Akira Oyama

Ohio Aerospace Institute, Brook Park, Ohio

Meng-Sing Liou

Glenn Research Center, Cleveland, Ohio

Multiobjective Optimization of Rocket Engine

Pumps Using Evolutionary Algorithm

NASA/TM—2001-211082

August 2001

AIAA–2001–2581

https://ntrs.nasa.gov/search.jsp?R=20010094059 2018-05-14T14:45:43+00:00Z
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Prepared for the

15th Computational Fluid Dynamics Conference

sponsored by the American Institute of Aeronautics and Astronautics

Anaheim, California, June 11–14, 2001

NASA/TM—2001-211082 AIAA–2001–2581

Akira Oyama

Ohio Aerospace Institute, Brook Park, Ohio

Meng-Sing Liou

Glenn Research Center, Cleveland, Ohio

Multiobjective Optimization of Rocket Engine

Pumps Using Evolutionary Algorithm
Acknowledgments

The work was supported under the NASA’s ISE and RAC programs. The authors would like to thank

Dr. J. Veres of NASA Glenn Research Center for providing the pump code.

Available from

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7121 Standard Drive

Hanover, MD 21076

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Available electronically at http://gltrs.grc.nasa.gov/GLTRS
NASA/TM—2001-211082 1

MULTIOBJECTIVE OPTIMIZATION OF ROCKET ENGINE PUMPS

USING EVOLUTIONARY ALGORITHM

Akira Oyama

Ohio Aerospace Institute

Cleveland, Ohio 44142

440–962–3148, aoyama@dino.grc.nasa.gov

Meng-Sing Liou1

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

216–433–5855, meng-sing.liou@grc.nasa.gov 

ABSTRACT

A design optimization method for turbopumps of

cryogenic rocket engines has been developed.

Multiobjective Evolutionary Algorithm (MOEA) is

used for multiobjective pump design optimizations.

Performances of design candidates are evaluated by

using the meanline pump flow modeling method based

on the Euler turbine equation coupled with empirical

correlations for rotor efficiency.

To demonstrate the feasibility of the present approach,

a single stage centrifugal pump design and multistage

pump design optimizations are presented. In both

cases, the present method obtains very reasonable

Pareto-optimal solutions that include some designs

outperforming the original design in total head while

reducing input power by 1%. Detailed observation of

the design results also reveals some important design

criteria for turbopumps in cryogenic rocket engines.

These results demonstrate the feasibility of the EAbased design optimization method in this field.

INTRODUCTION

While the budget for space development programs has

drastically shrunk in most countries, recent and future

space missions increasingly demand high performance

and reliable rocket engine systems and components,

such as turbopumps. Progress in computational fluid

dynamics (CFD) methods and development of

powerful computational facilities have contributed to

the reduction in required cost and time to develop

advanced turbopump designs. The design process still

largely depends on experienced designers. Therefore,

numerical design methods coupled with CFD, which

are capable of efficiently developing advanced

turbopump designs, can greatly reduce such

dependency.

Among numerical optimization algorithms, gradientbased methods are long-standing and most widely used

approaches1-3. These methods use the gradient of an

objective function with respect to changes in design

variables to calculate a search direction using steepest

descent, conjugate gradient, quasi Newton techniques,

or adjoint formulations. The solution obtained by these

methods will be a global optimum, only if the objective

and constraints are differentiable and convex4

.

Unfortunately, the distribution of an objective function

of real-world design problems is usually multimodal

and one could only hope for a local optimum

neighboring the initial design point. Therefore, to

determine the global optimum, one must optimize from

a number of initial points and check for consistency in

the optima obtained. In this sense, the gradient-based

methods are not robust.

