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

本文对氢燃料电池(HFC)和人工心脏(AH)进行了形状优化分析和计算。这个项目是我在渥太华大学建立形状分析和高性能最优形状计算中心的努力的一部分。对于HFC应用，我的最终目标是通过使用形状优化技术来提高HFC的效率。也就是说，我打算通过对气体扩散层(GDL)中的纤维形状、通道/GDL界面的孔形状和HFC通道几何形状的形状优化来改善水和气体的传输。对于AH应用，目标是找到最佳的AH形状，即最大限度地减少血液剪应力、漩涡和滞留的形状-这些都是致命的主要因素。从数学的角度来看，这些应用的动力学由一组非线性偏微分方程组(PDE)控制，包括Navier-Stokes方程和热和质量守恒定律(在AH的情况下加上一个微分方程)。最终目标是解决一个形状优化问题，即找到一个区域D*最小化一个形状泛函D->J(D)，其中D是一个可允许的形状，J(D)是一个度量期望和不期望效果作用的泛函。我打算研究这些偏微分方程解的存在性，分析它们的形状可微性，并寻找相关的形状最小化问题的解(分析和计算)。此外，我打算进行2D和3D形状计算。据我所知，这项建议是使用形状优化技术控制HFC动力学的首次尝试。对于HFC的应用，虽然在工程文献中已经有了一些结果，但该方案是第一个从严格的数学和数值形状优化的角度来考虑该问题的方案。由于我们的偏微分方程组的特殊性质(强非线性)，我们期望找到与其解和相关的形状优化问题相关的原始结果。

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.