Hybrid Fractal–Parametric Modeling and NSGA-II Optimization of Geometric Material-Volume Reduction in Generative Architectural Façades

This study develops a hybrid fractal–parametric modeling (HFPM) workflow integrating recursive affine iterated-function systems, finite-scale box-counting, fixed-thickness prism extrusion, and NSGA-II to explore trade-offs between geometric material volume and geometric complexity in generative architectural façades. A common 729 × 729 raster and four box-counting scales (3, 9, 27, and 81) are applied to both investigated recursion depths (r = 3 and r = 4). Across five independent NSGA-II runs, the pooled final non-dominated reference set yielded geometric material-volume reductions of 34.58% and 44.57% within 1% and 2% deviations, respectively, from the baseline finite-scale fractal dimension (Df). Mean hypervolume was 0.26182 ± 0.00113, evaluated in the original objective space against a common reference point (1.000000, 0.323566); mean inverted generational distance was 0.01732 ± 0.00244, evaluated using range-normalized distances to the empirical pooled reference front. Independent fixed-recursion optimization, simple branch-pruning controls, and density-seed, grid-origin, raster-resolution, and scale-window sensitivity tests further delimit these findings. Branch removal contributes substantially to the volume reductions, while the measured trade-offs and tolerance-feasible solutions depend on the chosen box-counting window. The results therefore establish a computational, protocol-dependent geometric trade-off for the investigated fixed-thickness prefactal prisms, not structural efficiency, equivalent façade functionality, fabrication feasibility, cost savings, or environmental benefits.