Evolutionary Algorithms (EAs, for example, see [5])

are emerging design optimization algorithms modeled

on the mechanism of natural evolution. EAs search

from multiple points, instead of moving from a single

point. In addition, they require no derivatives or

gradients of the objective function. These features lead

to robustness and simplicity in coupling with any

evaluation codes. Parallel efficiency also becomes very

high by using a simple master-slave concept for

function evaluations, if such evaluations consume most

of CPU time. Design optimization using CFD is a

typical case. Application of EAs to multiobjective

design problems is also straightforward because EAs

maintain a population of design candidates in parallel.

Due to these advantages, EAs are unique and attractive

approach to real-world design optimization problems.

Recently, EAs have been successfully applied to

aerospace design optimization problems5-9.
NASA/TM—2001-211082 2

The objective of the present study is to develop and

demonstrate a design optimization method for

turbopumps used in the cryogenic rocket engines. The

Multiobjective Evolutionary Algorithm (MOEA) will

be used for multiobjective optimization of pump

designs. Performances of design candidates will be

evaluated by using the meanline pump flow modeling

method based on the Euler turbine equation coupled

with empirical correlations for rotor efficiency. Present

approach will be applied to centrifugal and multistage

turbopump design optimization problems.

EVOLUTIONARY ALGORITHMS

EAs mimic mechanism of natural evolution, where a

biological population evolves over generations to adapt

to an environment by selection according to fitness,

recombination and mutation of genes (Fig.1). When

EAs are applied to optimization problems, individual,

fitness, and genes usually correspond to a design

candidate, an objective function value, and design

variables, respectively. One of the key features of EAs

is that it searches from multiple points in the design

space, instead of moving from a single point like

gradient-based methods do. Furthermore, these

methods work on function evaluations alone and do not

require derivatives or gradients of the objective

function. These features lead to the following

advantages:

1) Robustness: EAs have capability of finding a

global optimum, because they don’t use function

gradients that direct the search toward a local

optimum. In addition, EAs have capability to

handle any design problems that may involve nondifferentiable objective function and/or a mix of

continuous, discrete, and integer design

parameters.

2) Capability of sampling various Pareto-optimal

solutions in parallel: Real-world design

optimization problems typically involve multiple

and often competing objectives. The solution to

such problem is not a unique optimal solution, but

a set of compromised solutions, largely known as

Pareto-optimal solutions. Each of those solutions is

optimal in the sense that no improvement can be

achieved in one objective component that does not

lead to degradation in at least one of the remaining

components. Therefore, primary goal of a

multiobjective optimization problem is, unlike that

of a single objective optimization, to find various

Pareto-optimal solutions to show the precise

tradeoff information among the completing

objectives. By maintaining a population of

solutions and introducing the concept of Paretooptimality, EAs can uniformly sample various

Pareto-optimal solutions in parallel.

3) Suitability to parallel computing: Since EAs are

population-based search algorithms, all design

candidates in each generation can be evaluated in

parallel by using the simple master-slave concept.

Parallel efficiency is also very high, if objective

function evaluations consume most of CPU time.

4) Simplicity in coupling evaluation codes: As these

methods use only objective function values of

design candidates, EAs do not need substantial

modification or sophisticated interface to

evaluation codes. If an all-out re-coding were

required to every optimization problem like the

adjoint methods, extensive validation of the new

code would be necessary every time. EAs can save

such troubles.

The present MOEA uses floating-point representation,

where an individual is characterized by a vector of real

numbers. It is natural to use the floating-point

representation for real parameter optimization

problems instead of binary representation because it is

conceptually closest to the real design space, and

moreover, the string length is reduced to the number of

design variables. Fonseca’s Pareto-based ranking

method10 is used for fitness assignment where an

individual’s rank corresponds to the number of

individuals in the current population that are better than

the corresponding individual in every objective

function. To maintain diversity in the population, a

standard sharing function10 is incorporated. As the

elitism, the best-N selection1 1 is incorporated, where

the best N individuals are selected for the next

generation among N parents and N children based on

Pareto-optimality5

 so that the Pareto-optimal solutions

will be kept once they are formed. Parents are selected

from the best N individuals randomly. To generate new

design candidates from these parents, the blended

crossover (BLX-a) is used, which is the most common

approach for recombination of two parents represented

by a vector of real numbers proposed by Eshelman and

Schaffer12. In this approach, children are generated on a

segment defined by two parents, but the segment may

be extended equally on both sides determined by a user

specified parameter a. Thus, a child solution is

expressed as:

Child1= g ◊ Parent1+ (1-g )◊ Parent2 (1)

Child 2 = (1-g )◊ Parent1+g ◊ Parent2 (2)

where

g = ( 1 + 2a ) u - a (3)

Child1, Child2 and Parent1, Parent2 denote design

parameters of the children and parents, respectively. u

is uniform random number in [0,1]. Schematic view of

BLX-a is shown in Fig. 2. When an EA is applied to a

design optimization problem, what is important is

balance of two conflicting goals: exploiting good
NASA/TM—2001-211082 3

solutions and exploring the search space13. Thus, BLX0.5 is used in which both exploration and exploitation

are carried out equally. Since the strong elitism is used,

high mutation rate of 0.2 is applied and a random

disturbance is added to the parameter in the amount up

to ± 20% of the design space. Population size and

maximum number of generations are set to 100 and 90

for the centrifugal pump design, 100 and 120 for the

multistage pump design. Unbiased initial population is

generated by randomly spreading solutions over the

entire design space in consideration.

PUMP PERFORMANCE EVALUATION

Total head and required input power of pump design

candidates are evaluated by using a one-dimensional

meanline pump flow modeling method14, which

provides a fast capability for modeling turbopumps

within rocket engines. The components of the inlet and

exit fluid velocity triangles are calculated at the hub,

mean and tip locations along the rotor blades. The

meridional velocity of the fluid at the rotor leading

edge root-mean-square diameter CM1 [ft/sec] is defined

by equation (4).

1 1

1

144

A

m

CM = r

(4)

where

m: Mass flow [lbs/sec]

r1: Fluid density at leading edge [lbs/ft3]

A1: Flow area at leading edge [inch2

]

Flow area is calculated from the input flow path

dimensions.

A [ ] B( ) R R block = l p hub + tip - (5)

where

l: Boundary layer blockage factor

B: Blade span from hub to tip [inch]

Rhub: Radial distance from pump centerline

at hub [inch]

Rtip: Radial distance from pump centerline

at tip [inch]

The metal blockage of the rotor block is calculated by

eq. (6).

sin b

thk B Z block ◊ ◊ = (6)

where

thk: Normal blade thickness [inch]

Z: Blade number

b: Relative angle from tangential [degree]

The tangential component of velocity entering the rotor

is calculated in terms of the swirl angle of the flow a1

by equation (7).

/ tan( ) CU1 = CM1 a1 (7)

The meridional and tangential components of absolute

fluid velocity at the rotor trailing edge are calculated by

equations (8) and (9).

C M = m

A 2

2 2

144

r (8)

CU U 22 2 = U +W (9)

where

r2: Fluid density at trailing edge [lbs/ft3]

A2: Flow area at trailing edge [inch2]

U2: Blade tangential velocity at trailing edge

[ft/sec]

WU2: Tangential component of relative fluid velocity

at trailing edge [ft/sec]

Flow area at trailing edge is calculated by eq. (5). The

blade tangential velocity U and tangential component

of the fluid relative velocity WU2 are given by equations

(10) and (11), respectively.

720

2 R N U = p ◊ ◊ (10)

W = C +U (1- ) U 2 M 2 ◊ tan b 2 2 s (11)

where

R: Radial distance from pump centerline [inch]

N: Shaft rotative speed [rpm]

b2: Relative angle from tangential at trailing edge

[degree]

The slip factor s is defined by

s = - slip

U 2

1 (12)

The slip is the difference between the theoretical and

absolute fluid tangential velocities. For centrifugal

impellers, Pfleiderer correlation to geometry15 is used

to calculate the slip factor s. A default slip factor of

0.95 is used for inducers.

