Design space exploration (DSE) is a tool used to evaluate and compare designs as part of the design selection process. While evaluating every possible design in a design space is infeasible, understanding design behavior and response throughout the design space may be accomplished by evaluating a subset of designs and interpolating between them using surrogate models. Surrogate modeling is a technique that uses low-cost calculations to approximate the outcome of more computationally expensive calculations or analyses, such as finite element analysis (FEA). While surrogates make quick predictions, accuracy is not guaranteed and must be considered. This research addressed the need to improve the accuracy of surrogate predictions in order to improve DSE of structural responses. This was accomplished by performing comparative analyses of linear and nonlinear dimensionality-reduction-based radial basis function (RBF) surrogate models for emulating various FEA nodal results. A total of four dimensionality reduction methods were investigated, namely principal component analysis (PCA), kernel principal component analysis (KPCA), isometric feature mapping (ISOMAP), and locally linear embedding (LLE). These methods were used in conjunction with surrogate modeling to predict nodal stresses and coordinates of a compressor blade. The research showed that using an ISOMAP-based dual-RBF surrogate model for predicting nodal stresses decreased the estimated mean error of the surrogate by 35.7% compared to PCA. Using nonlinear dimensionality-reduction-based surrogates did not reduce surrogate error for predicting nodal coordinates. A new metric, the manifold distance ratio (MDR), was introduced to measure the nonlinearity of the data manifolds. When applied to the stress and coordinate data, the stress space was found to be more nonlinear than the coordinate space for this application. The upfront training cost of the nonlinear dimensionality-reduction-based surrogates was larger than that of their linear counterparts but small enough to remain feasible. After training, all the dual-RBF surrogates were capable of making real-time predictions. This same process was repeated for a separate application involving the nodal displacements of mode shapes obtained from a FEA modal analysis. The modal assurance criterion (MAC) calculation was used to compare the predicted mode shapes, as well as their corresponding true mode shapes obtained from FEA, to a set of reference modes. The research showed that two nonlinear techniques, namely LLE and KPCA, resulted in lower surrogate error in the more complex design spaces. Using a RBF kernel, KPCA achieved the largest average reduction in error of 13.57%. The results also showed that surrogate error was greatly affected by mode shape reversal. Four different approaches of identifying reversed mode shapes were explored, all of which resulted in varying amounts of surrogate error. Together, the methods explored in this research were shown to decrease surrogate error when performing DSE of a turbomachine compressor blade. As surrogate accuracy increases, so does the ability to correctly make engineering decisions and judgements throughout the design process. Ultimately, this will help engineers design better turbomachines.



College and Department

Ira A. Fulton College of Engineering and Technology; Mechanical Engineering



Date Submitted


Document Type





design space exploration, surrogate modeling, dimensionality reduction, principal component analysis, kernel principal component analysis, isometric feature mapping, locally linear embedding, finite element analysis, modal analysis, modal assurance criterion, turbomachinery, compressor blades



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