Pogo Accumulator Optimization Based on Multiphysicsof Liquid Rockets and Neural Networks

Kook Jin Park,∗ JeongUk Yoo,† SiHun Lee,‡ Jaehyun Nam,‡ and Hyunji Kim‡

Seoul National University, Seoul 08826, Republic of Korea

Juyeon Lee§ and Tae-Seong Roh¶

INHA University, Incheon 22212, Republic of Korea

and

Jack J. Yoh,** Chongam Kim,†† and SangJoon Shin**

Seoul National University, Seoul 08826, Republic of Korea

https://doi.org/10.2514/1.A34769

In this study, a numerical analysis of pogo instability in liquid propulsion rockets was conducted and

an optimization of the pogo suppressor was attempted. Pogo analysis was carried out using numerical results

obtained via the major models for fuselage structure, feedline, and propulsion systems. In the structural system,

the fuselage vibration modes were obtained and the relevant meta-model was constructed using the modal

superposition method. To obtain accurate results for the hydraulic transmission line modeling, cavitation effects

were also taken into account. Thus, a numerical analysis was performed on a pump inducer to provide the

quantitative information of the cavitation volume in the liquid-oxygen feedline. By employing the rocket

combustion equations, it was confirmed that the dynamic response was fed back to the longitudinal characteristics

of the fuselage structure. In addition, an accumulator was installed to suppress pogo instability. For design

optimization, an artificial neural network was suggested by performing Latin hypercube sampling. The sampling

verifies the convergence by the learning process. Finally, amulti-objective optimization for the pogo accumulatorwas

achieved with the present meta-model.

Nomenclature

�A� = fluid inertance matrixAj = plane of action

�B� = feedline geometry matrix�C� = fluid compliance matrixCf;i = fluid compliance

Cp;LOX = liquid-oxygen heat capacity at constant pressure

�D� = fluid resistance matrixeij = unit vector in the direction of point i from point jfij = transfer function coefficients determined from the

feedline characteristicsfi;j = minimization function for the local maximum of the

transfer functionf0 = mass objective functionGj = structural response

gij = transfer function coefficients of the meta-model fromcomputational fluid dynamics

H = feedline and engine feedbackhij = transfer function coefficients of the liquid propulsion

engine equation

Ii = fluid inertanceki = transfer function coefficients of combustion equa-

tion

Mnn= generalized mass

MW = molar massPLOX;0 = feedline-averaged pressure

p = pressure of the fluidpi = MFLUID pressureQLOX;0 = feedline-averaged volumetric flow rate

q = jet fuel heat of the combustionqn = generalized coordinateqnp = generalized coordinate of the pressure

qnq = generalized coordinate of the volumetric flow rate

Rfi = fluid resistance

T = temperature of the fluidu = x-direction velocity of the fluid_ui = MFLUID velocityVcav = cavitation volume ratioWi = weighting factor for the optimization processαLOX = liquid-oxygen thermal expansion coefficientγLOX = liquid-oxygen specific heat ratioδPLOX = variation of feedline pressureδpLOX = perturbation of δPLOX

δQLOX = variation of feedline volumetric flow rateδqLOX = perturbation of δQLOX

δrtank = variation of the local displacement at tankδT = variation of the engine thrustζn = structural damping ratioρ = density of the fluidσj = point source value of the fluid located at rjϕn = structural modeφ = local mode thrust position shapeφt = local mode tank shape_ω = reaction rateωi = natural frequency of the combined numerical modelωn = natural frequency of the fuselage structureωnp = eigenvalue of the pressure mode

ωnq = eigenvalue of the volumetric flow rate

Presented as Paper 2018-5416 at the AIAA SPACE and AstronauticsForum and Exposition, Orlando, FL, September 17–19, 2018; received 2February 2020; accepted for publication 16May 2020; published onlineOpenAccess 26 June 2020. Copyright © 2020 by the American Institute ofAeronautics and Astronautics, Inc. All rights reserved. All requests for copy-ing and permission to reprint should be submitted to CCC at www.copyright.com; employ the eISSN 1533-6794 to initiate your request. See also AIAARights and Permissions www.aiaa.org/randp.

*Contract Assistant Professor, 1 Gwanak-ro, Gwanak-gu.†M.S. Student, Interdisciplinary Program in Space System, 1 Gwanak-ro,

Gwanak-gu.‡Ph.D. Student, Department of Mechanical and Aerospace Engineering.§M.S. Student, Department of Aerospace Engineering, 100 Inha-ro,

Michuhol-gu.¶Professor,Department ofAerospaceEngineering, 100 Inha-ro,Michuhol-gu.**Professor, Department of Mechanical and Aerospace Engineering, 1

Gwanak-ro, Gwanak-gu. Associate Fellow AIAA.††Professor, Department of Mechanical and Aerospace Engineering, 1

Gwanak-ro, Gwanak-gu; also Institute of Advanced Aerospace Technology.

