Aerodynamic Shape Optimization with Output-Based Mesh Adaptation and Analytically Verified Sensitivities
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mcallister-nmmcall-sm-aeroastro-2026.pdf
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9.75 MB
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83bd0be3cc7b94929e167862fb2f06e8
Author(s)
McAllister, Noah M.
Advisor(s)
Galbraith, Marshall C.
Date Issued
May 2026
Publisher
Massachusetts Institute of Technology
Abstract
Gradient-based optimization is a scalable method for solving aerodynamic shape optimization problems; however, implementing and verifying correct shape gradients is non-trivial in computational fluid dynamics software. This work presents a method for systematically verifying shape gradients, employing a bottom-up approach to rigorously verify shape derivatives. The approach verifies all intermediate derivatives with finite-difference convergence checks and exact solutions to small discrete systems. The discrete total derivative is verified using continuous shape sensitivities, which are derived for two solutions to the linear advection-diffusion equation. For these cases, the discrete total derivatives are shown to converge asymptotically to continuous sensitivities at the expected discretization order of accuracy. Similarly, for Navier-Stokes (NS) and Reynolds-averaged Navier-Stokes (RANS) solutions around a flat plate and NACA-family airfoils, total shape derivative errors are shown to converge at a second-order rate to their Richardson extrapolate, demonstrating asymptotic consistency.
The NS and RANS shape sensitivities are further verified using finite-difference convergence tests on a given mesh, which show that the linearization of the discretization is correct. In these examples, output-based mesh adaptation is used to obtain asymptotic solutions.
The asymptotically correct derivatives allow for the study of two derivative approximations; namely, the use of an adjoint-inconsistent finite element discretization, and the impact of a non-differentiated distance function in the Spalart-Allmaras turbulence model. For the case of the adjoint-inconsistent scheme, the total shape derivatives do not converge at the rate implied by the discretization when high-order elements are used. The non-differentiated distance function is shown to impact the value of the asymptotic shape derivative on both structured and adapted meshes. However, there is greater difference in the derivative between the complete and non-differentiated distance shape sensitivity when computed on a non-asymptotic (i.e., coarse) adapted mesh. That is, the difference in the complete and approximate derivative is much larger for a coarse mesh on an output-based adapted mesh than on a structured grid.
Using both the verified derivatives and output-based mesh adaptation, two-dimensional shape optimization is performed for RANS flow around a NACA-family airfoil. To minimize computational cost, the optimization is performed on coarse meshes, and mesh adaptation is only triggered when discretization error dominates. However, because a reduction in discretization error can lead to an increase in the discrete objective (which may cause a failure in the optimization), the objective is modified to guarantee that it strictly decreases with mesh adaptation. Finally, the impact of parameters of the adaptation trigger is discussed in terms of their computational efficiency.
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
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