Deutsches Zentrum Kungliga Tekniska für Luft- und Högskolan Raumfahrt e.V.

XR-EE-SPP 2012:002

M. Johannsson

Optimization of Solid Rocket Grain Geometries

This report contains: 58 pages including 32 Figures 12 Tables 36 References

Deutsches Zentrum für Luft- und Raumfahrt e.V.

Institut für Raumfahrtsysteme

Systemanalyse Raumtransport (SART)

28359 Bremen

Master Thesis Report

Submitted to Royal Institute of Technology (KTH) in partial fulfilment of the requirements for the degree:

Master of Science in Aerospace Engineering

Supervised by: Etienne Dumont Nickolay Ivchenko

Abstract

Solid Rocket Motors (SRMs) are employed in many space launch applications from the booster rockets on the now retired Space Shuttle to the new European launch vehicle Vega. Preliminary design of these complex three dimensional SRMs, given certain requirements and limitations can be considered as an optimization process where a best geometrical solution is sought resulting in a desirable thrust profile. In this project, a derivate free direct search package titled NOMAD is employed together with the internally developed numerical burnback analysis tool SRP-GEO and ballistic solver SRP. An analytical model for the burnback analysis tool is also developed to take advantage of the support for surrogate functions within NOMAD. Due to the local nature of the optimizer, the results for complex geometries are shown to converge toward configurations different from the globally optimal geometries. Yet in most instances, the resulting calculated thrust profiles are shown to correlate well with the desired counterpart. This highlights the importance of carefully chosen initial values and boundaries while also emphasize the many possible solutions to a single problem.

Keywords: SRP, Solid rocket motor, grain geometry, derivative free optimization, internal ballistics, NOMAD, solid rocket propellant, numerical burnback analysis, analytical burnback analysis, thrust history

Acknowledgement

This is a master thesis project performed in DLR Bremen within the SART division during 2011-2012. During the course, I have acquired much knowledge in the field of preliminary design of Solid Rocket Motors and gained a firm understanding in many other related topics in rocket design. I have also acquired skills in software design and general optimization theory which are of use in all areas of engineering. Even though the project required slightly more time than expected, it can be considered a success. Following, I would like to show my appreciation to a few colleagues and friends.

I would like to express my deepest gratitude and respect to my DLR supervisor, Etienne Dumont for supporting me during the course of seven months while performing this thesis. Without his guidance and clever insights in all areas of the work, this thesis would not have been possible.

I would also like to express my sincere appreciation to my supervisor in KTH Sweden, Nickolay Ivchenko for the insights and comments given of the final report.

I would further like to thank Aaron Koch for the help in solving pesky integration issues between Fortran and C++ during late evenings.

I would likewise like to thank the whole SART group and Martin Sippel for providing me support and the uplifting spirit necessary for me to finish this thesis.

Lastly, I would like to give a special nod to the Awesome Study Group (ASG) of October 2011 - Mars 2012 for being awesome. SART TN-002/2012 i

Table of Contents

List of Figures iii List of Tables iv Nomenclature v Abbreviations vii

1 INTRODUCTION 1

1.1 Methodology 1

1.2 Scope 3

1.3 Contents 4

2 SOLID ROCKET MOTOR FUNDAMENTALS 5

2.1 Motor Components 5 2.1.1 Casing 5 2.1.2 Nozzle 5 2.1.3 Igniter 6 2.1.4 Insulation 6 2.1.5 Grain 7

2.2 Internal Ballistics 7 2.2.1 Specific Impulse and Losses 8 2.2.2 Burning Rate 9 2.2.3 Area Ratios 9 2.2.4 Thrust and Chamber Pressure 10

3 DIRECT SEARCH OPTIMIZATION ALGORITHM 11

3.1 NOMAD Overview 11

3.2 NOMAD Operation and Features 11 3.2.1 MADS Algorithm 11 3.2.2 Constraint Handling 12 3.2.3 Surrogate Functions 12 3.2.4 Variable Neighborhood and Speculative Search 13 3.2.5 Parallelis m 13 3.2.6 Additional Functionalities 14

4 GEOMETRY OPTIMIZATION PROGRAM 15

4.1 Pre-Processing 15 4.1.1 Input File and Propulsion Database 15 4.1.2 Initial Geometry Selection and Database 17 4.1.3 Geometry Modification 18 4.1.3.1 Dome Structures 18 4.1.3.2 Ignition System and Submerged Nozzle Assembly 20 4.1.3.3 Dome Grain Geometries 21 4.1.4 Constraint, Boundary and Initial Values 22 4.1.5 Miscellaneous 23

4.2 Optimization 24 SART TN-002/2012 ii

4.2.1 NOMAD Implementation 24

4.3 Blackbox 25 4.3.1 Burnback Analyses 25 4.3.1.1 Analytical Radial Burn Analysis 25 4.3.1.2 Analytical Axial Burn Analysis 28 4.3.1.3 Volume Calculations and Post Processing 29 4.3.2 Thrust Computation 29 4.3.3 Objective Value Determination 30

4.4 Post Processing 31

5 TESTING AND VALIDATION 32

5.1 Validation of Analytical Burnback Algorithm 32 5.1.1 Simple Cylindrical Tube 32 5.1.2 Simple Star 33 5.1.3 Complex Grain 34 5.1.4 Constraint Violation Analysis 34

5.2 Testing and Validation of Optimization Algorithm 35 5.2.1 Cylindrical Tube 35 5.2.2 Star 37 5.2.3 Three Segment Combination 39 5.2.4 P80 FW Derivative 40

5.3 Convergence Analysis 42

6 CONCLUSION 44

6.1 Future Work 44

REFERENCES 46

APPENDIX A 48

Main Input File 48

Propellant Database 50

SRP Input File 51

APPENDIX B 52

Derivation of Dome Equations 52 Length of Nose Dome 52 Nose Dome Outer Radius 52

Derivation of Modified Analytical Star Equations 53 Phase One 53 Phase Two 55 Phase Three 58 SART TN-002/2012 iii

List of Figures

Figure 1: Current and desired design process of a typical solid rocket motor in SART ...... 2 Figure 2: Basic structure of the optimization process ...... 3 Figure 3: Components of a typical SRM ...... 6 Figure 4: Classification of grain according to thrust-time characteristics ...... 7 Figure 5 Thrust/Pressure – time curve definitions ...... 8 Figure 6: Mesh configurations in MADS algorithm with ...... 12 Figure 7: SRP-GEOPT component structure and interaction chart ...... 16 Figure 8: SRP-GEOPT geometry parameter definitions with three segment example ...... 17 Figure 9: P80 FW first stage solid booster with hemispherical nose dome ...... 18 Figure 10: and variation with meters ...... 19 Figure 11: variation with meters ...... 20 Figure 12: Ignition system and submerged nozzle assembly ...... 21 Figure 13: Various modifications to nose dome grain geometry based on the first main segment...... 22 Figure 14: Validity range of SRP-ANLYT constraints ...... 24 Figure 15: Truncated star geometry and phase definition...... 26 Figure 16: Examples of axial segment burning evolutions based on neighboring inner radii ...... 28 Figure 17: Comparision of burnback results for single cylinder grain ...... 32 Figure 18: Comparision of burnback results for single star grain ...... 33 Figure 19: Comparision of burnback results for complex grain ...... 34 Figure 20: Comparision of burnback results for constraint violated star grain ...... 35 Figure 21: Desired and optimized thrust for single cylinder grain ...... 36 Figure 22: Desired and optimized thrust for single cylinder grain with advanced input configuration ...... 37 Figure 23: Desired and optimized thrust for single star grain ...... 38 Figure 24: Desired and optimized thrust for single star grain with advanced input configuration ...... 39 Figure 25: Desired and optimized thrust for three segment grain ...... 40 Figure 26: Desired and optimized thrust for an SRM derived from P80 FW ...... 41 Figure 27: Desired and optimized thrust for single cylinder grain with VNS strategy ...... 42 Figure 28: Nose cone geometry parameter definitions ...... 52 Figure 29: Phase One truncated star geometry breakdown ...... 54 Figure 30: Phase One to Phase Two switching criteria ...... 54 Figure 31: Phase Two truncated star analytical denotations ...... 55 Figure 32: Phase Three truncated star analytical denotations ...... 58

SART TN-002/2012 iv

List of Tables

Table 1: Overview of investigated Direct Search optimization algorithms ...... 11 Table 2: Single cylinder geometry dimensions in meters with m and ...... 32 Table 3: Single star geometry dimensions in meters with m and ...... 33 Table 4: Complex grain geometry dimensions in meters with m and ...... 34 Table 5: Constraint violated single star geometry in meters with m and ...... 34 Table 6: Initial values, boundaries and solutions for cylindrical tube geometry ...... 35 Table 7: Initial values, boundaries and solutions for cylindrical tube geometry with advanced configuration .36 Table 8: Initial values, boundaries and solutions for star geometry ...... 37 Table 9: Initial values, boundaries and solutions for star geometry with advanced configurations ...... 38 Table 10: Initial values, boundaries and solutions for three segment geometry ...... 39 Table 11: Initial values, boundaries and solutions for an SRM derived from P80 FW ...... 40 Table 12: Solutions for cylindrical tube geometry with and without VNS strategy ...... 42 SART TN-002/2012 v

Nomenclature

Burn Rate Coefficient [m/s Pa-n] SRP-ANLYT Global Resolution Parameter [m] 2 Burning Surface [m ] 2 Truncated Star Correction Term [m ] 2 Nozzle Exit Area [m ] 2 Truncated Star Rectangle Correction Term [m ] 2 Port Area [m ] 2 Throat Area [m ] SRP-ANLYT Burning Step Factor [-] SRP-ANLYT Internal Burning Step [-] Characteristic Exhaust Velocity [m/s]

Nozzle Thrust Coefficient [-] Throat Diameter Optimization Step Size [m] Nozzle Throat Diameter [m] Truncated Star Branch Secant Line [m]

Aft Dome Radius Fraction [-] Star Branch Thickness Diameter [m] Thrust [N]

Average Thrust [N] Maximum Thrust [N] 2 Gravitational Acceleration Constant [m/s ] Igniter Radius Fraction [-] Number of Main Segments [-] Total Number of SRP-ANLYT Segments [-]

Number of Dome Segments (Resolution) [-] Deliverable Specific Impulse [s]

Ideal Specific Impulse [s] Number of SRP-ANLYT Sub-Segments [-] MADS iteration number [-] Klemmung [-] Axial Segment Length [m] SRP-ANLYT Axial Sub-Segment Length [m] Total Solid Rocket Motor Axial Length [m] Submerged Nozzle Length [m] Number of Interpolation Steps [-] ̇ Mass Flow Rate [kg/s] Propellant Grain Mass [kg] Number of Optimization Constraints [-] Problem Space Dimension [-] Burn Rate Pressure Exponent [-] Star Branch Number [-] Objective Function Value [-] Objective Penalty Function [-] Objective Termination Target Value [-] Burn Rate [m/s] Grain Inner Radius [m] Star Branch Tip Radius [m] Igniter Radius [m] Submerged Nozzle Radius [m] Specific Gas Constant [J/(Kg K)] Main Outer Radius [m] Truncated Star Line Burning Perimeter [m] Truncated Star Inner Arc Burning Perimeter [m] Truncated Star Branch Slope [-] Objective Function Exponent [-] Chamber Pressure [Pa] Maimum Chamber Pressure [Pa] SART TN-002/2012 vi

Truncated Star Branch Intersection Point [m] Time [s] Action Time [s] Burning Time [s] Chamber Temperature [K] 3 Combustion Chamber Volume [m ] 3 Internal Static Volume [m ] 3 Final Combustion Chamber Volume [m ] 3 Total Volume of Nose Dome [m ] Axial Coordinate of Aft Dome [m] Axial Coordinate of Igniter [m] MADS Poll Center [-] Axial Coordinate of Nose Dome [m] Burn Distance [m] Radial Coordinate of Aft Dome [m] Stopping Burn Distance [m] Radial Coordinate of Nose Dome [m]

Greek Symbols

Objective Function Target Multiplier [-]

Nose Dome Shift Fraction [-] Nose Dome Scaling Factor [-] Heat Ratio [-] Nozzle Divergence Half Angle [deg]

Mesh Size Parameter [-]

Poll Size parameter [-] Constant Loss Percentage [-] Gradient Loss Percentage [-] Submergence Ratio [-] Floating Point Tolerance [-] Characteristic Velocity Efficiency [-] Thrust Coefficient Efficiency [-] Nozzle Divergence Correction Factor [-] Truncated Star Inner Arc Angle [rad] Truncated Star Main Arc Angle [rad] SRP-ANLYT Constraint Relaxation Coefficient [-] 3 Propellant Grain Density [kg/m ] 3 Propellant Gas Density [kg/m ]

SART TN-002/2012 vii

Abbreviations

BEM Ballistic Evaluation Motor BiMADS Bi-Objective Mesh Adaptive Direct Search COG Center of Gravity CONDOR Constrained, Non-linear, Derivative-free, parallel, Multi-Objective Optimization of continues, high computing load, noisy objective functions DLR Deutsches Zentrum für Luft- und Raumfahrt (German Aerospace Center) ESA European Space Agency GPS Generalized Pattern Search HOPSPACK Hybrid Optimization Parallel Search PACKage HR High Resolution IMSL International Mathematics and Statistics Library I/O Input/Output LR Low Resolution LT-MADS Lower-Triangular Mesh Adaptive Direct Search MADS Mesh Adaptive Direct Search MEOP Maximum Expected Operating Pressure MP Multi-Processing MPI Message Passing Interface NOMAD Nonlinear Optimization with the MADs algorithm OrthoMADS Orthogonal Mesh Adaptive Direct Search RMSE Root Mean Square Error RSRM Reusable Solid Rocket Motor (Space Shuttle Booster Rockets) SART Space Launcher Systems Analysis (Systemanalyse Raumtransport) STSM Space Transportation Systems Mass SRM Solid Rocket Motor SRP Solid Rocket Propulsion analysis SRP-ANLYT Solid Rocket Propulsion – Analytical geometry burnback analysis SRP-GEO Solid Rocket Propulsion – Geometry burnback analysis SRP-GEOPT Solid Rocket Propulsion – Geometry Optimizer TOSCA-(TS) Trajectory Optimization and Simulation of Conventional and Advanced space Transportation System PB Progressive Barrier PEB Progressive-to-Extreme Barrier PSD-MADS Parallel Space Decomposition of Mesh Adaptive Direct Search PSwarm Particle Swarm VNS Variable Neighbourhood Search SART TN-002/2012 1

1 Introduction

Solid Rocket Motors (SRMs) are utilized in many space launch applications, e.g. as the booster rockets on the now retired Space Shuttle and on the new European launch vehicle called Vega. The design of these rockets are therefore of great relevance to the global aerospace community and especially in the SART division of DLR in Bremen.

It can be recognized that the first step during preliminary design of SRMs for space launch applications is to define a number of requirements that the entire vehicle must fulfil. These typically include the trajectory, final orbit, payload performance, maximum loads and dynamic pressure. It is then possible to draw a flight profile in form of a desired thrust law and with the desired thrust law; suitable grain geometries, motor configuration and casing can be designed and sized.

DLR in Bremen currently utilizes several in-house developed programs for all stages of preliminary launch vehicle design [1]. For development specific to SRMs, a core suite containing the following programs are often employed:

1. STSM – Performs initial mass estimation and center of gravity (COG) calculations. The inputs to this program are masses or characteristic dimensions of known subsystems and the main output is a detailed mass breakdown of the total vehicle.

2. TOSCA-(TS) – Performs the ascent trajectory simulations and optimization. It is used concurrently with the output from STSM and a chosen thrust-time history. This enables manual optimization of the latter in respect of a desired trajectory and payload mass.

3. SRP-GEO – Performs internal ballistics burnback simulation in three dimensions of a given solid grain geometry. This results in the geometrical evolution of the solid grain as function of the burning distance.

4. SRP – Conducts comprehensive performance calculations based on the burning analysis supplied by SRP-GEO. Part of the result is a thrust-time history which can be utilized as input in TOSCA.

Indeed, the last two steps of designing the SRM can be improved by reversing the process. By instead utilizing the agreed thrust-time from TOSCA or any other desired source as input while automatically giving a compatible geometry as output, the design process would be greatly simplified and shortened. Figure 1 displays a basic outline of the current design process (left) and a possible future simplified variety (right). The intention of this project is thus to create the necessary component (bolded) to facilitate this streamlined design process for SRMs.

