Three Dimensional Voronoi Cell Finite Element Model for Microstructures - S. Ghosh and S. Moorthy

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Three dimensional Voronoi cell finite element model for microstructures

with ellipsoidal heterogeneties

S. Ghosh S. Moorthy

Abstract In this paper a three-dimensional Voronoi cellfinite element model is developed for analyzing hetero-geneous materials containing a dispersion of ellipsoidalinclusions or voids in the matrix. The paper starts with adescription of 3D tessellation of a domain with ellipsoidalheterogeneities, to yield a 3D mesh of Voronoi cells con-taining the heterogeneities. A surface based tessellationalgorithm is developed to account for the shape and size ofthe ellipsoids in point based tessellation methods. The 3DVoronoi cell finite element model, using the assumed

stress hybrid formulation, is developed for determiningstresses and displacements in a linear elastic materialdomain. Special stress functions that introduce classicalLame functions in ellipsoidal coordinates are implementedto enhance solution convergence. Numerical methods forimplementation of algorithms and yielding stable solu-tions are discussed. Numerical examples are conductedwith inclusions and voids to demonstrate the effectivenessof the model.

Keywords Voronoi cell Finite Element Model,Tessellation, Hybrid stress formulation, Ellipsoids

1IntroductionAdvanced heterogeneous materials are increasingly find-ing more use in various engineering applications. Thematerials may be metals or alloys with microscopic pre-cipitates and pores or composites containing a dispersionof fibers, whiskers or particulates. The heterogeneities inthe matrix play an important role on the overall structuralbehavior. Robust analytical and numerical models arenecessary for predicting effective properties and stressesand strains in the microstructure for these materials. Anumber of micromechanical studies have been reported inthe literature on heterogeneous materials containing in-clusions and voids. Analytical methods for determiningstress fields around a spherical cavity in an infinite domain

have been developed in Timoshenko and Goodier [1].Sadowsky and Sternberg have analyzed stress concentra-tion around an ellipsoidal cavity using ellipsoidalcoordinates in [2, 3], and around two spherical cavitiesusing bispherical coordinates in [3]. Chen and Acrivos [5]have utilized the Boussinesq-Papkovich stress functionsfor stress analysis of an infinite domain with two sphericalcavities and rigid inclusions. Chen and Young [6] haveproposed approximations using integral equations forvoids or inclusions of arbitrary shapes in an elastic med-

ium. While these analytical micro-mechanical models arepowerful, their effectiveness is generally limited to simplegeometries and low volume fractions.

Various numerical micromechanical approaches havebeen developed for a more versatile evaluation of micro-structural stresses and strains and overall behavior.Numerical unit cell models using the finite element methodor boundary element methods, have been proposed e.g. in[7,8]. Rodin and Hwang have numerically studied a finitenumber of spherical inhomogeneities in an infinite region[9]. Recently three dimensional multi-particle models havebeen developed by Gusev [10] for elastic particle re-inforced composites using tetrahedral finite elements, and

by Michel, Moulinec and Suquet [12] using Fast FourierTransform methods. Bohm et al. [13, 14], Segurado andLlorca [15] and Zohdi [11] have developed 3D elastic-plastic models for dispersion of multiple particles in metalmatrix composites with ductile matrix. Moes et. al. [16]have developed an elegant XFEM model for 3D elasticcomposite microstructures. A software package Palmyra[17] has been developed to design composite materials andto calculate physical properties of heterogeneous materials.

The 2-D Voronoi cell finite element model (VCFEM) hasbeen developed for elastic and elastic-plastic micro-mechanical problems in composite and porous materials in[18, 19, 20], and for damage initiation in reinforced com-

posites by particle cracking in [21, 20]. The model evolvesby Voronoi tessellation of the microstructure to generate amorphology based network of multi-sided Voronoi cells,each cell containing a heterogeneity. Each cell is treated as aFEM element and requires no additional discretization.VCFEM incorporates assumptions from micromechanicstheories, as well as adaptive enhancements. This model hasbeen shown to require significantly reduced degrees offreedom compared to displacement based FEM models andare hence computationally efficient.

The extension of VCFEM to 3D is a nontrivial enterprisedue to different characteristic micromechanical solutionsand differences in geometric considerations. In this paper,

Computational Mechanics 34 (2004) 510531 Springer-Verlag 2004

DOI 10.1007/s00466-004-0598-5

Received: 20 Feburary 2004 / Accepted: 17 May 2004Published online: 20 July 2004

S. Ghosh (&)Department of Mechanical EngineeringThe Ohio State University Columbus,OH USA e-mail: [email protected]

S. MoorthyDepartment of Civil EngineeringLouisiana State University, LA USA

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a 3D VCFEM model is developed for elastic materials withellipsoidal inclusions or voids. The paper starts with dis-cussion of the Voronoi tessellation algorithm for meshgeneration, accounting for shapes, sizes and locations ofheterogeneities. The VCFEM formulation is then devel-oped introducing special stress functions in ellipsoidalcoordinates to explicitly account for shape. Variousaspects of numerical implementation of the model are

subsequently discussed with attention on stability andaccuracy of the solutions. Finally a number of numericalexamples are solved for model validation and for modelingmulti-inclusion microstructures.

