Optimizing the Geometry of an L-Shaped Building to Enhance Energy Efficiency and Sustainability

TL;DR

This paper develops an analytical method to optimize L-shaped building geometries, reducing external surface area to improve energy efficiency while considering practical design constraints.

Contribution

It introduces explicit optimization formulas for L-shaped buildings, accounting for realistic constraints, and demonstrates practical energy-saving design guidelines.

Findings

Optimized geometries can significantly reduce external surface area.

Constraints prevent degenerate shapes, ensuring practical L-shaped designs.

Case studies confirm the method's effectiveness in real-world scenarios.

Abstract

The geometric form of a building strongly influences its material use, heat losses, and energy efficiency. This paper presents an analytical optimization of L-shaped residential buildings aimed at minimizing the external surface area for a prescribed volume. Both symmetric and asymmetric configurations are examined under realistic design constraints, including fixed or bounded wing aspect ratios and fixed building height. Using explicit optimization methods and Karush-Kuhn-Tucker conditions, closed-form expressions for the optimal geometric parameters and minimal envelope area are derived. The results show that unconstrained optimization leads to degenerate cuboid shapes, highlighting the importance of geometric constraints to preserve the L-shaped form. The obtained results provide practical design guidelines for architects and engineers, supporting informed early stage decisions that balance functional requirements, regulatory constraints, architectural intent, and energy performance. Case studies of existing houses demonstrate that the proposed approach can reduce external surface area or confirm near-optimality of practical designs, supporting energy-efficient early-stage architectural decisions.