Finite Element Analysis Based on Volume CAD

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Finite Element Analysis Based on Volume CAD∗

Tomoshi MIYAMURA∗∗,∗∗∗, Tomonori YAMADA∗∗∗, Hiroshi AKIBA∗∗∗∗,Akitake MAKINOUCHI∗∗∗ and Cristian TEODOSIU∗∗∗,†

Volume CAD, which is being developed at RIKEN, will be a common platform forstoring and handling volume data in manufacturing processes. In VCAD, both geometryand volume data are represented as attributes of a set of cells in a simple voxel or octree datastructure. In this paper, a method by which to conduct highly accurate structural analysisby directly using VCAD data is proposed. A degenerated hexahedral mesh representing theprecise geometry of an object is generated from VCAD data, and finite element analysis isthen conducted using the mesh. The proposed method is compared with the X-FEM, and theperformances of these two methods are shown to be approximately the same. The robustnessof the proposed method is demonstrated by presenting numerical examples of stamping tools.

Key Words: Manufacturing, CAD, VCAD, Mesh Generation, Finite Element Method,Voxel, Octree

1. Introduction

Three-dimensional (3D) solid modeler (3D CAD) hasbeen used widely in the design of artifacts. In 3D CADs,parametric surfaces such as the NURBS (NonUniform Ra-tional B-Spline) are used to represent surfaces of complexshape that define a solid model. Geometry data createdby 3D CADs are often used to generate input data forcomputer simulations that are important in product design.The finite element method (FEM) is one of the most com-mon methods for simulations dealing with stress analysisof solids and fluid analysis. Input data for FEM are a fi-nite element mesh and physical attributes that are attachedto the mesh, and output data are attributes that are alsoattached to the mesh. These input and output data are vol-ume data, i.e., data that are distributed in a 3D field insidean object. However, 3D CADs cannot treat such volumedata because they are designed for the generation of only

∗ Received 19th November, 2004 (No. 04-4239)∗∗ Department of Computer Science, College of Engineer-

ing, Nihon University, 1 Nakagawara, Tokusada, Tamura-machi, Koriyama, Fukushima 963–8642, Japan.E-mail: [email protected]

∗∗∗ Integrated VCAD System Research Program, Institute ofPhysical and Chemical Research, 2–1 Hirosawa, Wako,Saitama 351–0198, Japan

∗∗∗∗ Allied Engineering Corporation, 1–21–17 Oi, Shinagawa-ku, Tokyo 140–0014, Japan

† LPMTM-CNRS, University Paris 13, 93430 Villetaneuse,France

the surface geometry of objects.Volume CAD (VCAD), which is being developed at

RIKEN, will be a common platform for storing and han-dling volume data in manufacturing processes that use theinput and output data of conventional CAD, CAE, CAM,or CAT, for example(1), (2). In VCAD, both geometry dataand volume data are represented as attributes of a set ofcells. Each cell is an element of a simple voxel or octreedata structure. VCAD that uses only the voxel data struc-ture is referred to as the voxel version of VCAD; whereasVCAD that uses the octree data structure is called the oc-tree version of VCAD. In VCAD, cells that contain bothgeometry and volume data are referred to as boundarycells, and cells that contain only volume data are callednon-boundary cells. The surface geometry of an object isrepresented by triangles that are also stored as attributes ofthe boundary cells. In the following, the set of data storedin the cells of VCAD is called VCAD data.

It is desirable to utilize VCAD data for the simula-tions in order to seamlessly connect design, analysis andfabrication. The geometry data in the boundary cells andthe coordinates of the vertices of each cell can be used togenerate an analysis model such as a finite element mesh.In addition, as containers in which to store volume data,each cell can be used to transfer data in a series of dif-ferent analyses, for instance, thermal analysis and thermalstress analysis.

