hte

ologg Un

Moving least square (MLS)-based nite

immemeTheyrarm

2011 Elsevier B.V. All rights reserved.

t methor thes withd thend ex

tion. Therefore, the hexahedral elements are preferred for use inFEM if possible. Many works have been conducted on the modelingof arbitrary congurations with the hexahedral elements by way ofvarious approaches [14], such asmappedmeshing algorithms [5,6]including octree approaches, sweeping algorithms, andmulti-blockmethods; direct methods [713] including grid-based algorithm,advancing front or plastering algorithm, and whisker weavingalgorithm; and indirect methods [1,2,14] that combine several

Thus far, the grid-based methods and the advancing front meth-ods have been found to be useful with regards to the construction ofa mesh for complex domains. Nevertheless, it remains difcult toconstruct appropriate meshes around complicated geometries inan efcient manner. In the grid-based methods, as the nodes ofremaining elements are moved to the model boundaries, some ele-ments may be severely distorted and the shape of element may notbe maintained as a hexahedron. Otherwise, in the advancing frontmethods, the inner part may not be lled with the hexahedral ele-ments, because the elements are generated from the boundary ofmodel, layer by layer. Therefore, the additional treatments should

Corresponding author. Tel.: +82 63 850 6994; fax: +82 63 850 6691.

Comput. Methods Appl. Mech. Engrg. 201204 (2012) 208227

Contents lists available at

A

.eE-mail address: youngsamcho@wku.ac.kr (Y.-S. Cho).The meshing may not be straightforward if the problem domainhas a certain complex geometry. Although many auto-meshingalgorithms have been developed, the mesh generation is still trou-blesome in many cases.

In particular, it is very difcult to generate meshes with compat-ible hexahedral elements when modeling three-dimensional com-plex geometry. For this reason, generally, tetrahedral elements areused for three-dimensionalmodeling. However, the hexahedral ele-ments show better in performance than the tetrahedral elements,due to the presence of bilinear and trilinear terms in the shape func-

the given model. These methods remove the elements located out-side the model and on the model boundaries, and then t the outerremaining mesh to the boundary geometry and topology. In con-trast, the advancing front methods, also called outside-inside algo-rithms, start with a quadrilateral mesh formed on the modelboundaries. On the outer layer, the hexahedral elements are con-structed from the quadrilateral mesh. By advancing the front ofthe element layer from the boundary to the inner part, continu-ously, the hexahedral elements are generated over the entiredomains.elementsMarching cube algorithm

1. Introduction

In the framework of nite elemenation is a necessary pre-processing fanalysis of problem domains. Meshesatisfy the connectivity condition anin order to guarantee convergence a0045-7825/$ - see front matter 2011 Elsevier B.V. Adoi:10.1016/j.cma.2011.09.002od (FEM), mesh gener-mechanical or thermalnite elements shouldcompatibility conditionactness of the solution.

tetrahedral elements into a hexahedral element or decompose a tet-rahedral element into four hexahedral elements.

Especially, the grid-based methods [810,15] and the advancingfront methods [11,12,16] are widely known as the most successfulmethods to automatically generate unstructured quadrilateral andhexahedral meshes [17]. The grid-based methods, also called in-sideoutside algorithms, are based on the background grid overMesh generationPoly-pyramid elements

moving least square (MLS) approximation. Numerical results are presented to examine the performanceof the poly-pyramid elements and to demonstrate the effectiveness of the proposed scheme.A novel scheme to generate meshes withand poly-pyramid elements: The carving

Dongwoo Sohn a, Young-Sam Cho b,, Seyoung Im aaDepartment of Mechanical Engineering, Korea Advanced Institute of Science and TechnbDivision of Mechanical and Automotive Engineering, College of Engineering, Wonkwan

a r t i c l e i n f o

Article history:Received 17 September 2010Received in revised form 1 September 2011Accepted 2 September 2011Available online 10 September 2011

Keywords:

a b s t r a c t

A novel scheme for three-dthe aid of poly-pyramid eleerence mesh with brick elemarching cube algorithm.here are termed the poly-pated from the surface info

Comput. Methods

journal homepage: wwwll rights reserved.exahedral elementschnique

y (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 305-701, Republic of Koreaiversity, 344-2 Sinyong-dong, Iksan-si, Jeonbuk 570-749, Republic of Korea

ensional mesh generation, termed the carving technique, is proposed withnts. Soaking the geometry information of a given model into a regular ref-nts, the reference mesh is trimmed by the surface of the model using thetrimmed elements are reconstructed by the proposed elements, which

mid elements. Therefore, the nite element mesh is automatically gener-ation. Shape functions of the poly-pyramid elements are constructed by

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ppl. Mech. Engrg.

l sevier .com/locate /cma

l. Mebe introduced to generate the high quality elements over the entiredomains. In particular, it is difcult to resolve these issues for thecomplex geometry, e.g., domains with irregular surfaces, arbitraryinner voids, multiple materials. For three-dimensional structureswith an arbitrarily complex geometry, a general and efcientscheme is not yet developed [13].

The purpose of this work is to suggest a new meshing schemebased on the marching cube algorithm [1820] with the hexahe-dral elements. Our approach utilizes the similar concept of the con-ventional grid-based methods. However, the elements intersectingwith the model boundaries are not removed, but trimmed by themarching cube algorithm. In this work, we apply the trimming pro-cess with the marching cube algorithm and the non-conventionalelements to the regions of model boundaries, instead of the com-plicated procedures to t nodes on the outer layer to model bound-aries and to smooth the generated mesh over the entire domain.

The process of our proposed meshing scheme is as follows:First, a reference mesh with brick-shaped hexahedral elements isprepared. Subsequently, the elements intersected with the modelsurface are trimmed by the marching cube algorithm. Accordingto the marching cube algorithm, the active and non-active nodesare separated from the hexahedral elements. The active node de-notes a certain node which is included in the model. On the otherhand, the non-active node means a node which is not included inthe model. Due to the trimming process, the elements intersectingwith the model surface are cut off; in the next step, only the ele-ments located around surface are modied into or replaced withnew elements that properly represent the model surface. Most ofelements remain brick-shaped, as in the initial reference mesh.However, there are some limitations when reconstructing thetrimmed elements using the conventional types of nite elements,such as tetrahedral elements or hexahedral elements, because it isdifcult to meet the node connectivity condition and the compat-ibility condition at the interface with the intact hexahedral ele-ments that remain untrimmed. To resolve this issue, we proposenew types of elements termed poly-pyramid elements that in-clude pentagonal and hexagonal pyramid elements. In this paper,the pentagonal and the hexagonal pyramid elements are newlydeveloped by means of moving least square (MLS) approximation.MLS approximation has been used to construct the shape functionfor various meshfree methods [21]. Furthermore, Kim [22,23], Choet al. [2426], and Lim et al. [2729] applied MLS approximation toFEM to handle discontinuities due to nonmatching interfaces. Wewill show that the aforementioned combination of this new mesh-ing scheme with the poly-pyramid elements by MLS approxima-tion makes FEM process efcient. Once the geometry informationis prepared, the entire process, from mesh generation to analysis,can be automatically conducted without cumbersome techniques.Hereafter, the proposed scheme is termed the carving technique.

