CleanUp: Improving Quadrilateral Finite Element Meshes Paul Kinney MD-10 ECC P.O. Box 203 Ford Motor Company Dearborn, MI. 8121 (313) 28-1228 pkinney@ford.om Abstrat: Unless an all quadrilateral (quad) finite element mesher is of a high quality, the mesh it produes an ontain misshapen quads. This paper will desribe CleanUp, written to improve an all quad mesh. CleanUp looks at improving node onnetivity, boundary and flange patterns, quad shape, and to some extent, quad size. CleanUp is urrently used in onjuntion with the Paver algorithm developed by Sandia National Laboratories and is a part of their CUBIT software. Keywords: all quadrilateral meshing, mesh improvement Purpose: The quality of a finite element mesh affets the results of analysis done using that mesh. For example, the more the angles of a quadrilateral deviate from 90 degrees, the more unreliable the stress alulations beome. In ontrast, an all-quadrilateral mesher may, under some irumstanes, be doing a good job merely to put any mesh onto the surfae without being onerned with the quality of the mesh. A separate proess to review the mesh and improve it has proven to drastially redue the number of misshapen quads and the extent of the shape distortion. This paper desribes that proess and how it was developed. Definitions: The valene of a node is the number of edges that meet at a node (or the number of quads that have a orner at the node). A valent node has edges into the node and quads with a orner at the node. The valene of a boundary node is based on the valene of the node on the surfae being meshed and valene of the node on the adjaent surfae with the assumption that (1) there is another surfae there and (2) its mesh is ideal. These assumptions are base solely on the angle of the surfae boundary at the node. Fig. 1 has examples. Fig. 1 Add 2 to number of edges for ideal quads on adjaent surfae valene at boundary node Add 1 for ideal quads on next surfae No need to add to number of edges
An irregular node is one with a valene (regular or boundary) that is not. A permanent node is one that annot be moved. This refers to nodes on the surfae boundary, whih CleanUp assumes to be fixed as they were the starting point for the mesher. A permanent node may also refer to a hardpoint, whih might represent a weldpoint or other unmovable feature, or a hardline, whih is a srath or a pattern of nodes and edges required to exist in the mesh. A proteted quad is one that annot partiipate in the leanup proess. Examples of this are quads from neighboring faes and quads from a pattern mesh in diffiult geometry (suh as a surfae spur shown if Fig. 2) in whih the quads are poorly shaped but are the best that an be reated. Shape leaning would attempt to improve the middle two quads that have the large angles. Fig. 2 A surfae spur mesh Goals: How does one tell if a quad mesh is in good shape or if it needs some repairs? There are various quality metris available whih are desribed elsewhere [1]. The one used in evaluating a mesh for this projet put importane on skew (defined as the differene between the angle at the vertex and 90 degrees) and aspet ratio (defined as the ratio of the longest side to the shortest side of a quad). The quality metris suggest the following guidelines whih have been used in the development of CleanUp. All nodes, exept those expliitly marked as permanent may be moved or deleted. New nodes are added as needed. The pattern of quads around a permanent node may be hanged. All quads, exept those expliitly marked as proteted, are free to be adjusted or deleted. New quads are added as needed. Node valene of 2 and less or 6 and greater should be eliminated. A two valent node implies vertex angles of 180 degrees and a skew of 90 degrees. A 6 or higher valent node implies a vertex angle of 60 degrees or less and a skew of 30 degrees or more. The number of nodes with valene of 3 and should be minimized. These valenes are allowed as transitional meshes and some basi mesh patterns are impossible without them. They should be minimized as they imply angles of 120 degrees (and skew of 30) and of 72 degrees (and skew of 18). Quad angles greater than 160 degrees should be eliminated. Irregular nodes should be moved away from the surfae boundary as muh as possible. The number of edges on the surfae that meet at a boundary node should be appropriate for the angle of the boundary at the node as defined by boundary valene. Eah individual ation should improve the mesh by improving quad shape or reduing the number of irregular nodes. Method of development: The method of development has, unfortunately, been entirely empirial. The ases found and implemented have ome stritly from examining various test meshes to note areas that ould be improved and studying possible methods of improvement.
