Asymptotic models for optimizing the contour of multiply-connected cross-section of an elastic bar in torsion

An asymptotic solution is obtained for the problem of maximizing the torsional rigidity of elastic, multiply-connected cylindrical bars for a given area of cross-section. The shapes of the inner contours of the multiply-connected cross-section are specified while the outer contour is determined as a result of the shape optimization. We apply the method of matched asymptotic expansions to construct a first-order asymptotic model. The conditions for unique solvability of the asymptotic model have been established under some restrictions imposed on the location of the inner contours and their polarization matrices. The economy achieved by optimization is estimated.