1.1 Methodology

Limited numbers of studies are available in open literature concerning methods that can automatically and rapidly ascertain suitable solid rocket grain geometries in three dimensions, given certain criteria. Common amongst many is the use of derivative free optimization algorithms. These “heuristic” algorithms enable optimization of non-smooth functions where gradients are unavailable or unfavourable to evaluate due to computational costs. This is indeed the case with solid rocket grains where numerous design variables and combinations thereof results in a very complex search space. Instead, these methods rely strictly on function evaluations and through various algorithms drive the search towards a solution [2].

To escape local minima, most algorithms also incorporate randomness during the search process. Nonetheless, global optimum is not always guaranteed and good initial values and boundaries are therefore recommended [3]. Since the development of rockets often is evolutionary, i.e. newer designs are conceived by slightly modifying older designs; the starting point and search space tend to be defined and the latter issue can be minimized.

SART TN-002/2012 2

STSM Geometry STSM

TOSCA SRP-GEO TOSCA

Thrust Thrust SRP Time Time

no Thrust no Satisfied Satisfied Time

yes yes

no

Left/Right [New Program] Similar

yes

Finished Geometry Finished

Current Design Process Desired Design Process

Figure 1: Current and desired design process of a typical solid rocket motor in SART

The simplest forms of heuristic algorithms are called Direct Search methods, conceived initially by Robert Hooke and T. A. Jeeves in the 1960s. These methods are analogous with performing a random search on the design space, retaining favourable points while rejecting unfavourable ones until only the best point remains. This simple approach has then become the foundation for subsequent developments within this field [4]. Today, more complex versions of Direct Search methods exist that can intelligently guide the search towards the minimum. In this project, several algorithms employing these methods are explored. Based on their properties and features, the most suitable one is chosen and implemented.

A further benefit with direct search algorithms is their ability to treat problems as blackboxes. These blackboxes contain objective functions defining the state of the optimization problem. The inputs to these blackboxes are a set of optimization variables and potential constraints or boundaries , while the outputs are simple objective values that convey how near or far away the results are from optimum. Knowledge of the inner workings of these blackboxes is thus not required and they can be arbitrary complex. This latter property facilitates the usage of externally written simulation programs in the optimization process, for instance in this project, the burnback analysis tool SRP-GEO and the internal ballistics solver SRP.

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No

Initial Values Optimizer Minimzed? Yes Solution

Blackbox

Figure 2: Basic structure of the optimization process

Figure 2 displays the basic structure of the optimization process with results from the blackbox evaluations feeding back to the optimization algorithm. This process will repeat until a satisfactory objective value has been reached or the number of iterations has reached a certain maximum number.

To ensure compatibility and handling with existing DLR tools, the main programming language for this project is Fortran 95 and the execution environment is Linux. Likewise, the development occurs under the Linux distribution Ubuntu.

1.2 Scope

The intention is to construct the program, henceforth called SRP-GEOPT to be modular and expandable. The initial version (1.0) featuring in this report includes primarily core functionalities necessary for usage within SART. Future versions should then add additional functionalities and improvements in order to make the tool more versatile. The program utilizes a Direct Search optimization algorithm together with the burnback analysis tool SRP-GEO and the internal ballistics solver SRP in its operation. Thus, a sizable part of the work is aimed at constructing the framework necessary to automatically call these tools, manipulate the in-data and interpret the out-data. Lastly, emphasis is given to the stability of the program. This is due to the potentially long execution time encountered when complex geometries with many design variables are computed.

A quick overview of the core functionalities within SRP-GEOPT is followed below. Detailed description and explanation of each feature is available in the respective chapters.

. The ability to optimize grain geometries in three dimensions. In the radial direction, cross- section types can consist of cylindrical tubes, truncated stars and empty voids with no grains. Axially, the possibility exists to include conical shaped grains with variable radial burning cross- sections, purely straight counterparts or a combination of these. In addition, the overall length and width of the rocket can either be part of the optimization process or excluded.

. The ability to model and automatically scale the hemispherical dome structures at the top and bottom of a typical motor casing. Likewise, the grain elements inside the domes are modelled.

. The ability to model and automatically scale simple geometries representing the ignition system and submerged nozzle components within the three dimensional grain.

. A quicker but less accurate analytical burnback algorithm. This surrogate algorithm is used in tandem with SRP-GEO to accelerate the convergence of the Direct Search method.

. User defined specific or maximum chamber pressure (MEOP). For the latter case, automatic variation of the throat area is employed to control the pressure.

. A database of known solid fuel types for usage in the ballistic solver SRP.

. A structure for a database of initial grain geometries and associated algorithm. The intent is to further simplify the design process by providing users an automatic technique of determining suitable initial values for a given desired thrust-time curve.

SART TN-002/2012 4

1.3 Contents

This report contains five main chapters. In Chapter 2, an overview of basic solid rocket theory used in the construction of SRP-GEOPT is introduced. This includes typical components within such rockets and their purpose. Following, a breakdown of the internal ballistics and the relevant governing equations are also presented. Chapter 3 describes the selected Direct Search algorithm and the underlying theory behind its operation. In-depth explanation of SRP-GEOPT and its core functionalities are then presented in Chapter 4. In Chapter 5, the results from this project are validated through comparison with simple type cases, existing known grain geometries and thrust-time combinations. A simple performance and convergence analysis of the program is also made. Lastly, in Chapter 6, conclusions are drawn and a list of future improvements and additions compiled.

SART TN-002/2012 5

2 Solid Rocket Motor Fundamentals

In concept, SRMs are simple in design with few to no moving components. The propellant consists of fuel and oxidizer premixed into a solid, rubbery substance. An igniter creates the temperature and pressure conditions that ignite the exposed inner surface of the propellant. The resulting high thermal energy created by the combustion process can then be converted into kinetic energy through the nozzle, effectively creating thrust [5].

The simplicity makes SRMs reliable whilst also retaining a low development and production cost, ease of usage and ease of manufacturing [2]. They also make affordable and compact high thrust levels, which are very difficult to reach with liquid rocket engines. Further, the solid nature of the fuel renders the rockets exceptionally durable. They can be constructed, fuelled and then put to storage for numerous years without the performance characteristics changing noticeably when finally ignited [5]. These properties result in SRMs having excellent scalability and are thus used in a wide variety of applications and environments. This includes thrusters controlling attitude of nanosatellites to full- fledged booster rockets such as the Reusable Solid Rockets Motor (RSRM) on the Space Shuttle [6][7].

Downsides of SRMs include their inability to affect the thrust during operation. Once ignited, the fuel will burn until natural extinction. The only option is thus to predict the thrust-time history in advance during the design phase and shape the grain accordingly. Compared to liquid rocket engines, SRMs generally also have noticeably lower specific impulse making them a suboptimal choice as primary launch vehicles despite the simplicity and cost benefits [8].

2.1 Motor Components

In Figure 3, the principal components making up a solid rocket booster are shown. The next subsections detail each component in addition to a few comments regarding their context in the optimization program SRP-GEOPT.

2.1.1 Casing The casing makes up the external shell of an SRM. It acts as a supporting structure and combustion chamber with the grain occupying a majority of space inside. The latter implies that the casing also serves as a high pressure vessel. Casings are thus typically built of high-strength steel alloys or filament-reinforced composite materials capable of withstanding very high loads [5].

Common configurations include oblate spheroids and cylinders with either elliptical or hemispherical heads. Ratio between the length and diameter of cylindrical casings typically ranges between 2 and 5 in most launcher related applications. An overly low ratio has negative influence on the axial compressive drag loads exerted during ascent whereas a disproportionately high value adversely affects handling, stability and rigidity. This renders the rocket difficult to control whilst also induces buckling and bending problems [9]. In addition, pressure oscillation inside the combustion chamber are more likely to be severe.

For the purpose of SRP-GEOPT, only cylindrical outer casings with hemispherical dome shaped heads are included in the initial release. This configuration is used in solid rockets currently studied in SART, e.g. the Vega P80 FW first stage. Further, no restriction on the length-to-diameter ratio is currently implemented and the casing itself is considered thin in respect to the grain.

2.1.2 Nozzle SRM nozzles allow the expansion and acceleration of combustion gases through a converging- diverging nozzle. The geometry and mounting technique employed has a significant effect on the thrust produced and thus, a number of nozzle categories currently exist. For space launch applications and for projects currently studied in SART, submerged nozzles are used extensively. These nozzles have a large portion of the structure sunken into the combustion chamber that reduces the motor length and inert mass. This is especially beneficial for upper stage rockets due to the implied limitation of the length and mass of the inter-stage structure. SRP-GEOPT will therefore include this architecture in its operation [5].

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Figure 3: Components of a typical SRM [9]

Due to extreme temperature, high velocity and corrosiveness of typical combustion gases and accompanying impinging solid unburnt particles, nozzle material tend to erode during the burn sequence. This is particularly the case for areas around the throat where a change in diameter has in addition a noticeable impact on the overall performance [2]. Most nozzles incorporate this phenomenon into the design and use it as a method of cooling (Ablation cooling). This is accomplished by adding a layer of ablative liners to insulate the underlying metal structure and to provide controlled erosion. Naturally, the amount of lining must last through the burning sequence.

As the ablation phenomenon is modeled in the ballistic solver SRP, this functionality will also be transferred to SRP-GEOPT.

2.1.3 Igniter Two categories of igniters are found in SRMs, pyrotechnic igniters and pyrogen igniters. The first type utilizes explosives like black powder or small pellets of propellants that once ignited, results in a large surface burning area. The heat produced by this process then ignites the main grain. This type is commonly used for small to medium sized SRMs. The second type consists principally of miniature rocket motors with fast burning grains. Once ignited, the resulting flame spreads throughout the gas cavity area. The hot gases of the flame then interact and effectively ignite the main grain. This type of igniters is more common amongst larger SRMs [5].

For both types, igniters are often located at the forward end, opposite side of the nozzle. Although other configurations exist, the initial release of SRP-GEOPT will only include this configuration. The shape of the igniters varies to a greater extent depending on the ignition method and implementation utilized. However, as igniters tend to be small in relation to the main grain, a first approximation of modeling them as simple cylinders is acceptable.

2.1.4 Insulation With flame temperature of between 1500K-3000K in typical combustion chambers during combustion, insulation in-between the grain, internal surface of the casing and other vital components is vital [10]. Similar to nozzles, insulation inside the casing commonly employs ablation as the main method of cooling. A thorough presentation on insulation design and material choices is given in ref. [11].

The insulation in SRP-GEOPT is considered as “thin” in relation to the thickness of the grain. Thus, it is not accounted for during computation.

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2.1.5 Grain In a typical SRM, up to 96 percent of the total mass consists of propellant grain [9]. This mass of grain is commonly cast into the pre-insulated casing as a liquid. A subsequent curing process then bonds the liquid to the insulation, turning it to a solid form. This type of grain named case-bonded grain is often encountered on larger SRMs and launch vehicles. Another type is free-standing grains where the grains are manufactured separately by an external molding process and then assembled into the case. This latter type is often used in missile applications or smaller SRMs due to simplicity and cost. Conversely, free-standing grains suffer from lower performance, mass penalties due to additional support structures and lower stress tolerances [5].

While the motor is in operation, the solid grain burning surfaces recede as more propellant is turned into gas. This burnback process has a large influence on the resulting thrust-time. Depending on the starting grain geometry, a variety of thrust profiles can thus be acheieved.

Three categories of thrust profiles are commonly referred to. These include progressive burning thrust where thrust, pressure and exposed grain surface area increases, neutral burning thrust where thrust, pressure and exposed grain surface area remain generally within % and regressive burning thrust where thrust, pressure and exposed grain surface area decreases. To achieve these thrust profiles, two types of grain cross-section types are often encountered in space launcher applications. These are stars, cylindrical tubes or a combination of stars and cylindrical tubes. The advantages of these over other types of cross-sections are their ease of manufacture, natural support of their own weight and limited amount of unburned residual propellant or “slivers” [5]. Figure 4 displays the thrust profile categories and common cross-sectional types presented here.

As SRP-GEOPT is aimed at analysis of launch vehicles, only case-bonded grains with truncated star and tube cross-sectional types are considered in the initial version.

2.2 Internal Ballistics

The determination of SRM performance characteristics is known as the field of internal ballistics. Main governing parameters are related to the thrust or pressure – time curve of the motor. This curve must satisfy the mission objective and a number of constraints imposed by the launch vehicle. This includes instantaneous thrust, maximum acceleration and dynamic pressure. From Figure 5, a few important definitions related to the thrust/pressure – time curve can be distinguished. Of special note are the burn time interval and action time interval . The burn time is established as the interval between 10% of the thrust or pressure to the aft tangent bisector where burnout occurs, whereas the action time is defined as the interval between the initial and final 10% of the thrust or pressure. The total time is often denoted as the firing time whereas the ignition delay time signifies the delay encountered between when electric signals are sent to the igniter and when the propellant grain is actually ignited [12].

Figure 4: Classification of grain according to thrust-time characteristics (left) and typical grain cross-sectional types used in SRMs for space launch applications (right) [5]

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Figure 5 Thrust/Pressure – time curve definitions [5]

In the next few sections, a number of relevant internal ballistics principles are presented. Further, a few comments in the context of the ballistic solver SRP and the optimizer SRP-GEOPT are discussed.

2.2.1 Specific Impulse and Losses

Specific impulse, denoted measures the momentum change incurred from expending one unit of propellant mass, i.e. it gives an indication of total motor (propellant and nozzle) efficiency. The, is the ratio of motor thrust to mass flowrate. By cancelling out the mass flowrate, this relation can be written in a more general form as

(2.1)

where the characteristic exit velocity is used for measuring relative performance of the propellant. Further, a dimensionless thrust coefficient is used and is an indicator of the nozzle performance [13].

The ballistic solver SRP utilizes the ideal gas law to derive characteristic velocities and thrust coefficients for various conditions with the formulation of these equations available from Haidn (2008). Due to the idealistic nature of these formulations, additional efficiency factors are also taken into consideration [15]. These are available in SRP for the characteristic velocity in the form of a user defined constant percentage loss and a time-gradient percentage loss i.e.

(2.2) ( ) where is the resulting efficiency of the characteristic velocity and the time after ignition.

Similarly, a number of efficiencies are calculated in SRP for the thrust coefficient with the product of these constituting the total efficiency factor . One of these efficiencies is the nozzle divergence correction factor . This is applied to account for the radial turning momentum losses incurred on the exhaust gas. It is defined as the ratio between the exit-gas momentum of a real nozzle and that of an ideal nozzle with axial, uniform gas flow [16]. For conical nozzles treated in SRP, it is expressed as

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( ) (2.3)

where is the nozzle divergence half angle and is given as an input. Other efficiency factors for the thrust coefficient are calculated internally in SRP without requiring inputs. Governing formulas to calculate these additional efficiencies are available in the work by Reydellet [17].

Therefore, the actual deliverable specific impulse has the form of

(2.4)

of which is the ideal counterpart. From experience, typical values of for SRMs are generally in the range of around 280 seconds depending on combustion chamber pressure and nozzle expansion ratio.

2.2.2 Burning Rate

The rate of grain recession during motor operation is determined by the burning rate . This rate, which is in the direction perpendicular to the grain surface, can be expressed as a function of the chamber pressure and two empirical constants. Often referred to as the Saint Robert’s burn law, in its simplest form, it is expressed as

(2.5) where is the burn rate coefficient, the burn rate exponent and where is given in mm/s and in Bars. These two constants are typically determined from strand burning tests or Ballistic Evaluation Motors (BEM) at various pressures [13].

Apart from pressure and grain composition, the burning rate is also affected by a number of other factors. These include the solid grain storage temperature where higher temperature results in faster and fiercer combustion. Furthermore, if combustion gases inside the chamber have high kinetic energy and the gas cavity volume inside the grain is small, erosion of the solid grain surface might occur. This increases mass flow rate and thus the chamber pressure, resulting in higher thrust. Burning rate is also affected by the acceleration of the rocket with both longitudinal, spin stabilized rockets and high lateral acceleration resulting in slightly faster burn rates [5].

This burn rate law is an integral part of SRPs algorithm. The two empirical constants are given as part of the grain property and are thus supplied externally. However, additional influences from erosive burning, vehicle acceleration and solid grain temperature are currently not modeled in SRP.

2.2.3 Area Ratios Two important area ratios are required to evaluate SRM performance parameters: These are the subsonic area ratio ⁄ (inlet-to-throat) and supersonic area ratio (exit-to-throat). The inlet area is defined as the area leading into the nozzle and is often equivalent to the effective port area

, where the latter is the minimum area in case of a varying axial geometry. Similarly, the throat area is defined as the smallest area situated in the middle between the converging and the diverging part of the nozzle.

