PARAMETERIZED GEOMETRY FORMULATION FOR INVERSE … · 4th International Conference on Inverse Problems in Engineering Rio de Janeiro, Brazil, 2002 dependence, this way suggesting

4th International Conference on Inverse Problems in EngineeringRio de Janeiro, Brazil, 2002

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Helmut SobieczkyGerman Aerospace Center (DLR)

Bunsenstrasse 10, D-37073 Göttingen, Germany [email protected]

George S. DulikravichDepartment of Mechanical and Aerospace Engg.

The University of Texas at Arlington Arlington, TX 76019, U.S.A.

[email protected]

Brian H. DennisInstitute of Environmental Studies,

University of Tokyo7-3-1 Hongo, Bunko-ku, Tokyo 113-8656, Japan

[email protected]

PARAMETERIZED GEOMETRY FORMULATIONFOR INVERSE DESIGN AND OPTIMIZATION

ABSTRACTThis contribution focuses on the importance of

preprocessing tools for successful design and opti-mization in practice of turbomachinery engineer-ing. The development of problem-orientedcomputational geometry generation software isillustrated for the example of aerodynamic inversedesign of transonic flow elements which define thecompatible boundary conditions (surfaces) indetail. Resulting from learned sensitivity of highspeed flows to small changes in airplane wing orturbomachinery blade geometry, preprocessingsoftware is provided to create parametric shapes tobe varied for optimization cycles or numericalsimulation of mechanical adaptation processes.Supporting the need to design from a multidisci-plinary viewpoint, parameterized geometry com-ponents for aerodynamic, as well as for thermaland structural considerations are defined. Exam-ples for turbomachinery blade design and optimi-zation are given.

INTRODUCTIONIn the past years with rapid expansion of com-

puter speed and storage, and improvements inalgorithm speed and accuracy, optimization strate-gies have become affordable and reliable. Compu-tational analysis and simulation of physicalphenomena therefore become valuable designtools to improve technological performance of aproduct component. Here we focus on the complex

technology of coupling the aerodynamics, structural and thermal loading as occurring in turbomachinery component design. In this situation wneed realistic and flexible surface modelling tprovide boundary conditions produced systemacally and in rapid succession, with variations controlled by suitable and efficient sets of parameter

High speed aircraft design is posing similacoupled problems, as outlined in [1]. Here we ussome of the chapters in this book to be adapted afurther developed for turbomachinery problemslike aerodynamic blade design with thermal anstructural constraints.

With geometry data of a machine componenbeing the common database for desirable aerodnamic, thermodynamic and structural consideations, to name only the most important odisciplines relevant for successful product deveopment in the early engineering phase, we shouexplain some of the background of these fieldsfar as they have influenced parametrization of ogeometry preprocessor.

GASDYNAMIC PHENOMENA, INVERSEAND DIRECT DESIGN TOOLS

Flow machinery, just like aircraft wings andother free form shapes with a need of refined suface quality is sensitive to the physical phenomenespecially in the high speed domain. The knowedge base of transonic and supersonic gasdynaics tells us about regions of influence an


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Figure 1: Principle of inverse and direct supersonic design; application of both inverse and directmethods to redesigning parts of the contour for a turbine blade test case.

dependence, this way suggesting the definition ofsection geometry definition observing the lack ofupstream influence of any shape changes in certainregions.

Applied to the design of turbine blades we usesuch phenomena to perform flow computationsboth in a direct (downstream) or an inverse (cross-flow) marching procedure (Fig. 1). The latterallows to use certain given starting data to com-pute the flow along with a compatible boundarycondition: the shape results from this inverseapproach which, in practice, means that we maycontrol the flow quality to avoid or delay negativeeffects like separation and obtain design hints howto shape a blade to actually observe this desiredflow quality.

Figure 1 shows the principle, infinitesimallystarting from known data along (12), finding thesolution within a triangle (123), for potential flowcomputation, or with entropy updates alongstreamlines (14) for an Euler accurate CFD simu-lation. Also shown is the application to an experi-mentally tested turbine blade design: thesupersonic domain is re-constructed by startingfrom the given sonic line (AB) in the inversemode, then continuing downstream of (BC) in thedirect mode.

This transonic design method has been usfor turbomachinery cascades [2] and many aiplane wing design examples [3].

In the following we may not need to use thesmethods but we use geometry preprocessors whuse parametric airfoils and other component funtions which have been tailored using experiencwith this design concept. So we ensure to be cloto desirable conditions in the aerodynamic part othe many needed optimization steps in desigpractice.

