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2026_Article_RESILINK_ML
Commit bbc3bf0b by Joaquín Irazábal González
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Refactor manuscript title and enhance clarity in results discussion

- Updated manuscript title to reflect a focus on comparison of surrogate strategies for damage-aware optimization. - Improved clarity in the results section by refining descriptions of surrogate model performance and validation metrics. - Enhanced the discussion on the adaptive validation loop and its impact on optimization outcomes. - Clarified the interpretation of objective-function values and the implications of damage control in the optimization framework. - Added figures to illustrate the evolution of optimized window thicknesses and RBF objective surfaces during the optimization process.
parent 1ac5a43f
Showing with 3281 additions and 32 deletions
.gitignore
...@@ -22,6 +22,9 @@ Code/**/*.png ...@@ -22,6 +22,9 @@ Code/**/*.png
Code/**/*.html Code/**/*.html
Code/**/*.svg Code/**/*.svg
# Local browser used by Plotly/Kaleido static image export
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# GiD results # GiD results
*.post.lst *.post.lst
*.post.bin *.post.bin
Manuscript/Figures/OptimizedWindowThicknessEvolution/optimized_window_thickness_evolution.csv 0 → 100644
family,height_cm,width_cm,surrogate,iteration,source_model,window,window_index,thickness_mm,objective_surrogate
H30_B29,30,29,RBF,1,opt1,tw1,1,12.34,0.0
H30_B29,30,29,RBF,1,opt1,tw2,2,14.34,0.0
H30_B29,30,29,RBF,2,opt2,tw1,1,12.68,0.0
H30_B29,30,29,RBF,2,opt2,tw2,2,14.91,0.0
H30_B29,30,29,RBF,3,opt3,tw1,1,12.56,0.0
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H30_B29,30,29,Supervised ML,1,opt1,tw2,2,14.64,0.0
H30_B29,30,29,Supervised ML,2,opt2,tw1,1,12.53,0.0
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H30_B34,30,34,RBF,1,opt1,tw2,2,20.0,507.37762
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H30_B34,30,34,RBF,3,opt3,tw1,1,14.77,688.53041
H30_B34,30,34,RBF,3,opt3,tw2,2,18.95,688.53041
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H30_B34,30,34,Supervised ML,1,opt1,tw2,2,20.0,541.94744
H30_B34,30,34,Supervised ML,2,opt2,tw1,1,15.14,830.19116
H30_B34,30,34,Supervised ML,2,opt2,tw2,2,20.0,830.19116
H30_B34,30,34,Supervised ML,3,opt3,tw1,1,15.2,796.29365
H30_B34,30,34,Supervised ML,3,opt3,tw2,2,20.0,796.29365
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H60_B34,60,34,Supervised ML,2,opt2,tw5,5,5.0,16.96082
H60_B34,60,34,Supervised ML,3,opt3,tw1,1,5.74,16.90444
H60_B34,60,34,Supervised ML,3,opt3,tw2,2,7.46,16.90444
H60_B34,60,34,Supervised ML,3,opt3,tw3,3,8.5,16.90444
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H60_B34,60,34,Supervised ML,3,opt3,tw5,5,5.0,16.90444
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Manuscript/Figures/OptimizedWindowThicknessEvolution/plot_optimal_thickness_evolution.py 0 → 100644
#!/usr/bin/env python3
"""Plot optimized window thickness evolution in a single combined figure.
Input CSV expected columns:
family, surrogate, iteration, window, window_index, thickness_mm
The plot groups the x-axis by geometry family and shows adaptive iterations
inside each family block. Window color identifies each thickness variable, and
line style identifies the surrogate strategy:
solid = RBF, dashed = Supervised ML.