2Three dimensional mesh generation by tessellationinto Voronoi cellsDiscretization of 3D heterogeneous domains, containingparticles or voids, into a mesh of polyhedra is the firststep in the development of the 3D Voronoi cell finiteelement model. Each of the Voronoi cells in this con-struct contains a single heterogeneity at most. This sec-tion describes a 3D tessellation algorithm from thegeneral definition of Voronoi diagrams. The corre-sponding 2D algorithms have been described in [22].Following the mathematical description given in [23], theVoronoi diagram is used to geometrically subdivide aregion, based on a set of seed points. LetP fp1; ::pi:::;pn; 2 n < 1g represent a set of n in-dependent points dispersed in the 3D space with co-ordinates xi 6 xj 2


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first polyhedron face is a sub-domain of the bisectorplane Bij. Point pij in the bisector plane Bij, defined bythe position vector ri rj=2, is utilized in realizing theboundary of the face. The edge Lijl of the face Bij, shownin figure 1, corresponds to a boundary of the Voronoicell associated with the generator pi and is constructedfrom the 8N 1 6 possible neighbors. Here 6 corre-sponds to the number of bounding planes. For eachneighbor pll6 j and the 6 bounding planes, the edgeline Lijlrijl is formed as the intersection of the bisectoror boundary planes Bij and Bil i.e., Lijl : Bij \ Bil8j 6 l.The plane Bil that is chosen as the first face of the

Voronoi polyhedron, is the one for which the perpen-dicular distance "rjl rij rjl from the point pij to theedge Lijl is a minimum as shown in Fig. 1a. The inter-secting line Lijl is obtained by solving the simultaneousalgebraic equations for the bisector planes:

aijxG bijyG cijzG dij; ailxG bilyG cilzG dil6

The point pijl, which is the shortest distance on Lijl frompij, is obtained by solving an additional constraintequation:

nij nil rij rG 0 7

Here nij and nil are normals to bisector planes Bij and Bilrespectively and nij nil is the direction of the line Lijl.This yields the relation

bijcil cijbilxG xF cijail aijcilyG yF aijbil bijailzG zF 0 8

The coordinates of the point PGxG;yG; zG are obtainedby solving equations (6) and (8).

3. Delineate the limits of the first edge Lijl, correspondingto the vertices of the Voronoi cell for the generator pi.These are the intersection of the line Lijl with neigh-

boring bisector planes, generated from neighbors pmand pn. As illustrated in Fig. 1b, the vertices Vijln andVijlm, are obtained as the intersection of the line Lijl withthe bisector planes Bim and B in for m; n 6 i;j; l. Forany point pkk 6 i;j; l the vertex Vijkl is generated asthe intersection of bisector planes Bik, Bil and Bij bysolving the corresponding equations

aikx biky cikz dik; ailx bily cilz dil ;aijx bijy cijz dij 9These are constructed using the points pi, pk, pl and pj.Points pm and pn are chosen from

8N

2

6

points

Fig. 1. (a) Bisector and first plane and line of polyhedron i (b)Construction of the first edge (c) Location of the pivoting point. pI

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so that j rijlm rG j and j rijln rGj are the minimumdistances from PG to all vertices in both directions. Thepositive and negative directions along the line Lijl, fromPG to the vertices are chosen from the conditions

rF ri rijlm rij rG rij

0 10respectively. Each line Lijl, terminating with the vertices

Vijlm and Vijln constitutes an edge of a polygon desig-nated as Eijlmn. Of the subscripts, i corresponds to thegenerating point, j and lcorrespond to the neighboringpoints contributing to the edge Lijl, and m and n cor-respond to the generators of the limit plane.

4. Subsequently, construct the other edges in a sequentialmanner. Faces are constructed to close the polyhedronusing the property, that two faces share every edge ineach closed polyhedron.

5. The bisector point pij is considered as a central point inthe construction of all edges, e.g. Eijlmn, in the plane Bij.After constructing the first edge Eijlmn, subsequent edgesEijlmp are constructed in the direction of right-hand rule

with the thumb pointing from pj to pi. The next vertexVijnp for the generator pp is chosen such that the vectorrij ri rijnp rij rijnp rij has the same sign asrij ri rijlm rij rijlm rij

. Furthermore, the

vertex Vijnp is selected from all 8N 4 6 possiblecandidate bisector planes such that the distancerijln rijnp is a minimum. The process of generatingadjacent vertices and edges is continued until the lastvertex coincides with the starting vertex Vijlm.

6. Upon construction of the polygonal face Fij & Bij for thepoint pi, other faces of the Voronoi polyhedron, e.g.Fil & Bil, are generated by using the same algorithm.Note that the two vertices Vijlm and Vijln and edge Eijlmn

have already been determined. The other vertices arethen determined as the intersection of the plane Bil withanother candidate bisector plane following the stepsenumerated next and depicted in figure 1c.

