Multiple Approaches to Frame Field Correction for CAD Models
MMULTIPLE APPROACHES TO FRAME FIELDCORRECTION FOR CAD MODELS
Maxence Reberol ∗ Alexandre Chemin Jean-Fran¸cois Remacle Universit´e catholique de Louvain, Avenue Georges Lemaitre 4, 1348 Louvain-la-Neuve, Belgium
ABSTRACT
Three-dimensional frame fields computed on CAD models often contain singular curves that are not compatible withhexahedral meshing. In this paper, we show how CAD feature curves can induce non meshable 3-5 singular curvesand we study four different approaches that aims at correcting the frame field topology. All approaches consist inmodifying the frame field computation, the two first ones consisting in applying internal constraints and the twolast ones consisting in modifying the boundary conditions. Approaches based on internal constraints are shown notto be very reliable because of their interactions with other singularities. On the other hand, boundary conditionmodifications are more promising as their impact is very localized. We eventually recommend the 3-5 singular curve boundary snapping strategy, which is simple to implement and allows to generate topologically correct frame fields.
Keywords: frame field, hexahedral meshing, block decomposition
1. INTRODUCTION
In the last decade, frame field based approaches ( § § § § ∗ [email protected] of the approaches rely on extrusion of objects insidethe frame field: (i) the extrusion of boundary featurecurves ( § § § § a r X i v : . [ c s . G R ] D ec echniques would fail because of incorrect frame fieldtopology. Nevertheless, this approach is limited to 3-5singular curves close to the boundary, which are themost frequent in CAD, and cannot handle arbitrarycases of 3-5 singular curves.
2. RELATED WORK
The standard frame field based meshing approach con-sists in building a smooth and boundary-aligned framefield ( § § In dimension two, crosses are objects made of two or-thogonal directions, invariant by the four rotations ofrespectively 0, 90, 180 and 270 degrees. Cross fieldsare usually represented by 2D vector fields of the form f ( x ) = (cos(4 θ ( x )) , sin(4 θ ( x ))) [14]. To get a smoothvector field in a 2D domain Ω, the natural way is tominimize the Dirichlet energy (cid:82) Ω (cid:107)∇ f (cid:107) under Dirich-let boundary conditions. One issue with this approachis that the gradient of the frame field tends to infinityat singularities in the continuous setting. This is ei-ther ignored, as the energy stay finite after discretiza-tion, or addressed by using a scaling scalar field [15], ormore recently by turning to the Ginzburg-Landau the-ory [16, 17]. In practice, the frame fields obtained withthese different approaches are similar, with singular-ities whose indices are compatible with quadrilateralmeshing.In dimension three, frames are made of three orthog-onal directions and are invariant by the twenty-fourrotations of the cube. For the generation of bound-ary aligned smooth frame field, the ideas are similarto the 2D case: find a convenient representation of theframe field and minimize the Dirichlet energy. How-ever, finding a unique and continuous representation ismore tricky. The current best candidates are sphericalharmonics [18] and fourth-order tensors [19]. In bothcases, the spaces are of dimension nine and the frameslive on a manifold of dimension three. The Dirichletenergy is minimized while staying on the frame mani-fold, which in practice is done either by optimizing Eu-ler angles associated to frames [20] or iteratively witha non-linear solver and by using recurrent projectionson the frame manifold [19]. The resulting frame fieldsexhibit singularities made of internal curves, that is usually called the singularity graph. As in dimensiontwo, the norm of the gradient tends to infinity at thesingular curves, which can be seen as extrusions of thesingular nodes of a boundary cross field.Contrary to the 2D case, frame field singularity graphsdo not always correspond to feasible hexahedral meshconnectivity. This issue has been extensively describedin recent articles [1, 2, 3]. To our knowledge, thereexist no frame field generation approach that has anykind of guarantee to provide a singular graph that iscompatible with hex meshing. A simple 3D model exhibiting a fundamentally invalidsingularity graph is the