Algorithms for mixed continuous-discrete variable problems in structural optimization

The mixed variables (continuous-discrete) nonlinear optimization problem is defined and categorized into six different types. Methods for solution of these six types of problems are studied and characteristics of each method are catalogued. Some of the methods are implemented into a program for general applications. Enhancements for these methods are proposed. Several example problems are then solved to study performance of the methods.; Special attention is also given to the following two topics: (1) The need for an efficient and reliable local optimization algorithm when some mixed variable optimization methods are used. (2) Application of mixed variable methods to structural optimization problems.; For the first topic, the concept of the implicit sequential quadratic programming (ISQP) method is explained and an algorithm based on it is proposed. The basic idea of an ISQP algorithm for constrained problems is to use an approximate Hessian of the Lagrangian without explicitly calculating and storing it. The proposed method extends a similar algorithm for unconstrained problems where a two-loop recursion formula is used for the inverse Hessian matrix. The present research develops a similar algorithm for not only the constrained problem but also the direct Hessian updates. Several scaling procedures for the Hessian are also proposed and implemented. The basic method and some of its variations are evaluated using various problems--small to larger scale. The ISQP method performs much better than the full SQP method for larger scale problems. The results also show that an appropriate scaling of the Hessian can improve both efficiency and reliability substantially.; For the second topic, the methods for mixed continuous-discrete variable nonlinear optimization are studied for problems where the discrete variables are linked to other properties. Such variables are dependent on the discrete values of some properties but their relationships cannot be expressed by differentiable equations. In this research, a formulation of this type of problems is given. Examples of design of steel structures using the standard steel sections are presented. Three strategies for solving such problems are developed and tested.