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Xiaoping Qiane-mail: xpqian@umich.edu

Debasish Duttae-mail: dutta@umich.edu

Department of Mechanical Engineering,The University of Michigan,

Ann Arbor, MI 48105

Physics-Based Modeling forHeterogeneous ObjectsHeterogeneous objects are composed of different constituent materials. In these omaterial properties from different constituent materials are synthesized into oneTherefore, heterogeneous objects can offer new material properties and functionaThe task of modeling material heterogeneity (composition variation) is a critical issuthe design and fabrication of such heterogeneous objects. Existing methods cannciently model the material heterogeneity due to the lack of an effective mechaniscontrol the large number of degrees of freedom for the specification of heterogenobjects. In this research, we provide a new approach for designing heterogeneous obThe idea is that designers indirectly control the material distribution through the bouary conditions of a virtual diffusion problem in the solid, rather than directly in the natCAD (B-spline) representation for the distribution. We show how the diffusion probcan be solved using the B-spline shape function, with the results mapping directlyvolumetric B-Spline representation of the material distribution. We also extendmethod to material property manipulation and time dependent heterogeneous objecteling. Implementation and examples, such as a turbine blade design and prosthessign, are also presented. They demonstrate that the physics based B-spline momethod is a convenient, intuitive, and efficient way to model object heterogeneity.@DOI: 10.1115/1.1582877#

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1 IntroductionHeterogeneous objects are made of different constituent m

rials and/or have continuously varying material composition thproducing gradation in their material properties. They are sotimes known as functionally gradient materials~FGM!. In theseobjects, different materials can be mixed and varied to satmultiple or even conflicting design requirements, and they posmany desired material properties, which cannot be obtaiotherwise.

Over the past decade, the uses of heterogeneous objectsgrown into many fields. When the FGM concept was first intduced, two dissimilar materials such as metal and ceramic wcombined to relieve thermal stress. Now the applications of herogeneous objects span from aerospace, nuclear energy, cheplant, and biomedical engineering, to general commodities.example, a graded interface for bone in orthopedic implantshown in Fig. 1. Conventional methods of fixing an artificial boand joint prosthesis to bone include the following steps: total clcontact of the prosthesis to the bone, direct mechanical fixawith screws or spikes, and filling the space between the prosthand bone with polymethylmethacrylate~PMMA! bone cement.However, this causes pain to the patient during weight beasituations such as walking because there is micromotion ofprosthesis within the bone, and subsequently the prosthesiseven loosen in the bone. As a result, the breaking of such fixafrequently occurs. A more effective method for adhering prostsis to the bone is to coat it with a porous metal because new bingrowth into the pores occurs after implantation. A graded laof hydroxyapatite~HAp! is coated on the porous metal. It bondsthe bone physicochemically, thereby increasing the adhestrength and rate of binding to the bone. Therefore, porous mwith a HAp coating remedies the big drawbacks of cementlprosthesis. It prevents pain to the patient while walking cauby micromotion or loosening of a prosthesis fixed withobone cement, and also allows weight bearing earlier aimplantation@1#.

Contributed by the Design Automation Committee for publication in the JOUR-NAL OF MECHANICAL DESIGN. Manuscript received Aug. 2001; rev. Nov. 200Associate Editor: J. E. Renaud.

416 Õ Vol. 125, SEPTEMBER 2003 Copyright

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Figure 1 is a schematic structure of such an FGM interfaThis FGM region is composed of porous titanium plus hydroxypatite ~HAp!. Ti has good mechanical toughness and HAp hgood biocompatibility. The uniform combination of Ti and HAwould cause bio-incompatibility and weakened strength due tomaterial property differences. Such material property differenare resolved by using a mixture of Ti and HAp with varyinproportions. The sharp interface between Ti and HAp is elimnated through a graded zone of Ti/HAp. The bending strengththe resulting material is similar to human bone. The figure ashows typical variation in properties due to the variation in marial composition at the FGM region.

As evidenced in the prosthesis design example, heterogenobjects have many advantages over objects composed of unimaterials. Such advantages have brought to the forefront thesearch on heterogeneous objects realization. In the last decoptimal design techniques such as the homogenization demethod have been developed which create just such heterneous objects@2#. Also, in the last few years, layered manufactuing ~LM ! has matured significantly and is capable of fabricatiheterogeneous objects@3#. In layered manufacturing, a part is buiby selectively depositing materials layer-by-layer under compucontrol. A critical link between the design and fabrication of suheterogeneous objects is heterogeneous object modeling, acreating the heterogeneous object model that can be used fodesign, analysis and fabrication of heterogeneous objects.

