Discrete-Adjoint Free-Form Shape Optimisation for Choked Hyperloop Vehicle Aerodynamics: Verified Gradients and Numerical Robustness

Confined evacuated-tube aerodynamics can choke at subsonic vehicle Mach number when a high blockage ratio forces the displaced gas through a narrow annulus. We developed an in-house MATLAB axisymmetric finite-volume compressible-flow solver and a 30-variable discrete-adjoint free-form-deformation (FFD) framework for a β=0.36 vehicle at p∞=10 kPa, T∞=300 K and M∞=0.6–0.8. The inviscid discretisation was verified by uniform-flow, smooth-wave, manufactured-solution and shock-tube tests, while the complete FFD gradient agrees with central finite differences to 7.18×10−6. A matched-grid reassessment at M∞=0.6 gives pressure-drag reductions of 51.75%, 52.42% and 51.77% with first-order Rusanov, first-order HLLC and second-order HLLC/MUSCL, respectively. The integrated drag benefit is therefore insensitive to the tested flux/order choice, whereas the local supersonic topology is not: the dissipative Rusanov calculation gives a subsonic optimised peak, while HLLC-based calculations retain local supersonic flow. We therefore withdraw any claim that optimisation removes choking and report only a robust reduction in the integrated pressure load. The saved FFD geometry has Rv,max=1.2939 m (β=0.2679), with the same fixed reference area used for all coefficients. By contrast, the previously reported 11–20% fixed-radius length-only benefit is not grid-robust: on the 36,504-cell refinement it becomes −0.01%, 2.95% and 3.20% at M∞=0.6, 0.7 and 0.8. A pressure sweep from 10 kPa to 10 Pa reproduces identical CD,p and peak Mach to nine significant figures while dimensional drag scales linearly with p∞; at M∞=0.6 the matched Rusanov pair gives 45.78→22.09 kN and 9.54→4.60 MW at 10 kPa. These are Euler pressure-drag quantities, not total vehicle drag or total system energy.