BEM, propellers, wind turbines, rotorcraft, gradients, derivatives, Jacobians, sparsity, blade, optimization


Blade element momentum methods are widely used for initial aerodynamic analysis of propellers and wind turbines. A wide variety of correction methods exist, but common to all variations, a pair of residuals are converged to ensure compatibility between the two theories. This paper shows how to rearrange the sequence of calculations reducing to a single residual. This yields the significant advantage that convergence can be guaranteed and to machine precision. Both of these considerations are particularly important for gradient- based optimization where a wide variety of atypical inputs may be explored, and where tight convergence is necessary for accurate derivative computation. On a moderate-sized example optimization problem we show over an order of magnitude increase in optimization speed, with no changes to the physics. This is done by using the single residual form, providing numerically exact gradients using algorithmic differentiation with an adjoint, and by leveraging sparsity in the Jacobian using graph coloring techniques. Finally, we demonstrate a revised formulation for cases when no inflow exists in one of the directions (e.g., a hovering rotor or a parked rotor). These new residuals allow for robust convergence in optimization applications, avoiding the occasional numerical difficulties that exist with the standard formulation.

Original Publication Citation

Ning, A., “Using Blade Element Momentum Methods with Gradient-Based Design Optimization,” Structural and Multidisciplinary Optimization, May 2021. doi:10.1007/s00158-021-02883-6

Document Type

Peer-Reviewed Article

Publication Date



Springer Nature




Ira A. Fulton College of Engineering and Technology


Mechanical Engineering

University Standing at Time of Publication

Associate Professor