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.

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.

A procedure for generating a customized weight function for wing bending material weight of a High Speed Civil Transport (HSCT) is described. The weight function is based on HSCT configuration parameters. A response surface methodology is used to fit a quadratic polynomial to data gathered from a large number of structural optimizations. To reduce the time of performing a large number of structural optimizations, coarse-grained parallelization with a master-slave processor assignment on an Intel Paragon computer is used. The results of the structural optimization are noisy. Noise reduction in the structural optimization results is discussed. It is shown that the response surface filters out this noise. A statistical design of experiments technique is used to minimize the number of required structural optimizations and to maintain accuracy. Simple analysis techniques are used to find regions of the design space where reasonable HSCT designs could occur, thus customizing the weight function to the design requirements of the HSCT, while the response surface itself is created employing detailed analysis methods. Analysis of variance is used to reduce the number of polynomial terms in the response surface model function. Linear and constant corrections based on a small number of high fidelity results are employed to improve the accuracy of the response surface model. Configuration optimization of the HSCT employing a customized weight function is compared to the configuration optimization of the HSCT with a general weight function.

In this research we develop a mathematical construct for estimating uncertainties within the bilevel optimization framework of collaborative optimization. The collaborative optimization strategy employs decomposition techniques that decouple analysis tools in order to facilitate disciplinary autonomy and parallel execution. To ensure consistency of the physical artifact being designed, interdisciplinary consistency constraints are introduced at the system level. These constraints implicitly enforce multidisciplinary consistency when satisfied. The decomposition employed in collaborative optimization prevents the use of explicit propagation techniques for estimating uncertainties of system performance. In this investigation we develop and evaluate an implicit method for estimating system performance uncertainties within the collaborative optimization framework. The methodology accounts for both the uncertainty associated with design inputs and the uncertainty of performance predictions from other disciplinary simulation tools. These implicit uncertainty estimates are used as the basis for a new robust collaborative optimization (RCO) framework. The bilevel robust optimization strategy developed in this research provides for disciplinary autonomy in system design, while simultaneously accounting for performance uncertainties to ensure feasible robustness of the resulting system. The method is effective in locating a feasible robust optima in application studies involving a multidisciplinary aircraft concept sizing problem. The system-level consistency constraint formulation used in this investigation avoids the computational difficulties normally associated with convergence in collaborative optimization. The consistency constraints are formulated to have the inherent properties necessary for convergence of Proceedings of DETC’01

For complex systems, traditional methods of experiencebased design are ineffective: the design task must be supported by simulations. Conceptual design and system-level detailed design based on numerical simulation models is limited because of the difficulty in integrating disparate subsystem models to predict overall system behavior. A metamodel-based integration strategy allows simulation results from multiple submodels to be combined into a system-level simulation. The development of a metamodel-based integration strategy for system-level design depends on effective experiment design strategies for fitting and updating subsystem metamodels. 1

Simulation-based design has become a major tool in the design of automotive, aerospace and consumer products. Designers are faced with the continuous challenge of reducing manufacturing costs and design cycle times while improving the system's performance and reliability. Simulation-based design plays an increasingly prominent role in facilitating the conceptualization and realization of products under these competitive conditions. Single discipline simulations used for analysis are being coupled together to create complex coupled simulation systems. This investigation focuses on the development of methodologies that help designers reduce the cost of using optimization to manage the simulation based design process. The computational cost of executing a single complex coupled simulation and the total number of simulations required per iteration are the two factors which most influence an optimization framework's design. Original contributions