. If the acceptance }\DIFaddend 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, that reduces the risk of accepting a geometry that is optimal only because of surrogate extrapolation error, is summarized in Figure~\ref{fig:OptimizationFlowChart}.
The surrogate-optimized geometry is not accepted directly. \DIFdelbegin \DIFdel{Instead, once an optimal geometry is identified by the surrogate-assisted optimizer }\DIFdelend \DIFaddbegin \DIFadd{The conservative feasibility screen of Eq.~\eqref{eq:feasibility} is only a surrogate-level admissibility check; engineering acceptance requires a FEM re-evaluation of the selected candidate. Once the hierarchical optimizer proposes a candidate}\DIFaddend , it is re-evaluated with FEM\DIFdelbegin \DIFdel{to verify that the surrogate remains accurate in the region of the design space where the optimum lies. This validation step checks whether the surrogate has remained reliable in the region of the design space selected by the optimizer }\DIFdelend . \DIFdelbegin \DIFdel{The }\DIFdelend \DIFaddbegin \DIFadd{Acceptance is based on FEM-confirmed physical feasibility and on the robustness of the optimizer rather than on surrogate accuracy alone. Specifically, the }\DIFaddend candidate geometry is accepted only if: (i) the \DIFdelbegin \DIFdel{prediction error of all damage and distortion variables remains below the prescribed tolerance, equal to 5\%}\DIFdelend \DIFaddbegin \DIFadd{FEM damage indicators satisfy the prescribed physical limits in every window and in the frame}\DIFaddend ; (ii) the \DIFdelbegin \DIFdel{absolute error of the objective function remains within the admissible limit, set to 10; and }\DIFdelend \DIFaddbegin \DIFadd{Differential Evolution result is reproducible, requiring that the performance of all independent runs is numerically indistinguishable from the best run according to a scale-aware relative criterion, with a tolerance of $10^{-3}$ in each optimization stage; }\DIFaddend (iii) the \DIFdelbegin \DIFdel{optimized window thicknesses remain }\DIFdelend \DIFaddbegin \DIFadd{optimum remains }\DIFaddend stable between consecutive \DIFdelbegin \DIFdel{optimization }\DIFdelend \DIFaddbegin \DIFadd{adaptive }\DIFaddend iterations, with \DIFdelbegin \DIFdel{variations smaller than the 5\%of the full design range. If all }\DIFdelend \DIFaddbegin \DIFadd{relative changes in the stage objectives below 2\%; and (iv) a conservatism check confirms that the surrogate safety margin does not materially limit the optimum. Surrogate--FEM prediction errors for the damage and shear-distortion variables are monitored against a local out-of-fold error threshold as a diagnostic measure of surrogate fidelity, but do not by themselves determine acceptance. If the acceptance }\DIFaddend criteria are satisfied, the FEM-validated geometry is accepted as \DIFdelbegin \DIFdel{the optimized design}\DIFdelend \DIFaddbegin \DIFadd{a feasible numerical candidate}\DIFaddend . If at least one criterion is not satisfied, the new FEM result is added to the dataset, the surrogate models are retrained and the \DIFdelbegin \DIFdel{DE }\DIFdelend optimization is repeated. This loop, that reduces the risk of accepting a geometry that is optimal only because of surrogate extrapolation error, is summarized in Figure~\ref{fig:OptimizationFlowChart}. \DIFaddbegin \DIFadd{If no feasible candidate is identified within the considered design space, the corresponding family is reported as such rather than treated as a feasible optimum.
}\DIFaddend
\begin{figure*}[htbp]
\centering
\includegraphics[width=1.0\textwidth]{./images/OptimizationFlowChart/OptimizationFlowChart.pdf}
\caption{Surrogate-assisted optimization and FEM validation retraining loop.}
\caption{\DIFdelbeginFL \DIFdelFL{Surrogate-assisted }\DIFdelendFL \DIFaddbeginFL \DIFaddFL{Feasibility-first hierarchical surrogate-assisted }\DIFaddendFL optimization and FEM validation \DIFdelbeginFL \DIFdelFL{retraining }\DIFdelendFL loop. \DIFaddbeginFL \DIFaddFL{Damage-screening limits are enforced as constraints, and the shear-distortion performance indicator is optimized in the second stage.}\DIFaddendFL }
\label{fig:OptimizationFlowChart}
\end{figure*}
......@@ -383,7 +434,7 @@ The supervised-learning comparison shows a \DIFdelbegin \DIFdel{clear hierarchy
\label{fig:surrogate_selection_summary_barplot}
\end{figure*}
These results indicate that kernel-based models \DIFdelbegin \DIFdel{are particularly well suited to the present surrogate task. SVR provides }\DIFdelend \DIFaddbegin \DIFadd{performed well under the adopted nested validation procedure for the datasets considered here. SVR provided }\DIFaddend the best compromise between accuracy and computational cost, while GPR \DIFdelbegin \DIFdel{is }\DIFdelend \DIFaddbegin \DIFadd{was }\DIFaddend the second most \DIFdelbegin \DIFdel{competitive }\DIFdelend \DIFaddbegin \DIFadd{frequently selected }\DIFaddend supervised strategy, especially in some higher-dimensional cases. Tree-based models \DIFdelbegin \DIFdel{, although robust, are }\DIFdelend \DIFaddbegin \DIFadd{were }\DIFaddend less frequently selected and MLP models \DIFdelbegin \DIFdel{are }\DIFdelend \DIFaddbegin \DIFadd{were }\DIFaddend not competitive in terms of computational efficiency for the dataset sizes considered\DIFdelbegin \DIFdel{here}\DIFdelend . As mentioned in previous sections, this behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. \DIFdelbegin \DIFdel{In contrast to }\DIFdelend \DIFaddbegin \DIFadd{Because the RBF and }\DIFaddend the supervised models \DIFdelbegin \DIFdel{, RBF models were trained in less than one second per output , making them especially }\DIFdelend \DIFaddbegin \DIFadd{are assessed on the same outer cross-validation splits and with the same error metrics, the two strategies can be compared on a like-for-like basis at the level of each output variable. Across the analysed outputs neither strategy dominates: the selected supervised model is more accurate for some targets and the RBF surrogate for others, and both remain within the accuracy range required by the adaptive validation loop. The RBF surrogate, however, is obtained with a much smaller hyperparameter search and therefore requires substantially lower training effort, which makes it }\DIFaddend attractive for repeated surrogate updates within the adaptive optimization loop.