notch model [1, 2] (Figure 1),which is the boolean difference between a box and acylinder. The singularity graph produced by all ex-isting frame field methods is made of a single curvewhose hexahedral valence is three on one extremityand five on the second one, that will be called a in the rest of the paper. In a hexahedralmesh, it is not possible to have an interior vertex ad-jacent to only one valence three edge and one valencefive edge (other adjacent edges being regular, i . e . va-lence four). The only valid vertex configurations arethe ones which are topologically equivalent to spheretriangulations, as described in [3]. When a singularitygraph contains a different configuration, e . g . a 3-5 sin-gular curve, we say it is invalid or not hex-meshable.Given a valid singularity graph, there exist meth-ods that allow to compute a smooth frame field, e . g .[3, 21]. Thus, correcting a frame field can be achievedby generating a valid singularity graph, i . e . the set ofirregular edges of the associated block decomposition.But generating a valid singularity graph from scratchremains a totally open problem. An alternative ap-proach is to start from an initial frame field, possiblynon-meshable, and modify it in order to make it hex-meshable.In [2], a valid frame field for the notch model is builtby either manually extruding the concave feature curveinside the model (producing an internal surface) or bymanually adding a fillet to the feature curve. In thepresent paper, we propose two approaches ( § § § e . g . the zig-zag issue. They can be corrected with local opera-tions, as detailed in [22]. We are not interested in theseissues in this paper as they are artifacts of discretiza-tion choices, and it is possible to avoid them by usinga frame per vertex instead of one per tetrahedron.
3. BOUNDARY-ALIGNED SMOOTHFRAME FIELD
In this section, we first describe the energy mini-mization formulation of the boundary-aligned smoothframe field problem ( § T , bydefining one frame per vertex ( § § e . g . [18, 20, 23, 19]. The goal is to compute a frame field as smooth as pos-sible and that is aligned with the boundaries/featuresof the model. The natural approach is to translatethese requirements into a Dirichlet energy minimiza-tion problem. Minimizing the Dirichlet energy (1) en-sures the smoothness in the domain and the Dirichletboundary conditions (2) enforce the alignment withthe model boundary. Formally, the frame field f is thesolution to the problem: f = argmin f ( x ) ∈O (cid:90) Ω (cid:107)∇ f (cid:107) (1)subject to boundary conditions: (cid:40) f ( x ) = g ( x ) for x ∈ ∂ Ω d f ( x ) ⊥ n ( x ) for x ∈ ∂ Ω s (2)with O the space of frames, ∂ Ω s a smooth subset of themodel boundary ∂ Ω, where we want the frames to betangent to the boundary, n the boundary normal, and ∂ Ω d another subset of ∂ Ω where we impose the threedirections of the frames. Usually, ∂ Ω d corresponds tothe feature curves of the model, also called hard-edges or ridges. The symbol ⊥ means that one of the threeframe directions is parallel to a given vector (tangencyconstraint).Before going further, one needs to choose a represen-tation of the frames. Frames live in a space, denoted O , which is the quotient space of the space of rota-tions SO (3) with the octahedral group O [23]. Un-fortunately, there is no simple representation of theseobjects. To date, two representations have been pro-posed: spherical harmonics of degree four [18] andfourth-order tensors [19]. In both cases, frames live onmanifolds of dimension three immersed in R . Thereis an isomorphism between the two representations,so they are essentially equivalent. The continuity anduniqueness of these representations is a necessary con-dition to use directly their nine coefficients to computedistances and gradients, which is analogues to usingthe Euclidean distance on the unit circle instead ofthe circle arc length.In the continuous setting, the Dirichlet energy (1)blows up because of the presence of singular curvesand this singular behavior is the cause of many issuesfor the discretization. To find a numerical solution to the frame field problemintroduced previously ( § T and use a continuous piecewise linear approxima-tion of the frame coefficients, which can be coefficientsof either the spherical harmonics representation