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Robust Design of UnderwaterSubmersible using Inductive Design

Exploration Method (IDEM) A thesis submitted in fulfillment

of the requirements for the degree of

Bachelor of Technology In

Manufacturing Science & Engineering By

Ayan Sinha

[Roll No. 05MF1024]

Under the supervision of

Dr. C.S Kumar (IIT Kharagpur) &Dr. Janet K. Allen (GeorgiaTech)

Department of Mechanical EngineeringIndian Institute of Technology, Kharagpur

India 721302

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ABSTRACT

In this report, we introduce the construct of microstructure-mediated design of materialand product. We illustrate the efficacy of this construct via the integrated robust design of an underwater submersible and Al-based matrix composites. The integrated design iscarried out using an Inductive Design Exploration Method (IDEM) that facilitates robustdesign in the presence of model structure uncertainty (MSU).

Model structural uncertainty, originating from assumptions and idealizations in modeling processes, is a form of uncertainty that is often virtually impossible to quantify. In this paper, we demonstrate a method, the Inductive Design Exploration Method (IDEM) thatfacilitates robust design in the presence of model structural uncertainty. We achieverobustness by compromising the degree of system performance and the degree of reliability based on structural uncertainty associated with system models (i.e., models for performances and constraints). IDEM is demonstrated in the design of a shell of a roboticunderwater submersible. The material considered isin-situ Al metal matrix composites(MMCs) due to the advantages that thein-situ MMCs have over the conventional MMCs.This design task is a representative example of integrated materials and product design problems.

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Table of Contents

Nomenclature...7Introduction.8

In-Situ Al-Based MMCs9

Inductive Design Exploration Method (IDEM)...10

MICROSTRUCTURE-MEDIATED DESIGN.21

MODULE 1 (Precipitation modeling in Liquid Aluminum)...25MODULE 2 (Modeling Microstructure Evolution).26

MODULE 3 (Semi-solid Processing in MMCs) .27

MODULE 4 (Structure-Property Correlation of MMCs) ...29

MODULE 5 (Property-Performance Correlation of MMC ....30

MODULE 6 (Robust Design using IDEM) 32

SOLUTION STRATEGY USING IDEM ....35

DISCUSSION OF RESULTS..39

CLOSURE ...43

Inductive Design Exploration method accounting for Goals.43

REFERENCES 48

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Nomenclature

k y Strengthening coefficient in the Hall-Petch relation o Material constant related to lattice resistanced Grain diameter

y Yield stress calculated from the Hall-Petch relation Overall yield stress incorporating Orowan particle bypassT Semisolid processing temperature Density of the composite

2, ,TiB Cu Al Densities of TiB2, copper and aluminum respectively

2TiB Volume fraction of TiB2

Cuvf Volume fraction of copper

t Thickness of the shellOD Outer diameter of the shellID Inner diameter of the shellP External pressure

w Density of water d p Reinforcement sizeg Gravityh Depth of the submersible below water W Weight of the cylindrical shell L Length of the submersibleB Buoyant weight of the submersible

eff Efficiency of the batteryT opr Endurance time of the submersible y Output of a response surface model

ij Coefficients in a response surface model

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CHAPTER 1

INTRODUCTION AND LITERATURE REVIEW

Traditionally materials are selected from databases of experimentally determinedmaterials properties. However, the paradigm is shifting towards the concurrent design of materials and products, tailoring materials for specific performance required in specific products or processes.

In order to tailor materials, the approach taken by materials scientists is sequentialdeductive analysis, with a bottom-up mapping from processing path to nano- and micro-structure, material properties and performance. This corresponds to Olsons materialsdesign hierarchy [22] shown in Figure 1.1.

The microstructure of a material strongly influences physical, mechanical and chemical properties such as strength, toughness, ductility, corrosion resistance, high/lowtemperature behavior, etc., which in turn govern the application of these materials. Themicrostructure represents the interface between structure-property-performance relationsincluding systems design and process-structure relations. A microstructure-mediateddesign centered approach has been adopted for concurrent design of materials and product.

In our approach to the concurrent design, we move from the top-down in Olsonshierarchy after completing the bottom-up (deductive) analysis. To do this, a newsystems-based approach for materials design has been adopted that combines inductive(top-down) engineering with deductive (bottom-up) science. Fundamental to this designapproach is a system expressed in terms of variables, constraints, and models that embedrelevant aspects of the material microstructures through overall system configuration.

