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Copyright 2009 by ASME1

Proceedings of IDETC/CIE 2009ASME 2009 International Design Engineering Technical Conferences &

Computers and Information in Engineering ConferenceAugust 30-September 2, 2009, San Diego, CA, USA

DETC2009/DAC-87276

MICROSTRUCTURE-MEDIATED INTEGRATION OF MATERIAL ANDPRODUCT DESIGN UNDERSEA SUBMERSIBLE

Ayan SinhaUndergraduate student

Madhusudan ChakrabortyProfessor

Sudipto Ghosh, C.S. Kumar Associate Professors

Indian Institute of Technology, Kharagpur, India

Jitesh H. PanchalAssistant Professor

School of Mechanical and Materials EngineeringWashington State University, Pullman, WA, USA

Janet K. Allen, David L. McDowell, Farrokh MistreeProfessors

Woodruff School of Mechanical EngineeringGeorgia Institute of Technology, Atlanta, GA, USA

ContributorsS. Bagchi, S. Lenka, A.Patra, M. K. Singh, A. K. Srivastava T. K. Kundu

Undergraduate students Assistant Professor

Indian Institute of Technology, Kharagpur, India

ABSTRACTIn this paper, we introduce the construct of microstructure-mediated design of material and product. The microstructureof the material is controlled within feasible bounds to achievethe performance targets of the product. We illustrate theefficacy of this construct via the integrated robust design of asubmersible and Al-based matrix composites. The integrateddesign is carried out using an Inductive Design ExplorationMethod (IDEM) that facilitates robust design in the presenceof model structure uncertainty (MSU).

Model structural uncertainty (MSU), originating fromassumptions and idealizations in modeling processes, is a formof uncertainty that is often virtually impossible to quantify. Inthis paper, we demonstrate a method, the Inductive DesignExploration Method (IDEM), that facilitates robust design in

the presence of model structural uncertainty. We achieverobustness by trading off the degree of system performanceand the degree of reliability based on structural uncertaintyassociated with system models (i.e., models for performancesand constraints). IDEM is demonstrated in the design of ashell of a robotic submersible. The material considered is in-situ Al metal matrix composites (MMCs) due to theadvantages that the in-situ MMCs have over the conventional

MMCs. This design task is a representative example ofintegrated materials and product design problems.

Keywords: Microstructure mediated design, robust designinductive design exploration

Nomenclature

B Buoyant weight of the submersibled Grain diameterd Reinforcement sizeeff Efficiency of the batteryg Gravityh Depth of the submersible below water

ID Inner diameter of the shellky Strengthening coefficient in the Hall-Petch

relation

L Length of the submersibleOD Outer diameter of the shellP External pressureT Semisolid processing temperature

Topr Endurance time of the submersiblet Thickness of the shell

W Weight of the cylindrical shellxCu Volume fraction of Cu

2TiBx Volume fraction of TiB2

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Y Output of a response surface modelij Coefficients in a response surface model Density of the composite

TiB2, Cu, Al Densities of TiB2, copper and aluminumrespectively

w Density of water Overall yield stress incorporating Orowan

particle bypasso Material constant related to lattice

resistancey Yield stress calculated from the Hall-Petch

relation

1. FRAMING THE PROBLEMTraditionally materials are selected from databases ofexperimentally determined materials properties. However, theparadigm is shifting towards the concurrent design ofmaterials and products. This entails tailoring materials forspecific performance required in specific products orprocesses.

In order to tailor materials, the approach taken by materialsscientists is sequential deductive analysis, with a bottom-upmapping from processing path to nano- and micro-structure,material properties and performance. This corresponds toOlsons materials design hierarchy [22] shown in Figure 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 theapplication of these materials. The microstructure representsthe interface between structure-property-performance relations

including systems design and process-structure relations. Amicrostructure-mediated design-centered approach has beenadopted for concurrent design of materials and product.

A systems-based approach has been adopted. This combinesinductive (top-down) engineering with deductive (bottom-up)science; see Figure 1. Fundamental to this design approach isan interconnected system of modules (a design process chain)expressed in terms of variables, constraints, and models thatembed relevant aspects of the material microstructures throughoverall system configuration.

