Analysis of optimality criteria and gradient projection methods for optimal structural design

A general mathematical model for optimal design of structural and mechanical systems is defined. The finite element techniques for analysis of these systems are incorporated into the model. Optimality conditions for the model are derived. It is shown that an analytical solution of the optimality conditions is impossible except for trivial design problems. Numerical methods for solving these optimality conditions are presented. A direct method based on the gradient projection concept is derived. Numerical aspects for the method are discussed. These include step size selection, calculation of Langrange multipliers, identification of dependent design sensitivity vectors of constraint functions, and convergence criterion. Both direct and indirect approaches require total derivatives of cost and various constraint functions with respect to design variables. This is accomplished by integrating into both the approaches the state space design sensitivity analysis that has proven to be very general and efficient. It is shown that other major calculations of the two approaches are essentially the same. Thus there is potential for continuous transition between various algorithms for structural optimization. Optimal solutions for two design examples are obtained by hybrid methods to show the potential for further development of these methods.