"""
from __future__ import annotations
import csv
from pathlib import Path
import matplotlib.pyplot as plt
from matplotlib.lines import Line2D
plt.rcParams.update(
{
"font.family": "serif",
"font.serif": ["Times New Roman", "DejaVu Serif"],
"mathtext.fontset": "stix",
"font.size": 12,
"axes.labelsize": 12,
"axes.titlesize": 11,
"xtick.labelsize": 12,
"ytick.labelsize": 12,
"axes.linewidth": 0.7,
"axes.spines.top": False,
"axes.spines.right": False,
"figure.dpi": 300,
"savefig.dpi": 300,
}
)
BASE_DIR = Path(__file__).resolve().parent
INPUT_CSV = BASE_DIR / "optimized_window_thickness_evolution.csv"
FIG_PATH = BASE_DIR / "optimized_window_thickness_evolution"
SURROGATE_LINESTYLES = {
"RBF": "-",
"Supervised ML": "--",
}
SURROGATE_ORDER = ["RBF", "Supervised ML"]
WINDOW_COLORS = {
"tw1": "#1a6faf",
"tw2": "#E07060",
"tw3": "#4D9A57",
"tw4": "#8B6BBE",
"tw5": "#D49A2A",
}
COLOR_GRID = "#d8d8d8"
ITERATION_STEP = 0.78
FAMILY_GAP = 1.05
def load_rows(path: Path):
with path.open(newline="", encoding="utf-8") as f:
reader = csv.DictReader(f)
rows = []
for row in reader:
row["iteration"] = int(row["iteration"])
row["window_index"] = int(row["window_index"])
row["thickness_mm"] = float(row["thickness_mm"])
rows.append(row)
return rows
def style_axis(ax, *, include_x_grid: bool = False) -> None:
ax.grid(
True,
axis="both" if include_x_grid else "y",
color=COLOR_GRID,
linewidth=0.55,
alpha=0.75,
)
ax.tick_params(axis="both", which="major", direction="in", length=4, width=0.7)
ax.tick_params(axis="both", which="minor", direction="in", length=2.5, width=0.6)
ax.tick_params(axis="x", which="both", length=0)
ax.minorticks_on()
def save_figure_all_formats(fig, out_base: Path) -> None:
fig.savefig(f"{out_base}.png", format="png", bbox_inches="tight", dpi=300)
fig.savefig(f"{out_base}.svg", format="svg", bbox_inches="tight")
fig.savefig(f"{out_base}.pdf", format="pdf", bbox_inches="tight")
def family_sort_key(family: str) -> tuple[int, int, str]:
height_text, width_text = family.split("_")
return int(height_text.replace("H", "")), int(width_text.replace("B", "")), family
def window_sort_key(window: str) -> int:
return int(window.replace("tw", ""))
def plot_all_families(rows: list[dict], out_base: Path):
rows = sorted(
rows,
key=lambda r: (
family_sort_key(r["family"]),
r["window_index"],
SURROGATE_ORDER.index(r["surrogate"])
if r["surrogate"] in SURROGATE_ORDER
else len(SURROGATE_ORDER),
r["iteration"],
),
)
families = sorted({r["family"] for r in rows}, key=family_sort_key)
windows = sorted({r["window"] for r in rows}, key=window_sort_key)
surrogates = [s for s in SURROGATE_ORDER if s in {r["surrogate"] for r in rows}]
fig, ax = plt.subplots(figsize=(10.2, 4.85))
fig.subplots_adjust(left=0.075, right=0.985, bottom=0.3, top=0.95)
family_centers = {}
tick_positions = []
tick_labels = []
family_start_positions = []
x_by_family_iteration = {}
current_x = 0.0
for family in families:
iterations = sorted({r["iteration"] for r in rows if r["family"] == family})
family_start_positions.append(current_x)
family_positions = []
for offset, iteration in enumerate(iterations):
x = current_x + offset * ITERATION_STEP
x_by_family_iteration[(family, iteration)] = x
family_positions.append(x)
tick_positions.append(x)
tick_labels.append(f"it{iteration}")
family_centers[family] = (family_positions[0] + family_positions[-1]) / 2
current_x = family_positions[-1] + FAMILY_GAP
for family_start in family_start_positions[1:]:
ax.axvline(
family_start - FAMILY_GAP / 2,
color=COLOR_GRID,
linewidth=0.7,
zorder=0,
)
for family in families:
family_rows = [r for r in rows if r["family"] == family]
for window in sorted({r["window"] for r in family_rows}, key=window_sort_key):
for surrogate in surrogates:
subset = [
r
for r in family_rows
if r["window"] == window and r["surrogate"] == surrogate
]
if not subset:
continue
subset = sorted(subset, key=lambda r: r["iteration"])
ax.plot(
[x_by_family_iteration[(family, r["iteration"])] for r in subset],
[r["thickness_mm"] for r in subset],
marker="o",
color=WINDOW_COLORS.get(window, "#555555"),
markerfacecolor="white",
markeredgewidth=0.8,
linewidth=1.5,
markersize=4.2,
linestyle=SURROGATE_LINESTYLES.get(surrogate, "-"),
zorder=3,
)
ax.set_ylabel("Window thickness [mm]")
ax.set_xticks(tick_positions)
ax.set_xticklabels(tick_labels)
ax.tick_params(axis="x", labelsize=10, pad=3)
ax.set_xlim(min(tick_positions) - 0.35, max(tick_positions) + 0.35)
style_axis(ax)
y_values = [r["thickness_mm"] for r in rows]
y_margin = (max(y_values) - min(y_values)) * 0.06
ax.set_ylim(max(0, min(y_values) - y_margin), max(y_values) + y_margin)
for family, center in family_centers.items():
ax.text(
center,
-0.115,
family,
transform=ax.get_xaxis_transform(),
ha="center",
va="top",
fontsize=12,
clip_on=False,
)
window_handles = [
Line2D(
[0],
[0],
color=WINDOW_COLORS.get(window, "#555555"),
marker="o",
markerfacecolor="white",
markeredgewidth=0.8,
linewidth=1.5,
markersize=4.2,
label=window,
)
for window in windows
]
surrogate_handles = [
Line2D(
[0],
[0],
color="#444444",
linestyle=SURROGATE_LINESTYLES.get(surrogate, "-"),
linewidth=1.5,
label=surrogate,
)
for surrogate in surrogates
]
fig.legend(
window_handles + surrogate_handles,
[h.get_label() for h in window_handles + surrogate_handles],
loc="lower center",
ncol=len(window_handles) + len(surrogate_handles),
frameon=False,
bbox_to_anchor=(0.5, 0.035),
columnspacing=1.4,
handlelength=2.0,
)
save_figure_all_formats(fig, out_base)
plt.close(fig)
return out_base
def main():
rows = load_rows(INPUT_CSV)
created = plot_all_families(rows, FIG_PATH)
print(f"Saved: {created}.png/.svg/.pdf")
if __name__ == "__main__":
main()
Manuscript/Figures/RBFOptimizationSurfaceEvolution/plot_rbf_surface_evolution.py 0 → 100644
#!/usr/bin/env python3
"""Plot 3D RBF objective-surface evolution with one color per iteration.