The head rise through the rotor is calculated iteratively

from the Euler turbine equation coupled with empirical

correlations for rotor efficiency

( )

hyd

c

U U

g

U C U C H ◊h ◊ - ◊ = 2 2 1 1

2 (13)

where

H2: Head rise through the rotor [ft]

hhyd: Rotor hydraulic efficiency

gc: Gravitational constant, 32.174[lbm-ft/lbf-sec2

]

The rotor hydraulic efficiency is obtained from

empirical correlations to rotor-specific speed17. The

total pressure and static pressure at the rotor exit are

estimated from the rotor head rise by equations (14)

and (15).

+P 144

H P = t1

2

t2

1-2 ◊ r (14)
NASA/TM—2001-211082 4

2 144 g

C

P = P -

c

s t

◊ ◊

◊ 2

2

2

2 2

r (15)

where

Pt1: Total pressure at the leading edge [psia]

Pt2: Total pressure at the trailing edge [psia]

r1-2: Average density of the fluid from the leading

edge to the trailing edge [lbs/ft3

]

Ps2: Static pressure at the trailing edge [psia]

Fluid absolute velocity at trailing edge C2 [ft/sec] is

defined by

2

2

2 C2 = CM 2 +CU (16)

Total pressure at the discharge of the last stage Pt4

[psia] is given by the following equation

( ) Pt 4 Pt 2 2 4 Pt 2 - Ps2 = - ◊ w - (17)

where the design point total pressure loss coefficient of

the diffusion system is assumed to be known and is

input in terms of a normalized loss coefficient w2-4. The

total head rise through pump is calculated by

1 4

4 1

4

144 ( )

-

◊ - = rPt Pt H (18)

where

H4: Total head rise through pump[ft]

Pt4: Total pressure at the pump exit [psia]

r1-4: Average density of the fluid from the inlet

to the discharge [lbs/ft3

]

The input power required to drive the rotor is

calculated from the head rise through the rotor, mass

flow, rotor hydraulic efficiency, mechanical efficiency,

volumetric efficiency and disk pumping loss as

mech

disk

hyd vol

PL m Hinput

h h h

2 1 ◊

 ̃

 ̃

 ̄

ˆ

Á

Á

Ë

Ê + ◊

◊ = (19)

where

input: Input power [hp]

hmech: Mechanical efficiency

hvol : Volumetric efficiency

PLdisk: Disk Pumping loss [hp]

The mechanical efficiency is assumed to be 0.98 and

the volumetric efficiency is based on internal leakages

and is expressed as the ratio of leakage to the inlet

flow. The disk pumping loss is calculated from

empirical correlations to geometry, fluid density at

rotor trailing edge, and the shaft rotative speed16.

During the calculation, local static pressure at the rotor

tip is compared to the local vapor pressure to check for

the cavitation inception point.

To estimate off-design total head and required input

power, the empirically derived variation of slip factor

and rotor efficiency as a function of flow-speed ratio F

is used. Correction factor is also applied to the total

pressure loss coefficient of the diffusion system as a

function of loading parameter L.

2 3 1.534988 0.6681688 F 0.077472 F 0.0571508 F

design

= - ◊ + ◊ + ◊ s

s

(20)

2 3

,

0.86387 0.3096 F 0.14086 F 0.029265 F

hyd design

hyd = + ◊ - ◊ - ◊ h

h (21)

2 3 1.8151 1.83527 L 0.8798 L 0.18765 L

design

= - ◊ + ◊ + ◊ w

w

(22)

The loading parameter is defined in terms of the velocities

at the vaneless diffuser exit and the velocity at the

diffusion system throat.

2

3

2

U 3 M

throat

C C

C L

+

= (23)

where

Cthroat: Fluid absolute velocity at the diffusion system

throat [ft/sec]

CU3: Tangential component of fluid absolute velocity

at vaneless diffuser exit [ft/sec]

CM3: Meridional component of fluid absolute

velocity at vaneless diffuser exit [ft/sec]

The velocity at the diffusion system throat is defined by

the equations (24) and (25).

