Biomimetic Decussated Enamel Lattice

Figure Gallery — All figures from presentation_decussated_enamel.tex — Generated 2026-02-26

§ 1

Design, Architecture & Computational Pipeline

Biological Motivation & CAD
Natural enamel SEM
Slide: Motivation
SEM Micrograph of Natural Dental Enamel: Decussated Rod Bundles
CAD isometric
Slide: CAD Generation
CadQuery Isometric Render: 5 Concentric Rings, 91 Rods, Bridge Connections, End Plates
Geometry & Mesh
STEP cross section
Slide: CAD Generation
STEP Geometry: Side Cross-Section View
STEP with FEA overlay
Slide: CAD Generation
Same Geometry with FEA Mesh Overlay
Gmsh mesh
Slide: Meshing Strategy
Gmsh Unstructured Tetrahedral Mesh: Dense Element Packing at Bridge Junctions
FEA Results Visualization
FEA results isometric
Slide: FEA Results — Isometric Stress Visualization
Von Mises Stress Field: N=8 Lattice Under 10 mm Axial Compression (Red = High, Blue = Low)
§ 2

N=4 Baseline Analysis

Stress components N4
N=4 Baseline — Stress Component Distributions
Volume-Weighted Histograms of All Six Stress Tensor Components (σxx–τxz)
Strain components N4
N=4 Baseline — Strain Component Distributions
Volume-Weighted Histograms of All Six Strain Tensor Components (ϵxx–γxz)
Spatial stress N4
N=4 Baseline — Spatial Stress Distribution
2D Hexbin Heat Map: Mesh Analysis (Dashed Lines = Bridge Planes)
Regional comparison N4
N=4 Baseline — Regional Stress Comparison
Box Plots of Von Mises Stress by Structural Region: BB, RMS, RPJ, Plates, Rod Body
Failure metrics N4
N=4 Baseline — Failure Metrics
VM Stress CDF, Triaxiality Histogram, and Force–Displacement Curve for N=4 Baseline
Hotspot analysis N4
N=4 Baseline — Spatial Hotspot Analysis
Top-50 Hotspot Elements (r vs. z), Mean VM z-Profile, and Element Volume Histogram
Damage Tolerance & Energy Absorption
Damage tolerance N4
N=4 Damage Tolerance & Energy Absorption
Energy Budget by Region, Toughness Bar Chart, and 5-Axis Radar Scorecard for N=4
§ 3

Self-Optimizing Loop & Parameter Sweep Overview

Pareto front
Self-Optimizing Loop — Optuna NSGA-II
Pareto Front: SCF vs. Eeff Trade-Off and Specific Toughness vs. SCF
N bridge sweep
Parameter Sweep — Bridge Layer Count N
Six-Metric Sweep: Eeff, SCF, VM Peak, Toughness, Uniformity, Resilience vs. N
§ 4

N-Bridge Sweep Comparison Results (N = 2, 4, 6, 8)

N bridge comparison dashboard
Sweep Comparison — All Metrics Dashboard
Multi-Panel Dashboard: Eeff, SCF, VM Peak, Specific Toughness, and VM CDF for N=2, 4, 6, 8
N bridge regional
Sweep — Regional Stress by N
Box Plots of VM Stress per Structural Region (BB, RMS, RPJ, Plates) Grouped by N Value
N bridge toughness
Sweep — Toughness Metrics by N
Specific Toughness, Resilience Index, Toughness Uniformity κ, and Energy Absorption vs. N
N bridge hotspot
Sweep — Hotspot Distribution by N
Top-200 Hotspot Elements by Radial Position and z-Elevation for Each N Value
Radar chart all N
Sweep Radar Scorecard — All N Values
5-Axis Radar Chart: Stiffness, SCF, Toughness, Uniformity, Resilience — Score 1.0 = Best, 0.0 = Worst
Summary Dashboard
N bridge summary
Sweep Summary Dashboard
Combined Summary: Scalar Metric Bars, Force–Displacement Overlay, and Regional Stress Hierarchy for N=2,4,6,8
§ 5

Cross-N Comparison (Shared Axes)

