非圧縮粘性流れにおける部分的制御アルゴリズム及び気泡関数要素を用いた最適形状問題

抄録

The main purpose of this paper is to obtain the optimal shape of a body, which minimizes the fluid forces in an incompressible viscous flow. The main idea of the present paper is to add a process of partial control to formulation of the shape optimization and based on the hexahedral bubble element by the stabilized bubble function. The formulation of the shape optimization is based on the optimal control theory. An optimal state is defined by reduction of the fluid forces subjected to the body. the space discretization of the basic equations, the incompressible Navier-Stokes equations by the finite element approximations using the MINI element, in which hexahedral bubble element and bilinear element is employed, the temporal discretization by the fractional step method, calculation of the performance function gradient by the adjoint equation, and the minimization algorithm of the performance function by the weighted gradient method with the partial control algorithm. As a numerical example, a drag force minimization problem of a body in incompressible viscous flows in two and three dimension is carried out. As a result, each optimal shape at Re = 1, Re = 40, Re = 80 and Re = 100 is derived by the partial control algorithm. The present algorithm is shown to be useful for the optimal shape design of the incompressible viscous flow.