The areas that need improvement are found through the use of a mesh analyzer to report on situations that ould benefit from additional leanup. This mesh analyzer reports valene patterns (desribed below) with two or more irregular nodes and quads with angles more than 160 degrees. This was implemented first to give an idea of whih ases this mesher produes and in what frequeny. The implementation was geared to what a partiular mesher produes. The ation taken to improve a partiular pattern is determined mostly by experimentation, though some guidelines an help the proess. From an outline of the area surrounding the problem, an a standard pattern be used to fill it? Can a small hange (rotate an edge, replae 2 quads with 3) handle the problem? Does a replaement pattern have fewer irregular nodes? Even with these guidelines, many possible solutions might be drawn on paper before an aeptable one is found. Many times ases studied in isolation that improve the mesh in one area might ause degradation in an adjaent area. Beause of that, restritions are plaed on a ase as to when it an be used. For example, a ase may have a restrition that a ertain node must not be permanent as that would reate an irregularity on the boundary. The ases that are important for one mesher may not be the same as for another mesher. The author has worked with two different meshers. One tended to produe poorly shaped faes, the other tended to produe lusters of irregular nodes. A omplete general purpose leanup algorithm may be onsiderably larger than what is neessary for any one partiular mesher. The author was guided by a omment from another mesh improvement developer. One of the diffiulties in implementation: Finding and oding an exhaustive set of all permutations of ases (1000 s) [2]. A leanup implementation does not have to be omplete to be useful. Even if only a few of the ases are implemented, the mesh will be better than it was before. The implementer an onentrate on the ases determined to be most important, saving additional ases for later. The author knows of no mathematial proof or formula whih would reveal a finite set of ases and their proper resolution. The author has written a mathematial definition of the leanup proess and will make it available to anyone interested in pursuing a mathematial solution. Connetivity leanup: Eah node in the interior of the mesh is heked to improve its onnetivity. The nodes, edges, and quads surrounding the node being heked are ordered in a ounter-lokwise manner around the quad normal. The number of these neighbor edges and quads is the same as the node valene (by definition). The number of nodes is twie the valene as shown in Fig. 3. n n e2 n3 e0 - e3 are the neighboring edges to node n6 e3 e1 n2 n0 - n7 are the neighboring nodes to node n7 e0 n0 n1 Fig. 3 neighboring nodes and edges The valene of the node and eah of its neighbors is omputed and ompared against patterns that have a known leanup ation. Eah onnetivity leanup ase is doumented by the pattern of valenes that it heks. The valene of the entral node is listed first, followed by a dash. Then omes the valene of a node at the other end of an edge from the enter node. The remaining neighbor nodes are listed ounter-lokwise around the element normal. A - means a valene of or less. A + means a valene of or more. A means a valene of or more. A 0 means the valene is ignored and unhanged and is usually drawn as valene. Case -3000 is shown in Fig..
