Semi-Analytical Approach to Three-Dimensional Shape Optimization Problems

Abstract

Abstract : A semi-analytical approach to three-dimensional (3-D) shape optimization problems for a viscous incompressible fluid under the assumption of zero (low) Reynolds number has been developed. It couples the theory of generalized analytic functions with the adjoint equations-based method. A solution to Stokes equations governing the behavior of the fluid has been reduced to integral equations based on the Cauchy integral formula for k-harmonically analytic functions. The fluid velocity and boundary shape are the state and design variables, respectively, and a shape optimization problem is to find shape minimizing the energy dissipation rate. In contrast to the classical optimal control theory, the shape optimization problem has been formulated as an optimal control problem with constraints in the form of integral equations. The optimality conditions (state, adjoint and design equations) for the optimal control problem have been derived. The advantage of the suggested approach is that the state and adjoint variables are single-variable functions, which being represented analytically in the form of series with unknown coefficients, can be accurately determined from the state and adjoint integral equations, for example, by minimizing the total squared error. The optimal shape has been found iteratively by a gradient-based method, in which at each iteration, the state and adjoint variables have been determined for an updated shape and the gradient for the cost functional with respect to the shape has been obtained by the adjoint equations-based method. The suggested semi-analytical approach has been illustrated for the drag minimization problem for motion of a solid body of revolution in the viscous incompressible fluid and has proved to be efficient and accurate.