The outline of the paper is as follows: in Section 2, we describethe construction of the shape function for the poly-pyramid ele-ments by MLS approximation. Subsequently, the method of meshgeneration is discussed in Section 3. Numerical examples are givenin Section 4, and discussed there as well is the simplicity and ef-ciency of the proposed scheme. Finally, we close the paper withconcluding remarks in Section 5.

2. Development of poly-pyramid elements

The poly-pyramid elements are composed of one polygon faceand several triangle faces of which the number is the same as thatof edges of the polygon face. According to the shape of the polygon,the elements are named triangular, quadrilateral, pentagonal, and

D. Sohn et al. / Comput. Methods Apphexagonal pyramid elements, respectively. In this section, webriey describe the MLS approximation and then explain theprocess of constructing the shape functions of the pentagonal andthe hexagonal pyramid elements. Furthermore, the methods ofnumerical integration for the poly-pyramid elements are discussed.

2.1. Shape functions of poly-pyramid elements

When the variable n indicates the coordinate (n,g,f), the MLSinterpolant uh(n) of vector eld u(n) is written as

uhn XNBj1

pjnajn pTnan; 1

where NB is the number of the polynomial basis p(n), and a(n) is acoefcient vector to be determined by the MLS approximation. Inthree-dimensional problems, the linear basis is as follows:

pTn 1; n;g; f NB 4: 2The functional to be minimized in the least-square sense is given as

J XNPI1

xn nIpTnan uIn2; 3

where I indicates a nodal point, NP is the number of nodes havingnon-zero values of weight function x(n nI), and uI(n) is the valueof u at n = nI. The coefcient a(n) is calculated by minimizing thefunctional in Eq. (3)

@J@a

Mnan BnU 0; 4

with

Mn PTWnP and Bn PTWn;where P, W(n nI) and U in the case of NB = 4 are given as

P

1 n1 g1 f11 n2 g2 f21 n3 g3 f3... ..

. ... ..

.

1 nNP gNP fNP

266666664

377777775NPNB

;

W

x1n n1 0 0 00 x2n n2 0 00 0 x3n n3 0... ..

. ... . .

. ...

0 0 0 xNP n nNP

266666664

377777775NPNP

;

UT u1 u2 u3 uNPv1 v2 v3 vNPw1 w2 w3 wNP

264

3753NP

:

Substituting a(n) from Eq. (4) in Eq. (1), the interpolant is rewrittenas

uhn pTnM1nBnU XNPI1

/InuI; 5

where uI is the Ith column vector of UT. The shape function /I(n) canbe obtained as follows:

/In XNBj1

XNBL1

pjnM1jL BLI: 6

If the domain of inuence for the weight function dened at each no-

ch. Engrg. 201204 (2012) 208227 209dal point is limited to the interior of the poly-pyramid element, theshape functions by MLS approximation do not affect any elements,

except the elements including the node at which theweight functionis dened. In this case, the shape functions have the basic propertiesof nite elements, such as the partition of unity, the linear complete-ness, and the Kronecker delta condition [2430]. Therefore, thesetypes of elements satisfy the compatibility condition between theadjacent elements. These are known as MLS-based nite elementswith variable nodes [2729].

It is critical to dene the weight function in the development ofthe poly-pyramid elements. The pentagonal and the hexagonalpyramid elements in the master domain are shown in Fig. 1(a)and (b), respectively. First, considering the weight function usedto construct the shape function of the pentagonal pyramid ele-ment, the right of Fig. 1(a) shows the cross-section of the pentag-onal pyramid element on the ng-plane at a given value of f. Theweight functions of node I (I = 1,2, . . .,6) are dened as follows:

x1 f;

xI 1 fY3j1x 0; IAIjBIj ; IAIjBIj dAIjBIj

I1; 7

where IAIjBIjindicates the distance between the vertex I and the

edge AIjBIj which connects the vertex A

Ij to B

Ij, as shown in the right

of Fig. 1(a). The indices AIj and BIj can be expressed in a Matlab-likedenition as follows:

AIj I j 5 FLOORI j=7;BIj I j 1 5 FLOORI j 1=7; 8

where the command FLOOR returns the greatest integer less than

n (n,g,f) in the right of Fig. 1(a). The distance between an arbitrarypoint n (n,g,f) and the edge AIjB

Ij is indicated by dAIjBIj

.

The quartic spline function x used in Eq. (7) is given as

xx; x0; x1 1 6x2 8x3 3x4 for x xx0x1x0

6 1;0 for x xx0x1x0

P 1:

8>: 9

In the case of I 1, the weight functions in Eq. (7) are constructedby product of the quartic spline functions and the linear function inthe f-direction. Therefore, the value of the weight function is 1 fat its own vertex I, and 0 at others.

Similarly, the weight function of the hexagonal pyramid ele-ment is obtained through a combination of the quartic spline func-tions and the linear function in the f-direction. The cross-section ofthe hexagonal pyramid element on the ng-plane is shown in theright of Fig. 1(b). In the case of the hexagonal pyramid element,the weight function of node I (I = 1,2, . . .,7) can be dened in thesame form as Eq. (7), and the indices AIj and BIj are given as follows:

AIj I j 6 FLOORI j=8;BIj I j 1 6 FLOORI j 1=8: 10Due to the symmetric property of the hexagonal pyramid element,the value of x 0; 267 ; 267 d67

, which is equal to

x 0; 234 ; 234 d34

, is not used in the calculation of x2. Inthe same manner, all weight functions of the hexagonal pyramidelement are dened without duplication of the equal splinefunctions.