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I. Introduction

P OGO phenomenon is essentially the longitudinal instability ofliquid propulsion rockets caused by the coupling of the fuselage

structure, feedline, and propulsion systems. It has been observed thatthe natural frequencies of the fuselage structure increase as fuel isconsumed, whereas those of the feedline system change with thevarying cavitation volumes generated by the inducer. Because reso-nance occurs between the fuselage structure and the feedline system,multiphysics analyses of the structural vibration, feedline dynamics,pump cavitation, and engine dynamics are necessitated.Throughout thedevelopment history of space launchvehicles, pogo

instability has constantly endangered various manned or unmannedmissions, such as the Titan 2 [1], Diamond B [2], and Saturn 5 [3]. Inparticular, due to the fact that a large longitudinal instability mayjeopardizemanned spacecraft in amission-ending scale, pogoanalysisis required before a launch. A conventional solution to counter pogoinstability is to isolate the natural frequencies of each system andadjust the feedline frequencies with the aid of an accumulator [4,5].Anaccumulator is designed to change the feedline frequencies after

determining the longitudinal instability based on the flight test results.The general methodology is to induce the inertance, compliance, andresistance values based on the relevant dynamic equations of eachsubsystem, and to construct control equations based on the acquiredinformation. Rubin et al. [6,7] particularly suggested the lumpedparameter method, and Oppenheim [8] further developed a relevanttheory for its practical applications. The lumped parameter methodsuggested by Rubin et al. and Oppenheim has been widely used inlaunch vehicle instability analysis, and it provides accurate predic-tions with respect to pogo resonance. Zhao et al. [9] and Wang et al.[10] recentlymade improvements in pogo analysismodeling by usinglumped parameter methods. Their methods were efficient, but thetest result for each subsystem was required before assessing pogoinstability. Therefore, the lumped parameter method was suitablefor the practical design process, yet inapplicable at the preliminarydesign stage. Because the parametric data are only available after thegeometric specifications are determined, performing the modelingprocess based on the geometric information of each componentin the preliminary design phase would be more effective. To do so,numerical analysis needs to be performed for each of the majordynamics, and junction modeling is required to correlate the resultsof the analysis.In this study, a numerical analysis process is suggested for the

longitudinal dynamic system of liquid propulsion rockets. Further, animproved pogo resonance model is proposed to predict pogo insta-bility by using frequency response synthesis and fast Fourier trans-form (FFT). Each system is divided into the fuselage structure,feedline, pump inducer, combustor, and the liquid propulsion rocketengine (LPRE)dynamic system.The fuselage structure ismodeled by

the finite element method, and the feedline is analyzed using fluidtransmission linemodeling based on the geometric information avail-able in the preliminary design stage. Structural natural frequenciesaccurately reflect the change in pressure with varying fuel levels topredict frequency variations in the time domain [11,12]. The struc-tural vibration model of the launch vehicle is composed of a linearmodel, and a reduced-order rational transfer function is constructedby applying the modal superposition method.Meanwhile, by assuming small perturbations, the formulation of a

feedline system can be decoupled into the pressure and volumetricflow rate [13]. Using this advantage, each feedline can be modeledto ensure accurate inertance, compliance, and resistance values. Con-sequently, a rational transfer function similar to that of the fuselagestructural model can be constructed. To interpret the response ofthe liquid-oxygen (LOX) feedline accurately, it is essential to estimatethe cavitation volume and perform a numerical analysis of the pumpinducer. The pump inducer is analyzed using three-dimensional com-putational fluid dynamics (CFD) analysis, and the response of thepump outlet due to the perturbation of the LOX feedline is obtained.Engine components from the pump outlet to the combustion

chamber include the turbine, injector, valve, and the pump; theseare incorporated in the liquid propulsion rocket equation. This equa-tion allows the pressure and mass flow rate before entering thecombustion chamber (and in response to the pump inlet flow rate)to be derived. Finally, the junction of the combustion analysis forms afeedback loop with the structural displacement.The complete system is constructed based on the existing Atlas

launch vehicle for which test results are available. The transfer func-tion in response to the thrust perturbation is obtained, and sine sweepsimulation is performed to find the peak amplitudes to constructthe objective functions. Subsequently, multi-objective optimizationis carried out to minimize the size of the pogo accumulator. Forefficient optimization, a meta-model is constructed to find the optimalconfiguration via an artificial neural network (ANN). Finally, mini-mization of the frequency response function (FRF) is achieved at theoptimal configuration.

II. Longitudinal System Based on Multiphysicsof the Liquid Propulsion Rocket

A. Fuselage Structure–Feedline–Engine Dynamic System

In this section, the major dynamics of the longitudinal systemare specified to construct the numerical model. The interfaces forthe dynamics are joined to form a complete system, and the transfer-function synthesis is used to obtain the response. Figure 1 showsthe process for analyzing longitudinal instability. The main variablesare thrust, displacement, pressure, and volumetric flow rate. Bypostprocessing each analysis, the transfer function will be obtainedin terms of the aforementioned variables.

