Automation of reverse engineering process in aircraft modeling and related optimization problems

During the year of 1994, the engineering problems in aircraft modeling were studied. The initial concern was to obtain a surface model with desirable geometric characteristics. Much of the effort during the first half of the year was to find an efficient way of solving a computationally difficult optimization model. Since the smoothing technique in the proposal 'Surface Modeling and Optimization Studies of Aerodynamic Configurations' requires solutions of a sequence of large-scale quadratic programming problems, it is important to design algorithms that can solve each quadratic program in a few interactions. This research led to three papers by Dr. W. Li, which were submitted to SIAM Journal on Optimization and Mathematical Programming. Two of these papers have been accepted for publication. Even though significant progress has been made during this phase of research and computation times was reduced from 30 min. to 2 min. for a sample problem, it was not good enough for on-line processing of digitized data points. After discussion with Dr. Robert E. Smith Jr., it was decided not to enforce shape constraints in order in order to simplify the model. As a consequence, P. Dierckx's nonparametric spline fitting approach was adopted, where one has only one control parameter for the fitting process - the error tolerance. At the same time the surface modeling software developed by Imageware was tested. Research indicated a substantially improved fitting of digitalized data points can be achieved if a proper parameterization of the spline surface is chosen. A winning strategy is to incorporate Dierckx's surface fitting with a natural parameterization for aircraft parts. The report consists of 4 chapters. Chapter 1 provides an overview of reverse engineering related to aircraft modeling and some preliminary findings of the effort in the second half of the year. Chapters 2-4 are the research results by Dr. W. Li on penalty functions and conjugate gradient methods for quadratic programming problems

Advanced Robust Optimization With Interval Uncertainty Using a Single-Looped Structure and Sequential Quadratic Programming

Uncertainty is inevitable and has to be taken into consideration in engineering optimization; otherwise, the obtained optimal solution may become infeasible or its performance can degrade significantly. Robust optimization (RO) approaches have been proposed to deal with this issue. Most existing RO algorithms use double-looped structures in which a large amount of computational efforts have been spent in the inner loop optimization to determine the robustness of candidate solutions. In this paper, an advanced approach is presented where no optimization run is required for robustness evaluation in the inner loop. Instead, a concept of Utopian point is proposed and the corresponding maximum variable/parameter variation will be obtained just by performing matrix operations. The obtained robust optimal solution from the new approach may be conservative, but the deviation from the true robust optimal solution is small enough and acceptable given the significant improvement in the computational efficiency. Six numerical and engineering examples are tested to show the applicability and efficiency of the proposed approach, whose solutions and computational efforts are compared to those from a previously proposed double-looped approach, sequential quadratic program-robust optimization (SQP-RO)

Minimizing Nonconvex Quadratic Functions Subject to Bound Constraints

We present an active-set algorithm for finding a local minimizer to a nonconvex bound-constrained quadratic problem. Our algorithm extends the ideas developed by Dost al and Sch oberl that is based on the linear conjugate gradient algorithm for (approximately) solving a linear system with a positive-de finite coefficient matrix. This is achieved by making two key changes. First, we perform a line search along negative curvature directions when they are encountered in the linear conjugate gradient iteration. Second, we use Lanczos iterations to compute approximations to leftmost eigen-pairs, which is needed to promote convergence to points satisfying certain second-order optimality conditions. Preliminary numerical results show that our method is e fficient and robust on nonconvex bound-constrained quadratic problems