Phase-Intersected TPMS (PI-TPMS): Implicit Strut Networks from the Intersection Curves of Paired Level-Set Fields

Abstract

Triply periodic minimal surfaces (TPMS) are usually made solid by thickening one surface into a sheet or filling one side of it as a skeletal network. We introduce phase-intersected TPMS (PI-TPMS), in which the solid is a tube around the curve where two level-set fields vanish together, typically a TPMS and a phase-shifted copy. The construction is a single implicit inequality, and one surface pair yields a whole family of strut networks selected by the shift. The tube volume is exactly πr²L below the overlap scale; L is measured per shift for TPMS pairs and follows in closed form from the spectrum for stochastic fields (agreement 0.99 ± 0.01 over 43 pairs). Generic intersection curves never branch, so junctions arise only as crossings at symmetry-forced tangencies, which connect the network at any radius, or as radius-dependent contacts between nearby curves. Across 36 gyroid and Fischer–Koch S shifts, every shift without crossings gives separate strands; three-fold flat-point tangencies give six-branch junctions; and body-diagonal shifts make Young's modulus exactly isotropic across the shift axis. FFT homogenization of 35 cells reproduces independent finite-element data to within a median 5% and shows the connected PI-gyroid is bending-dominated (E ∝ ρ^2.1). At a matched thinnest feature it matches the direction-averaged stiffness of a skeletal gyroid at 1.7 times the solid fraction, with a shift-selected 4 : 1 anisotropy. The construction is released openly, and the software is freely available for research.