Fig. 2(a) and (b) show the weight functions of the pentagonal

210 D. Sohn et al. / Comput. Methods Appl. Mech. Engrg. 201204 (2012) 208227or equal to the value in parentheses. Here, superscript denotesthe vertices on a cross-section of f = constant, intersected by theedges connecting node 1 and nodes 26. The coordinate of I, whichis identical to that of node I when f = 0, is expressed as a function of

a

bFig. 1. Poly-pyramid elements and their cross-sections on the ngplane in the masterand the hexagonal pyramid elements, respectively, except x1,which is a linear function of f. For a constant f 2 [0,1], the valuesof the weight function in the master domain vary between 0 and1 f. Utilizing the weight functions dened as in Eq. (7), we obtaindomain: (a) a pentagonal pyramid element and (b) a hexagonal pyramid element.

l. MeD. Sohn et al. / Comput. Methods Appthe shape functions of the pentagonal and the hexagonal pyramidelements, as shown in Fig. 3(a) and (b). These shape functions arelinear at the element boundary and are compatible with adjacentelements. Inside the pentagonal and the hexagonal pyramid ele-ments, the shape functions are given in the rational type, not poly-nomial functions [24,28]. We will discuss the performance of thepresent poly-pyramid elements in Section 4 with the aid of numer-ical examples.

2.2. Gauss integration by partitioning the domain of a poly-pyramidelement

Considering the numerical integration for the aforementionedpoly-pyramid elements, there are several ways to conduct integra-tion in nite element ormeshfreemethods. A stabilized conformingnodal integration (SCNI) [30] was proposed to implement thenumerical integration in meshfree methods [21], and the smoothedintegration method [29,31] was developed to apply the concept of

Fig. 2. The weight functions for a constant f 2 [0,1]: (a) a pentagoch. Engrg. 201204 (2012) 208227 211SCNI to FEM. Furthermore, the generalized Gaussian quadraturerules [32] were developed for the two-dimensional domain.

In this paper, the domain of each poly-pyramid element is rstpartitioned into subdomains, and Gaussian integration is appliedto the individual subdomains. This refers to integration by parti-tioning for a polygon in a two-dimensional domain [33]. Althoughnodal integration and generalized Gaussian quadrature rules canbe utilized to improve the efciency and accuracy in the calcula-tion for the poly-pyramid elements, more cumbersome and com-plex procedures are required, in contrast to integration bypartitioning. Focusing on the development of poly-pyramid ele-ments and novel meshing scheme rather than the optimizationof the integration scheme, we choose the integration by partition-ing, which is a simple and straightforward scheme that can be usedto calculate the coordinates and weights of Gauss integrationpoints in the master domain.

The integration by partitioning procedure is illustrated in Fig. 4.First, the pentagonal pyramid element is partitioned into ve

nal pyramid element and (b) a hexagonal pyramid element.

. Me212 D. Sohn et al. / Comput. Methods Applquadrilateral pyramid-shaped subdomains in the master domain,as shown in Fig. 4(a). To obtain the location of the Gauss integra-tion points in the master domain, used is the mapping from thehexahedral element to each subdomain. The hexahedral elementis dened as the parental domain of each subdomain. Given thatGauss integration points in hexahedral elements are easily general-ized, it is straightforward to obtain the coordinates and weights ofthe Gauss integration points for higher order integration. Further-more, the Jacobian matrix, which represents the isoparametricmapping from the master coordinate (n,g,f) to the physical coordi-nate (x,y,z), is dened at Gauss integration points in the parentaldomain. In the same manner, the above procedure is applied to ahexagonal pyramid element with six subdomains, as illustratedin Fig. 4(b). When the order of Gauss integration is n, the numberof the integration points is n3 in the parental hexahedral element.The total number of integration points for the pentagonal and thehexagonal pyramid element is then 5n3 and 6n3, respectively.

Fig. 3. The shape functions obtained using MLS approximation with the weight functionch. Engrg. 201204 (2012) 208227Similarly, tetrahedral subdomains can be also used for the inte-gration by partitioning the elements. However, for the tetrahedralsubdomain, it is complicated to calculate the coordinates andweights of the integration points for higher order integration. Incontrast, the Gauss integration points of the hexahedral domaincan be easily calculated even for the higher order integration. Sincethe shape functions, which are derived by the MLS approximation,are given in the form of the rational functions inside the element,the higher order integration is required to obtain more accuratesolution. Therefore, in this paper, the integration scheme is limitedto the integration by partitioning the element into the quadrilat-eral pyramid-shaped subdomains, as shown in Fig. 4.

3. Mesh generation with the aid of poly-pyramid elements

In this section, a novel scheme to generate a mesh, termed thecarving technique, is proposed with the aid of the poly-pyramid

in Fig. 2: (a) a pentagonal pyramid element and (b) a hexagonal pyramid element.

Physical domainx

yz

a

Parental domain

1

32

4

5

6

Master domain

Subdomain

Physical domainx

yz

1

32

4

56

7

Master domain

Subdomain

b

Parental domain

Fig. 4. Numerical integration scheme by partitioning the poly-pyramid elements: (a) a pentagonal pyramid element and (b) a hexagonal pyramid element.

Fig. 5. Schematic diagram for the trimming process in the carving technique: (a) reference mesh with brick elements; (b) surface information soaked into the reference mesh;and (c) nal conguration of the model obtained by the trimming process.

D. Sohn et al. / Comput. Methods Appl. Mech. Engrg. 201204 (2012) 208227 213

Active nodeNon-active node (to be removed)

Triangulated surfaceswhich may not be co-planar

Fig. 6. Eight cases of the reference brick elements to be trimmed by the surface information.

Triangular, quadrilateral, pentagonal,and hexagonal pyramid elements

from faces of a hexahedral element

Triangular pyramid elementsfrom the trimmed face

intersecting with the model surface

Fig. 7. Reconstruction of the trimmed brick elements with the aid of poly-pyramid elements.

214 D. Sohn et al. / Comput. Methods Appl. Mech. Engrg. 201204 (2012) 208227

l. MeTable 1The procedure of the carving technique.

D. Sohn et al. / Comput. Methods Appelements discussed in Section 2. Furthermore, we discuss theadjustment of nodal positions to enhance the mesh quality gener-ated by the carving technique.

The procedure of the carving technique is divided into twosteps: the trimming and the splitting processes. The rst step is

Step 1Trimmingprocess

Loop over 12 element edges: Find intersection point (s) of each element edge and the g

End loop.If the intersection point exists on the element edge:

Loop over 8 element nodes: Check whether each node is active Nac or non-active N

End loop. Form the trimmed face Ft by connecting the nodes Nis, anGo to Step 2.

Else, end process to keep the hexahedral element untrimmed.