The recent research on heterogeneous object modeling hasprimarily focused on the representation schemes, i.e., using mematical model and computer data structures to represent theometry and material composition of heterogeneous objects. Notheless, limited means are available for specifying and controlthe material composition in the heterogeneous object model.of the main challenges lies in the fact there exist a large numbedegrees of freedom to completely define a heterogeneous obFor example, for a 3d object with m types of materials andnnumber of variations for each material, the modeling space isE3

3nm dimensional. To specify material composition within suchcomplex space, an effective and efficient method is crucial.

This paper presents a diffusion process based method for.

© 2003 by ASME Transactions of the ASME

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erogeneous object modeling. It provides a convenient, intuitand efficient way to specify material compositions within the herogeneous objects.

In the rest of this paper, Section 2 reviews the previous reseaon heterogeneous object modeling. Section 3 presents the vmetric B-spline representation for heterogeneous objects. Sec4 describes the mathematical model for a diffusion process. Stion 5 details the mathematical formation for the diffusion basB-spline heterogeneous object modeling. Section 5.2 extendsmethod to time dependent heterogeneous objects. Method forposing constraints on the diffusion model is presented in Sec6. The implementation and examples are shown in Section 7.nally, the paper is concluded in Section 8.

2 Previous Research. Kumar and Dutta proposed R-m secould be used for representing heterogeneous objects@4#. Jacksonproposed another modeling approach based on subdividingsolid model into sub-regions and associating the analytical coposition blending functions with each region@5#. Some othermodeling and representation schemes, utilizing either vomodel, implicit functions or texturing, have also been propos@6–8#.

Unlike the above research focusing on geometry/material rresentation schemes, in this paper, we specifically focus onmethods for specifying material variation. In each of the afomentioned representation schemes, they used different modemethods. In terms of material specification, we classify themethods into the following four categories: control points basanalytical functions based, implicit functions based, and vobased modeling methods. It should be noted that these method

Fig. 1 Schematic structure of an FGM interface withinprosthesis

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not necessarily associate with a particular representation schFor example, both control points based and analytical functibased methods can be applied in R-m set based represenscheme for heterogeneous objects. Therefore, the advantagedisadvantages of heterogeneity specification methods do not nrally reflect the advantages and disadvantages of each represtion scheme.

A comparison of these object heterogeneity specification mods is shown in Table 1.

In this table, model coverage refers to the coverage of geomand material variation each method can model. The convenierefers to the convenience level for users to specify and changematerial heterogeneity.

A control point based method models the object heterogenby specifying values of a set of control points and interpolatthem with the shape functions, such as B-spline@9#, Bezier@5,10#,or NURBS@6#. Due to the large number of control points, suchmethod has excellent representation coverage, but for the sreason it is not convenient to use.

An analytical function based method represents the objecterogeneity by explicit functions rather than control point basfunctions@4,11#. This method is particularly popular in functionally gradient materials research community where an expfunction, e.g., linear, parabolic, or exponential, is used to reprethe volume fraction at each point of an object@12#. Using such amethod, it is easy to manipulate the material composition buhas limited model coverage.

A voxel based method represents material heterogeneityterms of discretized voxel in the object. For example, voxel-bamulti-material object is proposed in@6#. Such a method has advantages, such as relatively wide representation coverage.also good for visualization. However, due to its large size of voels in 3 dimensional objects, it is inconvenient and inefficientmanipulate and control the heterogeneity.

An implicit function based method models the heterogeneitythe form of implicit functions. i.e.,f (m)50. For example, im-plicit functions can be constructed for an R-function method@7#.For complicated or arbitrary geometry boundary, it is difficultconstruct such implicit functions. Due to the implicit nature, sumethods are not good for anticipating the heterogeneity variaduring design modifications, a task that needs to be supporte

Therefore, the current methods for specifying material comsition face a trade-off between the model coverage and operaconvenience. To obtain overall desirable performance in mocoverage, convenience and efficiency of heterogeneous obmodeling, we propose physics-based heterogeneous object ming. In this paper, we use a virtual diffusion process to modelheterogeneity at each point. Mathematically, we extendB-spline to represent the heterogeneous objects. We use theond order differential equation for the diffusion process to genate the material composition distribution. Conceptually, onlyfew parameters, carrying physical implications, and constraare used to intuitively manipulate the material composition.

For the interests of simplicity, we refer to object heterogeneas material composition in this paper, even though the metpresented in this paper applies to other object heterogeneitwell, such as temperature distribution and stress field. It shouldnoted that this work was in part inspired by the physics bamodeling of geometry@13–15#, where the Lagrange dynamimodel is used for modeling the geometry deformation.

