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Steps of the main recovery algorithm on a simple 2D example: (a) edge insertion; (b) mesh entity splitting; (c) edge tracking; and (d) edge swapping and recovery.
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This study describes an algorithm for recovering an edge which is arbitrarily inserted onto a pre-triangulated surface mesh. The recovery process does not rely on the parametric space of the surface mesh provided by the geometric modeller. The topological and geometrical validity of the surface mesh is preserved through the entire recovery process....
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... The edges found in the above edge tracking process are swapped such that the edge recovery could be achieved. The basic steps of the algorithm are given for the case of a 2D mesh in Figure 1. The success of the recovery process depends on accurate insertions of the vertices of the edge into the triangulation. The mesh entity within which each vertex of the edge falls is found by means of a vertex insertion algorithm which is discussed in Section 2.1. Once the vertex is inserted into an existing mesh entity, the entity is split (if necessary) as depicted in Figure 1(b). To initiate the edge tracking process, a path between the two vertices of the edge is found and stored as a set of edges as explained in Section 2.2 and shown in Figure 1(c). The set of edges are then tried to be successively swapped to recover the edge as discussed in Section 2.3. A sample edge recovery after swap operations is depicted in Figure 1(d) as the ÿ nal phase of the algorithm. The recovery of an edge classi ÿ ed on a model edge needs speci ÿ c consideration to preserve conformity between mesh and model topology as discussed in Section 2.4. The overall pseudo-code for the recovery algorithm is given in Figure 2. The edge recovery procedure ÿ rst identi ÿ es where to insert the vertex points of the edge into the existing mesh. The insertion points de ÿ ning the end vertices of the edge may be at an ...
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... input to the edge recovery algorithm is an edge whose vertices are on the model geometry but not necessarily on the surface mesh. In other words, the edge can be considered to be dangling over the mesh freely as shown in Figure 1(a). In the case of edge recovery on a surface trian- gulation, the to-be-recovered edge M 1 r can be classiÿed on either a G 2 j or a G 1 j . ...
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... vertices of the edge are incorporated into the surface mesh: (b) The edges found in the above edge tracking process are swapped such that the edge recovery could be achieved. The basic steps of the algorithm are given for the case of a 2D mesh in Figure 1. The success of the recovery process depends on accurate insertions of the vertices of the edge into the triangulation. ...
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... mesh entity within which each vertex of the edge falls is found by means of a vertex insertion algorithm which is discussed in Section 2.1. Once the vertex is inserted into an existing mesh entity, the entity is split (if necessary) as depicted in Figure 1(b). To initiate the edge tracking process, a path between the two vertices of the edge is found and stored as a set of edges as explained in Section 2.2 and shown in Figure 1(c). ...
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... the vertex is inserted into an existing mesh entity, the entity is split (if necessary) as depicted in Figure 1(b). To initiate the edge tracking process, a path between the two vertices of the edge is found and stored as a set of edges as explained in Section 2.2 and shown in Figure 1(c). The set of edges are then tried to be successively swapped to recover the edge as discussed in Section 2.3. ...
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... set of edges are then tried to be successively swapped to recover the edge as discussed in Section 2.3. A sample edge recovery after swap operations is depicted in Figure 1(d) as the ÿnal phase of the algorithm. The recovery of an edge classiÿed on a model edge needs speciÿc consideration to preserve conformity between mesh and model topology as discussed in Section 2.4. ...
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... the possible conÿgurations are tried to collapse out the vertex as shown in Figures 9(a) and 9(b). However, if the collapse oper- ation could not be carried out successfully, because of topological and=or geometrical constraints, Figure 10. (a) Collapse onto edge e1 cre- ates a at face f2 and collapse onto edge e2 creates a at face f1: Collapse op- eration onto edge e3 and edge e4 is re- stricted. ...
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... The vertex is repositioned along edge e3 so that intersection with the recovered edge will always be on an edge. or the imposed restrictions over the collapse operation speciÿc to the application, then the vertex is perturbed along one of its edges so that the intersection will always be on an edge as depicted in Figure 10. One other problem is the possibility to get a path which wraps around itself or completes the circle from the other side of the mesh depicted in Figure 11. ...
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... the imposed restrictions over the collapse operation speciÿc to the application, then the vertex is perturbed along one of its edges so that the intersection will always be on an edge as depicted in Figure 10. One other problem is the possibility to get a path which wraps around itself or completes the circle from the other side of the mesh depicted in Figure 11. This situation usually results as a side eect of extending the projection vectors to ±∞ for the cases of no intersections. ...