Performance

Properties

Figure 1.1 Hierarchical Materials Design [22]

The shell of a typical underwater robotic vessel, namely and autonomous underwater vehicle [14 and 15], with both geometrical and material features is considered as a testapplication case for design in this paper (see Figure 1.2). The objective is to design theshell of the vessel for deep sea exploration with multifunctional requirements of minimizing the mass in walls (wall thickness) for given support superstructure for givenmaximum depth and associated pressure differential. Other design requirements includea) suitable factor of safety with respect to collapse at target maximum operating depth, b)a large endurance time satisfying the time of operation constraints under water without

Processing

Structure

Properties

Structure

Processing

Performance

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resurfacing/refueling/battery changes, c) satisfying geometric and weight constraints. The preferred design will have a) high strength to weight ratio and b) resistance againstenvironmental factors such as corrosion. Such requirements could be met by adequatedesign of material of the shell. Recent advances in material processing allow design of the materials with specific desired properties. Al-based metal matrix composites as a

promising material candidate is considered in this report for the design of the material for the shell.

Figure 1.2 Pressure Shell of an Underwater Submersible Robot

In situ Al based MMCs

Metal matrix composites (MMCs), in general, and Al-based MMCs in particular, have been the subject of intense research for the past two to three decades and are beingexploited for a range of commercial applications related to aerospace and automotiveindustries.

A metal matrix composite (MMC) is composite material with at least two constituent parts, one being a metal. The other material may be a different metal or another material,such as a ceramic or organic compound. The matrix is the monolithic material into whichthe reinforcement is embedded, and is completely continuous. This means that there is a path through the matrix to any point in the material, unlike two materials sandwichedtogether. In structural applications, the matrix is usually a lighter metal such asaluminum, magnesium, or titanium, and provides a compliant support for thereinforcement.The reinforcement material is embedded into the matrix. Thereinforcement does not always serve a purely structural task (reinforcing the compound), but is also used to change physical properties such as wear resistance, friction coefficient,or thermal conductivity.

MMCs are manufactured by mainly two methods: Ex-situ Method In-situ Method

In the ex-situ method, a separately formed reinforcement phase is added to the metalmatrix during its preparation in contrast toin-situ composites, where reinforcement phaseis generated as a result of reaction between added precursors and liquid metal during processing of the matrix.

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Compared to monolithic metals, MMCs have

Higher strength-to-density ratios Higher stiffness-to-density ratios Better fatigue resistance Better elevated temperature properties Higher strength Lower creep rate Lower coefficients of thermal expansion Better wear resistance

In ex-situ composites the reinforcements are added externally [16, 21, 24] whereas in in-situ composites the reinforcing particulates are formed by chemical reaction within theliquid melt. One of the important drawbacks during the processing of ex-situ MMCs isthe presence of interfacial impurities and oxides between reinforcement and matrixresulting in poor wettability and bonding. This has led to the development of in-situcomposites, wherein the reinforcements are generated in a metallic matrix via chemicalreactions between elements and/or compounds with Al alloy melt during the compositefabrication. The advantages that in-situ MMCs have over conventional MMCs includethermodynamically stable reinforcements in the matrix, clean reinforcement-matrixinterfaces resulting in a strong interfacial bonding, finer particle size yielding better mechanical properties and potential for lower cost of production. These advantages makeit a strong candidate for the design task at hand. On the other hand, the reinforcement particles in in-situ composites are subject to strong segregation effects and therefore postsolidification process strategies are necessary to more uniformly mix the particles.

In this project we will work with the Al matrix and TiB2 reinforcement with 5 % Copper based composites.

Inductive Design Exploration Method (IDEM)

We use IDEM for uncertainty modeling in the simulation models, management of uncertainty propagation and tools for design exploration in the presence of propagateduncertainty in the network .

Multi-Scale Design Traditionally, materials are selected from materials database. Paradigm changing to tailoring materials for a specific performance of a product

by virtue of continuous development of material science and computing. Significant advantage of designing products at multiple scales is increased design

freedom enabling designers to achieve performance not possible by designing at asingle level.

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Integrated Product and Materials Design

The Challenges faced in integrated materials design are: Decision based collaborative design Collaborative Computational Architecture Management of Uncertainty

Figure 1.3 Integrated Product & Materials Design

Uncertainty

Management of Uncertainty: Reducing Uncertainty: feasible when designer has large amount of data or

complete/better knowledge of the system. Robust Design: To design a system to be insensitive to uncertainty without

eliminating or reducing the sources in the system. Thus can be introduced todesign at a lower cost by sacrificing optimal performance.