These results indicate that kernel-based models \DIFdelbegin \DIFdel{are particularly well suited to the present surrogate task. SVR provides }\DIFdelend \DIFaddbegin \DIFadd{performed well under the adopted nested validation procedure for the datasets considered here. SVR provided }\DIFaddend the best compromise between accuracy and computational cost, while GPR \DIFdelbegin \DIFdel{is }\DIFdelend \DIFaddbegin \DIFadd{was }\DIFaddend the second most \DIFdelbegin \DIFdel{competitive }\DIFdelend \DIFaddbegin \DIFadd{frequently selected }\DIFaddend supervised strategy, especially in some higher-dimensional cases. Tree-based models \DIFdelbegin \DIFdel{, although robust, are }\DIFdelend \DIFaddbegin \DIFadd{were }\DIFaddend less frequently selected and MLP models \DIFdelbegin \DIFdel{are }\DIFdelend \DIFaddbegin \DIFadd{were }\DIFaddend not competitive in terms of computational efficiency for the dataset sizes considered\DIFdelbegin \DIFdel{here}\DIFdelend . As mentioned in previous sections, this behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. \DIFdelbegin \DIFdel{In contrast to }\DIFdelend \DIFaddbegin \DIFadd{Because the RBF and }\DIFaddend the supervised models \DIFdelbegin \DIFdel{, RBF models were trained in less than one second per output , making them especially }\DIFdelend \DIFaddbegin \DIFadd{are assessed on the same outer cross-validation splits and with the same error metrics, the two strategies can be compared on a like-for-like basis at the level of each output variable. Across the analysed outputs neither strategy dominates: the selected supervised model is more accurate for some targets and the RBF surrogate for others, and both remain within the accuracy range required by the adaptive validation loop. The RBF surrogate, however, is obtained with a simpler training procedure, based on a small grid search over the kernel and smoothing parameters, that does not require Bayesian hyperparameter optimization, which makes it }\DIFaddend attractive for repeated surrogate updates within the adaptive optimization loop.
The \DIFaddbegin \DIFadd{predictive assessment is constrained by the small datasets available for the two-window families ($N=8$) and the three-window families ($N=16$). Nested cross-validation separates hyperparameter selection from generalization assessment, and the outer folds provide out-of-sample predictions together with bootstrap confidence intervals for RMSE and MAE. For datasets with $N\leq20$ the outer loop is Leave-One-Out, so the reported errors are obtained from a single deterministic partition and their dispersion is estimated by resampling the out-of-sample predictions rather than by repeating the outer split. These results should therefore be interpreted as an assessment of predictive accuracy within the sampled design domain and for the analysed geometry families, under the adopted validation procedure, rather than as a general guarantee. No independent FEM test set was retained, and the uncertainty of the accuracy estimates for the smallest datasets remains a limitation of the present comparison.
}
......@@ -458,7 +509,7 @@ Figure~\ref{fig:optimized_window_thickness_evolution} shows the evolution of the
\label{fig:optimized_window_thickness_evolution}
\end{figure*}
From a methodological point of view, these results highlight the trade-off between surrogate complexity, accuracy and computational efficiency. Supervised models, particularly SVR and GPR, provide \DIFdelbegin \DIFdel{high predictive accuracy and robustness}\DIFdelend \DIFaddbegin \DIFadd{competitive predictive accuracy under the adopted validation procedure}\DIFaddend , but require hyperparameter optimization and cross-validation for every output variable and adaptive iteration. RBF interpolation, in contrast, has a much lower training cost and provides very competitive final predictions once the relevant regions of the design domain have been adaptively sampled. Therefore, the comparison does not identify a universally superior surrogate strategy. Instead, it suggests that RBF interpolation is especially suitable for low- to moderate-dimensional design spaces with well-distributed FEM samples and relatively smooth input--output relationships, as occurs for the response variables analysed in this work. Supervised ML surrogates remain valuable when greater robustness is required or when the response surface is expected to involve stronger nonlinear interactions, local irregularities or higher-dimensional dependencies.
From a methodological point of view, these results highlight the trade-off between surrogate complexity, accuracy and computational efficiency. Supervised models, particularly SVR and GPR, provide \DIFdelbegin \DIFdel{high predictive accuracy and robustness}\DIFdelend \DIFaddbegin \DIFadd{competitive predictive accuracy under the adopted validation procedure}\DIFaddend , but require hyperparameter optimization and cross-validation for every output variable and adaptive iteration. RBF interpolation, in contrast, \DIFdelbegin \DIFdel{has a much lower training cost and provides very }\DIFdelend \DIFaddbegin \DIFadd{requires a simpler training procedure and provides }\DIFaddend competitive final predictions once the relevant regions of the design domain have been adaptively sampled. Therefore, the comparison does not identify a universally superior surrogate strategy. Instead, it suggests that RBF interpolation is especially suitable for low- to moderate-dimensional design spaces with well-distributed FEM samples and relatively smooth input--output relationships, as occurs for the response variables analysed in this work. Supervised ML surrogates remain valuable when greater robustness is required or when the response surface is expected to involve stronger nonlinear interactions, local irregularities or higher-dimensional dependencies.
The objective-function values should also be interpreted carefully. The objective-function error measures the consistency between surrogate predictions and FEM validation, not whether the final objective value is necessarily close to zero. Some geometry families, such as $F_2$, retain non-negligible penalty contributions because the prescribed damage targets cannot be fully achieved within the admissible design bounds. Nevertheless, the close agreement between surrogate and FEM objective values indicates that the accepted designs are not artifacts of surrogate extrapolation, but FEM-consistent optimized candidates within the explored design space.
......@@ -473,23 +524,42 @@ This behaviour is illustrated in Figure~\ref{fig:rbf_surface_evolution}, which s
\section{Conclusions and future work}\label{sec:conclusions}
This work presents an adaptive surrogate-assisted optimization framework for \DIFaddbegin \DIFadd{the seismic window-thickness optimization of }\DIFaddend buckling-delayed shear-link dampers \DIFdelbegin \DIFdel{subjected to cyclic seismic loading}\DIFdelend \DIFaddbegin \DIFadd{within five predefined geometry families}\DIFaddend . The proposed methodology addresses the main limitation of direct FEM-based optimization, the high computational cost associated with repeatedly evaluating nonlinear cyclic simulations. By training surrogate models on FEM-generated datasets and validating the optimized candidates through additional FEM analyses, the optimizer can efficiently explore the design domain while remaining consistent with the mechanical response captured by the calibrated numerical model.
