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-

Abstract not found

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

The analysis of engineering systems must often be conducted using complex, non-hierarchic, coupled, discipline-specific methods. When the cost of performing these individual analyses is high, it is impractical to apply many current optimization methods to this type of system to achieve improved designs. Consequently, methods are being developed which attempt to reduce the cost of designing or optimizing non-hierarchic systems. This paper details the application of an extension of the Concurrent Subspace Optimization (CSSO) approach through the use of neural network based response surface mappings. The response surface mappings are used to allow the discipline designer to account for discipline coupling and the impact of design decisions on the system at the discipline level as well as for system level design coordination. The ability of this method to identify globally optimal designs is discussed using two example system design problems. Comparisons between this algorithm and full sys...

A multidisciplinary design optimization framework suitable for application to mixed continuous/discrete systems has been developed. This framework, called Concurrent Subspace Design, employs artificial neural networks to provide response surface approximations. A concise metric indicating how accurately a neural network is able to approximate the design space was defined and used to assess different networks, which were obtained by varying the amount of data considered in their construction and the means by which discrete design variables are represented in them. Results demonstrate that this framework is able to locate optimal designs and that its computational requirements are related to some degree to the database used in formulating the neural network approximations.

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.

An extension of the method of Concurrent Subspace Optimization (CSSO) has been developed to accomodate mixed continuous/discrete design problems. The mixed CSSO framework employs artificial neural networks to provide approximations to the design space, which are the means of coordinating design decisions in the individual disciplines. This approach is applied to a nonhierarchic test problem which contains continuous and discrete design variables. The results demonstrate that the mixed CSSO framework is able to locate optimal designs and did reduce the number of the complete system analyses required by conventional optimization techniques. Computational resources remain a concern, however, due to the large number of contributing (disciplinary) analyses required to perform mixed optimization at the discipline level. Results demonstrate that the database of design information assembled during CSSO can be exploited to enhance the efficiency of subsequent runs, even if the requirements of the system design problem are altered.

Design space approximations have proven useful as a means of coordinating individual discipline design decisions in the multidisciplinary design of complex, coupled systems. Artificial neural networks have been used to provide these parameterized response surface approximations. A method has been developed in which neural networks can be trained using both state and state sensitivity information. This allows for more compact network geometries and reduces the number of coupled system analyses required to develop useful design space approximations. This approach is applied to the Concurrent Subspace Optimization (CSSO) framework for a nonhierarchic test problem in which the sensitivity information is provided using the Global Sensitivity Equations (GSEs). I.

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.