Step 2Splittingprocess

Remove the non-active node (s). Add a node Nin in the interior of the trimmed element. Classify into the triangulated faces Ft newly formed in Step 1, a

In the case of the faces Ft, Generate triangular pyramid elements by connecting

In the case of the faces Fr, Generate triangular, quadrilateral, pentagonal, and he

a Exterior nod

3: Triangular pyramid element(Tetrahedral element)

4: Quadrilateral pyramid element 5: Pentagonal pyramid element 6: Hexagonal pyramid element

5

6

53

3

6

5

3

36

33

b

3

Fig. 8. The patch test for poly-pyramid elements: (a) the geometry of the model to be diand (c) the composition of outer part by 16 poly-pyramid elements.ch. Engrg. 201204 (2012) 208227 215based on the marching cube algorithm [1820]. Soaking thegeometry information, e.g., a stereolithography (STL) le, into a ref-erence mesh which consists of regular brick elements, the refer-ence mesh is trimmed with the geometry information, asdepicted in Fig. 5. The inner nodes of the model surfaces are active,

iven model surfaces, and add an additional node Nis on each intersection point.

na by a normal vector of the model surface.

d triangulate the faces Ft.

nd the pre-existing faces Fr from the hexahedral element.

Nis to Nin.

xagonal pyramid elements by connecting Nac and Nis to Nin.

e

Outer part

Inner part

3

3

3

33

33

33

3

3

3

3

4

6

c

vided into 7 parts; (b) the composition of inner part by 12 poly-pyramid elements;

. Meand the outer nodes are eliminated. The trimmed surfaces are new-ly formed by connecting the intersection points of the elementedge and the model surface. These may not be co-planar due tothe curvature of model surface. For this reason, the trimmed sur-faces should be triangulated to represent the given geometry infor-mation. Fig. 6 shows eight cases of the trimmed brick elements. InFig. 6, the closed and open circles indicate the active nodes and thenon-active nodes (to be removed), respectively. If all nodes of theelement are non-active, the element is excluded from the next pro-cedure. Otherwise, the element, which has only active nodes, re-mains as the brick element.

This trimming process for all regular brick elements may beaccompanied by an expensive cost. In particular, if the model is

The order of Gauss integration for each subdomain

Rel

ativ

eer

rors

and

abso

lute

erro

r

2 3 4 5 6

10-7

10-6

10-5

10-4

10-3Relative error in displacement normRelative error in energy normMaximum error norm (m)

Fig. 9. The errors of the patch test with respect to the integration order for eachsubdomain in the pentagonal and the hexagonal pyramid elements.

216 D. Sohn et al. / Comput. Methods Applcomplex, the computational cost may increase, because a largenumber of reference brick elements have to be used to representthe complex geometry of the model. Accordingly, it is worth con-sidering a way to improve computational efciency. In this work,only the elements located in the vicinity of the model surfacesare stored in advance of trimming process. By doing so, the trim-ming process is conducted for only a fraction of the entireelements.

In the second step, the non-conventional types of elements,which are carvedby themarching cube algorithm, are reconstructedby adding one node in the interior of each reference element andsplitting the reference element into several pyramid elements, asshown in Fig. 7. In this work, we have decomposed the non-conven-tional elements into the poly-pyramid elements. An additional nodeis introduced at the inner space of the non-conventional element,and the outer faces of the element are then connected to the addi-tional node to construct the poly-pyramid elements. The positionof the additional node could be determined according to the rule ex-plained in Appendix A. The trimmed surface, which is exposed toouter space and generated by the surface of domain, may not beco-planar, and then it should be triangulated as mentioned before.Meanwhile, the carved surfaces, which exist at the inside of domain,from the pre-existing hexahedral element are remained as planarsurfaces. Therefore, we could let them as triangle, quadrilateral,pentagon, or hexagon faces. For this reason, from the triangulatedfaces intersecting with the model surface, only triangular pyramidelements are generated by connecting the triangle faces and theadditional node. Moreover, from the carved pre-existing faces, thech. Engrg. 201204 (2012) 208227various poly-pyramid elements are reconstructed by the polygonand the additional node. Fig. 7 illustrates the procedure to decom-pose the non-conventional elements shown in Fig. 6 into severalpoly-pyramid elements. At this point, an additional effort is notrequired to formulate the shape functions of the triangular andquadrilateral pyramid elements, as the triangular pyramid elementis identical to the tetrahedral element and because the shape func-tions of the quadrilateral pyramid can be obtained by degeneratingthe hexahedral elements. Otherwise, the shape functions of thepentagonal and the hexagonal pyramid element are obtained byMLS approximation, as discussed in Section 2.

Fig. 10. The distribution of rmean for the meshes of a sphere, generated from: (a)10 10 10 reference brick elements; (b) 20 20 20 reference brick elements;and (c) 30 30 30 reference brick elements.

amid element or hexahedral element in a seamless way. The

a

The order of Gauss integration for each subdomain

Rel

ativ

eer

rori

ndi

spla

cem

entn

orm

2 3 4 5 6

10-7

10-6

10-5

10-4 From 10x10x10 reference elementsFrom 20x20x20 reference elementsFrom 30x30x30 reference elements

b

The order of Gauss integration for each subdomain

Rel

ativ

eer

rori

nen

ergy

norm

2 3 4 5 6

10-6

10-5

10-4

10-3

From 10x10x10 reference elementsFrom 20x20x20 reference elementsFrom 30x30x30 reference elements

c

The order of Gauss integration for each subdomain

Max

imum

erro

rnor

m(m

)

2 3 4 5 6

10-8

10-7

10-6

10-5

From 10x10x10 reference elementsFrom 20x20x20 reference elementsFrom 30x30x30 reference elements

Fig. 11. The errors of the sphere with respect to the integration order for eachsubdomain in the pentagonal and the hexagonal pyramid elements: (a) the relativeerror in displacement norm; (b) the relative error in energy norm; and (c)

l. Mech. Engrg. 201204 (2012) 208227 217compatibility condition is also kept on the pentagon face of thepentagonal element. The reason is that the pentagonal pyramidelement, from the trimmed element by the marching cube algo-rithm, meets always another pentagonal pyramid element. Simi-larly, the hexagonal pyramid element meets another element ofthe same type, and two hexagonal pyramid elements are compat-ible on the hexagon face. Due to these characteristics of the poly-pyramid elements, the framework of FE analysis does not change.Any additional process is not required in the coupling of twopoly-pyramid elements, and of the poly-pyramid element andthe hexahedral element. That is, the procedures of calculating thelocal stiffness matrices and assembling the global stiffness matrixare the same as those of conventional FE analysis. The stiffness ma-trix remains symmetric and positive denite.

When the intersection point on the element edge is very closeto the active node, the elements generated by the carving tech-nique could have poor quality. In this case, some poly-pyramid ele-ments become relatively small and skew, and hence they lead toill-conditioned stiffness matrix. Therefore, we introduce a simpleway to enhance the mesh quality. This way can be optionally in-serted between the trimming and the splitting processes. The pro-cedure of the adjustment of nodal positions is explained inAppendix B.

