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Peer-reviewedCC BY 4.0Fractal and Fractional

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

  • Golib Berdiev1
  • Shahzoda Anarova2
  • Navruz Khujakulov3
  1. 1Department of Information Technologies, University of Information Technologies and Management, Karshi 180100, Uzbekistan
  2. 2Department of Digital Technologies Convergence, Tashkent University of Information Technologies Named After Muhammad al-Khwarizmi, Tashkent 100084, Uzbekistan
  3. 3Department of Software and Hardware Support of Computer Systems, Karshi State Technical University, Karshi 180100, Uzbekistan
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@article{berdiev2026hybrid,
  title = {{Hybrid Fractal–Parametric Modeling and NSGA-II Optimization of Geometric Material-Volume Reduction in Generative Architectural Façades}},
  author = {Golib Berdiev and Shahzoda Anarova and Navruz Khujakulov},
  journal = {Fractal and Fractional},
  year = {2026},
  month = nov,
  doi = {10.3390/fractalfract10110771},
  url = {https://www.mdpi.com/2504-3110/10/11/771},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Hybrid Fractal–Parametric Modeling and NSGA-II Optimization of Geometric Material-Volume Reduction in Generative Architectural Façades
AU  - Golib Berdiev
AU  - Shahzoda Anarova
AU  - Navruz Khujakulov
PY  - 2026
DA  - 2026/11/05
JO  - Fractal and Fractional
DO  - 10.3390/fractalfract10110771
UR  - https://www.mdpi.com/2504-3110/10/11/771
AB  - 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.
SN  - 2504-3110
ER  -