AlAA 95 - 3886 Optimal Computational Methodology For HSCT Design: Interdisciplinary Coupling M. Human North Carolina A&T University Greensboro, NC

1 st AlAA Aircraft Engineering, Technology, and Operations Congress

September 19-21,1995/Los Angeles, CA For permission to copy or republish, contact the American institute of Aeronautics and Astronautics 370 L'Enfant Promenade, S.W., Washington, D.C. 20024

AIAA-95-3886,

OPTIMAL COMPUTATIONAL METHODOLOGY FOR HSCT DESIGN: INTERDISCIPLINARY COUPLING

Me1 Human' North Carolina A&T University

Greensboro. North Carolina

Abstract

A number of aircraft analysis and design problems can be performed in a "quasi-static" fashion where performance is evaluated at dif- ferent positions in the flight trajectory. A key element is determining the "state" of the air- craft - flow field response which involves an iterative coupling between the two appropri- ate analysis codes. The problem which is of concern here is that of optimally integrating a propulsion system within some airframe con- struct, a classic multidisciplnary design.

This paper discusses how such a problem can be classified in terms of the numerical pro- cedure which may "best" solve it. The ap- proach of a single level optimization, and nested analysis appears to be the appropriate solution methodology. Such an schema provides a con- nectivity architecture among these three disci- plines that can be generalized for a niimber of possible objective functions. Lift to drag ra- tio, and propulsion specific thrust are the two examples discussed.

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INTRODUCTION

Multidisciplinary Design and Optimization (MDO) techniques have enjoyed a number of applications, particularly in the aeroipace ve- hicle arena'-5. Many of these applications in- volve integration of syst-ems within an 'overall vehicle concept6-? High desees of interdisci- plinary coupling males such approaches almost essential in new system designs. There are a number of disciplines which enter into the de- sign of this type of vehicle; here we will con- cern ourselves with three - stri.ictiires, flow field analysis, and propulsion. Subsequently, there is a need to develop methodologies for analyzing such highly coupled systems

The diagram in Figure 1 shows the function- ality among the various discipline groups which would be involved in a siipersonic/hypersonic aircraft design proc.ess. The High Speed Civil- ian Transport (HSCT) is a prime example of such a highly integrated system. Each disci- pline exchanges data with the others via vec- tors of coupling variables. Depending on the problem, note that each exchange need not be both ways. w -

dsso. Prof. AIAA Sr. Mern. In this paper we will concern ourselves with " C o p y r i g h t @ 1 9 9 5 by t h e Amer ican I n s t i t u t e of A e r o n a u t i c s and 1 A s t r o n a u t i c s , 1nc:'All r i g h t s reserved. 'I

the engine airframe integration problem and related optimization issues. That is for some prescribed objective function, what is the besf engine box design relative to the airframr. In Figure 2, we show the three module subsets of interest - structures, flow field,and propiilsion. For example, a delta wing design with an nsso- ciated family of potential box designs is shown in Figure 3.

Let us attach physical relevance to t,he sit,ii- ation depicted in Figure 1. For a given set. of flight conditions, such as cruising at a level al- titude, we establish freestream velocity and at,- mospheric ambient density. The flow field nnal- ysis calculates pressure loadings acting o r 1 t,he vehicle, and also flow characteristics entrring the engine inlet. The former collection of tiat,a is direct input for the structural analysis rode, the latter directly affects the propulsion calcu- lations. The structaral module uses the load information to determine deformations arid de- flections; this in essence defines a new shape which must be reintroduced to the compiita- tional fluid dynamics (CFD) module. This pro- cess may be continued until convergencc. t,hat is when the values of the coupling variables is consistent with the results of both mudirles. There is also bilateral communication bet.uveen the fluid flow and propulsion modules. The flow field affects the engine performance which in turn is a factor in the flow calculation

For a matter of simplicity and wit,hoiit loss of generality, we define two global indeprndent variables, the longitudinal position of the en- gine intake and its spanwise edge, respedvely xe and ze .Note how this selection effectively de- couples the structure and propulsion mudirles; obviously structure design would generallv dic- tate the component layout of the engine con- figuration but maintaining the same base con- figuration eliminates this connectivity. ‘These couplings are bypassed by formulat,ing the hi- erarchy as shown.