4. Numerical results and discussion

To demonstrate the performance of the proposed scheme, wechoose several numerical examples. Firstly, two examples apatch test and a sphere under hydrostatic pressure are used tocheck the validity of the poly-pyramid elements in terms of itsaccuracy. Particularly, it is demonstrated that the proposed mesh-ing scheme is very simple in the example of the sphere underhydrostatic pressure. In the third and the fourth examples, we dis-cuss the modeling and analysis of a femur and a cube with the ran-domly distributed voids, whose geometries are very complex.Throughout all examples, the second-order integration is used forthe tetrahedral and the hexahedral elements. Meanwhile, varyingorders of integration are employed for the integration by partition-ing for the pentagonal and the hexagonal pyramids to examine itseffect on the convergence.

In the rst two examples, the exactness and convergence of theproposed scheme are veried in terms of the relative errors in thedisplacement norm and energy norm, as given in Eqs. (11) andThrough the above two steps, the surface information for thespecic model and the reference brick element are transformedinto a mesh appropriate for FEM. The entire procedure of the meshgeneration according to the present scheme is applicable to thefully automatic algorithm, and its usage is very simple. Consideringthe trimming and splitting processes for one hexahedral element,the procedure is described in Table 1.

Recall that the poly-pyramid elements satisfy the necessaryconditions to utilize in the framework of FEM. As the domain ofinuence for the weight function is limited to the interior of thepoly-pyramid element, the partition of unity and Kronecker deltacondition are satised. Since the linear basis is used in the MLSapproximation, the linear completeness is satised inside the ele-ments. Moreover, the linear interpolation is kept along all elementedges and on the triangle faces of the poly-pyramid elements,respectively. Therefore, the triangle faces of the poly-pyramid ele-ments are compatible to the triangular pyramid elements withoutany gaps or overlaps. The quadrilateral face of quadrilateral pyra-mid element is connected to the face of another quadrilateral pyr-

D. Sohn et al. / Comput. Methods App(12), respectively, as well as maximum error norm in Eq. (13) maximum error norm.

Ed Pnnode

i1 uexacti uhi

2Pnnode

i1 uexacti

2vuut ; 11

Ee RXeexact ehTCeexact ehdXR

XeexactTCeexactdX

vuut ; 12Em max

16i6nnodeuexacti uhi ; 13

where X is the entire domain and nnode is the total number ofnodes. The exact strain and the exact displacement correspondingto the problem are denoted by eexact and uexact, respectively. In addi-tion, eh and uh indicate the numerically calculated values.

Furthermore, the carving technique is compared with its varia-tional scheme. By additionally decomposing the pentagonal andthe hexagonal pyramid elements into several tetrahedral elements,the mesh can be reconstructed by only the tetrahedral and thequadrilateral pyramid elements, and the hexahedral elements.Throughout the comparison, we discuss the effectiveness of thecarving technique with the pentagonal and the hexagonal pyramidelements.

4.1. Patch test

A patch test was conducted to check the accuracy and conver-gence of the proposed method. The model of the patch test inFig. 8(a) was performed with a cube domain measuring4 m 4 m 4 m, Youngs modulus E = 1.0 106 Pa and Poissonsratio m = 0.3. The model is considered as a composition of sevenparts, and the inner part is surrounded by six outer parts. Each partconsists of several poly-pyramid elements, including triangular,

quadrilateral, pentagonal, and hexagonal pyramid elements.Fig. 8(b) and (c) show the composition of the inner part, and theupper outer part, which is one of the outer parts. The total numberof elements in the entire model is 93: There are 69 triangular pyr-amid elements, 12 quadrilateral pyramid element, 6 pentagonalpyramid elements, and 6 hexagonal pyramid elements.

We impose at the exterior nodes, as indicated in Fig. 8(a),the following linear displacement eld with constantstrain: u 1032x y z=2 m; v 103x 2y 2z=2 m; w 103x y 2z=2 m. Consequently, three error norms arecalculated by Eqs. (11)(13): the relative errors in displacementnorm and energy norm, and the maximum error norm. Their valuesare plotted in Fig. 9, with respect to the order of integration foreach subdomain by partitioning the poly-pyramid elements. Asmentioned in Section 2.2, the numbers of integration points ofthe pentagonal and the hexagonal pyramid elements are 5n3 and6n3, respectively, when the integration order is n. Fig. 9 shows thatthe errors decrease gradually as the integration order increases.Because the shape functions of the pentagonal and the hexagonalpyramid elements would be given in the form of the rational func-tions due to the MLS approximation, the result becomes moreaccurate as the order increases [24,28]. The lower order Gaussintegration is not enough to obtain an exact integral value of therational function. Therefore, we should implement the higher orderGauss integration for the acceptable magnitude of errors.

4.2. Sphere under hydrostatic pressure

To check whether or not the model obtained by the presentscheme represents uniformmean stress, a sphere is considered un-der hydrostatic pressure. The top of Fig. 5 shows the procedure to

218 D. Sohn et al. / Comput. Methods Appl. Mech. Engrg. 201204 (2012) 208227Fig. 12. The shape of femur and the boundary conditions imposed on top and bottom sides.

l. MeD. Sohn et al. / Comput. Methods Appmodel the sphere, whose radius r is 1 m. Soaking the surface infor-mation into the reference brick elements, the mesh for FEM is

Fig. 13. The meshes of the femur and the contours of von Mises stress obtainech. Engrg. 201204 (2012) 208227 219generated with the poly-pyramid elements. The displacements ofthe nodes on the surface are given to make the related surface node

d by: (a) the carving technique and (b) the use of Hypermesh and Abaqus.

.66 mm obtained by: (a) the carving technique with the poly-pyramid elements and (b)

125

126

The carving techniqueHypermesh & AbaqusReference solution

. Mech. Engrg. 201204 (2012) 208227moves to origin of the sphere, i.e., they are prescribed in the formof dr = (u2 + v2 + w2)1/2 = 0.01 m, where u, v, and w indicate the dis-placements in the x-, y-, and z-directions, respectively. The mate-rial properties are given as E = 1.0 106 Pa and m = 0.3.

In this example, the meshes are generated from 10 10 10,

Fig. 14. The contours of von Mises stress on the cross-section of the femur at z = 158the use of Hypermesh and Abaqus with the tetrahedral elements.