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Robust Design

Robust Design parameters are divided into: Noise factors Responses Control Factors

Figure 1.4 Robust Parameters

Type I Robust Design: Identify control factor (design variable) values that satisfya set of performance requirement targets despite variation in noise factors(uncontrollable variable). [Taguchi method]

Type II Robust Design: Identify control factor (design variable) values thatsatisfy a set of performance requirement targets despite variation in control andnoise factors. [I + II=RCEM]

Type III Robust Design : Identify adjustable ranges for control factors, that satisfya set of performance requirement targets and/or performance requirement rangesand are insensitive to the variability within the model. [I+II+III=RCEM-EMI]

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Comparison between Type I,Type II and Type III robust design

Figure 1.5 Comparison of Robust Design Techniques

Limitations of existing Robust Design methods

Intense information interface across the boundary of sub-system for estimatingfinal performance variation because of tightly coupled uncertainty analysis process and design exploration process.

Methods cannot employ the full power of parallel computing No consideration of unquantifiable model structural uncertainty in subsystems. If one model in the series is changed ,then whole process after the model needs to

be repeated for estimation of propagated uncertainty.

A new systems-based approach for materials design that combines inductive (top-down)engineering with deductive (bottom-up) science is essential. Hence IDEM!

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Finding Range Sets of Design Specification.

Figure 1.7 IDEM Structure

Information mapping in order of processing, structure, property and performance is possible; however, inverse mapping is limited. Basic idea of finding ranged sets is to passdown the feasible range in an inverse manner, from desired given performance ranges todesign space. For identifying feasible solution range, we use only material models for bottom-up calculation, not the materials model for inverse calculation.

Parallelizing Function Evaluation and Handling Uncertainty.

Figure 1.8 Passing of Feasible Spaces

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Taking into account resources and time required for uncertainty analysis, designexploration process and uncertainty analysis need to be decoupled. Design 2 is better thanDesign 1 because the projected range of design 2 in the interdependent variables space(y-space) is farther from the constraint boundary than Design 1.

Solution Technique for IDEM

Figure 1.9 IDEM Algorithm

Step 2Discrete Function Evaluation

It consists of: Seeding Splitting Projecting Merging

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Figure 1.11 HD-EMI Evaluation

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Step 3 (b) Generating Exact Boundary Points in Input space.

For generating the exact boundary points in Input Space we use Numerical root findingmethod like Bisection Method.

Exact border pts along axis 2Feasible pointsInfeasible pointsTrue constraint border

Exact border pts along axis 1

Input Space

1

2

Figure 1.12 Boundary Point Generation

Step 3 (c) IDCE using HD-EMIs

For selecting the best solution among feasible set of solutions, the required HD-EMI

value in each direction in each space should be statistically selected so that a designer gives more margin for the projecting function whose potential error is larger than

Figure 1.13 IDCE using HD-EMIs

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Step 4 HD EMIs Evaluation and cDSP

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Entities in the Graphical Robust Design Process Model (GRDPM)

GRDPM of Underwater Submersible Multi-level Design

G O A L S / M E A N S ( I N D U C T I V E D E S I G N

- I D E M ) C A U S E A N D E F F E C T ( D E D U C T I

V E )

Performance

Properties

Processing

Module5

Module4

Module3

Module2

Module1

GoalGiven Value or Parameter

Figure 2.1 Microstructure Mediated Design of Material and Product

Parameter to be determinedOutput ResponseParameter to be determined


Output Response


Output Response

Microstructure

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Figure 2.2 Analysis Diagram

\

Figure 2.3 Interface Diagram

Module 1 involves the prediction of the precipitation of liquid aluminum based on thecomposition and processing temperature. The output of Module 1 is the informationabout different phases formed, the size of precipitates and the time required to completethe reaction. This information is used in Module 2, which models the process of microstructure evolution and the effect of temperature and solutal fields on the resulting

Geometric Parameters

MODULE 4Structure property

correlation of MMCs

MODULE 5Requirement list,

microstructuremapping and design

Mechanical Properties

MODULE 6Robust design

using IDEM

Constraints 1.Stress conditions

2.Heat transfer 3.Shock response

Constraints Range of mechanical

properties

Perfor-mance

Phases,Ppt size

Init microstructure& ppt distribution

MODULE 1Pptn modeling inliquid aluminum

MODULE 2Modeling of

microstructureevolution in MMCs

MODULE 3Semisolid

processing ofMMCs

Constraints Max. volume fraction of

reinforcement

1. Temperature field 2. Solutal field

1. Composition2. Processing temp

3. Rxn time

Constraints 1.Range of working

temperature2.Shear stress

1.Rolling parameters2.Temperature

Constraints Mass transfer phenomenon

(convection )