The subsonic area ratio has a strong influence on axial gas velocities within the propellant cavity volume. When the ratio is large ( ), these velocities can be neglected. However, with a small ratio, the gases are able to expand as heat is added. The energy required for gas acceleration causes pressure to drop and further energy losses. This is especially true if gas velocity reaches the speed of sound before reaching the nozzle and a normal shockwave is formed. As a result, thrust and specific impulse are diminished [5]. Additionally, due to faster gas movement, propellant erosion phenomenon becomes noticeable and the burning characteristics are affected [8]. Even with a large subsonic area ratio, the maximum thrust and highest efficiency is only achieved when the exit pressure from the nozzle equals the ambient pressure [13]. This exit pressure is determined from the supersonic area ratio or expansion ratio as it is also referred to. However, as the rocket climbs through the atmosphere during the ascent, the ambient pressure will drop. Without an adaptive (variable) nozzle, optimum conditions will thus only occur at one specific altitude.

SART TN-002/2012 10

In SRP, these two area ratios, as well as the port and throat areas can be defined individually. Further, as previously mentioned, SRP is capable of modeling throat ablation effects. This is achieved through an additional term that is determined from a formula derived empirically from internal SRM modeling within SART.

( ) (2.6) where is the initial throat diameter before ignition and where has the unit of millimeters per second with chamber pressure in Bars.

2.2.4 Thrust and Chamber Pressure The pressure-time curves (pressure history) are characterized by a rapid change of pressure during the short period at start-up after ignition and final tail-off when the solid grain has been burnt. In-between, the pressure curve is dominated by a quasi-steady phase where its characteristics are determined primarily by grain geometry and its combustion.

The Governing equations for the chamber pressure are derived from mass conversation, ideal gas formulations and the definition of ideal characteristic velocity. One important equation describes the change of pressure with time and takes form of a first order differential equation

[ ] (2.7)

where is the instantaneous gas cavity volume that expands with time, is the exposed propellant burning area; and are the solid and gas density of the propellant; is the specific gas constant and is the chamber temperature, which is independent of pressure and often assumed to be constant [5],[8]. Further, as the propellant gas density is considerably less than its solid counterpart, i.e. , the gas density term is often neglected. With (2.7), it is thus possible to determine the complete pressure-time curve.

In the process of grain design, a maximum limit to is decided at an early stage. This constraint is often referred to as the Maximum Expected Operating Pressure or MEOP. In SRP-GEOPT, this is taken into consideration during optimization and is referred to in Subsection 4.3.3 [2].

With known chamber pressure, the thrust for the SRM can be calculated through the following simple relation

(2.8)

SRP utilizes alternative form of (2.7)-(2.8) and other comparable internal ballistic equations to evaluate thrust and pressure for SRMs. The purpose of this optimization program could thus also be formulated as the search for suitable values of and for each time unit during motor operation.

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3 Direct Search Optimization Algorithm

In present day, a number of derivate free direct search software packages exist and can be found on the Internet. A few were investigated in a previous study at SART [8] whilst others were found and analyzed during the current project. In the selection process, these software packages are categorized by their advantages and disadvantages in terms of performance, features, license type and documentation. In Table 1, a brief overview of the investigated software packages are shown.

From Table 1, NOMAD or Nonlinear Optimization with the MADs algorithm stood out particularly well. This package supports up to 50 variables in single-processor environments and up to 500 variables with multi-processor, parallel versions utilizing the MPI standard. Further, it is free and with open, accessible source code. Despite being designated a local optimizer, its abundant feature set and excellent documentation with numerous research papers, user guides, examples and default parameters makes it the most suitable choice for this project.

Table 1: Overview of investigated Direct Search optimization algorithms Name Type Environment Variables MPI/MP Source License Documentation CONDOR Local C++ <100 Yes No Limited Adequate [18] GLOBAL Global F77/Matlab/C “Low” No Yes Free Adequate [19] HOPSPACK Local C++ N/A Yes Yes Free Good [20] IMSL/DBCPOL Local F90/C/C# N/A No No Commercial Adequate [21] NLOpt/COBYLA Local C N/A No Yes Free Adequate [22] NOMAD Local C++ <500 Yes Yes Free Excellent [23] PSwarm Global C/Matlab N/A Yes Yes Free Adequate [24]

3.1 NOMAD Overview

NOMAD is a C++ implementation of the Mesh Adaptive Direct Search (MADS) algorithm. It has been under development since 2000 and has had three major version releases in its history. It is used both in academic research and industry, the latter including Airbus, ExxonMobil and GM.

Two modes of operation are supported by NOMAD, Batch and Library mode. In Batch mode, NOMAD is executed externally through system calls by the user created blackbox program. Inputs and outputs are written to temporary files and read between the programs. This mode is generally intended for basic optimization problems as it is easy to implement but lacks advanced customization options. In Library mode, a user defined C++ class containing the blackbox problem interfaces directly with NOMAD, its source code and its public methods. This mode supports the customization required for advanced projects with the added benefit of a single main executable.

Subsequent sections will provide a short overview of NOMAD and the MADS algorithm together with its features applicable for SRP-GEOPT.

3.2 NOMAD Operation and Features

3.2.1 MADS Algorithm MADS is an iterative algorithm derived from earlier work by Torczon (1997) and Davidson (1991). It evaluates a blackbox function through a mesh spanning the problem dimension . Given an iteration , two steps are performed on the mesh: poll and search.

In a poll, trial points are generated close to the current best feasible solution, designated poll center

and are evaluated through the blackbox function. If a trial point results in a new incumbent objective value, i.e. best feasible value, the poll center is moved to the new location and the mesh is either coarsened or kept at its current fidelity. Conversely, if the poll fails to find a better solution, the poll center will remain at its current location and the mesh fidelity is refined. In the next iteration, whether the iteration is a success or not, a new set of trial points are created and the process is repeated.

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In search, it is possible to incorporate optional custom strategies capable of analyzing trial points at arbitrary locations on the mesh. These are especially useful for specific applications where the user has more information regarding the direction towards the optimum than MADS. In NOMAD, two useful search strategies are incorporated by default, Speculative Search and Variable Neighborhood Search; these are presented in more detail in subsection 3.1.4.

In SRP-GEOPT, a specific subset of MADS algorithm named OrthoMADS 2n is used. This variant that was recently introduced in NOMAD utilizes orthogonal poll directions instead of previous randomly generated ones. According to Abramson et.al (2009), this revised strategy proved generally more efficient than previous implementations on a set of 45 test problems [23].

An example of the mesh configuration and mesh refining procedure is presented in Figure 6. Here, is the mesh size and is the trial point distance from poll center. Further, is always smaller or equal to . The reader is recommended to review ref. [27] and [28] for a more theoretical treatment including a convergence analysis of the MADS algorithm.

3.2.2 Constraint Handling Constraints in NOMAD can be of any form, including nonlinear or blackbox variants. The sole condition imposed is that the solutions must be negative or zero to signify constraint fulfillment with positive solutions implying constraint violation. Consequently, the dimension mesh spanned by MADS is divided into a feasible domain and an infeasible domain established by the constraint conditions.

In NOMAD, trial points within the infeasible domain are not necessarily rejected immediately. By utilizing the Progressive Barrier (PB) approach, constraints are relaxed and only at the final solution must they be satisfied. This is possible as the degree of constraint violation can be measured, i.e. more positive constraint solutions imply greater degree of violation. Naturally, direct rejection of infeasible points is likewise supported in form of the Extreme Barrier (EB) approach. A hybrid method of PB and EB, called Progressive-to-Extreme Barrier (PEB) was also introduced recently and can be referred to in Audet et al. (2010).

Additionally, NOMAD has support for unexpected hidden constraints. These constraints can occur in both the feasible and infeasible domain. Such constraints emerge when blackbox functions are unable to evaluate certain trial points for any reasons. This latter feature renders NOMAD versatile for use in complex simulations where certain trial points might fail due to unexpected errors within the blackbox [23].

3.2.3 Surrogate Functions If an evaluation of the blackbox is costly and/or time consuming, a surrogate replacement can be considered. This surrogate function shares similar characteristics as the main blackbox program but is cheaper and faster to evaluate. Often it takes the form of a simplified physical model or an approximation through interpolation from a finite set of points on the real model of the blackbox [29].

Figure 6: Mesh configurations in MADS algorithm with [23]

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Further, two categories of surrogates can be discerned. Non-adaptive surrogates are defined at initialization of the optimization, these surrogates then remain static throughout the execution. Conversely, adaptive surrogates are updated and modified automatically during execution based on past evaluations of the real model.

Currently, NOMAD only supports static, non-adaptive surrogates. These surrogates are utilized on the poll prior to the real evaluations for each iteration. Trial points from surrogate evaluations are sorted by their objective values such that the most promising points are then evaluated first during the true evaluation stage [23].

3.2.4 Variable Neighborhood and Speculative Search A heuristic method based on the Variable Neighborhood Search (VNS) strategy is embedded in MADS and NOMAD. This strategy aims at evading local minima where the optimizer otherwise might be trapped in. It conducts poll-like evaluations from a randomly selected perturbation surrounding the search space close to the incumbent solution. If it finds an improvement, a new neighborhood search is commenced from this point and the process is repeated. This will continue until a maximum number of evaluations have been reached. NOMAD will then utilize the best solution (if any) found during VNS as the poll center for the next iteration. Although VNS provides a solution against local minima at an expense of large numbers of additional evaluations, it does not guarantee global convergence. However, the performance penalty of utilizing VNS is partially circumvented by employing surrogate functions, if such a function is defined. For a more in-depth study of VNS and its effectiveness, the reader is referred to the work by Hansen and Mladenović (1999).

In addition to VNS, NOMAD also includes a Speculative Search strategy that is invoked when a new incumbent solution is found [23]. This strategy adds a single evaluation point in the direction of the last successful iteration and is beneficial when the direction is sufficient whereas the trial point distance is not.

3.2.5 Parallelism As the MADS algorithm generates polls with trial points at the beginning of each iteration, implementation of parallel versions of this algorithm is fairly straightforward and NOMAD supports a number of these. In the most basic method called P-MADS, each trial point in an iteration is simply evaluated in parallel. This can either occur synchronously where all trial points of the current poll is computed before a new iteration begins or asynchronously where the moment a new incumbent is found, the current iteration is terminated. This latter method does not cancel progressing evaluations. Instead, if an old evaluation finishes after an incumbent change and the old evaluation results in a better point, the algorithm reverses and will instead consider this point [23].

More advanced parallelization algorithms are also contained in the NOMAD package. This includes a method based on Parallel Space Decomposition or PSD-MADS. In PSD, the blackbox problem is divided into a finite number of lower dimension subproblems with a number of variables fixed. Each subproblem is assigned to a slave process that will apply MADS in an effort to improve the incumbent solution. A master process controls the slaves and has the task of assigning subproblems and interpreting results. This process is likewise asynchronous where results from each slave computation are evaluated immediately. If an improved solution is found, it becomes the new incumbent and the slave that found it is assigned a new sub-problem [31]. Numerical testing reveals that with this method, large problems with up to 500 variables are applicable.

NOMAD also has support for co-operative MADS or COOP-MADS that runs several instances of the basic MADS algorithm on the same optimization problem. These instances are performed with varying stochastic seeds that induces different behaviors. Through testing, this method is shown to be most suited for smaller problems and gives better results than the scalar version [32].

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3.2.6 Additional Functionalities Apart from the previously established functionalities within NOMAD, a number of additional features are worth mentioning briefly that are used or might be of use for future versions of SRP-GEOPT.

. Integer and binary type variables are supported natively in addition to continuous doubles. This is accomplished by simply setting the mesh size to 1.

. Categorical Variables are non-continuous variables that take a finite and predetermined set of specific values where each set of values do not constitute a system of numeration. E.g. a set of colors that includes red, blue and green. In NOMAD, extensions to MADS have been developed to treat these variables.

. The user can define certain variables to be fixed during the optimization process. This is automatically utilized with the parallel PSD-MADS method.

. Multiple starting points are supported in NOMAD. If several promising sets of initial values are available. NOMAD will evaluate each set and begin the optimization process with the best candidate.

. Besides OrthoMADS 2n, several older algorithms and configurations are supported in NOMAD, i.e. LT-MADS [28] and GPS [25]. Although OrthoMADS 2n surpasses these older algorithms; in certain applications, these might yet give better results and are thus included in NOMAD.

. Optimization problems with two objective functions are computable through the BiMADS algorithm. BiMADS calculates each objective function in series and combines the result through an approximation of the Pareto front [33].

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4 Geometry Optimization Program

The Solid Rocket Propulsion – Geometry Optimizer (SRP-GEOPT) is constructed through the integration of several program modules. These include the burnback analysis program SRP-GEO, the ballistic solver SRP and the derivate free optimizer NOMAD. Several routines are added to interpret and manipulate the input-output of these externally written modules. A number of routines are also added to simplify the user interaction and enhance the robustness of the program. Lastly, additional routines are built to enable support for the numerous capabilities available in NOMAD as described in Chapter 3.

The structure of SRP-GEOPT can be considered as modular with individual routines handling specific components. Through the interactions between these components, four distinct processes can be discerned. In Pre-processing, user defined input data and program settings are read and verified. Constraints, boundaries and initial values are also created. These are entered into NOMAD which commences the Optimization process. The optimization process utilizes the Blackbox routine for calculation of an objective function which is then minimized. Once the optimization process is terminated through fulfilment of one or more stopping criteria, the results are collected and printed out in Post-processing.

An overview of the main program components collected into the four processes and their interactions are presented in Figure 7 in some detail. In the subsequent sections, these processes, many of the included components and the theory behind their operation are presented.

4.1 Pre-Processing

4.1.1 Input File and Propulsion Database Many internally developed programs within SART utilize text based input files as a medium for program configuration and data insertion. This includes SRP, SRP-GEO and thus also SRP-GEOPT. In order to avoid requiring three separate files, configuration and data parameters from all program modules are collected in one main input file with the exception of propellant data. The latter is justified by the numerous types of propellants available for SRMs and is thus more suitable for a separate database.

The main input file contains all parameters necessary to initiate a standard SRP-GEOPT execution. A short description of each parameter group is described below whereas a more detailed treatment is made in the respective chapters for each parameter. Examples of the main input file and propellant database are available in Appendix A.

. Project, Motor and Propellant Name determines the naming and identification of several input and output files. Further, the propellant name should be identical to an available propellant in the propellant database.

. Configuration Switches are Boolean switches that influences a number of program modes within SRP-GEOPT. These include the usage of automatic initial grain selector (Subsection 4.1.2), VNS search strategy and burnback analysis routine (Subsection 4.2.1).

. Optimization Variable Switches are Boolean switches that controls whether the overall length and width of the SRM will be included in the optimization process.

. Motor Data contains a number of inputs related to the nozzle geometry and internal ballistic parameters. This includes the option to set either the nozzle throat diameter or maximum chamber pressure to a user defined value (Subsection 4.3.2). Also, constant and time-gradient losses for the specific impulse are specified together with nozzle submergence and area ratios (Subsection 4.1.4).

. Desired Thrust History is the user defined objective which SRP-GEOPT will attempt to obtain through the geometry optimization process. It is in the form of a column vector containing the time and associated vacuum thrust (Subsection 4.3.3).

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Pre-Processing

Main Input Primary Input Propellant File Verification Database l a n o i

t Geometry Initial Geometry

p GeoMod Database Selection O

Secondary Input Verification

Constraint Initial value Boundary Creation Creation Creation

Boundary Output Data

Optimization Post-Processing

Stopping Incumbent NOMAD Criterias yes Data Collection Output Reached?

no

yes

New Final Incumbent no Output Found? x o b

k Objective Function Constraint c Computation Computation a l B

Surrogate Burnback Analysis SRP-ANLYT Thrust Computation GeoMod SRP Main Burnback Analysis SRP-GEO

Figure 7: SRP-GEOPT component structure and interaction chart

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. Initial Geometry gives SRP-GEOPT the initial guess required for the optimization process. This is utilized only if the automatic initial grain selector is deactivated. It describes the grain geometry by segment with the help of following parameters for each segment: Main Outer Radius, Grain Type, Grain Ending Position, Radial (Inner) Beginning Position, Radial (Inner) Ending Position, Star Branch Number, Star Branch Thickness and Branch Tip Radius. These parameters are defined in Figure 8 and will be revisited in many of the subsequent chapters.

