Gappy proper orthogonal decomposition for flow reconstruction, flow sensing, sensor placement and structural assessment

Gappy POD for flow problems, e.g. airfoil surfaces

Relevant publications

Mainini, L. and Willcox, K. A surrogate modeling approach to support real-time structural assessment and decision-making, AIAA Journal, Vol. 53, No. 6, pp. 1612-1626, 2015.

Willcox, K., Unsteady Flow Sensing and Estimation via the Gappy Proper Orthogonal Decomposition, Computers & Fluids, Volume 35, Issue 2, February 2006, pp. 208-226.

Bui-Thanh, T., Damodaran, M. and Willcox, K., Aerodynamic Data Reconstruction and Inverse Design using Proper Orthogonal Decomposition, AIAA Journal, Vol. 42, No. 8, August 2004, pp. 1505-16.

Bui-Thanh, T., Damodaran, M. and Willcox, K., Proper Orthogonal Decomposition Extensions for Parametric Applications in Transonic Aerodynamics, AIAA Paper 2003-4213, presented at 15th Computational Fluid Dynamics Conference, Orlando, FL, June 2003.

Bui-Thanh, T., Proper Orthogonal Decomposition Extensions and their Applications in Steady Aerodynamics, Masters Thesis, Singapore-MIT Alliance, June 2003.

The gappy POD is a is a modification of the standard POD method that handles incomplete or “gappy” data sets. Gappy POD was developed by Everson and Sirovich in the context of reconstruction of images, such as human faces, from partial data. Our work has demonstrated the applicability of gappy POD in reconstructing fields in fluid and structural applications where distributed sensors provide only partial field measurements.

The gappy POD is relevant for flow problems where incomplete data is available. For example, in experiments, data may only be available on the airfoil surface or PIV may give an incomplete estimation of the flow field. Our research was the first to apply gappy POD to fluid flow applications. We demonstrated that gappy POD is an effective method to reconstruct both steady and unsteady flowfield data from limited surface pressure measurements.

Our research has also shown that the gappy POD is a powerful technique in the data-to-decisions flow. In the case of real-time structural assessment, we use the gappy POD to rapidly estimate the modal content of a strain field over a wing panel. The gappy POD estimates in turn drive the estimation of failure indices, which are used to support real-time decision-making to respond to wing structural damage.

Abstracts


Unsteady Flow Sensing and Estimation via the Gappy Proper Orthogonal Decomposition

Willcox, K. Computers & Fluids, Volume 35, Issue 2, February 2006, pp. 208-226.

The proper orthogonal decomposition (POD) has been widely used in fluid dynamic applications for extracting dominant flow features. The ‘‘gappy’’ POD is an extension to this method that allows the consideration of incomplete data sets. In this paper, the gappy POD is extended to handle unsteady flow reconstruction problems, such as those encountered when limited flow measurement data is available. In addition, a systematic approach for effective sensor placement is formulated within the gappy framework using a condition number criterion. This criterion allows for accurate flow reconstruction results and yields sensor configurations that are robust to sensor noise. Two applications are considered. The first aims to reconstruct the unsteady flow field using a small number of surface pressure measurements for a subsonic airfoil undergoing plunging motion. The second considers estimation of POD modal content of a cylinder wake flow for active control purposes. In both cases, using the dominant POD basis vectors and a small number of sensor signals, the gappy approach is found to yield accurate flow reconstruction results.

Aerodynamic Data Reconstruction and Inverse Design using Proper Orthogonal Decomposition

Bui-Thanh, T., Damodaran, M. and Willcox, K. AIAA Journal, Vol. 42, No. 8, August 2004, pp. 1505-16.

The proper orthogonal decomposition (POD) has been widely used in fluid dynamic applications for extracting dominant flow features. The ‘‘gappy’’ POD is an extension to this method that allows the consideration of incomplete data sets. In this paper, the gappy POD is extended to handle unsteady flow reconstruction problems, such as those encountered when limited flow measurement data is available. In addition, a systematic approach for effective sensor placement is formulated within the gappy framework using a condition number criterion. This criterion allows for accurate flow reconstruction results and yields sensor configurations that are robust to sensor noise. Two applications are considered. The first aims to reconstruct the unsteady flow field using a small number of surface pressure measurements for a subsonic airfoil undergoing plunging motion. The second considers estimation of POD modal content of a cylinder wake flow for active control purposes. In both cases, using the dominant POD basis vectors and a small number of sensor signals, the gappy approach is found to yield accurate flow reconstruction results.

Proper Orthogonal Decomposition Extensions for Parametric Applications in Transonic Aerodynamics

Bui-Thanh, T., Damodaran, M. and Willcox, K. AIAA Paper 2003-4213, presented at 15th Computational Fluid Dynamics Conference, Orlando, FL, June 2003.

Two extensions to the proper orthogonal decomposition (POD) technique are considered for steady aerodynamic applications. The first is to couple the POD approach with a cubic spline interpolation procedure in order to develop fast, low-order models that accurately capture the variation in parameters, such as the angle of attack or inflow Mach number. The second extension is a ”gappy” POD technique for the reconstruction of incomplete or inaccurate aerodynamic data. Gappy POD is shown to be an effective technique for reconstruction of full flow field data from limited surface measurements, and thus provides an effective way to combine experimental and computational data. A modification of the gappy POD is also shown to provide a simple, effective method for airfoil inverse design.

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