MODULE 4Structure property

correlation of MMCs


microstructuremapping and design

Mechanical Properties


using IDEM

Constraints 1.Stress conditions

2.Heat transfer 3.Shock response

Constraints Range of mechanical

properties

Perfor-mance

Phases,Ppt size

Init microstructure& ppt distribution

MODULE 1Pptn modeling inliquid aluminum

MODULE 2Modeling of

microstructureevolution in MMCs

MODULE 3Semisolid

processing ofMMCs

Constraints Max. volume fraction of

reinforcement

1. Temperature field 2. Solutal field

1. Composition2. Processing temp

3. Rxn time





1.Rolling parameters2.Temperature

Constraints Mass transfer phenomenon

(convection )

Init. Micro- structure, ppt.

distribution [Templates]

MODULE 1Precipitationmodeling in

liquid aluminum1.Phases

formed 2. Ppt size

MODULE 2Modeling

microstructureevolution in

MMCs

MODULE 3Semisolid

processing ofMMCs

Final microstructure after semisolid processing

[Templates]

MODULE 4Structure - Propertycorrelation of MMCs

1. Composite composition 2. Temp. of processing

3. Time of reaction [Templates]

Reqd mech. properties [Templates]

Obtained mech. properties [Templates]


using IDEM

Interfacevariables of Projects 1, 2, 3, 4

[Templates]

Design and uncertainty parameters [Text and Abaqus Output Files]

Modification parameters [Templates]

Ppt. info.

MATERIALS DESIGN

MECHANICALDESIGN


microstructuremapping & design

INTERFACE

Init. Micro- structure, ppt.

distribution [Templates]

MODULE 1Precipitationmodeling in

liquid aluminum1.Phases

formed 2. Ppt size

MODULE 2Modeling

microstructureevolution in

MMCs

MODULE 3Semisolid

processing ofMMCs

Final microstructure after semisolid processing

[Templates]

MODULE 4Structure - Propertycorrelation of MMCs

1. Composite composition 2. Temp. of processing

3. Time of reaction [Templates]

Reqd mech. properties [Templates]

Obtained mech. properties [Templates]


using IDEM

Interfacevariables of Projects 1, 2, 3, 4

[Templates]

Design and uncertainty parameters [Text and Abaqus Output Files]

Modification parameters [Templates]

MATERIALS DESIGN

MECHANICALDESIGN


microstructuremapping & design

INTERFACE

Ppt. info.

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Figure 2.5 Combined Flow Diagram

MODULE 1 (Precipitation modeling in Liquid Aluminum)A suitable route for the in situ Al / TiB2 composite manufacturing process utilizes thereduction of K 2TiF6 and KBF4 with aluminum, generally known as the halide salt process. K 2TiF6 and KBF4 are other precursors that dissolve in the aluminum melt toform intermediate phases Al3Ti and AlB2. The reaction between these intermediate phases has been studied to predict the particle size distribution of TiB2 phase thus formedin the matrix.

A model proposed by Anestiev and co-authors [1] has been used to investigate thediffusion reactions taking place between the intermediate phases. In this model, Al3Tiand AlB2 are allowed to react in liquid Al to form TiB2 particulates. A coordinate system

dividing a 2-dimensional space into strips of equal length has been used, half of whichcontains Al3Ti and the other half AlB2 dissolved in the Al melt, shown in Figure 2.6.When these intermediate phases react, random nucleation of TiB2 particulates is assumed.The kinetics of the formation of TiB2 particles are governed by unsteady state diffusionequations, which in turn depend on the concentration profile of the intermediate solute phases in the region. The solute consumption rate due to TiB2 formation is described byvolume fraction of the region transformed per unit time. Johnson-Mehl-Avrami analysis[2 and 12] is used to find the transformed volume fraction from the nucleation and

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temperature values at the specific points along the metal-mold interface, realistic fluxvalues at the metal-mold interface can be derived which can be fed into a ComputationFluid Dynamics (CFD) modeling tool to obtain accurate thermal fields across the castingdomain. These fields are used in the cellular automata model to predict the microstructureevolution as the solidification proceeds.

