Effect of Approximations of the Discrete Adjoint on Gradient-Based Optimization

Abstract

An exact discrete adjoint of an unstructured finite-volume solver for the RANS equations has been developed. The adjoint is exact in the sense of being based on the full linearization of all terms in the solver, including all turbulence model contributions. From this starting point various approximations to the adjoint are derived with the intention of simplifying the development and memory requirements of the method; considered are many approximations already seen in the literature. The effect of these approximations on the accuracy of the resulting design gradients, and the convergence and final solution of optimizations is studied, as it applies to a two-dimensional high-lift configuration.

Heuristic a posteriori estimation of error due to dissipation in finite volume schemes and application to mesh adaptation

“…Implementation details are to be found in [37,23]. Consider the Lagrangian: L(w, K, ψ) = J(w) + ψ T R(w, K), which always takes the value J provided the state equation R(w, K) = 0 is fulfilled.…”

Applications of a discrete viscous adjoint method for aerodynamic shape optimisation of 3D configurations

“…In fact, the gradients are more accurate than in the previous case since the design variables control parts where no shock occurs. It was indeed observed on 2D cases [17] that the adjoint computations based on frozen turbulence model gives gradients of good quality on flow without shock. Figure 9 shows a comparison of the region of corner separation before and after optimisation of fuselage shape.…”

“…A wide range of the spatial discretizations available in TAU have been differentiated, including the Spalart-AllmarasEdwards one-equation turbulence model. The effect of various approximations of the Jacobian was investigated and their impacts on the optimisation were analysed on several 2D optimisation problems [16,17]. It has been seen that approximating the adjoint by freezing turbulent quantities leads to gradients that are accurate in low-speed case, but exceptionally poor on configurations involving a strong shock.…”

CAD Integrated Multipoint Adjoint-Based Optimization of a Turbocharger Radial Turbine

“…To further reduce the development cost, the constant eddy viscosity (CEV) assumption is made, that is, the eddy viscosity is independent from changes of the design. The impact of this approximation was analyzed by [55,56] and is assessed in the course of this section for the present application.…”