A

m

CM =

3 3

3

144

r (24)

3

2 (1 ) 3 2 3 2 RR CU CU = - ◊ ◊ v - (25)

where the pressure loss coefficient at the diffuser exit w2-3

is assumed to be 0.1. Fluid velocity at the throat is given

by the equation (26).

A

m

C =

3 throat

throat

144

r (26)
NASA/TM—2001-211082 5

CENTRIFUGAL PUMP DESIGN

First, redesign of a single-stage centrifugal pump, M-1

oxygen scaled tester is demonstrated. Objectives of the

present design problem are maximization of total head and

minimization of input power at a design point. These

objectives are competing and therefore the solution to this

optimization problem is Pareto-optimal solutions.

The design point is shaft rotative speed of 5,416.7[rpm],

total temperature of the fluid entering the pump of 545

[Rankine], total pressure of the fluid entering the pump of

50.0 [psia], and mass flow into the pump of 188.7

[lbm/sec].

Design parameters are rotor leading edge tip radius (Rtip1),

rotor trailing edge radius (R2), volute tongue radius (R3),

blade span at trailing edge (B2), blade span at volute

tongue (B3), axial length of the blade at the root-meansquare diameter (S), number of blades (Zn), blade thickness

(thk), blade trailing edge angle at the hub, root-meansquare radius, and tip (bhub , bmid , btip) as shown in Fig. 3.

Table 1 presents present design spaces.

Total head and input power of Pareto-optimal designs,

original design, and all other design candidates are

illustrated in Fig. 4. Designs that have cavitation are

eliminated from the figure. Present Pareto-optimal

solutions successfully displays tradeoff information

between maximization of the total head and minimization

of the input power. Such tradeoff information is very

helpful to a higher-level decision-maker in selecting a

design with other considerations. Among the Paretooptimal solutions, some designs outperform the original

design in the total head while reducing the input power by

1%.

Figure 5 compares overall performance maps of the

original design, the highest total head design, the lowest

input power design, and a compromised design that

overcomes the original design in both objectives.

Moreover, the compromised design is seen to improve the

exit pressure in all off-design conditions. The design

parameters of these designs are shown in Table 2.

The absolute flow velocity at rotor exit hub is shown in

Fig. 6. This figure indicates the optimum designs have

small exit flow velocity, which contributes to minimizing

the total pressure loss in the diffusion system. By

minimizing the total pressure loss in the diffusion system,

designs can improve their total head rise. To minimize the

exit flow velocity, the optimum designs have small slip

factor values, i.e., large slip than others. Actually, all three

optimum designs in Table 2 have small axial length, while

the high head design and the compromised design

maximize R2 to increase slip due to the inertial effect (Low

input design minimize R2 to minimize input power). The

low input design and the compromised design also

minimize their blade angle at hub and tip to reduce

absolute fluid velocity at rotor exit (the high head design

maximize blade trailing edge angles to improve its total

head). However, it is known that non-uniform radial

velocity around the periphery of the impeller due to large

slip degrades its head rise. Therefore, this effect should be

counted in the future study.

Figure 7 shows the total pressure loss coefficient of the

diffusion system of the designs. Pareto-optimal solutions

successfully minimize it to increase their total head rise.

According to these detailed observations of the results,

present MOEA obtained reasonable Pareto-optimal

solutions, which ensure the feasibility of the present design

optimization approach in rocket engine pump designs.

MULTISTAGE PUMP DESIGN

Next, present design optimization method is applied to the

redesign of RL10A-3-3A liquid oxygen pump consisting

of one inducer and a single centrifugal impeller, followed

by a vaneless diffuser and conical exit volute. The

objectives are maximization of total head and

minimization of input power at the design point, which is

shaft rotative speed of 12,900[rpm], total temperature of

the fluid entering the pump of 175 [Rankine], total

pressure of the fluid entering the pump of 40.0 [psia], and

mass flow into the pump of 40.0 [lbm/sec]. Design

parameters and the corresponding parameter ranges are

shown in Fig. 8 and Table 3, respectively.