The source Plotly HTML files contain the objective surfaces as numerical
meshes. This script reads those meshes and overlays the 3D surfaces. Color
identifies the adaptive iteration, not the objective value.
"""
from __future__ import annotations
import base64
import csv
import json
import os
import shutil
from dataclasses import dataclass
from pathlib import Path
import numpy as np
import plotly.graph_objects as go
BASE_DIR = Path(__file__).resolve().parent
PROJECT_ROOT = BASE_DIR.parents[2]
SOURCE_DIR = (
PROJECT_ROOT / "Code" / "reports" / "width_optimization" / "2W" / "rbf_optimization"
)
FIG_PATH = BASE_DIR / "rbf_surface_tw1_tw2_B34_H30_TFD_W90_evolution_3d"
ITERATIONS = [0, 1, 2]
ITERATION_COLORS = {
0: "#1a6faf",
1: "#E07060",
2: "#4D9A57",
}
ITERATION_CONTOUR_COLORS = {
0: "#0d4f83",
1: "#9f4036",
2: "#2f6b38",
}
COLOR_GRID = "#d8d8d8"
COLOR_AXIS = "#222222"
OUTPUT_WIDTH = 900
OUTPUT_HEIGHT = 560
Z_AXIS_RANGE = [-1, 50000]
PREFERRED_BROWSER_BINARIES = [
str(PROJECT_ROOT / ".plotly_chrome" / "chrome-linux64" / "chrome"),
"/usr/bin/google-chrome-stable",
"/usr/bin/google-chrome",
"/usr/bin/chrome",
"/usr/bin/chromium",
"/usr/bin/chromium-browser",
]
@dataclass
class SurfaceData:
iteration: int
x: np.ndarray
y: np.ndarray
z: np.ndarray
optimum: tuple[float, float, float]
def save_figure_all_formats(fig: go.Figure, out_base: Path) -> None:
fig.write_html(f"{out_base}.html", include_plotlyjs="cdn")
fig.write_image(
f"{out_base}.png",
format="png",
width=OUTPUT_WIDTH,
height=OUTPUT_HEIGHT,
scale=2,
)
fig.write_image(
f"{out_base}.svg",
format="svg",
width=OUTPUT_WIDTH,
height=OUTPUT_HEIGHT,
)
fig.write_image(
f"{out_base}.pdf",
format="pdf",
width=OUTPUT_WIDTH,
height=OUTPUT_HEIGHT,
)
def configure_kaleido_browser() -> None:
"""Prefer a non-snap Chrome/Chromium for Plotly static image export."""
if os.environ.get("BROWSER_PATH"):
return
for browser in PREFERRED_BROWSER_BINARIES:
if Path(browser).exists():
os.environ["BROWSER_PATH"] = browser
return
for browser_name in ("google-chrome-stable", "google-chrome", "chrome"):
browser = shutil.which(browser_name)
if browser:
os.environ["BROWSER_PATH"] = browser
return
def find_argument_span(text: str, start: int) -> tuple[int, int]:
depth = 0
quote: str | None = None
escaped = False
for idx in range(start, len(text)):
char = text[idx]
if quote is not None:
if escaped:
escaped = False
elif char == "\\":
escaped = True
elif char == quote:
quote = None
continue
if char in ('"', "'"):
quote = char
elif char in "[{(":
depth += 1
elif char in "]})":
depth -= 1
elif char == "," and depth == 0:
return start, idx
raise ValueError("Could not find the end of the Plotly argument.")
