Optimal Design Problems for Elastic Bodies by Use of the Maximum Principle

A class of optimal design problems for elastic bodies requires minimization of a nonconvex energy functional with respect to both kinematical variables and a design variable. Such functionals for rod, beam, sphere, and cylinder problems are special cases of a single functional. Minimization of this functional using Pontryagin's Maximum Principle to handle the nonconvexity and the inequality constraints leads to the choice of the design variable as taking on only its upper and lower bound values, agreeing with some previous results found for specific problems. A generalization to extensible beam-column problems is discussed.