. Miscellaneous Advanced Parameters contains additional properties used in SRP-GEOPT and associated program modules. These include a number of stopping criteria for NOMAD (Subsection 4.2.1), coefficients and step sizes for the burnback analysis routines (Subsection 4.3.1) and coefficients defining the objective function (Subsection 4.3.3).

The propellant database file contains available propellant compositions and their characteristics. It includes support for up to three molecular components and five chemical elements for each propellant. Also, the burn rate coefficient, burn rate exponent and overall propellant density are specified. Only a single propellant is selected through the main input file during initialization. The database can and is intended to be easily updated with new propellant combinations.

4.1.2 Initial Geometry Selection and Database When selecting an initial geometry for the optimizer, two constraints have to be taken into consideration. Firstly, the user must be aware that NOMAD is predominantly a local optimizer and secondly, SRP- GEOPT inherently fixes the number of grain segments and grain type, these are thus not part of the optimization process. Consequently, a well-chosen initial geometry is important in order to attain adequate results.

Two approaches are available for defining the initial geometry. It can be provided directly through the main input file by the user or a suitable geometry can be determined automatically through examination of the provided thrust-time curve. The latter is accomplished through an expert system that compares the provided thrust-time with a database consisting of pre-calculated thrust-times and accompanied geometries. The database is intended to be expanded with time through user experience and operation of SRP-GEOPT, thus gradually improving its capability of detecting more suitable starting conditions and for a wider variety of thrust-times.

Three steps are performed in the automatic determination of a suitable starting geometry. A linear interpolation and nondimensionalization (normalization) operation with respect to the maximum thrust and firing time is first made on both the provided thrust-time and for each investigated thrust-time available in the database. Root Mean Square Error (RMSE) calculations are then made in form of

Tubular Segment Type

Radial Main Radial Branch Radial Radial Beg. Outer Beg./End. Tip Beg./End. End. Position Radius Position Radius Position Position Nose Dome End Dome

Star Branch Thickness

N=6 Star Segment Type Star Segment Type Axial Ending Axial Ending Axial Ending Position Position Position

Segment 1 Segment 2 Segment 3 Main Grain

Figure 8: SRP-GEOPT geometry parameter definitions with three segment example. This grain consits of two star segments with six branches and one tubular segment. Nozzle and igniter assemblies are not shown.

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√ ∑(| |) (4.1)

where is the nondimensional, desired thrust at time , is the database equivalent and is the number of interpolation steps. From these calculations, the entry with the lowest RMSE value is identified and chosen.

As a nondimensionalization procedure is conducted before the RMSE calculations, the intention of this algorithm is to find an initial geometry that shares similar thrust-time characteristics as the desired one while neglecting the absolute values. The latter are generally dependent on the geometrical dimensions of the chosen grain type and are thus determined during for the optimization routine.

4.1.3 Geometry Modification Initial geometries entered in the main input file or through the geometry database should only consist of the main grain depicted in Figure 8. Likewise, the optimization process conducted by NOMAD is applied to only this portion of the grain. Following its determination, additional components are automatically added and scaled accordingly through the GeoMod algorithm. These include the nose and aft domes, propellant ignition system and nozzle assemblies.

4.1.3.1 Dome Structures

The nose dome is constructed through approximation with additional segments placed in front of the first user defined segment. Higher values of are possible and increase the spatial resolution of the nose dome but at a cost of added computing time during the burnback analysis. These additional segments are represented as a two dimensional cross-section through the symmetry axis. Coordinates are calculated for such geometries and is derived for general cases in Appendix B. The results can be shown to follow

[ ( ) ] (4.2) √

√ ( ) (4.3)

where and where is a scaling factor describing the obliqueness of the dome structure whereas is a shift fraction describing a combination of obliqueness and flat surface size at the very front. Examples of the effects by selecting various and parameters are visualized in Figure 10. The radius at the very end of the dome, is thus the sole varying parameter and corresponds to the SRM main outer radius.

For the purpose of SRP-GEOPT, it is decided to create a dome resembling the P80 first stage used in the Vega rocket shown in Figure 9. As this dome is fully hemispherical with no flat front surfaces, is set to zero and a suitable value of is selected.

. Figure 9: P80 FW first stage solid booster with hemispherical nose dome

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variation 1

= 0.5  = 0.3  = 0.1  = 0 1 1 1 1

1.2

1

훽

0.8 y

0.6

0.4

0.2

0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x

variation 2

 = 1 = 5 = 2 = 1.2 2 2 2 2

1.2

1

훽

0.8 y

0.6

0.4

0.2

0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x

Figure 10: and variation with meters

The total volume within the nose dome could be considered as a stack of cylinders with diminishing or increasing radii, approximating a spherical shaped object:

∑ (4.4)

Indeed, when , an integration can be performed on equation (4.4). In the case of and , the equation of a sphere can be discerned.

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Similarly, the dome structure at the end of the rocket can be modeled with the help of equations (4.2) and (4.3). A single modification is required to account for the aft extrusion of the rocket nozzle. The result is a premature discontinuation of the dome forming a flat surface and is of similar appearance as with setting a non-zero value. The difference lies in the radius-curve at the discontinuation location where it is not intended to be parallel with the y-axis. From this, it can be concluded that the aft dome axial starting coordinate, should be considered as non-zero. This can be expressed from equation

(4.2) by replacing ( ) with ( ) , i.e.

[ ( ) ] (4.5) √

where is the aft dome radius fraction and determines the discontinuation location and where is linearly spaced between and . This is visualized in Figure 11 for a number of values. For the purpose of SRP-GEOPT, is decided which is comparable to the geometry of the P80 FW solid booster. Examples of aft domes with this configuration can be discerned in Figure 8 and Figure 9.

4.1.3.2 Ignition System and Submerged Nozzle Assembly The ignition system is modelled as a simple cylinder that stretches the length of the nose dome, i.e.

where is the coordinate at the end of the igniter. This is deemed sufficient in the preliminary design phase which SRP-GEOPT is intended for. Further, the igniter shares the same axial coordinate points as the nose dome, i.e. . The igniter radius can thus be considered as a fraction of the main outer radius through the relation

(4.6) where is the igniter radius fraction. It is decided to set this parameter to and thus once more mimicking the P80 FW solid booster configuration. Example of a modelled ignition system can be discerned in Figure 12. Of specific interest is the igniter radius which is marginally diminished at the front end of the nose dome in order to be fitted properly inside the casing.

The submerged nozzle assembly has similar properties as the ignition system in that it is modelled as a simple cylinder and shares same axial coordinates (segment divisions) as other components of the SRM. However, the axial length is determined by the user through setting a submergence ratio . This ratio is defined as

e variation 0

1.2

1

훽 e0 = 0.8 훽

0.8 y

0.6 e0 = 0.5

0.4 e0 = 0.2

0.2

e0 = 0.0

0 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 x

Figure 11: variation with meters

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(4.7)

where is the total axial length of the SRM including nose and aft domes. The radius of this cylinder is equivalent to the outer radius at the aft-dome discontinuation location. Thus, it can be expressed as

(4.8)

Special considerations are noted at the axial position where the submerged nozzle ends with the fundamental concept of these considerations displayed in Figure 12. From the figure, it is shown that if this ending position occurs at an arbitrary location far away from an existing, “natural” segment ending, two additional segments placed in close proximity to the submerged nozzle ending position are automatically created and overlaid on the existing geometry. This approximates a vertical discontinuation of the cylinder representing the submerged nozzle. If the submerged nozzle ends near an existing segment ending position, then only one additional segment is created. Lastly, if two existing segments end in close proximity to each other and the submerged nozzle ends in-between these, then no additional segments are added. This strategy reduces the addition of unnecessary segments that degrades the burnback algorithm performance and increases stability of SRP-GEOPT by eliminating risks of non-valid additions of segments.

4.1.3.3 Dome Grain Geometries Grain geometry within the dome sections are modeled as extrapolations of the first and last main segments. The GeoMod algorithm will thus attempt to continue this main geometry inside the domes. Indeed, with diminishing outer radii, it becomes progressively more difficult the farther inside the domes the grain reaches. Therefore, compromises to the grain are established based on certain rules. These are:

. If the first or last main segment is a star and continuation of this segment in the domes results in collision of the tip radius with outer radius, the grain will be altered to a cylinder. If the inner radius of this altered cylindrical geometry results in further collisions, it will be altered again to empty space.

Magnifcation

Two Add. Segments One Add. Segment One Add. Segment No Add. Segments

Real Subm. Nozzle Ending g g g g g g g g n n n n n n n n i i i i i i i i d d d d d d d d n n n n n n n n E E E E E E E E

...... g g g g g g g g e e e e e e e e S S S S S S S S

g g g g w w w w n n n n i i e e i e e i t t t t N N N N s s s s i i i i x x x x E E E E Nose Dome End Dome

Ignition System

Submerged Nozzle

Lsub

L

Figure 12: Ignition system and submerged nozzle assembly

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. Likewise, if the first or last main segment is a cylinder, the algorithm will immediately alter it to an empty space upon detecting a collision between the inner and outer radius.

. Algorithms are present to prevent the grain from passing through the igniter or submerged nozzle. Rather, the grain will be morphed to follow the outer contours of these.

. Algorithms are also present to prevent the grain from intersecting and passing through the main axial symmetry axis. It will instead follow the symmetry axis at a small distance. The grain will thus always have inner radii larger than zero.

. Inner radii are prohibited from intersecting the tip radius. Rather, these will follow the tip radius contours until or if the geometry is changed to cylinder type.

Figure 13 describes a number of geometries and the automatic modifications performed on the grains in the dome structures. Currently, manual definition of the grain within the domes is not supported. This is due to geometrical complexities arising when the outer radius and length of the rocket changes during the optimization process.

4.1.4 Constraint, Boundary and Initial Values The non-modified geometry entered in the input file or through the geometry database is collected in a single vector and inserted to NOMAD as initial values. Likewise, boundaries and constraints for each optimization variable are created and inserted. For compatibility the constraints are required to take the form of where are the constraints and must remain negative or zero to be fulfilled. To simulate strict inequalities, a small tolerance term is thus introduced. Following, descriptions of constraints and boundaries utilized for SRP-GEOPT are presented.

The axial ending positions denote the positively increasing x-coordinates at the end of each segment.

Thus the following equation must hold true and where is the ending position of segment .

(4.9)

Boundaries for where is the number of main grain segments are set to 0.01 meters at the lower bounds and infinity at upper bounds. The last coordinate influences the overall length of the main SRM. If chosen to be optimized, boundaries for this variable are set between and , per default.

For cylindrical and star-shaped segments, the inner radii are required to be smaller than the main outer radius and larger than the throat radius of the nozzle (to avoid high Mach numbers inside the combustion chamber), i.e.

(4.10)

Figure 13: Various modifications to nose dome grain geometry based on the first main segment

assuming . Identical modifications are performed on the aft dome.

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(4.11)

where is the inner radius and applies at both the beginning and ending of a segment. Boundaries for are set to between meters and infinity, thus relying primarily on the constraints while the boundaries for the outer radius are set between and if chosen to be optimized.

For star specific segments, a number of additional constraints are required. This includes a constraint limiting the thickness of the star branches at segment to twice the smallest inner radius of the same segment, i.e.

{ } (4.12)

with boundaries for set between and infinity. Further, two constraints are established for the branch tip radius which requires it to be smaller than the outer radius while larger than the maximum inner radius at the corresponding segment. Therefore, the following equations hold true.

(4.13)

{ } (4.14)

The default boundaries for are likewise set loosely to between and infinity. Lastly, additional two constraints are required for the analytical burnback algorithm. These constraints describe the upper validity limitations of the equations utilized during the burnback analysis and are clarified in more detail in subsection 4.3.1. The results are detailed below as

(4.15) √( { } ) ( )

( ) (4.16)

{ }

where is the number of star branches at segment and is bound restricted between 4 and 16. Further, is a stopping criterion for the burn process and is defined from Figure 15 as

[ ] (4.17) √

of which is a relaxation coefficient intended to ease these analytical burnback constraints and is selectable between and in the input file. Figure 14 shows the effects from various selections of this coefficient. It is shown that a value close to represents the strictest requirement and thus gives the most accurate calculations. Setting this relaxation coefficient to a low value is beneficial if the user does not intend to utilize the analytical burnback algorithm during optimization. Since the numerical counterpart, SRP-GEO is capable of handling abnormal geometries far better these constraints are not required to maintain program accuracy.

4.1.5 Miscellaneous All inputs entered through the main input file, the propulsion and geometry databases are verified during pre-processing. This allows the user to detect errors early in the execution. It also adds limitations to input parameters such that they remain within feasible ranges, thus avoiding program halts and unnecessary restarts of the optimization routine at a later time.

Two stages of input verifications are performed. In the primary verification, all parameters including the main grain geometry from the main input file (if applicable) are verified. Passing this stage, dome components, ignition systems and submerged nozzle assemblies are added to the selected initial geometry by GeoMod. This extended geometry is then verified through the secondary verification process, ensuring that no unexpected issues have surfaced after the modification.

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Validity Range of Star Branch Tip Radius With Varying  1.5 =1.00 1 =0.85 =0.50 =0.00 0.5 Invalid 0 Valid Variable ValidityVariable Range -0.5

-1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Star Branch Tip Radius r [m] p

Validity Range of Star Branch Thickness With Varying  0.8 =1.00 0.6 =0.85 0.4 =0.50 =0.00 0.2 Invalid 0

-0.2 Valid

-0.4 Variable ValidityVariable Range -0.6

-0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Star Branch Thickness Diameter f [m]

Figure 14: Validity range of SRP-ANLYT constraints. Upper graph plots (1.16) with m, m and . Whereas, the lower graph plots (1.17) with m, m, and m. Negative values represent valid constraints.

During the initial value creation, SRP-GEOPT will combine axial beginning and ending positions if these happen to have the same value, i.e. . Hence they will be optimized as a single variable, retaining the connection throughout the computation. Also, customizable boundaries are supported in SRP-GEOPT to give users finer control. These are modifiable in an automatically generated file during initial program execution but before the start of the NOMAD optimization process. As the default boundaries are indeed very relaxed, it is advisable to modify these for larger problems to avoid convergence toward local minima.

4.2 Optimization

4.2.1 NOMAD Implementation Before the optimization is launched, initial values, boundaries and constraints created during pre- processing are read by NOMAD. Following, a number of stopping criteria given in the main input file are also inputted. These include the maximum number of blackbox evaluations, a target value for the objective function and a maximum mesh size index giving the minimum size for the mesh spanning the search space. Lastly, settings concerning the usage of the VNS search strategy, burnback analyses modes and initial mesh index are defined.

During the optimization, NOMAD will repeatedly call the blackbox with new sets of variables with the output consisting of the corresponding objective value. If an improved incumbent is found, the objective value and the accompanied grain geometry are printed to a file. This process will then continue until a stopping criterion has been reached.

For constraint handling, the Extreme Barrier (EB) approach is utilized. This approach is chosen as constraint violations in SRP-GEOPT indicate invalid geometries that can cause program instabilities during burnback analyses. The algorithm will thus immediately reject solutions that violate any constraints with the returning objective value set to infinity.

Of note is that NOMAD in SRP-GEOPT is the sole program module implemented in C++ with the I/O communication between this module and others occurring transparently through Library mode. Further, in the current implementation, parallel computation of the NOMAD code is not supported.

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4.3 Blackbox

4.3.1 Burnback Analyses SRP-GEOPT is capable of utilizing the numerical burnback algorithm, SRP-GEO, the analytical counterpart, SRP-ANLYT or the combination of both algorithms during blackbox evaluations. In the latter case, SRP-ANLYT will perform as a surrogate “function” as described in Chapter 3.

SRP-GEO is an existing external program module and requires an input file with a given grain geometry while providing an output file with the analysis results. These results consist of the unburnt propellant mass, the burnt propellant mass and the gas port volume, all as functions of the grain burning distance/burnt web. SRP-GEO supports three-dimensional complex grain geometries with spatial definitions shown in Figure 8. The burnback calculations are performed by creating a cubical three- dimensional mesh approximation of the given grain geometry. During each iteration, an increasing part of this cubical mesh is considered as burnt until the entire mesh is expended. In the application of SRP- GEOPT, this numerical burnback routine can thus be considered as a blackbox component within the greater blackbox routine.

The advantage of SRP-GEO is its ability to handle complex grain geometries without explicit mathematical definitions. However, this approach also incurs long execution time. This arises since accurate modelling of the grain geometry necessitates very fine cubical mesh structures. As NOMAD may require thousands of blackbox evaluations to reach convergence, the long computation time is often prohibitive. However, with the analytical SRP-ANLYT, a compromise is reached. This algorithm mimics many of the functionalities, inputs and outputs of SRP-GEO though without the disadvantage of the long execution time. Following, this section will hence focus on detailing SRP-ANLYT.

