VARIABLE-COMPLEXITY MULTIDISCIPLINARY

OPTIMIZATION ON PARALLEL COMPUTERS

Progress Report for NASA Grant NAG-1-1562
10 October 1995 to 24 January 1997

Principal Investigators
Dr. Bernard Grossman, Dr. William H. Mason, Dr. Layne T. Watson
Multidisciplinary Analysis and Design (MAD) Center for Advanced Vehicles
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0203

and

Dr. Raphael T. Haftka
Department of Aerospace Engineering, Mechanics and Engineering Science
University of Florida, Gainesville, Florida 32611-6250

Graduate Research Assistants
Vladimir Balabanov, Anthony Giunta, Oleg Golovidov, Duane Knill
Multidisciplinary Analysis and Design (MAD) Center for Advanced Vehicles
Virginia Polytechnic Institute and State University
Blacksburg, Virginia 24061-0203

1. Introduction

This progress report covers work conducted under grant NAG-1-1562 for the NASA High Performance Computing and Communications Program (HPCCP) from October 10, 1995, to January 24, 1997. The objective of this research is to develop new multidisciplinary design optimization (MDO) techniques which exploit parallel computing to reduce the computational burden of aircraft MDO. The design of the High-Speed Civil Transport (HSCT) aircraft was selected as a test case to demonstrate the utility of our MDO methods. The three tasks identified in our HPCCP proposal are:
  1. development of parallel multipoint approximation methods for the aerodynamic design of the HSCT,
  2. use of parallel multipoint approximation methods for structural optimization of the HSCT,
  3. mathematical and algorithmic development including support in the integration of parallel computation for items (1) and (2).

During the third year of this grant funding we continued to refine our methods for what we have termed variable-complexity MDO. In addressing tasks (1) and (2) we applied statistical methods known as design of experiments (DOE) theory and response surface (RS) modeling to our HSCT optimization problem. Specifically, DOE and RS modeling were used to approximate subsonic and supersonic aerodynamic characteristics for HSCT optimization problems involving five and 10 variables. In addition, an RS model was created to incorporate high-fidelity supersonic drag estimates from an Euler flow solver into the five variable HSCT optimization problem. Task (2) efforts focused on continued refinements to a 25 variable HSCT optimization problem which employed an RS model for estimating wing bending material weight. The data for this RS model were obtained from approximately 3,000 structural optimizations using finite element analysis/optimization software. Without parallel computing, this large number of structural optimizations would not have been computationally affordable. Task (3) efforts concentrated on the development of Fortran 90 software to perform the efficient creation of D-optimal experimental designs. Additional task (3) efforts included the creation of theoretical models of the peak parallel efficiency possible in our MDO problem which were compared to actual parallel efficiencies as obtained on an Intel Paragon parallel computer. These research efforts are summarized below.

2. Variable-Complexity Modeling in Optimization

2.1 HSCT Design Problem

The HSCT optimization problem is to minimize the takeoff gross weight (TOGW) for a 250 passenger supersonic transport aircraft having a range of 5,500 nautical miles and a cruise speed of Mach 2.4. For these efforts we have developed a suite of conceptual level and preliminary level analysis and design tools which include several software packages obtained from NASA along with in-house software. The HSCT configuration and mission are defined using 29 variables. Twenty-six of these variables describe the geometric layout of the HSCT and three variables describe the mission profile. The HSCT design employs 69 nonlinear inequality constraints which consist of both geometric constraints (e.g., all wing chords >= 7.0 ft), and aerodynamic/performance constraints (e.g., C_L at landing <= 1, and range >= 5,500 naut.mi.). In formal optimization terms this problem may be expressed as
where x is the vector of 29 design variables, g(x) is the vector of 69 nonlinear inequality constraints, and TOGW is a nonlinear function of the 29 design variables.