Fig. 1 Integration procedure for the longitudinal instability of a liquid propulsion rocket.

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If the subsystem formulation is single input/single output, thetransfer function is simply configured in a series connection. In thecase of the multi-input/multi-output subsystem, additional couplingterms should be considered. In the fuselage structural analysis, andare connected in series:

δT

δrtank� fstruct�ϕ1;ϕ2; : : : ;ϕNs

� (1)

where ϕj is the jth structural mode. The tank displacement isdistributed into the feedline pressure PLOX;in and the volumetric

flow rate QLOX;in components. For each element, assuming a small

perturbation

δPLOX � PLOX;0 � δpLOX; δpLOX ≪ PLOX;0 (2)

δQLOX � QLOX;0 � δqLOX; δqLOX ≪ QLOX;0 (3)

where PLOX;0 and QLOX;0 are the feedline-averaged pressure and

volumetric flow rate, respectively. The perturbation of LOX feedlinepressure and volumetric flow rate can be expressed as shown in thefollowing equation:

δpLOX;in � 0; δqLOX;in � Afinδ_rtank (4)

Equations (2) and (3) can be analyzed using the pressuremode analysis and volumetric flow rate modal analysis, respectively.A transfer function matrix is constructed for the pressure and volu-metric flow rate at the outlet after passing through the feedline. Thetransfer function matrix is shown in Eq. (5):

�δpLOX;out

δqLOX;out

��

�f11 f12

f21 f22

��δpLOX;in

δqLOX;in

�(5)

where fij is the transfer function determined from the feedline

characteristics. The outlet of the LOX feedline becomes the inlet ofthe pump inducer, and the flow through the pump can be expressed asfollows:

264δPpump;out

δQpump;out

Vcav

375 �

264g11 g12

g21 g22

g31 g32

375"

PLOX;0 � δpLOX;out

QLOX;0 � δqLOX;out

#(6)

The variable gij is the transfer function of the meta-model derived

from theCFDanalysis, andVcav is the cavitationvolumeof the feedline.The feedline performance depends heavily on the cavitation, meaningthat fij depends on the cavitation volume. Meanwhile, the numerical

formation of pressure and volumetric flow rate of the pump outletLOX feedline include components, such as the turbine and injector.After passing through the LPRE equation, the following relations forthe combustor can be obtained:

"δpCombus

δqCombus

#�

"h11 h12 h13 h14

h21 h22 h23 h24

#26664δPLOXpump;out

δQLOXpump;out

PFuelpump;out

QFuelpump;out

37775 (7)

where hij is the LPRE transfer function. Finally, using the pressure and

volumetric flow rate of the combustor, thrust can be induced, as shownin Eq. (8):

δT � � k1 k2 �"δpCombus

δqCombus

#(8)

The variable ki is the transfer function of the combustion equation.Because the thrust perturbation feeds back into the thrust itself, pogoinstability will be determined by observing the attenuation. Here, thedifferent transfer functions can be combined in a series depending onthe model. Alternatively, a meta-model of the transfer functionthrough FFT can be extracted for the case of nonlinear analysis.Numericalmodeling is performed using the geometry of each system,and the relevant transfer function is obtained to derive the entiresystem. The pogo system composed of numerical analysis allowsfor the observation of the FRF depending on the geometric design.Such system information is modeled based on the geometry of the

Atlas-Centaur-Surveyor launch vehicle,where the numerical analysisof each component constitutes the transfer function of the integratedsystem. The conclusions are made based on whether the transferfunction can be controlled through the virtual design of the pogoaccumulator.

B. Structural Modal Analysis

Because the structural FRF fstruct of Eq. (1) is a transfer function ofthrust and local displacement δrtank, the rational function form can beobtained by analyzing the dynamic response. To accurately interpretthe response of the feedline inlet pressure and volumetric flow rate,detailed modeling of the local parts can be performed to improve theaccuracy of the tank displacement, as shown in Fig. 2. The modesuperposition can be expressed as Eq. (9):

δroδT

�XNs

n�1

φt�n�ϕn �XNs

n�1

φt�n�δqnδT

�XNs

n�1

φt�n�φ�n�Mnn�s2 � 2ζnωns�ω2

n�(9)

Here, ϕn denotes the structural mode, qn is the generalized coordi-nate,φ is the localmode thrust position shape, andφt is the localmodetank shape. From the results of the numerical mode analysis, the

generalized mass Mnn, damping ratio ζn, and natural frequencies ωn

can be obtained. Hence, the structural response is represented by alinearized equation. Among the many modes generated from finite

Fig. 2 Finite element model based on Atlas launch vehicle.

