Sweeping algorithms have become very mature and can create a semi-structured mesh on a large set of solids. However, these algorithms require that all linking surfaces be mappable or submappable. This restriction excludes solids with imprints or protrusions on the linking surfaces. The grafting algorithm allows these solids to be swept. It then locally modifies the position and connectivity of the nodes on the linking surfaces to align with the graft surfaces. Once a high-quality surface mesh is formed on the graft surface, it is swept along the branch creating a 2-D mesh.  Keywords: mesh generation, hexahedral meshing, refinement, sweeping, 2-D  *  This work was partly funded by Sandia National Laboratories, operated for the U.S. DOE under contract No. DE-AL0494AL8500.  Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S.  DOE.  # Scott Mitchell, samitch@sandia.gov, was supported by the Mathematical, Information and Computational ..

Jason F. Shepherd

Scott A. Mitchell

Steven E. Benzley

Steven R. Jankovich

Sweeping algorithms have become very mature and can create a semi-structured mesh on a large set of solids. However, these algorithms require that all linking surfaces be mappable or submappable. This restriction excludes solids with imprints or protrusions on the linking surfaces. The grafting algorithm allows these solids to be swept. It then locally modifies the position and connectivity of the nodes on the linking surfaces to align with the graft surfaces. Once a high-quality surface mesh is formed on the graft surface, it is swept along the branch creating a 2 3/4-D mesh

BENZLEY,STEVEN E.

JANKOVICH,STEVEN R.

MITCHELL,SCOTT A.

SHEPHERD,JASON F.