Even with a modest design problem the computational cost of MDO is prohibitive when using many traditional preliminary level analysis methods. For this reason, a variable-complexity modeling (VCM) technique is employed whereby both low fidelity and high fidelity analysis methods are used in the design optimization. In VCM, conceptual level analysis methods, typically algebraic models, are frequently used because of their low computational costs. More accurate, and more computationally expensive preliminary level methods, such as panel level aerodynamics and simple finite element models, are periodically used during optimization to provide more accurate analysis data. High fidelity analysis methods, such as Euler/Navier-Stokes aerodynamic analyses and detailed finite element structural analyses, also may be used in this VCM paradigm.

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. Numerical noise is typically manifested as a high frequency, low amplitude variation in the results obtained from the computer analyses as the design parameters vary. The existence of numerical noise in optimization problems inhibits the use of many gradient-based optimization methods in which convergence may be slowed or prevented. For this reason, we have developed an optimization technique which combines the VCM methods along with response surface modeling to facilitate HSCT optimization studies.

2.2 Response Surface Modeling Methods

Response surface models are formed using polynomial functions (typically quadratic) and are employed to approximate a selected function over a design region of interest. For polynomial functions, an RS model is created by performing a least squares curve fit (surface fit in multiple dimensions) to data sampled in the design region of interest. Design of experiments theory is the field of statistics which provides a mathematical foundation for selecting the number and location of the sample sites in the design region of interest. The choice of sample sites can significantly affect the properties of the RS model created from the sampled data. Once the RS model is created, statistical techniques including regression analysis and analysis of variance (ANOVA) are applied to the RS model to determine which design variables and their combinations most influence the predicted response values.

However, RS models are prone to what is commonly known as the curse of dimensionality in which the volume of the design region grows as 2^n where n is the number of dimensions (variables) to be modeled. Thus, as n becomes large (> 10), it becomes increasingly computationally expensive to sample the design region at the number of sites needed to create accurate RS models. One way to avoid the curse of dimensionality is to reduce the size of the design region of interest. Typically, there will be large portions of the design space where obviously unreasonable HSCT configurations exist. A major research contribution in this work stems from the use of what we term customized response surface models whereby we use our variable-complexity modeling approach to address the issue of very large design spaces. In particular, we use our inexpensive, conceptual level analysis methods to sample the original design region of interest using sample sites determined by a full-factorial experimental design (for n <= 10 ) or by a partially balanced incomplete block experimental design (for n > 10 ). By evaluating large numbers of candidate HSCT designs using these inexpensive analysis tools, regions in the design space are identified and excluded where the constraints are grossly violated or where nonsensical designs are likely to occur. We have termed the remaining design region the approximation domain or the reduced design space.

A second facet of our customized response surface approach involves using the HSCT analysis results from the inexpensive conceptual level analysis methods to determine the specific form of the response surface polynomial. With the data from the sample sites in the reduced design space, regression analysis and ANOVA are applied to identify the less significant (if any) terms in the polynomial RS models. This step is critical since the number of terms in the polynomial model determines the number of expensive preliminary level HSCT analyses needed in the next step of the VCM process. Since computational costs increase by several orders of magnitude for the preliminary level analysis tools versus the conceptual level analysis tools, it is desirable to analyze as few HSCT designs as necessary using the preliminary level analysis methods.

In the next step of the response surface modeling process, sample sites within the reduced design space are selected to create a D-optimal experimental design. The statistical reasoning behind the creation of a D-optimal design is that it leads to response surface models for which the uncertainty in the coefficients is minimized. Previously, a genetic algorithm was used to select D-optimal sample sites in the reduced design space. However this method proved to be too computationally expensive for n > 10 and a more efficient method employing Wynn's k-exchange algorithm [1] was developed during this past year.

Once the D-optimal sample sites are selected, they are evaluated using the preliminary level analysis methods. This data is then used to create the final form of the customized RS models that are employed during HSCT optimization. By using the RS models, numerical noise is reduced or eliminated and optimization may be performed without the convergence difficulties created by numerical noise.

This customized response surface modeling approach allows the development of accurate response surface models which have been tailored to the specific design problem. The computational expense of RS modeling is addressed by applying coarse-grained parallel computing to both the conceptual level and preliminary level analyses. Without parallel computing, the expense of using RS modeling for high dimensional optimization problems would be unacceptable.