The data structure in the voxel version of VCAD canbe used directly in FE analysis using a voxel mesh. Thismethod is often employed in image-based analysis(3) and

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is also suitable for analysis using VCAD data. Each cellin VCAD can be considered as a uniform hexahedral el-ement. However, the geometry data stored in the bound-ary cells are used only for the inside-outside check that isconducted in order to choose the voxels inside an objectto be analyzed. Nagashima et al. applied the extended fi-nite element method (X-FEM) proposed by Belytchko etal.(4) to analysis based on VCAD data(5) – (8). In X-FEM,discontinuities in a finite element can be represented, andthe input data for the method can be generated by VCADdata. However, the Dirichlet boundary conditions are dif-ficult to impose in X-FEM, which is a typical problemfor meshfree methods because the shape functions are notconsistent with the surface geometry. For the same reason,the contact problems, which are one of the most importantproblems in the simulations of manufacturing processes,may be difficult to consider in the X-FEM. In the fieldof fluid dynamics, Lei et al. proposed a method that usesVCAD data directly to make input data(9). The volume offluid (VOF) approach, based on the finite volume method,was employed in their study. The most widely used meshgeneration approach for octree description of geometry isthe octree mesh generation method(10). In this method,each octant is decomposed into tetrahedral elements bylocal Delaunay tessellation and smoothing of the finite el-ement mesh, in order to obtain a mesh with a better aspectratio. The bucket-by-bucket mesh generation approach isanother grid-based mesh generation technique for hierar-chical geometry description that is applicable to parallelcomputation(11). These mesh generation techniques, how-ever, were not specifically designed for the VCAD datastructure.

In this paper, a method for conducting highly accuratestructural analysis by directly utilizing VCAD data is pro-posed. The proposed method is based on the finite elementmethod. A mesh that represents the precise geometry of ananalysis model is generated using degenerated hexahedralelements from VCAD data. The proposed mesh genera-tion technique is designed particularly for being used inconjunction with VCAD data, in order to achieve fast androbust mesh generation.

In section 2, the mesh generation technique usingthe degenerated hexahedral elements is proposed. In sec-tion 3, parallel finite element analysis procedures makinguse of the mesh generated by the proposed method are ex-plained. Section 4 presents a number of illustrative exam-ples analyzed using the proposed method. The proposedmethod is compared with X-FEM based on the results oftwo test examples shown in this section. Concluding re-marks are presented in section 5.

2. Mesh Generation Using VCAD Data

2. 1 Data structure of VCAD dataIn volume CAD (VCAD), both geometry data and

volume data are stored as attributes of a set of cells(2).Each cell is an element of a voxel or octree data struc-ture. As described in section 1, a set of data stored inthe cells of VCAD is called the VCAD data. Cells thatcontains both geometry and volume data are referred toas boundary cells, whereas cells that contain only volumedata are called non-boundary cells. The geometry of anobject is represented by a set of triangular polygons; eachof which is confined within a boundary cell and stored asthe attributes of the cell.

A boundary cell with a number of triangles is gener-ated using the Kitta cube method developed by Kase andTeshima et al(12). In the Kitta cube method, some of theedges of the cell are cut by one vertex of a triangle. Thepoint at which the vertex is placed is called the cuttingpoint. There are two important rules for the placement ofcutting points: (1) the vertex of the cell cannot be a candi-date cutting point, and (2) only one cutting point can existon the edge of the cell. According to these rules, morethan one triangle can exist in a cell. The number of con-figurations of the triangles in the boundary cell generatedusing the Kitta cube method is very large, indicating thatsurfaces of complex shape can be approximated preciselyby the Kitta cube method. This feature distinguishes theKitta cube method from the marching cube method(13), inwhich the configurations of the surfaces in a cell can beclassified into only 15 patterns.

2. 2 Basic procedure for generating a degeneratedhexahedral mesh

In the following, a simple method for generating afinite element mesh by employing the VCAD data is pro-posed. In this method, the non-boundary cell of the VCADis adopted directly as a hexahedral element. The bound-ary cell is subdivided into a number of degenerate hexa-hedrons that can be used as degenerated hexahedral finiteelements. One critical aspect of this method is the subdivi-sion of the boundary cell. In case of the marching cube(13),a lookup table for conducting the subdivision can be pre-pared since the number of patterns for the marching cubeis only 14. On the other hand, as described in the previoussection, the number of patterns of the boundary cell gen-erated by the Kitta cube method is extremely large, andso making a lookup table for the subdivision is practicallyimpossible. However, the boundary cells are created us-ing some simple rules, and as a result, subdivision of thecell into degenerated hexahedrons can be performed by asimple procedure, as described in the following.

