Multivariable Look-up Table Construction for Aerodynamic Prediction Discrepancy Model Using Nested Grid Optimization of Linear and Nonlinear Methods with Non-smoothness Penalty

A technique was developed for multivariable look-up table construction to make aerodynamic prediction discrepancy model, which represents differences between predicted and measured aerodynamic parameter, by using a nested process of linear method such as least squares and gradient-based nonlinear optimization for coefficients and grid points of the look-up table, respectively. Test data covered only part of the multi-dimensional variable space and were distributed unevenly across the grid cells, so the optimization process had high probability of being ill-posed problem. To cope with this problem, non-smoothness penalty methods were introduced to consider variance error as well as bias error, one was ridge regression technique combined with the linear method and the other was curvature-based non-smoothness penalty involved into the loss function for the nonlinear grid optimization. First, a parameter study was conducted with the coefficient optimization by the linear method with a fixed and equally spaced grid to investigate effects of a weighting factor for the ridge regression and to acquire Pareto front for $${R}^{2}$$ and smoothness of optimized solutions for the various weighting factor values. Second, based on this parameter study results, a decision could be made about another weighting factor for the non-smoothness penalty term in the grid optimization. This two-step approach for the grid and coefficient optimization of multivariable look-up table was proved to work effectively and efficiently for the metamodel construction by appropriate trade-off between the bias and variance errors through a demonstration using real flight test data.