Adjoint CFD optimisation of fan/pylon interaction

David Radford

Inclusion of automatic optimisation techniques in the design process of engineering products can improve their quality while also freeing designers' time. However, when the performance of the design is assessed using high-fidelity modelling the overall commitment of computational resources needed to find an optimum in an acceptable timescale is often judged to be unacceptable.

For the common class of gradient-based optimisation methods, the main computational cost is that associated with calculating the derivatives of the objective function, the measure of design performance, with respect to each of the design variables being used. The simplest means of calculating this data is by finite differences between a base design and a series of perturbed geometries, but this needs at least one computation of the objective function for each design parameter. Recently the solution of"sensitivity equations" has gained ground. In this case a solver based on the original analysis code, but effectively differentiated with respect to the design geometry, is used, but again requires a separate solution for each design parameter.

These contour plots show the pressure field for the 10th Standard configuration compressor aerofoil, and a geometry derived by optimisation using the adjoint version of the Hydra CFD code. The objective function to be minimised was the total pressure loss through the cascade, which was reduced to negligible levels by reshaping the blade to eliminate the shock wave, over the course of 10 design iterations. Constraints were placed on the variation in the flow turning from the original design, and mass flow.

Adjoint methods provide an alternative, more efficient method. Rather than starting from the change in geometry produced by a design change and solving for the affect this has on the objective function, adjoint solvers start with the objective function itself and produce a set of "adjoint variables" representing its sensitivity to design inputs or intermediate quantities. This solution, of similar computational cost to the original analysis, is only required once for each objective function. In CFD, the adjoint variables commonly give the sensitivity of the objective function to the flux residuals at each point in the flowfield. Each design parameter is then represented by perturbing the boundaries of the domain according to its effect on the geometry, and calculating the first-order changes this causes in the flow residuals, which is a cheap operation equivalent to one iteration of the flow solver. By combining the set of residual perturbations representing the design parameters with the adjoint variables, the sensitivities of the objective function to the design parameters can be calculated, much more quickly than is possible for finite-difference based methods. As a result, it becomes practical to use hi-fidelity modelling as part of the design optimisation process.

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