CAREER: Optimization and continuation methods in fluid mechanics

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Shape optimization in fluid mechanics is the study of how to design a body, such as a wing or propeller, to maximize lift or thrust while minimizing drag. Tools developed for shape optimization can also be used to dramatically improve the performance of shooting methods for two-point boundary value problems governed by nonlinear partial differential equations. This project concerns the development and implementation of new algorithms for shape optimization of Newtonian and viscoelastic fluids in nearly singular geometries, and for the computation of time-periodic solutions in fluid mechanics. New adjoint-based optimal control methods will be developed to compute gradients of objective functions and constraints governed by partial differential equations featuring singular integrals, non-local operators, or hysteresis. The solutions of these optimization and boundary value problems occur in multi-parameter families that will be studied using numerical continuation algorithms capable of identifying bifurcations and following folds in the manifold of solutions.Shape optimization is fundamental in the design of a wide range of engineering systems such as fuel-efficient vehicles and aircraft or micro-fluidic "lab on a chip" devices used in chemistry and molecular biology. One of the goals of this research is to develop shape optimization techniques for materials that behave partly as viscous fluids and partly as elastic solids. Applications include designing inkjet printers, recycling and manufacturing plastics, and understanding bio-locomotion in invertebrates. Another goal of this research is to develop numerical methods for computing solutions of the equations of fluid mechanics that evolve to an exact copy of their initial state at a later time and then repeat themselves forever. These solutions are helpful in understanding complex phenomena such as ocean waves or the onset of turbulence in a pipe. They should also prove useful in dynamic shape optimization problems, where the goal might be to optimize the average speed over one cycle of a swimming stroke. Broader impacts of this project include course and curriculum development, organization of seminars and minisymposia, community outreach, advising of students and postdocs, and hosting of DOE Computational Science Graduate Fellows who wish to do a summer practicum at Lawrence Berkeley National Laboratory.