Figure 1. The MDO loop employing variable-complexity response surface modeling.
Graduate Research Assistants
Vladimir Balabanov,
Susan Burgee,
Anthony Giunta,
Matthew Kaufman
Multidisciplinary Analysis and Design (MAD) Center for Advanced Vehicles
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0203
In the second year of the grant we have significantly built upon the progress made in our first year of funding. Specifically, we have further refined our response surface modeling methods which use the multipoint approximations listed in tasks (1) and (2). Further, we have applied the response surface modeling technique to a four variable aerodynamic example design problem and to a 25 variable structural example design problem. Concurrent with these efforts has been the evaluation of GASP, an Euler/Navier-Stokes solver, and GENESIS, a structural finite-element code for use in the aerodynamic-structural design optimization of the HSCT. Efficient use of parallel computing is a necessity for our MDO efforts and we have made substantial strides in task (3) as we have developed a framework integrating our heterogeneous mixture of software into a streamlined MDO process. This was successfully implemented for the four variable aerodynamic example design problem using the Intel Paragon computers at both Virginia Tech and NASA Langley. Summaries of these research efforts are given below.
Optimization using the VCM procedure, although successful, was hampered by numerical noise in the aerodynamic analysis tools and by inaccurate estimates of the wing weight for HSCT-type configurations. For this reason, a response surface modeling method using the VCM techniques has been developed to facilitate our HSCT design optimization efforts.
One way to reduce the computational burden associated with response surface modeling is to reduce the size of the design space of interest. For a design problem involving n design variables the size of the total design space grows as 2^n. This is commonly known as the curse of dimensionality. Typically, there will be large regions of the design space where obviously unreasonable designs exist. A major research contribution in this work stems from the use of what we term customized response surfaces whereby we use our variable-complexity modeling approach to address the issue of very large design spaces. In particular, we use our inexpensive, conceptual design level tools to investigate the design space. By evaluating large numbers of candidate HSCT designs using these inexpensive design tools, we are able to identify regions in the design space where the constraints are grossly violated or where nonsense designs are likely to occur. We have termed the remaining reasonable design space the approximation domain.
A second facet of our customized response surface approach involves using the HSCT results from the simple, inexpensive design tools to determine the specific form of the response surface polynomial. With these data for the points in the approximation domain, regression analysis is applied to create candidate response surface models for quantities of interest. ANOVA is then used to identify the less significant (if any) terms in the polynomial response surface models. This step is critical since the number of terms in the polynomial model determines the number of detailed HSCT analyses needed for the next step in response surface construction. Since computational costs increase by several orders of magnitude for the detailed analysis tools versus the simple analysis tools, it is desirable to analyze as few HSCT designs as necessary using the detailed analysis methods.
In the next step of the response surface construction process, design points within the approximation domain are selected for detailed analysis using another statistical technique known as the D-optimality criterion. The D-optimality criterion arises from the linear system Y=Xc, where Y is an m by 1 vector of objective function values, c is a k by 1 vector of coefficients to be estimated, and X is an m by k matrix of constants having rank k. The rows of the matrix X are the response surface basis functions evaluated at the design points. For this system, the least squares estimate of c is c = (|X'X|^-1)X'Y . The goal is to find the m points from a set of l>m candidate points existing in the design space that will yield the best fidelity between the polynomial model and the actual objective function. The D-optimality criterion states that the m points to choose are those which maximize the determinant |X'X|. Conceivably, one could consider each of the l!/(m!(l-m)!) combinations of m points from the set of l candidate points, evaluate |X'X|, and identify the set with the largest determinant. However, this task is prohibitively expensive when using more than a small number of variables. For this reason, a genetic algorithm was developed to efficiently find a set of D-optimal points given a set of candidate points. In addition, the genetic algorithm allows D-optimal point selection for a design space of arbitrary shape [1].
The final forms of the response surface models are then created by performing a regression analysis on the detailed, more expensive HSCT analysis data. The customized response surface approach will allow us to develop accurate response surface models which have been tailored to the specific design problem and involve several hundred detailed analysis evaluations. The detailed analyses in this procedure may be evaluated efficiently using coarse-grained parallelization. Finally these response surface models are substituted into the original HSCT design optimization code in place of the noisy aerodynamic analyses and the inaccurate wing weight estimations, and a final HSCT configuration optimization is performed.
Using the VCM technique the design space for this example problem was investigated by analyzing approximately 1300 HSCT configurations using the simple analysis tools. After screening these data 157 reasonable HSCT designs remained. Of these, 50 designs were selected using the D-optimality criterion for detailed analysis and the resulting data were used to construct response surface models of three aerodynamic drag components. These models were then substituted into the HSCT analysis/optimization software in place of the noisy aerodynamic analysis methods. Finally, an optimization of the full HSCT configuration was conducted within the design space defined by the four variable problem. By using the response surface models several improved HSCT designs were identified. The results of this research were reported at the recent AIAA Applied Aerodynamics Conference [1].
Currently we are investigating an alternative formulation of the four variable design problem. Preliminary results show that the accuracy of the response surface models is improved by 10-30 percent when a different set of design variables is used. We will continue with this four variable design problem and apply the results to the research to be conducted during the third year of funding from this grant.
