Inexact Adjoint Gradients and Directional Tolerances for Full-Potential Airfoil Optimization
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arXiv
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| Natura: | Preprint |
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2026
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| author | Macedo, Humberto Gimenes Bueno, Luís Felipe |
| author_facet | Macedo, Humberto Gimenes Bueno, Luís Felipe |
| contents | This paper develops a framework connecting discrete adjoint gradient-error analysis with an optimization method that uses directional error tolerances, and applies it to airfoil shape optimization governed by a conservative full-potential flow solver on body-fitted structured meshes. The theoretical part derives the reduced discrete adjoint formula for scalar objectives constrained by a state equation and analyzes how inexact state and adjoint residuals propagate into the reduced gradient. For residuals that are affine in the state variable, the gradient error is bounded by a linear combination of the state and adjoint residual tolerances. On compact sets of decision variables, a uniform version of this bound is obtained, leading to a directional tolerance condition under which the inexact gradient satisfies an exact descent inequality. The resulting inexact general directions method inherits convergence properties under uniformly bounded, diminishing, and Armijo-type step-size rules. The computational part combines a parabolic initial grid generator, an elliptic mesh smoother, and a full-potential discretization with artificial-density stabilization and approximate-factorization iteration. The optimization problem is formulated as a pressure-matching problem in which a class-shape-transformation airfoil parametrization is adjusted so that the computed surface pressure coefficient approaches prescribed reference data, subject to mesh-generation and full-potential residual constraints. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17599 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Inexact Adjoint Gradients and Directional Tolerances for Full-Potential Airfoil Optimization Macedo, Humberto Gimenes Bueno, Luís Felipe Optimization and Control 49M41, 49K20, 35Q35, 65K10, 90C30, 49M05 This paper develops a framework connecting discrete adjoint gradient-error analysis with an optimization method that uses directional error tolerances, and applies it to airfoil shape optimization governed by a conservative full-potential flow solver on body-fitted structured meshes. The theoretical part derives the reduced discrete adjoint formula for scalar objectives constrained by a state equation and analyzes how inexact state and adjoint residuals propagate into the reduced gradient. For residuals that are affine in the state variable, the gradient error is bounded by a linear combination of the state and adjoint residual tolerances. On compact sets of decision variables, a uniform version of this bound is obtained, leading to a directional tolerance condition under which the inexact gradient satisfies an exact descent inequality. The resulting inexact general directions method inherits convergence properties under uniformly bounded, diminishing, and Armijo-type step-size rules. The computational part combines a parabolic initial grid generator, an elliptic mesh smoother, and a full-potential discretization with artificial-density stabilization and approximate-factorization iteration. The optimization problem is formulated as a pressure-matching problem in which a class-shape-transformation airfoil parametrization is adjusted so that the computed surface pressure coefficient approaches prescribed reference data, subject to mesh-generation and full-potential residual constraints. |
| title | Inexact Adjoint Gradients and Directional Tolerances for Full-Potential Airfoil Optimization |
| topic | Optimization and Control 49M41, 49K20, 35Q35, 65K10, 90C30, 49M05 |
| url | https://arxiv.org/abs/2605.17599 |