Table 1 Comparison of heterogeneity modeling methods

Methods Model Coverage Operation Convenien

Control points Excellent PoorAnalytical functions Poor ExcellentVoxel Excellent PoorImplicit functions Poor Excellent

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418 Õ Vol. 125,

Fig. 2 Tensor product B-spline volume

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3 B-Spline Representation for Heterogeneous ObjectsTensor product solid representation has been widely use

computer aided geometric design community. Under the conof heterogeneous objects, relevant proposals based on tensoruct volumes have also been reported@5,6,9#.

In this paper, B-spline tensor representation is used to reprematerial composition and material properties in heterogeneousjects. It is also extended to represent the time dependent hegeneous objects. We choose B-spline representation for heteneous objects simply to shorten the computational time. Therno technical difficulty to extend the methodology to a NURBvolume.

3.1 B-Spline Tensor Solid Representation for Heteroge-neous Objects. For each point (u,v,w) in the parametric do-main of a tensor product B-spline volumeV ~Fig. 2!, there is acorresponding pointV(u,v,w) at Cartesian coordinates (x,y,z)with material compositionM, noted as (x,y,z,M ). We define sucha B-spline volume as:

V~u,v,w!5(i 50

n

(j 50

m

(k50

l

Ni ,p~u!Nj ,q~v !Nk,r~w!Pi , j ,k (1)

WherePi , j ,k5(xi , j ,k ,yi , j ,k ,zi , j ,k ,Mi , j ,k) are control points for theheterogeneous solid volume.Ni ,p , Nj ,q, and Nk,r are thepth-degree,qth-degree, andr th-degree B-spline functions definedthe direction ofu, v, w respectively. For example,Ni ,p(u), thei-th B-spline basis function ofp-degree~order p11), is definedas:

Ni ,0~u!5H 1 if ui<u<ui 11

0 otherwise

Ni ,p~u!5u2ui

ui 1p2uiNi ,p21~u!1

ui 1p112u

ui 1p112ui 11Ni 11,p21~u!

3.2 Representation for Material Properties of Heteroge-neous Objects. Due to the material gradation in heterogeneoobjects, the material properties also exhibit variation. For desers, controlling heterogeneous object’s properties is more usand intuitive than controlling the material composition. In ordercontrol material properties, an effective representation is need

The relationship between material properties and the comption has been extensively studied@12#. For example, Eq.~2! andEq. ~3! give the approximate relationships of thermal conductities and mechanical strengths versus material compositionthese equations,Ma , Mb are volume fractions of two compositmaterials at each point,la , lb are the thermal conductivities, anSa , Sb are strengths for two materialsa andb, respectively.

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l5laMa1lbMb1MaMb

la2lb

3/~lb /la21!1Ma(2)

S5Sa•Ma1Sb•Mb (3)

Such property variations can be tailored to achieve useful futionality under various loading conditions as evidenced in mafunctionally gradient materials. We generalize the relationshipmaterial property versus material composition as follows:

E5 f ~Ea ,Eb ,Ma ,Mb! (4)

whereE is the material property of interest,Ea andEb are mate-rial properties of materiala and b and Ma , Mb are volumefractions.

Combining the material property equation and Eq.~1!, we canhave the B-spline representation for material properties:

E~u,v,w!5(i 50

n

(j 50

m

(k50

l

Ni ,p~u!Nj ,q~v !Nk,r~w!Ei , j ,k (5)

whereEi , j ,k is material property at each control point. It can bobtained from Eq.~4!.

In this paper, we assume the same set of geometric conpoints can be used for both material composition model and perty model. For a complex heterogeneous object, a single appmation function often does not hold. Several sub-functionsneeded to describe the relationship between property and volfraction. However, with the existing rich algorithms for contrpoints and knots inserting for B-spline representation, the sameof control points can still be used to represent both material coposition and property variation. Therefore, the assumptionholds.

3.3 Time Dependent Heterogeneous Objects.TheB-spline heterogeneous object representation can also be exteto represent time dependent heterogeneous objects. Time dedent heterogeneous objects are useful in at least the followingtypes of applications:~1! They can simulate dynamic physicaprocesses where material composition changes over time. Foample, when a bio-implant is inserted into a human body, itgrades over time, or takes a different shape over time.~2! Timedependent heterogeneous objects can offer a spectrum of hegeneity variations over time for one heterogeneous object. Fsuch a spectrum of heterogeneity, designers can choose a dematerial profile.