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element method analysis, it is necessary to select those that are the

most influential on longitudinal instability. Thus, the modal partici-

pation factors for the longitudinal displacement of the tank are

extracted to find the most influential modes.As fuel is consumed in the launch vehicle, the remaining fuel level

changes, thereby changing the characteristics of the entire fuselage

structure.With this consideration, it is necessary to consider the hydro-

elastic effect variation as a result of LOX and fuel consumption. This

can be achieved by the virtual mass method implemented in the

MFLUIDelement ofNASTRAN.The fluidmassmatrix is constructed

by the pressure and velocity derived by the following equations:

pi �Xj

Zd

Aj

ρ _σjeijjri − rjj

dAj (10)

_ui �Xj

Zd

Aj

σjeijjri − rjj2

dAj (11)

where σj is the point source value of the fluid located at rj, Aj is the

plane of action, and eij is the unit vector in the direction of point i frompoint j. MFLUID can be used to estimate the changes in the structural

frequencies in terms of fuel level.

C. Feedline and Pogo Accumulator Analysis from Fluid Transmission

Line Modeling

Numerical analysis of fij defined in Eq. (5) is carried out throughthe feedline containing the pogo accumulator. Dynamics of the

viscous compressible fluid in a circular transmission line are as shown

in the following equations:

Ii _Qi � Pi − Pi�1 − Rf;ijQ0i jQi (12)

Cf;i_Pi � Qi−1 −Qi (13)

where the variables are defined as follows: fluid inertance I, com-

pliance C, and resistance R. Equations (12) and (13) are nonlinear

equations, but they may be linearized by implementing the small

perturbation hypothesis mentioned in Eqs. (2) and (3).The systemwith the accumulator consists of branched pipes; Fig. 3

shows the LOX feedline with the accumulator. Equations (14) and

(15) give the results of the volumetric flow rate and pressure after

decoupling using the small perturbation assumptions:

�A� �q� �D� _q� �EQ�q � g1 (14)

�C� �p� �H� _p� �EP�p � g2 (15)

where �A�, [D], and �C� are the inertance, resistance, and compliance

matrices, with diagonal elements

�A� �

2666664

I1 0 0 0

0 I1 0 0

0 0 . ..

0

0 0 0 I1

3777775;

�D� �

2666664

2Rf1Q01 0 0 0

0 2Rf1Q01 0 0

0 0 . ..

0

0 0 0 2Rf1Q01

3777775;

�C� �

2666664

C1 0 0 0

0 C1 0 0

0 0 . ..

0

0 0 0 C1

3777775

(16)

where

Ii �ρLOXLi

Ai

; Rfi �ρLOX2A2

i

f

�Li � Le

Di

�;

Ci �1

Ks;LOX

� 1

ρLOXc2LOX

and consist of the element length Li, cross-sectional area Ai of theelement, density of the fluid ρLOX, diameter of the tube Di, and thefriction factor f. These parameters can be derived from the fluid proper-ties and the geometry of the pipeline. Also, �EQ� and �EP� are expressedas the product of the pipe geometry and the diagonal matrix as follows:

�EQ� � �B��C−1��BT � (17)

�EP� � �BT ��A−1��B� (18)

Because the preceding matrices are all symmetric, Eq. (14) for thevolumetric flow rate can be easily diagonalized by the modal analysis.The variables g1 and g2 are presented in Eqs. (19) and (20):

g1 � _f1 − �B��C−1�f2 (19)

g2 � _f2 � �BT ��A−1��f1 � �D��BT �−1f2� (20)

where

f1 �

2666664

Q0

0

..

.

0

3777775 and f2 �

2666664

0

0

..

.

−PN

3777775

Fig. 3 Fluid transmission line of LOX feedline with accumulator.

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�H� can be expressed as follows:

�H� � �BT ��A−1��D��BT �−1�C� (21)

Because the �H� matrix is asymmetric, deriving Eq. (15) will be a

complex process.The generalized coordinates and eigenpairs of the pressure mode

and thevolumetric flow rate can be expressed asqnp (ωnp ;PNp

i�1 xi;np )

and qnq (ωnq ;PNq

i�1 xi;nq ), respectively. The solution for the pressure

and volumetric flow rate at feedline inlet/outlet is obtained as follows:

qLOX;out �XNq

nq�1

xnq;Nqqnq ; qLOX;in �

XNq

nq�1

xnq;1qnq (22)

pLOX;out �XNp

np�1

xnp;Npqnp ; pLOX;in �

XNp

np�1

xnp;1qnp (23)

The generalized coordinate is obtained by the following equations:

�AnQnQs2 � DnQnQs� ω2

nQ�qnQ � G1�s� (24)

�CnPnPs2 � HnPnPs� ω2

nP �qnP � G2�s� (25)

G1�s� is Laplace transform ofg1�t� andG2�s� is Laplace transformof g2�t�.After performing the natural modal analysis of Eq. (15), we can

convert �H� into a symmetric matrix using the eigenvectors. If thebranched pipe of the accumulator has a sufficiently small radius

compared with the feedline pipe, the matrix �H� � �YT ��H��Y� sym-metrized about the pressure eigenvector matrix �Y� will become

diagonally dominant. Assuming a diagonal �H�, Eq. (25) can alsobe diagonalized, allowing qnP to be found.