The burnback computation performed by SRP-ANLYT could be divided into two elements: A radial analysis treating the cross-sectional evolutions of the individual segments and an axial analysis treating the third dimensional interaction between the various segments. The desired results for the complete three-dimensional grain can thus be discerned by combining the two computational elements.

4.3.1.1 Analytical Radial Burn Analysis

For cylindrical radial cross-sections, the empty port area evolution ( ) is modelled as an expanding circle, i.e.

( ) (4.18)

Expansion continues until , in which burnout is considered to have occurred. Similarly for empty cross-sections, equation (4.18) can also be utilized although with . Burnout thus occurs instantly and the equation can be reduced to

(4.19)

For truncated star cross-sections, analytical equations based on ref. [34] is utilized with the derivations of these equations presented in Appendix B. It can be discerned from Figure 15 that four phases exist where each change in phase requiring a modification to the underlying mathematical model for the burning process.

During Phase One, the initial grain structure is burnt. From Figure 15, the port area can be represented as the sum of a square (1) with side lengths , a circle sector (2) stretched by the arc , a triangle (3) defined by and a rectangle (4) of sides and .

The line burning perimeter can be shown to follow

√( ) ( ) (4.20)

where . It can be noted that must remain positive to ensure the validity of the analytical model. This places restrictions on the geometry and is considered during constraint formulation through equation (4.15). Equation (4.15) is thus a reformulation of (4.20) with and where is

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Phase 2 f/2

Ro 1 y R o f y e n 1 d Magnification se ‘ 4 a ri rp r p r p Ph S1

3 Phase 3 𝑠 S2 2

Phase 4

ri

Figure 15: Truncated star geometry and phase definition the stopping criterion occurring at a sufficient distance such that the resultant unburnt sliver comprising of Phase Four remains small (Figure 15). A similar analysis can also be conducted for the inner arc burning perimeter which takes the form of

( ) ( ) (4.21)

where the arc angle is

( ) (4.22)

Hence, must also remain positive as stipulated by the constraint equation (4.16). This gives the total port area for Phase One burning as

[( ) ( ) ( )( ) ( ) ] (4.23)

Phase One progresses until the square component (1) reaches the outer radius. From Appendix B, this phase shift is derived to occur when the burning distance reaches

(√ ) (4.24)

In Phase Two burning, the inclusion of a geometry correction term is necessary. This takes into consideration that part of the square component (1) will reside outside the curved outer radius. In Appendix B, this correction term is shown for a single truncated star section to take the form of

( ) ( ) [ √ ( ) ( )] (4.25)

with denoting the length of a perpendicular secant line intersecting the outer radius and follows

√ [ ( ) ] (4.26) ( )

where

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( ) ( ) ( ) (4.27)

Lastly, the angle is derived to be

( ) ( ) (4.28)

Thus, the total area evolution during Phase Two of the burning process can be expressed as

{( ) ( ) ( )( ) ( )

( ) ( ) (4.29) [ √ ( )

( )]}

It can be noted that with an increased burn distance, the secant line will decrease. When this line has reached zero, Phase Two is considered to have ended.

For Phase Three, the correction factor is rewritten with , thus

[( )√ ( ) ( ) ( )] (4.30)

where is equation (4.28) excluding the second term containing . An additional correction term is also required which considers the rectangle that forms from the enlarged square component (1) outside the outer radius. This correction term is expressed as

( )( ) (4.31) and is added to . Thus, the total area evolution during Phase Three is

{( ) ( ) ( )( ) ( )

[( )√ ( ) ( ) ( )] (4.32)

( )( )}

Phase Four is denoted the sliver phase and is not treated in the work by Hartfield et.al (2003) as it is considered “small”. This assumption holds as for the majority of this phase, the chamber pressure will be reduced to the limit of combustion and the grain will generally cease burning. In the current version of SRP-GEOPT, this last phase is thus not modelled explicitly. Instead, for compatibility with SRP-GEO, Phase Three equations are extended to Phase Four despite being not fully valid in this context. This is deemed appropriate as the constraints given by equation (4.15) and (4.16) insure the slivers will remain small. The radial burning process for a particular cross-section then ends when the port area equals the total area, i.e. when the entire grain cross-section is burnt.

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4.3.1.2 Analytical Axial Burn Analysis For the analytical axial analysis, it is first prudent to establish a few simplifications. These are recognized as:

. The burning process is considered strictly linear, thus no rounding of corners (fillets) occur at the junction between the grain segments.

. Fuel properties are identical for all segments, i.e. the grain evolves with same speed over the whole three-dimensional geometry.

. The outer radius is assumed to be constant and equals to . Consequently, the declining outer radii of the dome structures are not currently modeled.

The above simplifications enable calculation of the axial grain evolution based on Figure 16. In this figure, a grain segment can be considered to be axially expanding, contracting or remain static subject to the inner radius and the relationship to its neighbors and . When a segment has contracted to the length zero, it is considered as burnt out and is removed from the active roster. The inner radius neighborhoods are subsequently redefined to reflect this missing segment. Likewise, if the burning process reaches the ends of the SRM, further burning towards this direction is halted.

Cone shaped axial segments where are modeled as a series of automatically generated sub-segments with linearly interpolated constant inner radii, e.g. top-left example in Figure 16. The number of sub-segments created is dependent on a global resolution parameter which is coded with a value of m. If is denoted the initial axial length, in this instance, of a conical segment, then number of additional horizontal sub-segment each with the length of are created according to

⌊ ⌋ (4.33)

Indeed, if a segment does not have a conical shape, then is equal to zero. The total number of axial segments treated in SRP-ANLYT can thus be regarded as a sum of the main segments , the nose and aft dome segments and any automatically created conical sub-segments, i.e.

i+1 i i i-1 i-1 i+1

Burn Direction 1 + i r Burn Burn Burn i i Direction r Direction r Direction 1 1 1 - - + i i i r r r

ini y ini y ini ini y ini y ini l i-1 l i l i+1 l i-1 l i l i+1 i – Static i – Expanding

i-1 i+1 i i-1 i i+1

Burn Burn Direction Direction 1 1 - + i i r r 1 1 i i - + i i r r r r

ini y ini y ini ini ini ini l i-1 l i l i+1 l i-1 l i l i+1 i – Contracting i – Static Figure 16: Examples of axial segment burning evolutions based on neighboring inner radii

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∑ (4.34)

Additionally, a single burning step is utilized for both axial and radial computations. This step is added to the burning distance during each iteration, i.e. and is calculated as a fraction of the shortest axial segment, including sub-segments.

{ } (4.35)

where is a dimensionless scaling factor coded with a value of . Due to the typically short segments produced by the nose and aft domes, tends to be small. Conversely in SRP-GEO, as each iteration requires substantial computation time, the burning steps are also significantly larger. To preserve compatibility between the two algorithms and avoid erroneous oscillations during interpolation procedures in SRP, an additional linear interpolation step is performed on the analytical burnback result that equalizes the step distances.

4.3.1.3 Volume Calculations and Post Processing

The three-dimensional gas cavity volume evolves as a function of the burning distance. This evolution can be expressed as

( ) ∑ ( ) ( ) (4.36)

where is the static volume occupied by the ignition system and submerged nozzle assembly. Further, the number of segments is also noted to generally decrease during the computation process due to axially burnt out segments.

The unburnt propellant mass can then be calculated from

( ) ( ( )) (4.37)

where is the total internal volume of the SRM. In the current implementation, it can be noted that the dome structures are taken into consideration during calculation of as it gives more accurate results.

Lastly, the burnt propellant mass is determined through

( ) ( ( ) ( )) (4.38)

Results from (4.36), (4.37) and (4.38) are thus the outputs from SRP-ANLYT and corresponds to those from SRP-GEO.

4.3.2 Thrust Computation Rocket performance is predicted by the internal ballistic program SRP. It utilizes results from the burnback analysis, described in subsection 4.3.1, many of the motor properties defined in the main input file and propellant component data from the associated database. Further, the throat erosion speed is also calculated from (2.6). An additional input file is automatically created before each SRP execution containing most of the relevant information and can be reviewed in Appendix A. Similarly to SRP-GEO, SRP is thus also considered as a blackbox within the greater blackbox process.

SRP-GEOPT supports statically defined and dynamically determined initial throat diameters. Selecting the latter will initiate a custom optimization procedure separate from the main NOMAD routine. The objective of this procedure is to determine a throat diameter resulting in a maximum chamber pressure (MEOP) corresponding to the one chosen in the main input file. This optimization routine utilizes the

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same fundamental principles as NOMAD and similar direct search algorithms applied to the single variable . Specifically, it compares the maximum pressure given by SRP and drives the search towards a suitable direction. Thus it first requires an initial value for denoted . The determination

is based on the conservation of mass and a number of assumed internal ballistics parameters. Through this approach, the throat diameter can be expressed as

√ (4.39)

where

( ) ( ) ⁄ √ ( ) (4.40)

of which is the maximum desired thrust and the desired chamber pressure specified in the main input file. The assumed parameters include an internal heat ratio specified to , the real specific impulse s, the chamber temperature K and the specific gas constant set to J/(Kg K) with a molar mass of g/mol.

Further, an initial step size is required. This is determined from

(4.41)

where is the step length in meters. As there are only two directions for a single variable optimization procedure, each time the direction is changed, the step size is also halved. Following, the computation is ended if either the calculated pressure is within 1% of the target value or ten direction changes have occurred. It can also be noted that an increase in the throat area results in a decrease in chamber pressure due to higher outgoing mass flow and likewise vice versa. The direction of the search is thus intuitively known and the convergence speed is typically fast, often within ten iterations.

Further improvements can be made if the geometry difference between consecutive blackbox evaluations is recognized to be small. This holds especially true during the latter part of the computation where the mesh size index is large. To accomplish this, the optimized throat diameter from the previous blackbox evaluation is saved and is utilized as the initial value for the next iteration. Further, to take the narrower search space into consideration, the step size is also modified according to

(4.42)

where here, is the optimized throat diameter from the previous blackbox evaluation. This modification is applied to all executions except the first. The result is a further reduction in convergence time to generally within six iterations.

4.3.3 Objective Value Determination

The objective function is constructed from the -norm [35] weighed with the average desired thrust,

where is deduced from the thrust vector given by the user. In general form, it is expressed as

⁄

[ ∑| ( )| ] (4.43)

where is a multiplier coefficient and the norm exponent. Thus, equation (4.43) calculates the residual error between the desired thrust and the calculated thrust . Subject to various selections of and , the behaviour of the objective value calculation can be affected. In practice, the most

SART TN-002/2012 31

common implementation is the least square function, or the -norm with and but the option remains open for the user to redefine them. In order to discourage unrealistically high chamber pressures, especially with static throat diameters, a penalty function is added to (4.43). This penalty is activated if the MEOP exceeds 1% of the maximum pressure selected in the main input file and has the form of

( ) (4.44)

As holds in most circumstances, the addition of this penalty function will be considered equivalent to a constraint violation.

Due to dissimilar burning times, the vectors and are often not of the same element size and thus prevents the calculation of (4.43). To resolve this issue, additional zero-thrust elements are added to the shorter of the two vectors such that both are equalized. The drawback of this approach is that in certain circumstances when the computed burning time greatly exceeds the desired counterpart while the computed thrust level remains small after desired burnout, will also become small for any calculated thrust curve, i.e.

( ) ( ) (4.45)

To counteract this unwanted phenomenon, computed burning time exceeding the desired burning time is discouraged through an additional penalty function which takes the form of

√ ( ) (4.46)

where is the termination target value given trough the input file. The true objective function utilized by NOMAD is thus the sum of (4.43), (4.44) and (4.46).

4.4 Post Processing

When NOMAD computations are halted due to fulfilment of any stopping criteria, relevant data are sorted and printed to an automatically generated output file. This output file contains:

. Detailed statistics from the NOMAD optimizer, e.g., number of iterations, number of blackbox evaluations and execution time.

. Final objective function value, main outer radius, throat diameter and MEOP.

. The optimized geometry configuration given in the same format as in the SRP-GEO input file.

. A side-by-side comparison between the desired thrust supplied by the user and the computed final thrust as function of the burning time.

. Computed chamber pressure as function of the burning time.

To avoid accidental overwriting of this file, the filename is appended with the project name given by the user.

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5 Testing and Validation

Testing and validation of this project is conducted in three parts. The custom burnback analysis algorithm SRP-ANLYT is first verified through comparison with the existing numerical counterpart. This is done for a number of grain types and combinations thereof. Subsequently, the SRM geometry optimization program SRP-GEOPT is tested through examples utilizing varied program settings and input data. Results are then studied and briefly commented. Lastly, a simple convergence analysis of the applied optimization procedure is made and recommendations are put forth concerning input parameter choices.

All computations in this chapter are performed in DLR Bremen with a dedicated server running on an Intel Xeon X5570 processor with four cores and hyper-threading, i.e. eight threads. But due to current program limitations, only a single processing thread is utilized for the calculations.

Validation and analysis of previously existing program modules SRP-GEO, SRP and NOMAD are thus not performed in this report. The reader is recommended to review the existing documentations [15], [28] concerning each program for more in-depth assessments of these kinds.

5.1 Validation of Analytical Burnback Algorithm

5.1.1 Simple Cylindrical Tube The most rudimentary geometry applicable in SRP-GEOPT besides an empty casing is a lone cylindrical main grain segment with additional components added through the GeoMod algorithm. Such a grain with input dimensions given in Table 2 is thus investigated first.

Table 2: Single cylinder geometry dimensions in meters with m and Type Ending Radial Beg. Radial End. Star Branch Star Branch Branch Tip Position Inner Radius Inner Radius Number Thickness Radius Cylinder 10.0 0.45 0.45 0 0.00 0.00

Analytical Burnback Results for Single Cylinder Grain SRP-ANLYT 4 4 x 10 x 10 SRP-GEO (LR) 8 8 50 SRP-GEO (HR)

45 7 7

40 6 6

35

5 5

] 3

30

4 4

25

3 3 Volume [m Chamber

Burnt Propellant Mass Burnt Propellant [kg] Unburnt Propellant Mass Propellant [kg]Unburnt 20

2 2 15

1 1 10

0 0 5 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Burn Distance [m] Burn Distance [m] Burn Distance [m] Figure 17: Comparision of burnback results between SRP-ANLYT and SRP-GEO for single cylinder grain

SART TN-002/2012 33

The analytical results from this grain are compared to two SRP-GEO computations with different mesh resolutions. This enables investigation of the direction of convergence with respect to increasing resolution without conducting exceedingly long computations on very fine meshes.

Figure 17 shows the resulting chamber volume, unburnt and burnt propellant mass as function of the global burning distance . It can be discerned that with a crude SRP-GEO mesh, designated here as low-res (LR) with a mesh size of m, the difference between the two algorithms is noticeable. By increasing the resolution to m i.e. high-res (HR), the related curves are seen to move toward the solutions produced by SRP-ANLYT. As expected, this particular behaviour favours the validity of the analytical burnback algorithm. Indeed, such a grain geometry can easily and without simplification be treated analytically. It can be noted that the execution time for SRP-ANLYT in this particular case is less than one second whereas for HR calculation through SRP-GEO requires approximately 2 hours.

5.1.2 Simple Star Equations for calculation of star geometries presented in subsection 4.3.1 are validated by investigating a single element of such geometry. The input parameters of an example grain utilized for this purpose are listed in Table 3 with the SRP-GEO mesh size set to m. Further, this grain is likewise modified through the GeoMod algorithm before the burnback analyses are commenced.

Table 3: Single star geometry dimensions in meters with m and Type Ending Radial Beg. Radial End. Star Branch Star Branch Branch Tip Position Inner Radius Inner Radius Number Thickness Radius Star 10.0 0.45 0.45 4 0.05 1.10

From Figure 18, it can be discerned that the results correlate relatively well with those from the numerical SRP-GEO calculations. Only at the latter half of the burning sequence are the results between the two algorithms diverging slightly.

Analytical Burnback Results for Single Star Grain SRP-ANLYT 4 4 x 10 x 10 SRP-GEO (HR) 12 12 70

60 10 10

50

8 8

] 3 40

6 6

30

Chamber Volume [m Chamber

Burnt Propellant Mass Burnt Propellant [kg] Unburnt Propellant Mass Propellant [kg]Unburnt 4 4 20

2 2 10

0 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Burn Distance [m] Burn Distance [m] Burn Distance [m] Figure 18: Comparision of burnback results between SRP-ANLYT and SRP-GEO for single star grain

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5.1.3 Complex Grain To verify the axial burnback equations of SRP-ANLYT, a multi-segment, complex grain is tested. The input geometry of this grain is summarized in Table 4. It includes empty, cylindrical and star radial segment types in addition to straight and conical axial segments. Further modifications are also provided by the GeoMod algorithm.