Figure 9 shows total head and input power of Paretooptimal designs, original design, and all other design

candidates that have no cavitation. Though this design

optimization problem involves two stages and a large

number of design parameters, the present MOEA finds

reasonable Pareto-optimal solutions including some

designs that improve both total head and input power by as

much as 1%.

Figure 10 shows overall performance maps of the original

design and optimized designs. The compromised design

improves the exit pressure in all off-design conditions. The

design parameters of these designs are shown in Table 4.

Figures 11 and 12 illustrate the head rise and the required

input power of the first and the second stages. Because the

exit of the first stage connects with the inlet of the second

stage directly, the relation between the head rise and input

power becomes linear. The optimum designs increase their

head rise through the first stage because the slope of the

curve consisting of Pareto-optimal solutions in Fig. 12 is

steeper than that of the line in Fig. 11. To increase the head

rise through the first stage, the highest head design and the

compromised design increase Rtip, b 2hub, b2rms, and b2tip.

Another interesting thing is that the second stages of the

Pareto-optimal designs are not optimal by themselves,

especially in the high head region. This is due to the

nonlinear interactions between the first and the second

stages, which make a multistage pump design very

difficult.

Figure 13 shows fluid velocity at rotor exit hub. The

Pareto-optimal designs have a small fluid velocity at exit

like the single stage pump design. However, the optimum

designs that have the total head of 950-1200 inches do not
NASA/TM—2001-211082 6

minimize their total head. This is probably due to

complicated interactions between the first and second

stages.

Figure 14 is the total pressure loss coefficient of the

designs. This figure is also interesting because the optimal

designs in low total head region minimize their total

pressure loss coefficient but the optimal designs in the high

total head region maximize it. More work is necessary to

understand the multistage pump designs.

SUMMARY

In the present study, a design optimization method for

cryogenic rocket engine turbopumps has been developed.

Multiobjective Evolutionary Algorithm is used for the

multiobjective optimization of pump designs.

Performances of design candidates are evaluated by using

the meanline pump flow modeling method, which is based

on the Euler turbine equation coupled with empirical

correlations for rotor efficiency.

To demonstrate the feasibility of the present approach,

single stage centrifugal pump design and multistage pump

design optimizations are presented. In both cases, present

method obtains reasonable Pareto-optimal solutions that

include designs outperforming the original design in total

head as well as input power by 1%. Detailed observation

of the design results also reveals some important design

policies in turbopump design of cryogenic rocket engines.

These results ensure the feasibility of EA-based design

optimization method in this field.

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pp.206-213, 1993.

[12] Eshelman, L. J. and Schaffer, J. D., “Real-coded

genetic algorithms and interval schemata,”

Foundations of Genetic Algorithm 2, Morgan

Kaufmann Publishers, Inc., San Mateo, CA, pp.187-

202, 1993.

[13] Booker, L. B., “Improving Search in Genetic

Algorithms,” Genetic Algorithms and Simulated

Annealing, Morgan Kaufmann Publishers, Inc., San

Mateo, CA, pp.61-73, 1987.

[14] Veres, J. P., “Centrifugal and Axial Pump Design and

Off-Design Performance Prediction,” NASA TM

106745, Lewis Research Center, Cleveland, Ohio,

October 1994.

[15] Kovats, A., “Design and Performance of Centrifugal

and Axial Flow Pumps and Compressors,”

Macmillan, New York, 1964.

[16] Stepanoff, A. J., “Centrifugal and Axial Flow Pumps:

Theory, Design, and Applications,” Wiley, New

York, 1957.
NASA/TM2001-211082 7 

initial population 

evaluation of designs

selection according to fitness

recombination of selected designs

mutation applied to new designs

Next

generation

Figure 1. Flowchart of the present evolutionary 

algorithm. 