When pump cavitation occurs, bubbles are generated near thefeedline outlet, which affect the compliance and inertance terms.Using the rule of the mixtures model, the cavitation volume at theoutlet can be reflected in Eq. (26):

Icav �ρLOX;liquidLi

Ai

�1 − Vcav� �ρLOX;airLi

Ai

Vcav (26)

Ci �1

Ks;LOX

�1 − Vcav� �AiLi

ρLOX;airc2Vcav (27)

where c is the acoustic speed in oxygen gas, andVcav is the ratio of thecavitation volume to the element volume.From this model, and by including the feedline geometric informa-

tion, the transfer functionwill be formulated.Depending on the size ofthe pogo accumulator, the compliance, resistance, and inertance of theentire system are affected, which will affect the transfer function.

D. Pump Cavitation Analysis

Computational fluid dynamics analysis is performed to calculatethe cavitationvolumeVcav used in Eqs. (26) and (27) and to derive thetransfer function of the pressure and volumetric flow rate to theengine dynamic model. The inducer configuration was provided bythe Korea Aerospace Research Institute. It is an experimental modelof the turbopump inducer of RD-151 engine, which is of the sameseries as the RD-180, the first stage engine for Atlas V. To meet theworking conditions of theAtlas Vinducer, the target geometry for thesimulation was acquired by scaling up the experimental model.A multiblock hexahedral grid, modeling all the three blades, was

generated for the simulation, with 20,835,135 cells in total. In thiscomputation, AUSMPW+_N flux scheme [14,15] with a fifth-ordermultidimensional limiting process limiter [16] was adopted for accu-rate and robust simulation. For convergence, the lower–upper sym-metric Gauss–Seidel method [17] was applied and the effect of theturbulence was modeled using Menter’s k − ω shear-stress transportmodel [18]. All thermodynamic properties of both liquid and vaporphases of oxygen were obtained from the Standard Reference Data-base 23, available at the National Institute of Standards and Technol-ogy [19]. Finally, the present simulation describes the nonequilibrium

Fig. 4 Inducer CFD simulation result.

Fig. 5 Pump outlet pressure response in terms of inlet perturbation.

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phase change process based onMerkle’s cavitationmodel [20]. Before

the actual simulations for LOX, the flow solver was validated by

comparing the numerical results with the experimental results of the

water test [14].

Figure 4 shows an example of the inducer simulation results.

Because unsteady analysis is computationally heavy, a quasi-steady

solution is used to obtain the transfer function and the cavitation

volume. Figure 5 shows the results of the induced LOX pump outlet

mass flow rate and pressure, in response to the inletmass flow rate and

pressure perturbations. It can be observed that the range of inducer

resonance frequency is greater than that of pogo instability, and the

response is almost constant in the lower frequency region of 0–30Hz.

The pressure and mass flow rate outputs are derived assuming 5%

perturbation. As a result, the compression ratio of the inducer, the

cavitation volume, and the time lag of the transfer function can be

calculated. Figure 6 describes the cavitationvolume regressionmodel

in terms of the pressure and mass flow rate perturbations.

E. Liquid Propulsion Rocket Engine Dynamic Model

The transfer function from the pump to the combustor chamber can

be formulated by establishing an equation for LPRE. A pipe system

for RP-1 and LOX, in which a gas generator is incorporated, can then

be constructed. Static analysis [21] allows design modifications

to bemade following pressure andmass flow changes based on target

thrust. The properties of each RD-180 engine system, based on the

Atlas type LV, are shown in Fig. 7. Table 1 shows the static analysis

results of the engine performance.

Fig. 6 Cavitation volume regression model in terms of the pressure/

mass flow rate perturbation.

Fig. 7 RD-180 engine type properties based on Atlas launch vehicle.

Table 1 Performance analysis result of the RD-180 engine type

Input Output

Thrust 780 kN Specific impulse 235.70 sCombustionchamber pressure

35 bar Fuel mass flow rate atcombustion chamber

214.83 kg∕s

LOX/fuel ratio 2.25 LOX mass flow rate atcombustion chamber

480.07 kg∕s

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Dynamic analysis predicts the response of the combustion cham-ber to the pump inlet perturbation, and a well-known linearizationmodel exists [22]. Using this model, a dynamic system in the form ofa Simulink block diagram is constructed to secure the transfer func-tion for Eq. (7). As shown in Fig. 8, the pump inlet pressure pertur-bation causes the perturbation of the engine nozzle. The input and theresponse have the same frequency, but with different magnitudes andphases. The amplitude of the perturbation at the inlet is attenuatedwhen it goes through the engine section, and finally only 0.2% of theinitial perturbation reaches the combustor. However, although theperturbation reached by combustion seems negligible, the effect canstill be fatal due to the closed-loop formulation.