3. MDO Using Aerodynamic Response Surface Models

Building upon experience gained in the first two years of this grant funding, the use of VCM and RS modeling was applied to sample HSCT optimization problems involving five and 10 variables. In both sample problems, the geometric definition of the HSCT fuselage, vertical tail, and mission parameters was held constant while variables associated with the wing shape were allowed to vary. The five variable problem included four wing shape variables and the mission fuel weight, whereas the 10 variable problem included eight wing shape variables, mission fuel weight, and a single variable for the spanwise location of the engine nacelles. In the five variable problem three RS models were constructed; supersonic volumetric wave drag and two components of supersonic drag-due-to-lift. When optimization cases were performed with these RS models, the optimized HSCT configuration was 14,000 lb lighter (2.1% of TOGW) than the typical optimal HSCT configurations obtained without the RS models. This occurred because numerical noise in the optimization problem was smoothed out by the RS models and the optimizer did not become trapped in artificial, locally optimal HSCT configurations. Similar results were obtained for the 10 variable problem which used four RS models; three supersonic drag RS models as in the five variable problem and an RS model for the subsonic lift curve slope. HSCT optimizations conducted with the RS models resulted in a savings of 17,000 lb (2.7% of TOGW) (Figs. 1, 2). Without the RS models the optimization cases became trapped in various local optima created by the existence of numerical noise. Results for the five and 10 variable HSCT optimization problems were documented in a paper presented at the 1996 Multidisciplinary Analysis and Optimization Symposium [2] and the results of the 10 variable HSCT optimization study will be submitted for publication in the Aeronautical Journal [3].


Figure 1. Locally optimal HSCT obtained without using aerodynamic RS models.


Figure 2. Globally optimal HSCT obtained using aerodynamic RS models.

In conjunction with the RS modeling efforts for low and medium fidelity analyses, research has progressed on the incorporation of high fidelity aerodynamic analysis methods, from the Euler/Navier-Stokes solver GASP [4], into the HSCT optimization process. Comparisons of Euler and linear theory aerodynamic loads show good agreement at cruise conditions, but significant variations at high-lift conditions where shocks form on the wing. The wing-bending stresses are sensitive to these differences in wing loading, but the optimal wing bending-material weights show only small differences up to 1,750 lb (0.3% of TOGW). Studies comparing Navier-Stokes, Euler, and linear theory drag predictions show linear theory underestimates the cruise drag by approximately two counts. While this may appear to be a fairly good correlation, the aircraft range and optimal TOGW are very sensitive to drag differences of this magnitude. A two count underestimate in the cruise drag produces a 120 naut.mi. overprediction in the aircraft range and a 56,000 lb (7.2%) underprediction of the optimal TOGW. The majority of this weight difference is attributable to a reduction in fuel weight which is made possible by the erroneous drag prediction. Clearly, Euler/Navier-Stokes solutions are necessary to provide drag values which give accurate aircraft weight and performance predictions. Currently we are evaluating the use of response surface models created from Euler drag estimates in the five variable HSCT optimization problem.

Another area of recent research is the evaluation of Bayesian estimation for use in HSCT optimization in place of RS models. The approximating functions are taken from the statistical literature for the design and analysis of computer experiments (DACE). They differ from RS models in that the DACE interpolating model is not restricted to a predefined shape such as a quadratic polynomial. Research on the application of DACE models to the five and 10 variable HSCT optimization problems is in progress.

4. MDO Using Structural Response Surface Models

When structural optimization is integrated into aircraft configuration optimization, the structural weight is inherently a non smooth function of the configuration shape variables. Additional noise may be created due to noise in aerodynamic loads and incomplete convergence of the structural optimization. For our model of the HSCT, we have found noise of the order of 30% of the wing bending material weight obtained from structural optimization. A small part of this noise was eliminated by improving the convergence of the optimization procedure. Most of the noise however, was traced to noise in the aerodynamic loads. Careful implementation of camber optimization allowed us to reduce the amount of noise significantly. In this study the finite element analysis/optimization code GENESIS [5] was used to perform structural optimizations of the HSCT configuration.