It, many structural optimization problems, there exist a ”library” of possible designs from which to build from, the end product often some combination of members from the collec- t,ion. It would be useful to know how changes of one discipline affects the performance of oth- ers. Another key issue is what designs are most sensitive to parameter changes; obviously it is easier to control processes which do not exhibit large variability as design quantities change.

A legitimate question now becomes what are the possible approaches of optimizing such a system, and whether some methods are supe- riur to others. It is this issue we wish to explore

FORMULATION METHODOLOGY -. Optimization problem In general we are concerned with obtaining

the best design or combination of subsystems for optimal overall performance. An optimiza- t,iun problem would require the definition of some objective (may be more than one) func- tiun. An important factor in aero-vehicle de- sign is the lift/drag ratio L / D ; we may wish to optimize its value for the lengthy cruise phase of t,he mission. Note that this function implic- itly depends on the control variables as for level constant velocity flight at vehicle relative free stream velocity u,

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L I D = Mg/T = F[Mi, f , mor uait; u,] The instantaneous mass (f) and nozzle exit

velocity is a propiilsion module output, and the air flow rate is dependent on the flow field, both of which depend on the ( zc ,ze ) values. For a predetermined set of engine conditions (nozzle exit velocity and fuel air ratio) and flight con- dition u,, this can be rewritten as

[‘[ma; CL, GI, Afl where all four quantities are functions of xe

and z,. We have included the drag influencing frontal area which is merely x,z,.

The complete optimization problem would include inequality constraints snch as lower bounds on quantities like thrust or lift (every L

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'd flight condition must meet marginal reqnire- ments although off design will be suboptimal), equality constraints which relate functionality among variables or specific functional require- ments, and side conditions which places limits on variables.

This can be an involved optimization as it is a nonhierarchic structure, that is t.he disci- plines are coupled to each other rather than to only a top level module in a tree like fash- ion. However, traditional hierarchic methods are applicable to these systems after some re- formulation.

Nomenclature Before actually formulating an optimization

problem, we must define the required nomen- clature. We will use the convention defined in the excellent paper by Balling and Sobieski" where the following are vector quantities.

: global system design variables XI: structure design variables Xz: flow field design variables x 3 : propulsion design variables ,?I: structure state variables 92: flow field state variables $3: propulsion state variables

: struct,ure flow coupling TIS: structure propulsion coupling l&: flow structure coupling Y..3: flow propulsion coupling Y31: propulsion structure coupling &z :propulsion flow coupling Design variables are defined as input to the

various modules, the system design vector con- tains parameters which are utilized in all mod- ules. The state variables are quantities associ- ated within each discipline. Coupling variables are quantities which are calculated in the i th module and is required by the j t h modules, as represented by the convention y Z j . Finally, we also define vectors of "residual" variables & which are essentially the difference between the common values calculated by the ith and j t h

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'd'

modules. Accordingly converged solutions have residual values near zero or beneath some pre- scribed tolerance.

As the stated problem is to investigate the influence of engine box geometry, the system design variables are defined as earlier stated. Of course in general this vector may be of dif- ferent size and content. The structure module will require information necessary to perform a finite element analysis (FEA). This would be airframe geometry with the engine box loca- tors referenced accordingly. Similarly, the flow module needs specifications for a CFD calcula- tion, and the propulsion modules needs its own set on initialization input. Suitable for these functions would be freestream flight conditions such as velocity and ambient air properties. A potential design input for the engine would be the overall pressure ratio.

Subsequently, key state variables are inter- nally calculated within each module; note that these computations may occur in parallel. Re- sulting state variables would be those associ- ated with describing the aircraft shape, sur- rounding flow field, and engine performance, respectively.

The structure would couple into the flow field and propulsion modules via any deflec- tion/deformation changes. Flow will influence structure and propulsion by resulting pressure coefficients and air mass capture rate, respec- tively. The third pair of coupling quantities originating in the propulsion module, would be propulsive force on the airframe and the ex- haust jet velocity characteristics.

At this point, it would be useful to review the situation as exhibited in Figure 2. Any optimization problem concerning these three disciplines would be nonhierarchic in nature. To actually execute propulsion calculations, the "subproblem" of a consistent fluid-structure so- lution must be solved. The question exists as what is the Lest computation procedure to ef-

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fectively address such a problem.

