求解Stokes方程形状优化问题的无网格方法  

Moving Mesh Method and Its Applications in Solving Optimal Control Problems of Navier-Stokes Equation

作　　者：段献葆[1] 党妍 秦玲 DUAN Xian-bao;DANG Yan;QIN Ling(School of Sciences,Xi'an University of Technology,Xi'an 710048,China)

机构地区：[1]西安理工大学理学院,陕西西安710048

出　　处：《数学的实践与认识》2020年第20期250-256,共7页Mathematics in Practice and Theory

基　　金：国家自然科学基金(11601410,11971377);陕西省自然科学基金(2017GY-090,2019JM-284)。

摘　　要：研究流体流动的最优形状设计问题,以流体力学中典型的Stokes方程为状态方程,目标泛函使得总势能达到最小.将水平集方法与优化问题的灵敏度分析结果相结合以保证精确捕获优化形状的界面.用径向基函数来求解Stokes方程和演化水平集函数的Hamilton-Jacobi方程.由于径向基函数方法是真正无网格的,因此与网格依赖的数值求解方法相比,所提算法具有更好的适应性也更容易实施.数值算例表明,所提算法可以高效、稳定地求解流体力学中的形状优化问题.In this work,we focus on the optimal shape design of fluid flow,with Stokes problem as the state equation and the objective is to minimize total potential energy of the fluid.In order to capture the interface,the level set method was integrated with sensitivity analysis results of the optimization problem.The radial basis function was introduced to solve the Stokes equations and the Hamilton-Jacobi equation that evolves the level set function.Since the radial basis function method is truly mesh free,the proposed algorithm has better adaptability than the mesh-dependent technique.The numerical example provided shows that our algorithm is efficient and stable.

关 键 词：最优形状设计 STOKES方程 无网格方法 径向基函数 

分 类 号：O35[理学—流体力学] O175[理学—力学]