To smooth out the noise and facilitate the integration of the structural optimization in the overall HSCT optimization process we have explored the use of response surface models for representing the optimal structural weight. To ensure good accuracy of the response surface models, conceptual level analyses were employed to identify and to eliminate infeasible regions of the design space, thus creating the reduced design space. The original FLOPS weight equation was used to identify important parameters for determining bending weight which allowed a reduction in the number of variables from 25 to 10. To create response surface models 1,000 sites were sampled in the reduced design space. Instead of using a D-optimal experimental design to select the sample sites, two different experimental design criteria were investigated; a minimum variance experimental design and a minimum bias experimental design.

The accuracies of the revised response surface models created from the structural optimization data with reduced noise levels were compared to the original response surface models created from very noisy data. With 10 variables, the 1,000 sample sites were sufficient to filter out the large amount of numerical noise even for the structural optimization data. With 25 variables, and a much larger number of coefficients, it was not possible to obtain the same modeling accuracy with only 1,000 sample sites. For that reason, the reduction of noise in the wing bending material weight did not have a significant effect on the apparent accuracy of the 10 variable response surface model. However, for the case of 25 design variables with the noisy data, the response surface model proved to be useless for HSCT optimization. Results from this study (Figs. 3, 4) showed that the optimal HSCT configurations obtained using the original FLOPS weight equation and the response surface model were similar in planform shape. However, the optimal designs differed in TOGW by approximately 4,000 lb.


Figure 3. Twenty-five variable initial and optimal HSCT configurations. Optimal HSCT obtained using the original FLOPS estimates for structural weight.


Figure 4. Twenty-five variable initial and optimal HSCT configurations. Optimal HSCT obtained using the structural weight RS model.

In addition to the continued work on the 25 variable HSCT optimization problem, progress occurred during the third year of grant funding in developing higher fidelity finite element models for HSCT wing/fuselage configurations. The new finite element model employs 3,000 degrees of freedom and 70 variables are available for the structural optimization, whereas the previous finite element element model had 1,000 degrees of freedom and 40 structural optimization variables. Note that the variables used in the structural optimization (e.g., wing skin panel thicknesses, spar thicknesses) are separate from the 29 variables used to define the HSCT geometry and mission profile.

Current research is focusing on methods to combine structural optimization data from both the low and high fidelity finite element models for use in constructing response surface models. As with the varying fidelity aerodynamic data, DACE models are also being explored as a method to combine the varying fidelity data.

5. Parallel Computing and Algorithmic Development

Coarse-grained parallel computing methods developed during the first two years of this grant funding were extensively applied in performing the numerous structural optimizations needed to create response surface models for the 25 variable HSCT optimization problem. Approximately 1,000 GENESIS structural optimization cases (sample sites) were needed to create an accurate response surface model for the wing bending material weight in the 25 variable HSCT optimization problem. With each structural optimization requiring 30-40 CPU minutes on an Intel Paragon, a serial execution of 1,000 GENESIS runs would require 500-600 CPU hours, or over three weeks of wall-clock time to complete. However, a parallel execution of the 1,000 GENESIS runs using 20 processors required only about 40 CPU hours. Clearly, performing this number of structural optimizations is not realistic without the use of parallel computing.

Additional developments in parallel computing focused on the creation of theoretical models for estimating peak parallel efficiency when performing the conceptual level and preliminary level aerodynamic analyses. The theoretical models predicted parallel efficiency as a function of the number of available processors on the Paragon through estimates of the amounts of parallel computation, serial computation, and data input/output which comprise the aerodynamic analyses. Comparisons between actual parallel efficiency and theoretical peak parallel efficiency were nearly identical over a range of three to 37 processors (Fig. 5) which demonstrates that our parallel implementation of the aerodynamic calculations is nearly ideal. Results of this study are documented in a journal article to be published in late 1996 or early 1997 [6].