Figure 1 shows a cube that is cut into two solids bya triangle. Each of the subdivided solids is surrounded bypolygons. Because a polygon with more than four ver-tices can be subdivided into triangles and/or quadrilater-als, the solid can be converted into a number of solidsthat are surrounded by several triangles and/or quadrilat-erals. When this solid is convex, it can be subdivided into

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Fig. 1 A cube cut into two solids by a triangle

Fig. 2 A convex solid subdivided into tetrahedrons and/orpyramids

Fig. 3 Subdivision of a cell with a hanging node

a number of tetrahedrons and/or pyramids, which are rep-resented as degenerated hexahedrons, by simply placinga vertex (node) in the center of the solid and connectingthe newly added vertex with the vertices of the trianglesand/or quadrilaterals, as shown in Fig. 2. The coordinatesof the added vertex are calculated by averaging the coor-dinates of the vertices of the solid. This procedure canalso be used to treat hanging nodes (vertices) in an octreemesh, as shown in Fig. 3.

When the geometry of the subdivided solid is a tetra-hedron, pyramid, prism or hexahedron, then the geometrycan be represented by degenerated hexahedrons, and thusthe technique described in Fig. 2 can be skipped. Thisstrategy is also implemented in our mesh generator, andcan be chosen as an option.

2. 3 Modification of the geometry of the boundarycells for mesh generation

In order to generate degenerated hexahedral elementsin the boundary cell, modification of the geometries ofsome boundary cells is performed in the mesh generationprocedure, as described below.

One important technique is the modification of theposition of the cutting point that is very close to a vertexof the cell, as shown in Fig. 4. Surface triangles havingvery large aspect ratios can be removed by this technique.

Fig. 4 Modification of cutting point positions

Fig. 5 Typical examples of the modification of cutting pointpositions

(a) (b)

Fig. 6 Elimination of triangles with large aspect ratios bymodification of cutting point positions

In addition, tiny degenerated hexahedral elements, whichsometimes cause numerical troubles in finite element anal-ysis, are eliminated. Figure 5 shows typical examples of aboundary cell modified by this technique. After this mod-ification, some triangles may be collapsed to an edge orvertex, and some solids may have zero volume. Thesecollapsed triangles and solids are removed after modifica-tion of the cutting points. In Fig. 6 (a), some triangles inthe mesh having very large aspect ratios are identified, andin Fig. 6 (b), these triangles are removed by applying theabovementioned modification technique.

Another important technique is the swapping of a di-agonal edge in a quadrilateral. When a solid that is gen-erated by cutting a cell is not convex, the position of a di-agonal edge that splits a quadrilateral into two triangles isswapped, as shown in Fig. 7. By employing this technique,the subdivided solid is converted into a convex solid.

After applying the abovementioned techniques, in-consistencies with respect to the positions of cutting pointsamong neighboring boundary cells may occur. These in-consistencies are resolved by post processing, as shown inFig. 8. Since the number of patterns of the inconsistencyis limited, this post processing is implemented by defin-


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Fig. 7 Swapping of diagonal edges in order to make a convexsurface

(a) Original (Kitta cube) (b) After elimination of somecutting points and swap-ping of triangles

(c) After modification

Fig. 8 Modification of the inconsistencies of cutting pointpositions among the neighboring boundary cells

Fig. 9 Patterns of inconsistencies that are stored in the lookuptable for the modification of cutting points.

ing a lookup table, as shown in Fig. 9, for classifying thepatterns. The inconsistencies of the first three patterns areencountered most frequently.

