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Periodic microstructure interface reference benchmark

First dossier, 9 October 2026 Australia/Brisbane. Initial decision: Prototype formula; conditional simulation integration. Buyer demand, commercial novelty and profitability remain unvalidated.

Research finding

Family 354 claims the exact isotropic perimeter profile of the unit cubic flat three-torus: with v=min(V,1−V), the minimum is min((36π)^(1/3)v^(2/3), 2sqrt(πv), 2). Minimizers are balls, tubes around shortest geodesics or coordinate slabs, plus complements. The transitions are 4π/81 and 1/π, with adjacent types both minimizing at equality.

Problem and buyer

A periodic materials-simulation team needs a simple reference to distinguish an expected morphology transition from finite-interface, discretization or optimization effects in an idealized two-phase model. A controlled benchmark can be useful when changing a solver or area estimator.

What the finding could enable

A regression library can expose the candidate phases, transition fractions and area scaling, and compare a sharp-interface numerical result against the claimed global optimum. A custom app could visualize the volume regimes and attach explicit model assumptions. It would validate an ideal reference case, not design every realistic microstructure.

Technical and commercial limits

The profile uses a cubic flat periodic cell and isotropic continuum perimeter at fixed volume. Anisotropic interfacial energy, mechanics, composition-dependent bulk energy, rectangular cells and diffuse-interface widths define different problems. A voxel face-count area can be anisotropic and should not be compared as if it were continuum perimeter without a justified estimator.

Minimal architecture

Volume fraction and cubic cell units -> explicit profile evaluation -> phase/transition record -> solver-run import -> validated area and volume estimates -> convergence study -> scope-aware regression report. periodic_interface_reference.py implements the formula and dimensional area scaling in floating point; no solver or numeric certification is supplied.

Existing alternatives and differentiation

MOOSE’s Phase Field Module already provides tools for phase-field simulation, anisotropy and multiphysics coupling. A new simulator is unnecessary for the initial hypothesis. The proposed contribution is a precise ideal sharp-interface reference and a comparison adapter, with finite-interface limits stated. MOOSE Phase Field Module.

Monetization hypothesis

Hypothesis: AUD 4,000–12,000 for a solver-validation integration, perhaps bundled with a broader simulation assurance service. At a hypothetical AUD 7,000 fee, 30 specialist hours costed at AUD 150/hour leave AUD 2,500 before compute, support and overhead. The tiny formula calculator alone does not justify a high-priced subscription.

Validation experiment

The local tests verify both transition equalities, complementary volume symmetry, quadratic side-length area scaling and agreement with independently evaluated sphere/tube geometric area formulas. Next use one periodic solver and refine mesh/interface width across a transition, recording volume conservation and estimator bias. A phase-field energy must first be related to the sharp perimeter model.

Conditions to reject or defer

Reject if the target software never uses an isotropic cubic reference case or its existing tests already provide the same diagnostic. Defer realistic microstructure optimization and certified area comparisons until a suitable model bridge and numerical validation exist.

Next concrete action

Select a public periodic two-phase example, define units and the sharp-interface limit, and measure whether this reference catches a regression an existing test misses.

Pinned research sources