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.
Benchmarks

Benchmarks

Sep 13, 2025

Validation and Benchmark Protocol

This page specifies planned comparisons. It does not present a completed benchmark or a demonstrated performance advantage.

Separate Tasks

Measure circumference and area evaluation separately from boundary generation, rasterisation, and three-dimensional modelling. Equal sector counts alone do not make different tasks comparable.

Accuracy and Cost

Compare runtime at the same achieved error, and error at the same runtime. Report arithmetic format, rounding, memory, hardware, software versions, warm-up, repetitions, timing variability, setup cost, and precomputation.

Baselines

Include geometric bisection with inner–outer bounds, comparable weighted or extrapolated kernels, and direct analytical evaluation where appropriate. Give every method the same target and include its full computational cost.

Data and References

Begin with the circle kernels whose hypotheses are documented in the current formulation. Validate other shapes and three-dimensional methods separately. Publish reference precision and how it is obtained.

Convergence

The current regular-sector proofs distinguish second-order arithmetic kernels from fourth-order weighted kernels. These rates must not be transferred automatically to Cartesian trapezoids, endpoint-singular integrands, general shapes, or finite-precision implementations.

Reporting

Publish executable scripts, inputs, raw CSV or JSON, version identifiers, plots, and the conditions of every comparison. Former sample timing rows and schematic plots are not retained as measured results.

Complexity

Complexity

Sep 12, 2025

Computational Complexity

Complexity must identify the task, algorithm, arithmetic precision, and output representation. This page distinguishes costs rather than assigning a single complexity to all GRG methods.

Scalar Kernel Evaluation

Given local geometric ratios, the current circle kernels require a fixed number of arithmetic operations. Obtaining the ratios has its own cost. A binary-bisection evaluation performs a number of refinement stages proportional to the binary depth; square-root cost depends on precision and implementation.

Explicit Geometry

Generating and storing all sector nodes has a cost proportional to the output count. Streaming may reduce storage, but does not remove output work.

Raster and Volume Tasks

Boundary drawing, disk filling, surface generation, and voxel filling have different output sizes. Their cost statements must specify whether every output element is generated.

Comparisons

Report setup, precomputation, precision, memory, and runtime at equal achieved accuracy under the benchmark protocol. Earlier repeated cost models have been consolidated here.

Selector

Selector

Sep 12, 2025

Choosing a GRG Method

Choose a method by its documented geometry, required output, error model, and implementation evidence.

Current Circle Measures

For regular binary sectors, the current formulation supplies arithmetic and weighted kernels, explicit area bounds, and exact-arithmetic convergence proofs. Use the stated domains and preserve the distinction between geometric approximation and rounding.

Historical Cartesian Methods

Projection and trapezoidal constructions are separate methods. Their endpoint behaviour, stopping rules, and error rates require independent validation.

General Shapes and Adaptivity

A circular chord–sagitta ratio alone is not a demonstrated universal curvature or stability selector. An adaptive rule must define the boundary, local error indicator, limits, and termination behaviour before it is recommended.

Performance

No automatic selector is claimed as validated here. Compare actual implementations under the benchmark protocol.

Methods

Methods

Sep 12, 2025

Methods and Their Scope

Current Binary Radial Kernels

The current formulation defines regular equal sectors, local chord–tangent lengths, triangular areas, and weighted kernels. Its proofs concern those definitions.

Earlier Methods

Algorithm Specifications

A runnable method must identify initial values, refinement or direct evaluation, arithmetic format, outputs, and stopping conditions. A geometry identity alone is not a runtime benchmark.

Extensions

Adaptive sampling, general curves, multiple seeds, and three-dimensional constructions require their own definitions and validation. No error order or performance advantage is transferred automatically from the regular-circle proof.