(a) Generate a plane Fthat is perpendicular to the planeBil (F? Bil) and a bisector of the edge Eijlmn(F? Eijlmn), satisfying the conditionrm rijlmrijln2 rijln rijln 0 resulting inamx bmy cmz dm 11where

am

xijlm

xijln; bm

yijlm

yijln; cm

zijlm

zijln;

dm 12

x2ijlm x2ijlny2ijlm y2ijlnz2ijlm z2ijlnh i

12(b) From pi, drop a perpendicular on the plane F, in-

tersecting the latter at point pH. The point pH maybe obtained from a directional line Eq. [34], bysolving the equations

xH xiam

yH yibm

zH zicm

;

amxH

bmyH

cmzH

dm

13

where am; bm; cm; dm are determined in Eq. (11)and (12).

(c) From pH, drop a perpendicular on plane Bil to in-tersect at the point pI.

(d) The location ofpI with respect to the edge Eijlmn isimportant in determining the directional sequenceof vertices of the face Fil. A reference point p

0I on the

line containing point pI and the middle point of

edge Eijlmn is constructed very close to the edgeEijlmn and on the same side as the point pI. The signof the vector operation

rI0 ri rijlm rI0 rijln rI0

is used to determine the directional sequence. Therest of the construction follows steps 1 5.

7. After the first face of polyhedron is constructed aboutgenerator pi, the other faces are constructed from edgesof the first face till the entire polyhedron is closed.

2.2Surface based Voronoi tessellation for ellipsoidal

heterogeneitiesThe VCFEM mesh is based on Voronoi tessellation of themicrostructure consisting of finite sized heterogeneities. Ifthe heterogeneities are equi-sized non-intersectingspheres, the resulting Voronoi polyhedra are the same asthose generated by the point generation methods withtheir centroids as generators. However, most physicaldomains contain nonuniform heterogeneities of differentsizes and shapes. Intersection of the bisector planes withheterogeneities may be a common occurrence with thecentroid-based tessellation algorithms for these cases.Consequently, a surface-based algorithm is developedbased on bisector planes between closest points on thesurface of two neighboring ellipsoids. In this algorithm,closest points p0ix0i;y0i; z0i and p0jx0j;y0j; z0j between twoneighboring ellipsoids are first obtained by solving aconstrained minimization problem:

Minimize : fx0i;y0i; z0i; x0j;y0j; z0j x0j x0i2 y0j y0i2 z0j z0i2

wrt x0i;y0i; z

0i; x

0j;y

0j; z

0j

such that the points belong to the ellipsoidal surfaces:

x2ia2i

^y2i

b2i z

2i

c2i 1; x

2j

a2j

^y2j

b2j z

2j

c2j 1 14

where

xi^yizi

8 represents the L2 vector inner product.Consequently, ea

M=I is positive for all r 6 0, provided thematrix [H] is positive definite. From the definition of [H]in Eqs. (51) and(52), the necessary condition for it to bepositive definite is that the compliance tensor [S] bepositive definite, which is true for elastic problems. Asecond condition is that the finite-dimensional subspacesTM=Ie be spanned uniquely by the basis functions[PMx;y; z] and [PIx;y; z]. This is satisfied by assuringlinear independence of the columns of basis functions[PMx;y; z] and [PIx;y; z], which also guarantees theinvertibility of [H]. Furthermore, additional stabilityconditions should be satisfied to guarantee non-zero stressparameters bM=Ie in ea

M=I for all non-rigid body boundarydisplacement fields u

E=Ie . This is accomplished by careful

choice of the dimensions of the stress and displacementsubspaces. From Eqs. (18) and (20), the bilinear forms ofthe energy functional eb

M=IE=I

may be represented in terms ofthe stress and displacement parameters as

ebME rMe ; uEe < GEqEe ;bMe >;

ebMI rIe; uIe < GMIqIe;bMe >; and

ebIIrIe; uIe < GIIqIe;bIe > 8rM=Ie 2 TM=Ie and

8uE=Ie 2 VE=Ie 56It is necessary that all these functionals eb

M=IE=I

0 for rigidbody displacement modes uE=I on the element boundaryand interface. Thus, displacement fields in ?

VE=Ie that are

orthogonal to the subspace of rigid body modes, shouldstrictly produce positive strain energies. This discreteL-B-B condition [45,46], ensures stability of multi-fieldvariational problems such as in VCFEM.

3.3.1Voronoi cell elements with voidsThe strain energy for a Voronoi cell element with a void isexpressed from Eqs. (20), (56) and (52) as

SEMe e aMe rMe ; rMe e bME rMe ; uEe e bMI rMe ; uIe< GEqEe ; bMe > < GMIqIe;bMe >< GEqEe GMIqIe; bMe >

< GE GMI qEe

qIe

& ';

HM1 GE GMI qEe

qIe

& '

< Gvoid

Qf g; HM1 Gvoid

Qf g > 57For the porous element,

Gvoid

is a nMb

nEq

nIq

rec-

tangular matrix, where nMb dimTMe , nEq dimVEe andnIq dimVIe. Since HM is positive definite, the strainenergy in the Voronoi cell element vanishes for zero stressfields in the matrix, and consequently in Eq. (57)

SEMe 0 , Gvoid

Qf g 0 58The necessary condition of stability is written from Eq.(58) as

GvoidfQg UkVfQg 6 0; 8Q\

Qrb ; 59where Qrb correspond to the six rigid body modes of

displacement. The matrices U and V, whose columnsare the eigenvectors of GvoidGvoidT and GvoidTGvoidrespectively, are orthonormal matrices obtained by sin-gular value decomposition of Gvoid. [k] is a rectangularmatrix with positive entries on the diagonal correspondingto the square roots of the non-zero eigenvalues of bothGvoidGvoidT and GvoidTGvoid. Premultiplying both sidesof Eq. (59) by U1 yieldskVfQg kfQg 0 60Since the columns of V are linearly independent, theabove equation can only be satisfied for either trivial orrigid body solutions of the boundary displacement.