GEOMETRY MODELS WITH PARAMETRICSHAPE CONTROL

Results from the above cited inverse approacin aerodynamics have taught us about shape sentivities [4] and consequently about the neederefined parameter definition for the following morerecent and future optimization efforts. In practicadesign, there will be a more multidisciplinaryapproach trying to optimize aerodynamics, structure, thermal properties etc. in a synchronized waHere we call for a setup of parameters for controling the complete set of boundaries to vary thshape for each discipline effectively. Restrictinour illustrations to turbine blade technology, whichmight be resulting from the above illustrated


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design process, now needs to be created includingits structure of coolant passages to allow for adesign optimization including structural and ther-mal loads.

Without knowledge of some physical proper-ties of aerodynamics leading to a suitable parame-trization, the size and shape of the mathematicalspace that contains all the design variables (forexample, coordinates of all blade surface points) isvery large and complex in a realistic cooled bladegeometry. Only when it is possible to use fastflow-field analysis codes could it be affordable tohave an ideal optimization situation where eachsurface grid point on the optimized configurationis allowed to move independently. Otherwise, thedesigner is forced to somewhat restrict the designspace by working with a relatively small numberof the design variables by performing parameter-ization - if not by a specialized software like theone introduced here, for example, by fitting poly-nomials - of either the 3-D surface geometry or the3-D surface pressure. The optimization code thenneeds to identify the coefficients in these polyno-mials. Since it is often necessary to constrain andsometimes not allow motion of certain parts of the3-D surface, the most promising choices for the 3-D parameterization appear to be different types ofBezier functions [5] and the geometry preprocess-ing tools used here which is based on a library ofsuitable analytical functions and successivemanipulations and integrations in 3D cartesiancoordinates ([1], pp 123-136).

This approach allows to vary the airfoil param-eters as found suitable from 2D design (Fig. 1)into the third dimension, to compose a 3D bladewith drastically changing sections as occurringbetween the root and tip sections of a realistic tur-bine blade. Moreover, mathematical description ofevery surface point without any interpolation anditeration to approximate given data, allows for aneasy construction of parallel surfaces as needed tomeet wall thickness constraints. These are crucialwhen the inner structure of the blade needs tohouse a coolant flow passage reducing the heatload on the blade and still maintain structural stiff-ness to support the forces produced by the flowand through structure transferred to yield shafttorque.

A starting geometry for subsequent simula-tions and optimization is illustrated in Fig 2.

In the following chapter, some of our first

results on optimization, will be commentedobtained prior to the availability of the fullyparameterized blade geometry introduced herThe goal is, to learn from bi-disciplinary (aerodynamic-thermal, aerodynamic-structural, thermastructural) optimization, before a truly multidisci-plinary, automated optimization will be feasible.

Finally, the fully parameterized geometry obasic blade with coolant flow passages serves atest bed for varying the parameters following thsuggestions of a structural optimization strategy.

MULTIDISCIPLINARY DESIGN TASKS INTURBOMACHINERY TECHNOLOGY

With presently available materials such anickel-based alloys, gas turbine blades cannwithstand metal temperatures in excess of appromately 1300 K. Internal coolant flow passageaugmented with heat transfer enhancements, suas trip strips or turbulators, impingement coolingbanks of pin fins and miniature heat exchangecan provide significant enhancements of convetion heat transfer. For example, when neededthe initial turbine stages, cooling air can be madto impinge on the leading and trailing edge internal cooling passage surfaces in order to enhanconvection. Impingement cooling schemedemand large leading and trailing edge diametebut this creates thicker blades that can substatially increase aerodynamic losses. Complex heexchangers have two major drawbacks. First, thinduce early transition to turbulence and greatincrease the coolant passage effective frictiowhile moderately increasing the convective hetransfer. Second, manufacture of such compleinternal configurations requires special machininprocesses.

The design variable set defines the geometrythe turbine blade including the external turbine aifoil shape definition, thermal barrier coating thickness, blade wall thickness distribution, and bladinternal strut configurations. The blade stackinaxis, twist, and taper are incorporated into thdesign variable set for three-dimensional bladeWith the execution of this geometry generatioprogram, a set of optimization design variable(the parametric model) is used to represent a vtual (electronic) prototype of the turbine blade ovane. The optimization design variable set controlled the internal coolant passage configuratiothickness variation of the coolant passage wapositions and thicknesses of the internal ribs, an


