粘性不可压缩流体形状优化的快速水平集和自适应方法

项目名称： 粘性不可压缩流体形状优化的快速水平集和自适应方法

项目编号： No.11201153

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 朱升峰

作者单位： 华东师范大学

项目金额： 22万元

中文摘要： 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。

中文关键词： 形状优化；水平集方法；流体；自适应有限元；

英文摘要： 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 e

英文关键词： shape optimization；level set method；fluid；adaptive finite element；