UNT Digital Library

THE GRAFT TOOL:4@-sAN ALL-HEXAHEDRAL TRANSITION ALGORITHM FORCREATING A MULTI-DIRECTIONAL SWEPT VOLUME MESH*Steven R. Jankovich’, Steven E. Benzlef, Jason F. Shepherd3, Scott A. Mitchell+lBri$ham Young Universi~r Provo, UT U.S.A. jankovic@et.byu. eduBrigham Young University, Provo, UK U.S.A. seb @byu,edu3Sandia National Laboratories, Albuquerque, NM, U.S.A. jfsheph @sandia.govABSTRACTSweeping algorithms have become very mature and can create a semi-structured mesh on a large set of solids. However, thesealgorithms require that all linking surfaces be mappable or submappable. This restriction excludes solids with imprints orprotrusions on the linking surfaces. The grafting algorithm allows these solids to be swept. h then locally modifies the positionand connectivity of the nodes on the linking surfaces to align with the graft surfaces. Once a highquahty surface mesh is formedon the grafi surface, it is swept along the branch creating a 2%-D mesh.Keywords mesh generatio~ hexahedral meshing; refinemen~ sweeping, 2%-D1. INTRODUCTIONThree-dimensional finite element analysis (FEA) is animportant design tool for physicists and engineers. Beforethe analysis can begin, a mesh needs to be generated on themodel. During the last several decades, much research hasbeen devoted to mesh generation. Tetrahedral meshgenerators are well developed and many have beenimplemented in software packages. Only recently has theresearch focus shifted to hexahedral meshes.For most applications, hexahedral elements are preferred overtetrahedral elements for meshing 3-D solids [1,2].Unfortunately, a high quality mesh of hexahedral eIemertts ismore difficult to generate. Minimally, the mesh needs to beconforrnal between adjoining solids and have high qualityelements at the bounding surfaces. Because of the constraintson hexahedral elements, automatic generation of high qualityhexahedral meshes on arbitrary 3-D solids has proven elusive[3].Over the last several years much work has been put intosweeping algorithms. These algorithms can mesh a widerange of 2%-D (prismatic) solids. The sweeping algorithmsgenerally take a 2-D unstructured quadrilateral mesh from thesource surface and project it through the volume to the targetsurface. Sweeping algorithms have matured to handle non-planar, non-parallel source and target surfaces and variablecross-sectional area [4] as well as multiple source and targetsurfaces [5,6].To maintain the structured mesh in the sweep direction,sweeping algorithms require the linking surfaces (those thatconnect the source to the target) to be mappable orsubmappable. llris constraint limits the number of soIids that‘This work was partly funded by Sandia National Laboratories, operated for the U.S. DOE under contract No. DE-AL04-94AL8500. Sandia is a mtdtiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the U.S.DOE.‘Scott Mitchell, samitch@sartdia.~ov, was supported by the Mathematical, Information and Computational Sciences Division ofthe U.S. Department of Energy, Office of Energy Research and works at Sandia National Laboratories..DISCLAIMERThis report was prepared as an account of work sponsoredby an agency of the United States Government. Neither theUnited States Government nor any agency thereof, nor anyof their employees, make any warranty, express or implied,or assumes any legal liability or responsibility for theaccuracy, completeness, or usefulness of any information,apparatus, product, or process disclosed, or represents thatits use would not infringe privately owned rights. Referenceherein to any specific commercial product, process, orservice by trade name, trademark, manufacturer, orotherwise does not necessarily constitute or imply itsendorsement, recommendation, or favoring by the UnitedStates Government or any agency thereof. The views andopinions of authors expressed herein do not necessarilystate or reflect those of the United States Government orany agency thereof.DISCLAIMERPortions of this document may be illegiblein electronic image products. Images areproduced from the best available originaldocument.t,, ,.~.can be meshed with these algorithms. They specificallyexclude solids with imprints or protrusions on the linkingsurfaces. This paper introduces a new grafting algorithm thatlifts this constraint on linking sutiaces. The gratlingalgorithm locally modifies the structured mesh of the linkktgsurfaces allowing the mesh to conform to additional surfacefeatures. Thus, the grafting algorithm can provide atransition between multiple sweep directions extendingSW&phg algorithms to 2%-D solids.Atler a brief definition of terms, the algorithm is presentedwith a simple example. The paper concludes with examplescreated using the algorithm.2. DEFINITION OF TERMSThe grafting algorithm can be used for a single solid or for aset of connected solids that require conformal meshesbetween them. For ease of presentation, the 2%-D soliddiscussed is assumed to be a single solid. The solids thatgenerally benefit from the gratiing algorithm have one majorsweep direction with imprints or protrusions cluttering thelinking surfaces. F@re 1 shows a sample 2%-D solid.TrunkBase Surface Y-/\ ( -.A /b-z ““(s”’’”BranchFigure 1. Definition of terms for a 2%-D solid.Usually the central and largest part of the solid is the majorSWfXp direction and will be known hereafter as the rr-wrk.The trunk often has protnrsions from the linking surfaces thatare sweepable subvohrmes. These subvohrmes are termedbranches. The linking surface on the trunk that contains oneor more branches is termed a base suface and begins with astructured mesh. The intersection of the trunk and branch isdefined as a graft su~ace.3. THE GRA~lNG ALGORITHMThe goal of the grafting algorithm is to create a conformalmesh between the trunk and the branches composed of highquality elements. The