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 P, Schwartz D, and Schoen 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 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.Comment: 28 pages, 16 figures, 3 table