T. Bui-Thanh, M. Damodaran. Singapore-Massachusetts Institute of Technology Alliance (SMA). School of Mechanical and Production Engineering. Nanyang ...
Inverse Airfoil Design Using Proper Orthogonal Decomposition on Computed Flowfield Snapshots T. Bui-Thanh, M. Damodaran Singapore-Massachusetts Institute of Technology Alliance (SMA) School of Mechanical and Production Engineering Nanyang Technological University, Nanyang Avenue, Singapore 639798 K. Willcox Aerospace Computational Design Laboratory Massachusetts Institute of Technology, Cambridge, MA 02139
The inverse design of a Korn airfoil whose pressure distribution is specified as the design target is used to illustrate this approach. Thirty one Hicks-Henne bump functions are added to each of the upper and lower surfaces of the baseline RAE 2822 airfoil to parameterize the airfoil shapes. The flow solutions for arbitrary airfoils inclined at zero angle of attack and a freestream Mach number of 0.5 are computed using a compressible flow solver solving the Euler equations to construct the collection of snapshots. Figures 1 and 2 compare the exact Korn airfoil shape and the target pressure to the POD design results using one and 29 POD modes, respectively. It can be seen that as the number of modes is increased, the predicted shape and its pressure distribution agree more closely with the exact solutions. Using 29 POD modes, which accounts for 99.97% of the snapshot energy, the design airfoil matches the exact one almost perfectly. A new extension of the POD has been proposed for inverse design of airfoil shapes. Given a database of airfoil shapes and pressure distributions, it has been shown that the gappy POD approach can be used to
Inverse design with 1 POD eigenfunctions Design Exact
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Fig. 1 Top: The exact Korn airfoil (solid) and the design airfoil (dash) with one POD mode; Bottom: The corresponding pressure coefficients. Inverse design with 29 POD eigenfunctions Design Exact
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First a collection of flowfield snapshots (in practice, these could also be obtained from computational simulations, experimental results, tabulated data, or a combination thereof) is constructed by choosing a set of arbitrary airfoil shapes and obtaining their corresponding surface pressure distributions using compressible flow solvers. In order to solve the inverse airfoil design problem using the gappy POD method, these snapshots are redefined to contain not only the flowfield variables but also the corresponding airfoil coordinates. The goal, then, is to use the gappy POD method to determine the optimal airfoil shape that produces a given target airfoil surface pressure distribution, which is not contained in the snapshot collection.
design an airfoil to match a specified target airfoil surface pressure distibution. It has also been shown in1 that a systematic restart procedure can be used to obtain accurate results even when the target airfoil shape is different from those contained within the original snapshot database.
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The gappy proper orthogonal decomposition (POD) proposed and explained in detail in Bui-Thanh et al.1 which allows one to cast an subsonic and transonic inverse airfoil design task as a gappy data problem simply and efficiently is outlined here.
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Fig. 2 Top: The exact Korn airfoil (solid) and the design airfoil (dash) with 29 POD modes; Bottom: The corresponding pressure coefficients.
References 1 Bui-Thanh,
T. and Damodaran, M. and Willcox, K., “Proper Orthogonal Decomposition Extensions for Parametric Applications in Transonic Aerodynamics”, AIAA Paper 20034213, 21th AIAA Applied Aerodynamics Conference, Orlando Florida”, 2003.
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