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Project 2 : Modelling of Microstructural Evolution In MMCs

Clarification of Task

Modeling ofmicrostructure of as-

cast Al-TiB2composite

Design of Experiment

Computationalsimulations based ona theoretical model

InfrastructureProgramming tool

(MATLAB, FORTRAN) &FLUENT

Discrete Constraints Evaluation

GivenPrecipitate size and distributionPhases present in the system

FindAs-cast microstructure with least amount of

segregation and dendritic regiondAchieved HD-EMI Desired

BoundsX>=X

criConstraintsT>=850 K

Integrated Metamodel andPrediction Interval Estimation

Estimated modelCellular Automatabased model to

simulate as-castmicrostructure

To be determined

EstimatePrediction Interval

To be determined

HD-EMI_1

InitialDesignSpace

FeasiblDesignSpace

Requirements and Constraints

Figure 2.8 cDSP for Module 2

MODULE 3 (Semi-solid Processing in MMCs)The present module deals with the simulation of the semi-solid processing [9] of metalmatrix composites. The actual process [10, 11 and 17] consists of passing slabs of as-castcomposite material through rollers [Figure 14] at such a temperature that part of it is insemi-solid or mushy state. Two-high mill rollers of diameter 120 mm and 125 mm barrel width are used in this process. The sample is heated to temperatures between 610to 633C to obtain 10 to 30% liquid in the material. When the slab is passed through therollers, the grains deform and rearrange and a nearly homogeneous distribution of TiB2 particles is obtained. Multiple passes are performed to refine the grain size. Such a process enhances the properties of the MMC and homogenizes its composition.

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Figure 2.9 Schematic of Semi-Solid Processing

Since this is a novel process and its physics are not yet fully understood, an empiricalmodel is used, based on data taken form a large number of experiments. The model takesas input the processing conditions of semi-solid processing, including ratio, and then predicts the final average grain size and also gives an approximate microstructure. To predict the final grain size it takes in the experimental details and interpolates the grainsize. After processing, the TiB2 particulates rearrange themselves to achieve a moreuniform spatial distribution, which is also reflected in the model. Using a geneticalgorithm based Voronoi and Monte Carlo code [8], equiaxed globular grains are created.It forms in 100 x 100 matrix grains differentiated by different color codes which can bethen be interpreted to render the final microstructure after semi-solid processing.

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Project 3 : Semi Solid Processing of Composite


Modeling andexperimental study of

the evolution of themicrostructure duringsemi solid processing

of Metal MatrixComposites.


Computationalsimulations and

experiments.


(MATLAB) &CFD tools


GivenMapping of as cast microstructure from project2.

Experimental data.Find

Final microstructure and particle distribution.Achieved HD-EMI

SatisfyRolling speed=.283m/s, .1/s

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MODULE 4 (Structure-Property Correlation of MMCs)

Yield Stress : The matrix yield stress is assumed to obey the Hall-Petch relation, i.e.,

0 y

y

k

d = +

(1)

Where k y is the strengthening coefficient (a constant unique to each material; for pure Al,k y = 3.4 MPa-mm), o is a material constant related to lattice resistance (for pure Al, o=2.95 MPa), d is the grain diameter, and y is the yield stress. The constants correspondingto matrix properties are assumed to be that of pure Al. The calculation of overall yieldstress ( ) also incorporates Orowan particle bypass via dislocation looping [30], i.e.,

( )( )( )1 21 1 1 y f f f = + + + orowan (2)where f 1 takes the effect of volume fraction of particles, f d takes into account the thermalexpansion coefficient mismatch between matrix and reinforcement, and f orowan takes intoaccount the effect of particle size (d) and spacing. It receives input from outputs of

Modules 1 and 3, specifically reinforcement size (d p, grain size (d ), semisolid processingtemperature (T) and volume fraction of TiB2 particles.

Density : The determination of density is based on the average property of each of theconstituent phases, i.e.,

( )2 2 21TiB TiB Cu Cu Al TiB Cuvf x vf = + + (3)where

2, , ,TiB Cu Al

2TiB

are the densities of the composite, TiB2, copper and aluminumrespectively. Also, x is the volume fraction of TiB2 and is the volume fraction of copper (typically 6%).

Cuvf

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Computationalsimulations on efficientmodels.


(MATLAB,ABAQUS) &Codes i.e. OOF


GivenMicrostructural specifications (in terms of statistical

descriptors) and mappings of mechanical behavior tomicrostructures.