or ofthe 4-th order tensor one. The discretized frame field f h is now entirely defined by its coefficients f i ∈ R at each vertex v i ∈ V of the tetrahedral mesh. It isworth noting that inside a tetrahedron, the linearlyinterpolated coefficients do not correspond to a frame,but the closest frame can obtained by projection onthe frame manifold O . Compared to the one frameper tetrahedron discretization, the piecewise linear ap-proximation is more efficient: less unknowns for thesame mesh, better accuracy [20], possibility to use lin-ear finite elements.If each f i is a frame, then the singularities, which cor-respond to infinite gradients in the continuous formu-lation, cannot be represented at vertices. With thisapproach, one possibility is to detect singularities bylooking at axis permutations along edge loops. Thesmallest loops to look at are the internal faces of thetetrahedral mesh, which are simply triangles. The sin-gularity graph is then made of connected chains oftetrahedra. In practice, it forces the singularity graph )a) c) d) e) Figure 1 : a) Natural frame field singularity graph, made of one non-meshable 3-5 singular curve. b) When replacing theconcave feature curve by a fillet, the new singularity graph is made of three valence-five and one valence-three singularitycurves, which are compatible with hexahedral meshing. c) Via the streamlines (green), the feature curve is extruded insidethe volume and the resulting singularity graph is made of two singular curves (one valence five and one valence three). d)Tracing streamlines from boundary singular nodes generates two valid singular curves. e) The initial invalid 3-5 singularcurve of a) is snapped to the boundary surface, creating a new valence three feature curve.
Bottom row: schematic viewof the corrections, in the diagonal cut view. to be locally very distorted as it must be containedinside tetrahedra, whose facets are randomly orientedin the mesh.Another way to deal with singularities is to allowthe coefficients f i to represent objects which are notframes. This is analogous to letting 2D crosses tendto zero at singularities instead of staying unit vectors.The advantage is that the singularities are smoother,because they are less affected by the tetrahedral mesh,but they are also more diffuse and there is no longera clear localization of the singularities.Frame field solvers usually work in two stages : ini-tialization via Laplacian smoothing of the frame coef-ficients, without enforcing the frame constraint f ∈ O ,followed by a smoothing of the frames, where theframes f i must lie on O or stay close to it. The processusually converges to a local minimum which is not farfrom the initialization [20].The frame coefficients can be forced to stay on theframe manifold O by either recurring projections [19]or by optimizing the Euler angles of the associatedrotations [20].In any case, the Dirichlet energy associated to theframe field tends to infinity with mesh refinement,but stays finite because of the discretization. Con-sequently, these approaches only work on uniformmeshes. On non-uniform meshes, the singularitiesmove to areas of coarse elements as the same singu-larity graph topology can be represented while costingless energy.We use 4th order tensors to represent the frames, and we allow the coefficients f i to deviate from O at sin-gularities. This particular choice is not important forthe rest of the paper: the correction techniques westudy can be applied to all energy-minimizing framefield solvers. Boundary aligned frame fields produced by energyminimizing methods are interesting because they ex-hibit singularities that form a graph, which is topo-logically similar to the singular edges of a hexahedralmesh. Unfortunately, as described in the literature( § splitting leads to theapparition of singular nodes on the surface, which arenecessary to have coherent cross fields that respect thePoincar-Hopf theorem. But from the point of view ofthe volume frame field, the boundary singular nodesmust be the extremities of singular curves, as there areno isolated singular nodes in a 3D frame field. simple example exhibiting this behavior is a boxwith a circular arc imprinted on one face, see Fig-ure 2.a.. This example can be seen as a simplificationof the notch model. In this example, the two