3 ase -3000 Fig. Case -3000 is a andidate for leanup as the list inludes more than two irregular nodes. Transitions may require both a -valent and a 3-valent node but patterns with more than two an usually be improved. The first hoie for a replaement pattern is one that results in one of the standard mesh patterns. These are shown in Fig. If none of those fit, the replaement pattern is one that redues the number of irregular nodes. An ation routine will hange the old pattern into the new one. no transition, opposite sides have the same edge ount Fig. Standard mesh patterns single transition, ount on one pair of opposite sides varies by two double transition, ount on both pairs of opposite sides vary by one The most frequent ations are a ombine with neighbor operation. A quad and a neighbor (one that shares a ommon edge) are deleted along with that ommon edge. This reates a hole surrounded by 6 nodes. The hole is filled with two, three, or four quads, shown in Fig. 6. Fig. 6 fill_2 fill_3 fill_ Filling a 6 node hole
The simplest of these is the swith diagonal CW and -CCW. Fig. 7 is an example of how swith diagonal CCW would be used. The edge marked A is the one swithed. The quads on either side of edge A are deleted and fill_2 is done to fill the hole suh that the two new faes have a different onnetivity. The valene pattern in this example is -33000000. Three irregular nodes in the old pattern are replaed with one irregular node in the new pattern. A A Fig. 7 swith diagonal CCW Another example of a ombine with neighbor is an open quad operation. In this ase, the two quads - on either side of edge B - are deleted and a fill_3 is used to lose the hole. This is shown in Fig. 8. Again, three irregular nodes are replaed with one irregular node. B Fig. 8 open quad operation Many ases also have a mirror image ase. The mirror pattern to -3000 is -0003 (or, if starting from the top edge, -3000) and is shown in Fig. 9. This means there are frequently pairs of ation routines to handle a ase and its mirror. ase -3000 ase -0003 3 3 Fig. 9 mirror ases
If no math is found against the known ases, the lists are adjusted to orrespond to starting with the next edge and the ases are heked again. Many ases also hek for permanent nodes in various positions beause their orresponding ation would disturb the permanent node, or would not be appropriate lose to a surfae boundary as it would reate a quad with one node on the boundary, or would not have enough room for proper smoothing. One an ation has been done for onnetivity leanup, the neighbor nodes now have a different valene, so they are put bak into the list to be heked again. There are urrently 6 onnetivity ases in the ode. These invoke 27 different ation routines. The ation routines range from simple operations to rebuilding a small area of the mesh. Additional examples of onnetivity leanup ases and their resolution are shown in Fig. 10 and 11. In Fig. 10, all three quads around the enter node are deleted and a fill_2 is used to fill the hole. Four irregular nodes are replaed with zero irregular nodes. + 3 Fig. 10 + Case 3-+3+000 In Fig. 11, three quads around the enter node and a fourth outer quad are deleted and the hole is losed with a fill_2. Four irregular nodes are replaed with two irregular nodes. 3 Fig. 11 Case 3-3 While some ases hange a small number of quads, other ases an hange a large number of quads as shown in Fig.12. In this ase, 10 quads, 1 edges, and of 6 interior nodes are deleted. The replaement is 6 quads.
3 3 Fig. 12 Case -33 10 quads replaed with 6 quads Boundary leanup: The boundary ases fall into three types. The first is a boundary node with only two edges, both on the surfae boundary, at a plae where the boundary forms an angle larger than 10 degrees. This forms a triangular (or nearly so) shaped quad against the boundary. Examples of replaing a triangular quad are shown in Fig 13. one row transition two row transition Fig. 13 replaing triangular shaped quads The seond type of boundary leanup is similar to onnetivity leanup. The nodes, edges, and quads around the boundary are again put in order, but with a definite start and end to the list. Most of these ases are designed to lean up a flange and some of the ations may work their way along the flange. They are a part of boundary leanup as there isn t a pattern detetable from onnetivity leanup of interior nodes. They an be found by using a valene pattern similar to onnetivity leanup, though there are some differenes. The valene pattern annot rotate around the entral node and many ases must also inlude areful heks for permanent nodes or the angles in a quad. There are over a dozen boundary ases. Fig. 1 shows some examples of boundary ases and their resolutions.
