Roadmap

Roadmap

Sep 13, 2025

Research Roadmap

This page records work to be completed rather than retaining expired milestones or unsupported delivery promises.

Documentation

Align the current formulation, area-kernel article, notation, historical methods, and references. Consolidate duplicate content and preserve versions.

Implementation

Trace dependencies in every interactive tool. Specify which operations construct geometry, evaluate measures, compute comparison references, and draw the visualisation.

Verification

Review domains, normalisations, error proofs, rounding, cancellation, and stopping conditions. Validate general shapes and three-dimensional methods independently.

Benchmarks and Releases

Publish actual scripts, datasets, environments, licences, and measured results. Assess equal-accuracy performance before reporting a speed advantage.

Publication

List manuscripts and releases with verified status and persistent links. No release date, throughput target, or completed milestone is implied until evidence is available.

Integration

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.