220 D. Sohn et al. / Comput. Methods Appl20 20 20, and 30 30 30 reference elements. Fig. 10 dis-plays the mean stresses dened as rmean = (rxx + ryy + rzz)/3 forthese three models, using the sixth-order integration for each sub-domain of the pentagonal and the hexagonal pyramid elements.The mean stress of the exact solution is 25,000 Pa, which is uni-formly distributed throughout the entire domain. It can be seenfrom Fig. 10 that the results by the proposed scheme are very sim-ilar to the exact values. Furthermore, the relative errors and max-imum error were calculated according to Eqs. (11)(13) to examinethe accuracy in a strict sense. Fig. 11(a) and (b) display the relativeerror in the displacement norm and energy norm versus the inte-gration order in each subdomain. The maximum error norm is alsoplotted in Fig. 11(c) with respect to the integration order. These g-ures show that the errors decrease in the case of higher order inte-gration, consistent with the tendency in the rst example.

4.3. Femur with complicated surfaces

As another example of modeling with general and complexgeometry, a static analysis of a femur was carried out. Due to thefact that the shape of the femur is very complex as shown inFig. 12, the conventional meshing schemes are associated withtroublesome procedures involving an extremely large number ofelements to ensure the accuracy of the solution. Moreover, thequality of the meshes may not be guaranteed by the traditionalscheme. In contrast, the carving technique proposed in this paperhas advantages of simplicity and efciency in the modeling ofthe complicated geometry.

The boundary conditions are imposed as follows: As shown inFig. 12, the bottom face of the model is xed, and displacementsin the negative z-directionwith amagnitude of 1 mmare prescribedat the nodes whose z-coordinate is greater than 169.23 mm. The

Table 2The time spent on the modeling and analysis of the femur.

Scheme The number ofnodes

The number ofelements

Wall-clock time (s)

Modeling Analysis

The carvingtechnique

6,101 13,421 0.437 10.76612,752 26,202 0.828 24.71922,917 44,472 1.500 45.84455,612 98,254 4.453 128.781

Hypermesh andAbaqus

6,683 32,486 1.203 712,809 64,923 2.013 1422,860 119,249 3.370 2756,047 302,921 8.507 95

The number of nodes

Tota

lstra

inen

ergy

(mJ)

10000 20000 30000 40000 50000 60000

122

123

124

Fig. 15. The total strain energy of the femur versus the number of nodes.

material properties of the femur are given as E = 15 GPa andm = 0.25. Setting the allowable relative errors to 106, as in the twoprevious examples, we deduce that the sixth-order integration forthe integration by the partitioning of the pentagonal and the hexag-onal pyramid elements may be needed to obtain the same level ofaccuracy. Therefore, all results in this example were obtained usingthe sixth-order integration of the poly-pyramid elements, whichcomprises only a fraction of the entire elements.

According to the procedure of the carving technique as shownin the bottom of Fig. 5, several meshes were prepared by varyingthe numbers of reference elements. Fig. 13(a) shows the meshesfor each model with the total number of nodes and elements.The distributions of von Mises stress are also shown in Fig. 13(a).To verify whether or not these results by the carving techniqueare reasonable, we conducted modeling and analysis of this prob-lem using Hypermesh 9 and Abaqus 6.10, commercial packagesfor FE analysis. The meshes with four-node linear tetrahedral

elements were generated by Hypermesh, and then FE analyseswere conducted by Abaqus. The mesh congurations and the vonMises stress contours are shown in Fig. 13(b). It can be seen fromFig. 13(a) and (b) that the results by the carving technique andAbaqus are very similar to each other. In particular, nearly thesame distributions of von Mises stress are observed in the rightbottoms of Fig. 13(a) and (b) which display the results with the -ner meshes by the carving technique and by the use of Hypermeshand Abaqus, respectively. Fig. 14 displays the distribution of vonMises stress on the cross-section of the ner models in Fig. 13(a)and (b), where the z-coordinate equals 158.66 mm. It is observedthat the stress contour by the carving techniques is in good accordwith the result by Abaqus, as is evident from Fig. 14.

Furthermore, the total strain energy was calculated to check thetendency of convergence. Fig. 15 displays the curves of the totalstrain energy versus the total number of nodes, according to threesolutions: red indicates the solution from the present carving

D. Sohn et al. / Comput. Methods Appl. Mech. Engrg. 201204 (2012) 208227 221Fig. 16. The model description of a cube with voids: (a) the random distribution of various sized voids and (b) the geometry and boundary conditions.

. Me222 D. Sohn et al. / Comput. Methods Appltechnique, while blue from Hypermesh and Abaqus models withfour-node linear tetrahedral elements, and orange from anothertype of the Hypermesh and Abaqus model with ten-node quadratictetrahedral elements. Note that the total strain energy from theHypermesh and Abaqus models with the quadratic elements is ob-tained as the converged value, 122.09 mJ, with 424,910 nodes and302,921 elements. In this context, this converged solution is em-ployed as a reference solution to which other solutions may becompared. It is noticed that our solution from the carving tech-nique approaches the reference solution faster than the solutionfrom the Abaqus models with four-node linear tetrahedral ele-ments, as the renement is enhanced with the increasing totalnumber of nodes.

To verify the efciency of the proposed scheme, we evaluatedthe wall-clock time to generate nite element meshes and to ob-tain the solutions. Using Intel(R) Core(TM)2 Extreme CPU X9770

Fig. 17. The distributions of von Mises stress in the cube with voids obtained by: (a) thelements; and (c) the use of Hypermesh and Abaqus with quadratic tetrahedral elemench. Engrg. 201204 (2012) 208227(3.2 GHz), the time for mesh generation and analysis was mea-sured as shown in Table 2. When the models are constructed withthe similar number of nodes, it is observed from Table 2 that thecarving technique is cheaper than Hypermesh in the aspect of com-putational cost of modeling. In the case of analysis, Abaqus pro-vides better performance than the carving technique, becauseAbaqus is the commercial tool which adopts the advanced tech-niques to reduce the computational time. It is noteworthy thatthe carving technique gives the accurate solutions with low costof modeling.

4.4. Cube with the randomly distributed voids

A cube including sphere-shaped voids, illustrated in Fig. 16(a),is chosen as an example of the model with inner trimmed surfaces.Inside the 100 mm 100 mm 100 mm cube, total 30 voids are

e carving technique; (b) the use of Hypermesh and Abaqus with linear tetrahedralts.

meshes faster than Hypermesh, while Abaqus gives the solutionfaster than the carving technique. These are consistent with thetendencies of the previous example.

4.5. Numerical performance of pentagonal and hexagonal pyramidelements

In the splitting process of the carving technique, the trimmedelements are decomposed into the poly-pyramid elements includ-ing the pentagonal and the hexagonal elements. We now considera variation of the carving technique, with additional decompositionof the pentagonal and the hexagonal pyramid elements into sev-eral tetrahedral elements. By doing so, the mesh is composed ofonly the tetrahedral and the quadrilateral pyramid elements, andthe hexahedral elements.