Figure 1 Hierarchical Materials Design [22]

In this paper, the method is illustrated through the design ofthe shell and design of the material from which the shell of ansubmersible is made. The shell is characterized by bothgeometrical and material features; see Figure 2 and Refs. 14and 15. The objective is to design the shell of a roboticsubmersible for deep sea exploration with the multifunctionarequirements of minimizing the mass in walls (wall thickness)for given support superstructure for given maximum depth andassociated pressure differential. Other design requirementsinclude a) suitable factor of safety with respect to collapse attarget maximum operating depth, b) a large endurance timesatisfying the time of operation constraints under waterwithout resurfacing/refueling/battery changes, c) satisfyinggeometric and weight constraints. The preferred design musthave a) high strength to weight ratio and b) resistance againstenvironmental factors such as corrosion. Recent advances inmaterial processing allow designing the material to attainspecific desired properties.

Figure 2 Pressure Shell of a Submersible Robot

Al-based metal matrix composite is used to illustrate theproposed method. Metal matrix composites (MMCs), ingeneral, and Al-based MMCs in particular, have been thesubject of intense research for the past two to three decadesand are being exploited for a range of commercial applicationsrelated to aerospace and automotive industries. Al-based meta

matrix composites can be divided into two classes,, namelyex-situ and in-situ. In ex-situ composites the reinforcementsare added externally [16, 21, 24] whereas in in-situ compositethe reinforcing particulates are formed by chemical reactionwithin the liquid melt. One of the important drawbacks duringthe processing of ex-situ MMCs is the presence of interfaciaimpurities and oxides between reinforcement and matrixresulting in poor wettability and bonding. This has led to thedevelopment of in-situ composites, wherein thereinforcements are generated in a metallic matrix via chemicareactions between elements and/or compounds with Al alloymelt during the composite fabrication. The advantages that insitu MMCs have over conventional MMCs includethermodynamically stable reinforcements in the matrix, clean

reinforcement-matrix interfaces resulting in a stronginterfacial bonding, finer particle size yielding bettermechanical properties and potential for lower cost ofproduction. These advantages make it a strong candidate forthe design task at hand. On the other hand, the reinforcementparticles in in-situ composites are subject to strong segregationeffects and therefore post solidification process strategies arenecessary to more uniformly mix the particles.

Processing

Structure

Properties

Performance

Processing

Structure

Properties

Performance

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2. MICROSTRUCTURE-MEDIATED DESIGNThe design approach is based on systems-based integrated top-down (inductive) and bottom-up (deductive) multilevel designas illustrated in Figure 3. Multilevel design for the shell designproblem involves two activities, namely, process path -structure relationships and structure-property-performancerelationships. These two design objectives interact via themicrostructure. While on one hand the processing conditionsinfluence the obtained microstructure, the performance of theproduct depends on the mechanical properties which in-turnare mapped from the microstructure.

In the present study, two major aspects of the design problem,namely, the materials design (rather than just materialsselection) and structural design, are combined. The materialsdesign aspect has been divided into three parts based on thedifferent processing steps of the material. The interfacebetween materials design and structural design is the mappingof the processed microstructure to the required mechanicalproperties.

The Inductive Design Exploration Method (IDEM) is used toeffect solution. The design process chain for this applicationconstitutes of six interconnected modules. Five modulesaccount for the modeling of the behavior of the material andthe structure. The sixth is used to address uncertaintyembodied in the simulation models, the management ofuncertainty propagation and tools for design exploration in thepresence of propagated uncertainty in the design processchain. Based on the materials processing steps involved andmechanical design requirements, the interconnected modulesthat constitute the design process chain for this application are(see Figure 3):

MODULE 1: Precipitation modeling in liquid Al.MODULE 2: Modeling of microstructure evolution in MMCs.MODULE 3: Evolution of microstructure during semisolid

processing of MMCs.MODULE 4: Structure - property correlations of MMCs.MODULE 5: Requirement list, microstructure mapping and

system-level design.MODULE 6: Robust design strategy using IDEM to address

model structure uncertainty and propagateduncertainty among levels of models.

