Optimization of elastic bars in torsion

The paper considers the problem of optimization of mechanical systems described by partial differential equations. The shape of the region of integration of these equations is not specified beforehand but is to determined from the condition that a certain integral functional attains an extremal value. The mathematical optimization problem is reduced to a variational one having no differential constraints and the necessary optimality conditions are derived. The latter are used for seeking the cross-sectional shape of elastic bars of maximum torsional rigidity. Exact and approximate analytical solutions are given and the effectiveness of the optimal solutions is estimated.