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Figure 2: Turbine blade with coolant flow duct: Parametric outer shape definition plus meandering ductwithin the blade observing local shape control of duct cross section shape and wall thickness. Blade sur-

face partly removed (a), three-view (b)

die pull angles of the ribs [6].In our first exercise in multidisciplinary designoptimization of internally cooled gas turbineblades, a turbulent compressible flow Navier-Stokes solver was used to predict the hot gas flow-field outside of the blade subject to specified real-istic hot surface temperature distribution. As abyproduct, this analysis provides hot surface nor-mal temperature gradients thus defining the hotsurface convection heat transfer coefficient distri-bution. This and the guessed coolant bulk temper-ature and the coolant passage wall convection heattransfer coefficients create boundary conditions for

the steady temperature field prediction in the bladand thermal barrier coating materials using faboundary element technique. The quasi-ondimensional flow analysis (with heat addition anfriction) of the coolant fluid dynamics is coupledto the detailed steady heat conduction analysisthe turbine blade material. By perturbing thedesign variables (especially the variables defininthe internal blade geometry) the predicted thermboundary conditions on the interior of the bladwill be changing together with the coolant flowparameters. As the optimization algorithm runs,also modifies the turbine inlet temperature. Onc


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the turbine inlet temperature changes significantly,the entire iterative procedure between the thermalfield analysis in the blade material and the compu-tational fluid dynamic analysis of the external hotgas flow-field will be performed again to find abetter estimate for thermal boundary conditions onthe blade hot surface. This global coupling pro-cess, so far, was performed only a small numberof times during the course of the entire optimiza-tion. This semi-conjugate optimization uses sec-tional 2-D blade hot flow-field analysis and asimple quasi 1-D coolant flow-field analysis (Fig.3).

This design methodology was successful atgenerating a wide range of realistic internallycooled turbine blades and vanes, while the surfacemeshing, grid generation, and boundary conditionswere automatically mapped between the interfa-cial surfaces. This information was transferredbetween the various design, optimization, andnumerical analysis tools without user intervention.A constrained hybrid optimization algorithm [7]controls the overall operation of the system andguides the multidisciplinary internal turbine cool-ing design process towards the objectives of cool-ing effectiveness and turbine blade durability.Design variable sets which had generated an infea-sible or impossible geometry, were restored to afeasible shape automatically using a constraintsub-minimization.

There are also possibilities for further improve-ment in the design of cooled turbine blades. Theexternal turbine blade shape could be modified inan effort to make the external aero-thermodynam-ics reduce the amount of heat absorbed by theblade. Each new design of the external airfoilwould require a fully conjugate viscous three-

dimensional steady-state CFD analysis of the hgas flow field and the temperature field inside thblade [8]. This CFD solution would then be useto predict new external heat transfer coefficientas well as provide an aerodynamic constraint funtion so that the efficiency and work of the turbinerow could be fixed [9], [10]

RESULTS ON TURBINE BLADE STRUC-TURAL ANALYSIS

The geometry preprocessing tool based on anlytical functions was already used to model boundary conditions for the automatic structural analysof internally cooled turbine blades. The preprocessing tool can quickly generate realistic coolapassage shapes within a specified outer blade. Tpassage shapes are controlled by a set of paramters that the users provide as input. When combined with automatic grid generation and finiteelement analysis tools, the system is ideal for autmatic parameter studies as well as for design opmization. In the current structural analysis systemthe geometry preprocessing tool generates a mublock structured grid that represents the turbinblade geometry. Another program then automacally generates a surface triangulation [11] anthen another code makes a volume grid composof tetrahedrons [12]. A typical surface mesh ishown in Fig. 4. Once a mesh is generated, a strutural analysis is performed. The current structuranalysis system uses a parallel finite element anysis (FEA) code that can do both linear and nonlinear structural analysis [13]. This code also hathe capability of doing automatic partitioning ofthe mesh as well as automatic FEA. Figureshows an example finite element linear stress anysis result for a turbine blade with coolant passages spinning at 3000 RPM. In this case, thnumber of degrees of freedom was aroun100,000. Two Pentium II 333 MHz processorwere used to compute the solution in roughly 1minutes. With this system, the user only needsinput the parameters that govern the shape of tblade and start the system. Once completed, tsystem provides the detailed stress and displacment field for the turbine blade without any furtheinteraction with the user. It is hoped that whecombined with optimization this automatic geometry generation/analysis system will be a powerfutool for turbine blade design.