Table 4: Complex grain geometry dimensions in meters with m and Type Ending Radial Beg. Radial End. Star Branch Star Branch Branch Tip Position Inner Radius Inner Radius Number Thickness Radius Cylinder 2.00 0.15 0.45 0 0.00 0.00 Cylinder 4.00 0.45 0.45 0 0.00 0.00 Star 5.00 0.45 0.45 4 0.05 1.10 Star 7.50 0.45 0.50 4 0.05 1.15 Cylinder 9.50 0.50 1.30 0 0.00 0.00 Empty 10.0 1.40 1.40 0 0.00 0.00

The resulting mass and chamber volume evolution are shown in Figure 19. It can be discerned that the difference between SRP-ANLYT and SRP-GEO is small for the initial half of the burning sequence with a slight divergence during the latter half.

5.1.4 Constraint Violation Analysis As the two analytical burnback constraints given during pre-processing are relaxable by setting in the input file, it is of interest to investigate the behaviour of SRP-ANLYT during instances of constraint violation. Table 5 lists the inputs of an extreme case where both constraints are unfulfilled.

Table 5: Constraint violated single star geometry in meters with m and Type Ending Radial Beg. Radial End. Star Branch Star Branch Branch Tip Position Inner Radius Inner Radius Number Thickness Radius Star 10.0 0.15 0.15 8 0.05 0.90

Analytical Burnback Results for Complex Grain SRP-ANLYT 4 4 x 10 x 10 SRP-GEO (HR) 10 10 70

9 9

60 8 8

7 7

50 ]

6 6 3

5 5 40

4 4

Chamber Volume [m Chamber

Burnt Propellant Mass Burnt Propellant [kg] Unburnt Propellant Mass Propellant [kg]Unburnt 30 3 3

2 2 20

1 1

0 0 10 0 0.5 1 1.5 0 0.5 1 1.5 0 0.5 1 1.5 Burn Distance [m] Burn Distance [m] Burn Distance [m] Figure 19: Comparision of burnback results between SRP-ANLYT and SRP-GEO for complex grain

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Analytical Burnback Results for Constraint Violated Star Grain SRP-ANLYT 4 4 x 10 x 10 SRP-GEO (LR) 8 8 50

45 7 7

40

6 6

35

5 5 ]

3 30

4 4 25

20

3 3 Volume [m Chamber

Burnt Propellant Mass Burnt Propellant [kg] Unburnt Propellant Mass Propellant [kg]Unburnt

15

2 2

10

1 1 5

0 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 0 0.1 0.2 0.3 0.4 0.5 Burn Distance [m] Burn Distance [m] Burn Distance [m] Figure 20: Comparision of burnback results between SRP-ANLYT and SRP-GEO for constraint violated star grain

From Figure 20, it is shown that the burnback results between SRP-GEO and SRP-ANLYT differ noticeably through the burning sequence. This validates the importance of these constraints and highlights the limitations of an analytical approach when calculating complex star geometries (as opposed to simpler cylinders)

5.2 Testing and Validation of Optimization Algorithm

5.2.1 Cylindrical Tube Initial validation of SRP-GEOPT is performed by revisiting the basic one segment main cylindrical grain. In this test example, the simplest program settings are also utilized which excludes the optimization of the outer radius, overall length and the usage of the VNS search strategy. Further, static throat diameter and only the analytical burnback algorithm are employed with a default relaxation coefficient of . Table 6 lists the geometrical target values that give the desired thrust history together with a set of “displaced” Initial values and corresponding boundaries. It is shown that this problem only includes two optimization variables, the beginning and ending inner radii.

Table 6: Initial values, boundaries and solutions for cylindrical tube geometry Parameter Lower Initial Upper Target Optimized Difference in [m] Bound Value Bound Values Value Percentage Outer Rad. - 1.20 - 1.20 - - Throat Diam. - 0.60 - 0.60 - - End. Pos. - 10.0 - 10.0 - - Radial Beg. 0.10 0.35 0.90 0.60 0.60 0.00 Radial End. 0.10 0.75 0.90 0.60 0.60 0.00

Additional NOMAD configuration settings (Section 4.2) include the termination target value set to ; an initial and maximum mesh size index of and respectively, the maximum number of blackbox evaluations set to , the objective function target multiplier set to and the exponent set to .

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Thrust Time Curve for Single Cylinder Grain 3500 Desired Thrust Calculated Thrust Initial Guess 3000

2500

2000 Thrust [kN] 1500

1000

500

0 0 50 100 150 Time [s] Figure 21: Desired and optimized thrust for single cylinder grain

Default motor data are utilized for this example with a submergence ratio of , nozzle half angle of degrees, ISP loss gradient and constant loss of and % respectively and a subsonic/supersonic area ratio of / . The propellant is AP68/HTPB14/Al18 with the chemical compositions and properties listed in Appendix A.

Figure 21 depicts the desired and optimized thrust time curve with the optimized geometry listed in Table 6. From these results, it can be concluded that the optimization process was successful and the global optimum was found. Thus, the objective target function stopping criterion was reached with an end result of . This can be compared to an objective value of for the first iteration. The NOMAD routine required 85 blackbox evaluations and a total running time of 35 minutes

More advanced input configuration utilizing variable outer radius and SRM length is also investigated. In this configuration, SRP-GEO is activated with SRP-ANLYT acting as the surrogate function. Further, the throat diameter is set to dynamic with an optimized MEOP value specified to MPa. Table 7 summarizes the geometrical parameters, initial values, boundaries and optimized results for this test example. Previous NOMAD configuration, motor data and propellant compositions are utilized with the addition of the SRP-GEO mesh resolution set to , i.e. LR. In order to safeguard good convergence results, the boundaries are also narrower than in the previous test example.

Table 7: Initial values, boundaries and solutions for cylindrical tube geometry with advanced configuration Parameter Lower Initial Upper Target Optimized Difference in [m] Bound Value Bound Values Value Percentage Outer Rad. 1.10 1.35 1.40 1.20 1.27 5.80 Throat Diam. - 0.596 (calc.) - 0.60 0.56 6.43 End. Pos. 9.00 9.00 11.00 10.0 9.01 10.4 Radial Beg. 0.40 0.40 0.75 0.60 0.61 1.99 Radial End. 0.40 0.75 0.75 0.60 0.67 11.1

It can be noted that the desired thrust profile displayed in Figure 22 is shown to be dissimilar to the one in Figure 21 despite being described by the same set of geometrical parameters. This is due to the noticeable difference in burnback analysis results between LR SRP-GEO and SRP-ANLYT. By employing higher resolution in the numerical calculations, the difference can be diminished but with the downside of drastically higher computation time.

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Thrust Time Curve for Single Cylinder Grain With Advanced Configuration 4500 Desired Thrust Calculated Thrust 4000 Initial Guess

3500

3000

2500

Thrust [kN] 2000

1500

1000

500

0 0 20 40 60 80 100 120 Time [s] Figure 22: Desired and optimized thrust for single cylinder grain with advanced input configuration

The resulting optimized thrust profile is shown to correlate fairly satisfactory with the desired counterpart although the geometry is relatively far from the target. This behaviour highlights the local nature of the NOMAD optimizer and is analysed further in section 5.3. The computation for this example was concluded when the mesh index reached above 11 after 2 days and 34 minutes of execution time. A total number of 375 blackbox evaluations were performed with SRP-GEO and an additional 393 evaluations with SRP-ANLYT. The final objective value was recorded to and can be compared to an objective value of in the first iteration. The MEOP for the optimized grain and throat configuration was calculated to MPa.

5.2.2 Star A simple star grain is investigated with its geometry given in Table 8 together with initial values and boundaries for the optimization process. In this example, the total length and outer radius remain fixed and the throat diameter is set to a static value of m. Only the analytical burnback algorithm with a relaxation coefficient of is utilized and the VNS search strategy remains off. Default NOMAD configuration values, motor data and propellant compositions are also specified.

Table 8: Initial values, boundaries and solutions for star geometry Parameter Lower Initial Upper Target Optimized Difference in [m] Bound Value Bound Values Value Percentage Outer Rad. - 1.40 - 1.40 - - Throat Diam. - 0.65 - 0.65 - - End. Pos. Seg. 1 - 10.0 - 10.0 - - Radial Beg. 0.40 0.45 0.80 0.65 0.65 0.65 Radial End. 0.40 0.75 0.80 0.65 0.65 0.07 Branch Number 4 4 6 5 5 0.00 Branch Thick. 0.04 0.06 0.06 0.05 0.05 0.38 Branch Tip Rad. 1.00 1.30 1.35 1.20 1.20 0.00

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Thrust Time Curve for Single Star Grain 7000 Desired Thrust Calculated Thrust Initial Guess 6000

5000

4000 Thrust [kN] 3000

2000

1000

0 0 50 100 150 Time [s] Figure 23: Desired and optimized thrust for single star grain

The desired and optimized thrust profiles are shown in Figure 23. It can be discerned that the optimization process was indeed successful giving a solution close to the global optimum. The NOMAD computation was ended with the mesh index reaching above 11 giving a final objective value of and resulting in 699 blackbox evaluations with a total computation time of 4 hours and 25 minutes.

An identical grain configuration with variable outer radius and SRM length is also tested. The Target geometry, initial values and boundaries are listed in Table 9. In this example, SRP-GEO with a mesh size of acts as the main burnback routine whereas SRP-ANLYT is relegated to surrogate status. The throat diameter is set to dynamic with a MEOP specified to MPa. Default configuration settings are utilized for NOMAD, the motor and propellant with the exception that the objective target value is increased to . The VNS search strategy remains likewise offline in this example.

Table 9: Initial values, boundaries and solutions for star geometry with advanced configurations Parameter Lower Initial Upper Target Optimized Difference in [m] Bound Value Bound Values Value Percentage Outer Rad. 1.00 1.50 1.60 1.40 1.38 1.71 Throat Diam. - 0.63 (calc.) - 0.65 0.65 0.08 End. Pos. Seg. 1 9.00 9.00 11.00 10.0 10.2 1.98 Radial Beg. 0.40 0.45 0.80 0.65 0.61 6.35 Radial End. 0.40 0.75 0.80 0.65 0.65 0.00 Branch Number 4 4 6 5 5 0.00 Branch Thick. 0.04 0.06 0.06 0.05 0.04 19.8 Branch Tip Rad. 1.00 1.30 1.35 1.20 1.18 1.71

Figure 24 displays the resulting optimized thrust compared to the desired and initial profiles. As seen, the optimization process was again successful with a final geometry close to the desired counterpart. It can be noted though that the resulting thrust history differs in many respect to the one given by the analytical burnback routine in Figure 23. In this instance, it is prudent to conclude that the numerical burnback routine gives a more accurate interpretation of the thrust history for this particular star shape. The NOMAD routine was manually ended after 2 days, 1 hour and 26 minutes of computing time. A final mesh index of 3 and an objective value of was recorded at the time of abortion. The recorded maximum pressure for the optimized solution was MPa.

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Thrust Time Curve for Single Star Grain With Advanced Configuration 8000 Desired Thrust Calculated Thrust Initial Guess 7000

6000

5000

4000 Thrust [kN]

3000

2000

1000

0 0 20 40 60 80 100 120 140 160 Time [s] Figure 24: Desired and optimized thrust for single star grain with advanced input configuration

5.2.3 Three Segment Combination SRP-GEOPT is capable of handling more complex grain geometries. To validate this capability, a three segment test grain is investigated. This grain contains two cylindrical segments at the sides with a star segment in the middle. The target geometry, boundaries and initial values are listed in Table 10. In this example, the outer radius and overall length of the rocket is fixed at optimum. Further, the throat diameter is set to a static value of m, only the analytical burnback algorithm with default relaxation coefficient is utilized and the VNS search strategy remains offline. Likewise, default configurations for NOMAD, the motor and propellant are utilized in this test example.

Table 10: Initial values, boundaries and solutions for three segment geometry Parameter Lower Initial Upper Target Optimized Difference in [m] Bound Value Bound Values Value Percentage Outer Rad. - 1.40 - 1.40 - - Throat Diam. - 0.60 - 0.60 - - End. Pos. Seg. 1 2.00 2.30 3.00 2.50 2.50 0.07 Radial Beg. 0.20 0.35 0.60 0.45 0.52 14.6 Radial Comb. 0.40 0.55 0.80 0.65 0.68 4.17 End. Pos. Seg. 2 7.00 7.20 8.50 7.50 7.44 0.79 Radial Comb. 0.40 0.55 0.80 0.65 0.68 4.84 Branch Number 4 4 6 5 5 0.00 Branch Thick. 0.04 0.04 0.06 0.05 0.06 18.1 Branch Tip Rad. 1.00 1.10 1.35 1.20 1.20 0.09 End. Pos. Seg. 3 - 10 - 10.0 - - Radial End. 0.50 0.65 0.90 0.75 0.78 3.49

Figure 25 displays the optimized thrust profile compared to the desired counterpart and the initial guess. The results are shown to correlate well with the desired profile although the optimized geometry differentiates in certain areas as shown in Table 10. The optimization process was ended with the mesh index exceeding 11. A total execution time of 5 hours and 26 minutes were required resulting in 1307 blackbox evaluations. An initial and final objective function value of and respectively were recorded. Due to the considerable longer computation time associated with SRP-GEO, analysis with numerical burnback mode is not performed for this particular example.

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Thrust Time Curve for Three Segment Diverse Grain 4500 Desired Thrust Calculated Thrust 4000 Initial Guess

3500

3000

2500

Thrust [kN] 2000

1500

1000

500

0 0 20 40 60 80 100 120 140 160 180 Time [s] Figure 25: Desired and optimized thrust for three segment grain

5.2.4 P80 FW Derivative Final validation of the optimizer is conducted on the P80 FW first stage of the Vega launch vehicle. Unfortunately, only a crude approximation of the grain geometry is currently available for this rocket, provided from a previous study in SART. Nevertheless, for the purpose of validating SRP-GEOPT it is deemed sufficient.

Modifications are also performed on the grain to facilitate the usage within SRP-GEOPT. These include a slight reduction in the number of main grain segments and the removal of segments associated with the nose and aft domes. This reduces the pool of optimization variables to a more manageable size. Further, as axial-conical star tips are not currently supported in SRP-GEOPT, the geometry is modified such that these are avoided. Table 11 lists the geometrical target; initial values and boundaries for this modified P80 FW SRM with the overall length and main radius held static. The burnback analysis is handled by SRP-GEO with a mesh size of and with no surrogate function defined. VNS search strategy also remains offline.

Default NOMAD configuration settings are employed with the exception that the minimum and maximum mesh indexes are set to 1 and 5 respectively. Motor data and propellant compositions are gained in ref. [15]. Accordingly, an initial static throat diameter of m, a nozzle half angle of degrees, an Isp loss gradient and constant loss percentage of and respectively and a subsonic/supersonic area ratio of / are specified. The propellant is AP69/HTPB12/Al19 with the compositions and properties defined in Appendix A.