MODULEs 1, 2 and 3 provide the simulated microstructureafter processing. The resulting mechanical properties are

estimated in MODULE 4, whereas MODULE 5 maps therequired mechanical properties based on the system designconsiderations.

Given the complexity inherent in the design process chain, it isimportant to define the variables, the interface and the designconstraints between the different modules. In Figures 4, 5 and6 we show the analysis, interface and the respective integratedflow diagrams for this design process chain. In the analysisdiagram [Figure 4] we show the various independent and

dependent variables in the six modules of the design processchain. In the interface diagram [Figure 5] we map theconnectivity and flow of information between the modules.

Figure 3 Microstructure Mediated Design of Material and

Structure

Figure 4 Analysis Diagram

Figure 5 Interface Diagram

GeometricParameters

MODULE 4Structure property

correlation of MMCs

MODULE 5Requirement list,microstructure

mapping and design

MechanicalProperties

MODULE 6Robust design

using IDEM

Perfor-mance

Constraints1.Stress conditions

2.Heat transfer3.Shock response

ConstraintsRange of mechanical

properties

Phases,Ppt size

Init microstructure

& ppt distribution

MODULE 1Pptn modeling inliquid aluminum

MODULE 2Modeling of

microstructureevolution in MMCs

MODULE 3Semisolid

processing ofMMCs

ConstraintsMax. volume fraction of

reinforcement

1. Temperature field

2. Solutal field

Constraints1.Range of working

temperature2.Shear stress

1.Rolling parameters

2.Temperature

ConstraintsMass transfer phenomenon

(convection)

1. Composition2. Processing temp

3. Rxn time

GeometricParameters

MODULE 4Structure property

correlation of MMCs


mapping and design

MechanicalProperties


using IDEM

Perfor-mance

Constraints1.Stress conditions

2.Heat transfer3.Shock response

ConstraintsRange of mechanical

properties

Phases,Ppt size

Init microstructure

& ppt distribution

MODULE 1Pptn modeling inliquid aluminum

MODULE 2Modeling of

microstructureevolution in MMCs

MODULE 3Semisolid

processing ofMMCs

ConstraintsMax. volume fraction of

reinforcement

1. Temperature field

2. Solutal field





1.Rolling parameters

2.Temperature

ConstraintsMass transfer phenomenon

(convection)

1. Composition2. Processing temp

3. Rxn time

Init. Micro-structure, ppt.

distribution[Templates]

MODULE 1Precipitationmodeling in

liquid aluminum1.Phasesformed

2. Ppt size

MODULE 2Modeling

microstructureevolution in

MMCs

MODULE 3Semisolid

processing ofMMCs

Final microstructure aftesemisolid processing

[Templates]

MODULE 4Structure - Propertycorrelation of MMCs

1. Composite composition2. Temp. of processing

3. Time of reaction[Templates]

Reqd mech. properties[Templates]

Obtained mech. properties [Templates]


using IDEM

Interfacevariables ofProjects 1, 2, 3, 4

[Templates]

Design and uncertainty parameters[Text and Abaqus Output Files]

Modification parameters [Templates]

Ppt. info.

MATERIALSDESIGN

MECHANICALDESIGN


mapping & design

INTERFACE

Init. Micro-structure, ppt.

distribution[Templates]

MODULE 1Precipitationmodeling in

liquid aluminum1.Phasesformed

2. Ppt size

MODULE 2Modeling

microstructureevolution in

MMCs

MODULE 3Semisolid

processing ofMMCs

Final microstructure aftesemisolid processing

[Templates]




Reqd mech. properties[Templates]



using IDEM

Interfacevariables ofProjects 1, 2, 3, 4

[Templates]


Modification parameters [Templates]

Ppt. info.

