An Adjoint Method for a High-Order Discretization of Deforming Domain Conservation Laws for Optimization of Flow Problems

TL;DR

This paper develops a high-order adjoint method for discretized conservation laws on deforming domains, enabling efficient gradient computations for flow optimization problems involving complex domain transformations.

Contribution

It introduces a fully discrete adjoint approach for high-order discretizations on deforming domains, combining ALE transformations with discontinuous Galerkin and Runge-Kutta schemes for accurate gradient calculations.

Findings

Reduced energy consumption in airfoil trajectory optimization by nearly an order of magnitude.

Achieved high-precision satisfaction of impulse constraints in flapping motion optimization.

Demonstrated the effectiveness of the adjoint method in complex, deforming domain flow problems.

Abstract

The fully discrete adjoint equations and the corresponding adjoint method are derived for a globally high- order accurate discretization of conservation laws on parametrized, deforming domains. The conservation law on the deforming domain is transformed into one on a fixed reference domain by the introduction of a time-dependent mapping that encapsulates the domain deformation and parametrization, resulting in an Arbitrary Lagrangian-Eulerian form of the governing equations. A high-order discontinuous Galerkin method is used to discretize the transformed equation in space and a high-order diagonally implicit Runge- Kutta scheme is used for the temporal discretization. Quantities of interest that take the form of space-time integrals are discretized in a solver-consistent manner. The corresponding fully discrete adjoint method is used to compute exact gradients of quantities of interest along the manifold of solutions of the fully discrete conservation law. The adjoint method is used to solve two optimal shape and control problems governed by the isentropic, compressible Navier-Stokes equations. The first optimization problem seeks the energetically optimal trajectory of a 2D airfoil given a required initial and final spatial position. The optimization solver, driven by gradients computed via the adjoint method, reduced the total energy required to complete the specified mission nearly an order of magnitude. The second optimization problem seeks the energetically optimal flapping motion and time-morphed geometry of a 2D airfoil given an equality constraint on the x-directed impulse generated on the airfoil. The optimization solver satisfied the impulse constraint to greater than 8 digits of accuracy and reduced the required energy between a factor of 2 and 10, depending on the value of the impulse constraint, as compared to the nominal configuration.