In the area of algorithmic development, a significant effort was involved in the implementation of Wynn's k-exchange algorithm into a Fortran 90 software package used to create D-optimal experimental designs. This was acutely needed as both commercial statistical analysis software and our previously developed genetic algorithm-based D-optimal experimental design software proved inadequate for the 10 and 25 variable HSCT optimization problems.


Figure 5. Theoretical efficiency (solid line) and actual parallel efficiency (circles) for performing aerodynamic analyses on an Intel Paragon.

6. Future Research Directions

Some avenues for future work relating to this grant have been identified above. These include the integration of Euler drag estimates into the five and 10 variable HSCT optimization problems, a study of DACE modeling schemes for use in both aerodynamic and structural modeling, and the continued application of coarse-grained parallel computing methods to reduce the computational expense of HSCT optimization. Furthermore, we also are currently applying both aerodynamic and structural response surface models to the full 29 variable HSCT optimization problem.

A major challenge is the incorporation of simple and more complex analyses in a single response surface model. We envision, for example, a response surface model that will integrate the results of tens of thousands of algebraic models, hundreds or thousands of panel models and dozens of CFD models for drag calculations. Both weighted least squares polynomial approximations as well as DACE approximations may be used for this problem.

7. References

  1. Craig, J. A., Jr., ``D-Optimal Design Method: Final Report and User's Manual,'' General Dynamics Report FZM-6777 for Air Force Contract F33615-78-C-3011, 1978.

  2. Giunta, A. A., Balabanov, V., Haim, D., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Wing Design for a High-Speed Civil Transport Using a Design of Experiments Methodology,'' Proceedings of the 6th AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA, 1996, pp. 168-183.

  3. Giunta, A. A., Balabanov, V., Haim, D., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Aircraft Multidisciplinary Design Optimization Using Design of Experiments Theory and Response Surface Modeling,'' (to be submitted) Aeronautical Journal, 1997.

  4. McGrory, W. D., Slack, D. C., Applebaum, M. P., and Walters, R. W., GASP Version 2.2 Users Manual , Aerosoft Inc., Blacksburg VA, 1995.

  5. Burgee, S., Giunta, A. A., Balabanov, V., Grossman, B., Mason, W. H., Haftka, R. T., and Watson, L. T., ``A Coarse Grained Parallel Variable-Complexity Multidisciplinary Optimization Paradigm,'' (to appear) Intl. J. Supercomputing Appl. , 1996.

Highlights of Research Progress 1994-1997

7 December 1993 - 3 October 1994

4 October 1994 - 9 October 1995

9 October 1995 - 24 January 1997

List of Publications 1997-1997

1994

  1. Burgee, S., Watson, L. T., Giunta, A. A., Grossman, B., Haftka, R. T., and Mason, W. H., ``Parallel Multipoint Variable-Complexity Approximations for Multidisciplinary Design,'' Proceedings of the IEEE Scalable High-Performance Computing Conference, 1994, pp. 734-740.

  2. Giunta, A. A., Dudley, J. M., Narducci, R., Grossman, B., Haftka, R. T., Mason, W. H., and Watson, L. T., ``Noisy Aerodynamic Response and Smooth Approximations in HSCT Design,'' Proceedings of the 5th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Panama City Beach, FL, 1994, pp. 1117-1128. (AIAA Paper 94-4376)

1995

  1. Burgee, S., ``A Coarse Grained Variable-Complexity MDO Paradigm for HSCT Design,'' M.S. Thesis, VPI\&SU, 1995.

  2. Burgee, S., Giunta, A. A., Grossman, B., Haftka, R. T., and Watson, L. T., ``A Coarse Grained Variable-Complexity Approach to MDO for HSCT Design,'' Proceedings of the Seventh SIAM Conference on Parallel Processing for Scientific Computing, Editors: Bailey, D. H., Bjorstad, P. E., Gilbert, J. R., Masagni, M. V., Schreiber, R. S., Simon, H. D., Torczon, V. J., and Watson, L. T., SIAM, Philadelphia, PA, 1995, pp. 96-101.

