Techniques for improving the accuracy of the globalapproximations used in various Multidisciplinary DesignOptimization (MDO) procedures, while reducing the amount of designspace information required to develop the approximations, werestudied in this research. These improvements can be achieved by incorporating gradient or sensitivity information into the existing approximation techniques. An approach to develop response surface approximations based upon artificial neural networks trained using both state and sensitivity information is developed. Compared to previous approaches, this approach does not require weighting the residuals of the targets and gradients and is able to approximate gradient-consistent response surfaces with a relatively compact network architecture. Numerical simulation on selected problems shows that this approach possesses the capability to develop improved response surface approximations compared to the non-gradient neural network training approach.
One issue that this approach cannot address properly, however, is to determinethe step size for the design variables in the Taylor Seriesexpansion that is used to utilize sensitivity-based, approximateinformation. It is also a common challenge associated with a particulargradient-enhanced approach, Database Augmentation. This research develops another gradient-enhanced approach based on Kriging models to solve the problem by including the step size as one of model parameters. This approach can also characterize the uncertainty of approximations, which is another goal of this research. Based on Database Augmentation, the approach develops Krigingmodels by minimizing the Integrated Mean Squared Error (IMSE)criterion instead of the Maximum Likelihood Estimation (MLE) process often used.
Numerical simulation on selected, small-scale problemsshows that this IMSE-based gradient-enhanced Kriging (IMSE-GEK) approach can improve approximation accuracy by 60~80% over the non-gradient Kriging approximation. An analytical approach to compute IMSE was developed to reduce the prohibitive computing cost associated with applying the IMSE-GEK approach to high-dimensional problems. Some additionalimplementation issues associated with the approach, such as thedatabase augmenting scheme, the use of variable step sizes andthe inclusion of nugget effects at added points, are also presented.