Reaction–diffusion level set: A B‑spline route to triply periodic minimal surfaces

This paper presents a reaction–diffusion B-spline level-set (RDBLS) framework for generating and exploring triply periodic surfaces under prescribed volume-fraction and crystallographic-symmetry constraints. The geometry is represented implicitly on a periodic tensor-product B-spline grid, providing a smooth representation for evaluating surface area, curvature, and their shape gradients. Area- and curvature-based objectives are combined with reaction–diffusion regularization to stabilize the evolution. An inner bisection procedure enforces the prescribed global constraint, while symmetry projection restricts the evolving surface to the selected space-group class. Numerical examples demonstrate accurate reconstruction of the classical Primitive, Diamond, and Gyroid surfaces from different initial configurations, and control-grid studies confirm systematic improvement with increasing resolution. The framework also generates three complex high-genus configurations without using reference triangulations or surface-specific analytical expressions. Comparison with periodic-surface catalogues using genus, connectivity, crystallographic symmetry, signed-field correlation, and relaxation behaviour associates these configurations with the N14, Manta, and F-RD families. Numerical homogenization shows that their effective stiffnesses are comparable to those of classical TPMS at the same volume fraction, while their elastic responses span coordinate-axis-dominant, nearly isotropic, and body-diagonal-dominant cubic-anisotropy regimes. The proposed framework therefore provides a constraint-aware approach for reconstructing known periodic surfaces and exploring complex symmetry-consistent design regions beyond commonly used low-order nodal representations.