Equation (58) also leads to the L-B-B condition forrank sufficiency of a Voronoi cell element with a void.Positive singular values of k imply that the strainenergy associated with the stress field solution rMe uEe ; uIeassociated with non-rigid body displacement fields

uE=Ie 2? VE=Ie 8?VE=Ie

Trb VE=Ie ; is strictly non-zero. FromEqs. (20) and (58), the L-B-B condition may be stated as:

9c > 0 such that sup8uE=Ie 2?VE=Ie

ebME rMe ; uEe e bMI rMe ; uIe

k uEe uIe k! c k rMe k 8rMe 2 TMe 61

where k k are metric norms defined in the respectivesubspaces. The corresponding necessary condition forstability in terms of the matrix dimensions become

nMb > nEq nIq 6 62

The sufficient condition for stability is established byensuring that the eigen-values in k are positive, which isenforced at the solution stage.

3.3.2Voronoi cell elements with inclusionsFor a composite Voronoi cell element with an embeddedinclusion, positiveness of the total strain energy

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SEe SEMe SEIe can be stated in a similar manneras:

9c > 0 such that sup8uEe 2?VEe

ebME rMe ; uEe k uEe k

! c k rMe k 8rMe 2 TMe and

sup8uIe2?VIe

ebIIrIe; uIek u

Ie k !

c

krIe

k 8rIe

2 TI

63

The corresponding L-B-B condition or the necessarycondition for stability are (see [18,19])

nMb > nEq 6 and nIb > nIq 6 64

These conditions are sufficient to guarantee the existenceof solution and its convergence for multi-field saddle pointproblem posed by the Voronoi cell FEM with elastic con-stituents [46].

4Numerical implementation

4.1Scaling of the stress functionIt is desirable that matrices HM and HI have goodcondition numbers and are invertible. Global cartesiancoordinate representation with varying exponents makedisparate contributions to these matrices. For example, forx;y; z >> 1, different exponents n can make big differ-ences in the matrix components that can lead to badconditioning with poor invertibility. Scaling of stressfunctions have been proposed in [18,19] through localelement coordinates n; g; f. Coordinates x;y; z arelinearly mapped as

n x xcl

; g y ycl

; f z zcl

65where xc;yc; zc are the center coordinates of the Voronoicell element and l is a scaling length determined as:

l max j maxxe xc; maxye yc;maxze zcj8xe;ye; ze 2Xe

The scaled coordinates are in the range of1 to 1 for mostVoronoi cell elements. The corresponding matrix stressfunctions in Eq. (49) have the form

Uij Xm

p

q

r

1

npgqfrbpqr Xm

p

q

r

1

npgqfr

Xnk1

1

apqrk1

bmpqrk; i j 1; 2; 3 66

4.2Numerical integration schemes for G and H matrices

4.2.1 Integration of [G] matricesIn Eqs. (52) and (54), the matrices GMI and GII arenumerically integrated over the interface and the matrixGE over the element boundary. All numerical integrations

are executed using Gaussian quadrature. VCFEM elements

have polygonal boundaries, which are divided into 6noded quadratic triangular elements as shown in Fig. 4a.For each polygonal face the triangular elements are con-structed with one node at the centroid and two otherscoinciding the vertices of the edges. For the ellipsoidalinterface, subdivision to construct 9-noded quadratic ele-ments is done in the following sequence (see Fig. 4c).

1. A bounding box with its edges parallel to the principle

axes of the ellipsoids and completely encompassing theellipsoids is constructed. The ratio of the three edges ofthe box is the same as the ratio of the three axes of theellipsoid.

2. 9 nodal points are inscribed on each face of thebounding box. This includes 4 corner nodes, 4 middlenodes and 1 center node.

3. Each of the 9 nodes are joined with the center of thebounding box and the corresponding intercepts withthe interface form the quadrilateral surface element.This is repeated for all the 6 faces of the bounding boxfaces. It provides smaller elements in regions of highercurvature

The GII matrix, requiring integrating over ellipsoidalsurface segments of the interface, is sensitive to the surfaceelements used for the integration. Standard 9 noded-biquadratic elements with isoparametric shape functionscan result in significant deviation from the actual surfacearea especially in regions of high curvature. To overcomethis, a parametric equation for the ellipsoid is expressed asx a cos h sin/;y b sin h cos/; z ccos/ 67where a; b; c are the semi-axes and 0 h 2p , 1

2p / 12 p correspond to the angular range of thesurface. The nodal coordinates are represented as ha;/awith a 1 . . . 9. The Gauss integration points in thismapping are interpolated from the spherical coordinatesof the nodes

h X9a1

Naha/ X9a1

Na/a

where Na is the interpolation function for a 9 nodedbiquadratic element. Subsequently, the global Cartesiancoordinates of the integration points x;y; z are expressedin terms of h;/ and the semi-axes a; b; c. In the in-tegration scheme, the integral of a function over a segmenton the interface

Rs fx;y; zds

is written as

R11

R11 f J1J2dndf

. J1 and J2 are the determinants of the

Jacobian operators relating the spherical to cartesian

coordinates and natural (master) to spherical coordinatesrespectively.

