Using Black-Box Optimization Techniques to Solve Shape Optimization Problems in Fluid Mechanics

Drumm, Derek

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Using Black-Box Optimization Techniques to Solve Shape Optimization Problems in Fluid Mechanics

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Shape optimization problems are common in engineering applications. These problems arise in fluid dynamics through the need to control internal and external flow properties. We are interested in a problem of passive flow control, where we would like to determine optimal flow obstacles which generate pre-specified Lagrangian Coherent Structures. Performing numerical optimization on this type of problem can be computationally challenging; difficulties arise when numerically approximating gradient information from flow simulation data. These difficulties can be avoided by using derivative free and black-box optimization techniques. Towards the development of numerical solutions to our shape optimization problem of interest, we modify an existing derivative free optimization algorithm, the mesh adaptive direct search, by introducing a means for the algorithm to utilize derivative information. We test this algorithm for robustness on a variety of prototypical optimization problems. We also test this algorithm on a staple optimization problem in fluid mechanics: the minimization of drag around an obstacle in a Stokes flow. The smoothness constraints we enforce upon the obstacles lead to a unique class of Stokes drag minimizers, with which we perform a shape analysis upon. With our modified algorithm sufficiently tested, we analytically describe our shape optimization problem of interest using the variational theory of Lagrangian Coherent Structures. Utilizing techniques from abstract shape optimization, we are able to prove existence of solutions to our problem of interest. We then use our modified algorithm to study numerical solutions to this shape optimization problem in a Navier-Stokes channel flow.

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