rdgorithm has three major steps:meshing of the trunk, modification of the base surface meshat the graft surface, and meshing of the branch. Each of thesesteps will be described in the following sections.3.1 Meshing the TrunkThe first step in the grafting algorithm is to obtain a mesh onthe trunk. The tmnk is generally defined such that astructured meshing algorithm can create a successful mesh onit. In the examples used in this paper, the trunks are meshedwith either volume mapping, submapping [7], or sweepingalgorithms. It is not necessary, however, for the trunk tohave a structured mesh on it. For severely complicatedsolids, it may not be easy to find a simple trunk. In thesecases, it may be desirable to use an unstructured algorithm(e.g. Whisker Weaving [8], Plastering [9], or Hex-Tet [10]) tomesh the more complicated trunk and then use the graftingalgorithm to create a transition to a more structured branch.Regardless of the mesh scheme, special care must be takenwhen assigning the element size to the trunk. The mesh mustbe fine enough to resolve all small features of the trunk andthe graft surface.3.2 Creating the GraftOnce the trunk is meshed, the branches are grafted into thebase surface mesh one at a time. Figure 2 shows a samplegraft surface with the underlying structured mesh of the basesurface.The first step is to locate each graft surface on the mesh. Thegraft surface is separated into individual loops of curves thatdefine the surface boundary. Each loop is temporarilymeshed with a one-dimensional mesh that is twice as fine asthe underlying mesh on the base surface. This is done toapproximate the curve loop with a closed set of linear linesegments.Figure 2. An example graft surface with underlyingstructured mesh from the base surface.The mesh elements on the loop are traversed to find the set ofmesh edges on the base surface that intersect the loop. Asimple three-dimensional linear intersection routine is used todetermine where each of these intersections occurs. In Figure2, these intersected edges are highlighted with bold lines.Once the intersected edges of the base surface are located, thetemporary mesh on the loop is deleted.3.2.1 Smoothing the Mesh to the Loop,’ r,,. “ .The next step is to adjust the surface mesh to conform to theloop. Using the set of intersecting mesh edges and thecorrespond@ intersection locations, the closest node on eachedge is moved to the loop. If two nodes are comparablyclose, the adjacent intersecting edges are checked todetermine which node to move. The node that produces thehighest quaJity quadrilaterals is moved. Figure 3 shows themodified surface mesh after the nodes are moved.Figure 4. The base surface mesh is completelysmoothed to the loop.Figure 3. The nodes of the intersecting edges aremoved to the loop.By moving the nodes of the intersecting edges, occasionallythe loop will be made to span diagonally across aquadrilateral. In Figure 3, eIement 17 is intersecteddiagonally by the loop. When this happens, a third node ofthe quadrilateral is moved to the loop. Again, which node tomove is determined by the final mesh quality. Figure 4shows the mesh that results from moving the nodes to theloop.The final quality of the quadrilateral mesh inside the loop islimited by the resolution of the original mesh on the basesurface and by the number of mesh faces that are diagonallyintersected by the loop. Unfortunately, this smootilngprocedure results in the poorest quafity mesh faces at the loopboundary.Artother important consideration is the quality of thehexahedral mesh of the trunk immediately under the basesurface. Generally, the further the surface nodes are moved,the poorer the quafity of the underlying hexahedrat mesh. Inmany cases; this smoothing process produces hexes withnegative Jacobians. Though, the quality can be improvedslightly by a smooth on the volume, the benefits in quality arenot worth the computational expense of the smooth. Alone,smoothing is not sufficient to improve the mesh quality.3.2.2 Refining tnside the LoopAs mentioned previously, it is important to have high quafityelements near the surfaces of the solid. ‘f?te quality of theelements at the surface of the branch is determined by thequafity of the quadrilaterals immediately inside the boundingloops of the graft surface. In Figure 4, the poorest qualityelements are at the bounding loop of the graft surface. TOimprove the quafity of these elements, a refinement scheme isused that modifies the mesh connectivity locally.The refinement scheme is best understood by inspection ofthe Spatial Twist Continuum (STC) [11], or dual of the mesh.A complete STC sheet is inserted directly inside thebounding loops of the graft surface. The sheet passes behindthe first layer of hexes in the trunk creating a pillow of newhexes inside the loop [12]. Thus, the connectivity of theinterior side of the hexes remains unchanged. This insuresthat the connectivity modification is Iocal, especially on thinsolids..’ ,.. ,“,.from the trunk. The results of the grafting algorithm can beseen in Figure 7. The branch has been cut away to show thedetails of the base surface. Notice the high quality elementsinside the graft surface. Further refinement was done outsidethe graft surface to improve the quality of the mesh.Figure 5. A pillow of elements is inserted directlyinside the loops to improve element quality.This process was applied to the mesh of Figure 4 and theresulting surface mesh is shown in F@re 5. Though only thesurface mesh is shown, the new layer of hexes wraps aroundbehind the existing hexes using the comer primitivesuggested by Murdoch in [11]. lle new layer of elementsshows an improvement in quality and moves the lowerquality elements to the center of the graft surface..3.2.3 improving Quality Outside the LoopBefore leaving the base surface, the elements immediatelyoutside the loop are surveyed for poor quality. As mentionedabove, some of these elements will have negative Jacobiansdue to the movement of the nodes. These quality issues canbe corrected by inserting another STC sheet away from theloop. Finally, the mesh on the base surface is smoothed tooptimize the node locations.3.3 Meshing the BranchesWhen the quality of the mesh on the base surface has beenimproved, the branch is ready to be meshed. The set ofquadrilateral elements inside the graft surface is defined asthe source mesh for a sweeping algorithm. This mesh is thenswept through to the end of the branch.Previously it was mentioned that the trunk could be meshedwith any voIume mesh scheme, though most ofien astructured scheme was chosen. The same is true with thebranches. Most often the branches are sweepablesubvolumes with the graft surface as the source surface.However, any scheme, structured or unstructured, can beused to mesh the branches using the existing mesh on theI graft surface.4. EXAMPLESFigure 6. Mapped trunk mesh with cytinder branchbefore grafting.Figure 7. Cut away view of the base surface aftergrafting.Figure 8 shows a slice from the center of the volume meshfrom Figure 7. It is easy to see the layer of hexes that wereadded directly beneath the graft surface. Notice that thevolume mesh is completely conformable. Additionally, allthe hexes are of acceptable quality.Shown below are three simple examples of the graftingatgorithm. Figure 6 shows a block-shaped trunk with amapped volume mesh. The branch is a cylindrical protrusionFigure 8. A slice of the volume of Figure 7 to showthe internal hexes.The trunk in Figure 9 was meshed with a volume submapalgorithm. The branch is in the shape of a figure eight toshow how the grafting algorithm handles the cusps. Tleresults of the algorithm can be seen in F@re 10. Thegrafting algorithm again produced high quafity elements atthe bounding loop of the graft surface. This time there wereno quafity issues outside of the graft surface.Figure 9. Submapped trunk with figure eightbranch before grafting.Figure 10. Cut away view of the base surface aftergrafting.Finally, F@re 11 shows a trunk with a swept volume mesh.Notice that there is a through hole down the center of thetrunk. The heart-shaped branch protruding from the trunkcomplicates the meshing of the solid. It cannot be easilymeshed with any of the structured meshing algorithms. Thegrafting afgorithm is able to produce a high quality mesh onthe solid. Figure 12 shows the base surface after grafting themesh.Figure 11. Swept volume with through hole andheart-shaped branch before grafting.. ,.“ . *. -“- -“, ●Figure 12. Cut away view of base surface aftergrafting.5. CONCLUSIONSweeping algorithms have become very mature over the lastseveral years. However, these algorithms require that thelinking surfaces be mappabIe or submappable relying on theunstructured techniques to mesh the rest. With the grafting~,gorithm, ~ addition~ set of solids cm be meshed in astructured way. The grafting algorithm modifies the linkingsurfaces of a swept trunk to create a high quality transition tothe sweepable branches.REFERENCES[1][2][3]Benzley, S. E.; Perry, E.; Merkley, K.; Clark, B.; andSjaardema, G., “A Comparison of AH-Hexahedral andAll-Tetrahedral Finite Element Meshes for Elastic andElasto-Pkistic Analysis: Proceedings, 4’h InternationalMeshing Roundtable, Sandia Natioml Z.uboratories95, pp. 179-191, October 1995.Cifhentes, A. O. and Kalbag, A., “A PerformanceStudy of Tetrahedral and Hexahedrrd Elements in 3-DFinite Element Structural Analysis; Finite Elements inAnalysis and Design, Vol. 12, pp. 313-318, 1992.Mitchell, S. A., “A Characterization of theQuadrilateral Meshes of a Surface Which Admit aCompatible Hexahedral Mesh of the EnclosedVohtme,” Proceedings, 13’hAnnual Symposium onTheoreticaiAspects of Computer Science (STACS ‘%),Lecture Nodes in Computer Science 1046, Springer,pp. 465476, 1996.[4][5][6][7[8[9][10][11][12]Staten, M. L., Canann, S. A., and Owen, S. J.,“BMSweep: Locating Interior Nodes DuringSweeping,” Proceedings, ? International MeshingRoundtable 98, pp. 7-18, October 1998.Blacker, T., ‘The Cooper Tool,” Proceedings, 5rkIntemationai A4eshtig Roundtabie 96, pp. 13-29,October 1996.Mingwu, L. and Bet@ey, S. E., “A Multiple Sourceand Target Sweeping Method for Generating AllHexahedral Finite Element Meshes: Proceedings, 5’*International Meshing Roundtable %, pp. 217-225,October 1996.White, D.R., “Automatic, Quadrilateral andHexahedral Meshing of Pseudo-Cartesian Geometriesusing Virtual Decomposition,” Master’s Thesis,Brigham Young Universiy, August 1996.Tautges, T.J., BIacker, T.D., and ,MitcheII,S.A., ‘TheWhisker Weaving Algorithm A Connectivity-basedMethod for Constructing A1l-hexahedralFiniteElement Meshes; International Joumalfor NumericalMe!hods in Engineering, Vol. 39, pp. 3328-3349,1996.Cartarm, S.A., “Plastering: A New Approach toAutomated, 3-D Hexahedral Mesh Generatioru”American Institute of Aeromutics and Astronics, 1992.Meyers, R.J., Tautges, T.J., and Tuchinsky, P.M., ‘The“Hex-Tet” Hex-Dominant Meshing Algorithm asImplemented in CUBIT,” Proceedings, #’International Meshing Roundtable 98, pp. 151-158,October 1998.Murdoch, P. and Benzley, S. E., “The Spatial TwistContinuum”, Proceedings, 4& International MeshingRoundtable 95, pp. 243-251, October 1995.Mitchell, S.A. and Tautges, T-J., ‘pillowing Doublets:Refining a mesh to ensure tha~faces share at most oneedge,” http:/!endo.sandia. gov/-samitch/pillowing-doublets.pdf.

The Graft Tool: An All-Hexahedral Transition Algorithm for Creating a Multi-Directional Swept Volume Mesh

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The Graft Tool: An All-Hexahedral Transition Algorithm For Creating A Multi-Directional Swept Volume Mesh

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