patcheson top must have singular nodes to accommodate theirboundary conditions, but the frame field solver, whichis minimizing the Dirichlet energy, do not propagatethese singularities inside the model as this would costa lot of energy, but merge them as soon as possiblein the volume. Models with such configurations are,for example, the ones containing boolean operationsinvolving spheres or cylinders that do not go throughthe whole model, which are common in CAD mod-elling.Our interpretation is that the energy-minimizationformulation ( § − ∆ f = 0, even with the additional constraint f ( x ) ∈ O . Hence, this is essentially a smoothing ker-nel that act locally when possible. Constraints fromthe boundary that cost energy (non-zero frame gradi-ent) do not propagate far in the volume.On the other hand, hexahedral mesh topological con-straints (chords, sheets) propagate on arbitrary longdistances. Thus, hexahedral meshes associated toframe fields are often different from the ones that auser would produce by manually building a block de-composition.An ideal answer to this issue would be a new framefield problem formulation that allows better propaga-tion of boundary constraints, but unfortunately nonehas been successfully developed up to now.Another important observation is that even if theframe field have a wrong topology, one of the frame di-rections is not affected (the vertical ones in the notch and box with arc examples). At the transition fromvalence three to valence five in the 3-5 singular curve,the frame field stable direction along the singularityis no longer tangent to the curve. Exploiting this sta-ble direction is the basis of the feature curve extru-sion ( § singularity extrusion ( §
4. FRAME FIELD CORRECTIONTECHNIQUES
We present and discuss various strategies to automat-ically correct frame fields, for them to be suitable forfull hexahedral meshing. We focus on removing the3-5 singular curves. Feature curve extrusion ( § § notch model in [2].Singularity extrusion ( § § § § § § To recap the previous frame field observations ( § Tracing streamlines from feature curves
Thesimplest approach would be to propagate the featurecurves straight into the model, following the directionsof the initial boundary normals. It would work forsimple blocky models, but not when the boundariesare curved, because the new internal surfaces wouldnot follow the curvature of the model boundaries.Instead, we extrude the concave feature curves by us-ing certain directions in a initial frame field. In prac-tice, we achieve this extrusion by tracing streamlines,that start from the feature curves and end on the firstreached boundaries. In our case, the tracing is doneby using a standard fourth order Runge-Kutta explicitscheme, very similar to the one used with vector fields.The only difference is that at each new position, we ex-tract the frame direction which is closest to the previ-ous direction. The reader can refer to Algorithm 1 fora more detailed description of the streamline tracing b c AB Figure 2 : Model A: box with imprinted circle arc, simplified version of the notch model . Model B: same model butthe box has curvature, which causes two additional singular lines.
Col. a: singularities of the initial frame field.
Col. b: feature curve extrusion, works for A but fails for B because of interaction with other singularities, which split the streamlinetrajectories.
Col. c: singularity extrusion, works for A but fails for B because the streamlines hit another singularity. a) b) c) d) e)
Figure 3 : Half-sphere on top of a box. a) Natural frame field singularity graph, containing four non-meshable 3-5 singularcurves. b) When replacing the concave feature curve by a fillet, the new singularity graph contains four additional singularcurves of valence five, making the graph compatible with hexahedral meshing. c) With internal constraints (green), thefeature curve is extruded inside the volume and the resulting singularity graph is valid. d) Tracing streamlines from boundarysingular nodes replace the four 3-5 singularities by four singularities of valence five, but four valence three singular lines aremissing. These ones should be traced from the internal singular nodes. e) The four 3-5 singular curves are snapped to theboundary, forcing snapping of the other singularities.