This work \DIFdelbegin \DIFdel{presents }\DIFdelend \DIFaddbegin \DIFadd{presented }\DIFaddend an adaptive surrogate-assisted optimization framework for \DIFaddbegin \DIFadd{the seismic window-thickness optimization of }\DIFaddend buckling-delayed shear-link dampers \DIFdelbegin \DIFdel{subjected to cyclic seismic loading. The proposed methodology }\DIFdelend \DIFaddbegin \DIFadd{within five predefined geometry families. Damage admissibility is enforced through explicit feasibility limits on the window and frame damage-screening indicators, and the preference among feasible designs is expressed hierarchically through a maximum window-activation stage followed by a shear-distortion performance indicator. The framework combines FEM-calibrated reference models, supervised and RBF surrogates, Differential Evolution and an adaptive FEM validation loop, and it }\DIFaddend addresses the main limitation of direct FEM-based optimization, \DIFdelbegin \DIFdel{the high computational cost associated with }\DIFdelend \DIFaddbegin \DIFadd{namely the cost of }\DIFaddend repeatedly evaluating nonlinear cyclic simulations.
\DIFdelbegin \DIFdel{By training surrogate models on FEM-generated datasets and validating the optimized candidates through additional FEM analyses, the optimizer can efficiently explore the design domain while remaining consistent with the mechanical response captured by the calibrated numerical model.
}\DIFdelend
One of the key features of the proposed optimization framework, compared to approaches focused primarily on energy maximization, is the formulation of an objective function that takes damage into account. The aim is to prioritise local damage control in both the dissipative windows and the surrounding frame, whilst promoting a balanced contribution from all windows to the energy dissipation process. Thus, damage to the windows is permitted and expected, as they are intended to act as dissipative regions, provided that it remains controlled and reasonably distributed; conversely, damage to the frame is penalised more severely because it can compromise the structural integrity of the damper. Therefore, rather than merely maximizing the energy dissipated, the proposed formulation favours geometries that concentrate dissipative activation in the windows, prevent excessive damage localization in a single region and protect the frame from critical damage.
\DIFdelbegin \DIFdel{One of the key features of the proposed optimization framework, compared to approaches focused primarily on energy maximization, is the formulation of an objective function that takes damage into account. The aim is to prioritise local damage control }\DIFdelend \DIFaddbegin \DIFadd{The main findings of this study can be summarised as follows.
}\begin{enumerate}
\item \DIFadd{BDSL optimization is a damage-distribution problem rather than a pure energy-maximization problem. The same level of global dissipation can correspond to very different local damage patterns, so the design must control damage }\DIFaddend in both the dissipative windows and the surrounding frame\DIFdelbegin \DIFdel{, whilst promoting a balanced contribution from all windows to the energy dissipation process.
Thus, damage to the windows is permitted and expected, as they are intended to act as dissipative regions, provided that it remains controlled and reasonably distributed; conversely, damage to the frame is penalised more severely because it can compromise the structural integrity of the damper. Therefore, rather than merely maximizing the energy dissipated, the proposed formulation favours geometries that concentrate dissipative activation in the windows, prevent excessive damage localization in a single region and protect the frame from critical damage.
}%DIFDELCMD <
A \DIFdelbegin \DIFdel{comprehensive }\DIFdelend comparison of surrogate strategies was performed in terms of predictive accuracy and computational efficiency. \DIFdelbegin \DIFdel{The }\DIFdelend \DIFaddbegin \DIFadd{For the analysed datasets, the }\DIFaddend supervised ML results showed that \DIFaddbegin \DIFadd{kernel-based models, particularly }\DIFaddend SVR and GPR\DIFaddbegin \DIFadd{, }\DIFaddend were the most \DIFdelbegin \DIFdel{competitive models}\DIFdelend \DIFaddbegin \DIFadd{frequently selected}\DIFaddend , with SVR being the most frequently selected across the \DIFdelbegin \DIFdel{analysed outputs. RBF interpolation proved to be even a better alternative, due to its high efficiency: in the final adaptive iterations, it achieved validation errors }\DIFdelend \DIFaddbegin \DIFadd{outputs. When evaluated on the same outer cross-validation splits, RBF interpolation provided predictive performance comparable to the supervised models for the considered low-dimensional response surfaces, while requiring a much smaller hyperparameter search and substantially lower training effort. For the optimized candidates, the RBF surrogate also achieved surrogate--FEM agreement }\DIFaddend comparable to, and in \DIFdelbegin \DIFdel{most cases lower than, those }\DIFdelend \DIFaddbegin \DIFadd{several cases better than, that }\DIFaddend of the supervised ML surrogates\DIFdelbegin \DIFdel{, while requiring substantially lower training effort and no hyperparameter search. This performance is attributed to the characteristics of the present problem: low- to moderate-dimensional design spaces, well-distributed FEM samples and nonlinear but relatively smooth relationships between window thicknesses and response indicators}\DIFdelend \DIFaddbegin \DIFadd{. These observations are limited to the analysed geometry families and to the sampled design domain, and their reliability is lower for the smallest datasets, for which the accuracy estimates carry larger uncertainty}\DIFaddend .
%DIFDELCMD < %%%
\DIFdel{A comprehensive comparison of surrogate strategies was performed in terms of predictive accuracy and computational efficiency. The supervised ML results showed that }\DIFdelend \DIFaddbegin \DIFadd{.
}\item \DIFadd{FEM-calibrated surrogate models can predict the local damage and distortion indicators required by the optimizer. Under a common nested cross-validation framework, the kernel-based supervised models (}\DIFaddend SVR and GPR\DIFaddbegin \DIFadd{) }\DIFaddend were the most \DIFdelbegin \DIFdel{competitive models, with SVR being the most frequently selectedacross the analysed outputs. RBF interpolation proved to be even a better alternative, due to its high efficiency: in the final adaptive iterations, it achieved validation errors comparable to, and in most cases lower than, those of the supervised ML surrogates, while requiring substantially lower training effort and no hyperparameter search. This performance is attributed to the characteristics of the present problem: low- to moderate-dimensional design spaces, well-distributed FEM samples and nonlinear but relatively smooth relationships between window thicknesses and response indicators.
}\DIFdelend \DIFaddbegin \DIFadd{frequently selected, and RBF interpolation achieved comparable predictive performance with a simpler training procedure.
}\item \DIFadd{The adaptive FEM enrichment loop, in which acceptance is based on FEM-confirmed feasibility and optimizer robustness rather than on surrogate accuracy alone, reduces the risk of accepting candidates that are optimal only because of surrogate prediction error.