3. Finite Element Analysis Using the DegeneratedHexahedral Mesh

3. 1 Use of degenerated hexahedral elements in ex-isting finite element codes

Tetrahedrons, pyramids and prisms can be obtainedby simply degenerating (collapsing) some of the ver-tices of a hexahedron into one vertex. The vertices of adisplacement-based linear hexahedral finite element canalso be collapsed (degenerated), i.e., the element stiff-ness matrix for tetrahedral, pyramidal or prismatic finiteelements can be calculated by degenerating a number ofnodes of a hexahedral element into one node(14). For in-stance, the tetrahedral element that is generated by the de-generation procedure is the same as a constant strain lin-ear tetrahedral finite element. Thus, it is not necessary toimplement specific element libraries for tetrahedral, pyra-midal or prismatic elements.

From the viewpoint of input data, a degenerated hex-ahedral element is defined by assigning the same globalnode number to all of the nodes to be collapsed. There-fore, it is not necessary to modify the code of the finiteelement solver in order to treat degenerated hexahedral el-ements. This is one of the advantages of the proposedmethod compared to other methods, such as X-FEM. Thatis, the resources of existing finite element solvers can beused without modification. As such, parallel finite elementcodes can be used for the analysis using the VCAD data.A great advantage here is that the finite element analysisprocedures with degenerated hexahedral elements gener-ated by the method described in section 2 can be immedi-ately parallelized using existing codes because the numberof degrees of freedom of the finite element meshes of in-dustrial products of complex shape becomes very large.

3. 2 Parallel computation based on domain decom-position

In the domain decomposition method (DDM), ananalysis domain is decomposed into a number of subdo-mains(18). When a problem is discretized based on FEM,the decomposition of the analysis domain corresponds tothe subdivision of a finite element mesh. This subdivisionis conducted using the software called the domain decom-poser before starting the analysis. The DDM is suitablenot only for shared-memory type parallel computers, butalso for distributed-memory systems. There are a numberof parallel finite element analysis codes. In this study, theADVENTURE system(15), (16), which is a freely availablesystem of software tools for parallel finite element analy-sis, and the ADVENTURECluster(17), which is the com-mercial version of the ADVENTURE system, are usedfor analysis using the mesh generated as described in sec-tion 2. The domain decomposer included in the systemrequires only minor modification in order to conduct theanalysis using the degenerated hexahedral mesh.


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3. 3 Application to contact problemsA solid model created by a conventional CAD is made

up of a number of parametric surfaces. When two solidmodels that represent two objects to be analyzed are ini-tially in contact, the parametric surfaces of the two solidsat the contact region should be the same. The VCADmodels generated from these solid models have almost thesame boundary cells at the contact region, because theseboundary cells are generated from the same parametricsurface. Therefore, two finite element meshes that havealmost the same location of nodes at the initial contact re-gion can be generated from the two sets of VCAD datausing the method proposed in section 2. In this case, anode-to-node contact model, which is the simplest con-tact model in the finite element contact analysis, can beadopted.

Many industrial products consist of assembled partsthat are initially in contact. Therefore, the abovemen-tioned procedure is useful in the structural analysis of in-dustrial products. This procedure was also applied to theanalysis of a crust structure having discontinuous surfacesbetween plates. The solid models of the crust structure arecreated by the solid modeler that was developed especiallyfor the modeling in the CHIKAKU project, which is partof the earth simulator research(19); the VCAD data is thengenerated from the models. Finally, the degenerated hex-ahedral mesh with the contact surfaces is generated fromthe VCAD data.

4. Illustrative Examples

4. 1 Examples of mesh generation by the proposedmethod

Figure 10 shows a mesh of a tube. In order to demon-strate the robustness of the proposed method, the centeraxis of the tube is inclined 45 degrees with respect to theVCAD frame. VCAD data with 32× 32× 32 voxels is

Fig. 10 Degenerated hexahedral mesh of a tube generated by the proposed method (32×32×32voxels, center axis of the tube is inclined 45 degrees)

created. Figure 11 shows two meshes of a sphere. Themeshes in Fig. 11 (a) and (b) are generated from VCADdata with 32×32×32 voxels and 64×64×64 voxels, re-spectively. Figure 12 illustrates the degenerated hexahe-dral mesh for a simple beam with octree VCAD data. Aclose-up illustration of the area connecting different octreelevels is shown in Fig. 13. The degenerated hexahedralmeshes of a set of stamping tools are shown in Fig. 14.These meshes were used in the sheet metal forming anal-ysis considering the deformation of the tools.

