Dual estimates in problems of design of elastic structures

The problem of minimizing the volume of two- and three-dimensional structures, subject to certain stress constraints, which are known as the conditions of strength theory and are used in practice for various materials, is considered. The control is achieved by adjusting the shape of the boundary. Cavities inside the design region are allowed, and the shape of the cavities is also optimized. Dual problems, constructed for such optimal design problems, can be used for estimates of optimal or nearly optimal designs. Examples of dual estimates for three optimal design problems are considered.