Department of Aerospace and Ocean Engineering
HPCCP Progress Report
1994 - 1995
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TABLE OF CONTENTS
- Introduction
- Variable-Complexity MDO
- CFD and FEM Tools for MDO
- Parallel Computations
- Future Research Directions
- Highlights of Present Progress
- References
This progress report covers work conducted under grant
NAG1-1562 for the NASA High Performance Computing and Communications
Program (HPCCP) from October 4, 1994, to October 9, 1995.
The objective of this research is to develop new multidisciplinary
design optimization (MDO)
techniques which employ parallel computing. The design of the High
Speed Civil Transport (HSCT) aircraft is used as a testbed for these
methods. The three tasks identified in our HPCCP proposal are
(i) development of parallel multipoint approximation methods
for the aerodynamic design of the HSCT,
(ii) use of parallel multipoint approximation methods for
structural optimization of the HSCT,
(iii) mathematical and algorithmic development including support in
the integration of parallel computation for items (i) and (ii).
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 (i) and (ii). 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 (iii)
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.
The combined aerodynamic-structural design optimization of the HSCT
employs an efficient parameterization of the aircraft design using
25 design variables to specify the aircraft configuration
along with three design
variables to define an idealized mission profile. Sixty-six
constraints are used to evaluate the aerodynamic, structural, and
performance criteria influencing the HSCT design. Even with a
modest design problem the computational cost of MDO is prohibitive
when using many traditional preliminary design level tools. For
this reason, a variable-complexity modeling (VCM) technique is
employed whereby both simple and detailed models are used in the
design optimization. In VCM, simple conceptual design level methods,
typically algebraic models,
are frequently used because of their low computational costs.
More accurate, and more computationally expensive,
preliminary design level methods, such as panel level aerodynamics
and simple finite-element models, are periodically used during
optimization to provide updated results.
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.
Customized Response Surface Modeling Methods
Response surfaces are smooth models, typically formed using
polynomial functions, which are used to approximate a selected
function over a design region of interest. A response surface
model is created by performing a statistical technique known as
regression analysis, which primarily consists of a least-squares
curve fitting procedure. Various inferences about the response model
may be made by using other elements of regression analysis along with
another statistical technique known as analysis of variance (ANOVA).
For example, by examining the coefficients of the terms in the
polynomial response surface model and selected ANOVA data, one may
determine which terms in the polynomial model, and thus the associated
design variable combinations, which most affect the
predicted function values.
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^T X)^(-1) X^T Y. The goal is to find the
m points from a set of 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^T X|. Conceivably, one could
consider each of the possible combinations
of m points from the set of l candidate points, evaluate
|X^T 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 [8].
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.
MDO Using Aerodynamic Response Surface Models
During the second year of this grant funding we have applied response surface
modeling along with the VCM technique to a four variable
problem involving HSCT
wing design. Here, the four design variables are leading-edge sweep angle, root
chord, tip chord, and a wing thickness-to-chord ratio scaling factor. The
objective of this study was to further investigate the sources of numerical
noise in the aerodynamic analysis tools and to determine if response surface
modeling of the noisy drag predictions would eliminate the convergence
difficultities previously encountered.
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.
MDO Using Structural Response Surface Models
In previous studies we employed an experience-based statistical weight equation
to account for wing bending material weight. However, since the HSCT is a
new class of aircraft, the weight equation does not account for all features of
the design. A comparison of the weight equation with structural optimization,
based on finite-element analysis, revealed that the weight equation is
suitable for predicting general trends in structural weight, but is unable to
accurately model all effects of changing aircraft geometry.
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[14].
GASP Evaluation
As described above, our VCM method combines simple algebraic relations
and panel level codes to evaluate the aerodynamic characteristics of
the HSCT. To supplement these tools, we have investigated the
feasibility of introducing Euler computations as the next level
in the hierarchy of aerodynamic prediction
techniques for HSCT optimization. As a candidate we selected GASP (General
Aerodynamic Simulation Program) written by Aerosoft, Inc.
We have conducted a study [15]
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.
GENESIS Evaluation and Implementation
In late 1994 we obtained a version of the GENESIS finite-element structural
optimization code which incorporated a reduced file input/output (I/O) format.
Since file I/O severely limits parallel performance it was critical to find
such a program to carry out the numerous structural optimizations needed for
the HSCT design 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.
During the first year of this grant the computer science efforts
centered on gaining
familiarity with the parallel computing environment, the HSCT code, and the
practical issues of MDO. The second year saw considerable progress in
parallel computing, with the parallelization and automation of the entire
MDO loop (Figure 1).
Figure 1. The MDO loop employing variable-complexity
response surface modeling
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.