def extract_plotly_data(html_path: Path) -> list[dict]:
text = html_path.read_text(encoding="utf-8")
call_start = text.rfind("Plotly.newPlot(")
if call_start < 0:
raise ValueError(f"Plotly.newPlot call not found in {html_path}")
cursor = call_start + len("Plotly.newPlot(")
while text[cursor].isspace():
cursor += 1
_, first_arg_end = find_argument_span(text, cursor)
cursor = first_arg_end + 1
while text[cursor].isspace():
cursor += 1
data_start, data_end = find_argument_span(text, cursor)
return json.loads(text[data_start:data_end])
def decode_plotly_array(value) -> np.ndarray:
if isinstance(value, list):
return np.asarray(value, dtype=float)
if not isinstance(value, dict) or "bdata" not in value:
raise TypeError(f"Unsupported Plotly array encoding: {type(value)!r}")
dtype = np.dtype(value.get("dtype", "f8"))
data = np.frombuffer(base64.b64decode(value["bdata"]), dtype=dtype)
shape = tuple(int(part.strip()) for part in value["shape"].split(","))
return data.reshape(shape)
def load_optimum_from_csv(csv_path: Path) -> tuple[float, float, float]:
values = {}
with csv_path.open(newline="", encoding="utf-8") as f:
for row in csv.DictReader(f):
values[row["Parameter"]] = float(row["Value"])
return (
values["tw1_optimal"],
values["tw2_optimal"],
values["Objective_score"],
)
def load_surface(iteration: int) -> SurfaceData:
it_dir = SOURCE_DIR / f"it{iteration}"
html_path = it_dir / "surface_tw1_tw2_B34_H30_TFD_W90.html"
csv_path = it_dir / "optimization_results_2W_B34_H30_TFD_W90.csv"
traces = extract_plotly_data(html_path)
surface = next(trace for trace in traces if trace.get("type") == "surface")
return SurfaceData(
iteration=iteration,
x=decode_plotly_array(surface["x"]),
y=decode_plotly_array(surface["y"]),
z=decode_plotly_array(surface["z"]),
optimum=load_optimum_from_csv(csv_path),
)
def solid_colorscale(color: str) -> list[list[object]]:
return [[0.0, color], [1.0, color]]
def axis_style(title: str) -> dict:
return {
"title": {
"text": title,
"font": {"family": "Times New Roman, DejaVu Serif", "size": 17},
},
"showbackground": True,
"backgroundcolor": "white",
"gridcolor": COLOR_GRID,
"gridwidth": 1,
"linecolor": COLOR_AXIS,
"linewidth": 1,
"mirror": True,
"ticks": "inside",
"tickfont": {"family": "Times New Roman, DejaVu Serif", "size": 15},
"zeroline": False,
}
def z_axis_style() -> dict:
style = axis_style("Objective score")
style.update({"tickvals": [10000, 20000, 30000, 40000, 50000]})
return style
def main() -> None:
configure_kaleido_browser()
surfaces = [load_surface(iteration) for iteration in ITERATIONS]
fig = go.Figure()
for surface in surfaces:
color = ITERATION_COLORS[surface.iteration]
contour_color = ITERATION_CONTOUR_COLORS[surface.iteration]
fig.add_trace(
go.Surface(
x=surface.x,
y=surface.y,
z=surface.z,
surfacecolor=np.zeros_like(surface.z),
colorscale=solid_colorscale(color),
showscale=False,
opacity=0.44,
name=f"it{surface.iteration}",
showlegend=True,
legendgroup=f"it{surface.iteration}",
contours={
"z": {
"show": True,
"usecolormap": False,
"color": contour_color,
"width": 3,
"start": 0,
"end": Z_AXIS_RANGE[1],
"size": 500,
}
},
hovertemplate=(
f"it{surface.iteration}<br>"
+ "tw1=%{x:.2f} mm<br>"
+ "tw2=%{y:.2f} mm<br>"
+ "Objective=%{z:.3e}<extra></extra>"
),
)
)
for surface in surfaces:
color = ITERATION_COLORS[surface.iteration]
fig.add_trace(
go.Scatter3d(
x=[surface.optimum[0]],
y=[surface.optimum[1]],
z=[surface.optimum[2]],
mode="markers",
marker={
"size": 5.5,
"color": color,
"line": {"color": "white", "width": 1.0},
},
name=f"it{surface.iteration} optimum",
legendgroup=f"it{surface.iteration}",
showlegend=False,
hovertemplate=(
f"it{surface.iteration} optimum<br>"
+ "tw1=%{x:.2f} mm<br>"
+ "tw2=%{y:.2f} mm<br>"
+ "Objective=%{z:.3e}<extra></extra>"
),
)
)
fig.add_trace(
go.Scatter3d(
x=[surface.optimum[0] for surface in surfaces],
y=[surface.optimum[1] for surface in surfaces],
z=[surface.optimum[2] for surface in surfaces],
mode="lines",
line={"color": "#444444", "width": 4, "dash": "dot"},
name="Optimum path",
hoverinfo="skip",
)
)
fig.update_layout(
width=OUTPUT_WIDTH,
height=OUTPUT_HEIGHT,
margin={"l": 0, "r": 0, "b": 0, "t": 0},
paper_bgcolor="white",
plot_bgcolor="white",
font={
"family": "Times New Roman, DejaVu Serif",
"size": 16,
"color": COLOR_AXIS,
},
legend={
"orientation": "h",
"x": 0.5,
"xanchor": "center",
"y": 0.0,
"yanchor": "bottom",
"font": {"family": "Times New Roman, DejaVu Serif", "size": 18},
"bgcolor": "rgba(255,255,255,0)",
},
scene={
"xaxis": {**axis_style("t<sub>w,1</sub> [mm]"), "range": [9.6, 20.4]},
"yaxis": {**axis_style("t<sub>w,2</sub> [mm]"), "range": [9.6, 20.4]},
"zaxis": {**z_axis_style(), "range": Z_AXIS_RANGE},
"camera": {
"eye": {"x": 1.28, "y": 1.23, "z": 0.7},
"center": {"x": 0, "y": 0, "z": -0.08},
