A methodology to reduce dimensionality of a commercial supersonic transport design space using active subspaces

Abstract

As the commercial aviation industry continues to grow, the next technological leap is speed, and commercial supersonic transports are reappearing from multiple companies. Although this problem has been solved before, supersonic design is still difficult as it is highly interdisciplinary, lacks historical data, and requires additional design considerations earlier in the design cycle. Without historical data, higher fidelity analysis is needed early in the design process. The large number of design variables and the need for high fidelity analysis creates large computational costs, limiting design space exploration. To address this, the dimensionality of the design space needs to be reduced without removing the effects from the design variables. A recent technique called Active Subspaces has accomplished this goal by rotating a design space into the most active direction and taking surrogates in this active direction. Through rotation, the effects of each design variable are still present, but less impactful directions can be removed from the surrogate model, reducing dimensionality. This research applies this method to a commercial supersonic design space and asks additional questions about active subspace implementation into a design methodology. These questions address the gradient oversampling needed for good active subspace surrogate fits, if a better active subspace could be found in a partition of the full design space, and how the goodness of an initial surrogate, used to calculate gradients, affects the active subspace surrogate. Finally, the research compares computational cost between a traditional surrogate and an active subspace surrogate. These questions were addressed using aerodynamic data of various aircraft configurations at supersonic cruise conditions. Beginning with a design of experiments of 20 planform variables, the configurations were input into Engineering Sketch Pad to generate the geometry. The geometry was taken into an inviscid computational fluid dynamics (CFD) tool to calculate coefficients of lift and drag at the cruise condition, and these were tabulated. The results were post processed, and a traditional surrogate was created. From this surrogate, gradients were taken to develop active subspace variables. These variables were used to generate a sweep of active subspace surrogates starting from a single variable to a surrogate made from all 20 variables. From these surrogates, it was concluded that oversampling gradients beyond the published range does not decrease error while undersampling increases error at a lower significance than expected. An active subspace in a local partition of a design space initially reduced error, but error reduction decreased as more variables were included in the active subspace surrogate. The number of cases per design variable of an initial surrogate used to calculate gradients was significant. The error of the active subspace surrogate created from these gradients decreased until 50 cases per design variable, when the decrease in error plateaued. Finally, active subspaces saw a large potential to reduce computational time. A small reduction in dimensionality could greatly reduce computational time, especially if gradients are found within a tool. Using these results, a design methodology was presented incorporating active subspaces into the design loop.