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... the swap could not be realized then the to-be-recovered edge is replaced with a set of edges forming a path between its end vertices and this list is returned to the application. These edge pieces are then tried to be recovered by the same swap algorithm given in pseudo- code form in Figure 12. The recovery of each of the edges in the returned path is guaranteed since insertion of their vertices will naturally result in their recovery. ...
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... of edges to recover the edge is illustrated on a planar mesh in Figure 13. The dotted edges which are items of swappable edges list are found by edge tracking algorithm (see Section 2.2). ...
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... dotted edges which are items of swappable edges list are found by edge tracking algorithm (see Section 2.2). The ÿrst and the second edges are swapped without any diculty consecutively as depicted in Figure 13(b) and 13(c). The third edge however could not be swapped due to the validity check of swap operation. ...
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... the third edge is skipped unswapped. The edge denoted by arrowed line is recovered by swapping of the third edge whose swap is made available after the swap of the fourth edge as shown in Figure 13(e). ...
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... investigate the case of algorithmic constraint on the same example in Figure 13, the swapping orders of the edges should be changed. If the second edge is tried ÿrst to swap, it would be rejected by the algorithm since the new conÿguration would have intersected with the to-be-recovered edge. ...
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... all the vertices in the list are collapsed then the edge is recovered and the algorithm is exited successfully. An example of how the vertex collapsing operations are tried is shown in Figure 14. If the recovery could not be realized then a similar methodology is utilized as in the case of edge recovery on a model face. ...
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... these cases, the edge must be split into two or more edges. Figures 15 and 16 show the recovery of very long edges to illustrate the capability of the procedure to deal with large changes in the mesh face normal over the span of the edge. Note that in the case of planar faced objects, recovery is always possible. ...
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... that in the case of planar faced objects, recovery is always possible. This is not true for the curved surface case of the cylinder in Figure 15 which demonstrates the ability of the algorithm to deal with an extreme case. The recovery of extremely long edges onto curved surfaces may result in surface distortions as shown in Figure 15. ...
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... is not true for the curved surface case of the cylinder in Figure 15 which demonstrates the ability of the algorithm to deal with an extreme case. The recovery of extremely long edges onto curved surfaces may result in surface distortions as shown in Figure 15. The recovery of an edge classiÿed on a model edge is illustrated in Figure 16. ...
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... recovery of extremely long edges onto curved surfaces may result in surface distortions as shown in Figure 15. The recovery of an edge classiÿed on a model edge is illustrated in Figure 16. Note that the intention of showing these distorted meshes is to depict the capability of the algorithm and not to use them for numerical simulations. ...
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... need for the edge recovery algorithm is prompted for the construction of boundary layers over pre-existing surface meshes. The steps of surface boundary layer construction process utilizing the proposed edge recovery are briey summarized below and depicted in Figure 17. The details can be found in Reference [12]. ...
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... details can be found in Reference [12]. The result of the surface mesh of a particular model geometry before and after the recovery process is shown in Figures 18(a) and 18(b). The insertion of boundary layer edges by means of the edge recovery process for dierent models is also illustrated in Figures 19-21. ...
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... result of the surface mesh of a particular model geometry before and after the recovery process is shown in Figures 18(a) and 18(b). The insertion of boundary layer edges by means of the edge recovery process for dierent models is also illustrated in Figures 19-21. The process has been applied to the recovery of boundary layer mesh entities (with aspect ratios greater than 10,000) into the surface mesh of arbitrarily complex, general non-manifold models. ...
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... process has been applied to the recovery of boundary layer mesh entities (with aspect ratios greater than 10,000) into the surface mesh of arbitrarily complex, general non-manifold models. To demonstrate Figure 18. Recovery of boundary layers: (a) surface mesh before the insertion of boundary layers; (b) surface mesh after the insertion and recovery of boundary layers. ...
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... of boundary layers: (a) surface mesh before the insertion of boundary layers; (b) surface mesh after the insertion and recovery of boundary layers. Figure 19. Recovery of boundary layers: (a) surface mesh before the insertion of boundary layers; (b) surface mesh after the insertion and recovery of boundary layers. ...
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... Transfer of user attributes to new geometrical topologies. 16. Locally remesh to improve quality around the intersection graph. ...
... However, it may happen after the first split as there will be more faces due to the prior splits inside the master face. In such a case, we move to the next adjacent face of the edge whose area coordinate is negative, and start localizing the vertex within the next adjacent face [16]. If only one barycentric area is less than the edge relative tolerance, then the vertex is localized on the corresponding face edge. ...
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