J1 deti ^j k

oxo/

oyo/

ozo/

oxoh

oyoh

ozoh

2664

3775

sin/ cos hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b2c2 sin2 / a2c2 sin2 / a2b2q

J2 detP9

i1 N0ihhi

P9i1 N

0ih/i

P9i1 N

0i/hi P

9i1 N

0i//i

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This mapping scheme guarantees that all integrationpoints are on the actual surface.

4.2.2Integration of [H] matrices:For accurate domain integration of matrices HM and HI ,the matrix and inclusion volumes XMe and X

Ie are sub-

divided into 3D brick and tetrahedral elements respec-

tively. For the matrix domainX

m, the following algorithmis adopted.

Each face on the element boundaryoXEe is subdividedinto triangles by joining the face edges to the center ofthe element as shown in Fig. 4a. The triangles arerepresented using 9-noded biquadratic elements withcollapsed nodes at the central vertex. Each node of theabove triangular element is projected on the interfaceoX

Ie. The projected point is the intersection of the line

joining the node with ellipsoid centroid, with the in-terface. This results in a 9-noded element at the inter-face as shown in Fig. 4b. The element-pair at theelement face and interface are used to generate

18-noded brick elements for volume integration. It ispossible that the projected element on the interface istoo large due to the relative positioning of the interfacein the Voronoi cell. To correct this problem, each of thetriangle pair is subdivided into 3 sub-triangles beforegenerating the brick elements. The subdivision iscarried out for the following conditions:

area of projected triangle

interface area> specified tolerance

orarea of face triangle

element surface area> specified tolerance

The value of the tolerance is set to 4.5%, which is

slightly greater than the area ratio generated by a cubicelement with a spherical inclusion at the center. The18 noded brick elements are further subdivided to en-hance the accuracy of integration of the reciprocalfunction in PM, particularly near the interface. To ac-complish this, the projection line from the face node tothe inclusion boundary is subdivided into four seg-ments using the ratio of the in ellipsoidal coordinates:a11 : a

21 : a

31 : a

41 1:1 : 1:2 : 1:3 : 1:4. The resulting 4

brick elements become progressively larger as theymove away from the interface is shown in Fig. 4b. Gauss quadrature rules are used in each brick elementfor numerical integration.

For volume integration in the inclusion to evaluate HI,tetrahedral elements are used. As shown in 4d, these ele-ments are constructed by joining interface element nodeswith the inclusion centroid.

4.3Implementation of conditions for stabilityLinear independence of the columns of PM and PI isnatural for pure polynomial expansions. However, whenreciprocal functions are used, some of the reciprocal termsmay be linearly dependent on the polynomial terms. Therank of matrices like

PM

is determined apriori from the

diagonal matrix resulting from a Cholesky factorization ofthe square matrix

HM ZXe

PMTPMdX 68

Nearly dependent terms in the columns of PM will resultin very small pivots during the factorization process.Corresponding terms in the stress function are dropped to

prevent numerical inaccuracies in the inversion HM.In VCFEM, the interface nodes are in general, not topo-logically connected to the element boundary nodes. It isnecessary to specify rigid-body modes for the displacementfield fqIg on the interface. A simple procedure, corre-sponding to the constraining selected displacement modesbased on the singular value decomposition of matrix GI, isperformed for Voronoi cell elements with inclusions. Sin-gular value decomposition of the matrix GE GMI, andmatrices GE and GII are performed for Voronoi cell ele-ments with voids and inclusions respectively to satisfy thediscrete L-B-B conditions. The number of degrees of free-dom nMb and n

Ib in the stress functionsU

M andUI are chosen

to satisfy the Eqs. (62) and (64). Zero singular values in thediagonal of the resulting k matrix are removed by enrich-ing the corresponding stress function with polynomialterms. Additionally, extremely small eigen-values in k mayresult in inaccurate displacements. This is averted by con-straining selected displacements based on the singular valuedecomposition ofGMI or GII. The procedure involvesre-writing the matrix multiplication as:

GfqIg UkVfqIg UkfqIgcalt GaltfqIgalt69

Elements in fqIaltg corresponding to small eigen-values in

k

are pre-constrained to zero. The process decreases the

dimensions of VIeH and results in a loss of accuracy. Thisprocedure constrains rigid-body modes on the inclusioninterface and also extremely small eigen-values in kwhich causes inaccurate displacements. The rotated Galtmatrix from singular value decomposition is used in thestiffness matrix calculation and the corresponding dis-placement vector at the interface is fqIaltg.

