Aerodynamic Shape Optimization Using the Adjoint-Based Truncated Newton Method

This paper presents the development and application of the truncated Newton (TN) method in aerodynamic shape optimization problems. The development is made for problems governed by the laminar flow equations of incompressible fluids. The method was developed in OpenFOAM\(^{\copyright }\) with the aim to stress its advantages over standard gradient-based optimization algorithms. The Newton equations are solved using the Conjugate Gradient (CG) method which requires the computation of the product of the Hessian of the objective function and a vector, escaping thus the need for computing the Hessian itself. The latter has a computational cost that scales with the number of design variables and becomes unaffordable in large-scale problems with many design variables. A combination of the continuous adjoint method and direct differentiation is used to compute all Hessian-vector products. A grid displacement PDE (Laplace equation) is also used to compute the necessary derivatives of grid displacements w.r.t. the design variables. The programmed method is used to optimize the sidewall shapes of 2D ducts for minimum total pressure losses.