},
"aspectmode": "manual",
"aspectratio": {"x": 1.0, "y": 1.0, "z": 0.68},
"domain": {"x": [0.0, 1.0], "y": [0.16, 1.0]},
},
)
save_figure_all_formats(fig, FIG_PATH)
print(f"Saved: {FIG_PATH}.html/.png/.svg/.pdf")
if __name__ == "__main__":
main()
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Manuscript/RESILINK_surrogate_optimization_manuscript.tex
...@@ -25,7 +25,7 @@ ...@@ -25,7 +25,7 @@
\begin{document} \begin{document}
\title{Damage-aware surrogate optimization of buckling-delayed shear-link dampers with adaptive finite element validation} \title{Comparison of surrogate strategies for damage-aware optimization of buckling-delayed shear-link dampers with adaptive finite element validation}
\author[1]{J. Irazábal} \author[1]{J. Irazábal}
\author[1,2]{J. Ramírez} \author[1,2]{J. Ramírez}
...@@ -36,7 +36,7 @@ ...@@ -36,7 +36,7 @@
\author[4]{L. Bozzo} \author[4]{L. Bozzo}
\authormark{IRAZÁBAL \textsc{et al.}} \authormark{IRAZÁBAL \textsc{et al.}}
\titlemark{DAMAGE-AWARE SURROGATE OPTIMIZATION OF SHEAR-LINK DAMPERS} \titlemark{COMPARISON OF SURROGATE STRATEGIES FOR DAMAGE-AWARE OPTIMIZATION OF SHEAR-LINK DAMPERS}
\address[1]{\orgname{Centre Internacional de Metodes Numerics en Enginyeria (CIMNE)}, \orgaddress{\city{Barcelona}, \country{Spain}}} \address[1]{\orgname{Centre Internacional de Metodes Numerics en Enginyeria (CIMNE)}, \orgaddress{\city{Barcelona}, \country{Spain}}}
\address[2]{\orgname{Universitat Politecnica de Catalunya (UPC)}, \orgaddress{\city{Barcelona}, \country{Spain}}} \address[2]{\orgname{Universitat Politecnica de Catalunya (UPC)}, \orgaddress{\city{Barcelona}, \country{Spain}}}
...@@ -315,7 +315,7 @@ If all criteria are satisfied, the FEM-validated geometry is accepted as the opt ...@@ -315,7 +315,7 @@ If all criteria are satisfied, the FEM-validated geometry is accepted as the opt
\section{Numerical results and discussion}\label{sec:results} \section{Numerical results and discussion}\label{sec:results}
The supervised-learning comparison shows a clear hierarchy among the candidate surrogate models. A total of 100 output-specific training problems were considered, corresponding to the $2W+1$ target variables required for each geometry family and adaptive iteration. The H30\_B29 family required two optimization iterations, whereas the remaining families required three. Across all geometry families, iterations and output variables, SVR was the most frequently selected model and also the model that most often achieved the lowest cross-validated RMSE. As summarized in Figure~\ref{fig:surrogate_selection_summary_barplot}, SVR was selected in 71 cases, followed by GPR, GBR, XGBoost and MLP, while Random Forest was not selected in any case. This dominance was particularly clear for the two-window families, where only one output was assigned to a different model, namely GPR, and for the damage-related outputs, for which SVR was selected in 47 out of 57 cases. For more complex devices, those with three and five windows, and for distortion-related outputs, the model selection became more heterogeneous, although SVR still provided the best overall performance. From the computational point of view, SVR also offered one of the lowest median training times among the supervised models, whereas MLP required substantially longer training times without providing gain in accuracy. The supervised-learning comparison shows a clear hierarchy among the candidate surrogate models. A total of 100 output-specific training problems were considered, corresponding to the $2W+1$ target variables required for each geometry family and adaptive iteration. The H30\_B29 family required two optimization iterations, whereas the remaining families required three. Across all geometry families, iterations and output variables, SVR was the most frequently selected model and also the model that most often achieved the lowest cross-validated RMSE. As summarized in Figure~\ref{fig:surrogate_selection_summary_barplot}, SVR was selected in 71 cases, followed by GPR, GBR, XGBoost and MLP, while Random Forest was not selected in any case. This dominance was particularly clear for the two-window families, where only one output was assigned to a different model, namely GPR, and for the damage-related outputs, for which SVR was selected in 47 out of 57 cases. For more complex devices, those with three and five windows, and for distortion-related outputs, the model selection became more heterogeneous, although SVR still provided the best overall performance. From the computational point of view, SVR also offered the lowest median training times among the supervised models, whereas MLP required substantially longer training times without providing gain in accuracy.
