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，以最大限度地提高升力或推力，同时最小化阻力。 为形状优化开发的工具也可以用于显着提高由非线性偏微分方程控制的两点边值问题的射击方法的性能。 该项目涉及的发展和实施的新算法的形状优化牛顿和粘弹性流体在近奇异的几何形状，并在流体力学的时间周期的解决方案的计算。 新的基于伴随的最优控制方法将被开发来计算目标函数的梯度和由具有奇异积分、非局部算子或滞后的偏微分方程所支配的约束。 这些优化和边界值问题的解决方案发生在多参数的家庭，将使用能够识别分叉和以下折叠的解决方案的流形的数值延拓算法进行研究。形状优化是在设计的基础上，广泛的工程系统，如燃油效率的车辆和飞机或微流体“芯片实验室”的化学和分子生物学设备。 这项研究的目标之一是开发形状优化技术的材料，部分作为粘性流体和部分作为弹性固体。 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计算流体力学方程的数值方法，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。这些解决方案有助于理解复杂的现象，如海浪或管道中湍流的发生。 它们在动态形状优化问题中也应该被证明是有用的，其中目标可能是优化游泳划水的一个周期的平均速度。 该项目的更广泛的影响包括课程和课程开发，研讨会和minisymopsia的组织，社区推广，学生和博士后的建议，并希望在劳伦斯伯克利国家实验室做暑期实习的DOE计算科学研究生研究员的托管。

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.