In light of the deficiencies in the weight equation, a full integration of the structural and configuration optimization was considered. However, this approach was found to be difficult for two reasons. First, results from the structural optimization do not produce smooth functions with respect to the configuration shape parameters. Equally as important, the structural optimizations were too expensive to implement within the configuration design process, which requires structural weight information at a large number of design points.
To resolve this problem, a response surface model of the wing bending material weight is used to address the concerns of function smoothness and cost, while also increasing the accuracy of the statistical weight equation. Instead of performing the structural optimizations during the configuration design process, a large number of aircraft geometries are evaluated prior to the HSCT configuration design. These results are then used to create a response surface model of the wing bending material weight.
The construction of the response surface for the wing bending material weight requires data from a large number of structural optimizations. To minimize the cost of this procedure 10 geometric and loading parameters which contribute to the wing structural weight were identified. In this way, the number of design variables for this example problem was reduced from 25 to 10. This significantly decreased the number of structural optimizations required to construct the response surface model for wing bending material weight. The results of this research will be presented in an upcoming AIAA conference [2].
We have conducted a study [3] for the verification, validation, and certification of this Euler solver to obtain aerodynamic forces and moments in the HSCT optimization procedure. Forces, moments, and pressure distributions on analytic forebodies, wings, and wing-fuselage combinations were computed and compared with experimental data and/or other valid computational results. Grid refinement studies were performed in order to assure converged results for the Euler computations. In a multidisciplinary design optimization, the constraints for the structural optimization are evaluated at a number of load cases. Low and high supersonic cruise conditions as well as a high-speed pull-up load case have been investigated.
For supersonic cruise conditions, linear theory and Euler predictions of the drag were surprisingly close, but there is an overprediction in the linear theory lift curve slope and pitching moment slope magnitudes. In the 2.5g pull-up load case, while there were significant differences in the wing pressure distributions between linear theory and Euler calculations, these differences tended to decrease in the integration to total forces. The stresses computed from the aerodynamic loads using a rigid wing structural model displayed a 15-20 percent discrepancy between loads estimated using the Euler solver and linear theory. The certification procedure involves not only comparing the accuracy of the results, but also comparing the computational time required and the ease with which the computations can be implemented in an optimization procedure. The slight increase in accuracy of the Euler solver over panel methods on HSCT configurations for the supersonic load cases studied was overshadowed by the tremendous increase in computational cost. Further studies of bodies in the transonic regime are expected to reveal better opportunities for improvement using the Euler solver in a variable-complexity optimization.
For the GENESIS parallelization a master/slave paradigm has been applied. Here, the master node oversees the structural optimizations carried out on the individual processors. When a slave processor completes one structural optimization, the master node assigns another structural optimization to that processor. This continues until all required structural optimizations are completed.
To date we have achieved parallel efficiencies of approximately 80 percent with the reduced I/O version of GENESIS. With the original version of GENESIS parallel efficiencies of approximately 30 percent were obtained. Thus far GENESIS has been used to perform nearly 1000 HSCT structural optimizations in support of the response surface modeling efforts for the wing bending material weight calculations. The application of parallel computing has allowed us to perform a large number of structural optimizations which would not have been possible using a serial computer.
In addition to these efforts, a more efficient finite-element mesh generator was developed to exploit coarse-grained parallelization. This new mesh generator works with GENESIS and allows more general HSCT geometry descriptions than were used previously. Further, the mesh generator was refined to enable the input of aerodynamic loads from an external program.
A coarse-grained parallelization of the aerodynamic analysis modules within the HSCT analysis code was developed and was applied to the four variable aerodynamic example design problem described above. The parallelization uses a master/slave paradigm on the Paragon whereby one designated master node controls the data transfer and file I/O for the remaining slave nodes. This coarse-grained approach is used for both the simple and detailed HSCT analyses. With this approach parallel efficiencies for the simple and detailed aerodynamic analyses are approximately 80 percent and 90 percent, respectively.
In the automation of the MDO loop all communication between the subtasks (with the exception of the weight response surfaces used in the configuration optimization) is automatic, controlled by a UNIX shell script. Using the automated MDO loop, the response surface construction and the configuration optimization were performed for the aerodynamic example design problem. In conjunction with this four variable problem, some data visualization in 4-D was achieved.
In conjunction with this work, we will continue to evaluate the Euler flow solver GASP for use as the third level of analysis in the VCM framework. In particular, we will consider the use of GASP for the prediction of aerodynamic loads under transonic and supersonic maneuver conditions. Because of the considerable computational expense of GASP compared to our current conceptual/preliminary design tools, data from GASP may be best assimilated into the HSCT design optimization software through one or more response surface models.
An issue, not in the original proposal, that has become central is how to efficiently blend data of variable accuracy, and where, and of what quality, to seek further information. Certain aspects of this are studied in statistics, but the precise situation of variable complexity modeling in MDO remains largely unexplored. We need to address the theoretical issues in variable complexity modeling. We will seek ways of incorporating some very accurate data along with larger numbers of less accurate (detailed) analysis results in the development of our customized response surfaces. The very accurate data will come from Euler/Navier-Stokes results, more detailed finite-element structural optimization, and perhaps even experimental results.