First, a three-dimensional cantilever beam is considered undertransverse loading condition as shown in Fig. 19(a). The materialproperties are given as E = 210 GPa and m = 0.3. Fig. 19(a) displays40 8 8 reference brick elements of the cantilever beam. Toexamine the performance of the carving technique and its varia-tional version, the reference brick elements are trimmed by fourinner surfaces and decomposed as shown in Fig. 19(b). Moreover,the mesh is constructed with only the tetrahedral elements, bydecomposing a reference brick element into ve tetrahedral ele-ments as shown in Fig. 19(c). In the same manner, based on60 12 12, 80 16 16 and 100 20 20 reference brick ele-

Scheme The number ofnodes

The number ofelements

Wall-clock time (s)

Modeling Analysis

The carvingtechnique

11,213 19,734 0.453 38.57931,961 50,698 1.641 92.20369,134 102,210 4.484 216.782

Hypermesh andAbaqus

10,028 48,796 1.830 1138,528 200,804 6.293 7472,235 385,286 11.833 187

l. Merandomly distributed, and radii of voids vary from 10 mm to20 mm. Using conventional meshing algorithms, the modeling isaccompanied by cumbersome procedures if there are the trimmedsurfaces inside the model. Therefore, this problem can be regardedas an appropriate example to examine the effectiveness of thecarving technique.

The displacement boundary conditions are imposed on bottomand top faces of the cube, as shown in Fig. 16(b). The displacementsin the z-direction are prescribed as zero at the nodes located on thebottom face, and the displacements in the x- and y-directions areprescribed at two nodes to avoid rigid body motions. Otherwise,on the top face, the displacements in the negative z-direction areimposed with a magnitude of 0.1 mm. The material of the modelis aluminum with E = 70 GPa and m = 0.32. Similar to the previousexample, the sixth-order integration is used for the integrationby the partitioning of the pentagonal and the hexagonal pyramidelements.

Fig. 17(a) displays the distributions of von Mises stress, whichare obtained by using the meshes from the carving technique. Eventhough the voids are located inside the cube and on the face of thecube, the meshing procedure is the same as that for the case with-out the inner trimmed surface. To conrm the accuracy of the re-sults with poly-pyramid elements, we again conducted themodeling and analysis using Hypermesh and Abaqus with four-node linear tetrahedral elements because there is no analytic solu-tion for this example. The analysis results from Hypermesh andAbaqus are shown in Fig. 17(b). The distributions of von Misesstress from the Hypermesh and Abaqus models are very similarto the results from the carving technique.

In addition, the maximum stress of each model appears at thesame position, and it becomes higher as the number of nodes in-creases, as shown in Fig. 17. The maximum values by the carvingtechniques are higher than those by the use of Hypermesh andAbaqus when the similar number of nodes is used in the analyses.However, all these values are bounded by the maximum stress of areference solution, 370.859 MPa, which were calculated by usingAbaqus with ten-node quadratic tetrahedral elements. As depictedin Fig. 17(c), this value was obtained with 544,780 nodes and385,286 elements, and may be regarded as the reference solution.The carving technique gives a smaller discrepancy between theirresults and the reference solution than the use of Hypermeshand Abaqus.

In the same manner as the previous example, we also evaluatedthe tendency of convergence in terms of the total strain energy.The results from the carving technique are compared with thosefrom Hypermesh and Abaqus in Fig. 18. It is observed that the val-ues of the total strain energy from the two meshing schemes areconverged in almost the same ratio, as the number of nodes in-creases. However, the values from the carving technique are lowerthan those from Hypermesh and Abaqus. To check which valuesare more accurate, the results are compared with the convergedvalue, 22.050 J, of the reference solution with the quadratic tetra-hedral elements. From the fact that the result from the carvingtechnique is closer to the reference solution than the result fromHypermesh and Abaqus, it is found out that the carving techniquewith the hexahedral and the poly-pyramid elements provides bet-ter solutions than Abaqus with the four-node linear tetrahedralelements.

Finally, the carving technique is compared with the use ofHypermesh and Abaqus in terms of computational cost. Thewall-clock time spent on the mesh generation and analysis is mea-sured using the identical computer systemmentioned in the previ-ous example, and the values are summarized in Table 3. Comparing

D. Sohn et al. / Comput. Methods Appthe carving technique with the use of Hypermesh and Abaquswhen the numbers of nodes or degrees of freedom are on a similarlevel, it can be seen that the carving technique generates theThe number of nodes

Tota

lstra

inen

ergy

(J)

10000 20000 30000 40000 50000 60000 70000

22.0

22.2

22.4

22.6

22.8

23.0

23.2 The carving techniqueHypermesh & AbaqusReference solution

Fig. 18. The total strain energy of the cube with voids versus the number of nodes.

Table 3The time spent on the modeling and analysis of the cube with the randomlydistributed voids.

ch. Engrg. 201204 (2012) 208227 223ments, more rened meshes were obtained. Using the variousmeshes, convergence tests were carried out for the problem ofthe cantilever beam.

. Me(a) u = v = w = 0

100 mm

x

y

z

224 D. Sohn et al. / Comput. Methods ApplFig. 20(a) shows tip deections in the y-direction at the node,indicated by A in Fig. 19(a), with respect to the number of nodes.It is observed that the deections for the hexahedral elements,which are the best result, converge much faster than those for theothers, as the number of nodes increases. Otherwise, the deectionsfor the tetrahedral elements are relatively small with a low conver-gence ratio. Between the results of the hexahedral and the tetrahe-dral elements, there are the results from the carving techniques, aswell as those from the variation of the carving techniques withoutthe pentagonal and the hexagonal pyramid elements. Meanwhile,the values of the total strain energy are plotted in Fig. 20(b). The ten-dency of convergence in terms of the total strain energy is consistentwith that in terms of the tip deections.

Subsequently, the variation of the carving technique is appliedto the previous two examples. For the modeling and analysis ofthe femur, the meshes are reconstructed from those in Fig. 13(a)by decomposing the pentagonal and the hexagonal pyramid ele-

Additional decomposition for a va

A hexagonalpyramid element

Mesh generation by th

Triangularpyramidelements

A pentagonalpyramid element

(b)

(c)

x

y

z

x

y

z

Triangular pyramid elements over the entir

Fig. 19. A cantilever beam under transverse loading condition: (a) the geometry and bocarving technique and its variation; and (c) the mesh with tetrahedral elements over thA

20 mm

20 mm

P = 10 MPa

ch. Engrg. 201204 (2012) 208227ments into the tetrahedral elements. In the same manner, for thecube with the randomly distributed voids, the meshes are recon-structed from those in Fig. 17(a) without the pentagonal and thehexagonal pyramid elements. Fig. 21(a) and (b) show the valuesof the total strain energy from the variation of the carving tech-nique with the results in Sections 4.3 and 4.4. It can be seen thatthe carving technique provides slightly better results, closer tothe reference solution, than its variation.