\begin{figure}[htbp] \begin{figure}[htbp]
\centering \centering
...@@ -324,9 +324,9 @@ The supervised-learning comparison shows a clear hierarchy among the candidate s ...@@ -324,9 +324,9 @@ The supervised-learning comparison shows a clear hierarchy among the candidate s
\label{fig:surrogate_selection_summary_barplot} \label{fig:surrogate_selection_summary_barplot}
\end{figure} \end{figure}
These results indicate that kernel-based models are particularly well suited to the present surrogate task. SVR provides the best compromise between accuracy and computational cost, while GPR is the second most competitive supervised strategy, especially in some higher-dimensional cases. Tree-based models, although robust, are less frequently selected, and MLP models are not competitive in terms of computational efficiency for the dataset sizes considered here. As mentioned in previous sections, this behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. In contrast to the supervised models, which require Bayesian hyperparameter optimization and cross-validation, RBF models were trained in less than one second per output, making them especially attractive for repeated surrogate updates within the adaptive optimization loop. These results indicate that kernel-based models are particularly well suited to the present surrogate task. SVR provides the best compromise between accuracy and computational cost, while GPR is the second most competitive supervised strategy, especially in some higher-dimensional cases. Tree-based models, although robust, are less frequently selected, and MLP models are not competitive in terms of computational efficiency for the dataset sizes considered here. As mentioned in previous sections, this behaviour motivated the additional evaluation of RBF interpolation as a simpler surrogate alternative. In contrast to the supervised models, RBF models were trained in less than one second per output, making them especially attractive for repeated surrogate updates within the adaptive optimization loop.
The FEM validation of the optimized geometries is summarized in Table~\ref{tab:final_surrogate_comparison}. For each geometry family and surrogate strategy, the table reports the final accepted adaptive iteration, the optimized window thicknesses, the surrogate-predicted objective value, the corresponding FEM-recomputed objective value, and the associated validation errors. The optimization process required between two and three adaptive iterations depending on the geometry family and surrogate type, with most cases converging after three iterations. No systematic difference in the number of iterations was observed between RBF and supervised ML surrogates. The maximum variable error, $e_{\max}$, is defined as the largest relative error among all quantities entering the objective function, namely ${\Exy}_i$, $\TFD_i$ and $\TFD_f$, whereas the objective-function error, $|e_J|$, is reported in absolute value. The FEM validation of the optimized geometries is summarized in Table~\ref{tab:final_surrogate_comparison}. For each geometry family and surrogate strategy, the table reports the final accepted adaptive iteration, the optimized window thicknesses $\mathbf{t}_w^{\star}$, the surrogate-predicted objective value ($J_{\mathrm{surr}}$), the corresponding FEM-recomputed objective value ($J_{\mathrm{FEM}}$) and the associated validation errors ($|e_J|$ and $e_{\max}$). The optimization process required between two and three adaptive iterations depending on the geometry family and surrogate type, with most cases converging after three iterations. No systematic difference in the number of iterations was observed between RBF and supervised ML surrogates. The maximum variable error, $e_{\max}$, is defined as the largest relative error among all quantities entering the objective function, namely ${\Exy}_i$, $\TFD_i$ and $\TFD_f$, whereas the objective-function error, $|e_J|$, is reported in absolute value.
\begin{table*}[ht!] \begin{table*}[ht!]
\centering \centering
...@@ -380,51 +380,54 @@ H60\_B34 & Supervised ML & 3 ...@@ -380,51 +380,54 @@ H60\_B34 & Supervised ML & 3
\end{tabular} \end{tabular}
\end{table*} \end{table*}
The final validation results show that both surrogate strategies provide FEM-consistent optimized geometries after few adaptive iterations. RBF surrogates generally lead to lower objective-function discrepancies, with an average $|e_J|$ of 1.36, compared with 5.04 for the supervised ML surrogates. The average maximum variable error is also lower for RBF, with 1.79\% compared with 2.67\% for supervised ML. The RBF surrogate provides particularly accurate predictions for the two-window devices, with $e_{\max}$ below 0.3\%, while remaining below 4.3\% for the three- and five-window families. The supervised ML surrogates also satisfy all validation criteria, although they show larger discrepancies in some cases, especially for the H60\_B34 family, where the objective-function error reaches 9.97. The final validation results show that both surrogate strategies provide FEM-consistent optimized geometries after only a few adaptive iterations. RBF surrogates generally lead to lower objective-function discrepancies, with an average $|e_J|$ of 1.36, compared with 5.04 for the supervised ML surrogates. The average maximum variable error is also lower for RBF, with 1.79\% compared with 2.67\% for supervised ML. The RBF surrogate provides particularly accurate predictions for the two-window devices, with $e_{\max}$ below 0.3\%, while remaining below 4.3\% for the three- and five-window families. The supervised ML surrogates also satisfy all validation criteria, although larger discrepancies are observed in some cases, especially for the H60\_B34 family, where the objective-function error reaches 9.97.