MATERIALSDESIGN

MECHANICALDESIGN


mapping & design

INTERFACE

GOALS/MEANS (INDUCTIVEDESIGN-IDEM)

CAUSEANDEFFECT (DEDUCTIVE)

Performance

Properties

Processing

Module

5

Module

4

Module

3

Module

2

Module

1Parameter to be determined

Goal

Given Value or Parameter

Output ResponseParameter to be determined

Goal


Output Response

Goal


Output Response

Microstructure

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MODULE 1 involves the prediction of the precipitation ofliquid aluminum based on the composition and processingtemperature. The output of MODULE 1 is the informationabout different phases formed, the size of precipitates and thetime required to complete the reaction. This information isused in MODULE 2, which embodies the process ofmicrostructure evolution and the effect of temperature andsolutal fields on the resulting microstructure. The next step isthe semi-solid processing of the Al-MMCs through a rollingoperation which modifies the materials microstructure. InMODULE 3 the effect of the rolling parameters on theresulting microstructure is predicted. In MODULE 4, thismicrostructure is used to predict the mechanical propertiesinherent in the material. These mechanical properties are usedin the system-level MODULE 5 to predict the effects ofdifferent AUV geometries on overall system performance. Ascan be seen from the integrated flow diagram [Figure 6], themicrostructure is the essential link between the design of thematerial and the design of the undersea submersible.

Init. Micro-

structure, ppt.distribution[Templates]

MODULE 1Precipitation

modeling inliquid aluminum

1.Phases

formed2. Ppt size

[Templates]

MODULE 2Modeling

microstructure

evolution inMMCs

MODULE 3Semisolid

processing ofMMCs

Final microstructure aftersemisolid processing

[Templates]




Reqd mech. Prop. [Templates]



using IDEM

Interface variables ofProjects 1, 2, 3, 4

[Templates]


Modification parameters[Templates]

Ppt. info.

MATERIALS

DESIGN

MECHANICALDESIGN


mapping & design

INTERFACE

Range of Mech.

Properties

Constraints

Convection

StressHeat transferShock response

Max. vol. frac.TiB2

Analysis FlowVariables

Indep. Parameters

Rolling ParametersTemperature

Syn. Flow Variables

Figure 6 Integrated Flow Diagram

In this application, the strength is principally determined bythe sizes, shapes and distribution of TiB2 precipitates inother words the microstructure of the material. Themicrostructure is determined by processing methods in thiscase, it is initially created by precipitation and followed by theevolution of the precipitate size and distribution during thesemi-solid rolling. The structural design can be modified intwo ways, namely, 1) by changing the processing conditions tomodify the microstructure, which has an effect on the overall

system performance and 2) by changing the geometry of theshell, which in turn not only affects structural performance,but also puts constraints on required mechanical properties ofthe material. Hence, the material microstructure needs to bedesigned in such a way that the constraints on the materialproperties, imposed by the structure, are satisfied. Since thematerial microstructure acts as the interface between thematerial and structure, we have adopted the phrasemicrostructure mediated design. Having defined the design

variables and the connectivity within the design process chainthe modules described in the sections that follow.

2.1 MODULE 1 (Precipitation Modeling in LiquidAluminum)

A suitable route (Mixed-Salt route) for the in situ Al / TiB2composite manufacturing process utilizes the reduction ofK2TiF6 and KBF4 with aluminum, generally known as thehalide salt process. Yang and coauthors [31] proposed adiffusion mechanism wherein Al3Ti is formed in the melinitially by a very fast reaction. Boron then diffuses into Al3Tparticles in the melt, thus forming TiB2 particles according tothe reaction, Al3Ti + 2B = 3Al + TiB2.

The liquid-state processing techniques to produce in-situcomposites include self propagating high temperaturesynthesis (SHS), exothermic dispersion (XD), reactive hotpressing (RHP), flux assisted synthesis (FAS) and rapidsolidification processing (RSP). Any of these processes couldbe used. K2TiF6 and KBF4 are other precursors that dissolve in

the aluminum melt to form intermediate phases Al3Ti andAlB2. The reaction between these intermediate phases hasbeen studied to predict the particle size distribution of TiB2phase thus formed in the matrix.

