Shape optimization for a link mechanism

This paper presents a numerical solution for shape optimization problems for link mechanisms, such as a piston‐crank mechanism. The dynamic behavior of a link mechanism is described by a differential‐algebraic equation (DAE) system consisting of motion equations for each single body and constraints of linkages and rigid motions. In a shape optimization problem, the objective function to maximize is constructed from the external work done by a given external force, which agrees with the kinetic energy of the link mechanism, for an assigned time interval, and the total volume of all the links forms the constraint function. The Fréchet derivatives of these cost functions with respect to the domain variation, which we call the shape derivatives of these cost functions, are evaluated theoretically. A scheme to solve the shape optimization problem is presented using the H ¹ gradient method (the traction method) proposed by the authors as a reshaping algorithm, since it retains the smoothness of the boundary. A numerical example shows that reasonable shapes for each link such that mobility of the link mechanism is improved are obtained by this approach.