Primary Cross-N Figures
Cross-N stress
Cross-N — Stress Component Histograms
All Six Stress Components for N=2,4,6,8 on Shared Axes — Visual Width Encodes Magnitude Directly
N=4 has tightest distributions; N=8 shows broadest tails
Cross-N spatial
Cross-N — Spatial Maps (Z vs. Von Mises)
Spatial Hexbin Maps (z vs. VM Stress) for Each N — Dashed Orange Lines Mark Bridge Elevation Planes
N=4: isolated peaks with stress-relief gaps; N=8: planes merge into plateau
Cross-N regional
Cross-N — Regional Analysis
Regional VM Stress Box Plots for All N Values on a Shared y-Axis; 6 Structural Regions
BB median rises sharply N=6→N=8; rod mid-span protected across all N
Cross-N radar scorecard
Cross-N — Radar Scorecard
Radar Scorecard: 5-Axis Performance Normalised to Min–Max Range Across All N Values
N=4 is Pareto-optimal — the only design with no weak axis
Cross-N strain
Cross-N — Strain Component Distributions
All Six Strain Tensor Components for N=2,4,6,8 on Shared Axes — Axial Mean Identical for All N
Identical ϵzz confirms FEA validity; growing shear widths show bridge-junction coupling
Cross-N failure
Cross-N — Failure Metrics Comparison
VM CDF, Triaxiality Histogram, and Force–Displacement Curve for All N Values on Shared Axes
Invariant η = −1/3 peak confirms compression-dominated failure mode for all N
Cross-N hotspot
Cross-N — Hotspot Spatial Analysis
Top-200 Hotspot Elements (r vs. z) for Each N — Banded Pattern Signals Functional Bridge Planes
Discrete 4-band pattern in N=4 vs. merged column in N=8 = over-bridging failure signature
Cross-N damage tolerance
Cross-N — Damage Tolerance Bar Charts
Elastic Energy Absorbed per Structural Region for Each N Value — BB Share Grows Fastest with N
N=4: highest total absorption AND best distribution; N=8: BB >50% over-reliance
§ 6

Overlaid Curves & Distributions (N = 2, 4, 6, 8)

Overlaid F-D curves
Overlaid Force–Displacement Curves
Force–Displacement Curves for All N Values Overlaid — Slope = Structural Stiffness k
N=8 steepest (1.28 MN/mm); N=4 best energy-per-SCF ratio (1.02 MN/mm)
Overlaid VM CDF
Overlaid Von Mises Stress CDF
Volume-Weighted Cumulative Distribution of VM Stress for All N — Steeper = More Uniform Stress Sharing
N=4 steepest mid-range rise; N=8 extended high-stress tail
Overlaid VM z-profile
Overlaid Axial Stress Z-Profile
Mean VM Stress vs. Specimen Height for All N — Peaks Mark Bridge-Band Elevations
Valley depth between peaks measures bridge stress-relief effectiveness — deepest in N=4
Overlaid triaxiality
Overlaid Triaxiality Distributions
Volume-Weighted Triaxiality Histograms Overlaid for All N — Invariant η = −1/3 Peak Confirms Uniaxial Failure Mode
N only changes stress magnitude, not failure mode
Energy Budget
Overlaid energy budget
Overlaid Energy Budget by Region
Stacked-Bar Energy Budget: Fraction of Total Strain Energy per Structural Region for Each N
N=4: BB ≈31% (optimal sacrificial fuse — “Goldilocks zone”); N=8: BB >50% (structural over-reliance)
§ 7

Printing Process Direction Study

Material sweep
Printing Process Direction Study
Process Comparison: Eeff and Peak Stress Scale with E; SCF, Uniformity, Resilience Are Process-Independent
A

Appendix — Per-N Full Analysis (N = 2, 6, 8)

