Shape optimization problems - analysis and computations. application to hydrogen fuel cells and artificial hearts

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This research proposal deals with the optimal shape analysis and computations applied to hydrogen fuel cell (HFC) and artificial hearts (AH). This project is part of my efforts to build at University of Ottawa a center of shape analysis and high performance optimal shape computations.    For HFC application my final objective is to enhance HFC efficiency by using shape optimization techniques. Namely, I intend to improve water and gas transport by shape optimization of fiber shapes in gas diffusive layers (GDL), pore shape of channel/GDL interface and the HFC channel geometry.    For AH application the goal is to find the best AH shape, i.e. the shape minimizing the blood shear stress, vortices and stagnation - which are major factors of fatalities.    From mathematical standpoint, the dynamics of these applications are governed by a system of nonlinear partial differential equations (PDE), including Navier-Stokes equations and heat and mass conservation laws (coupled with a differential equation in the case of AH). The final objective is to solve a shape optimization problem, i.e. to find a domain D* minimizing a shape functional D->J(D), where D is an admissible shape and J(D) is a functional measuring the action of desirable and undesirable effects.    I intend to study the existence of solutions to these PDEs, to analyse their shape differentiability and to look for the solution of the associated shape minimization problem (analysis and computation). Moreover, I intend to carry 2D and 3D shape computations.    To my knowledge, this proposal represents a first attempt to control HFC dynamics using shape optimization techniques. For the HFC application, though there exist some results in Engineering literature, this proposal is a first one to consider the problem from a rigorous mathematical and numerical shape optimization viewpoint. Due to the specific nature of our system of PDEs (strongly nonlinear) we expect to find original results related to their solution and the associated shape optimization problems.