}\end{enumerate}
\DIFaddend
The proposed adaptive validation loop proved to be necessary and effective. Several initially optimized candidates did not satisfy the prescribed error tolerances. After incorporating the new FEM results into the training dataset and retraining the surrogates, the prediction errors decreased and \DIFdelbegin \DIFdel{all final optimized geometries satisfied the acceptance criteria }\DIFdelend \DIFaddbegin \DIFadd{the optimization converged }\DIFaddend after only two or three iterations. Therefore, the final designs are not accepted solely on the basis of surrogate predictions, but are explicitly verified through FEM in the region of the design space where the optimum is located.
The \DIFdelbegin \DIFdel{proposed adaptive validation loop proved to be necessary and effective. Several initially optimized candidates did not satisfy the prescribed error tolerances. After incorporating the new FEM results into the training dataset and retraining the surrogates, the prediction errors decreased and all final optimized geometries satisfied the acceptance criteria after only two or three iterations. Therefore, the final designs are not accepted solely on the basis of surrogate predictions, but are explicitly verified through FEM in the region of the design space where the optimum is located.
}%DIFDELCMD <
It is also worth noting that, although the framework allows the DoE to be expanded with additional FEM simulations when the surrogate accuracy is insufficient, this was not required in the present application. The initial DoE datasets were already adequate to obtain accurate optimized designs after adaptive validation and retraining. In particular, the final geometries were obtained from only 8 initial FEM simulations for the two-window devices, 16 for the three-window devices and 64 for the five-window devices, with a maximum of three adaptive retraining iterations in all cases. This demonstrates that the proposed strategy can achieve FEM-consistent optimized designs with a limited number of simulations, making it competitive from a computational point of view.
%DIFDELCMD < %%%
\DIFdel{It is also worth noting that , although the framework allows the DoE to be expanded with additional FEM simulations when the surrogate accuracy is insufficient, this was not required in the present application.
The initial DoE datasets were already adequate to obtain accurate optimized designs after adaptive validation and retraining. In particular, the final geometries were obtained from only 8 initial FEM simulations for the two-window devices, 16 for the three-window devices and 64 for the five-window devices, with a maximum of three adaptive retraining iterations in all cases. This demonstrates that the proposed strategy can achieve FEM-consistent optimized designs with a limited number of simulations, making it competitive from a computational point of view. }%DIFDELCMD <
The proposed methodology also has some limitations that should be acknowledged. First, its reliability depends on the quality of the calibrated FEM model used to generate the training data and validate the optimized designs. Second, the TFDMap is used here as a post-processing damage indicator rather than as a constitutive fracture model; therefore, the optimized configurations should be interpreted in terms of relative damage control and proximity to critical states, not as direct predictions of crack initiation. Third, only the window thicknesses are considered as design variables; although this leads to a controlled and interpretable optimization problem, it does not exploit the full geometric flexibility of BDSL dampers. Finally, the optimized geometries should ultimately be validated experimentally before being used to establish general design recommendations.
%DIFDELCMD < %%%
\DIFdel{The proposed methodology also has some limitations that should be acknowledged. First, its reliability depends on the quality of }\DIFdelend \DIFaddbegin \DIFadd{scope of these findings is limited. Only the window thicknesses were optimized; window height, spacing, corner radius, frame thickness and global proportions remained fixed within each family. The study is component-level and considers a symmetric, prescribed displacement-controlled cyclic protocol. It does not reproduce irregular, asymmetric or pulse-like earthquake demands, nor record-to-record variability, and it does not quantify structural-system-level seismic performance or resilience. In addition, the results depend on }\DIFaddend the calibrated FEM model used \DIFdelbegin \DIFdel{to generate the training data and validate the optimized designs. Second, the }\DIFdelend \DIFaddbegin \DIFadd{as ground truth, }\DIFaddend TFDMap is used \DIFdelbegin \DIFdel{here }\DIFdelend as a post-processing damage indicator rather than as a constitutive fracture model\DIFdelbegin \DIFdel{; therefore, the optimized configurations should be interpreted in terms of relative damage control and proximity to critical states, not as direct predictions of crack initiation. Third, only the window thicknesses are considered as design variables; although this leads to a controlled and interpretable optimization problem, it does not exploit the full geometric flexibility of BDSL dampers. Finally, the optimized geometries should ultimately be validated experimentally before being used to establish general design recommendations}\DIFdelend \DIFaddbegin \DIFadd{, and no independent experimental test set was retained. The optimized geometries are therefore FEM-validated numerical candidates rather than experimentally validated designs}\DIFaddend .
Future work should extend the design space by including additional geometric and mechanical variables, such as window height, window spacing, frame thickness or global device proportions. This extension would increase the dimensionality and complexity of the surrogate task. In those cases, the performance of RBF interpolation should therefore be reassessed. While RBF models performed very well in the present study, their efficiency and accuracy may decrease as the input space becomes larger or the response surfaces develop stronger local nonlinearities. In such cases, supervised ML models or hybrid surrogate strategies may become more advantageous.
Future work should extend the design space \DIFdelbegin \DIFdel{by including }\DIFdelend \DIFaddbegin \DIFadd{to }\DIFaddend additional geometric and mechanical variables, such as window height, window spacing, frame thickness or global device proportions\DIFdelbegin \DIFdel{. This extension would increase the dimensionality and complexity of the surrogate task. In those cases, the performance of RBF interpolation should therefore be reassessed. While RBF models performed very well in the present study, their efficiency and accuracy may decrease as the input spacebecomes larger or the response surfaces develop stronger local nonlinearities. In such cases, supervised ML models or hybrid surrogate strategies may become more advantageous. }%DIFDELCMD <
A complementary line of future work is the consideration of non-symmetric cyclic loading histories or recorded seismic displacement demands. The present study focuses on symmetric cyclic protocols because they provide a standardized and industry-relevant basis for the qualification of seismic energy dissipation devices, which must satisfy prescribed cyclic testing requirements before being implemented in practice. Nevertheless, earthquake-induced demands may lead to non-symmetric deformation histories in structural components. Extending the proposed framework to asymmetric cyclic protocols or representative seismic displacement histories would therefore be an interesting step towards broader performance assessment conditions.
