Integration

Sep 13, 2025

Integration: Structural Optimization & Machine Learning

Category: Generative Radial Geometry (GRG)

GRG integrates with structural optimization (SIMP/level set) and with machine learning for parameter calibration. Angles are measured in turns with dyadic step Δu = 2^{-n} on the full turn; a quarter has M = 2^{\,n-2} steps. Formulas remain π-free at finite resolution.

$$ n \in \mathbb{N}_{\ge 0},\qquad \Delta u = 2^{-n},\qquad M = 2^{\,n-2}\ \text{(steps per quarter)}. $$

Topology Optimization Bridge

We expose GRG’s radial density as design variables for SIMP/level-set frameworks. Let x denote per-sector densities, mapped from radial weights:

$$ x(u_k)\;=\;\sigma\!\big(w(u_k;\,\theta)\big),\qquad u_k = k\,2^{-n},\ \ k=0,\ldots,2^{n}-1, $$

with a smooth logistic σ and parameters θ controlling w(u). A standard compliance-minimization model uses:

$$ \min_{x}\; C(x)=\mathbf f^\top \mathbf u \quad\text{s.t.}\quad \mathbf K(x)\,\mathbf u=\mathbf f,\;\; \sum_i x_i \le \bar{v}\,m,\;\; x_{\min}\le x_i\le 1. $$

By parameterizing x via GRG weights, the optimizer searches a low-dimensional, center-seeded space that preserves radial structure while meeting volume and stiffness targets.

Level-Set Coupling

Alternatively, drive a level-set ϕ by GRG fields: initialize ϕ from radial strips and evolve with curvature-regularized motion under structural sensitivities. GRG provides stable seeding and symmetric updates across quarters.

ML-Assisted Calibration

Given exemplars (e.g., bones, shells, corals), regress GRG parameters θ to match silhouettes or performance. Use mixed geometric + physical loss:

$$ \mathcal L(\theta)\;=\;\lambda_{\text{geom}}\,d_{\text{Chamfer}}\!\big(\partial S_{\text{GRG}}(\theta),\,\partial S^\*\big) \;+\; \lambda_{\text{phys}}\!\left(\alpha\,\big|A(\theta)-A^\*\big|+\beta\,\big|P(\theta)-P^\*\big|\right) \;+\;\lambda_{\text{reg}}\|\theta\|_2^2. $$

Perimeter and area use GRG quarter aggregation with Δu=2^{-n}:

$$ P(n)\;=\;4\sum_{k=1}^{M} 2\,s\!\left(2^{-(n+1)}\right),\qquad A(n)\;=\;4\sum_{k=1}^{M}\tfrac{1}{2}\,s\!\left(2^{-n}\right), \quad M=2^{\,n-2}. $$

Pipeline

  1. Radial seed & step: pick n so Δu=2^{-n}; choose rule for chords/strips (midpoint, averaged, Δx trapezoidal).
  2. Design variables: set θ for w(u) or direct densities x(u_k).
  3. Solver coupling: run SIMP/level-set; back-propagate sensitivities to θ.
  4. Calibration (optional): fit θ to exemplars using the mixed loss.
  5. Export: reconstruct full shape from quarter; output polylines/meshes.

Safety & Regularization

  • Step cap: enforce a minimum Δu_{\min} to prevent local collapse.
  • Curvature damping: penalize large changes of w(u) across adjacent sectors.
  • Volume guard: soft penalty if \sum x_i drifts from target fraction \bar v.

Interoperability

  • Inputs: seed, n, rule, parameters θ or densities x.
  • Outputs: perimeter P, area A, nodes/strips, optional field samples.
  • Formats: .OBJ polylines, .STEP (via downstream CAD), JSON config for reproducibility.

Defaults

  • Rule: chord — midpoint for perimeter; Δx trapezoidal for area.
  • n: start with n=10 (Δu=2^{-10}) and refine dyadically.
  • ML loss: set \lambda_{\text{geom}}=1, \lambda_{\text{phys}}=0.3, \lambda_{\text{reg}}=10^{-3} as safe starters.