Curvature-Constrained Geometric Optimization of Solar Thermal Collectors Via Rayleigh–Ritz Variational Design

Abstract

The geometric configuration of solar thermal collectors plays a critical role in determining their energy absorption capacity, optical efficiency, and structural feasibility. While traditional designs rely on fixed geometries such as flat plates or parabolic troughs, these do not account for site-specific conditions or manufacturing constraints. This study introduces a variational optimization framework for generating solar absorber profiles that maximize energy absorption while explicitly constraining curvature to ensure manufacturability. The collector geometry is formulated as the solution to a constrained variational problem, discretized via the Rayleigh–Ritz method using orthogonal sine basis functions. A curvature penalization term is integrated into the objective functional to limit structural complexity and enhance mechanical integrity. The numerical optimization is performed using the Sequential Least Squares Programming (SLSQP) algorithm, and the entire process is implemented in Python. The complete source code and workflow are made openly available for full reproducibility. Results demonstrate that the emergent profile achieves the same total absorbed energy as classical geometries under uniform irradiance while improving local energy distribution and reducing curvature concentration. Sensitivity analysis confirms the robustness and tunability of the approach with respect to the regularization parameter and basis resolution. Despite the promising outcomes, the model currently assumes two-dimensional geometry and idealized irradiance conditions, which may limit real-world applicability without further extension. Future work will address 3D geometries and thermal-fluid coupling to bridge simulation and deployment.