[FEATURE] Katz method for load calculations

[camUrban]

Problem Statement

There are two primary algorithms for the load-calculation step in an unsteady vortex lattice method: the Joukowski method and the Katz method. Each has tradeoffs in terms of convergence behavior and computational speed. In their most basic ("vanilla") formulations, the Joukowski method tends to be accurate in a wider range of edge cases, which is why it was originally selected for Ptera Software.

However, recent literature more commonly uses the Katz method, and as a result, many extensions, particularly those related to modeling leading-edge separation, are built on top of it. Unfortunately, experimenting with or adopting these techniques in Ptera Software has been difficult due to our original decision to exclusively use the Joukowski formulation.

Proposed Solution

Introduce a force_method optional parameter to UnsteadyVortexLatticeMethodSolver.run(). The default value would be Joukowski to preserve backward compatibility. Users could alternatively set this parameter to Katz, which would compute loads using a new Katz-based algorithm.

At this stage, it is undecided whether a similar option should be added to the steady VLM solvers.

Alternative Solutions

Maintain the status quo. This is not ideal for the reasons described in the problem statement.

Additional Context

• Thorough discussion of tradeoffs between the two methods: Lambert, Thomas. “Modeling of Aerodynamic Forces in Flapping Flight with the Unsteady Vortex Lattice Method.” University of Liege, 2015.
• Example of an extension to the Katz method for modeling separation: Nguyen, Anh, Joong Kim, Jong Han, and Jae-Hung Han. “Extended Unsteady Vortex-Lattice Method for Insect Flapping Wings.” Journal of Aircraft 53, no. 6 (2016): 1709–18. https://doi.org/10.2514/1.C033456.
• Prior discussion of the need for a separation model in Ptera Software: [FEATURE] Separation Modeling #17