We represent the time dependent heterogeneous objecEq. ~6!:

V~u,v,w,t !5(i 50

n

(j 50

m

(k50

l

Ni ,p~u!Nj ,q~v !Nk,r ~w!Pi , j ,k~ t ! (6)

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The control pointsPi , j ,k(t)5(x(t),y(t),z(t),M (t)) are now func-tions of timet. So the volumeV(u,v,w,t) becomes a time dependent heterogeneous object.

4 Mathematical Model for Diffusion ProcessesWith the B-spline as the representation for heterogeneous

jects, we can now proceed to the investigation of material coposition distribution generation. In this section, we describe hovirtual diffusion process generates different material composiprofile. Diffusion is a common physical process for the formatiof material heterogeneity:

• in integrated circuit fabrication, diffusion has been the pmary method for introducing impurities such as boron, phphorus, and antimony into silicon to control the type of mjority carrier and the resistivity of layers formed in the waf@16#.

• in biological mass transport, diffusion is a typical waytransport the solute across the membrane due to the dience in chemical potential for the solute between two sidethe membrane@17#.

• in the drug delivery from a polymer, the drug release candescribed in most cases by diffusion@18#.

In all these diffusion processes, the volume ratios~or particleconcentration in some context! throughout the problem domaiare controlled to be certain profiles to achieve the respectivejectives. Such is the rationale we chose the diffusion procesthe underlying physical process for the heterogeneous object meling. We can use a virtual diffusion process to intuitively contthe volume ratios in the heterogeneous objects.

The mathematical modeling of controlled material compositin these processes is based on the Fick’s laws of diffusion.

Fick’s first law of diffusionstates that the particle flow per unarea,q ~called particle flux!, is directly proportional to the concentration gradient of the particle:

q52D•

]M ~x,t !

]x(7)

whereD is the diffusion coefficient andM is the material compo-sition ~particle concentration!.

Fick’s second law of diffusioncan be derived using the contnuity equation for the particle flux:]M /]t52]q/]x. That is, therate of increase of concentration with time is equal to the negaof the divergence of the particle flux. Combining these equatiowe have Fick’s second law:

]M

]t5D•

]2M

]x2(8)

For a given volume, the amount of material is conserved durthe diffusion process. In other words, the rate of change ofmaterial M within the volume must equal to the local productiof M within the volume plus the flux of the material across tboundary. Mathematically,

d

dt EVMdV5E

VQdV2E

SqinidS (9)

whereQ is the material generated per unit time.Applying Fick’s laws into the above equation and using t

divergence theorem, we have d/dt*VMdV5*VQdV2*V]qi /]xjdV. After dropping the integral over volume, whave

dM

dt5Q1

]

]xiS Di j •

]M

]xjD (10)

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5 Diffusion-Based Heterogeneous Object ModelingIn this section, we combine B-spline representation and di

sion equations and derive the equations for diffusion baB-spline heterogeneous object modeling.

To model a heterogeneous object using a diffusion processfirst concentrate on the steady state of the diffusion proces~astatic volume!. We then extend it to the time dependent object

In a static volume, there is no material change over time, idM/dt50. So Eq.~10! becomes

Q1]

]xiS Di j •

]M

]xjD50 (11)

which has essentially the same format as many other steady-field problems governed by the general ‘‘quasi-harmonic’’ eqution, the particular cases of which are the well-known Laplace aPoisson equations. The range of physical problems falling ithis category is large. To name a few, there are heat conducand convection, seepage through porous media, irrotationalof ideal fluid, distribution of electrical or magnetic potential, antorsion of prismatic shaft@19#.

5.1 Finite Element Approximation for Steady StateEquation. If we abbreviate Eq.~1! asV(u,v,w)5N•P, and wesubstitute the material composition M component from Eq.~1!into Eq. ~11!, we have the diffusion equation for the B-splinmodel:

Q1]

]xiS Di j •

]~N•P!

]xjD50 (12)

For any given B-spline volumeV, there are (n11)3(m11)3( l 11) control points. In Eq.~12!, we assume the position(x,y,z) of each control point for the solidV is given. Our objec-tive is to calculate the material compositionM at each point. Thatis, there are (n11)3(m11)3( l 11) degrees of freedom foEq. ~12!.

Solving Eq.~12! leads to the solution to material compositiofor a heterogeneous object. Direct analytical solution for sucsecond order differential equation is difficult to obtain. Instead,use finite element technique to solve the above equation. Wdiffers this method from most standard FE methods are:~1! Theunknowns are control points rather than points on the volumeshown in Fig. 3, the points on the B-spline volume and the conpoints are different.~2! The shape functions are B-spline raththan typical FEM shape functions.~3! The elements are formedaccording to the knot span in the parametric domain as opposeby an extra meshing process. The following is a brief presentaof finite element approximation for diffusion based B-spline herogeneous object modeling.