N = 2
N2 stress
Appendix N=2
Stress Component Distributions (N=2)
N2 strain
Appendix N=2
Strain Component Distributions (N=2)
N2 failure
Appendix N=2
Failure Metrics (N=2)
N2 spatial
Appendix N=2
Spatial Stress Distribution (N=2)
N2 regional
Appendix N=2
Regional Stress Comparison (N=2)
N2 damage
Appendix N=2
Damage Tolerance & Energy Budget (N=2)
N = 6
N6 stress
Appendix N=6
Stress Component Distributions (N=6)
N6 strain
Appendix N=6
Strain Component Distributions (N=6)
N6 failure
Appendix N=6
Failure Metrics (N=6)
N6 spatial
Appendix N=6
Spatial Stress Distribution (N=6)
N6 regional
Appendix N=6
Regional Stress Comparison (N=6)
N6 damage
Appendix N=6
Damage Tolerance & Energy Budget (N=6)
N = 8
N8 stress
Appendix N=8
Stress Component Distributions (N=8)
N8 strain
Appendix N=8
Strain Component Distributions (N=8)
N8 failure
Appendix N=8
Failure Metrics (N=8)
N8 spatial
Appendix N=8
Spatial Stress Distribution (N=8)
N8 regional
Appendix N=8
Regional Stress Comparison (N=8)
N8 damage
Appendix N=8
Damage Tolerance & Energy Budget (N=8)
T

Optimization Iteration Metrics

8 iterations of the Optuna NSGA-II self-optimizing loop. All metric improvements/regressions relative to the Baseline (Iter 0).

#LabelNotes E_eff (MPa)SCFVM_max (MPa) VM_P95 (MPa)BB_hotspot_frac RPJ_VM_maxsolid_frac spec_toughnessresilience uniformityenergy_abs (mJ)Time (s)
0Baseline Original design, uniform mesh 0.5 mm 1.00 48040.31 6.78 186388.80 63676.48 0.485 40908.57 —— —— -0.305
1Jcn Refined Mesh 0.5× mesh refinement at bridge elevations z=4,8,12,16 0.50 (−50%) 47994.65 (−0.1%) 7.22 (+6.5%) 198342.62 (+6.4%) 63628.93 (−0.1%) 0.44 (−9.3%) 40867.75 (−0.1%) ———— -0.304
2Jcn Spheres 0.7× 1.4 mm spheres at 360 bridge-rod junctions + 0.5× mesh refinement 0.50 (−50%) 47996.20 (−0.1%) 6.68 (−1.5%) 183586.50 (−1.5%) 63680.54 (+0.0%) 0.51 (+4.1%) 40869.07 (−0.1%) ———— -0.304
3Thick Bridge+Spheres Bridge 1.5 mm (was 1.0), sphere factor 0.8 (1.6 mm diam), jcn refine 0.5× 0.50 (−50%) 49573.31 (+3.2%) 5.40 (−20.4%) 152024.54 (−18.4%) 67039.54 (+5.3%) 0.00 (−100%) 42567.78 (+4.1%) ———— -0.309
4—Optuna trial 1.00 (0.0%) 50338.75 (+4.8%) 5.06 (−25.5%) 121399.04 (−34.9%) 53426.33 (−16.1%) 0.24 (−49.5%) 33001.79 (−19.3%) ———— -0.299
5—Optuna trial 1.00 (0.0%) 51313.63 (+6.8%) 7.47 (+10.2%) 181592.89 (−2.6%) 55267.82 (−13.2%) 0.12 (−74.2%) 33576.96 (−17.9%) ———— -0.300
6—Optuna trial 1.00 (0.0%) 52247.11 (+8.8%) 7.16 (+5.6%) 175946.41 (−5.6%) 57365.21 (−9.9%) 0.10 (−78.4%) 34018.59 (−16.8%) ———— -0.302
7Taper 0.2 + Spheres 0.8 bridge=1.5 mm, jcn_sphere=0.8, rod_taper=0.2, plate_overlap=1.0, all fixes applied 47197.77 (+4,719,677%) 5.70 (−100%) 134949.44 (+1,989,212%) 48401.51 (−74.0%) 0.29 (−100%) 28558.59 (+5,888,269%) 0.44 (−100%) 378.658283.720.0353 51,098,482 0.0

Best values: SCF = 5.70 • VM_max = 5.06 MPa • BB_hotspot_frac = 0.29 • E_eff = 47,197.77 MPa • Specific Toughness = 378.65 mJ/MPa • Resilience = 8,283.72 • Uniformity = 0.035 — all achieved by Taper 0.2 + Spheres 0.8