2021 年 87 巻 901 号 p. 21-00191
In this study, we present a parameter-free shape optimization method for preform design in forging process. The objective functional is the total plastic work of forging process. We formulate this shape optimization problem as distributed-parameter optimization system with both constraint conditions of the rigid-plastic equilibrium equation and the volume. The shape gradient function for this problem is theoretically derived by applying the Lagrange multiplier method, the adjoint variable method and the material derivative method. The shape gradient function is numerically determined by the Cauchy stress tensor obtained from the state equation and the strain rate tensor obtained from the adjoint equation with the stiffness variation term. The shape gradient function determined is applied to the H1 gradient method, which is a gradient method in the function space. In this method, the optimal shape variation is determined by applying the fictitious distributed load proportional to the shape gradient function in the normal direction to the boundary, and then the shape variation obtained is superposed to the previous shape, which results in reducing the objective functional while maintaining the smoothness of the design boundary. Several numerical examples are presented to demonstrate the effectiveness and practical utility of the proposed method and the shape optimization system developed.