5.1.1 Weak form of Quasi-Form Equation.We present Eq.~12! in the following form for finite element approximation:

Governing field equation: Q1]

]xiS Di j •

]M

]xjD50 (13)

~ i! Forced boundary condition: f5f0 on G1 (14)

Fig. 3 B-spline control point and point on the B-spline volume

SEPTEMBER 2003, Vol. 125 Õ 419

420 Õ Vol. 125,

Fig. 4 Element in B-spline solid

-

irFt

d

,

is

s

the

e

:

~ ii ! Natural boundary condition: qn5qW •n̂ on G2 (15)

To create the finite element approximation for Eq.~13!, weneed to convert the partial differential equation into its weak forThe 2nd order differential equation can be turned into its weform for an arbitrary functionC by using integration by parts.

EV

Di j ~¹C!•]M

]xjdV52E

G2

CqndG1EV

CQdV (16)

5.1.2 Galerkin Approximation. By the Galerkin approxima-tion, the weighting functionC is defined with the aid of the element interpolation function~B-spline function! Na such thatC5NaCa , whereCa is the value of the functionC at the controlnodea. So we have

EV

Na,iDi j

]M

]xjdV5E

VNaQdV2E

G2

NaqndG (17)

5.1.3 B-Spline Heterogeneous Volume Discretization.Werepresent the B-spline heterogeneous solid as a set of elemwhere each element is associated with a set of degrees of freeThe element boundaries are defined by the known locations wthe underlying B-spline basis functions and the degrees of fdom are the B-spline’s geometric and material control points.example, in Fig. 4, a cubic in parametric domain corresponds3d volume in the physical coordinate system.

Let us note

M ~u,v,w!5(i 50

n

(j 50

m

(k50

l

Ni ,p~u!Nj ,q~v !Nk,r~w!Mi , j ,k (18)

asM (u,v,w)5NMi , j ,k . Here we letN be the shape function, anM be the material composition control points.

Through finite element approximation, Eq.~17! becomes

F EV

]Nam

]xi•Di j •

]Nbm

]x jdVGM5E

VNa

mQdV2EG2

NamqndG

(19)

whereM is control pointMi , j ,k5(x,y,z) i , j ,k .In matrix form, Eq.~19! becomes

KM5BW 2SW (20)

where

KeN3N

5FkEVe

S ]Ni

]x•

]Nj

]x1

]Ni

]y•

]Nj

]y1

]Ni

]z•

]Nj

]z DdVG(21)

BeWN31

5F EVe

NmQdVG , SeW

N315F E

Ge

NmqndGG (22)

SEPTEMBER 2003

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With functionQ andq interpolated in terms of its nodal valueswe have Be

WN31

5@*VeNi

mNjmdV#Qj , and Se

WN31

5@*GeNi

mNjmdG#qj .

Ke is the element stiffness matrix, andBeW is the element body

force andSeW is the element surface force.

It is not difficult to prove that the element stiffness matrixsymmetric and positive definite. Solving Eq.~20! would give theunknown valuesMi , j ,k—the material composition control pointin the solid.

5.1.4 Transformation of Differential Operator.To calculatethe partial differential equation, e.g.,]Ni /]x in the matrixK, weneed to use differential property of B-spline basis function andchain rule of partial differential.

Let P5( i 50n ( j 50

m (k50l Ni ,p(u)Nj ,q(v)Nk,r(w)Pi , j ,k , from

B-spline basis function property@20#, we know

5]N

]u5(

i 50

n

(j 50

m

(k50

l

Ni ,p8 ~u!Nj ,q~v !Nk,r~w!

]N

]v5(

i 50

n

(j 50

m

(k50

l

Ni ,p~u!Nj ,q8 ~v !Nk,r~w!

]N

]w5(

i 50

n

(j 50

m

(k50

l

Ni ,p~u!Nj ,q~v !Nk,r8 ~w!

