CAREER: Optimization and continuation methods in fluid mechanics

流体力学中的形状优化是研究如何设计一个物体，如机翼或螺旋桨，使升力或推力最大化，同时阻力最小。为形状优化开发的工具也可以用来显著提高由非线性偏微分方程组支配的两点边值问题的打靶法的性能。本项目涉及牛顿流体和粘弹性流体在几乎奇异几何结构中形状优化的新算法的开发和实现，以及流体力学中时间周期解的计算。新的基于伴随的最优控制方法将被开发来计算由具有奇异积分、非局部算子或滞后的偏微分方程组控制的目标函数和约束的梯度。这些优化和边值问题的解出现在多参数家族中，将使用能够识别分叉和跟随解流形中折叠的数值连续算法进行研究。形状优化是广泛的工程系统设计的基础，例如节能型车辆和飞机，或用于化学和分子生物学的微流控“芯片实验室”设备。这项研究的目标之一是为部分表现为粘性流体、部分表现为弹性固体的材料开发形状优化技术。应用包括设计喷墨打印机、回收和制造塑料，以及了解无脊椎动物的生物运动。这项研究的另一个目标是开发数值方法来计算流体力学方程的解，这些方程在稍后的时间演化到其初始状态的精确副本，然后永远重复。这些解有助于理解复杂的现象，如海浪或管道中的湍流开始。它们在动态形状优化问题中也应该被证明是有用的，目标可能是优化游泳一个周期的平均速度。该项目的更广泛影响包括课程和课程开发，组织研讨会和小型研讨会，社区推广，为学生和博士后提供建议，以及接待希望在劳伦斯伯克利国家实验室进行暑期实习的美国能源部计算科学研究生。

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.