5Numerical examplesA number of linear elastic boundary value problems arenumerically solved by the 3D Voronoi cell finite element

model to understand its effectiveness in analyzing het-erogeneous microstructures. Heterogeneities in themicrostructure are in the form of either voids or inclusionsof ellipsoidal shapes. The problems solved are divided intotwo different categories, namely comparison of micro-scopic VCFEM solutions with: (i) known analytical solu-tions for simple unit cells; (ii) results using commercialcodes for more complex microstructures.

5.1Stress distribution around a spherical voidThe analytical solution for the three-dimensional stressesaround a spherical void in an infinite medium under

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uniaxial tension (rzz 1) has been provided in Ti-moshenko and Goodier [1] (section 137). The stress fieldfrom a special stress function is superposed on the so-lutions of a solid bar in tension for this solution. Thespecial stress field matches the stress field for the solidbar on the surface of the sphere and vanishes to zero atinfinity. In the VCFEM analysis, a L L L domainwith a single spherical void of radius rd L5 is modeledusing a single cubic element. The Possions ratio of thematerial is relevant to the solution and is taken to bem 0:3. In the VCFEM implementation, lineardisplacement fields are assumed on the triangular sub-domains on each face, while quadratic triangular ele-ments are used for displacement fields on the voidsurface. The matrix stress function UMpolyij in equation(25) is taken as a fifth order polynomial stress functionp q r 0 5; npolyb 336. The reciprocal stressfunctions in Eq. (49) is constructed with i 1 5 forp q r 0 2. The axisymmetric stress functionused in [1] can be proved to be equivalent to 3DMaxwell stress functions of the form:

U11 x2y2a5

2z2a5

c 2m1 x2y2

a3 2m2 z2

a3

b

2m1 x2y2 2m2 z2 2m1 x2z2

a3

rd

2m2 y2z2y2a3

rdr0y2

U22 x2y2a5

2z2

a5

c 2m1 x

2y2 a3

2m2 z2

a3

b

2m1 x2y2 2m2 z2x2z2

a3

rd

2m1 z2

y2

2m2 x2

a3

rd

U33 x2y2a5

2z2

a5

c x

2y2a3

b

2m1 x2y2 2m2 x2 2m1 x2z2

a3

rd

2m2 y2x2y2a3

rd 70

where a ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x2y2z2p

rdcorresponds to the ellipsoidal

coordinate and a; b; c are material constants that can beexpressed as,

a r0 1 5m 4 7 5m 1 2m ; b

5m

2 7 5m ; cr0

2 7 5m For this case, the VCFEM stress interpolation function inEq. (49) matches the theoretical stress function in Eq. (70)exactly. The solution error can therefore be attributed tothe error in displacement interpolations on the void andelement boundaries produced by the triangular elementsand solution error. Different stress components along aline passing through the center of the sphere are plotted inFig. 5. The dominant stress along this line, perpendicularto the loading direction, is the normal stress in the loading

direction. The VCFEM solutions closely match the stressesin [1].

5.2Stress distribution around an ellipsoidal voidSadowsky and Sternberg [2] have presented an analyticalsolution to the problem of stress field around a smallellipsoidal void under uniaxial tension in an infinitemedium. The exact solution for stresses is expressed interms of elliptic functions. In this example, the stressdistribution generated by the VCFEM is compared withthat in [2]. The ellipsoid has an aspect ratioa : b : c 9 : 3 : 1 in a matrix cube of dimensionsL L L, with L 5a. The material properties and thestress and displacement interpolation fields in this pro-blem are same as in the previous example. Stress dis-tributions along the centroidal major axis of the ellipsoid,that is perpendicular to the loading direction are shown inFigs. 6a and b. Concentration of the dominant stress rzzoccurs near the tip of the void on the major axis. The Fig.6b shows a zoom-in of the stresses near this region. The

concentration is very well represented by the VCFEMsolution. The slight deviation from the analytical solution,away from the tip, is because of the displacement inter-polations on the ellipsoidal surface.

5.3Effect of interaction of spherical heterogeneitiesThe interaction between two heterogeneities, which aresources of stress concentration, is of considerable in-terest to the composites community. Semi-analyticalsolutions to these problems have been provided in [4]for cavities using bispherical coordinates, and in [5] forrigid inclusions and cavities based on the Boussinesq-

Fig. 5. Comparison of stress distribution along the center linez 0 for the cubical domain with a spherical void


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Papkovich stress functions. The solutions in the lattermethod are expanded in series of spherical harmonicswith respect to the centers of the heterogeneities. TheVCFEM implementation involves a mesh of two cubicelements, with each element containing a sphericalinclusion or void. The problem is analyzed with theheterogeneities approaching each other and hence thecommon edge shared by the elements. The stressconcentration at the interface increases with decreasingdistance. For improved accuracy, the adaptive schemedeveloped in [19] is implemented to enhance thedisplacement interpolation using h p enrichment. Theerror indicator for adaptation is based on the tractiondiscontinuity along the element boundary and theheterogeneity-matrix interface. Once identified forrefinement, the boundaries and interfaces aresuccessively subdivided into smaller triangles till the

traction reciprocity error is within acceptable tolerance.The first problem solved using VCFEM involves two

voids of radius r, whose centroids are separated by adistance R. The distance is set to R 4r in this problem.The boundary condition corresponds to a far field

hydrostatic tension ofr1xx r1yy r1zz 1. Stresses gen-erated by VCFEM at the equators and poles of the spheresare compared with analytical solutions of [4] in table 1.The maximum difference between the two solutions isfound to be less than 1% .

