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A stabilized finite element method for shape optimization in low Reynolds number flows
Author(s) -
Srinath D. N.,
Mittal S.
Publication year - 2007
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1432
Subject(s) - finite element method , parametrization (atmospheric modeling) , mathematics , reynolds number , shape optimization , mathematical optimization , finite difference , optimal design , inverse , finite volume method , flow (mathematics) , function (biology) , mathematical analysis , geometry , mechanics , physics , engineering , structural engineering , turbulence , statistics , quantum mechanics , radiative transfer , evolutionary biology , biology
A gradient‐based optimization procedure based on a continuous adjoint approach is formulated and implemented for steady low Reynolds number flows. A stabilized finite element formulation is proposed to solve the adjoint equations. The accuracy of the gradients from the adjoint approach is verified against the ones computed from a simple finite difference procedure. The validation of the formulation and its implementation is carried out via flow past an elliptical bump whose eccentricity is used as a design parameter. Shape design studies for the elliptical bump are then carried on with a more complex 4th order Bézier parametrization of the bump. Results for, both, optimal design and inverse problems are presented. Using different initial guesses, multiple optimal shapes are obtained. A multi‐objective function with additional constraints on the volume and the drag coefficient of the bump is utilized. It is seen that as more constraints are added to the objective function the design space is constrained and the multiple optimal shapes become progressively similar to each other. The study demonstrates the usefulness of this tool in obtaining multiple engineering solutions to a given design problem and also providing a framework to impose multiple constraints simultaneously. Copyright © 2007 John Wiley & Sons, Ltd.