A variational level set approach for surface area minimization of triply-periodic surfaces

In this paper, we study triply-periodic surfaces with minimal surface area under a constraint in the volume fraction of the regions (phases) that the surface separates. Using a variational level set method formulation, we present a theoretical characterization of and a numerical algorithm for computing these surfaces. We use our theoretical and computational formulation to study the optimality of the Schwartz primitive (P), Schwartz diamond (D), and Schoen gyroid (G) surfaces when the volume fractions of the two phases are equal and explore the properties of optimal structures when the volume fractions of the two phases are not equal. Due to the computational cost of the fully three-dimensional shape optimization problem, we implement our numerical simulations using a parallel level set method software package.