Aerodynamic Analysis
For the third year of this grant our efforts will focus on the
application of the response surface modeling and VCM techniques
to the full 28 variable HSCT design problem. As was
done in the four design variable example problem, response
surface models
will be used to replace the aerodynamic analysis methods which
introduce numerical noise into the optimization problem. Further,
response surface models will be used to provide more detailed
HSCT configuration data where the current analysis tools are
inaccurate, e.g., response surface modeling of the wing bending
material weight factor.
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.
Structural Analysis
For the 28 variable HSCT design problem we will
continue to improve the accuracy of the response surface
models for the HSCT wing bending material weight. This will be
accomplished through the use of a more refined finite-element
model of the HSCT in GENESIS. Currently the GENESIS models employ
approximately 1000 members whereas the refined models will use
on the order of 5000 members.
Parallel Computation
Goals for the third year are the parallelization and (automatic) incorporation
of the weight response surface into the shell script, the fine-grained
parallelization of the configuration optimization, and visualization of
higher dimensional feasible sets. Fine grained parallelization
of the Euler and finite-element analysis codes will be investigated.
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.
Response Surface Modeling
- We established that numerical noise pervades many aspects
of the calculation, not only in the drag components that we knew about
before. We found ways to eliminate some of these noise sources and we used
response surface models to smooth out the others.
- We have established
a four design variable problem that allowed us to
explore various aspects of response surface methodology for aerodynamic
design. We have similarly used the FLOPS weight equation to explore
response surface methodology for predicting structural weight.
- We have established
that simple constraints may be used to reduce the
design space where response surface models are constructed,
thus greatly improving the accuracy of the response surface models.
- We have established
that the use of intervening or intrinsic variables
instead of design variables, can help in both the accuracy and cost of the
response surface modeling.
- We have shown that ANOVA can be used to reduce the number of
coefficients in a response surface model with minimal loss of accuracy.
Parallelization
- We have employed a
master/slave paradigm to allow efficient parallel calculation of both
the aerodynamic analyses and structural optimizations. With this parallel
computing strategy, we routinely perform approximately ten
thousand aerodynamic analyses and to date we have performed
one thousand structural optimizations of the HSCT.
- We have generalized the
structural mesh generator, and integrated the
various programs used to generate input for the structural optimization to
allow for a more general and more parallelizable program. With that we have
achieved good parallel efficiencies with the process.
Miscellaneous
- We have taken a first step towards visualization of the design space with
a four-dimensional example problem.
- We have undertaken and almost completed a thorough evaluation of the
relative accuracy of the HSCT aerodynamic analysis code and an Euler code.
- 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," in Proceedings of the
IEEE Scalable High-Performance Computing Conference,
May, 1994, pp. 734--740.
- 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,"
AIAA Paper 94-4376, Sept., 1994.
- 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," in
Proceedings of the Seventh SIAM Conference on Parallel Processing
for Scientific Computing, Feb., 1995.
- 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," in Proceedings of
the NASA Computational Aerosciences Workshop, Moffett Field, CA,
March, 1995.
- 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,"
in Proceedings of ICASE/LaRC Workshop on
Multidisciplinary Design Optimization, Hampton, VA, March, 1995.
- MacMillin, P.E., Huang, X., Dudley, J.,
Grossman, B., Haftka, R.T., and Mason, W.H.,
"Multidisciplinary Optimization of the High-Speed Civil Transport,"
in Proceedings of ICASE/LaRC Workshop on
Multidisciplinary Design Optimization, Hampton, VA, March, 1995.
- 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,"
in Proceedings of the First World Congress on Structural and
Multidisciplinary Optimization, Goslar, Germany, May, 1995.
- 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," AIAA 95-1886, June,
1995.
- 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.
- 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,"
Internat. J. Supercomputing Appl., submitted, July, 1995.
- Burgee, S., and Watson, L. T., ``The promise (and reality) of
multidisciplinary design optimization'', in Proceedings of the IMA
Workshop on Large-Scale Optimization, Minneapolis, MN, July, 1995.
- Burgee, S. "A Coarse Grained Variable-Complexity MDO Paradigm for
HSCT Design," M.S. Thesis, VPI&SU, Aug., 1995.
- 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," (to appear) in Proceedings
of the 1995 SIAM Annual Meeting, Oct., 1995.
- 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," (to appear) AIAA 96-0089, Jan., 1996.
- Knill, D.L., Grossman, B., Mason, W.H., and Haftka, R.T.,
"Certification of an Euler code for High-Speed Civil Transport Optimization,"
(to appear) AIAA 96-0330, Jan., 1996.
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Last Update 20 October 1995
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