Gradient Enhanced Kriging Using Modal Sensitivity Approximations in a Reduced Basis Space for As-Manufactured Airfoil Analysis

Abstract

This work develops a new process to efficiently predict the effect of as-manufactured geometry variations on airfoil modal response. A gradient enhanced kriging approach is formulated that uses two sets of results at the training model sites in order to reduce the total number of models required. The first set of results are the frequency or mode shape values and the second are their respective gradients. Computational efficiency is achieved through analytical calculations of eigenvalue and eigenvector sensitivities. This approach avoids the need to recompute full eigensolutions for finite difference sensitivity approximation. Efficiency in mode shape approximation is further enabled through transformation of displacement variations into a principal component reduced-basis space. This allows prediction of the full mode shape variation without emulating each degree of freedom independently. In order to perform gradient enhanced kriging with this space, a novel approach is developed to calculate reduced basis space gradients. The process projects the modal displacement principal component vectors onto first-order Taylor series expansions of the physical mode shape variation at each training site. The resulting perturbed principal component coordinates are used in finite difference estimates of sensitivity that make gradient enhanced kriging in this space possible. This new process is demonstrated on three cases, and it is shown that the new approach provides significant improvements in accuracy and efficiency compared to traditional kriging methods.