A model proposed by Anestiev and coauthors [1] has beenused to investigate the diffusion reactions taking placebetween the intermediate phases. In this model, Al3Ti andAlB2 are allowed to react in liquid Al to form TiB2particulates. A coordinate system dividing a 2-dimensionaspace into strips of equal length has been used, half of whichcontains Al3Ti and the other half AlB2 dissolved in the Amelt, shown in Figure 7. When these intermediate phases

react, random nucleation of TiB2 particulates is assumed. Thekinetics of the formation of TiB2 particles is governed byunsteady state diffusion equations (Solute redistributiontheory), which in turn depends on the concentration profile ofthe intermediate solute phases in the region. The soluteconsumption rate due to TiB2 formation is described byvolume fraction of the region transformed per unit timeJohnson-Mehl-Avrami analysis [2,12] is used to find thetransformed volume fraction from the nucleation and growthrates of the particles as: = 1 exp(-ktn) where is is thevolume fraction transformed, k = _ N G 3/3 and n = 4, N and Gare Nucleation and Growth Rate respectively.

The Nucleation rate is primarily a function of the Gibbs

energy change associated with the formation of the particlewhile the Growth rate also depends on its surface energy. Thethermodynamic models predicting the Gibbs free energies ofthe involved phases in the current system are described inRefs. [19, 26-28]. The kinetics of reinforcement particles canbe mathematically described by the following set of partialdifferential equations:

X1/t= D(2X1/x2) - X1S(/t),

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X2/t= D(2X2/x2) X2S(/t)

where,X1 andX2 are the mol fractions of the dissolved Ti andB in the Al matrix respectively, t is the time, D is the diffusioncoefficient, X1SandX

2Sare the mol fractions of Ti and B in

the solid phase (TiB2). The complex diffusion equations are

solved numerically to compute the TiB2 particle sizedistribution across the matrix.

Figure 7 Schematic of Coordinate SystemUsed in MODULE 1

2.2 MODULE 2 (Modeling Microstructure Evolution)Microstructural evolution of materials during various material

processes relates key properties such as mechanical strengthand electrical properties to the average grain size and the grainsize distribution, which are direct consequences of themicrostructure evolution. In MODULE 2, microstructureevolution during solidification depends on the thermal and thesolutal fields. The mathematical description of the dendriticsolidification process of a three component alloy in twodimensional square solidification domain () is:

The S/L interface evolves in time and has to be found as partof solution. The solidification of a three component alloy isgoverned by the evolution of temperature T(t,x,y) andconcentration field C(t,x,y) ,where = 1,2 which satisfiesseveral boundary conditions at the moving S/L interface aswell as the initial and the boundary conditions. The equationsthat describe the physics of solidification process follow.

Temperature T in (heat transfer equation):

Cp( T/ t) = KT + L( fs/ t)

where t is time, (x,y) is the domain co-ordinates, is thedensity, Cpis the specific heat, Kis the thermal conductivity,

L is the latent heat of solidification and fs is solid fraction. Forsimplicity the notation fL= 1 - fs, denotes the liquid fraction.

The concentration (C_) for the solute (solute diffusionequation)

CL/ t = DL

CL For liquid phase

CS/ t = DS

CS ...For solid phase

where = 1,2, DL and DS are liquid and solid diffusioncoefficients of solute , respectively. The cross diffusion isneglected and zero flux boundary conditions are applied tofour wall of simulation domain .

Fluid flow due to forced or natural convection also influencesthe microstructure evolution. The present module involves thenumerical solution of continuum equations for thermal fieldsand coupling it with a cellular automata model that computesthe evolution of grain structure with solidification time. Themeasured flux values are used to derive the evolution of thethermal fields with solidification time. Using measuredtemperature values at the specific points along the metal-mold

interface, realistic flux values at the metal-mold interface canbe derived which can be fed into a Computation FluidDynamics (CFD) modeling tool to obtain accurate thermafields across the casting domain. These fields are used in thecellular automata model to predict the microstructureevolution as the solidification proceeds.