Parent1 Parent2

α⋅d d α⋅d

Possible sampling region

Figure 2. Blended crossover. 

R2 R3

R1tip S

B2

B3

thk βtip

βhub

βrms

Zn : number of blades

Figure 3. Design parameters of the centrifugal pump 

design problem. 

Table 1. Design parameter ranges of the centrifugal 

pump design problem. 

design variables R1tip R2 R3 B2 B3 S

[inch] [inch][inch][inch][inch][inch]

Upper boundary 4.00 5.60 6.20 0.85 1.00 4.30

Lower boundary 3.40 5.00 5.60 0.70 0.85 3.70

design variables β hub β rms β tip thk Zn

[degs.][degs.][degs.][inch]

Upper boundary 25.0 25.0 25.0 0.03 6

Lower boundary 45.0 45.0 45.0 0.10 18

350

400

450

500

550

600

650

700

750

600 700 800 900 1000 1100 1200

all design candidates

Pareto-optimal designs

Input [HP]

Head [inch]

Low input design

Compromise

High head design

Original design

Figure 4. Objective function values of the centrifugal 

pump designs. 

0

100

200

300

400

500

600

0 500 1000 1500 2000

low input design

compromise

high head design

original design

EXIT PRESSURE[PSIA]

FLOW[GPM]

N=5416.7

N=4875.0

N=4333.4

N=3791.7

N=3250.0

N=2708.4

N=2166.7

N=1625.0

N=1083.3

N=541.7

Figure 5. Pump overall performance map. 
NASA/TM2001-211082 8 

Table 2. Pareto-optimal designs of the centrifugal 

pump design problem. 

design variables R1tip R2 R3 B2 B3 S

[inch] [inch][inch][inch][inch][inch]

High head design 3.50 5.60 6.05 0.846 0.871 3.74

Low input design 3.71 5.00 6.08 0.701 0.859 3.74

Compromised design 3.92 5.59 5.63 0.730 0.878 3.76

Original design 3.66 5.34 5.91 0.814 0.908 4.00

design variables β hub β rms β tip thk Zn

[degs.] [degs.][degs.][inch]

High head design 44.9 36.0 44.6 0.100 18

Low input design 25.3 33.2 26.1 0.065 6

Compromised design 26.7 37.2 26.4 0.042 7

Original design 35.0 35.0 35.0 0.050 12

Figure 6. Flow velocity at second stage exit hub. 

Figure 7. Total pressure loss coefficient of the 

centrifugal pump designs. 

R2hub

R3

R1tip

S

B3

B4

β2hub β2rms β2tip thk2

R4-R3

β3 thk3 Zn3

Figure 8. Design parameters of the multistage pump 

design problem.

Table 3. Design parameter ranges of the multistage 

 pump design problem. 

design variables Rtip R2hub β 2hub β 2rms β 2tip thk2

[inch] [inch] [degs.][degs.][degs.][inch]

Upper boundary 1.18 0.53 50.0 40.0 35.0 0.08

Lower boundary 1.08 0.43 35.0 25.0 20.0 0.03

<1st stage>

design variables R2 (R4-R3) B2 B3 thk3 S Zn3 β 3

[inch] [inch][inch][inch] [inch] [inch] [degs.]

Upper boundary 2.20 0.15 0.35 0.50 0.08 1.00 16 90.0

Lower boundary 2.00 0.05 0.15 0.30 0.03 0.80 8 60.0

<2nd stage>

Figure 9. Objective function values of the multistage 

pump designs. 

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

600 700 800 900 1000 1100 1200 1300

Total pressure loss coefficient

head [inch]

Compromise

Original design

Low input design

120

140

160

180

200

220

240

260

600 700 800 900 1000110012001300

C2hub [ft/sec]

Head [inch]

Low input design

High head design

Original design

Compromise

80

90

100

110

120

130

140

150

160

800 900 1000 1100 1200 1300 1400 1500

all design candidates

Pareto-optimal designs

Input [HP]

Head [inch]

Compromise

Low input design

Original design

High head design

High head design
NASA/TM2001-211082 9 

Figure 10. Pump overall performance map. 