Approaches Balling and Sobieski distinguishes bet.xeen

system level and discipline level analyses which then may be of either an evaluative or optimiza- tion mode. There is also a further distinct,ion as whether an optimization is being performed simultaneously, either on the system or disci- pline level. For the class of problems WP are discussing here, there is no actual optimiz a t ' .ion occurring at the discipline levels as we are ron- cern with a system wide optimum - the ovwall optimal design. This is referred to as single level optimization as only the external syst,em level routine determines all design variablrs.

Because of the highly connected nat,iire of the design and state variables in the thrw dis- ciplines of interest, most procedures will have their determination occurring sirnultanr~~isly. This is referred to as simultaneous analysis and design (SAND). Finally on the system Icvel, the optimization procedure could operatr ei- ther on the system design variables alonr. or these quantities along with the coupling vari- ables. Thus we may employ either a ShKD 01- nested analysis and design (NAND) ap- proach. An example to illiistrate this tliffer- ence is that the fluid structure coupling ral- culation appears somewhat independent from the propulsion module computations, in which cme the SAND approach would appear t,lir~ best (as it would be simpler). However, a s l o ~ con- vergence rate between the first two inudriles could be addressed by incorporating t,hc roil- pling variables between the two in the optirniza- tion algorithm, a NAND procedure.

According, these appear to be the tnru clas- sifications for attacking this problem as stated. They are summarized by the notations, single- SAND-SAND and single-SAND-NAND. Ilow- ever, certain objective functions might involve a nested approach within the disciplines; hence the single-NAND-NAND formulation s h d d

also be examined. Obviously, the optimization v proidem statement will affect the formulation; accordingly, test problems will have to be de- fined before further discussion.

OPTIMIZATION EXAMPLE PROBLEMS Lift to drag ratio As mentioned before, a critical parameter in

aircraft performance is the lift to drag ratio. This is a parameter which we like to have max- imized throughout as much of the flight path as possible. We could then formulate the problem:

maximize L / D si1bjec.t to J ( x ) = 0 !J(2) 5 0 pi , nhere f and g are equality and inequality con-

straints respectively, and p are lower and upper boiinds on selected variables.

In this problem, the objective function must inwlve an interdisciplinary calculation. For any given position in the trajectory, we must es- tablish a consistent flow field - structure state, a resiilt of the iterations between these two mod- ides. Once this is established, propulsion per- formance can be computed.

Observe the sequence of calculations. A con- verged solution between the structure and fluid modules must be obtained; coupling variables siich as structure boundary locus are passed be- tveen the two. No optimization is occurring at, this level. Once we have a converged solution, me may determine the lift and drag coefficients (despite high degrees of structural flexibility, we do not expect changes in frontal and lift area t,o be significant). Appearing in the constraints will be requirements such as thrust levels and dynamic pressure values corresponding to the desired trajectory. At this point constraint fea- sildity will be checked by the propulsion mod- iile.

It can be seen from Figiire4,that this is a single-NAND-NAND approach. The "master" cuntrol program meshes the structure and fluid

xi 5 pi2 for various 2's

L

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calculations, calls the propulsion constraint check, and proceed to implement the system optimization.

Propulsion performance As stated in the previous section, the pri-

mary influences upon the propulsion plant are the mass flow decrement due to forebody drag and temperature changes of the inlet air due to boundary layer interaction with the vehicle sur- face. This is is readily calculated at the engine inlet by

ma = SpudA Thrust is then computed by the usual expres-

sion of

T = %[(I + f ) U e z i t - u,] The optimization problem for specific thrust

max Tlm, with the constraints following the same form

as in equations, although a different set of vari-

In this set of calculations, we must still estab- lish the state of the system, structure - flow field compatibility. Now the objective function is calculated and subject to optimization directly from the propulsion module. This procedure can easily be fitted in the single-SAND-NAND approach, but once again because of the struc- ture fluid coupling, the presence of the con- trol program is required, suggesting as before a single-NAND-NAND design.

A key point should be made here that de- pending on the flight conditions, the conver- gence to the system state could potentially be slow. A method to address this would be inror- porating a gradient optimization in that, loop for the purpose of speeding the convergence. In this case the met.hodology is t,hat of the single- NAND-NAND; note that the optimizer is not concerned with the st,ate of t.he disciplines but only in the design and convergence of the coli- pling variables.

could now be defined as

’--/ ables will be included.

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COMPUTATIONAL COSTS

We shall conclude with a brief discussion about the optimal program design with respect to computer analysis costs. Each module will have a number of calculations per ”module call” which is dependent on the type of analysis be- ing performed.