2.3 MODULE 3 (Semi-solid Processing in MMCs)The present module deals with the simulation of the semi-solidprocessing [9] of metal matrix composites. The actual process[10, 11 and 17] consists of passing slabs of as-cast compositematerial through rollers [Figure 8] at such a temperature thapart of it is in semi-solid or mushy state. Two-high milrollers of diameter 120 mm and 125 mm barrel width are usedin this process. The sample is heated to temperatures between610 to 633C to obtain 10 to 30% liquid in the material. Whenthe slab is passed through the rollers, the grains deform andrearrange and a nearly homogeneous distribution of TiB2particles is obtained. Multiple passes are performed to refinethe grain size. Such a process enhances the properties of theMMC and homogenizes its composition.

Figure 8 Schematic of Semi-Solid Processing

Since this is a novel process and its physics are not yet fullyunderstood, an empirical model is used, based on data takenfrom a large number of experiments. The model takes as inputhe processing conditions of semi-solid processing, includingratio, and then predicts the final average grain size and also

InterfaceAl + AlB2Al + Al3Ti

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gives an approximate microstructure. To predict the final grainsize it takes in the experimental details and interpolates thegrain size. After processing, the TiB2 particulates rearrangethemselves to achieve a more uniform spatial distribution,which is also reflected in the model. Using a genetic algorithmbased Voronoi and Monte Carlo code [8], equiaxed globulargrains are created. It forms in 100 x 100 matrix grainsdifferentiated by different color codes which can be then beinterpreted to render the final microstructure after semi-solidprocessing.

2.4 MODULE 4 (Structure-Property Correlation ofMMCs)2.4.1 Yield Stress: The matrix yield stress is assumed to obeythe Hall-Petch relation, i.e.,

y = 0 + ky (d)-0.5 (1)

where ky is the strengthening coefficient (a constant unique toeach material; for pure Al, k

y= 3.4 MPa-mm),

ois a material

constant related to lattice resistance (for pure Al, o= 2.95MPa), d is the grain diameter, and y is the yield stress. Theconstants corresponding to matrix properties are assumed tobe that of pure Al. The calculation of overall yield stress ()also incorporates Orowan particle bypass via dislocationlooping [30], i.e.,

= y (1 +f1) (1 +f2) (1 +forowan) (2)

where f1 takes the effect of volume fraction of particles, f2takes into account the thermal expansion coefficient mismatch

between matrix and reinforcement, andf

orowan takes intoaccount the effect of particle size (d) and spacing. It receivesinput from outputs of MODULEs 1 and 3, specificallyreinforcement size (dp, grain size (d), semisolid processingtemperature (T) and volume fraction of TiB2 particles.

2.4.2 Density: The determination of density is based on theaverage property of each of the constituent phases, i.e.,

= TiB2xTiB2 + CuxCu + Al (1-xTiB2 xCu) (3)

where , TiB2, Cu, Alare the densities of the composite, TiB2,copper and aluminum respectively. Also, xTiB2 is the volumefraction of TiB2 and xCu is the volume fraction of copper

(typically 6%).

2.5 MODULE 5 (Property-Performance Correlation ofMMCs)MODULE 5 acts as an interface between the materials designaspect and the design of the structure of the submersible. Theperformance parameters considered are depth, time ofoperation and weight of the outer shell of submersible. Theobjective is to maximize the depth and time of operation while

minimizing the weight of the outer shell of the submersibleThe formulas used for the calculation of these performanceparameters are stated in what follows [13, 25 and 29].

