Shape Optimization Using Adjoint Variable Method for Reducing the Surface Force of an Object under Unsteady Flow

Abstract

To obtain the optimal shape of a 3D object minimizing the fluid surface force, an adjoint variable method based on the variational principle is formulated and applied to the finite element method. This method is a type of sensitivity analysis (the sensitivity is the gradient of the Lagrange function with respect to spatial coordinates), and is based on the calculus of variations. The constrained optimization problem of the cost function is converted into an unconstrained optimization problem of the Lagrange function by introducing Lagrange multipliers called adjoint variables. The optimality condition of the adjoint variable method consists of the state equations, the adjoint equations, and the sensitivity equations. The equations for reducing the fluid surface force under a constant volume condition are formulated. By using the 3D shape optimization system, the surface force of a object located in Reynolds number 1000 can be reduced by about 57.6% .