Table 11: Initial values, boundaries and solutions for an SRM derived from P80 FW Parameter Lower Initial Upper Target Optimized Difference in [m] Bound Value Bound Values Value Percentage Outer Rad. - 1.5120 - 1.5120 - - Throat Diam. - 0.5000 - 0.5000 - - End. Pos. Seg. 1 0.15 0.15 0.18 0.1656 0.15 7.91 Radial Beg. 0.25 0.30 0.45 0.3456 0.27 24.6 Radial Comb. 0.35 0.40 0.55 0.4464 0.49 9.31 End. Pos. Seg. 2 4.50 4.75 5.00 4.9248 4.50 9.01 Radial Comb. 0.40 0.45 0.60 0.5040 0.45 11.3 End. Pos. Seg. 3 5.10 5.20 5.50 5.3568 5.50 2.64

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Radial Comb. 0.55 0.55 0.75 0.5904 0.61 3.27 End. Pos. Seg. 4 5.60 5.85 6.00 5.6088 5.84 4.04 Radial Comb. 0.60 0.60 0.80 0.6336 0.60 5.45 Branch Number 10 13 14 12 10 18.2 Branch Thick. 0.30 0.35 0.40 0.3355 0.37 9.78 Branch Tip Rad. 0.55 0.70 0.75 0.6624 0.70 5.52 End. Pos. Seg. 5 6.10 6.25 6.30 6.1488 6.28 2.11 Radial Comb. 0.55 0.60 0.75 0.6408 0.59 8.25 Branch Number 10 13 14 12 10 18.2 Branch Thick. 0.30 0.35 0.40 0.3355 0.38 12.4 Branch Tip Rad. 0.55 0.75 0.75 0.6900 0.75 8.33 End. Pos. Seg. 6 6.35 6.40 6.45 6.3720 6.43 0.91 Radial Comb. 0.55 0.60 0.75 0.6408 0.55 15.3 Branch Number 10 13 14 12 12 0.00 Branch Thick. 0.30 0.35 0.40 0.3355 0.32 4.73 Branch Tip Rad. 0.70 0.80 0.90 0.9000 0.84 6.90 End. Pos. Seg. 7 6.50 6.90 7.00 6.5088 6.80 4.38 Radial Comb. 0.55 0.65 0.75 0.6408 0.55 15.3 Branch Number 10 13 14 12 13 8.00 Branch Thick. 0.30 0.35 0.40 0.3355 0.40 17.5 Branch Tip Rad. 1.00 1.10 1.25 1.2096 1.25 3.29 End. Pos. Seg. 8 7.10 7.10 7.35 7.2288 7.33 1.32 Radial Comb. 0.60 0.65 0.80 0.7128 0.60 17.2 Branch Number 10 13 14 12 10 18.2 Branch Thick. 0.30 0.35 0.40 0.3355 0.30 11.2 Branch Tip Rad. 1.10 1.20 1.30 1.2960 1.30 0.31 End. Pos. Seg. 9 - 7.3584 - 7.3584 - - Radial End. 0.60 0.75 0.80 0.7128 0.60 17.2 Branch Number 10 13 14 12 10 18.2 Branch Thick. 0.30 0.35 0.40 0.3355 0.35 4.23 Branch Tip Rad. 0.80 1.00 1.00 0.9200 0.98 8.33

Figure 26 displays the resulting optimized thrust profile compared with an initial assumption and the target profile. It can be shown that with a total of 36 optimization variables, the resulting thrust profile

Thrust Time Curve for Approximate P80 SRM Grain 2500 Desired Thrust Calculated Thrust Initial Guess

2000

1500 Thrust [kN]

1000

500

0 0 50 100 150 Time [s] Figure 26: Desired and optimized thrust for an SRM derived from P80 FW

SART TN-002/2012 42

correlates well with the desired counterpart. The optimized geometry on the other hand shows large deviations from the target. This was previously expected due to the local nature of the NOMAD optimizer and the large number of variables employed in this test example. The optimization process was ended with the mesh index exceeding 5. A total execution time of 13 days 18 hours and 52 minutes was required resulting in 6983 blackbox evaluations. A final objective function value of was recorded and can be compared to the initial objective function value at the first iteration of .

5.3 Convergence Analysis

From the previous test results, it can be discerned that for more complex geometries or when the length and the outer radius is part of the optimization process in addition to rather poorly chosen initial values, the geometry tends to converge towards a local minimum. As a potential solution to this problem, the VNS strategy can be employed. To analyse the effect of this search strategy, it is tested on the one segment cylindrical tube grain with geometrical data given in Table 7. Identical settings and initial values as the previous example are utilized except the surrogate function is disabled and a static throat diameter of m is selected.

The results are shown in Table 12 and Figure 27 where it can be concluded that the addition of the VNS strategy greatly improved the resulting thrust profile. However, the desired global minimum for the geometry was still not found. The final objective function value with VNS activated was recorded to which can be compared to the case without VNS of . With the inclusion of VNS, the execution time is however increased to 6 days 2 hours and 23 minutes with a total of 1863 blackbox evaluations.

Table 12: Solutions for cylindrical tube geometry with and without VNS strategy Parameter Target Optimized Difference in Optimized Difference in [m] Values Value (VNS Off) Percentage Value (VNS on) Percentage Outer Rad. 1.20 1.27 5.80 1.17 2.58 Throat Diam. 0.60 0.56 6.43 - - End. Pos. 10.0 9.01 10.4 10.4 3.98 Radial Beg. 0.60 0.61 1.99 0.58 4.02 Radial End. 0.60 0.67 11.1 0.57 5.98

Thrust Time Curve for Single Cylinder Grain with VNS 4500 Desired Thrust Calculated Thrust 4000 Initial Guess

3500

3000

2500

Thrust [kN] 2000

1500

1000

500

0 0 20 40 60 80 100 120 140 Time [s] Figure 27: Desired and optimized thrust for single cylinder grain with VNS strategy

SART TN-002/2012 43

Due to the substantially increased number of blackbox evaluations associated with activating VNS, it is thus only recommended if computation time is not an issue or if tightening of the boundaries is not possible. Further, the optimization of the outer radius and overall length of the rocket should also be avoided if possible in order to reduce the number of local minima.

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6 Conclusion

The work conducted in this study includes the development of a piece of software capable of determining and optimizing three dimensional SRM grains. This is accomplished by utilizing a derivative free Direct Search optimization algorithm, an internally developed ballistic solver and a numerical burnback analysis tool. Additional algorithms are also developed to support these programs and extend their capabilities.

From the results presented in this report, a consensus can be reached that the implemented optimization procedure is generally successful for cases even with a substantial amount of optimization variables (>30). Though for most cases, convergence occurred towards a geometrical configuration different than from the target. This behaviour emphasizes that a good understanding of the boundaries, initial values and grain shape should be available before commencing the optimization routine. Fortunately, due to the evolutionary approach of rocket design, this criterion can often be fulfilled. Furthermore, it also highlights the many possible geometrical solutions applicable for a single thrust profile, thus a “single correct answer” often does not exist. Nevertheless, further testing with existing SRMs is desirable in order to better demonstrate the effectiveness of the optimization tool in real-world applications. Testing with different boundaries and initial values for a single geometry is also desirable in order to more thoroughly analyse the convergence behaviours.

Addition of the VNS strategy is shown to support problem cases where the results converge toward unsatisfactory local minima at the expense of a radical increase in the number of blackbox evaluations. It is thus up to the user to decide whether this approach should be taken instead of improving the initial conditions and/or shrinking the boundaries. Naturally, the latter cannot be performed excessively or a potential fitting solution might be neglected.

It is also important to recognize the limitations of both the ballistic solver and the numerical burnback analysis tool utilized in this project. For the former, the mesh resolution has an important role for accuracy. Unfortunately, high resolution computations require hours to days in its current implementation. Thus, it is not practical to run these computations as part of a blackbox where thousands of such evaluations might be required. An effort was thus made to create a faster analytical burnback analysis algorithm to alleviate this problem to some extent. But due to the limitations of this analytical approach in respect to applicable geometries, it is unfortunately not a perfect solution. Further, simplifications are made on the grain geometry compared to a real world SRM, e.g. the modelling of the igniter and submerged nozzle components as simple cylinders. These simplifications and inaccuracies then carry over to the ballistic solver and manifests as errors in the resulting thrust profiles. The most visible error due to low resolution analysis can be glanced from the peak in Figure 27 for the desired and optimized thrust which in reality should not exist given the grain geometry. Indeed, with the analytical burnback algorithm, such a peak is not present as seen in Figure 21. The user of this software must therefore be aware of the limitations and not follow the results explicitly. Further manual tweaking of the resulting geometry is thus recommended.

The current implementation of the optimization software relies on a single processing core during blackbox evaluations but the option remains open to implement parallel versions of the same software, potentially resulting in greatly reduced execution time. Another benefit with utilizing a parallel approach, e.g. with COOP-MADS and the usage of randomly generated seeds is the ability to more thoroughly scan the search space and thus circumvent local minima. By combining this capability with the VNS strategy for each thread, it is expected that good results are to be followed. A further extension to parallel computing would naturally lead to massive parallel computing, i.e. supercomputing. The direct search approach to optimization is particularly suitable for this task due to the usage of polls and random seeds. A supercomputing approach can thus conceivably scan a much larger search space with looser boundaries and many variables in order to create radical and complex grain geometries.

6.1 Future Work

As the intention of this project is to create a piece of software containing only core functionality for usage internally within DLR, much work remains in order to generalize and expand the software; with parallelization being the most pressing but yet only a small part of a larger to-do list. A list of other functionalities yet to be implemented is thus displayed below based roughly on priority.

SART TN-002/2012 45

. Static Variables: By holding variables constant, the user is capable of more tightly controlling the optimization process. This feature is especially useful in instances where the grain geometry inside the nose and aft domes must take a specific shape. By creating a static grain with specific properties and inner radius slope very close to the domes, the user can indirectly define the grain structures within these domes.

. Groupable Variables: Similarly, by grouping variables together such that many variables share the same polling step and direction during optimization, weird resulting geometries can be avoided. A typical example is the number of star branches for several star segments. This also reduces the number of optimization variables without losing grain fidelity.

. Custom Grain inside Domes: By manually defining a static grain structure within the SRM domes, another layer of customization can be added. This will likely only work in instances where the outer radius and overall length is not part of the optimization process.

. Improved Analytical Burnback Algorithm: The current algorithm share many limitations both in the axial and radial direction. For the axial direction, it is desirable to improve the analytical equations such that a wider range of star geometries are applicable. For the radial direction, better treatments of variable outer radii are wanted.

. Support for conical star tip radii: In the current implementation of SRP-GEOPT, only straight star tip radii are supported in the main program whereas in the numerical burnback algorithm, conical tip radii similar to conical inner radii are supported. Adding support for this feature will increase the capability and range of the optimization program.

. Improved Numerical Burnback Tool: Improvements in execution time can be made to the numerical burnback tool by treating simple cylinders with analytical equations while preserving the numerical algorithm during calculation of star segments. The development of a new version of this numerical burnback tool is already underway.

. Custom Search Algorithms: The NOMAD optimizer supports custom search algorithms . By defining such algorithms for specific grain type that are based on heuristics, it is most likely possible to attain faster convergence rates toward good solutions.

. Additional Geometries: In the current version of SRP-GEOPT, only truncated stars, cylindrical tubes and empty segments can be treated. Future versions should expand on the capabilities of the program by adding more treatable geometries of which end burning segments are the most interesting.

. Visualization: Currently, no graphical visualization of the optimized grain or the optimizing evolution of the grain is available. It has been identified that an external program module referred to as Xplot might be suitable for this task. Xplot is written in C++ and can similarly to NOMAD be directly integrated into the current Fortran codebase. It supports three dimensional, colored graphs that are interactive and can be simply manipulated [36].

. Geometry and Propellant Database Entries: The geometry and propellant database is currently populated by example entries. A wider selection of real data is desirable if SRP-GEOPT is to be used in actual project.

. Integration with SRP: The final goal of this project is to integrate SRP-GEOPT into SRP as an additional mode selectable in a main input file. Due to the structure of SRP-GEOPT and the usage of automatically generated SRP input files, another option would be to integrate the other execution modes in SRP into SRP-GEOPT and rename the program.

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References

[1] Sippel, M.: Concurrent Launcher Engineering at DLR. Concurrent Engineering Workshop, ESTEC, Noordwijk, Holland, 2004. [2] Acik, S.: Internal Ballistic Design Optimization of a Solid Rocket Motor. Department of Mechanical Engineering, Middle East Technical University, 2010. [3] Bairstow, B.K.: Effectiveness of Integration of System-Level Optimization in Concurrent Engineering for Rocket Design. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 2006. [4] Lewis, R., Torczon V., Trosset, M. W.: Direct Search Methods: Then and Now. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 2000. [5] Sutton, G.P., Biblarz, O.: Rocket Propulsion Elements. 7th Ed. Wiley Interscience, USA, 2001. [6] Larangot, B. et al.: Solid Propellant MicroThruster: an alternative propulsion device for nanosatellites. Aerospace Energetic Equipment Conference, Avignon, France, 2002. [7] Michener, J.A.: The Space Shuttle Reusable Solid Rocket Motor. ATK, 2012. [8] Antonov, A.: Solid Rocket Motors Performance and Grain Geometry Optimization. SART TN- 003/2011, German Aerospace Center, Bremen, Germany, 2011. [9] Whitfield, H.K., Keller Jr., R.B. ed.: Solid Rocket Motor Metal Cases. SP-8025, NASA, 1970. [10] Zandbergen, B.T.C.: Solid Rocket Propellants and Their Properties. Department of Space Engineering, Delft University of Technology, 2004. [11] Twitchell, S.E., Knauer, R.G., Keller Jr, R.B. ed.: Solid Rocket Motor Internal Insulation. SP-8093, NASA, 1974. [12] Brooks, W.T., Keller Jr., R.B. ed.: Solid Propellant Grain Design and Internal Ballistics. SP-8076, NASA, 1972. [13] Nakka, R.: Richard Nakka’s Experimental Rocketry Web Site. Available at http://www.nakka- rocketry.net/articles/nakka_theory_pages.pdf, 2007, accessed 25th January 2012. [14] Haidn, O.J.: Advanced Rocket Engines. RTO-EN-AVT-150, Advances on Propulsion Technology for High-Speed Aircraft, Rhode St. Genèse, Belgium, 2007. [15] Dumont, E., Kopp, A., Sippel, M.: SRP 1.2: Updates and Simulation Results. SART TN013/2009, German Aerospace Center, Bremen, Germany, 2009. [16] Braeunig, R.A.: Rocket Propulsion. Available at: http://www.braeuning.us/space/propuls.htm, 2009, accessed 22nd January 2012. [17] Reydellet, D.: Performance of Rocket Motors with Metallized Propellants. AGARD PEP WG-17, AGARD AR-230, 1986. [18] Berghen, F.V, Bersini, H.: CONDOR, a New Parallel, Constrained Extension of Powell's UOBYQA Algorithm: Experimental Results and Comparison with the DFO Algorithm. Journal of Computational and Applied Mathematics, Volume 181, Issue 1, Pages 157-175, 2005. [19] Csendes, T. et al.: The GLOBAL Optimization Method Revisited. Optimization Letters, Volume 2, Issue 4, Pages 445-454, August 2008. [20] Plantenga, T.D.: HOPSPACK 2.0 User Manual. SAND2009-6265, Sandia National Laboratories, Albuquerque, USA, 2009. [21] Anon.: IMSL: Fortran Subroutines for Mathematical Applications Vol. 2, Visual Numerics, Inc. 1997. [22] Johnson, S.G.: The NLopt nonlinear-optimization package. Available at: http://ab-initio .mit.edu/ nlopt, 2008, accessed 25 January 2012. [23] Digabel, S.L.: Algorithm 909: NOMAD: Nonlinear optimization with the MADS algorithm. ACM Transactions on Mathematical Software, Volume 37, Issue 4, Pages 44-59, 2011. [24] Vaz, A.I.F., Vicente, L.N.: PSwarm: A hybrid solver for linearly constrained global derivative-free optimization. Optimization Methods and Software, Volume 24, Issue 4-5, Pages 669-685, 2009.

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[25] Torczon, V.: On the Convergence of Pattern Search Algorithms. SIAM Journal on Optimization, Volume 7, Issue 1, Pages 1-25, 1997. [26] Davidson, W.C.: Variable Metric Method for Minimization. SIAM Journal on Optimization, Volume 1, Issue 1, Pages 1-17, 1991. [27] Abramson, M.A. et al.: OrthoMADS: A deterministic MADS instance with orthogonal directions. SIAM Journal on Optimization, Volume 20, Issue 2, Pages 948-966, 2009. [28] Audet, C., Dennis Jr., J.E.: Mesh Adaptive Direct Search Algorithms for Constrained Optimization. SIAM Journal on Optimization, Volume 18, Issue 1, Pages 188-217, 2006. [29] Booker, A.J. et al.: A Rigorous Framework for Optimization of Expensive Functions by Surrogates. Structural and Multidisciplinary Optimization, Volume 17, Issue 1, Pages 1-13, 1999. [30] Hansen, P., Mladenović, N.: Variable neighborhood search: Principles and applications. European Journal of Operational Research, Volume 130, Issue 3, Pages 449-467, 2001. [31] Audet, C., Dennis Jr., J.E., Le Digabel, S.: Parallel Space Decomposition of the Mesh Adaptive Direct Search Algorithm. SIAM Journal on Optimization, Volume 19, Issue 3, Pages 1150-1170, 2008. [32] Le Digabel, S. et al.: Parallel Versions of the MADS Algorithm for Black -Box Optimization. Available at www.gerad.ca/Sebastien.Le.Digabel/talks/2010_JOPT_25mins.pdf, 2010, accessed 31st January 2012. [33] Audet, C., Savard, G., Zghal, W. Multiobjective optimization through a series of single objective formulations. SIAM Journal on Optimization, Volume 19, Issue 1, Pages 188-210, 2008. [34] Hartfield, R. et al.: A Review of Analytical Methods for Solid Rocket Motor Grain Analysis. AIAA 2003-4506, 39th Joint Propulsion Conference and Exhibit, Huntsville, USA, 2003. [35] Nassef, A.O., ElMaraghy, H.A.: Determination of Best Objective Function for Evaluating Geometric Deviations. International Journal of Advanced Manufacturing Technology, Volume 15, Issue 2, Pages 90-95, 1999. [36] Del Rio, M.S.: Xop/Xplot User’s Guide. Available at http://ftp.esrf.eu/pub/scisoft/xop2.0/ XopTutorial/XplotManual.pdf, 2001, accessed 14 Mars 2012.