In the second problem set considered, the two hetero-geneities are assumed to be either voids or rigid inclu-sions. The rigid material is simulated in VCFEM with avery high modulus, corresponding i.e. Einclusion 50Ematrix.Two different applied far field strains are considered forgenerating the solutions suggested in [5]. They are:

(i) A far field hydrostatic tension, represented by the strainfield 1xx 1yy 1zz 1.

(ii) A far field in-plane tension and out-of-plane com-pression, represented by the strain field 1xx 1yy

1zz

1. The R=r ratio is varied from 0 to 3 in this

problem. Figures 7 shows the comparison of VCFEMresults with those in [5] for the normalized stress fieldrzz along a line joining the centers of the spheres forR=r 3. A good agreement of the results is observedwith less than 1% .

Table 1. Comparison ofVCFEM generated stresseswith [?] for two spherical voidsin an infinite medium at (a)adjacent pole (b) equator (c)remote pole

Stress Near Pole Equator Remote Pole

[4] VCFEM [4] VCFEM [4] VCFEM

rxx 1.570 1.5610 0.000 0.0000 1.510 1.4963ryy 1.570 1.5673 1.470 1.4810 1.510 1.4988rzz 0.000 0.0000 1.500 1.4922 0.00 0.00

Fig. 6. Comparison of stress distribution along the center liney 0; z 0 for a cubical matrix with an elliptical void: (a) fordifferent values of x and (b) near the tip of the elliptical void

6


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The convergence rate of the VCFE model using purelypolynomial stress functions and the combined polynomialand reciprocal terms are examined for rigid inclusionswith R=r 3 in table 2. The rate is very slow with purely

polynomial based stress function enrichments. Howeverthe addition of the reciprocal terms significantly enhancesthe convergence rate.

Table 2. Convergence rate of the normalized stressrzzr1zz

l0for purely polynomial (poly) vs polynomial+reciprocal (VC) stress function

at point A for 2 spherical inclusions shown in fig(??) with R=r 3 with 1xx 1yy 1zz 1n

polyb 336 npolyb 468 npolyb 620 nVCb 574 [6]

Normalized Stress )7.567 )7.614 )7.693 )8.343 )8.3588

Fig. 7. Comparison of stressditribution along the centerline bewteen the two spheres,for various values of sepera-tion distance and macroscopicloads with (a) 1ij dij and (b)1ij

di1dj1

di2dj2

2di3dj3


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5.4Comparison with ANSYS for random microstructuresA microstructure consisting of randomly dispersed 20spherical voids in a 10 10 4 cuboidal matrix(5:0 x 5:0;5:0 y 5:0; 0:0 z 4:0), is mod-eled in this example. The microstructure and the Voronoicell mesh are shown in Fig. 8a, while Fig. 8b shows themesh with commercial code ANSYS. The VCFEM mesh

contains twenty elements corresponding to the number ofvoids, with a total of 144 nodes on the element boundariesand 1,480 nodes on the matrix-inclusion interfaces. Thecorresponding converged ANSYS mesh contains 84,123ten-noded tetrahedron SOLID92 elements and 124,655nodes. The matrix material has a Youngs modulus ofE 200GPa and Poissons ratio m 0:3. The boundaryconditions are: (i) Symmetry conditions on faces withx 5;y 5; and z 0;; (ii) Displacement uz 4 onthe face z 4, corresponding to an overall strain zz 1:0in the zdirection. The other two faces (x 5;y 5) aretraction free. The VCFEM solutions for microstructuralstresses are compared to those generated by the highly

refined ANSYS model. The tensile stressr

zz along threelines parallel to the coordinate axes x;y; z, and through theorigin are plotted in Fig. 9. Stress concentrations of upto 4are observed along the x and y directions. The VCFEMmodel is able to capture the important features in thestress distribution with an accurate representation of thepeak stresses along the void surface. The bumps and peaksin these plots are due to the unsmoothened representationof matrix stresses resulting in small discontinuities acrosselement boundaries.

5.4.1Parallel implementation of the VCFEM codeWhile the 3D VCFEM is accurate for heterogeneousmicrostructures, it has high requirements of computingtime, mainly because of numerical integration using a

large number of integration points. A multi-level parallelprogramming approach is implemented to significantlyenhance the computational efficiency of the 3D VCFEM in[47]. The parallelization is conducted for a cluster ofsymmetric multi-processor (SMP) workstation nodes. MPIis used for data decomposition at a coarse level betweenthe nodes and OpenMP is used for multi-threaded paral-lelism on each node. The multi-level parallelism combines

benefits of improved loop timings and domain decom-position methods to obtain optimized solution times usingSMP cluster systems. The code is scalable to any numberof multiprocessor nodes such that any number of elementscan be solved simultaneously with the only limit being theavailable hardware resources. The addition of OpenMPdirectives into the VCFEM model allows for loop levelparallelization to occur in an efficient manner. The com-putations for each element can be performed across mul-tiple processors in a shared memory environment. For the20-element microstructure, the timings for the multi-levelprogram using different number of nodes of the cluster,with each node running four OpenMP threads, are pro-vided in Fig. 10. Details of the parallelization scheme areprovided in [47].

