提出了一种基于水平集的自适应网格方法,并将其应用于求解由 Stokes 方程控制的不可压缩流体阻力最小问题。推导出了目标泛函的形状灵敏度分析。在优化过程中,采用在整个计算区域上定义的用于演化水平集函数的均匀粗网格和细网格。均匀的细网格是以水平集函数作为细化指标,由含界面的粗网格进一步划分得到,从而使得计算主要集中在界面附近。因此,与为实现相同数值精度把整个计算区域均匀细分的网格相比,该方法计算成本大大降低, 特别是边界上的形状导数值可以隐式求得,这在经典的形状优化设计问题中是一项非常困难的任务。

We present a level-set-based adaptive mesh method for solving the drag minimization problem of incompressible flow governed by the Stokes equations. A shape sensitivity analysis of the cost functional is presented. Two levels of meshes are employed during the optimization. A uniform coarse mesh for evolving the level set function is defined over the entire computational domain. Additionally, the level set function serves as a refinement indicator. The coarse mesh comprising the interfaces is then further divided into a uniform fine mesh. The computation is performed mainly near the interfaces. Therefore, the computational cost is significantly reduced compared with the uniform refined mesh over the whole domain that achieves the same resolution. Furthermore, the shape derivative on the boundary can be obtained implicitly, which is a very challenging task in classical optimal shape design problems.