F. Combustion Analysis

Pressure and mass flow at the combustor inlet can be converted tothrust output by the combustion analysis. To do so, numerical analy-sis of the combustion equation should be performed. In this study,because only the longitudinal and lower-frequency instabilities aretaken into account, one-dimensional formulation can provide a rel-atively accurate response. The following governing equations areapplied to analyze the one-dimensional reactive flow:

continuity:∂ρ∂t

� ∂�ρu�∂x

� 0 (28)

momentum:∂u∂t

� u∂u∂x

� −1

ρ

∂p∂x

� μ

ρ

∂2u∂x2

(29)

energy:Cp

�∂T∂t

� T∂u∂t

�� k

ρ

∂2T∂x2

� _Q (30)

species:∂∂t�ρYi� �

∂∂x

�ρYiu� �∂∂x

�ρD

∂Yi

∂x

�� _Si (31)

equation of state:p � ρRT (32)

where μ is the dynamic viscosity, k is the thermal conductivity, andDis the mass diffusivity.

_Q � qMW_ωKERO; _S � MW_ωi (33)

where q is the jet fuel heat of combustion,MW is themolar mass, and_ω is the reaction rate.Equation (33) can be solved via the pressure-based method, and

the approximated Poisson equation is derived without solving the

continuity directly:

−1

ρ

�∂2p∂x2

� ∂2p∂y2

��

�∂u∂x

�2

� 2∂u∂x

∂u∂y

��∂u∂y

�2

(34)

Using a one-dimensional approximation of the combustor

−1

ρ

�∂2p∂x2

��

�∂u∂x

�2

(35)

The kerosene combustion reaction is based on a mechanism

consisting of two Arrhenius reaction rates. Figure 9 shows the

response of the combustion chamber perturbations, simulated using

the combustor solver.

Fig. 8 Engine nozzle pressure perturbation induced by pump inlet

pressure perturbation.

Fig. 9 Inlet/outlet perturbations in the combustion chamber, simulated using combustor solver.

Fig. 10 Liquid rocket closed-loop system.

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III. Multi-objective Optimization of a PogoAccumulator

A. Problem Description

Design optimization of a pogo accumulator should be performedbased on meta-modeling in the preliminary design stage. The objec-tive function is a multi-objective problem consisting of the transfer

function of the integrated system and the weight reduction functionof the pogo accumulator. Pogo resonance can be observed from thetransfer function of the integrated system, and the optimal design isachieved by minimizing the local maximum point. Because thenatural frequencies of the fuselage structure vary with fuel consump-tion, the fuel consumption model is considered. For the longitudinalsystem, the initial four modes of the synthesis function can cover upto 60Hz, and the fuel level controls theMFLUIDmodel. This aims toachieve an optimal design by using the six representative models,which are in 20% intervals from 0 to 100%. The definition of themulti-objective function is as follows:

f0 � minM

fi;j � min

�Gj�jωi�

1�Gj�jωi�H�jωi��

i � 1; 2; 3; 4; j � 1 − 6

(36)

subject to

x1;min ≤ x1 ≤ x1;max; x3;min ≤ x3 ≤ x3;max; x4;min ≤ x4 ≤ x4;max

whereM is the mass of the accumulator, and fi;j is the minimizationfunction for the local maximum of the transfer function. Gj is the

structural responsewhen the fuel level is f100 − 20�j − 1�g%, andωi

is the frequency of each model at the local maximum point. Afterobtaining the transfer function for the entire function as the rationalfunction, the objective function matrix �fi;j� will be constructed by

aligning the local maximum points and selecting the four smallestmodes. The transfer function forms a closed-loop system thatincludes feedback on the feedline analysis, engine dynamic analysis,and combustion analysis. When the isolated system is configuredduring closed feedback, the time lag of these three feedback parts willbe more significantly affected than that of the open loop. Figure 10shows the present control flow chart for the closed-loop system, withthe feedline and engine feedback H�s� for the structural vibrationfunction G�s�.

B. Meta-modeling by the ANN

The problem in Sec. III.A is a multi-objective optimization prob-lem of 25 output layers with four structural design input layers, eachwith a minimize condition. The meta-model for the multiphysicssystem should be selected for lower-dimension input data and higher-dimension output data. Because these models should be character-ized by nonlinear regression, an ANN will be suitable for this case.Because of the small number of input layers, a sufficient number ofsamples can be provided. Furthermore, because of significant modelnonlinearity, the number of hidden layers should be sufficientlyprovided.

Fig. 11 Elapsed time for 0.1 performance vs number of neurons.

Fig. 12 Convergence of ANN performance.

Fig. 13 Multi-objective optimization procedure for the meta-model of the longitudinal dynamic model.