2.5.1 Model for Depth (h): We use Roarks formula [29] forthickness (t) to outer diameter (OD) ratio.

=

P211

2

1

OD

t(4)

where tis the thickness of the shell, OD is the outer diameterof the shell; P is the external pressure and (from eq. 2) is theyield stress of the metal matrix composite. Substituting for Pas wgh where w is the density of water (1025 kg/m

3), g is thegravitational attraction (9.81 m/sec2) and h is the depth ofsubmersible below water. Solving for h we get:

2

w

2th 1 1

2 g OD

=

(5)

2.5.2 Model for Weight (W): The weight of a cylindrical shellwith spherical end caps is calculated.

W = L (OD2 ID2) + (4/3) (OD

3 ID3) (6)

where in eq. (3) is the density of the composite, L is thelength of the submersible, OD is the outer diameter and ID isthe inner diameter of the cylindrical shell with spherical end-caps [Figure 2]. We shall fix the outer diameter (OD) at 260mm and the length (L) at 1.6 meter. Thickness (t) can varyfrom 5 mm to 15 mm as representative parameters of a typica

Autonomous Underwater Vehicle as described in [14, 15].2.5.3 Model for Endurance Time (Topr)

LoadpropulsionFixedLoad

DensityEnergyeffWBT

opr+

=

)(8.0(7)

whereB is the buoyant weight of the submersible, W(eq.6) isthe weight of the cylindrical shell, effis the efficiency of thebattery. The efficiency of a Lithium-Ion battery is typically60% and its energy density is 128 Watt-Hour/Kg. For theinitial design, assuming a slow moving submersible andsubmergence/surfacing rates, we shall ignore propulsion loadin our calculations and assume a fixed electrical load of 400Watt-Hour which is typical of the control computers andelectronics payloads in a small underwater robotic submersible[14, 15].

2.6 MODULE 6 (Robust Design using IDEM)We employ IDEM to achieve a robust multi-level design thattraverses process-structure, structure-property and property-performance relationships; see Figure 1. IDEM includes

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parallel discrete function evaluation, Inductive DiscreteConstraints Evaluation (IDCE) based on Hyper-DimensionalError Margin Indices (HD-EMIs), and the CompromiseDecision Support Problem (cDSP) for finding the best solutionunder MSU [3-7 and 23]. The overall procedure for the IDEMis schematically illustrated in Figure 9.

Figure 9 Schematic of IDEM [3]

IDEM is exercised to identify adjustable ranges of controlfactor (design variable) values in a system with uncertaintypropagation in a design/analyses process chain and to accountfor uncertainty in downstream activities and uncertaintypropagation. With IDEM, a designer can maximize ormaintain ranges of values for design variables or performanceparameters that are shared or linked with another designersrobust design process. Thereby, design freedom is preservedfor another collaborating designer who can make changes tothe designwithin specified rangeswithout compromisingdesign requirements.

IDEM facilitates multi-level design, the management ofuncertainty inherent in the models and the propagation ofuncertainty through the design process chain shown in Figure4. In IDEM, we deal with the propagation of uncertainty indesign and analysis modules that constitute the design processchain for a particular application. We start with performanceand traverse sequentially to process; see Figure 1. At eachlevel we identify a ranged set of feasible specifications.

IDEM, embodies the concept of Error Margin Indices (EMIs).EMIs are indicators of the degree of reliability of a decisionthat it will satisfy the prescribed system constraints or bounds.The procedure for obtaining the EMI is as follows: (a) obtainthe upper and/or lower deviation of a response (URL andLRL) and (b) calculate the EMI from this deviation. The EMIis calculated by including the response mean (y) andupper/lower deviations (Yupper and Ylower) from a combineddistribution of a system model and error bounds. The EMIincludes the response deviations of a system model due tovariations in design variables and the response deviations oferror bounds as well as the system model. The mathematicalformulations of EMI corresponding to a goal i are:

( ( )) / i i i upper EMI URL f x Y = for minimization problems;

( ( ) ) / i i i lower

EMI f x LRL Y = for maximization problems;

i i i i

i

i i

| f URL | | f LRL |EMI Min{ , }

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

=

=

Y

Y

upper upper y

lower y lower

Y

Y

=

=

As shown in Figure 9, the objective is to find the best rangedset of design specifications in the space x consideringuncertainty in mapping functions (f) and propagateduncertainty through a design process. IDEM involves findingranged sets of design specifications by passing feasiblesolution spaces from performance requirements by way of aninterdependent response space to the design space whilepreserving the feasible solution space as much as possible. Theprocedure includes the following steps [3]. Step 1: Conduct parallel discrete function evaluation:

Define rough design and performance spaces (hyper

dimensional x, y, and z spaces) and generate discretepoints in each of these spaces.