3 3 Fig. 1 boundary onnetivity patterns 3 The third type of boundary leanup is removal of boundary diamonds. These are quads that have only one node on the surfae boundary. There are some ases where these quads an be ollapsed. There are others where empirial evidene indiates they annot be, at least not without more researh. Some examples are shown in Fig. 1. a diamond that an be ollapsed a diamond best left alone Fig. 1 boundary diamonds Shape leanup: There are two types of misshapen quads. The first type merely has a large angle - greater than 160 degrees. When these poorly shaped quads our, they are frequently alongside a surfae boundary or hardpoint. The usual method of resolution is to do a ombine with neighbor. There are several ases to deide whih neighbor is best and what pattern
should be used to fill it. Fig. 16 ontains an example. Fig. 17 shows a fae ut in two and a ombine with neighbor done on eah half. Fig. 16 ombine with neighbor for large angles Fig. 17 split a fae in half and ombine with neighbor on both sides The seond type of misshapen quad inludes both the hevron (or arrowhead) and the bowtie. Both have a large angle (over 200 degrees). The differene is that two of the edges of the bowtie interset, forming a twist in the quad. The hevron may be a poorly shaped quad, but the bowtie is illegal in that it violates the topology of the mesh. The hevron and bowtie are proessed separately from simple large angles as both of these types of quads must be arefully analyzed to determine the bad angle as the quad normal may be inverted. Again, the bad quad is removed along with a neighbor and the hole is filled with better quads. These types of quads annot be resolved by adjusting the node at the bad angle as it is frequently a hardpoint. Fig. 18 shows a hevron and its resolution. Fig. 19 shows a bowtie and its resolution. Fig. 18 leaning up a hevron
Fig. 19 leaning up a bowtie Sine shape leanup is dependent on angle measurements, smoothing is done as a part of eah ation, so that the angles represent an atual situation in the mesh. A major dead end was enountered during development of shape leanup. The first implementation used a generi ombine with neighbor routine. It would hoose what seemed to be a good neighbor, delete the two quads, and fill the hole based only on the shape of the hole. This did not take any ues from the surrounding area and frequently filled the hole with the same pattern of quads that had just been deleted. This was replaed by a more thoughtful look at the ases involved. Size leanup: An edge that is onsiderably larger or smaller than it is supposed to be implies a misshapen quad or one that is out of step with an underlying sizing or adaptivity funtion. Size leanup heks eah edge on the surfae (not the boundaries), omparing its length against the underlying size funtion. If the size is more than 2. times what it should be, the quads on either side are removed. If two opposite nodes an be joined with a shorter edge, that is done. Otherwise the hole is filled with three quads. Fig. 20 shows examples. Long edge replaed with a shorter edge. Fig. 20 replaing a long edge Long edge replaed with 3 quads to redue the size of the quads. Some leanup of quads that are too small an be done as part of onnetivity leanup. The standard transition patterns of -3 and -3 (zeroes assumed for the rest of these patterns) an be moved if the quad size is too small or too large. In the ase of the -3 pattern, if the quads adjaent to the valent node opposite of the 3 valent node are too large, the transition an be shifted into these quads to make them smaller. Likewise if the nodes adjaent to the 3 valent node and opposite of the valent node are too small, the transition an be shifted toward these quads to make them larger. This transition an work its way aross a surfae. These ases are shown in Fig. 21.
a -3 transition transition shift for smaller quads Fig. 21 adjusting a transition for size transition shift for bigger quads Other than shifting transitions, there is urrently no leanup done if the edge is onsiderably smaller than it is supposed to be beause obvious solutions (ollapsing a quad, for instane) introdue problems with onnetivity and shape. General algorithm flow: The leanup proess starts with a mesh topology inspetion. Dangling nodes, edges, and quads are deleted. Smoothing of the mesh is done at the start and after eah major part of the algorithm. Loal smoothing is also done after eah shape leanup ation. The smoother that is used during leanup is the same one used during the reation of the mesh so that the same goals are maintained. In this projet, a Laplaian smoother is used. A more omplete desription of smoothing an be found elsewhere [3,, ]. An initial pass is made through the quads to eliminate hevron and bowtie quads as onnetivity leanup around a bowtie usually makes the situation worse. Again, this is from empirial evidene. There is a loop over the major leanup proesses - onnetivity, boundary, shape, size. This loop is exeuted up to three times. More than three passes probably won t improve the mesh and there is a hane of an infinite loop. Sine the omponents of leanup, primarily onnetivity and shape, have different goals and different riteria for what should be done, an ation by one ase may degrade the mesh aording to the rules of another ase. A onnetivity fix may introdue a poorly shaped quad. A shape fix may introdue a bad onnetivity situation. It is possible for two ations to yle between two patterns of quads. This situation ours beause the ases and ations are derived empirially and then may be applied in a situation in whih it isn t appropriate. Only areful examination of a large number of test examples an resolve suh situation, and the size of data from those test examples limits how many ases an be arefully traked. Results: The CleanUp proess an make a onsiderable differene in a mesh. Fig. 22 shows a mesh for a single surfae without any CleanUp performed. The entire part was meshed in 2 seonds. The skew rating used here is based on the deviation of the angle at eah vertex from 90 degrees. The mesh shown here has an average deviation of 27 degrees.