LGORITHM 1:
Streamline tracing in frame field
Input:
Initial position p Initial direction v Step length h for the explicit scheme Output:
Streamline S described by an ordered listof points ( p l ) l and their associateddirections ( v l ) l p = p f ( p ) = interpolate frame field at( p ) v = closest direction( v , f ( p )) repeat append ( p , v ) to S p = p + h ( v + 2 v + 2 v + v ) with v = closest direction( v , f ( p )) v = closest direction( v , f ( p + h v )) v = closest direction( v , f ( p + h v )) v = closest direction( v , f ( p + h v )) v = v until p outside tetrahedral mesh T ;process.To determine the initial directions for the propaga-tion, we look at the appropriate hexahedral valenceassociated to the feature curve. It is usually one orthree, and occasionally two for user-inserted curveson smooth surfaces, such as the circular arc on theFigure 2, or valence four for model with cuts whosedihedral angles are close to 360 degrees. For valenceone, there is no need to propagate the curves as theframe field directions are already imposed by the sur-face normals at both sides of the curve. For valencetwo, three and four, we respectively extend the curvein one, two and three directions.Successful applications of the streamline tracing pro-cess can be seen on Figures 1.c., 2.A.b and 3.c., wherethe extruded curves are shown with green points. Internal constraints for frame field
To computea new frame field that respects the extruded curve, weadd internal constraints in the boundary conditions(2) of the frame field formulation ( § S = { ( p k , v k ) } k , made of ( p k ) k the points, ( v k ) k the associated directions, startedfrom the edge e , of tangential direction t e . Then theassociated internal conditions are: ∀ k, f ( p k ) ⊥ ( t e × v k ) (3)To apply these internal conditions precisely, we com-pute a new tetrahedral mesh containing the points ( p k ) k associated to all the streamlines traced from theconcave feature curves.In our examples, the singularity graphs of the newframe fields are displayed on Figures 1.c., 2.A.b and3.c.. They correspond to valid hexahedral meshes,similar to the ones a user would generate by manu-ally building the block decompositions. Failure cases and limitations
Extruding a fea-ture curve inside a frame field is equivalent to tracinga sheet. If the sheet encounters another singularity or-thogonally, then it is sheared in multiple parts. Whenthe multiple parts can stay inside a future valid sheetof the hexahedral mesh, as for the horizontal sheetin Figure 1.c., then our approach still works. But ifthe sheet is sheared into parts that are sent in differ-ent arbitrary directions, then the new internal surfacemakes no sense and should not be used to constrainthe frame field, as it would definitely produce an in-coherent frame field. This failure case is illustrated onFigure 2.B.b..The feature curve extrusion is also not applicable tomodels where the frame field contains limit cycles,such as the Nautilus example in Figure 1 of [2]. Insuch cases, which are rare, the streamlines will spiralindefinitely without reaching any boundary. They canbe detected by monitoring the streamline lengths.Regarding the model on Figure 2.B., other frame fieldsolvers could produce a frame field without the pairof valid singularities that accommodates the curva-ture. In that case, the feature curve extrusion correc-tion would work. However, it is still possible to addfeatures to the model, e . g . boolean difference with acylinder, that would introduce singularities to perturbthe extrusion of the feature curve.The initial 3-5 singular curve is a local solution of theframe field solver to the boundary constraints. By ex-truding the feature curve using the stable directions,we are trying to generate global constraints, as theytypically propagate through the whole model. Wethink this approach cannot work in the general case, asthere is no reason that the feature curve conserves itsshape during the extrusion process in the frame field. Another possible extrusion approach is to propagatethe boundary singular nodes via the frame field stabledirections, instead of propagating the concave featurecurves. This approach has been implemented in [12] toseparate the contour of dual sheets, in the context ofblock decomposition. The additional step in our caseis to compute a new frame field that respect theseforced singularities.he hope with this approach, compared to the featurecurve extrusion ( § § § Limitations
This approach suffers from the samefailure case than the feature curve extrusion, whichis shown on Figure 2.B.c.. Both streamlines hit anexisting singularity during the tracing process, leadingto an incoherent singularity graph.The singularity extrusion correction technique mayhave a better success rate than the feature curve ex-trusion on specific models (less risk of collision), butit suffers from the same fundamental flaw (interactionwith other singularities) and does not work in the gen-eral case.
