NASA Technical Reports Server (NTRS) 19980006552: Shape Optimization by Bayesian-Validated Computer-Simulation Surrogates

176 Pages·6.6 MB·English

by  NASA Technical Reports Server (NTRS)

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Final Report
NASA Grant NAG 1-1613
"Shape Optimization by Bayesian-Validated Computer-Simulation Surro_,aO"tes _T
Principal Investigator:
Professor Anthony T. Patera
Department of Mechanical Engineering
Massachusetts Institute of Technology
December 30, 1997

SUMMARYOFWORK
The application of large-scale simulation to problems in shape optimization is very difficult
for (at least) three reasons. First, the objectives can not always be stated a priori, and thus
there is reluctance to expend resources. Second, simulations are still very expensive, at
least for most complex physical problems, and thus optimization -- which calls for
repeated appeal to the forward problem -- is often prohibitively costly. Third, many
interesting problems in shape optimization also require variations in topology: the latter
introduces singularities that frustrate many classical optimization procedures.
In this grant, we have proposed an approach to shape optimization that addresses, or at
least mitigates, these difficulties. First, as regards how the problem is posed, we have
applied concepts from multi-criterion optimization theory, in particular, Pareto theory. The
Pareto formulation permits the designer to identify (monotonic) preferences in certain
performance metrics (e.g., lift and drag), and then obtain a trade-off curve between these
variables; all points on this trade-off curve are optimal in the sense that there is no design
point at which both metrics can be improved (e.g., both lift increased and drag decreased).
This trade-off curve can then be used to help the designer identify the right balance between
different preferences, and determine the optimal operating point once these preferences are
established.
Pareto optimization has been used only rarely (if ever) for complex problems in fluid
dynamics because the determination of the trade-off curve constitutes a difficult rain-max
problem that requires many appeals to the (expensive) simulation. Our approach to
alleviating this computational bottleneck is to replace the simulation with a simulation
surrogate, In particular, we first construct a model, or surrogate, of the input-output
behavior of the simulation -- based on some small set of input-output pairs -- and then use
this surrogate, not the original simulation, in the resulting (Pareto) optimization. The
surrogate, unlike the original simulation, is very inexpensive to evaluate, and thus
extensive optimization can be performed; of course, the surrogate may also be significantly
less accurate than the originating simulation.
To address the accuracy question, we have developed an extensive statistical validation
procedure and associated theory for understanding how well a surrogate is performing, and
whether any particular surrogate-predicted result can be trusted. The latter is, in fact,
facilitated by the Pareto framework, since the region of input space of interest is narrowed
to the pre-image of the trade-off curve. In particular, if the number of performance metrics
of interest (e.g., lift, drag) is small, the Pareto pre-image will typically be a low-
dimensional manifold, even if there are many design variables (inputs).
Finally, in order to treat the topology issue, we have implemented a level-set geornett'y
description within the surrogate context. [n this approach, shapes are described as level
sets of parametrized functions such that (discontinuous) changes in geometu can be
described by continuous changes in parameters which, in turn, correspond to continuous
changes in the performance metrics, The latter thus permits the use of gradient
information, critical to any efficient search algorithm.
The ingredients described above have been demonstrated in the simulation-based
optimization of a compact (laminar-flow) heat exchanger. The performance metrics in the
Pareto analysis are the pumping power and heat removal; the simulation is a full unsteady
Navier-Stokes calculation, on the basis of which a simple (radial-basis function) surrogate
input-output model is constructed; level sets are used to describe the geomet_T of the eddy
promoters through which the flow is excited and the thermal-hydraulic performance
improved.

The results of this study indicate that our approach can prove quite effective in practice,
permitting us to address problems not amenable to other techniques. However simple
("connect-the-dot") surrogate methods are fundamentally limited to rather low-dimensional
inputs spaces; although the Pareto approach permits us to validate a surrogate over a low-
dimensional manifold, we must first construct the surrogate over the entire input space. As
is well-known, approximation in many dimensions is plagued by the "curse of
dimensionality" _ there are simply too few points in each coordinate to represent the
function unless a prohibitively expensive (exponential) number of input-output pairs is
used.
One promising approach to improving the dimensionality scaling of surrogates is to exploit
state-space based models -- models that contain not only simple input-output information,
but also information originating in the underlying mathematical system that describes the
phenomenon. In a follow-on grant we will be studying this possibility, using for our
"surrogate" a low-dimensional numerical approximation, and for our validation procedure a
new form of a posteriori error estimation theory. The latter also replaces the statistical
results of our earlier work with purely deterministic bounds (albeit at some loss in
generality).
A detailed description of the results of the current grant are included in Appendix A in the
form of a recently completed Ph.D. thesis by John Otto.

PUBLICATIONS
1. M.E. Kambourides, S. Yesilyurt, and A.T. Patera, Nonparametric-validated computer-
simulated surrogates: A Pareto formulation. International Journal of Numerical
Methods in Engineering, to appear.
2. J.C. Otto, D. Landman, and A.T. Patera, A surrogate approach to the experimental
opthnization ofmultielement airfoils. Technical Report, AIAA Paper 96-4138CP,
1996.
3. J.C. Otto, M. Paraschivoiu, S. Yesilyurt, and A.T. Patera. Bayesian-validated
computer-simulation surrogates for optimization and design. In N.M. Alexandrov and
M.Y. Hussaini, eds., Multidisciplinary Design Optimization: State of the Art, pp. 368-
392, 1995.
4. J.C. Otto, M. Paraschivoiu, S. Yesilyurt, and A.T. Patera, Computer-simulation
surrogates for optimization: Application to trapezoidal ducts and axisymmetric bodies.
Technical Report, Proceedings of the Forum on CFD for Design and Optimization,
ASME International Mechanical Engineering Conference and Exposition, 1995.
5. J.C. Otto, M. Paraschivoiu, S. Yesilyurt, and A.T. Patera. Bayesian-validated
computer-simulation surrogates for optimization and design: Error estimates and
applications. IMACS Journal of Mathematics and Computers in Simulation, to appear.
6. S. Yesilyurt, C. Ghaddar, M. Cruz, and A.T. Patera, Bavesian-vali_htted surrogates
for noisy comDuter simulations," application to random n_edia. SIAM Journal of
Scientific Computing, 17 (1996), pp. 973-992.
7. S. Yesilyurt and A.T. Patera, Surrogates for numerical simulation: optimization of
eddy-promoter heat exchangers'. Comp. Methods Appl. Mech. Engr., 121 (I 995), pp.
231-257.
8. M. Paraschivoiu and A.T. Patera, A hierarchical duality approach to bottnds for the
outputs of partial differential equations, Comp. Meth. Appl. Mech. Engrg., to appear.
9. M. Paraschivoiu, J. Peraire, and A.T. Patera, A posteriorifinite elemeltt botoldsfor
linear-functional outputs of elliptic partial d!fferential equations, Comp. Meth. Appl.
Mech. Engrg., 150 (1997), pp. 289-312.
I0. M. Paraschivoiu, J. Peraire, Y. Maday, and A.T. Patera, Fast botmdsfor OlttptttS of
partial differential equations, in J.A. Burns, E.M. Cliff, and B. Grossman, eds.,
Proceedings AFOSR Workshop on Optimal Design and Control, Washington, D.C., to
appear.
11. J. Peraire and A.T. Patera, Bounds for linear-functional outputs of coercive partial
differential equations: local indicators and adaptive refinement, in P. Ladeveze and J.T.
Oden, eds., Proceedings of the Workshop on New Advances in Adaptive
Computational Methods in Mechanics, Cachan, France, Elsevier, 1997.

APPENDIX A