From the above results in this section, we can deduce as fol-lows: In the framework of the carving technique, the pentagonaland the hexagonal pyramid elements do not signicantly affectthe results in the simple problems such as the cantilever beam.However, in more practical problems, these elements play a roleto enhance the accuracy of solutions. Besides, when various typesof the poly-pyramid elements are directly used in the splitting pro-cess, we can reduce an additional cost to decompose them into thetetrahedral elements. Therefore, it is worthwhile to apply the pen-

riation of the carving technique

e carving technique

x

y

z

e domain

undary conditions for the reference brick elements; (b) the mesh generated by thee entire domain.

lst

l. MeTip

defle

ctio

n(m

m)

0.470

0.475

0.480

0.485

The carving techniqueThe variation of the carving techniqueHexahedral elementsTetrahedral elements

a

D. Sohn et al. / Comput. Methods Apptagonal and the hexagonal pyramid elements to the carvingtechnique.

5. Conclusion

In this paper, a novel meshing scheme, termed the carving tech-nique, is proposed. The scheme is based on the marching cubealgorithm in combination with the poly-pyramid elements byMLS approximation. The carving technique simplies the meshgeneration procedure. In particular, a strong advantage is gainedwhen it is applied to structures with a complicated geometry. Fur-thermore, the entire process, from modeling to analysis, can beimplemented automatically.

Through several numerical examples, the performance of thepoly-pyramid elements developed in this paper was veried andefcient applications of the carving techniques were shown. Theproposed scheme is expected to be useful for various applications,such as three-dimensional crack propagation, physics-based com-puter graphics, and others. Future work may include the proposedscheme applied to more complex problems.

The number of nodes0 10000 20000 30000 40000 50000

The number of nodes

Tota

lstra

inen

ergy

(mJ)

0 10000 20000 30000 40000 50000

935

940

945

950

955

960

965

970

The carving techniqueThe variation of the carving techniqueHexahedral elementsTetrahedral elements

b

Fig. 20. The tendency of convergence according to the elements used in thecantilever beam problem, in terms of: (a) tip deections and (b) the total strainenergy.Tota

122

123rain

ener

gy(m

J)

124

125

126

The carving techniqueThe variation of the carving techniqueHypermesh & AbaqusReference solution

a

ch. Engrg. 201204 (2012) 208227 225Acknowledgements

Y.-S. Cho gratefully acknowledges the support from Mid-careerResearcher Program through NRF grant funded by the MEST (No.2009-0083774). S. Im and D. Sohn also appreciate the support fromNRF for the Grant (No. R0A-2007-000-20115-0).

Appendix A. Determination of the position of an additionalnode for the proper element splitting

In the splitting process, an additional node is introduced in thetrimmed non-conventional elements. The position of the addi-tional node should be appropriately determined to avoid negativevalues of the Jacobian of the split elements. In this paper, we adopta simple way to determine the physical coordinate xa (xa,ya,za) ofthe additional node, as follows:

xa 1NelXNelI1

xI; A:1

The number of nodes

Tota

lstra

inen

ergy

(J)

10000 20000 30000 40000 50000 60000 70000

22.0

22.2

22.4

22.6

22.8

23.0

23.2 The carving techniqueThe variation of the carving techniqueHypermesh & AbaqusReference solution

The number of nodes10000 20000 30000 40000 50000 60000

b

Fig. 21. The tendency of convergence in terms of the total strain energy, accordingto the elements used in: (a) the example in Section 4.3 and (b) the example inSection 4.4.

(2010) 405415.

t

elem

. Mewhere Nel is the number of nodes in a trimmed element, and xI is thephysical coordinate of node I (I = 1,2, . . .,Nel) in the trimmed ele-ment. In the majority of cases, the center in an average sense is va-lid, because the trimmed elements remain as the convexpolyhedrons or they have little geometrical non-convexity, asshown in Fig. 6.

However, the averaged center may lead to the negative Jacobianof the split elements, when the reference brick elements are notsufciently ne or there is extremely sharp curvature in the modelsurface. In this case, the trimmed elements may be severely non-convex, and then another scheme should be considered to splitthem without any elements which have the negative Jacobian.For example, the nodal position can be determined through trialand error. The position is iteratively shifted with a small perturba-tion from the center, until all of the split elements have the positiveJacobian. Alternatively, the additional node can be placed on thetrimmed surface, not inside the non-convex elements.

Appendix B. Adjustment of nodal positions for small and skew

The intersection pointsvery close to the active nodes

Pentagonalpyramid element

Pentagonalpyramid elemen

Pentagonalpyramid element

a

Active nodeIntersection pointAdditional node in an element

Fig. B.1. Adjustment of nodal positions to improve the mesh quality: (a) the trimmednodes to the intersection point.

226 D. Sohn et al. / Comput. Methods Applelements

Consider the trimmed elements which have the intersectionpoint close to the active node, as shown in Fig. B.1(a). In this case,the distorted poly-pyramid elements may be generated with theoriginal nodal positions of the reference brick element in the split-ting process. Therefore, if the distance between the intersectionpoint and the active node is less than the preset value of tolerance,the active node is moved to the intersection point. In this work, thetolerance is preset as 10% of the length of an element edge.Fig. B.1(b) illustrates an example of the adjustment of nodal posi-tions. Through this process, the carved shapes from the pre-exist-ing faces of hexahedral element are changed: from quadrilateralto triangle; from pentagon to triangle or quadrilateral; from hexa-gon to triangle, quadrilateral, or pentagon. By replacing the coordi-nate of the active node to that of the intersection point, thesplitting process is conducted without small and skew elements.It is shown in Fig. B.1(b) that the mesh quality is improved bythe adjustment of nodal positions.

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A novel scheme to generate meshes with hexahedral elements and poly-pyramid elements: The carving technique1 Introduction2 Development of poly-pyramid elements2.1 Shape functions of poly-pyramid elements2.2 Gauss integration by partitioning the domain of a poly-pyramid element

3 Mesh generation with the aid of poly-pyramid elements4 Numerical results and discussion4.1 Patch test4.2 Sphere under hydrostatic pressure4.3 Femur with complicated surfaces4.4 Cube with the randomly distributed voids4.5 Numerical performance of pentagonal and hexagonal pyramid elements

5 ConclusionAcknowledgementsAppendix A Determination of the position of an additional node for the proper element splittingAppendix B Adjustment of nodal positions for small and skew elementsReferences