Evaluate the generated discrete points using themapping models (fand g in Figure 9) that include alquantified amount of uncertainty.

Store the evaluated data sets, including discrete inputpoints and output ranges, in a database.

Step 2: Inductive Discrete Constraints Evaluation (IDCEprocess: Using information from Step 1, sequentiallyidentify feasible regions in y and x spaces with a giveninitial requirement range in z space

Step 3: Solve the Compromise Decision Support Problem(cDSP): Find the best robust solution under MSU by

performing Step 2 with adjusted HD-EMIs.As HD-EMI increases for a particular model, the output rangemoves farther from the constraint boundary. This means thedecision becomes more reliable under potential shift of theoutput range due to MSU. In the IDCE process, thespecifications, the performance ranges and the initial HD-EMvalues for the discrete constraint evaluation are specified bythe designer. To determine the best solution among feasiblesets of solutions the required HD-EMIs for each space shouldbe anchored in statistics. Values of HD-EMIs are important indetermining the most desirable robust solution against modestructural uncertainty, because HD-EMIs represent the amounof margin for potential errors in the mapping models for

estimating output range. A designer may leave wider marginsbetween an output range and constraint boundaries byincreasing the HD-EMI for the mapping model whose MSU islarger than others. An additional constraint is that all HD-EMIs should be greater than or equal to one so that the entireoutput range can satisfy the constraints Depending on thevalue of required HD-EMI, the identified feasible range maybe large, small, or unattainable. The solution strategy for thiapplication is outlined in the next section.

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3. SOLUTION STRATEGY USING IDEMThe solution strategy for this application is illustrated inFigure 9. The modeling in MODULE 2 has presented manychallenges and these have yet to be resolved. Hence, it isbypassed in illustrating our method via this application.

Figure 10 Modules Used in This Application

In Figure 10,f1, f3, f4, f5, f7, f8 and f9 represent the theoreticalor empirical models considered at the different levels ofdesign. The inputs to MODULE 1 are the volume fraction ofTiB2 (xTiB2) and temperature of processing in degree K (T).Theoutput of MODULE 1 (f1) is the average TiB2 particle size (dp)which is one of the inputs to MODULE 4. The independentinputs to MODULE 3 are volume fraction of TiB2 (xTiB2) andpercentage of liquid in processing (%L) and the output ofMODULE 3 (f3) is the average grain size (d) ofmicrostructure. MODULE 4 receives inputs from the outputsof MODULE 1 and 3 along with the independent inputs of

volume fraction of TiB2 (xTiB2) and temperature of semi-solidprocessing (temp). MODULE 4 deals with the structure-property relationships andf4 gives the density () [eq. 3]andf5gives yield stress ( [eq. 2]) as outputs. Finally, MODULE 5deals with the property-performance relationship of thedeveloped microstructure and f7 evaluates the performancevariable of depth of operation (h), f8 evaluates the weight ofthe outer shell (W) andf9 evaluates the time of operation (Topr)of the submersible. The independent parameter in this level ofdesign is the thickness of the shell (t) and the dependentparameters are density () and yield stress ().

The solution scheme for this application is illustrated in Figure11. We observe that the that the feasible design spaces areinductively passed from MODULE 5 to MODULE 4 andsubsequently to MODULES 3 and 1 of design.

We note that the volume fraction of TiB2 is an input toMODULE 1, MODULE 3 and MODULE 4 of design. Theresponses of MODULE 1 and MODULE 3 are influenced bymultiple variables and hence we use response surfacemethodology for modeling and analysis of the design task at

these levels. The Response Surface Methodology employedembodies second order models [20]:

k k2

0 i i ii i ij i j

i 1 i 1 i k

Y x x x x = =

As Idetc2009

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