Zhu Hao,Tian Hui,Cai Guobiao

School of Aeronautics Science and Engineering,Beihang University,Beijing 100083,China

Hybrid uncertainty-based design optimization and its application to hybrid rocket motors for manned lunar landing

Zhu Hao*,Tian Hui,Cai Guobiao

School of Aeronautics Science and Engineering,Beihang University,Beijing 100083,China

Hybrid rocket motor;Hybrid uncertainty-based design optimization;Manned lunar landing;Uncertainty analysis

Design reliability and robustness are getting increasingly important for the general design of aerospace systems with many inherently uncertain design parameters.This paper presents a hybrid uncertainty-based design optimization(UDO)method developed from probability theory and interval theory.Most of the uncertain design parameters which have sufficient information or experimental data are classified as random variables using probability theory,while the others are defined as interval variables with interval theory.Then a hybrid uncertainty analysis method based on Monte Carlo simulation and Taylor series interval analysis is developed to obtain the uncertainty propagation from the design parameters to system responses.Three design optimization strategies,including deterministic design optimization(DDO),probabilistic UDO and hybrid UDO,are applied to the conceptual design of a hybrid rocket motor(HRM)used as the ascent propulsion system in Apollo lunar module.By comparison,the hybrid UDO is a feasible method and can be effectively applied to the general design of aerospace systems.

1.Introduction

Deep space exploration has attracted considerable attentions in China in recent years and manned lunar landing will be the first step.The development of HRMs1–6which are less complex than liquid rocket motors(LRMs)and more easily throttled and restarted than solid rocket motors(SRMs)provides an alternative solution to satisfy the need for green,nontoxic and cheap propulsion system used in manned lunar landing.

With the rapid development of aerospace vehicle technology,there has been an increasing interest in the UDO at the early design phases of aerospace systems.7–10In the process of UDO,the design uncertainties arising from the aerospace system itself,as well as the involved environmental and operational conditions,need to be characterized with some mathematical theory to express their uncertain information and characteristics.Then the uncertainty analysis method is usedto obtain the uncertainty propagation from the design parameters to the system responses.For example,probabilistic UDO is one of the most widely used methods to solve UDO problems based on probability theory.11–13At this time,the uncertainties are defined as random variables and quantitatively measured with mean,standard deviation,probability density function(PDF)or cumulative distribution function(CDF).However,aerospace systems are often complex and have a multitude of uncertain design parameters.Most of them have sufficient information or experimental data and they are appropriate to be classified as random variables using probability theory.On the other hand,there are also a few inevitable uncertain design parameters which have insufficient information or experimental data,for example only the uncertain boundaries are known.They are unsuitable to be expressed with probability theory.Thus,it is necessary to model them with other appropriate theories,such as interval theory which defines uncertainties as interval variables using their boundary values.

For this reason,this paper proposes a hybrid UDO method,based on our previous works14which handled the conceptual design of HRMs with probabilistic UDO.In this method,the probability theory is used to define the uncertain design parameters with sufficient information,and at the same time the interval theory is together introduced to quantify the other uncertain design parameters.Then a hybrid uncertainty analysis method is developed based on Monte Carlo simulation(MCS)and Taylor series interval analysis.The conceptual designs on a HRM substituting for the LRM of ascent propulsion system(APS)in Apollo lunar module are carried out with this method and the design results are compared with those generated by DDO and probabilistic UDO.

2.Background theories

The general UDO process includes three main procedures:uncertain system modeling,uncertainty analysis and optimization under uncertainty.15In uncertain system modeling,the system design mathematical model is established,and then the design uncertainties are described and quantified with some mathematical theory such as probability theory and interval theory.

2.1.Probabilistic uncertainty-based design optimization

In probability theory,the uncertain design parameters are considered as random variables as above mentioned.They can be defined with statistic method,if the information or experimental data of the uncertainties is sufficient enough.The mean l and the standard deviation r of the system responses can be obtained using MCS.In probabilistic UDO,the system design robustness is achieved through minimizing l and r.In addition,the reliability of constraints is accomplished by the formulation that confidence level constraints are met with a higher probability.The mathematical model can be expressed as

2.2.Interval uncertainty-based design optimization

2.2.1.Basic theory

In interval theory,the interval numberXis de fined as

The definition of interval function can be obtained in the same way,which is shown as follows:

whereZis a real number.In interval theory,the uncertain design parameters are considered as interval variables and defined using interval middle and radius,when only the boundaries and no other data of the uncertain design parameters are known.Therefore,the system design robustness is achieved through minimizing the interval middle and radius of the output responses,and the reliability of constraints is accomplished through magnitude probability that constraints are met with a higher level in interval UDO.The mathematical model can be expressed as