Although the computation time required for makingthe VCAD data using the current version of VCAD is notnegligible compared to the computation time for the meshgeneration, the development of VCAD will be continuedand the performance of VCAD will be certainly improved.

4. 2 Comparison with X-FEMAs described in section 1, X-FEM is a method in

which VCAD data can be utilized directly. In X-FEM,some shape functions that are multiplied by a so-called en-riched function are added to the conventional shape func-tions of the finite element method in order to enrich thesolution, for instance, for a crack propagation analysis. Inthe case of VCAD application, the enriched terms can beused to treat the boundary of an object that passes throughan element. Therefore, the method is suitable for import-ing VCAD data having voxel or octree data structures withthe boundaries of an object represented as triangles in eachcell. Nagashima et al.(5) – (8) used a set of the following en-riched shape functions to treat VCAD data:

uh(x)=NE∑

I=1φI(x)V(x)uI , (1)

where uh(x) is an interpolated displacement vector at apoint x, NE is the number of nodes of an element, φI(x)designates the conventional shape functions of an isotropicelement, uI is the nodal displacement vector at node I,


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(a) 32×32×32 voxels (b) 64×64×64 voxels

Fig. 11 Degenerated hexahedral meshes of a sphere

Fig. 12 Degenerated hexahedral mesh for a simple beam withoctree VCAD data

Fig. 13 Close-up of area connecting different octree levels

and V(x) is an enriched function. Here, V(x) is a stepfunction that represents the discontinuity of the displace-ment field at the boundary of an object: V(x) is 1.0 insidethe object and 0.0 outside the object. In the following il-lustrative examples, 8× 8× 8 subvoxels are generated inthe boundary-cell for the numerical integration of the en-riched element. The Neumann boundary conditions in theenriched element are represented as an equivalent nodalforce vector, and the Dirichlet boundary conditions in theelement are treated as multipoint constraints for the nodesof the enriched elements. This method is described in de-tail in Refs. (5) – (8).

A rectangular hexahedron with a rectangular hole is

Fig. 14 Degenerated hexahedral meshes of stamping tools

solved by the present method, the X-FEM, and a commer-cial finite element analysis code ABAQUS with a hexa-hedral mesh. In the ABAQUS analysis, C3D8 elementsare used. The dimensions of the analysis model are shownin Fig. 15. The Young’s modulus and Poisson’s ratio are21 GPa and 0.3, respectively. The solid model shown inFig. 15 is converted into VCAD data. The boundariesof the solid model do not coincide with the boundariesbetween the voxels in the VCAD model; therefore, theboundaries of the solid model are represented by the tri-angles in the boundary cells. The number of cells is64× 8× 128. Figure 16 shows the boundary conditions.The total numbers of degrees of freedom of the X-FEMmodel, the degenerated hexahedral model, and the hexahe-dral model for ABAQUS are 84 660, 82 986, and 15 330,respectively.

The results of the analyses are compared in Table 1and are found to be approximately equivalent. The compu-tation time for the translation of VCAD data into finite ele-ment analysis data of X-FEM and that for the degenerated


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Fig. 15 Dimensions of a rectangular hexahedron with arectangular hole

Fig. 16 Boundary conditions for a rectangular hexahedron witha rectangular hole

Table 1 Comparison of results (deghex: degenerated hexa-hedral model, X-FEM: X-FEM model, ABAQUS:ABAQUS with hexahedral mesh)

hexahedral mesh are 27 seconds and 13 seconds, respec-tively, using a Pentium4 (1.7 GHz) processor. The com-putation times for both analyses are approximately equal:140 seconds for X-FEM and 133 seconds for the proposedmethod. In the X-FEM analysis, elimination of the depen-dent conditions in the multipoint constraints for represent-ing the Dirichlet boundary conditions is conducted beforestarting the analysis. Because this procedure is so timeconsuming, the translation time for X-FEM is twice thatof the degenerated hexahedral model, even for this simpleanalysis model. In the case of the stamping tool shown

Fig. 17 VCAD data for a stamping tool for the bumper of anautomobile (courtesy of Tsubamex Co., Ltd.)