6ConclusionsA three-dimensional Voronoi cell finite element model(VCFEM) is developed in this paper for analyzing micro-structural stresses in elastic domains containing ellipsoidalinclusions or voids. The paper begins with the develop-ment of a 3D domain tessellation method for generatingthe Voronoi cell mesh, in which each Voronoi cell containsone heterogeneity at most. To account for the shapes andsizes of heterogeneities in the domain discretization pro-cedure, a surface-based tessellation algorithm is proposedas a modified form of the point-based tessellation. Forplanar faces, the surface based tessellation may give rise to

Fig. 8. (a) VCFEM and (b) ANSYS meshes for 20 spherical voidsin a cuboidal material domain

8


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non-coinciding triple points. Local adjustments are

implemented to avoid such incongruence. The meshgeneration algorithm is successfully tested for differentmicrostructures with various shapes, sizes and spatialdistribution of heterogeneities.

The Voronoi cell finite element model for smalldeformation elasticity is subsequently developed using anassumed stress hybrid formulation. In this model, equili-briated stress fields are constructed from symmetricMaxwell or Moreras stress functions. Complete poly-nomial representation of the stress functions guaranteesinvariance of stresses with respect to coordinate trans-formations. A necessary condition for stability is that thecolumns of the stress interpolation function [P(x1; x2; x3)].

A special procedure of selective elimination of the

dependent modes is invoked to restore this condition.Stress functions comprised of pure polynomials yield poorconvergence characteristics and consequently specialaugmentation functions are developed to improve accu-racy and efficiency. These functions account for the shapeof the interface in its vicinity, but decay with increasingdistance from it. Creation of these functions in terms ofelliptic integrals and using ellipsoidal harmonics is a majorcontribution of this paper. The development follows fromthe derivation of stresses from the general solutions to theNaviers equation. Numerical implementation of thealgorithms is presented and especially the methods of fil-tering out the rigid body modes and enhancing stability

Fig. 9. Comparison of tensile stress distribution along centerlinesof the cuboidal domain


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and convergence of the resulting finite element model.Various numerical examples are solved in this paper tovalidate the model. Comparison of microstructural stressresults generated by the 3D VCFEM with analyticalsolutions in the literature for smaller number of hetero-

geneities confirm the accuracy of the model. Stress dis-tribution results are also compared with a highly refinedFEM model using ANSYS using multiple voids. Theaccuracy of the VCFEM predictions in these simulationsprovide adequate validation to the robustness of theformulation. A multi-level code parallelization usingOpen-MP and MPI adds significant efficiency to theVCFEM simulations.

Three dimensional stress analysis in complex micro-structures has currently become a necessity in the designof advanced materials. Despite the disadvantages asso-ciated with conventional modeling tools like FEM inefficiently model real microstructures, serious and novel

attempts are being made to incorporate three dimensionalanalyses into practice [8, 9, 10, 13, 14, 15, 16]. The presentpaper is developed to propose an alternative approach tothese developments by way of 3D VCFEM. While thismethod of modeling with direct interface to the micro-structure has considerable promise, a difficulty that iscurrently faced with, is the large number of integrationpoints needed for the special functions in Gauss quad-rature methods. This is a topic of future investigation andreduction.

7Appendix

7.1Coefficients for lame and stress functionsThe coefficients A

ji for the lame functions C1; S1&D1 in

equation (40) are given as:

A11 3h2;A12 3h2;A13 0

A21 1 h2

k2

3hkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p ;A22

3hkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 h2p ;

A23 3h3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 h2

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 k2

pa1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21

h2p

A31 3h2kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 h2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 h2

pa1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 k2

p ;A32

3h2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik

2

h2

p ;A33 0 71

The coefficients Bijk ; C

ik for the stress functions in equation

(44) are given as

B111 h; B121 k2h

k2 h2 ; B131

h3

k2 h2B211

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 h2p h

k; B221


k;

B231 h3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p kB311

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 h2

p; B321

k2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 h2p ; B

331 0

C11

0; C21

0; C31

0

B112 h; B122 h; B132 0B212


k; B222

hkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2 h2p ; B

232 0

B312 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p

; B322 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p

; B332 h2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p

C12 h3; C22 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p h3k

; C32 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p

h2

B113 0; B123 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 k2

ph3kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 h2p

k2 h2 a1;

B133 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 h2p h3kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 k2p k2 h2 a1B213 0; B223


ph3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 h2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p a1;

B233 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 h2

ph3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 k2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p a1B313 0; B323



a21 h2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p a1;

B333 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia21 h2


a21 k2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2 h2p a1C13 0; C23 0; C33 0 72

Fig. 10. Speedup with multi-level parallelcode with additional computing nodes.

0


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Three Dimensional Voronoi Cell Finite Element Model for Microstructures - S. Ghosh and S. Moorthy