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To extract high-quality samples, Latin hypercube sampling isperformed to conduct even sampling throughout the entire domain.Thus, 17 sampleswere evenly chosen to simulate the gradient of eachvariable.To perform the ANN, an appropriate number of neurons should

be clarified. For a total of 100,000 samples, 70% training data, 15%validation data, and 15% testing datawere specified. Figure 11 showsthe learning rate of the proposed numerical model according to theneuron number. As it has lower-dimension input data, almost noconvergence is observed in the 10 hidden-layer model used for thehigher-dimension input data. Thus, 40 hidden layers are consideredappropriate for the transfer function; overfitting with more neuronscan rather reduce the performance of the function.Figure 12 shows the maximum convergence condition of ANN

performance for an appropriate value of the hidden layer. Least time iselapsed to reach 0.1 performance when the number of neurons is 40.Because nonlinearity and excess computational time are requiredwhen learning data increase, scaled conjugate gradient with GPUcalculation is used to allow the derivation of accurate models. Also,because the peak point network has to be constructed and the noise ofthe result value increases, the learning tolerance is set to 5%. In thiscase, about 80,000 out of 100,000 samples can perform accurate peakprediction within a 1% tolerance range.

C. Optimization for Combined Systems

Optimization of the multi-objective function system is carriedout alongside the optimization of the meta-model. This process isshown in Fig. 13, in which sequential optimization is performed onthe meta-model. Because the number of objective functions is quitelarge, pareto-optimization will be required. In it, the solution isobtained by providing theweighting factor for the objective function.The variable wi represents the weighting factor for the ith objectivefunction, and four weighting models are used for the simulation. Forthe accumulator mass transfer function, the weighting factor is fixedat w0 � 1. The combination of weighting factor models is shown inFig. 14. The first is an equivalent model, in which the optimal designis carried out after assigning a weighting factor of 1 to both massratio and frequencies. This is shown in Fig. 14a. The secondmodel isthe weighting factor for the large response mass ratio, as shown in

Fig. 14b. The highest weighting factor is given to the 60% remainingfuel level among the sixmass ratio cases,where the largest response isobserved for the same sine sweep input. Figure 14c shows the case,where a weighting factor is assigned to a mode with a large modalparticipation factor. Figure 14d shows the case, where the weightingfactor is given to themodewith a large response considering both fuellevel andmodal participation factor. Four optimal points are analyzedby changing the weighting factor of the multi-objective functions.The optimal points are analyzed and compared to minimize the massof the pogo suppressor.

IV. Results

A. Feedline and Pogo Accumulator Analysis by Fluid Transmission

Line Modeling

Pogo instability can be evaluated by analyzing the longitudinaldynamic system. Figure 15 shows the structural response in terms

Fig. 14 Weighting factor distribution for the present multi-objective function.

Fig. 15 Structural response changes following fuel and LOX consump-

tion in the tank.

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of fuel consumption considering the junction coupling: the relationship

between the mass flow rate perturbation δqLOX;in at the feedline inletin Eq. (4) and the thrust perturbation δT in Eq. (1). Natural frequency

increaseswith fuel consumption, thus changing the structural response.

According to the LOX and fuel ratio, the condition values are provided

in the tank skinvariables, and natural-frequency analysis is performed,

as shown in Fig. 15. The natural frequency obtained for 100% fuel

level is 9.3 Hz, increasing to 16.2 Hz when the fuel is exhausted

to 0%.The mass flow rate perturbation of the output in response to the

input perturbation at the LOX inlet is shown in Fig. 16. The response

is shown in terms of the cavitation volume, and f11 from Eq. (5) is

shown. As the cavitation volume increases, the natural frequency

of the feedline decreases. As the cavitation volume increases, so does

the response of the system. Therefore, cavitation is one of the

important factors that greatly influence the evaluation of longitudinal

instability. Resonance frequency also decreases in a manner similar

to the peak points of the structural response.Figure 17 shows the response of the integrated system by the sine

sweep simulation. Assuming a 10% thrust perturbation, the response

Fig. 17 Sine sweep results for the combined longitudinal system of the launch vehicle.

Fig. 16 Transfer function results in terms of the cavitation volume.

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of theG�s�H�s� series system is plotted in terms of the remaining fuel

level. Because the system is connected in a series, the structural

response G�s� increases even with the fixed feedback function.

Figure 17c shows that the response increases dramatically with 60%

of fuel remaining. This confirms that the pogo resonance has been

accurately simulated.

B. Dynamic Response and Structural Optimization of the

Structure–Feedline–Engine–Accumulator System

Installing a pogo accumulator significantly attenuates the total

system response. The performance of the accumulator constructed

in this study during the initial design stage is evaluated under sine

sweep simulation, as shown in Fig. 18. Damping performance sig-

nificantly improved compared with the case without the pogo accu-

mulator; however, with an acceleration of more than 1 g, an optimal

design is required.

