Three-Dimensional Large-Scale Aerodynamic Shape Optimization Based on Shape Calculus

Abstract

Large scale three dimensional aerodynamic shape optimization based on the compressible Euler equations is considered. Shape calculus is used to derive an exact surface formulation of the gradients, enabling the computation of shape gradient information for each surface mesh node without having to calculate further mesh sensitivities. Special attention is paid to the applicability to large scale three dimensional problems like the optimization of an Onera M6 wing or a complete blended wing-body aircraft. The actual optimization is conducted in a one-shot fashion where the tangential Laplace-operator is used as a Hessian approximation, thereby also preserving the regularity of the shape.

Euler-Equation-Based Drag Minimization of Unconventional Aircraft Configurations

“…Applications of CFD-based ASO to wing design have, up until recently, largely focused on the Euler equations [21][22][23][24][25][26][27][28]. Contrary to circulation-distribution and panel methods, the Euler equations do not require the user to prescribe the starting location and shape of the wake.…”

Efficient shape optimization for certain and uncertain aerodynamic design

“…the compressible Navier-Stokes equations without turbulence modeling, is straight forward [10]. However, the popular Reynolds Averaged Navier-Stokes Equations (RANS) with turbulence modeling pose some difficulties for the shape calculus approach, as most turbulence models have elements that make the derivation of the analytic adjoint equations needed for shape calculus non-standard.…”

“…The idea presented here from [9,10] is to study the symbol of the control to state mapping of the dynamic part only. Considering a sinusoidal perturbationqðxÞ ¼qe ixx of some control q, the pseudo-differential operator nature of the Hessian H can be seen by comparing the inputq with the output Hq.…”

Airfoil design for compressible inviscid flow based on shape calculus

“…For the inviscid compressible case considered here, such an analysis is conducted in Arian and Ta'asan (1996), Arian and Vatsa (1998), showing the Hessian is a second order differential operator. For the incompressible viscous case (Schmidt 2010;Schmidt and Schulz 2009) the Hessian is a true pseudodifferential operator of order 1. Thus, we use the following reduced Hessian approximation…”