The adaptive validation loop is essential to reach these levels of agreement. In several cases, the first surrogate-optimized candidate did not satisfy the error tolerances, particularly for the three- and five-window devices. After incorporating the additional FEM results and retraining the surrogates, the prediction errors decreased significantly. For instance, the maximum variable error of the RBF surrogate decreased from 13.70\% to 1.70\% in the H45\_B29 family, and from 11.44\% to 2.66\% in the H60\_B34 family. A similar behaviour was observed for the supervised ML surrogates, whose final candidates also satisfied all acceptance criteria. This confirms that the adaptive loop reduces the risk of accepting geometries that appear optimal only because of surrogate extrapolation errors. The adaptive validation loop is essential to reach these levels of agreement. In several cases, the first surrogate-optimized candidate did not satisfy the prescribed error tolerances, particularly for the three- and five-window devices. After incorporating the additional FEM results and retraining the surrogates, the prediction errors decreased significantly. For instance, the maximum variable error of the RBF surrogate decreased from 13.70\% to 1.70\% in the H45\_B29 family, and from 11.44\% to 2.66\% in the H60\_B34 family. A similar behaviour was observed for the supervised ML surrogates, whose final candidates also satisfied all acceptance criteria. This confirms that the adaptive loop reduces the risk of accepting geometries that appear optimal only because of surrogate prediction errors in sparsely sampled regions of the design space.
The optimized geometries obtained with RBF and supervised ML surrogates are also very similar in most geometry families. In general, the difference in the optimized window thicknesses is below 0.1 mm and always below 0.2 mm, except for the H30\_B34 family. In this case, the differences reach 0.43 mm and 1.05 mm for the first and second windows, respectively. This family also presents the largest objective-function values, indicating that the prescribed damage targets, $\TFD_i$ and $\TFD_f$, cannot be fully maintained below the desired threshold within the admissible thickness range. Consequently, the optimizer is forced to find the best compromise between damage control and dissipative activation, which may amplify the differences between surrogate predictions. Even in this more demanding case, the FEM validation confirms that both surrogate strategies remain within the prescribed error limits. The optimized geometries obtained with RBF and supervised ML surrogates are very similar in most geometry families. In general, the differences in the optimized window thicknesses are below 0.1 mm and always below 0.2 mm, except for the H30\_B34 family. In this case, the differences reach 0.43 mm and 1.05 mm for the first and second windows, respectively. This family also presents the largest objective-function values, indicating that the prescribed damage targets, $\TFD_i$ and $\TFD_f$, cannot be fully maintained below the desired threshold within the admissible thickness range. Consequently, the optimizer is forced to find the best compromise between damage control and dissipative activation, which may amplify the differences between surrogate predictions.
The optimized geometries are stable between consecutive adaptive iterations. The maximum change in any window thickness between the last two iterations remains below 5\% of the corresponding design range for all validated cases, with a maximum observed variation of 4.44\%. This indicates that the optimization process has converged not only in terms of surrogate prediction accuracy, but also in terms of the geometry proposed by the optimizer. Figure~\ref{fig:optimized_window_thickness_evolution} shows the evolution of the window thicknesses during the adaptive optimization process. In most cases, the largest adjustment occurs between the first and second iterations, while the changes between the second and third iterations are considerably smaller, indicating convergence of the proposed geometry. According to the acceptance criteria, the maximum variation in any window thickness between the last two iterations must remain below 5\% of the corresponding design range. This condition is satisfied in all validated cases, with a maximum observed variation of 4.44\%. Therefore, the adaptive process converges not only in terms of surrogate prediction accuracy, but also in terms of the optimized geometry proposed by the optimizer.
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 high predictive accuracy and robustness, 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, whereas supervised ML models remain valuable when greater robustness is required or when the response surface is expected to be more complex. \begin{figure}[htbp]
\centering
\includegraphics[width=1.0\textwidth]{./Figures/OptimizedWindowThicknessEvolution/optimized_window_thickness_evolution.png}
\caption{Evolution of the optimized window thicknesses during the adaptive optimization process.}
\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 high predictive accuracy and robustness, 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.
The objective-function values should 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 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 prediction errors, but FEM-consistent optimized candidates within the explored design space. 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 H30\_B34, 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.
The proposed methodology addresses a central limitation of direct FEM-based damper optimization: the high cost of evaluating many nonlinear cyclic simulations. By training surrogate models on FEM-generated datasets and validating the optimized candidates through additional FEM analyses, the optimizer can explore the design domain efficiently while retaining consistency with the underlying mechanics of the device. The objective function also reflects the mechanical hierarchy of the BDSL damper. Damage in the windows is not intrinsically undesirable, since it is associated with the intended dissipative mechanism, provided that it remains below the adopted threshold and is reasonably distributed among windows. By contrast, damage in the frame is structurally more critical, as it may compromise the integrity of the device. The stronger frame penalty therefore guides the optimizer towards geometries that preserve the frame while using the windows as sacrificial dissipative regions. This behaviour is illustrated in Figure~\ref{fig:rbf_surface_evolution}, which shows the evolution of the RBF objective surface during the adaptive optimization process for the two-window families. The response surfaces remain relatively smooth, supporting the suitability of RBF interpolation for this problem. The evolution between adaptive iterations also shows how the surrogate surface is progressively corrected as new FEM information is incorporated near the optimized region.
