A common engineering practice is the use of approximation models in place of expensive computer simulations to drive a multidisciplinary design process based on nonlinear programming techniques. The use of approximation strategies is designed to reduce the number of detailed, costly computer simulations required during optimization while maintaining the pertinent features of the design problem. To date the primary focus of most approximate optimization strategies is that application of the method should lead to improved designs. This is a laudable attribute and certainly relevant for practicing designers. However to date few researchers have focused on the development of approximate optimization strategies that are assured of converging to a solution of the original problem. Recent works based on trust region model management strategies have shown promise in managing convergence in unconstrained approximate minimization. In this research we extend these well established notions from the literature on trust-region methods to manage the convergence of the more general approximate optimization problem where equality, inequality and variable bound constraints are present.The primary concern addressed in this study is how to manage the interaction between the optimization and the fidelity of the approximation models to ensure that the process converges to a solution of the original constrained design problem. Using a trust-region model management strategy, coupled with an augmented Lagrangian approach for constrained approximate optimization, one can show that the optimization process converges to a solution of the original problem. In this research an approx1 Graduate Research Assistant.

The use of Response Surface Approximation (RSA) within an approximate optimization framework for the design of complex systems has increased as designers are challenged to develop better designs in reduced times. Traditionally, statistical sampling techniques (i. e., experimental design) have been used for constructing RSA's. These statistical sampling techniques are designed to be space filling, so that the response surface approximations are predictive across the range of the design sample space. When used in sequential approximate optimization strategies, a portion of the samples can be in the infeasible and/or ascent regions of the design space. These samples can bias the resulting RSA and make it less predictive in the usable feasible region where the optimization takes place. In the response surface based concurrent subsace optimization approach the design sampling strategy for RSA construction is optimization based. This optimization based sampling has proved to be effective due to the fact it samples in the linearized usable feasible region. In the present research, an experimental design strategy for projecting data points in the linearized usable feasible region is developed for constructing RSA's. The technique is implemented in a Sequential Approximate Optimization framework and tested in application to two multidisciplinary design optimization (MDO) test problems. Results show that the proposed technique pro-

Sequential Approximate Optimization (SAO) is a class of methods available for the multidisciplinary design optimization (MDO) of complex systems that are composed of several disciplines coupled together. One of the approaches used for SAO, is based on a quadratic response surface approximation, where zero and first order information are required. In these methods, designers must generate and query a database of order O(n²) in order to compute the second order terms of the quadratic response surface approximation. As the number of design variables grows, the computational cost of generating the required database becomes a concern. In this paper, we present an new approach in which we require just O(n) parameters for constructing a second order approximation. This is accomplished by transforming the matrix of second order terms into the canonical form. The method periodically requires an order O(n²) update of the second order approximation to maintain accuracy. Results show

Applying nonlinear optimization strategies directly to complex multidisciplinary systems can be prohibitive when the complexity of the simulation codes is large. Increasingly, response surface approximations(RSAs), and specifically quadratic approximations, are being integrated with nonlinear optimizers in order to reduce the CPU time required for the optimization of complex multidisciplinary systems. RSAs provide a computationally inexpensive lower fidelity representation of the system performance. The curse of dimensionality is a major drawback in the implementation of these approximations as the amount of required data grows quadratically with the number of design variables.

Rapid turn-around time for investigating new design concepts is a primary force driving design productivity initiatives across the industry. An integration framework focusing on the collaborative nature of rapid design automation at the preliminary and detailed design stage would ensure higher quality designs from the beginning of the product design cycle. As a result producing reliable, robust optimum designs from the preliminary design phase would enable companies to reduce the overall design cycle time.

Response Surface Approximations (RSA's) are widely used in the design community to provide designers with an approximate representation of a system. The use of RSA's allow designers to query the system while avoiding the high computational costs associated with today's advanced simulation codes. Sequential Approximate Optimization (SAO) methodologies have proved to be effective in managing the optimization of multidisciplinary design problems. In SAO the sampling required to build the RSA's often takes place within the same bounds as imposed on the current optimization iterate. This assures a good representation of the system in the region where it will be optimized. However it may restrict the approximation from extrapolating beyond the design space, and therefore improve the convergence rate of the algorithm. In this research a decoupling of the sampling region from the trust region is proposed.

For design problems involving computation-intensive analysis or simulation processes, approximation models are usually introduced to reduce computation time. Most approximation-based optimization methods make step-by-step improvements to the approximation model by adjusting the limits of the design variables. In this work, a new approximation-based optimization method for computation-intensive design problems — the adaptive response surface method (ARSM), is presented. The ARSM creates quadratic approximation models for the computation-intensive design objective function in a gradually reduced design space. The ARSM was designed to avoid being trapped by local optimum and to identify the global design optimum with a modest number of objective function evaluations. Extensive tests on the ARSM as a global optimization scheme using benchmark problems, as well as an industrial design application of the method, are presented. Advantages and limitations of the approach are also discussed.

A Multidisciplinary Design Optimization framework called Concurrent Subspace Design (CSD) has been applied to the design of an aircraft brake assembly. This application entailed an interactive implementation of CSD in which design information was obtained using existing industrial analysis software. The optimization problem statement in this study included a number of performance requirements associated with a brake that has been produced for a commercial aircraft. The results indicated that the CSD framework was able to e#ciently identify improved designs which met all the constraints imposed on the problem. The designs do not represent practical alternatives, however, because considerations related to production and maintenance costs, which are paramount in industry, were not incorporated into the optimization problem statement.

The use of optimization in a simulation based design environment has become a common trend in industry today. Computer simulation tools are common place in many engineering disciplines, providing the designers with tools to evaluate a design's performance without building a physical prototype. This has triggered the development of optimization techniques suitable for dealing with such simulations. One of these approaches is known as sequential approximate optimization. In sequential approximate minimization a sequence of optimizations are performed over local response surface approximations of the system. This paper details the development of an interior point approach for trust region managed sequential approximate optimization. The interior point approach will insure that approximate feasibility is maintained throughout the optimization process. This facilitates the delivery of a usable design at each iteration when subject to reduced design cycle time constraints. In order to deal with infeasible starting points, homotopy methods are used to relax constraints and push designs toward feasibility. Results of application studies are presented, illustrating the applicability of the proposed algorithm.

These notes describe some recent ideas for distributed design and their application to large-scale aerospace systems. In this type of multidisciplinary optimization, design tasks are decomposed into domain-specific subproblems, and coordinated to achieve an optimal system. Focusing on collaborative optimization, one form of design decomposition, the notes detail the methods, summarize recent results, and suggest new variants of these approaches that improve performance. 2.