Table 4. Pareto-optimal designs of the multistage 

 pump design problem. 

design variables Rtip R2hub β 2hub β 2rms β 2tip thk2

[inch] [inch] [degs.][degs.][degs.][inch]

High head design 1.17 0.517 49.9 38.5 34.9 0.0788

Low input design 1.17 0.525 42.9 37.3 28.0 0.0378

Compromised design 1.17 0.514 48.3 40.0 33.4 0.0520

Original design 1.13 0.480 43.0 27.3 21.6 0.0400

<1st stage>

design variables R3 (R4-R3) B3 B4 thk3 S Zn3 β 3

[inch][inch] [inch][inch] [inch] [inch] [degs.]

High head design 2.20 0.1350 0.340 0.301 0.0311 0.855 16 87.4

Low input design 2.00 0.0548 0.152 0.393 0.0798 0.804 8 64.4

Compromised design 2.16 0.0608 0.333 0.370 0.0439 0.812 9 77.8

Original design 2.10 0.0950 0.251 0.400 0.0300 0.878 12 90.0

<2nd stage>

Figure 11. First stage performances of the multistage 

pump designs. 

Figure 12. Second stage performances of the 

 multistage pump designs. 

0

100

200

300

400

500

600

700

800

-50 0 50 100 150 200 250 300

original design

low input design

high head design

compromise

EXIT PRESSURE[PSIA]

FLOW[GPM]

N=12900

N=11610

N=3870

N=5160

N=6450

N=7740

N=9030

N=10320

N=1290

N=2580

4

6

8

10

12

14

16

20 40 60 80 100 120 140 160 180

First stage input [HP]

First stage head [inch]

High head design

Compromise

Low input design

Original design

60

70

80

90

100

110

120

130

140

600 800 1000 1200 1400

Second stage input [HP]

Second stage head rise [inch]

Compromise

High head design

Low input design

Original design
NASA/TM2001-211082 10 

Figure 13. Flow velocity at second stage exit hub. Figure 14. Total pressure loss coefficient of the 

multistage pump designs. 

140

160

180

200

220

240

260

800 900 100011001200130014001500

C3hub [ft/sec]

Total head [HP]

Low input design

High head design

Compromise

Original design

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0.25

800 900 100011001200130014001500

W

3-4

Total head [inch]

High head design

Compromise

Original design

Low input design
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16

Multiobjective optimization; Evolutionary algorithm; Pumps

Unclassified -Unlimited

Subject Categories: 05 and 07 Distribution: Nonstandard

Prepared for the 15th Computational Fluid Dynamics Conference sponsored by the American Institute of Aeronautics and

Astronautics, Anaheim, California, June 11–14, 2001. Akira Oyama, Ohio Aerospace Institute, 22800 Cedar Point Road,

Brook Park, Ohio 44142; and Meng-Sing Liou, NASA Glenn Research Center. Responsible person, Meng-Sing Liou,

organization code 5880, 216–433–5855.

Akira Oyama and Meng-Sing Liou

Multiobjective Optimization of Rocket Engine

Pumps Using Evolutionary Algorithm

A design optimization method for turbopumps of cryogenic rocket engines has been developed. Multiobjective Evolutionary Algorithm (MOEA) is used for multiobjective pump design optimizations. Performances of design candidates are

evaluated by using the meanline pump flow modeling method based on the Euler turbine equation coupled with empirical correlations for rotor efficiency. To demonstrate the feasibility of the present approach, a single stage centrifugal

pump design and multistage pump design optimizations are presented. In both cases, the present method obtains very

reasonable Pareto-optimal solutions that include some designs outperforming the original design in total head while

reducing input power by 1 percent. Detailed observation of the design results also reveals some important design criteria

for turbopumps in cryogenic rocket engines. These results demonstrate the feasibility of the EA-based design optimization method in this field.