Abstract:

A new algorithm is presented for the solution of structural optimization problems in which the stress constraints are nonsmooth. The allowable stresses of structural members may be governed by one of three types of behavior: yielding, inelastic buckling, or elastic buckling. Consequently, the strength of members is defined by a piecewise function that depends on the crosssection and other design parameters. Some of these allowable stress functions are discontinuous, while some are continuous but nonsmooth. The allowable stress functions are sometimes defined by nonsmooth envelope functions, wherein the strength is determined by the controlling failure mechanism. Absolute values of stresses are compared to positive allowable stresses for simplicity, and the absolute value function is nonsmooth at zero.
Optimization of structural members with such nonsmooth constraint functions is limited, because derivativebased algorithms assume that the objective and constraint functions are smooth. Typical approaches in current practice are to oversimplify the constraints, use slower, derivative free methods, apply adhoc solutions, or ignore the problem altogether. In the approach taken here, the causes of the nonsmooth constraints in a typical design code are systematically identified and replaced with nearly equivalent alternatives so that the problem can be solved using readily available and powerful derivativebased optimization methods. Theoretical models, finite element models, and experimental data are used as benchmarks to predict the behavior. These are used to fit an appropriate set of curves for use in design.
The new optimization algorithm presented uses a combination of a continuation method, then a judicious choice of added "secondary constraints" to transform the original nonsmooth problem to an equivalent smooth one. First, a solution is obtained for a smooth approximation of the original problem. It is used as a starting value to successively solve more and more nonsmooth, but closer approximations until a reasonably close solution to the original problem is determined. This solution is used to constrain the variables governing each of the piecewise defined functions. The original problem is thus transformed to a smooth problem with the added secondary constraints, and is solved using a standard derivativebased optimization method. 