  3. Burgee, S., and Watson, L. T., ``The Promise (and Reality) of Multidisciplinary Design Optimization,'' Abstract, IMA Workshop on Large-Scale Optimization, Minneapolis, MN, 1995.

  4. Giunta, A. A., Balabanov, V., Kaufman, M., Burgee, S., Grossman, B., Haftka, R. T., Mason, W. H., and Watson, L. T., ``Variable-Complexity Response Surface Design of an HSCT Configuration,'' Proceedings of ICASE/LaRC Workshop on Multidisciplinary Design Optimization, Hampton, VA, 1995. (to be published by SIAM)

  5. Giunta, A. A., Narducci, R., Burgee, S., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Aerodynamic Optimization of a High Speed Civil Transport on Parallel Computers,'' Proceedings of the First World Congress on Structural and Multidisciplinary Optimization, Goslar, Germany, 1995, pp. 765-769.

  6. Giunta, A. A., Narducci, R., Burgee, S.,Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Variable-Complexity Response Surface Aerodynamic Design of an HSCT Wing,'' Proceedings of the 13th AIAA Applied Aerodynamics Conference, San Diego, CA, 1995, pp. 994-1002. (AIAA Paper 95-1886)

  7. Giunta, A. A., Balabanov, V., Burgee, S., Grossman, B., Haftka, R. T., Mason, W. H., and Watson, L. T., ``Variable-Complexity Multidisciplinary Design Optimization Using Parallel Computers,'' in Proceedings of the International Conference on Computational Engineering Science (ICES), Mauna Lani, Hawaii, July, 1995. see also Computational Mechanics '95 - Theory and Applications, Editors: Alturi, S. N., Yagawa, G., and Cruse, T. A., Springer, Berlin, 1995, pp. 489-494.

  8. MacMillin, P. E., Huang, X., Dudley, J., Grossman, B., Haftka, R. T., and Mason, W. H., ``Multidisciplinary Optimization of the High-Speed Civil Transport,'' Proceedings of ICASE/LaRC Workshop on Multidisciplinary Design Optimization, Hampton, VA, 1995. (to be published by SIAM)

  9. Watson, L. T., Burgee, S., Balabanov, V., Giunta, A. A., Grossman, B., Mason, W. H., Narducci, R., and Haftka, R. T., ``Software Engineering of Parallel Disciplinary and MDO Codes,'' Abstract, 1995 SIAM Annual Meeting, 1995.

1996

  1. Balabanov, V., Kaufman, M., Giunta, A. A., Haftka, R. T., Grossman, B., Mason, W. H., and Watson, L. T., ``Developing Customized Wing Weight Function by Structural Optimization on Parallel Computers,'' Proceedings of the 37th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference and Exhibit, Salt Lake City, UT, 1996, pp. 113-125. (AIAA Paper 96-1336)

  2. Balabanov, V., Kaufman, M., Knill, D. L., Haim, D., Golovidov, O., Giunta, A. A., Haftka, R. T., Grossman, B., Mason W. H., and Watson, L. T., ``Dependence of Optimal Structural Weight on Aerodynamic Shape for a High Speed Civil Transport,'' Proceedings of the 6th AIAA/NASA/USAF Multidisciplinary Analysis and Optimization Symposium, Bellevue, WA, 1996, pp. 599-612. (AIAA Paper 96-4046)

  3. Burgee, S., Giunta, A. A., Balabanov, V., Grossman, B., Mason, W. H., Haftka, R. T., and Watson, L. T., ``A Coarse Grained Parallel Variable-Complexity Multidisciplinary Optimization Paradigm,'' (to appear) Intl. J. Supercomputing Appl., 1996.

  4. Giunta, A. A., Balabanov, V., Burgee, S., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Parallel Variable-Complexity Response Surface Strategies for HSCT Design,'' Proceedings of the NASA Ames Computational Aerosciences Workshop 95, Editors: Feiereisen, W. J. and Lacer, A. K., NASA CD Conference Publication 20010, Moffett Field, CA, 1996, pp. 86-89.