In the frame field analysis section ( § e . g . the rockerarm in Figure 10 of [3], they aremuch less widespread than with CAD models in ourexperience. We think it is worth trying to exploit the smoothing of feature curves to correct frame fields. Afeature curve (made of hard-edges) can be replaced bya smooth transition between the two surface patchesadjacent to the curve. In terms of CAD modeling,this is equivalent to placing a fillet on the curve, asillustrated on Figure 1.b., Figure 3.b. and Figure 5.b..Smoothing a convex feature curve (hex-valence of one)induces an internal singular curve of valence threein the frame field and smoothing a concave featurecurve (hex-valence of three) induces an internal singu-lar curve of valence five. If the feature curve that isreplaced by a fillet was responsible for a 3-5 singularcurve, the newly introduced singular curves join the in-valid curve at its transition between valence three andfive, creating a hex-meshable singular node. In thecase of the notch model (Figure 1.a. and b.), the newsingular node connects one valence-three and threevalence-five singular curves after the insertion of thefillet. The associated block decomposition is shown onFigure 4.a.. Another application is illustrated on Fig-ure 3.b., where the frame field of the smoothed modelcontains four additional valence-five singular curves,making it suitable for full hexahedral meshing.From the point of view of the block decomposition,the fillet correction introduces a new hexahedral layer(the colored blocks on Figure 4.a.). This layer waspartially present in the initial frame field, which con-tained the 3-5 singular curve, but it was pinched and itdid not correspond to topological blocks (six quadran-gular faces). The smoothing of the hard-edges allowsto recover valid blocks.Another example of singularity graph obtained aftertransforming the feature curves into fillet is illustratedon Figure 5. This model is a volume version of the box with arc model, but with the arc replaced by agroove. As with the notch model, the smoothing ofthe feature curves introduces new singular curves thatconnect with the 3-5 singular curves, making all thesingular nodes hex-meshable.This approach is interesting because it is a local ap-proach, which implies local modifications of the framefield, well in accordance with the spirit of the energy-minimizing frame field formulation. Limitations
Unfortunately, it is not straightfor-ward to implement it in an automatic way. Once ahex-meshable frame field, or equivalently its singular-ity graph, is computed on the model with fillets, itmust be brought back to the initial model with hard-edges. Before trying to develop an automatic tech-nique for building such mapping, it is interesting todo it manually and observe the result.By carefully looking at the block decomposition on thesmoothed notch model on Figure 4.a., we can see that ) b) c)
Figure 4 : Block decomposition of the notch model after correction. a) Feature curve transformed to a fillet. The blockdecomposition is valid. b) 3-5 singular curve snapped to boundary. The block decomposition is topologically valid but thecolored block has a flat corner. c) Refinement of the block with a flat corner. The decomposition is topologically andgeometrically valid. a) b)
Figure 5 : a) Box with a circular rectangular groove on the top. The singularity graph is made of two 3-5 singularcurves. b) The feature curves of the rectangular groove are transformed into fillets, making the geometry smooth. Thischange introduces singular curves along the groove, that connect with the previously invalid 3-5 singular curves. The newsingularity graph, which contains four internal nodes, is compatible with hex meshing. he valence-three boundary singular node has been po-sitioned at the center of the fillet, where the curvatureis maximal. In the initial model, this node has to bemapped on the center the concave feature curve. Thisimplies that the topological block in green will havetwo adjacent edges on this curve, forming a flat angle.This geometry is not valid from a numerical analysispoint of view, as the jacobian of the hexahedra wouldbe zero on this corner.Instead of pursuing this approach, which would re-quire lot of engineering, we focus on a similar but sim-pler one: the boundary snapping of 3-5 singular curves( § The previous fillet approach ( § fillet cor-rection on Figure 4.a., we can see that there is a layerof blocks close to the fillet (the colored ones). If weremove this layer, we get a new block decompositionwhere the singular curve is on the boundary, see Fig-ure 4.b.. We propose to mimic this process by directlysnapping 3-5 singular curves to the boundary, skip-ping the fillet correction (Figure 4.b.). A drawbackis the resulting block decomposition has blocks withzero jacobian at some corners, but it was already thecase with the fillet correction, and the final geometrycan be improved by refining some blocks in a post-processing phase (Figure 4.c.). Singularity snapping