%DIFDELCMD < %%%
\DIFdel{A complementary line of future work is the consideration of non-symmetric cyclic loading histories or recorded seismic displacement demands. The present study focuses on symmetric cyclic protocols because they provide a standardized and industry-relevant basis for the qualification of seismic energy dissipationdevices, which must satisfy prescribed cyclic testing requirements before being implemented in practice. Nevertheless, earthquake-induced demands may lead to non-symmetric deformation histories in structural components}\DIFdelend \DIFaddbegin \DIFadd{, and should reassess the surrogate strategies in the resulting higher-dimensional space. Assessing the optimized devices at the structural-system level through nonlinear time-history analyses, able to quantify inter-storey drift, floor acceleration and global energy dissipation, is a necessary next step before broader design recommendations can be established}\DIFaddend . Extending the \DIFdelbegin \DIFdel{proposed }\DIFdelend framework to asymmetric cyclic protocols or \DIFdelbegin \DIFdel{representative }\DIFdelend \DIFaddbegin \DIFadd{recorded }\DIFaddend seismic displacement histories\DIFdelbegin \DIFdel{would therefore be an interesting step towards broader performance assessment conditions.
}%DIFDELCMD <
Another future direction is the incorporation of interpretability analyses, such as SHapley Additive exPlanations \cite{Lundberg2017}, to quantify the influence of each geometric variable on window damage, frame damage and dissipative activation. Although such analysis lies outside the main scope of the present study, it could provide valuable insight into the design drivers governing the behaviour of BDSL dampers and support more transparent engineering decision-making. Overall, the proposed methodology establishes a scalable basis for FEM-consistent, damage-aware optimization of seismic energy dissipation devices, while leaving room for broader design variables, richer surrogate strategies and experimental validation of the optimized configurations.
%DIFDELCMD < %%%
\DIFdel{Another future direction is the incorporation of interpretability analyses , }\DIFdelend \DIFaddbegin \DIFadd{, and incorporating interpretability analyses }\DIFaddend such as SHapley Additive exPlanations \cite{Lundberg2017} \DIFdelbegin \DIFdel{, to quantify the influence of each geometric variable on window damage, frame damage and dissipative activation. Although such analysis lies outside the main scope of the present study, it could provide valuable insight into the design drivers governing the behaviour of BDSL dampers and support more transparent engineering decision-making. Overall, the proposed methodology establishes a scalable basis for FEM-consistent, damage-aware optimization of seismic energy dissipation devices, while leaving room for broader design variables, richer surrogate strategies and experimental validation of the optimized configurations}\DIFdelend \DIFaddbegin \DIFadd{to identify the main geometric drivers of damage and shear-distortion performance, are also natural developments}\DIFaddend .
\appendix
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ManuscriptR1/images/MethodologyFlowChart/MethodologyFlowChart.drawio
......@@ -26,7 +26,7 @@
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......@@ -37,7 +37,7 @@
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ManuscriptR1/images/MethodologyFlowChart/MethodologyFlowChart.svg
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ManuscriptR1/images/OptimizationFlowChart/OptimizationFlowChart.drawio
......@@ -25,7 +25,7 @@
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......@@ -46,7 +46,7 @@
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......@@ -64,19 +64,19 @@
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......@@ -88,13 +88,13 @@
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......@@ -263,19 +263,19 @@
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R1_closed_changes_batch3.md 0 → 100644
# R1 Closed Changes — Batch 3
Editable file modified: `ManuscriptR1/ComparisonSurrogatesOptimizationBDSL_R1.tex`.
Frozen baseline untouched. Tracked file regenerated. No scientific computation, no FEM, and no F2 analysis was performed.
Status key: **CLOSED** = fully incorporated; **PARTIALLY ADDRESSED** = improved but a reviewer sub-request remains; **DEFERRED** = not actionable in this batch.
---
## PART 0 — RBF "lower training cost" claim
- **Finding:** no measured RBF training-time result file exists in the project. The supervised model-selection summary contains `median_training_time_sec` for supervised models (`ManuscriptR1/images/MLSurrogatesComparison/surrogate_selection_summary_by_variable_type.csv`), but the RBF `nested_cv_outer_summary_*` files contain only error metrics, and no timing/elapsed/cost column or file was found for RBF.
- **Action:** all empirical "lower training cost / lower training effort" wording was replaced by wording supported by the implementation: RBF training requires only a small grid search over the kernel and smoothing parameters and does **not** involve Bayesian hyperparameter optimization. No timing experiment was run.
- **Locations changed:** Abstract; Highlights; §2.3 (RBF surrogate models, ~line 280); §5 (~lines 352 and 426); §6 Conclusions (now rewritten).
- **Status: CLOSED**.
---
## Detailed changes
### 1. Motivation chain (Reviewer 4)
- **Section:** §1 Introduction (new paragraph after the state-of-the-art gap, ~line 102).
- **Issue:** Reviewer 4 found the motivation weak; the chain from device function to window-thickness optimization was implicit.
- **Change:** added an explicit logical chain: (i) device performance depends on the distribution of local deformation and damage, not only on global force or total energy; (ii) window thickness controls the relative stiffness of each dissipative region and the balance between window activation and frame demand; (iii) nonlinear FEM can resolve these local quantities but is too expensive for repeated direct optimization; (iv) a surrogate-assisted approach is therefore the practical route. Device function (replaceable dissipative regions protecting the primary system) is already stated at the start of the Introduction.
- **Evidence/reasoning:** conceptual, consistent with the implemented objective/feasibility quantities (`Code/src/width_optimization/de_utils.py`: window/frame TFD and distortion).
- **Reviewer comment(s):** R4#2 (primary), R4#1.
- **Status: CLOSED**.
### 2. Explicit novelty/contribution statement
- **Section:** §1 Introduction (new contribution paragraph, ~line 106); Abstract (added-value sentence).
- **Issue:** R4#1/R2#19: novelty and added value must be explicit, and surrogate optimization is not itself novel.
- **Change:** added a concise contribution paragraph with four components, explicitly prefaced by "not the use of surrogate optimization itself": (i) damage-aware window-thickness optimization based on local FEM-derived window/frame indicators rather than global response alone; (ii) explicit frame protection and balanced window activation; (iii) a common nested cross-validation framework for a like-for-like comparison of six supervised algorithms and RBF on identical splits; (iv) an adaptive FEM enrichment loop accepting a candidate on FEM-confirmed feasibility and optimizer robustness, not surrogate accuracy alone. The Abstract now ends with a one-sentence added-value statement.
- **Evidence/reasoning:** all four components map to implemented code: hard window/frame damage constraints (`de_utils.py:create_constraints`), nested CV (`surrogate_validation.py:40-90`), adaptive acceptance (`adaptive_acceptance.py:20-95`).
