Multi-Objective Shape Optimization of Viscous Flow Fields for Prescribing Flow Velocity Distribution and for Minimizing Dissipation Energy (Topology optimization theory and applications toward wide fields of natural sciences)

"Topology optimization theory and applications toward wide fields of natural sciences". May 7～9, 2014. edited by Takashi Nakazawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.This paper presents a numerical solution to multi-objective shape optimization problem in steady-state viscous flow fields. In this study, a multi-objective shape optimization problem using normalized objective functional is formulated for the flow velocity distribution prescribed problem and the total dissipated energy minimization problem in the viscous flow fields. In addition, another multi-objective shape optimization problem is formulated for the flow velocity distribution prescribed problem, while the total dissipated energy is constrained to less than a desired value, in the viscous flow fields. Shape gradients of these multi-objective shape optimization problems are derived theoretically using the Lagrange multiplier method, adjoint variable method, and the formulae of the material derivative. Reshaping is carried out by the traction method proposed as an approach to solving shape optimization problems. The validity of proposed method is confirmed by results of 2mathrm{D} numerical analysis