Our snapping strategy issimple. For each 3-5 singular curve, we snap both ex-tremities. If an extremity is a boundary singular node,it is snapped to the closest feature curve, if it is an in-ternal singular node, it is snapped to the closest pointof the boundary surface. Other singularities (initiallyvalid) may have only one of their extremity snapped,if so the other is also snapped. This process is appliediteratively until all necessary singularity extremitieshave been snapped.Once the extremities are snapped, the geometry of thecurve on the boundary must be recomputed. A simpleapproach is to take the shortest path between bothextremities on the boundary triangle edges, e . g . Fig-ure 1.e. and Figure 3.e., but a more accurate one is tobuild a new triangular mesh of the boundary that in-cludes smooth curves joining the snapped extremities. Corrected frame field
To generate a frame fieldthat respect the snapped curves, the boundary con-ditions (2) of the frame field formulation ( § e . g .locally flat.On the snapped curves, instead of imposing tangencyto the boundary normal, we impose frames (Dirichletboundary conditions) that are tangent to the curveand 45-degrees rotated from the boundary normal(along the curve axis).Feature curves that received a snapped extremity aresplit and new boundary conditions are obtained bylinearly interpolating the frames at both extremitiesof the split curves.To avoid incompatible boundary conditions, we alsoremove the boundary alignment constraints on thevertices close to the snapped curves. These framesbecome free, allowing a smooth transition from theframes of the snapped curves to the frames on the restof the model boundary.While the new frame field is no longer boundaryaligned everywhere, the affected areas remain local-ized and the resulting frame field is still similar to theinitial one, minus the 3-5 singular curves that havebeen snapped. Applications
The singularity snapping correctionis applied successfully on the notch model, see Figure1.e., on the union of a cube and a half-sphere, see Fig-ure 3.e. and on three more complicated CAD models,see Figure 6.On the boxes with imprinted arcs (models A and B onFigure 2), the 3-5 singular curve would be reduced toa single node on the feature curve and the frame fieldwould be as if there were no imprinted curves.
Geometry and block refinement
One drawbackof this approach is that it produces hexahedral blockswith an invalid geometry (zero jacobian at some cor-ners), e . g . the bottom right block on Figure 4.b..We propose to refine the affected blocks in a post-processing step, as shown on Figure 4.c.. For morecomplex cases, the post-processing refinement can fol-low a template-based strategy, such as [25]. To pre-serve the topology of a hexahedral mesh, the refine-ment must be propagated to adjacent blocks whenthey share a refined quadrangular face. This is equiv-alent to sheet insertion. b c d ABC
Figure 6 : Model A: model built from the boolean union of two cylinders.
Model B: model built from boolean operationsbetween cylinders and a sphere, with two fillets on the bottom.
Model C: model with various CAD features, from [24].
Col. a: initial models.
Col. b: singularity graphs of the initial frame fields, which contain many 3-5 singular curves.
Col.c: snapping of the 3-5 singular curves, the snapped curves are shown in dark green.
Col. d: valid singularity graphs of thecorrected frame fields, whose boundary conditions have been changed according to the snapped curves (in green). imitations
This approach is only applicable whenthe 3-5 singular curves are close to the model bound-ary, as they are snapped on it. When dealing withCAD models, this is often the case because the in-valid singularities are mostly caused by curved surfacepatches, e . g . from a boolean operation with a cylinder.That being said, there are 3-5 singular curves that liefar inside the volume and whose cause is global, e . g .the non-meshable singularity graph of the rockerarm model shown in the Figure 10 of [3]. Our snappingtechnique does not handle such case.
5. CONCLUSIONS
To deal with non-meshable 3-5 singular curves, whichare induced by CAD feature curves, we studied fourheuristic-based frame field correction strategies.The feature curve extrusion ( § boundarysingular node extrusion ( § § § fillet correction. However,as the frame field is no longer aligned with the bound-ary everywhere, some blocks of the associated decom-position may have flat corners. Fortunately, these ge-ometric defects can be removed via a post-processingblock refinement procedure.Short of having a better frame field formulation thatdo not produce 3-5 singular curves, we recommendusing the singular curve snapping correction ( § Acknowledgement
This research is supportedby the European Research Council (project HEX-TREME, ERC-2015-AdG-694020).
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