The structural analysis will in general be some form of finite element analysis. If we as- sume that each node in the model has three degrees of freedom, the analysis has 3N degrees of freedom (dof). In general for larger models, the solution cost is proportional to d o f 2 , so we have a 9N2 cost per finite element evaliiation.

Finite difference based analysis which many fluids are based is also predicated on solving a matrix equation, although a general rule of thumb is not as simple as structural analysis. Indeed, we make the assumption that the for- mer is linear, bnt fluid equations are inherently nonlinear which demand that the solution re- quires a number of iterations. Note that the order of the fluid equation system will be dif- ferent than that of the structure, denoted as

The state of the flow field can be defined at each grid point with three velocity compo- nents, two thermodynamic properties, and also value for turbulent kinetic energy and dissipa- tion rate, a total of seven quantities. If we re- quire N I f iterations to attain a flow field solu- tion, then the number of calculations per solu- tion can be approximated by (73/3)NIf N/”.

Accordingly, attaining a converged strncture fluid solution will require 9N2 + (343/3)NI f (S f )Nj . The number of iterations will be state dependent, that is some function of trajectory position (density and temperature), pressure and density gradients, among other effects. Because we are assuming to be in the linear domain for the structural responses, only a single pass is required here.

The propulsion calculation will involve the

N f .

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solution of Np state variables depending on t,he number of components in the system. The cal- culation cost here is proportional to N ; / 3 .

Finally, the computations in an optimization analysis is proportional to the cube of tJie num- ber of design variables. Depending on t,he niim- ber of optimization iterations No, the tot,nl com- putational cost per design position (flight. pat,h) becomes

No[9N2 + (343/3)NI,(Sf)N,3 + q / 3 . In a typical model, the number of st,l-iict,nral

and fluid grid nodes will be much largrr than the number of engine components, SO these dominate the cost.

SUMMARY The problem of arriving at an integrnt,ed ve-

hicle design has been discussed from the st,and- point of classifying the computational method- ology of approaching such problems. Kut, only can the numerical scheme be systemat,ically classified, but general conclusions can be ap- plied to such schema, resulting in possible man- agerial tools for conducting such analyscs. The estimation of program cost with respect ti, com- puter resources is such an issue and this is dis- cussed. The analysis approach suggeskd here could be applied to a number of analysis and optimization objectives.

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2. Barthelemy, J., G.A. Wrenn. A.R. Dovi, P.G. Coen: and L.E.Hal1, "Siipcrsonic Transport Minimum Weight Design Int,egrat- ing Aerodynamics and Structures", Joiirnal of Aircraft, V. 32, # 2, March-April, 1994. pp.

3. Dovi, A. et. al., "Multidisciplinary Design Integration System for A Supersonic Transport Aircraft," 4th Symposium on MDO, AIAA pa-

330-338.

per # 92-4841, Cleveland, Ohio, September, L 9 1994.

4. Bowen, Z., 61 Q. Zhide, "Multiobjec- iive Optimization Design of Transonic Air- foils," 19th Congress of International Council of Aeronautical Sciences (ICAS), # ICAS-94- 21.1, Los Angeles, Ca. September, 1994.

5. Haftka, R.T., "On Options for Interdis- ciplinary Analysis and Design Optimization", Structiiral Optimization, V. 4, #2, June, 1992.

6. Berry, D. L., "The Boeing 777 En- gine/Airframe Integration Aerodynamic De- sign Process". 19th Congress of ICAS #ICAS- 9.1-6.4.4, Los Angeles, Ca. Sept,. 1994.

7. Rossow, C.C. et. al., "Investigation of Propulsion Airframe Integration Interference Effects on a Transport Aircraft Configuration", AIAA-92-3092, 1992

8. Chen, A. et. al., "TRANAIR Applica- t,ions to Engine/Airframe Integration", AIAA-

9. Tinoco, E.N. and A.W. Chen, "Transonic CFD Application to Engine/Airframe Integra- t,ion," AIAA-84-381.

10. Balling, R.J., and Sobieszczanski- Sobieski, "Optimizat,ion of Coupled Systems: A Critical Overview of Approaches," AIAA 94- 4330-CP.

89-2165CP, August, 1989. 'LJ

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Figure: 1 Iiitcrdisciplinory Flow C h a r t

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