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Appendix A

Main Input File

#------# #--SOLID ROCKET PROPULSION - GEOMETRY OPTIMIZER (SRP-GEOPT) v1.0 Beta--# #------MAIN INPUT FILE------# #------# #1 -Project Name [String]-# Internal Testing Project

#2 -Motor Name [String]-# Test Motor

#3 -Propellant Name [String: Same Name as in propellants.dat]-# AP68/HTPB14/Al18

#------#

#4 -Configuration Switches [Boolean: T|F]-# Automatic Initial Grain : F VNS Search Strategy : F Analytical Burnback (SRP-ANLYT): T Numerical Burnback (SRP-GEO) : T

#------#

#5 -Variable Optimization Switches [Boolean: T|F]-# Outer Radius : T Length : T

#------#

#-Motor Data-# #6 -Throat Diameter [String: static|dynamic, Double: pressure (MPa), # Double: throat diameter if static (m)]-# static, 9.0, 0.30 #7 -Nozzle Half Angle [Double: 8-18 (Deg)]-# 16.5 #8 -ISP Loss Gradient [Double: 0-100 (%/s)]-# 0.0 #9 -ISP Constant Loss [Double: 0-100 (%)]-# 3.5 #10-Nozzle Submergence Ratio [Double: 0-0.25]-# 0.015 #11-Subsonic Area Ratio [Double: 1.5-3.5]-# 1.5 #12-Expansion Ratio [Double: 10-1000]-# 16

#------#

SART TN-002/2012 49

#13-Desired Vacuum Thrust History-# #------# #| Time | Thrust |# #|Double| Double |# #| | |# #| [s] | [kN] |# #------# 0.0 29.841 2.0 237.295 4.0 522.386 6.0 832.418 8.0 1137.107 10.0 1418.545 12.0 1667.150 14.0 1879.174 16.0 2054.819 18.0 2198.826 #------#

#-Initial Geometry-# #14-Main Outer Radius [Double: 0.1-1000 (m)]-# 1.74 #15-Initial Grain Geometry-# #------# #| Type |Ending |Radial |Radial |Star |Star |Branch |# #| 0=empty |Position|Begin. |Ending |Branch |Branch |Tip |# #| 1=cyl/cone | |Position|Position|Number |Thickness |Radial |# #| 2=star | Double | Double | Double | Double | Double | Double |# #| | | | | | | |# #| | [m] | [m] | [m] | | [m] | [m] |# #------# 2 4.2 0.25 0.25 3 0.05 0.7 #------# #Comment: A nose dome, aft dome, igniter and submerged nozzle is added automatically to the above geometry. Please do not manually add it. Star branch numbers can take values of type double, but will automatically round up/down to nearest integer.

#-Miscellaneous Parameters-# #-[FOR ADVANCED USERS ONLY]-# #16-SRP GEO Resolution [Double: 0.001-0.1]-# 0.05 #17-Objective Function Target Multiplier [Double: 0.01-9999]-# 1.0 #18-Objective Function Exponent [Double: 1-9999]-# 2.0 #19-SRP ANLYT Relaxation Coefficient [Double: 0-1]-# 1.0 #20-NOMAD Maximum Blackbox Evaluations [Integer: 10-INF, INF if > 10000]-# 2000 #Comment: Includes cache hits and true evaluations. Does not include surrogate evaluations. #21-NOMAD Termination Target Value [Double: 0.001-INF]-# 0.001 #22-NOMAD Minimum Mesh Index [Integer: 0-35, -INF if = 0]-# 0 #23-NOMAD Maximum Mesh Index [Integer: 1-35]-# 11

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Propellant Database

Each propellant in the database is assigned a number of lines. Line 1 denotes the propellant name. The same name must be added to the input file to identify the desired propellant for SRP-GEOPT. Line 2 to Line 4 are used for the molecular component of the propellant. For each line, a number of characters are reserved for propellant parameters; these are from left to right:

1. Two characters for the chemical element symbol. If the chemical consist of only a single symbol, space is used for the second symbol. 2. Seven characters for the amount of each element in the molecular component. 3. Seven characters for the relative weight of the component in the propellant. 4. Nine characters for the enthalpy of formation given in kCal/kgmol. 5. One character identifying the phase of the component, i.e. S for solid. 6. Seven characters for the idle temperature of the grain in K. 7. One character identifying if the component is an oxidizer, O or fuel, F. 8. Eight characters for the component density in kg/dm3.

The last three lines denotes the pressure coefficient , pressure exponent and overall propellant density respectively. Note that the values for the overall propellant density are currently only a

placeholder.

0

00 00

0.00S 298.15 F 2.69800 F 298.15 0.00S

13800.00S 298.15 F 0.92000 F 298.15 13800.00S 1.950 O 298.15 S 70690.0 70690.00S 298.15 O 1.95000 O 298.15 70690.00S 0.920 F 298.15 S 13800.0 1.91000 O 298.15 17196.90S

- - - - -

9.00 0.0 S 298.15 F 2.6980 F 298.15 S 0.0 9.00

.00000O 4.00000 68.000 68.000 4.00000 .00000O

e exponent: 0.37 exponent: e

AP68/HTPB14/Al18 14.000 0.06300 0.22300N 10.6500O 7.07500H C 1.92 coefficient: Pressure 69.00 4. O CL1. 4. H 1. N 1 AL1. 1.8 Density: HNF62/GAP20/Al18 18.000 AL1.00000 1.92 coefficient: Pressure N 1.00000H 4.00000CL1 1.00000H N 2.69800 F 298.15 0.00S 18.000 AL1.00000 Pressur 1.8 Density: AP69/HTPB12/Al19 12.00 .063 N .223 O 10.65 H 7.075 C 3.0846 coefficient: Pressure 0.232 exponent: Pressure 62.000 6.00000 3.00000O 1.00000N 5.00000C 2.00000H N 1.30000 F 298.15 27992.70S 20.000 1.00000 3.00000O 5.00000N 3.00000H C 0.6 exponent: Pressure 1.8 Density:

SART TN-002/2012 51

SRP Input File

Dport[m] ]

-

0, 0.50 0,

density

-

.000, 0.000, 0.000, 0.000, 0.000, 0.000, .000, 0.000, 0.000,

Fuel/Ox

-

T[K]

-

Phase 298.152.69800 F 0S

-

mol)]

70690.00S 298.1570690.00S1.95000 O 298.1513800.00S0.92000 F

- -

Energy[kCal/(kg.

-

Wt Fraction Wt

-

tio (SUBAR, tio(SUBAR, I=1,13)

Chemicalformula

-

1.92000, 0.37000 1.92000, 0.000,0.000, 0.000, 0.000, 0.000,0.000, 0.000, 0.000, 0.000, 0.000, 1.500, 0.000, 0.000, 0.000,0.000, 0.000, 0.000, 0 0.000, 0.000, 0.000, 16.000, T , T ,T ,,T ,T T T ,T ,T ,T

0.20000000, 0.50, 1.63, 0.015 16.50, 0.20000000, 3.50 0.00

TYPE of analysis ofhistory 1:pressure analysis input inputhistory 2:vacuum 3:geometryTYPE thrust input thrust vacuum [kN] Minimum VS. TIMEGRIDTHRUST * 0.1 10000 EQL,FROZ,SUBM,DIV,TBL,TP,KIN * Dthroat nozzleDinlet[m][m] deltaDt initial angle[deg] * half[mm/s] ratio[ Submergence lossgradient [%/s] Isp constant Isp loss[%] * BurningPropellant rate:r=a*Pc[bar]**n [mm/s] * #Internal Testing Project #Internal REACTANTS 4.00000CL1.00000O 1.00000H N 4.00000 68.000 10.6500O 0.22300N7.07500H C 0.06300 14.000 AL1.00000 0.0 18.000 3 0.001d0 timestep * iteration number * switches *Calculation Logical) (Type Pressure coeff.:a * Pressure exp.:n Subsonic area * ra Supersonicarea (SUBAR,ratio * I=1,13) *&ENDR

SART TN-002/2012 52

Appendix B

Derivation of Dome Equations

Length of Nose Dome Utilizing the equation for a circle

(8.1)

The objective is to find an x-coordinate for the nose dome such that the radial y-coordinate is ( ) with denoting the axial shift fraction of the dome. By increasing , thus becomes smaller and this will affect the location of . Further, the circle can be scaled by setting the outer radius as where and is thus the circle scaling factor. Lastly, the coordinate-axis for is also reversed such that begins at the outer boundary of the circle and moves towards the center with increasing value. Thus equation (8.1) can be written as

( ) ( ) ( ) (8.2) which can be further simplified to

( ) [ ( ) ] (8.3)

This gives the length of the nose dome as

( ) ( √ ) (8.4)

( ⁄ ) where for . Figure 28 visualizes the geometrical parameters utilized for these derivations.

Nose Dome Outer Radius As for the dome length, the equation for the circle is utilized to find the dome outer radius. In this case, the position for each between 0 and is calculated. This is done by setting the outer radius in the second left term in (8.2) to , thus

β

2

R R

0

0

(

1

-

β

1 )

xn

Figure 28: Nose dome geometry parameter definitions

SART TN-002/2012 53

( ) ( ) ( ) (8.5)

Simplifying this expression by developing ( ) leads to

( ) ( ) (8.6)

which gives the outer radius as function of the axial position as

√ ( ) (8.7)

With and , it is possible to describe the dome structures at the front of the SRM.

Derivation of Modified Analytical Star Equations

In the work by Hartfield et.al (2003), the star branch endings are modelled as arcs whereas in SRP- GEO, these endings are modelled as simple straight lines. Although the difference is small, this change necessitates a number of modifications to the equations presented in the paper. Below, these modified equations are derived in some detail.

It is also worth reiterating that the grain evolution due to burning is assumed to oc cur at the same speed on all exposed surface areas. All burning perimeters will thus recede at a unique speed in the entire combustion chamber, i.e. a single burning distance is applicable to the whole geometry.

Phase One For Phase One, the empty port area is simply the sum of the sub-geometries denoted 1-4 in Figure 29. These are shown to be described by

( ) (8.8)

( ) ( ) (8.9)

( )( ) (8.10)

( ) (8.11)

Utilizing Pythagoras’ theorem on sub-geometry (3) and subtracting the resulting branch tip radius gives the line burning perimeter as

√( ) ( ) (8.12)

Likewise, the angle can be derived from sub-geometry (3) with the hypotenuse equalling .

( ) (8.13)

The total port area is thus

( ) (8.14)

SART TN-002/2012 54

y f Phase 2 + /2 /2 + f f/2 y 1

1 y R o y e n 1 d se ‘ 4 a r p r p Ph S1

3 2 Phase 3 𝑠 S2 2 𝑠

Phase 4 ri+y ri

f/ f/ 2 2 + + y y 3

1 ’ -S r p 4

S 1

Figure 29: Phase One truncated star geometry breakdown

As the branch ending is a straight line, an edge of the square will first reach the outer casing.

According to Figure 30, the length can be calculated by first applying the law of sines on the triangle stretched between , and a known angle ⁄ . This gives the remaining angles and as

( ) (8.15) √

( ) (8.16) √

a 1 -> 2

R o 4

‘ r p

Figure 30: Phase One to Phase Two switching criteria

SART TN-002/2012 55

The length is thus one side of the right triangle stretched by the hypotenuse and the angle , i.e.

[ ( )] (8.17) √

which can be simplified to

(√ ) (8.18)

The burning distance where sub-geometry (1) reaches the outer casing must then be , or

(√ ) (8.19)

Phase Two

Further burning signifies that . As visualized in Figure 31, part of the geometry will then reside outside the casing and must be considered invalid. It is possible to deduct this invalid component

of the geometry denoted by considering the line as a secant line. From the figure, this line is shown to become shorter with increasing burning distance until it eventually disappears. When this occurs, Phase Two is also deemed to have ended. The first task is thus to model this secant line as a function of the burning distance, i.e. ( ).

From Figure 31, the slope of ( ) is shown to remain constant with the distance being

( ) (8.20)

The slope is thus

( ) (8.21)

e

M b a gn g ific at δ h ion a‘ θg

Acorr i

Υ

c ′ Χ θ

Figure 31: Phase Two truncated star analytical denotations

SART TN-002/2012 56

Similarly, the y-intercept point is determined from the equation of the line and (8.21) which jointly gives

Υ Χ Υ [ ( )] Χ Υ ( ) Χ (8.22)

Where Υ and Χ are coordinates representing the movement of the secant line, i.e.

Χ ( ) ( ) Υ ( ) ( ) (8.23)

Inserting (8.23) into (8.22) and simplifying gives

( ) ( ) [ ( )] (8.24)

The intersection points between the secant line and the outer radius can be determined by solving the system of equations containing

Χ Υ (8.25) Υ Χ

Reducing (8.25) to contain a single variable Χ results in a second order equation in form of

( )Χ Χ (8.26)

Which has the general solution

Χ √( ) (8.27)

Assuming real roots, (8.27) must have two solutions during Phase Two denoted Χ and Χ . One of the solutions is visualized in Figure 31. The second solution must then be the intersection point at the mirror side of the same geometry. These two solutions will thus move toward each other until the end of Phase Two where they are merging.

Following, it is of use to calculate the sum and product of the two solutions gained from (8.27). These are after a number of simplifications

Χ Χ Χ Χ (8.28)

With (8.28), the difference squared between Χ and Χ can be determined from

(Χ Χ ) (Χ Χ ) Χ Χ (8.29) or

(Χ Χ ) [ ( ) ] (8.30) ( )

Similarly, the difference squared between Υ and Υ is

(Υ Υ ) [ Χ ( Χ )] (8.31) or

SART TN-002/2012 57

(Υ Υ ) (Χ Χ ) (8.32)

The absolute length squared between the intersection points ( ) is thus

( ) (Χ Χ ) (Υ Υ ) ( )(Χ Χ ) (8.33)

Inserting (8.30) into (8.33) and taking the square root results in

( ) √ [ ( ) ] (8.34) ( )

Where is a function of .

The next task is to determine the small triangle given by the areas as depicted in Figure 31. The hypotenuse of this triangle, denoted meets the outer radius at two locations. If two lines are drawn from these locations to the centre of the grain, an isosceles triangle is formed. The main angle of this triangle is

(8.35)

Where and are given by the geometry as

( ) ( ) (8.36)

This isosceles triangle can be further split into two separate right angle triangles with the short sides equalling , the hypotenuses and the main angle ⁄ . Thus, must be

( ) (8.37)

With Pythagoras’ theorem, the short side of the triangle can thus be determined as

√ (8.38)

Where . Consequently, the total area of the triangle is

( )√ ( ) ( ) (8.39)

The small sliver area can be calculated by subtracting the isosceles triangle from the circle sector spanning i.e.

( ) (8.40)

Combining (8.40) with (8.37) and then inserting it to (8.39) gives the final result for as

( )√ ( ) ( )

(8.41)

( ) ( )

SART TN-002/2012 58

which simplifies to

( ) ( ) [ √ ( ) ( )] (8.42)

Thus for Phase Two, the area evolution is modelled as

(8.43)

It can also be noted that Phase Two is ended when the burning distance has reached .

Phase Three Phase Three is initiated when the secant line has turned zero. Equation (8.42) is thus rewritten to reflect this change. Further, an additional rectangle is formed outside the outer radius as visualized in Figure 32. This rectangle must also be deducted from and is shown to conform to

( )( ) (8.44)

The area evolution during Phase Three is then

(8.45) with ( ( ) ).

Phase Three equations are also extended to Phase Four despite being not fully valid in this context. This is applicable as the last phase has negligible influence on the resulting thrust. The radial burning process then ends when the port area is equal the total area, i.e.

(8.46)

f/ 2 + y

Arect R o

‘ r p

Acorr

θg

Figure 32: Phase Three truncated star analytical denotations