Figure 19 shows the response function for the closed-loop situa-

tion. Figure 19a describes the response without the accumulator,

and Fig. 19b is the response equipped with an accumulator at its

initial design stage. The results for the four models are described

in Figs. 19c–19f after the optimal design is carried out using the

weighting factor configured in Sec. III.C. For each of the fourweighting factors, different optimal design points are obtained. Thefirst model (case 1), with the equally distributed weighting factors,has been optimized to contain sufficiently small mass and response ata frequency range of 0–60Hz. The secondmodel (case 2) gives higherweighting factors for higher response frequencies, and the response iswell bounded below –60 dB for the given frequency range. However,compared with the first model, the accumulator is much heavierin spite of its negligible response improvement. For the third case,a smaller accumulator mass is achieved compared with those ofthe other three; yet, the peak point is highest among them. The lastcase has the smallest response of all and is found to be the mostconservative design in terms of the local maximum of the systemresponse.Figure 20 shows the sine sweep response analysis results

for the four optimized models from 0 to 60 Hz with 10% thrustperturbation. The results are described for the 60% fuel level con-dition, which generally shows a large response. The initial designstage of the system determines whether the launch vehicle is mannedor unmanned, and thus the acceleration condition may changeaccording to system specifications. If the design criteria are low foracceleration, all of the following cases show satisfactory results in

Fig. 18 Sine sweep results for the combined longitudinal system with initial accumulator design.

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terms of response, and thus can be used to design a manned mission

with maximum efficiency. In terms of mass optimization, case 3 has

the smallest mass but a relatively large response at 10 Hz. For cases 2

and 4, highweighting factors are assigned for the 60% remaining fuel

level condition. In these cases, where the structure and the feedline

frequencies are likely to resonate, almost no longitudinal response

can be observed after the optimized pogo accumulator design is

applied. Therefore, it can be concluded that a higher weighting factor

for the resonance-expected fuel level gives the optimal optimization

results.

Fig. 19 Closed-loop response function for the launch vehicle longitudinal dynamics.

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V. Conclusions

In this study, frequency response analysis for pogo resonancewas performed. It was verified that pogo resonance is predicted byderiving system properties both geometrically and numerically,and the accumulator was designed to suppress pogo resonance.The dynamics considered for the optimal design of the pogo sup-pressor in this study are as follows:1) Finite element analysis of structural vibration and acquisition of

the transfer function in response to thrust perturbationwere achieved byconsidering the local response from the modal superposition method.2) Implementation of numerical analysis, including the feedline

geometry, was ensured through the feedline transmission line mod-eling and consideration of the volumetric and mass flow rate transferfunction between the feedline inlet and outlet.3) Cavitationvolume near the feedline outlet was considered based

on the CFD analysis of an inducer.4) The time lag induced by the thrust generation was accurately

reproduced by considering engine dynamics via the LPRE equation.5) Analysis of a combustor was carried out to include thrust

perturbation in terms of the temperature increase.In this way, the transfer matrix method based on multiphysics

is constructed to implement the overall system transfer function. Thefeedback loop based on numerical analysis is defined to form theentire system in a closed loop. Databases of the response functionare constructed using the numerical pogo accumulator model that ischaracterized by the branched pipe. Multi-objective structural opti-mization is performed by building a meta-model of the transferfunction for the accumulator geometry.To build a meta-model, the depth of the neuron that can be used to

construct anonlinearmodel is evaluated.Astudyof25outputdatavectorswas conducted for four designvariables, and the nonlinear learning of thepogo suppressor size was accomplished. For the meta-model construc-tion, the appropriate neuron depth was evaluated to build the nonlinearmodel. Nonlinear learning of the pogo suppressor sizewas conducted bylearning the output data vectors for the four design variables.The sine sweep function was used to evaluate the acceleration

that is experienced by the projectile payload under the same thrust

perturbation conditions as that for Atlas-Centaur class projectiles.The attenuation was confirmed by the pogo accumulator installation.Furthermore, the results of the multi-objective suppression of thepogo suppressor are capable of improving the result, enabling thesearch for the required structural size of the pogo suppressor.The sine sweep function is used to evaluate the random vibration

that the Atlas-Centaur class launch vehicle will experience under thesame thrust perturbation conditions. Thepogo accumulator is installedfor the attenuation purpose. Furthermore, the present multi-objectiveoptimization results reduce the mass of the pogo suppressor by morethan 50%, further improving performance.As a result, further researchon the required dimensional specification for the pogo suppressor isexpected.

Acknowledgments

This work was supported by the Advanced Research CenterProgram (NRF-2013R1A5A1073861) through a grant from theNational Research Foundation of Korea, which was funded by theKorean government (Ministry of Science, ICT and Future Planning)and contracted through the Advanced Space Propulsion ResearchCenter at Seoul National University.

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E. BladesAssociate Editor

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