FindDesired mechanical properties and microstructural

changes needed for the t ailoring.Satisfy

Constraint: The minimum requirement of differentmechanical properties for the robust designBounds: Volume fraction of TiB2 {2.5,10}

Reinforcement particle size {1.0,2.5} mGrain size of Al-Cu matrix {20,55} m

Integrated Metamodel and

Prediction Interval Estimation

EstimatePrediction IntervalTo be determined

HD-EMI_4,5,6

InitialDesignSpace

FeasibleDesignSpace


Project 4 : Structure-property correlation of MMCs

Modeling of thecorrelation between final(semi-solid processed)as well as the as castmicrostructure andmechanical behavior ofin-situ Al-Cu based TiB2reinforced composite.

EstimateMetamodelMechanicalresponse asfunctions of

microstructuralspecifications


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MODULE 5 (Property-Performance Correlation of MMCs)

Module 5 acts as an interface between the materials design aspect and the mechanicaldesign of the submersible. The performance parameters considered are depth, time of

operation and weight of the outer shell of vehicle. The objective is to maximize the depthand time of operation while minimizing the weight of the outer shell of the submersible.The formulas used for the calculation of these performance parameters are stated in whatfollows [13, 25 and 29]

MODEL FOR DEPTH (h): We use Roarks formula [29] for thickness (t) to outer diameter (OD) ratio.

1 21 12

t P OD

=

(4)

where t is the thickness of the shell,OD is the outer diameter of the shell; P is theexternal pressure and (from equation 2) is the yield stress of the metal matrixcomposite. Substituting for P as w gh where w is the density of water (1025 kg/m

3), gis the gravitational attraction (9.81 m/sec2) and h is the depth of submersible below water.Solving for h we get:

221 12 w

t h

g OD

= (5)

Final Model to be provided by Ashish base on FEM and ABAQUS analysis.

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Figure 2.12 Depth Analysis

For External Pressure only:

Putting and solving for h, we get:

Inputs Required: where:

Alternative Model:

Substituting for P and solving for h we get:

Inputs Required:

MODEL FOR WEIGHT (W): The weight of a cylindrical shell with spherical end caps iscalculated.

( ) ( )2 2 343W L OD ID OD ID3 = + (6)

where in Eq. (3) is the density of the composite, L is the length of the submersible, ODis the outer diameter and ID is the inner diameter of the cylindrical shell with sphericalend-caps [Figure 2]. We shall fix the outer diameter (OD) at 260 mm and the length (L)at 1.6 meter. Thickness (t) can vary from 5 mm to 15 mm as representative parameters of a typical Autonomous Underwater Vehicle as described in [14, 15].

MODEL FOR ENDURANCE TIME (T opr )

( )0.8 * *EnergyDensityFixed Load + Propulsion Loadopr

B W eff T

= (7)

where B is the buoyant weight of the submersible,W (eq.6) is the weight of thecylindrical shell, eff is the efficiency of the battery. The efficiency of a Lithium-Ion battery is typically 60% and its energy density is 128 Watt-Hour/Kg. For the initial

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design, assuming a slow moving vehicle and submergence/surfacing rates, we shallignore propulsion load in our calculations and assume a fixed electrical load of 400 Watt-Hour which is typical of the control computers and electronics payloads in a smallunderwater robotic vehicle [14, 15].

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Project 5 : Requirements List and System-level Design


Prepare the geometricmodel of the UW

vehicle in accordancewith the mechanicalproperties obtained

and to satisfy therequirements list


Computationalsimulations based ona theoretical model


(ABAQUS) &CFD tool (Fluent)


GivenLength, Yield Stress, Youngs Modulus, Poissons ratio,

Thermal conductivity coefficient and PressureFind

Feasible range of thicknessAppropriate range of Outside diameter

Appropriate WeightSatisfy

5 mm t 15 mm260 mm OD 300 mm

Heat Transfer = [1.5 KW, 10 KW]No buckling for P 60 MPa

Integrated Metamodel andPrediction Interval Estimation

EstimateMeta model

= f (t, OD, W)Response Surface

ModelTo be determined

EstimatePrediction IntervalTo be determined

HD-EMI_5

InitialDesignSpace

FeasibleDesignSpace



MODULE 6 (Robust Design using IDEM)We employ IDEM to achieve a robust solution over the various levels of process-structure, structure-property and property-performance relations. IDEM includes paralleldiscrete function evaluation, Inductive Discrete Constraints Evaluation (IDCE) based onHyper-Dimensional Error Margin Indices (HD-EMIs), and the Compromise DecisionSupport Problem (cDSP) for finding the best solution under MSU [3-7 and 23]. Theoverall procedure for the IDEM is schematically illustrated in Figure 2.14.