Table 2 Comparison of displacement at loading point (deghex:degenerated hexahedral model, X-FEM: X-FEMmodel)

in Fig. 17, as described in the following subsection, thetranslation for X-FEM lasts approximately 12 hours.

In addition, a cantilever model (50×50×250 mm) isanalyzed using X-FEM and the proposed method, respec-tively. Young’s modulus is 205.8 GPa, and Poisson’s ratiois 0.3. A VCAD model with 13×13×63 voxels is gen-erated. The boundaries of the solid model do not coin-cide with the boundaries between the voxels in the VCADmodel. As shown in Table 2, the results of the two meth-ods are approximately equivalent.

4. 3 Analysis of a stamping toolFigure 17 shows the VCAD data for a stamping tool

for the bumper of an automobile (courtesy of TsubamexCo., Ltd.). This model is generated using VCAD from anIGES file exported from a conventional CAD. Figure 18shows a degenerated hexahedral mesh generated by theproposed method. The number of nodes is 36 919, and thenumber of elements is 71 013, among which 56 266 degen-erated elements are included. The computation time forgenerating this mesh from the VCAD data is 25 secondsusing a Pentium4 (1.5 GHz) processor. Stress analysiswith this mesh is conducted using the ADVENTUREClus-ter(17). Figure 19 shows the result of the analysis visual-ized using the VCAD.

5. Concluding Remarks

A finite element mesh can be generated quickly andautomatically from VCAD data using degenerated hexa-hedral finite elements. This mesh can be used as input


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Fig. 18 Degenerated hexahedral mesh of the stamping toolgenerated by the proposed method

Fig. 19 Stress distribution of the analysis visualized usingVCAD

data for various kinds of finite element solvers for struc-tural analysis, including parallel finite element analysiscodes such as the ADVENTURE system and its commer-cial version ADVENTURECluster. This indicates that theproposed method enables parallel finite element analysisof industrial products of complex shape using the existingcodes.

The accuracy of the solutions obtained using the de-generated hexahedral mesh is compared to that of the so-lutions obtained by X-FEM, and the accuracy of the solu-tions obtained by these methods is shown to be approxi-mately equivalent.

The data structure of the degenerated hexahedralmesh generated by the proposed method corresponds di-rectly to VCAD data. Therefore, connecting the analysisprocedure using the proposed method to other applicationsthat make use of VCAD data is easy.

Acknowledgement

This study was conducted as part of the integratedVCAD program at RIKEN. The authors would like to ac-knowledge the support of Mr. Y. Hanamoto, who devel-oped the computer code and performed the computation

for the present study. The authors also would like to thankDr. K. Kase (RIKEN), and Prof. T. Nagashima (SophiaUniversity), and the members of the Mitsubishi ResearchInstitute and the manufacturing process simulation teamfor their kind discussion and help.

References

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( 4 ) Belytschko, T., Moes, N., Usui, S. and Parimi, C., Arbi-trary Discontinuities in Finite Elements, Int. J. Numer.Methods Eng., Vol.50 (2001), pp.993–1013.

( 5 ) Niyama, K., Ishihara, Y., Nagashima, T., Yamada, T.and Makinouchi, A., Development of X-FEM AnalysisSystem Using V-CAD Data Structure, Proceedings ofthe 15th Computational Mechanics Conference, JSME,(in Japanese), No.02-02, (2002), pp.349–350.

( 6 ) Nagashima, T., Ishihara, Y., Niiyama, K. andMakinouchi, A., Development of Stress AnalysisMethod Based on Voxel-Type X-FEM, Proceedings(CD-ROM) of WCCM-V, Vienna, Austria, (2002).

( 7 ) Nagashima, T., Basic Study for Elastic Analysis byX-FEM, Trans. Jpn. Soc. Mech. Eng., (in Japanese),Ser.A, Vol.67, No.662 (2001), pp.1569–1575.

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Finite Element Analysis Based on Volume CAD

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