2.2.2.Interval uncertainty analysis method

3.Hybrid uncertainty-based design optimization

3.1.Definition of system response uncertainties

3.2.Mathematical model of design optimization

Referring to the mathematical model of probabilistic UDO,the robustness of the optimum solution is achieved by minimizing the mean value of interval middle l(fC(x,p)),the standard deviation of the interval middle r(fC(x,p))and the mean value of interval radius l(fR(x,p))of the target function.At the same time,the reliability of the optimum solution is evaluated through the probability that boundaries of constraints are met withahigherlevel.Themathematicalmodelcanbeexpressedas

3.3.Procedures of hybrid uncertainty-based design optimization

Corresponding to the mathematical model,the design optimization is progressed(Fig.1).In this process,the experiment design and parameter analysis14on the uncertain parameters are carried out to filter the ones that have important effect on the system performance.Then the Taylor series-based method is selected as the solution for the interval uncertainty analysis to reduce the computing time and ensure the computational precision at the same time.In addition,a Latin hypercube sampling method18is used to take 1000 samplings from the random variables.

4.Design optimization of HRM

The ascent propulsion system of the Apollo lunar module is a typical LRM,which consists of an oxidizer tank,a fuel tank,two gas bottles and a thrust chamber.19–21Our former study worked on the UDO and applied HRMs to an alternative propulsion system for APS to analyze the advantages and disadvantages of HRMs,and all the uncertain design parameters were considered as random variables.In this paper,the hybrid UDO method is used in the conceptual design of HRMs and the uncertain design parameters are classified as random or interval variables corresponding to their uncertain information characteristics.

4.1.HRM system design

A typical HRM with a nitrogen gas pressure feed system and the98% hydrogen peroxide(H2O2)/hydroxyl-terminated polybutadiene(HTPB)propellant combination is adopted in this paper.A submerged cone-shape nozzle with a 20?half expansion angle is used in this study.In order to achieve that the working time is close to that of APS,a single circle grain which is suitable for long time combustion is used.The HRM system design model,in which the motor responses can be computed from design variables x and model parameters p,is established and the uncertain design parameters are analyzed.Three grain dimension parameters(outer diameterDf,lengthLfand initial web thicknessef),a nozzle parameter(nozzle expansion ratio e),a chamber parameter(initial chamber pressurePci)and an oxidizer feed system parameter(oxidizer mass flow rate _mo)are selected as input variables x(Table 1).Seven uncertain model parameters,which are concluded to be the most important model parameters for the response performance of HRM through sensitive analysis,are selected as model parameters p(Table 2).Both the design variables and the model parameters are uncertain and the detailed analysis processes were given in our former work.12The subjective uncertain model parameters are handled as interval variables in this paper,since their uncertainty information is insufficient.

Fig.1 Procedures of the hybrid UDO method.

4.2.Design optimization

The conceptual designs are carried out to develop an alternative propulsion system using HRM.In order to provide enough velocity increment and be comparable with the original design of APS,the optimization target is to minimize the motor massMand the constraints are the total vacuum impulseIv,the chamber lengthLand the engine operation timet.The mathematical model of DDO,probabilistic UDO and hybrid UDO are shown as Eqs.(24)–(26):

whereIv1andt1are the total vacuum impulse and final operation time of APS,H1is the height of Apollo lunar module,andLb,LtandLcare the gas bottle length,oxidizer tank length and chamber length in HRM.

5.Results and discussion

The performance parameters of different motors,including LRM of APS and HRMs,are shown in Table 3.And those of HRMs are generated by different design optimization methods.As a result of comparison,hybrid UDO is a feasible method for the general design of aerospace systems.

Table 1 Design variables of HRM system model.

Table 2 Model parameters of HRM system model.

Table 3 Performance parameters of the LRM of the APS and HRMs.

Table 4 Design optimization results.

Fig.2 PDF of target and CDF of constraints in different design results.

6.Conclusions

This paper presents a hybrid UDO method based on probabilistic and interval theory.Then the conceptual designs of HRMs substituting for the LRM of APS in Apollo lunar module were carried out using this method together with DDO and probabilistic UDO methods.The results show that both of the two UDO methods produce optimal designs that achieve reliability requirements at higher confidence levels than the deterministic approach.The comparison of the two UDO methods shows that hybrid UDO gives a design result with higher reliability and robustness and it is a feasible method that can be effectively applied to the general design of aerospace systems.

Acknowledgements

This study was supported by the National Natural Science Foundation of China(No.51305014).

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4 January 2016;revised 2 September 2016;accepted 21 November 2016

Available online 16 February 2017

*Corresponding author.

E-mail address:zhuhao@buaa.edu.cn(H.Zhu).

Peer review under responsibility of Editorial Committee of CJA.