An Aerodynamic Optimization Method Based on the Inverse Problem Adjoint Equations

Abstract

An adjoint optimization method, based on the solution of an inverse flow problem, is proposed. Given a certain performance functional, it is necessary to find its extremum with respect to a flow variable distribution on the domain boundary, for example, pressure. The adjoint formulation delivers the functional gradient with respect to such a flow variable distribution, and a descent method can be used for optimization. The flow constraints are easily imposed in the parameterization of the distributed control, and therefore those problems with several strict constraints on the flow solution can be solved very efficiently. Conversely, the geometric constraints are imposed either by additional partial differential equations, or by penalization. By adequately constraining the geometric solution, the classical limitations of the inverse problem design can be overcome. Several examples pertaining to internal flows are given.

A pseudo-compressibility method for solving inverse problems based on the 3D incompressible Euler equations

“…Nevertheless, it offers the main advantage of requiring a fewer number of flow-field evaluations to determine the enquired geometry, and also it allows for an easier imposition of the flow constrains (e.g. no flow separation and adverse pressure gradient control [5,6]). Approaches to the inverse problem solution are based on the potential flow theory and conformal mapping techniques, [4,7,8] on stream-function base formulations, [9][10][11] on boundary elements replacing the body surface [12] and on direct design methods.…”

“…Optimization techniques based on the adjoint method can be adopted to drive inverse problems towards the maximization or minimization of target functionals. [6] From this point of view, the numerical solution of an inverse problem becomes attractive, as an alternative route to shape optimization and to automated design.…”

A viscous inverse method for aerodynamic design

“…A typical inverse problem solves for the geometry that realizes a specified wall pressure distribution. By a proper selection of this distribution one can, for instance, control the wall pressure gradient to avoid the flow separation [2], or to obtain shock-free flowfields [3], or to reduce aerodynamic noise [4]. In a multidisciplinary design environment the solution of an inverse problem should be considered an alternative approach of fulfilling various and heterogeneous requirements and constraints dictated by different disciplines.…”

“…In a multidisciplinary design environment the solution of an inverse problem should be considered an alternative approach of fulfilling various and heterogeneous requirements and constraints dictated by different disciplines. Optimization techniques based on the adjoint method have recently been adopted to drive inverse problems towards the maximization or minimization of target functionals [2].…”

“…Initial (0), transient(1),(2) and final (3) shape of the boundary contour, i.e. the plate itself, compared with the exact solution Y = 0.…”