Inexact Adjoint Gradients and Directional Tolerances for Full-Potential Airfoil Optimization

TL;DR

This paper introduces a framework for inexact adjoint gradient analysis with directional error tolerances, applied to full-potential airfoil optimization, ensuring convergence despite residual errors.

Contribution

It develops a theoretical connection between gradient error bounds and directional tolerances, enabling reliable inexact gradient-based optimization for airfoil shape design.

Findings

Gradient error bounds are linear in residual tolerances.

Directional tolerance conditions ensure descent and convergence.

The method successfully optimizes airfoils with residual constraints.

Abstract

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.