According to the chain rule of partial differential equation, wcan write

5]N

]u5

]N

]x•

]x

]u1

]N

]y•

]y

]u1

]N

]z•

]z

]u]N

]v5

]N

]x•

]x

]v1

]N

]y•

]y

]v1

]N

]z•

]z

]v]N

]w5

]N

]x•

]x

]w1

]N

]y•

]y

]w1

]N

]z•

]z

]w

If we note Jacobian matrix as

J5]~x,y,z!/]~u,v,w!5@]x/]v]x/]w

]x/]u

]y/]v]y/]w

]y/]u

]z/]v]z/]w

]z/]u

#,

we have

F ]N

]u]N

]v]N

]w

G5F ]x

]u

]y

]u

]z

]u

]x

]v

]y

]v

]z

]v

]x

]w

]y

]w

]z

]w

G •F ]N

]x

]N

]y

]N

]z

G5J•F ]N

]x]N

]y]N

]z

G (23)

So ]N/]x, ]N/]y, ]N/]z can be obtained from the following

Transactions of the ASME

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n-d

F ]N

]x]N

]y]N

]z

G5J21•F ]N

]u]N

]v]N

]w

G (24)

5.2 Time-Dependent Heterogeneous Objects.In Eq. ~11!,we dropped the time dependent itemdM/dt to model the hetero-geneous objects. However, as described earlier, there are stions where the object heterogeneity varies over time. In this stion, we extend the diffusion equation to model time dependB-spline heterogeneous objects.

The most commonly used time integration method for eqtions such as Eq.~10! is normally referred to as theu method. Itapproximates the time derivative by the difference:

dM

dt'

1

Dt~Mn112Mn! (25)

whereMn5M (x,y,z,tn) is the material heterogeneity at the timtn . The heterogeneity can be defined by the relaxation paramu, i.e.,

M5uMn111~12u!Mn (26)

u is a value between 0 and 1 and is used to control the accuand stability of the algorithm. Substituting the two equations inEq. ~10!, we have

F 1

DtC1KGMn115F 1

DtC2~12u!KGMn1uQn111~12u!Qn

(27)

with C5*VcNiNjdV wherec is a coefficient for controlling thespeed of heterogeneity variation over time.

Therefore, for any given initial condition, Eq.~27! gives a one-step method for time-dependent heterogeneous object modeli

theing

ow-

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6 Heterogeneous Object Modeling by ImposingConstraints

The above formulation has provided a methodology to calcuthe material composition for a diffusion process. It can be genalized to manipulate material composition of B-spline heteroneous solid objects by imposing constraints, i.e., boundary cotions. The constraints that are imposed on the B-solid includeheterogeneity information on the boundary, or the heterogeneispecific location (u,v,w), or any other type of constraints that cabe transformed into a set of equations.

Equation~20! can be mathematically transformed into a lineaconstrained quadratic optimization problem:

minp

I1

2MTKM2MTBI (28)

whereP is the control point set for the B-spline solid.Here, we consider a set of linear constraints:

A•M5E (29)

To accommodate the constraints in Eq.~29!, solution methodsgenerally transform this to an unconstrained system@21#:mini1/2M̄TKM2M̄TB̄i , in which solutionsM̄ , when transformedback toM, are guaranteed to satisfy the constraints. The uncstrained system is at a minimum when its derivatives are 0, twe are led to solve the systemKM5B̄.

Specifically, we introduce a Lagrange multiplier for each costraint row Ai , and we then minimize the unconstraineminpi1/2MTKM2MTB1(AM2F)Gi . Differentiating with re-spect to M leads to the augmented system:

FK AT

A 0 G•FMG G5FBF G (30)

Solving the above linear equations leads to the solution toconstrained system. Note, in this paper, for the sake of savcomputational time, only linear constraints are considered. H

Fig. 5 Flowchart of Physics based heterogeneous object modeling

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Fig. 6 Material composition from a diffusion process

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ever, it is not difficult to generalize the method to accommodnonlinear constraints by enforcing Lagrange multipltechniques.

7 Implementation and Examples

7.1 Prototype System. A prototype system of a diffusionprocess based heterogeneous object modeling is implementedSUN Sparc workstation. Figure 5 shows the flowchart ofphysics-based B-spline heterogeneous object modeling proThe input of the system is a B-spline solid, consisting of a secontrol points with known geometry. The output is the matervariation at each control point. The user interacts with systemtwo ways. First, the user can change system parameters, suQ, the material source~material/unit time! and D, the materialdiffusion coefficient. Second, the user can impose constraints.two types interaction processes continue until the user is satiswith the result. When constraints are changed, the system matremain the same. Only when the system properties are chanshould the system stiffness matrix and body force matrix becalculated.

Corresponding to the flowchart in Fig. 5, this system includthe following modules: input module, display module, matrix cculation module, and equation solving module.

The input for the system is a B-spline solid, which gives tcontrol points of the volume, and B-spline knot vectors. At eacontrol point, the material value may or may not be known. If it

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it serves as an initial value. If not, the system can automaticderive it when the users interact with the system. The data stture for the B-spline volume also records the system parameand imposed constraints.

The display module essentially discretizes the B-spline voluinto a field model and connects the output to the Applicationsualization System@22#, which displays the resulting geometrand material heterogeneity.