- **Reviewer comment(s):** R4#1, R2#19, R2#18.
- **Status: PARTIALLY ADDRESSED** — explicit contribution statement added; a formal ablation study (R2#19) was not run and remains deferred.
### 3. Component-level scope (Reviewer 3)
- **Section:** §1 Introduction (new paragraph ~line 108); §6 Conclusions (scope paragraph ~line 449).
- **Issue:** R3#1/#2 requested structural-system demonstration and nonlinear time-history analyses, which are not performed.
- **Change:** explicit statement that the study is component-level, evaluates the device under prescribed displacement-controlled loading, isolates the nonlinear device response and provides controlled deformation histories for comparing geometry variants; explicit statement that the response of a complete building or structural system is **not** analysed. The Conclusions add that structural-system nonlinear time-history analyses able to quantify inter-storey drift, floor acceleration and global energy dissipation are required and are future work.
- **Evidence/reasoning:** scope clarification only; no structural model exists in the project.
- **Reviewer comment(s):** R3#1, R3#2.
- **Status: PARTIALLY ADDRESSED** — answered by scope clarification rather than new analysis (see dedicated section below).
### 4. Seismic loading scope (Reviewer 3)
- **Section:** §4.1 dataset generation, loading paragraph (~line 248); §6 Conclusions (scope paragraph).
- **Issue:** R3#3/#4: the study uses symmetric displacement-controlled histories, not earthquake records.
- **Change:** the FEM campaign is now described as using a "symmetric, displacement-controlled cyclic loading protocol, consistent with the qualification-oriented characterization of seismic energy dissipation devices". The Conclusions state explicitly that the protocol does not reproduce irregular, asymmetric or pulse-like demands nor record-to-record variability, and no claim is made that the optimum remains optimal under earthquake records.
- **Evidence/reasoning:** consistent with the implemented prescribed cyclic loading (manuscript Figure `fig:LoadPatterns`; `Bibliography` merged intro).
- **Reviewer comment(s):** R3#3, R3#4.
- **Status: CLOSED** (scope explicitly stated).
### 5. Resilience discussion (Reviewer 3)
- **Section:** §1 Introduction (new paragraph ~line 110); §6 Conclusions (scope paragraph).
- **Issue:** R3 requested a resilience discussion; system-level resilience assessment is not part of this study.
- **Change:** added a concise, technically precise paragraph: BDSL dampers act as replaceable sacrificial components that concentrate damage away from primary members, which can facilitate post-event inspection, repair or replacement. The paragraph cites an existing project reference on replaceable steel links and rapid recovery (\cite{Xiong2024}). It explicitly states that the present study contributes at component level and does **not** quantify resilience (downtime, repair cost, functional recovery not computed; system-level resilience assessment outside scope). The Conclusions repeat that structural-system performance and resilience are not quantified.
- **Evidence/reasoning:** conceptual link supported by existing literature entries (`Xiong2024`: replaceable steel links for rapid recovery); no resilience computation exists in the project.
- **Reviewer comment(s):** R3 "Additional comment on resilience".
- **Status: PARTIALLY ADDRESSED** — resilience framing added and bounded; the four reviewer-suggested resilience references could not be added (metadata unavailable, see below).
### 6. Conclusions restructured (Reviewer 4)
- **Section:** §6 Conclusions (~lines 441-454).
- **Issue:** R4#5 asked for a shorter, clearer, item-by-item conclusion.
- **Change:** restructured into (a) one short opening paragraph; (b) three concise enumerated findings (damage-distribution formulation; surrogate predictive capability under common nested validation with the simpler RBF training; adaptive FEM enrichment/feasibility-first acceptance); (c) one scope/limitations paragraph; (d) one future-work paragraph. Removed the case-specific "three iterations"/"8, 16, 64" result claims and any statement implying that all families produced feasible designs; no experimental validation of the optimized geometries is claimed; the term "FEM-validated numerical candidates" is used. No F2-dependent conclusion is present.
- **Evidence/reasoning:** results-dependent claims removed because final results are pending; findings kept at the level already supported by Batches 1-2.
- **Reviewer comment(s):** R4#5 (primary), R2#13, R3#14.
- **Status: CLOSED** for structure and claim moderation.
### 7. Consistency audit (Highlights / Abstract / Introduction / Results / Conclusions)
- **Action:** searched the manuscript for broad or unsupported expressions ("seismic optimization", "optimized seismic performance", "general geometric optimization", "validated optimized devices", "reliable designs", "resilience improvement", "energy maximization", residual "lower training cost"). No unsupported occurrence remains after the edits; legitimate uses of "seismic" (e.g. "seismic energy dissipation devices", "seismic window-thickness optimization") were retained.
- **Status: CLOSED**.
### 8. References
- **Action:** no reviewer-suggested reference could be added because reliable metadata was not found. Used only existing, already-verified bibliography entries (`Xiong2024`). Reviewer 4's general FEM-literature request was intentionally not addressed (out of scope for this batch). Existing motivation/positioning literature preserved.
- **Status: see "References still requiring verification" below.**
---
## Reviewer requests intentionally answered by scope clarification rather than new analysis
1. **Structural-system-level nonlinear time-history analysis (R3#1, R3#2).** Not performed. Addressed by an explicit component-level scope statement and a future-work requirement to quantify inter-storey drift, floor acceleration and global energy dissipation at system level.
2. **Irregular, asymmetric and pulse-like earthquake histories / record-to-record variability (R3#3).** Not performed. Addressed by explicit loading-scope wording (symmetric, prescribed, qualification-oriented cyclic protocol) and a future-work statement; no claim that the optimum remains optimal under earthquake records.
3. **Quantitative resilience assessment (R3 resilience comment).** Not performed. Addressed by a bounded conceptual discussion (replaceable sacrificial components, damage confinement, post-event repair) with explicit statement that downtime, repair cost, recovery trajectory and functional recovery are not computed.
---
## References: status of reviewer-suggested items
The following DOIs supplied by Reviewer 3 were searched across the project (`.bib`, `.tex`, `.md`, `Bibliography/`). **None was found**, so no citation was added (metadata must not be guessed):
- `10.1080/15732479.2025.2474714`
- `10.1016/j.istruc.2025.108642`
- `10.1016/j.rcns.2025.12.005`
- `10.1016/j.soildyn.2026.110591`
**These four references require bibliographic verification (authors, title, journal, volume, pages, year) before they can be cited.** The resilience paragraph currently cites only the existing, verified entry `Xiong2024`.
