Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 209780, 22 pages
Research Article

Shape Sensitivity Analysis in Flow Models Using a Finite-Difference Approach

1Interdisciplinary Center for Applied Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
2Industrial Materials Institute, National Research Council, Boucherville, Qc, J4B6Y4, Canada

Received 8 September 2009; Accepted 25 November 2009

Academic Editor: José Balthazar

Copyright © 2010 Imran Akhtar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Reduced-order models have a number of practical engineering applications for unsteady flows that require either low-dimensional approximations for analysis and control or repeated simulation over a range of parameter values. The standard method for building reduced-order models uses the proper orthogonal decomposition (POD) and Galerkin projection. However, this standard method may be inaccurate when used “off-design” (at parameter values not used to generate the POD). This phenomena is exaggerated when parameter values describe the shape of the flow domain since slight changes in shape can have a significant influence on the flow field. In this paper, we investigate the use of POD sensitivity vectors to improve the accuracy and dynamical system properties of the reduced-order models to problems with shape parameters. To carry out this study, we consider flows past an elliptic cylinder with varying thickness ratios. Shape sensitivities (derivatives of flow variables with respect to thickness ratio) computed by finite-difference approximations are used to compute the POD sensitivity vectors. Numerical studies test the accuracy of the new bases to represent flow solutions over a range of parameter values.