粘性不可压缩流体形状优化的快速水平集和自适应方法

Researches on shape optimization of fluids have significance for natural sciences and engineering. This project is concerned with algorithm analysis and numerical simulation for shape optimization models in viscous imcompressible Newtonian and non-Newtonian fluids. We consider shape optimization problems for stationary and unstationary Navier-Stokes equations as states and the enery dissipation/tracking type of velocity as cost functionals. .Most existing level set algorithms are gradient type. Based on the improved level set approaches, shape Hessian and multigrids, we plan to design fast second-order algorithms of Newton type, present theoretical anlysis and show numerical examples in two and three dimensions. .Moreover, the analysis and application of adaptive methods for shape optimization of fluids are novel, interesting and challenging attempts. Under the framework of gradient/Newton methods, we will construct　theoretically the dynamic a posteriori error estimators to locate the local domains where the fluid velocity influences the cost functional largely and optimize the shapes of these places. Meanwhile, we will seek a rule to decide whether the singular points appearing during the optimization process are real for the optimal shape or artificial．Finally, we will provide numerical examples to show the effectiveness of our adaptive algorithms.

流体形状优化的研究对科学和工程都有重要意义。本项目拟研究粘性不可压缩牛顿和非牛顿流体形状优化的快速水平集和自适应方法。我们考虑以定常和非定常Navier-Stokes方程为控制方程、以能量耗散型/速度追踪型为目标泛函的形状优化问题。.目前流体形状优化中已有的水平集算法大多是梯度型算法，牛顿型方法的研究还很少。我们基于改进的水平集方法、形状Hessian和多重网格，将设计二阶牛顿型快速算法，给出理论分析并计算二维、三维例子。.另外，自适应方法在流体形状优化中的分析与应用是新颖、有趣和具有挑战性的尝试。我们将在梯度法/牛顿法的框架下，理论上构造动态的后验误差估计指示子对目标泛函影响大的局部区域定位，并对这些地方进行优先优化形状。同时我们将找到准则来判断优化过程中产生的奇点是最优形状固有的还是虚假的。最后给出数值例子表明自适应算法的有效性。

本课题主要研究了流体形状优化的数值求解方法，包括水平集方法和自适应有限元方法。基于改进的水平集方法和形状导数计算式，我们为进一步推进形状优化的基础理论研究提供了新的数值方法。我们在梯度法框架下，采用自适应有限元方法计算状态方程和伴随状态方程。理论上构造动态的后验误差估计指示子，用于形状优化的梯度算法中。本课题的主要创新之处在于把水平集方法、自适应有限元方法与形状优化理论很好地结合起来，为进一步推进形状优化的基础理论研究提供了新的近似计算方法，也为其它形状优化问题的研究提供了新思路。