\begin{figure}[htbp]
\centering
\includegraphics[width=1.0\textwidth]{./Figures/RBFOptimizationSurfaceEvolution/rbf_surface_evolution.png}
\caption{Evolution of the RBF objective surface during the adaptive optimization process for the two-window families. Left: H30\_B29. Right: H30\_B34.}
\label{fig:rbf_surface_evolution}
\end{figure}
Several limitations should also be acknowledged. First, the reliability of the proposed framework depends on the quality of the calibrated FEM model used to generate the training data and validate the optimized geometries. Second, the TFDMap is used here as a post-processing damage indicator rather than as a constitutive fracture model, so the optimized designs should be interpreted in terms of relative damage control rather than direct crack prediction. Third, the present optimization considers only the window thicknesses as design variables. Additional parameters, such as window height, spacing, frame thickness or filler properties, could be incorporated in future work, although this would increase the dimensionality of the problem and require a larger FEM dataset. Finally, the optimized configurations should ultimately be validated experimentally before being used to derive general design recommendations. \section{Conclusions and future work}\label{sec:conclusions}
\section{Conclusions}\label{sec:conclusions} This work presented an adaptive surrogate-assisted optimization framework for buckling-delayed shear-link dampers subjected to cyclic seismic loading. 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 optimisation framework, compared to approaches focused primarily on energy maximisation, 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 maximising the energy dissipated, the proposed formulation favours geometries that concentrate dissipative activation in the windows, prevent excessive damage localisation in a single region, and protect the frame from critical damage.
As a result, the main contribution of this work lies in the development of a robust, scalable and physically informed design methodology that explicitly accounts for the trade-off between energy dissipation and damage. To summarize, the main contributions of this work are: A comprehensive comparison of surrogate strategies was performed in terms of predictive accuracy and computational efficiency. The supervised ML results showed that SVR and GPR were the most competitive models, with SVR being the most frequently selected across the analysed outputs. RBF interpolation also proved to be a highly efficient alternative: 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.
\begin{itemize}
\item generation of high-fidelity FEM datasets for BDSL dampers with increasing geometric complexity and different numbers of dissipative windows;
\item geometric optimization using surrogate models, including supervised ML techniques and RBF interpolants;
\item systematic comparison of surrogate strategies in terms of predictive accuracy, computational cost and practical suitability for optimization;
\item a damage-aware objective function that combines window damage control, severe frame-damage penalization, window-to-window damage balancing and dissipated-energy maximization;
\item an adaptive FEM validation and retraining strategy based on explicit tolerances for surrogate error, objective-function error and stability of the optimized geometry between successive iterations.
\end{itemize}
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 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.
This work presents an adaptive surrogate-assisted optimization framework for BDSL dampers under cyclic seismic loading. The methodology combines FEM-calibrated numerical simulations, supervised ML models, RBF interpolation, DE and FEM-based validation of the optimized geometries. The following conclusions can be drawn from the proposed formulation: 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.
\begin{enumerate} Future work should extend the design space by including additional geometric and mechanical variables, such as window height, window spacing, frame thickness, global device proportions, or filler properties. This extension would increase the dimensionality and complexity of the surrogate task, and the performance of RBF interpolation should therefore be reassessed under those conditions. 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.
\item The BDSL optimization problem must be formulated as a damage-aware design problem rather than as a pure energy-maximization problem. The same energy level may correspond to different local damage distributions and different safety margins in the frame.
\item FEM simulations provide the necessary ground truth for training because they supply internal indicators such as TFDMap and local distortion, which cannot be obtained experimentally with the same spatial resolution.
\item Supervised ML models and RBF interpolants provide complementary surrogate strategies. SVR and GPR are expected to be highly accurate based on preliminary tests, while RBF models offer a faster alternative for well-sampled design domains.
\item The proposed objective function explicitly encodes the desired mechanical behaviour: controlled window damage, severe penalization of frame damage, balanced participation of windows and preference for higher dissipated energy when damage performance is comparable.
\item The adaptive FEM validation and retraining loop is a key component of the framework. A candidate geometry is accepted only if surrogate predictions match FEM results within the defined tolerances and if the optimized thicknesses remain stable between consecutive iterations.
\end{enumerate}
The numerical campaign reported in Section~\ref{sec:results} provides the final model-selection tables, RBF validation metrics, optimized geometries and FEM-validation comparisons. Another relevant future direction is the incorporation of interpretability analyses, such as SHapley Additive exPlanations, to quantify the influence of each geometric variable on window damage, frame damage, and dissipative activation. Although such an 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.
%\backmatter %\backmatter
\bmsection*{Author contributions} \bmsection*{Author contributions}
Conceptualization: J. Irazábal, J. Ramírez, J. M. González, G. Bozzo, L. Bozzo. Methodology: J. Irazábal, J. Ramírez, J. M. González. Software and surrogate optimization: J. Irazábal. FEM modelling and validation: J. Ramírez, J. M. González, L. Lázaro, F. Rastellini. Writing--original draft: J. Irazábal and J. Ramírez. Writing--review and editing: all authors. Conceptualization: J. Irazábal, J. Ramírez, J. M. González, G. Bozzo, L. Bozzo. Methodology: J. Irazábal, J. Ramírez, J. M. González. Software and surrogate optimization: J. Irazábal. FEM modelling and validation: J. Ramírez, J. M. González, L. Lázaro, F. Rastellini. Writing--original draft: J. Irazábal. Writing--review and editing: all authors.
\bmsection*{Acknowledgments} \bmsection*{Acknowledgments}
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