John Dennis, David Serafini, Rice University; Virginia Torczon, The College of William & Mary and the Institute for Computer Applications in Science and Engineering

Researchers from the CRPC Optimization Group are applying numerical optimization techniques to the design of industrial products to improve performance, reduce manufacturing costs, and avoid time-consuming testing and evaluation procedures. They are working with the Boeing Company and other collaborators to develop new methods to improve the utility of optimization for general industrial design applications. Recently, they successfully applied numerical optimization technology developed at Rice University to a major problem in helicopter design.

The helicopter problem involves the design of rotor blades. The blades are made of advanced composite material and have to withstand extreme loads. The problem is to design the rotor blades to minimize their weight, vibration, and noise while maximizing strength and safety, without requiring more power than the engines can deliver. The objective function for this problem is a multidisciplinary analysis involving aerodynamics, structures, and propulsion. The computation for a single design point can take up to several hours on a vector supercomputer, although a simple version only takes a few minutes on a high-performance workstation.

A revised implementation of the Parallel Direct Search (PDS) software has been applied to a simple version of this problem. The revised PDS software solves nonlinear constrained optimization problems using a family of methods originally developed by Virginia Torczon (see "Parallel Profile," page 8) working with Optimization Group Leader John Dennis. Unlike most optimization methods, PDS does not require derivative information; only the values of the objective function are needed. This makes PDS applicable to complex or ill-behaved problems where derivatives are impractical to compute. The PDS software can be run in either a serial or distributed computing environment as its performance scales well with increasing numbers of processors.

There were 56 design variables in the problem solved using PDS. As part of a Cooperative Research Agreement that includes Rice, Boeing, IBM, and NASA, the PDS software was run on the IBM SP2 parallel computer at NASA/Ames Research Center. From a starting point specified by Boeing, PDS reduced the objective function by more than 50 percent, an achievement that is as good as or better than any other optimization method that has been applied to this problem. The amount of computer time used to solve the problem was significant: roughly 30 wall-clock hours using 32 processors of the NASA SP2. In this time, about 30,000 different rotor blade designs were evaluated. However, because PDS is robust and easy to use, the time spent setting up the problem and running was small: roughly two weeks, including time spent waiting for the computation to finish.

Although PDS can be applied to expensive problems like this one, the large amount of computer time needed is not practical for everyday use in an industrial design process. A new methodology being developed by Rice, Boeing, and IBM is designed to address this issue. In this approach, called "model management," an optimization method like PDS is applied to an approximation of the expensive objective function, called a "model," instead of to the actual objective. This allows the optimization to be performed at greatly reduced cost since typically the model is less expensive to evaluate than the actual objective. The management scheme determines how much improved the actual objective function is at the optimizer of the model and decides what to do next. The usual next step is to adjust the model to account for the difference between it and the actual objective function and optimize the model again. In this way, the model is brought closer to the actual objective. This should eventually lead to a point where the optimizers of the model and the actual function are nearly the same. In the final stage, the actual function is optimized directly to get rapid local convergence to the minimum of the actual objective function.

David Serafini of Rice ( is the focal point for the model management framework. For more information on optimization model management, see "Optimizing Using Approximation Models," Winter 1996 Parallel Computing Research, page 10, or "Managing the Use of Approximation Models in Nonlinear Programming," Summer 1996 Parallel Computing Research, page 9. For more information about PDS, see

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