Figure 2.14 Schematic of IDEM [3]

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IDEM is a method used to identify adjustable ranges of control factor (design variable)values in a system with uncertainty propagation in a design/analyses process network andto account for uncertainty in downstream activities and uncertainty propagation. WithIDEM, a designer can maximize or maintain ranges of values for design variables or performance parameters that are shared or linked with another designers robust design

process. Thereby, design freedom is preserved for another collaborating designer whocan make changes to the designwithin specified rangeswithout compromising designrequirements.

IDEM facilitates hierarchical design while accounting for uncertainty in models and its propagation through the chains of modules, such as the one shown in Figure 1.4. Weconsider propagation of uncertainty in design and analysis process networks andsequentially identify a ranged set of feasible specifications, instead of an optimal point, ineach domain of a hierarchical design process in an inductive way. The IDEM method is based on the concept of Error Margin Indices (EMIs) which indicates the degree of reliability of a decision satisfying system constraints or bounds. The procedure for

obtaining the EMI is as follows: (a) obtain the upper and/or lower deviation of a response(URL and LRL) and (b) calculate the EMI from this deviation. The Error Margin Index(EMI) is calculated by including the response mean (y) and upper/lower deviations(Yupper and Ylower ) from a combined distribution of a system model and error bounds.The EMI includes the response deviations of a system model due to variations in designvariables and the response deviations of error bounds as well as the system model. Themathematical formulations of EMI corresponding to a goali are:

( ( )) /i i i upper EMI URL f x Y = for minimization problems;( ( ) ) /i i i low EMI f x LRL Y = er for maximization problems;

| | |{ ,

( 1, 2,...,Number of the goals)

i i i ii

i i

|} f URL f LRL EMI MinY Y

i

=

=

Y

Y

upper upper y

lower y lower

Y

Y

=

= As shown in Figure 2.14, the objective is to find the best ranged set of designspecifications in the space X considering uncertainty in mapping functions (f) and propagated uncertainty through a design process. IDEM involves finding ranged sets of design specifications by passing feasible solution spaces from final performance require-ments by way of an interdependent response space to the design space while preservingthe feasible solution space as much as possible. The procedure includes the followingsteps [3].

Step 1: Conduct parallel discrete function evaluation: Define rough design and performance spaces (hyper-dimensionalx, y, and z

spaces) and generate discrete points in each of these spaces. Evaluate the generated discrete points using themapping models (f and g in

Figure 2.14) that include all quantified amount of uncertainty.

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Store the evaluated data sets, including discrete input points and output ranges, ina database.

Step 2: Inductive Discrete Constraints Evaluation (IDCE) process: Using informationfrom Step 1, sequentially identify feasible regions iny and x spaces with a giveninitial requirement range inz space

Step 3: Solve the Compromise Decision Support Problem (cDSP): Find the bestrobust solution under MSU by performing Step 2 with adjusted HD-EMIs.

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CHAPTER 3

SOLUTION STRATEGY USING IDEM

The model description has been outlined in the previous section. Now we focus on themodel development with respect to IDEM [Figure 3.1]. The modeling in MODULE 2 has presented many challenges in view of the complexity of the phenomena and theimmaturity of available models; accordingly, it is bypassed in the following.

Figure 3.1: Solution Strategy Using IDEM

In Figure 16, f1, f3, f4, f5, f7, f8 and f9 represent the theoretical or empirical modelsconsidered at the different levels of design. The inputs to MODULE 1 are the volumefraction of TiB2 (xTiB2) and temperature of processing in degree K (T).The output of MODULE 1 ( f1) is the average TiB2 particle size (d p ) which is one of the inputs toMODULE 4. The independent inputs to MODULE 3 are volume fraction of TiB2 (xTiB2)and percentage of liquid in processing (%L) and the output of MODULE 3 ( f3) is theaverage grain size (d) of microstructure. Module 4 receives inputs from the outputs of Module 1 & 3 along with the independent inputs of volume fraction of TiB2 (xTiB2) andtemperature of semi-solid processing (temp). MODULE 4 is the module that deals withthe structure-property relationships and f4 gives the density () [eq. 3]and f5 gives yieldstress ( [Equation 2]) as outputs. Finally MODULE 5 deals with the property- performance relationship of the developed microstructure and f7 evaluates the performance variable of depth of operation (h),f8 evaluates the weight of the outer shell(W) and f9 evaluates the time of operation (T opr ) of underwater submersible. Theindependent parameter in this level of design is the thickness of the shell (t ) and thedependent parameters are density ( ) and yield stress ().The final solution for IDEM will take the structure as shown in Figure 3.2. We observethat the that the feasible design spaces are inductively passed from MODULE 5 toMODULE 4 and subsequently to MODULES 3 and 1 of design