The matrix calculation module includes the element matrix cculation and assembly processes. In its implementation, to athe ambiguous physical implication ofq under the context of het-erogeneous object modeling, we substituted natural boundaryforced boundary. That is, we only calculateK, B, andC ~for thetime dependent heterogeneous objects!.

Gaussian integration is used in the element matrix calculatFor example, assuming the parametric domain of the elemen@u0 ,u1#3@v0 ,v1#3@w0 ,w1#, theK matrix for each element is

Ki j 5Eu0

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Gaussian integration is used trice to calculate the abintegration. That is, the above equation under some tra

Fig. 7 Heterogeneous object modeling by diffusion equations

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Fig. 8 Manipulating both geometry and material composition

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whereI, J, andK are the number of Gaussian points inu, v, andw directions. Given these numbers, we can find Gaussian weiwi , wj , wk and abscissasj i , h j , zk .

Matrix calculation is a time consuming process. Since allmatrices are symmetric~and positive definite!, only half of theelements in the matrices are needed.

All the element matrices are then assembled and turned inset of linear equations. An equation solver then solves the etions and gives the control point values for the B-spline volum

7.2 Examples. In all of the following examples, the diffu-sion coefficientD are assumed to be uniform and has a unit valThe color variation in the pictures reflects the material compotion variation.

Example 1: Diffusion processesTwo diffusion processes are shown in Fig. 6, one with conc

tration source from top face, one with concentration source frtop/right edge. We obtained the heterogeneous objects by iming the concentration source constraints on the face and therespectively.

Example 2: Modeling heterogeneous objects by diffusionequations

Figure 7 shows two examples of heterogeneous objects meled by changing system parameters and imposing constrainboth Figs. 7~a! and 7~b! two B-spline volumes undergo differenconstraints, which lead to different material heterogeneity disbutions. In the left of Fig. 7~a!, two boundary face constraints anone point constraint are imposed. In the right of Fig. 7~a! theseconstraints are replaced with a new point constraint. SimilaFig. 7~b! shows how a turbine blade model changes under difent constraints. Figure 7~b! also includes a meshed model for thturbine blade.

Example 3: Manipulating both geometry and materialcomposition

Figure 8 shows an example of changing both geometrymaterial composition. In the top of Fig. 8 is an initial B-splinsolid, imposed with constraints on two boundary surfaces. Fig

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8~a! is the result. We can change the geometry of the solid andthe new solid in Fig. 8~b! and then impose composition constraints. This leads to a new solid in Fig. 8~d!. We can also imposethe material composition constraints first~Fig. 8~c!! and then ma-nipulate the geometry~Fig. 8~d!!. This alteration of geometry andmaterial composition manipulation sequence leading to the sresult demonstrates the robustness of the diffusion based hegeneous object modeling method.

Example 4: Torsion bar modelingAn optimal torsion bar was modeled by the system~Fig. 9!. It

has been proved that the cross- section of a torsion bar like Fgives the optimal rigidity. In this case, there are two materials wtwo different elastic moduli.

During the modeling of this torsion bar, four constraints aimposed at the four corners and one constraint is imposed acenter. By adjusting the constraint values and the system paeters, one can get the desired profiles of the volume fraction oftwo materials.

Example 5: Time dependent heterogeneous object modelingFigure 10 shows a time dependent heterogeneous object. F

Fig. 9 Cross section of torsioned bar designed to maximizethe rigidity

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Fig. 10 Time dependent heterogeneous object

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10~a! shows the initial state and Fig. 10~h! the final state. Basedon Eq.~27!, such time-dependent heterogeneous objects are amatically derived.

Example 6: Heterogeneous object design„turbine blade…An ideal turbine blade is designed to possess the follow

properties: heat resistance and anti-oxidation properties on

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high temperature side, mechanical toughness and strength olow temperature side, and effective thermal stress relaxathroughout the material@23#.

A heterogeneous object~turbine blade! with materials, SiC andAl6061 alloy, is shown in Fig. 11. The thermal conductivitiesthe two materials are 180 W/mK and 25 W/mK. The strengths

Fig. 11 Turbine blade

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Journal of Mechanical

Fig. 12 Turbine blade after modification

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1145/2145 Mpa and10/28300 Mpa. Using Eq.~2!, Eq. ~3! andEq. ~5!, we can have the thermal conductivity and tensicompression stress for each control point. These propertiesrespectively shown in Fig. 11. Figure 11 also shows the valuethe tip. Note, the notationa/(b,c) in the figure means the value athe tip point isa while the minimal value of the whole volume ib, and the maximum value isc.