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Figure 3. Comparison of external wall temper-ature variations computed at the quarter-rootspan of the second HPT blade of the F100

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4th International Conference on Inverse Problems in EngineeringRio de Janeiro, Brazil, 2002 6

Fig. 4. View of triangular surface mesh from blade tip and from blade root

undeformed deformed

Fig. 5. Result from linear stress analysis of spinning turbine blade with passages, (for illustration, deformations have been multiplied by a factor of 20)


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CONCLUSIONWe have shown some first results of what is

going to be a software system for multidisciplinaryoptimization for turbomachinery components likecascades, stators and rotors. Other applications foraerospace and ground vehicle design seemstraightforward and rather may be less compli-cated: A very close coupling of high speed areody-namics, thermal and extreme structural loadingmay occur only in high speed aircraft design.While showing several results for monodisci-plinary design and first results of bidisciplinaryoptimization, we come to the conclusion and havestressed the fact that fast, flexible and realistic sur-face modelling for practical components is effec-tively supporting any future multidisciplinaryapproach to optimize product components observ-ing advantages and constraints of all mayor disci-plines involved in the operation of the component.Optimization of turbomachinery blades poses first,but strong test cases challenging all aspects of thesimulation software.

REFERENCES1. Sobieczky, H., (editor): New Design Concepts

for High Speed Air Transport. CISM Coursesand Lectures Vol. 366. Wien, New York:Springer (1997)

2. Sobieczky, H., Dulikravich, D. S.: A Computa-tional Design Method for Transonic Turboma-chinery Cascades. ASME paper 82-GT-117,(1982)

3. Sobieczky, H., Seebass, A. R.: SupercriticalAirfoil and Wing Design.Ann. Rev. FluidMech. 16, pp. 337-63 (1984)

4. Klein, M., Sobieczky, H.: Sensitivity of aero-dynamic optimization to parameterized targetfunctions. In: M. Tanaka, G.S. Dulikravich,(Eds.), Inverse Problems in EngineeringMechanics, Proc. Int. Symp. on Inverse Prob-lems in Engineering Mechanics (ISIP2001),Nagano, Japan (2001)

5. Farin, G., Curves and Surfaces for ComputerAided Geometric Design, Second Edition,Academic Press, 1990.

6. Dennis, B. H., Dulikravich, G. S. and Han, Z.-X., 2001, “Constrained Optimization of Turbo-

machinery Airfoil Shapes Using a Navier-Stokes Solver and a Genetic/SQP Algorithm”AIAA Journal of Propulsion and Power, Vol.17, No. 5, 2001, pp. 1123-1128.

7. Dulikravich, G. S., Martin, T. J., Dennis, B. H.and Foster, N. F., 1999, “MultidisciplinaryHybrid Constrained GA Optimization,” Chap-ter 12 in EUROGEN’99 - Evolutionary Algo-rithms in Engineering and Computer ScienceRecent Advances and Industrial Applications,(editors: K. Miettinen, M. M. Makela, P. Neit-taanmaki and J. Periaux), John Wiley & SonsLtd., Jyvaskyla, Finland, May 30 - June 31999, pp. 231-260, 1999...

8. Han, Z.-X., Dennis, B. H. and Dulikravich, G.S., 2001, “Simultaneous Prediction of ExternaFlow-Field and Temperature in InternallyCooled 3-D Turbine Blade Material,”Interna-tional Journal of Turbo & Jet-Engines, Vol. 18,No. 1, pp. 47-58.

9. Martin, T. J. and Dulikravich, G. S., “Aero-Thermo-Elastic Concurrent Design Optimization of Internally Cooled Turbine Blades”,Chapter 5 inCoupledField Problems,Serieson Advances in Boundary Elements (eds:Kassab, A. J. and Aliabadi, M. H.), WIT PressBoston, MA, 2001, pp. 137-184.

10. Martin, T. J. and Dulikravich, G. S., Analysisand Multi-disciplinary Optimization of Inter-nal Coolant Networks in Turbine BladesASME IMECE’01, New York, November 11-16, 2001.

11. Shewchuk, J.R., “Triangle: Engineering a 2DQuality Mesh Generator and Delaunay Triangulator,” First Workshop on Applied Computa-tional Geometry, Philadelphia, Pa, 1996

12. Marcum, D. L. and Weatherhill, N. P.,“Unstructured Grid Generation Using IterativePoint Insertion and Local Reconnection,”AIAAJournal, Vol. 33, No. 9, 1995, pp. 1619-1625.

13. Yoshimura, S. “Development of Computational Mechanics System for Large Scale Anaysis and Design,” Annual Report of JSPSRFTF CS&E Project, 1999, pp.45-56 (in Japanese).http://adventure.q.t.u-tokyo.ac.jp/.

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PARAMETERIZED GEOMETRY FORMULATION FOR INVERSE … · 4th International Conference on Inverse Problems in Engineering Rio de Janeiro, Brazil, 2002 dependence, this way suggesting