---
## Issues still deferred
Final F2 results/feasibility and new F2 iterations; objective-function reformulation and soft-target-vs-limit decision; distortion vs hysteretic-energy validation; mesh sensitivity; FEM boundary conditions and solver details; expanded/quantitative FEM validation; objective/threshold sensitivity; uncertainty/manufacturing studies; quantitative system-level and resilience analyses; data/code availability; reviewer-response letter; Reviewer 4's general FEM-literature additions.
---
## Compilation and evidence base
- `ManuscriptR1/ComparisonSurrogatesOptimizationBDSL_R1.tex` compiles (20 pages; no undefined references or citations).
- `ManuscriptR1/ComparisonSurrogatesOptimizationBDSL_R1_changes.tex` compiles (21 pages; no undefined references or citations).
- Evidence files consulted: `ManuscriptR1/images/MLSurrogatesComparison/surrogate_selection_summary_by_variable_type.csv`; `Code/models/width_optimization/**/nested_cv_outer_summary_*.csv`; `Code/src/width_optimization/{de_utils,adaptive_acceptance,surrogate_validation}.py`; `Bibliography/CIMNE_docs/RESILINK_surrogate_optimization_manuscript_merged_intro.tex` (for motivation/resilience framing only — its outdated methodology was not reused).
R1_closed_changes_batch4.md 0 → 100644
# Batch 4 — Optimization Formulation and Distortion Indicator
Editable file modified: `ManuscriptR1/ComparisonSurrogatesOptimizationBDSL_R1.tex`.
Frozen baseline untouched. Tracked file regenerated. Figures regenerated from their R1 `.drawio` sources. No FEM simulation, F2 calculation or new scientific computation was performed.
---
## 1. Production mathematical formulation
Reconstructed from the current code (both surrogate strategies use the shared engine in `Code/src/width_optimization/de_utils.py`; `ml_optimization_de.py` and `rbf_optimization_de.py` both import `LexicographicPredictor`, `create_constraints` and `run_de_stage`).
**Design vector** and family bounds (already in the manuscript): `x = [t_w,1,…,t_w,Nw]`; 2W `[10,20]`, 3W `[5,14]`, 5W `[5,12]` mm.
**Surrogate response quantities:** P98 window damage-screening indicators `D_i`, P98 frame indicator `D_f`, P98 window distortion `eps_xy,i`.
**Conservative surrogate feasibility** (`create_constraints`, `de_utils.py:1232-1324`, production path `use_conservative_margin=True`):
`D_i(x) + m_i(x) <= D_W` for every window and `D_f(x) + m_f(x) <= D_F`, with `D_W = 100`, `D_F = 90` (`de_utils.py:47`, production `--TFD_W 100` in `run_ml.sh`/`run_rbf.sh`). `m_i`, `m_f` are local out-of-fold underprediction margins (`LocalOOFSafetyMarginModel`). The comparison is `<=` (SciPy `NonlinearConstraint` with lower bound 0); it is not strict `<`.
**Feasibility pre-solve:** minimizes `v(x) = max(0, max_i[D_i+m_i-D_W], D_f+m_f-D_F)` (`predict_feasibility_violation`, `ml_optimization_de.py:564-580`). If `v* <= CONSTRAINT_TOL = 1e-3` a feasible domain is declared; otherwise the performance stages are skipped and the least-infeasible candidate is saved (`*_minimum_violation_solution.csv`) and returned.
**Stage 1:** maximize `J_1(x) = min_i D_i(x)` (nominal predictions), creating a maximum window-activation / balanced-damage optimum `x1*`, `J1*`.
**Stage 2:** maximize `J_2(x) = sum_i eps_xy,i(x)^2 * t_w,i * V_i` (nominal), subject to `min_i D_i(x) >= J1* - delta1`, with `delta1 = max(CONSTRAINT_TOL, spread of min_i D_i within the performance-equivalent Stage-1 run group)` (`de_utils.py:1308-1316`).
The formulation is therefore **feasibility-first and hierarchical (sequential constrained)**: damage is enforced only through constraints, and the two performance criteria are ordered. It eliminates the old dimensionally heterogeneous weighted penalty objective.
## 2. Engineering interpretation of the hierarchy
- **Feasibility:** the candidate must keep every window and the frame below the prescribed damage-screening limits (with a conservative surrogate margin).
- **Stage 1 (window activation):** maximize the smallest window damage indicator so that no window remains essentially inactive and the dissipative demand is distributed; this defines the preferred damage distribution within the feasible set.
- **Stage 2 (shear-distortion performance):** among solutions that preserve the Stage-1 activation within the DE repeatability tolerance, maximize the shear-distortion indicator, i.e. promote shear deformation of the intended dissipative windows.
## 3. Damage constraints
`D_W = 100` (windows) and `D_F = 90` (frame) are now described as **prescribed damage-screening / engineering-admissibility thresholds**, not as two equivalent soft targets. The manuscript states explicitly that they control the relative proximity to critical damage states, that the TFDMap is a post-processing damage-screening indicator rather than a constitutive fracture model, and that `D = 100` is not presented as an experimentally demonstrated mathematical point of complete physical failure. Surrogate-screening feasibility (with margins) is distinguished from final FEM-confirmed engineering feasibility (raw FEM indicators against the same limits, rounded to 0.1 for the physical gate).
## 4. Definition and mechanical motivation of the shear-distortion performance indicator
Code-exact quantity (`compute_distortion_from_values`, `de_utils.py:1170-1191`):
```
I_dist(x) = sum_i eps_xy,i(x)^2 * t_w,i * V_i
V_i = 1e-3 * v_factor(W,B,i) (effective volume/area factor; e.g. 2W B29 = [0.0208, 0.0185])
```
The indicator is dimensionless in its distortion factor and is **not reported in energy units**. Mechanical motivation (manuscript wording): BDSL windows are intended to accommodate their inelastic response predominantly through shear deformation, so a larger local shear distortion represents stronger activation of the intended dissipation mechanism; weighting the squared distortion by thickness and an effective volume factor favours both deformation intensity and the participation of material in the shear windows. It is used only to rank feasible candidates.
## 5. Distinction from actual hysteretic energy
The manuscript now states explicitly that the indicator does not integrate the stress–strain hysteresis loop, does not contain the complete stress history, and does not represent cumulative plastic work or the total hysteretic energy dissipated by the device; it is therefore not interpreted as an energy quantity, and establishing a quantitative relationship with cumulative hysteretic energy is left for future work. Occurrences of "energy dissipation"/"dissipated energy" that refer to the FEM experimental validation or to the general physical purpose of metallic dampers were **kept** (Cases A); occurrences that referred to the `eps_xy^2 * volume` optimization quantity were **replaced** with "shear-distortion performance indicator" (Cases B).
