Shape optimization of a body located in incompressible flow using adjoint method and partial control algorithm

Abstract

The purpose of this study is to obtain an optimal shape of a body located in an incompressible viscous flow. The optimal shape of the body is defined so as to minimize the fluid forces acting on it by determining the surface coordinates based on the finite element method and the optimal control theory. The performance function, which is used to judge the optimality of a shape, is defined as the square sum of the drag and lift forces. The minimization problem is solved using an adjoint equation method. The gradient in the adjoint equation is affected by the finite element configuration. The use of a finite element mesh whose shape is appropriate for the procedure is important in shape optimization. If the finite element mesh used is not suitable for computations, the exact gradient is not calculated. Therefore, a structured mesh is used for the adjacent area of the body and all finite element meshes are refined using the Delaunay triangulation at each iteration computation. The weighted gradient method is applied as the minimization technique. Using an algorithm in which all nodal coordinates on the surface of the body are employed and starting from a circle as an initial shape, a front-edged and rear-round shape is obtained because of the vortices at the back of the body. To overcome this difficulty, we introduced the partial control algorithm, in which some of the nodal coordinates on the surface of the body are updated. From four cases of computational studies, we reveal that the optimal shape has both sharp front and sharp rear edges. All computations are conducted at Reynolds number Re=250. The minimum value of the performance function is obtained. Copyright © 2010 John Wiley & Sons, Ltd.