  5. Giunta, A. A., Balabanov, V., Haim, D., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Wing Design for a High-Speed Civil Transport Using a Design of Experiments Methodology,'' Proceedings of the 6th AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA, 1996, pp. 168-183. (AIAA Paper 96-4001)

  6. Giunta, A. A., Golovidov, O., Knill, D. L., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Multidisciplinary Design Optimization of Advanced Aircraft Configurations'', keynote paper at the 15th International Conference on Numerical Methods in Fluid Dynamics, Monterey, CA, 1996, (to appear in Lecture Notes in Physics, Springer-Verlag). Also available as MAD Center Report 96-06-01, Virginia Tech, Dept. of Aerospace and Ocean Engineering, Blacksburg, VA, 1996.

  7. Kaufman, M., ``Variable-Complexity Response Surface Approximations for Wing Structural Weight in HSCT Design,'' M.S. Thesis, VPI\&SU, April 1996.

  8. Kaufman, M., Balabanov, V., Burgee, S., Giunta, A. A., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Variable Complexity Response Surface Approximations For Wing Structural Weight,'' AIAA Paper 96-0089, 1996.

  9. Kaufman, M., Balabanov, B., Burgee, S. L., Giunta, A. A., Grossman, B., Haftka R. T., Mason W. H., and Watson, L. T., ``Variable-Complexity Response Surface Approximations for Wing Structural Weight in HSCT Design,'' J. Computational Mechanics, Vol. 18, No. 2, 1996, pp. 112-126.

  10. Kaufman, M., Balabanov, V., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Multidisciplinary Optimization via Response Surface Techniques,'' Proceedings of the 36th Israeli Conference on Aerospace Sciences, Israel, 1996, pp. A-57 to A-67.

  11. Knill, D. L., Balabanov, V., Golovidov, O., Grossman, B., Mason, W. H., Haftka, R. T., and Watson, L. T., ``Accuracy of Aerodynamic Predictions and its Effects on Supersonic Transport Design,'' MAD Center Report 96-12-01, Virginia Tech, Dept. of Aerospace and Ocean Engineering, Blacksburg, VA, 1996.

  12. Knill, D. L., Balabanov, V., Grossman, B., Mason, W. H., and Haftka, R. T., ``Certification of a CFD Code for High-Speed Civil Transport Optimization,'' AIAA Paper 96-0330, 1996.

  13. MacMillin, P., Golovidov, O., Mason, W. H., Grossman, B., and Haftka, R. T., ``Trim, Control and Performance Effects in Variable-Complexity High-Speed Civil Transport Design,'' MAD Center Report 96-07-01, Virginia Tech, Dept. of Aerospace and Ocean Engineering, Blacksburg, VA, 1996.

1997

  1. Balabanov, V., Giunta, A. A., Grossman, B., Mason, W. H., Watson, L.T., and Haftka, R.T., ``Parallel Computing and Variable-Complexity Modeling Strategies for HSCT Design,'' (to appear) Proceedings of the Computational Aerosciences Workshop 96, Moffett Field, CA, 1997.

  2. Giunta, A. A., Balabanov, V., Haim, D., Grossman, B., Mason, W. H., Watson, L. T., and Haftka, R. T., ``Aircraft Multidisciplinary Design Optimization Using Design of Experiments Theory and Response Surface Modeling,'' (to be submitted) Aeronautical Journal, 1997.

  3. Haim, D., Giunta, A. A., Holzwarth, M. M., Mason, W. H., and Watson, L. T., ``Suitability of Optimization Packages for an MDO Environment,'' (submitted to) Engineering Computation, 1996.

  4. MacMillin, P. E., Golovidov, O., Mason W. H., Grossman, B., and Haftka, R. T., ``An MDO Investigation of the Impact of Practical Constraints on an HSCT Configuration,'' AIAA Paper 97-0098, 1997.