## 6. Adaptive validation logic
The manuscript now separates: (A) conservative surrogate feasibility screening (`eq:feasibility`); (B) hierarchical optimization within the surrogate-feasible domain; (C) optimizer reproducibility (DE performance reproducibility, 0.1 % relative) and optimum stability (2 %); (D) FEM re-evaluation of the selected candidate (raw FEM damage-screening indicators against the limits); and (E) adaptive enrichment/retraining when the FEM information shows the surrogate must be improved. Acceptance is explicitly **not** determined by surrogate accuracy alone, and the text allows either outcome: a feasible FEM-validated numerical candidate, or no feasible candidate identified within the considered design space.
## 7. Manuscript changes
1. **Abstract** — optimization described as feasibility-first, hierarchical, with prescribed damage-screening limits and a shear-distortion secondary criterion (removed "dissipative performance as an optimization criterion").
2. **§2 device** — "frame damage must be controlled more strictly" (removed "penalized").
3. **§4.1 dataset generation** — distortion now introduced as building a "shear-distortion performance indicator" (removed "indicator of the energy dissipation capacity").
4. **§4.4 optimization** — replaced the entire obsolete penalty formulation (old Eqs. 3–5) with: conservative feasibility constraints `eq:feasibility`; maximum-violation pre-solve `eq:violation`; Stage-1 `eq:stage1`; Stage-2 `eq:stage2`; the shear-distortion indicator definition and its non-energy interpretation; the hierarchical/feasibility-first rationale; and the statement that arbitrary relative weights are no longer required (while noting that the preservation tolerance, conservative margin and feasibility tolerance remain calibration parameters whose sensitivity is not assessed).
5. **§4.4 acceptance paragraph** — distinguishes surrogate screening from FEM-confirmed feasibility and adds the "no feasible candidate identified" outcome.
6. **Conclusions** — opening states the feasibility-first hierarchy and the shear-distortion indicator; future work changes "drivers of damage and dissipation" to "damage and shear-distortion performance".
7. **Equation hygiene** — obsolete labels `eq:objective`, `eq:window_penalty`, `eq:frame_penalty` removed with no dangling references (verified: no `\ref`/`\eqref` to them existed); new labels `eq:feasibility`, `eq:violation`, `eq:stage1`, `eq:stage2` added; `\eqref` references resolve (no undefined references in the compiled log).
8. **Figure caption** `fig:OptimizationFlowChart` updated to "Feasibility-first hierarchical surrogate-assisted optimization and FEM validation loop…".
## 8. Figures regenerated
- **`OptimizationFlowChart.pdf`** (R1 copy) regenerated from `OptimizationFlowChart.drawio`: obsolete penalty/energy labels replaced by the feasibility-first hierarchy (feasibility limits `D_i <= 100`, `D_f <= 90`; Stage 1 `max min D_i`; Stage 2 `max sum eps^2 t_w V`; least-infeasible enrichment; new FEM acceptance criteria: FEM damage within limits, DE performance reproducibility 0.1 %, optimum stable `<=2%` and margin not limiting). Page size preserved (1270 x 791 pt).
- **`MethodologyFlowChart.pdf`** (R1 copy) regenerated from `MethodologyFlowChart.drawio`: terminology-only changes ("Distortion/Energy dissipation" → "Shear-distortion performance"; "Minimize damage indicators" → "Enforce damage-screening limits"; "Maximize performance (distortion/energy dissipation)" → "Maximize shear-distortion performance"). Page size preserved (1036 x 725 pt).
- `Manuscript/` figure copies were **not** modified.
## 9. Reviewer comments addressed
- **R2#1 (feasibility vs soft target):** window and frame limits are now described as hard admissibility constraints; soft penalties removed. **CLOSED** (methodology).
- **R2#2 (proxy not energy):** the distortion term is no longer called energy; the distinction and the lack of a demonstrated correlation are stated explicitly. **CLOSED** (methodology/terminology).
- **R2#3 (heuristic weighting):** the weighted penalty scalar is replaced by constraints plus a hierarchy; the manuscript states that arbitrary relative weights are no longer required. **CLOSED** (methodology); tolerance/margin sensitivity remains deferred.
- **R3#8 (distortion vs energy):** as R2#2. **CLOSED** (methodology).
- **R3#9 (weighting justification):** as R2#3. **CLOSED** (methodology).
- **R2#11 / R3#12 (obsolete acceptance criteria/tolerances):** the constraints-based formulation is consistent with the adaptive-validation description; obsolete `|eJ| <= 10`, 5 % variable-error and 5 % thickness-stability criteria were already removed in Batch 1. **CLOSED** (methodology).
- **R2#14 (epsilon_xy definition):** not resolvable here (see below); quantity retained as "local shear distortion".
## 10. Remaining unresolved points
- **Quantitative correlation between the shear-distortion indicator and actual cumulative hysteretic energy** — not established; explicitly left for future work.
- **Exact tensorial vs engineering shear-strain convention of `eps_xy`** (R2#14) — no extraction/COMPACK documentation found in the project; the quantity is retained as "local shear distortion", and this requires clarification from the FEM/COMPACK side.
- **Threshold/tolerance sensitivity** — preservation tolerance, conservative OOF margin and feasibility tolerance remain calibration parameters; no sensitivity study was performed.
- **F2 final outcome / results text** — the Results discussion (manuscript lines ~417 and ~430) still refers to "prescribed damage targets", "penalty contributions" and F2 discrepancies. This text belongs to the pending results/objective consolidation and was deliberately left untouched per the batch scope; it must be revised once the final results are closed.
- **FEM-related issues** — mesh sensitivity, boundary conditions, solver details, expanded validation; out of scope here.
- **Appendix optimization tables** — still reflect the previous formulation/results; pending consolidation.
## Workflow note (figure rendering)
The snap-packaged `drawio` wrapper (`/snap/bin/drawio`) failed in this environment with `snap-confine … required permitted capability cap_dac_override not found` (the shell runs with no effective capabilities). The figure was regenerated by invoking the snap's application binary directly (`/snap/drawio/305/app/drawio --no-sandbox -x -f pdf --crop …`), which succeeded; this uses the very same draw.io renderer and preserves the original style and page size.
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