Suppose the designer is not satisfied with the strength at thof turbine blade, the designer can choose to strengthen the tiimposing constraints at the tip. The revised model is shown in12, where the thermal conductivity has been changed from 2140.94, tensile strength from 0 to 116.92 and compressstrength from 8300 to 1724.37.

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Using this method, the designers directly interact with the stem with familiar concepts~the material properties! rather thanmaterial composition. We believe this direct quantitative feedbof material properties is particularly useful for a designer durthe design evolution process.

Example 7: Constructive design of a prosthesisPhysics based B-spline volume provides an efficient way

model object heterogeneity. It can not only model material gration within each separate volume. It can also be extended toconstructive design approach for complex heterogeneous obdesign. The following example of a prosthesis design demstrates such a design process.

Figure 13 shows a flowchart for the prosthesis design proc

Fig. 13 Flowchart of design process for a new prosthesis

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Fig. 14 Graded interface within a prosthesis

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It is a flowchart enhanced from the current development proc@23#. In our approach, starting from design functions, users sematerials and form the heterogeneous material features, eacwhich is a B-spline volume. A feature combination algorithcombines these features into a heterogeneous object@24#. Afterthe mechanical and biological properties are known from thetabase for each individual material, these properties at each pin this prosthesis can then be evaluated. If users are not satiwith the properties, they can select new material for each voluor change volume fractions. These steps of changing materiaeach feature in a constructive process form a feature based dprocess. After the property evaluation, property in vitro testsanimal tests are then conducted.

In the example of Fig. 14 is a prosthesis designed followingflow chart in Fig. 13. Titanium is selected as the base materialto its better overall mechanical strength and biocompatibilGraded pore is needed for the bone ingrowth into Titanium sotitanium can be bonded to bone. Graded HAp helps to bondbone chemically to titanium. It also precludes the pain for patidue to a connectivity tissue membrane between the Titaniumbone. Each of these design intents is represented as a sepB-spline volume~heterogeneous feature!, such as in Region 2 and

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Region 7 in Fig. 14. In these two regions, pore and HAp amodeled as one material, while the Titanium is the other mateOnce the volume fraction for pore and HAp is known, anothfraction is used to separate pore and HAp. This fraction is cstant throughout the region. Figures 14~a! and 14~b! show thegraded porous structure and graded HAp respectively withM pore /MHAp50.5. Figure 14~c! shows the construction historyThe partition in the construction history is similar to union opetion but with the intersection region’s material redefined. A direface neighborhood alteration method is adopted during the cstructive operations@24#.

After the heterogeneous prosthesis model is constructed,mechanical and biological properties are evaluated. We usebiofunctionality (BF) BF5sb /E, a quotient of the~bending!fatigue strengthsb for Young’s modulusE, as a characterizationof mechanical strength for biomedical purposes@25#. We havesb5550 MPa andE31035105 MPa@23#. The Young’s moduluscan be calculated fromEm5E0(121.21M2/3) for the graded re-gion, whereEm is the modulus of the graded material andE0 isthe modulus of the bulk material. If we do not consider the being strength of HAp, we can calculate theBF for any point in the

Fig. 15 Variation of Young’s modulus and biofunctionality due to Q change

Transactions of the ASME

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prosthesis, as shown in Fig. 15. Modification to the material coposition can lead to different Young’s modulus andBF distribu-tion throughout the region. In Fig. 15, we show the propertvariation due to the change ofQ ~material generation source!.These values are measured at different distance points frominner surfaces of the graded regions.

This example demonstrates that the physics based B-spmodeling method not only provides an intuitive way to control tmaterial composition but also provides means to directly conthe material properties. This is an enhancement to the exisdesign methods for prosthesis design where material composdesign and material property evaluation are conducted separand sequentially.

8 ConclusionThis paper presents a virtual diffusion process based B-Sp

heterogeneous object modeling method. It enables designeuse only a few parameters to intuitively control material hetegeneity. It uses B-spline function to represent heterogeneousjects and diffusion equations to generate the material heterogity. The finite element method is utilized to solve the constraindiffusion equations with the control points of the B-spline solidunknown.

The implementation of the physics based heterogeneous omodeling demonstrates that this synergy of B-spline represetion and the diffusion process leads to a convenient, efficienteffective scheme for heterogeneous object modeling. Even furit enables the direct modeling and manipulation of material prerties of heterogeneous objects. It allows object heterogeneibe modeled in a time dependent fashion. It also supports thestructive design approach as exemplified in the example of a pthesis design.

AcknowledgmentWe gratefully acknowledge the financial support from NS

~grant MIP-9714751!. The authors are also thankful for feedbafrom the anonymous reviewers.

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