Depth

Weight

Geometric ParametersOD and t

MODULE 1

Grain sizeafter

semisolid processing

Average TiB 2 grain size

f3

% Liquid Reduction xTiB 2

f1

Dependent Variables

Time of Operation

Independent Variables

T

Temp

f5

f4 f7

f8

f9

MODULE 3 MODULE 4 MODULE 5

Depth

Weight

Geometric ParametersOD and t

MODULE 1

Grain sizeafter

semisolid processing

Average TiB 2 grain size

f3

% Liquid Reduction xTiB 2 Temp

f1

Dependent Variables

Time of Operation

Independent Variables

T

f5

f4 f7

f8

f9

MODULE 3 MODULE 4 MODULE 5

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HD-EMI_5


Range of vol.fraction TiB 2

HD-EMI_3

Design space of:% liquid reduction

Volume fraction TiB 2

HD-EMI_1

Design space of:% TiB 2

T

Vol. fractionTiB 2

T

Required RangeDesign SpaceHD-EMIsSolution

f3

f1

Range of grain size distribution

HD-EMI_4

f4 f5

Range of depth (D) Range of weight (W) Range of time of operation (T)

HD-EMI_7 HD-EMI_8 HD-EMI_9

Range of

f7 f9f8

Design Space of:Geometric

ParametersID, t, , and rho

Range of

Design space of:TiB 2 size distributionVolume fraction TiB 2

Grain size distribution

% liquid reduction Average grain size, d

HD-EMI_5


Range of vol.fraction TiB 2

HD-EMI_3

Design space of:% liquid reduction

Volume fraction TiB 2

HD-EMI_1

Design space of:% TiB 2

T

Vol. fractionTiB 2

T

Required RangeDesign SpaceHD-EMIsSolution

f3f3

f1f1

Range of grain size distribution

HD-EMI_4

f4f4 f5f5

Range of depth (D) Range of weight (W) Range of time of operation (T)

HD-EMI_7 HD-EMI_8 HD-EMI_9

Range of

f7 f9f8f7f7 f9f9f8f8

Design Space of:Geometric

ParametersID, t, , and rho

Range of

Design space of:TiB 2 size distributionVolume fraction TiB 2

Grain size distribution

% liquid reduction Average grain size, d

Figure 3.2 Solution Strategy Using IDEM

Range of Depth (D)Range of Payload (P)

Range of Heat Gen. (H)

Geometrical Parameters

f7 f8

Design Space of GeometricalParameters (ID ,OD ,t, L) andYield Stress, Y, K and

HD-EMI_7 HD-EMI_8 HD-EMI_9 Range of YieldStress

Range of Y Range of

HD-EMI_5HD-EMI_4 HD-EMI_6

D.S ofFinal MS

D.S ofAs cast MS

D.S of Prep.

Range of Final MS

HD-EMI_3

D.S of Rolling Par.D.S of Temperature

Rolling Par.Temperature

D.S ofAs cast MS

Range of As cast MS

Range of As cast MS

HD-EMI_2

D.S of Temp .Field

D.S of Solutal FieldD.S of Precipitate size

Temp.Field

Solutal Field

HD-EMI_1

Range ofPrep. Size &Distribution

D.S of Composition

D.S of Temperature

Comp.

Temp .

f9

f6f4 f5Range ofPrep. Size &Distribution

f3

f2

HD-EMIs

Solution

f1Design Space

Required Range

Figure 3.3 Complete Solution Strategy Using IDEM

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We see that the volume fraction of TiB2 is an input to MODULE 1, MODULE 3 andMODULE 4 of design. The responses of MODULE 1 and MODULE 3 are influenced bymultiple variables and hence we use response surface methodology for modeling andanalysis of the design task at these levels. The Response Surface Methodology that will be employed is the second order model [20]:

20

1 1

k k

i i ii i ij i ji i i k

Y x x x x = =

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