This mesh has quads that have 3 olinear nodes, a 6 valent node, and 2 irregular nodes. Fig.22 a mesh without CleanUp Fig. 23 shows the same surfaes with CleanUp performed. The entire part was meshed and leaned in 6 seonds. This mesh has an average deviation of 23 degrees. There are no quads with 3 olinear nodes, no 6 valent nodes, and only 28 irregular nodes. Fig. 23 a mesh with CleanUp
Conlusions The mesh leanup proedure desribed here an be adapted to work on the results of any quad mesher. The atual valene, boundary, shape, or size ases that are implemented depend on what the mesher produes. This partiular implementation of CleanUp annot be applied to a triangular mesh or to a mesh that is a mix of trias and quads. Methods of improving triangular meshes have been desribed elsewhere [6, 7]. The author has had limited experiene with leaning a mesh with a mix of quads and trias. The mesher produed a small perentage of trias and a large perentage of quads. This mesh was leaned by examining the mesh for ases where trias and quads ould be ombined for more quads and fewer trias or rearranged for better shaped elements. Then the trias were treated as holes in the mesh with virtual surfae boundaries around them and CleanUp was applied to just the quads. The mixed element leaner ontained only a few ases as it (1) dealt only with ases involving only trias or a mix of trias and quads and (2) the perentage of trias in the mesh was small. Even so, the ases were determined empirially. If the perentage of trias was higher and the trias ould not effetively be isolated from the rest of the mesh, the author feels the omplexity and number of the ases for a mixed element mesh may prove daunting. Areas of future diretion: Additional work in mesh leanup would likely be done in these areas: (1) Additional ases for eah of onnetivity, boundary, shape, and size would be implemented. The analysis reports now generated are by no means lean. (2) Explore ways to do leanup on elements that are too small. (3) Consider ways to do leanup on quads that have angles that are too small, instead of too big. Aknowledgments: This work was done as part of the Sandia National Laboratories CUBIT projet and its Mesh Generation Consortium. This work builds on work done by Blaker and Stephenson [3], Sott Canann, and Roger Cass. The author wishes to thank the members of the CUBIT team for the initial work and for ongoing support. Referenes: [1] John Robinson, CRE Method of Element Testing and the Jaobian Shape Parameters, Engineering Computers, 1987, Vol.. [2] Sott Canann, Sella Muthukrishnan, Bob Phillips, Topologial Improvement Proedures for Quadrilateral and Triangular Finite Element Meshes, Proeedings, 3rd International Meshing Roundtable pp. 9-88. (199) [3] T. D. Blakler and M.B. Stephenson, Paving: a New Approah to Automated Quadrilateral Mesh Generation, International Journal for Numerial Methods in Engineering, Vol. 32, pp 811-87 (1991). [] L.R. Herrimann, Laplaian-Isoparametri Grid Generator Sheme, J. Eng. Meh. Div. ASCE, 102, EM 79-76 (1976). [] R. E. Jones, QMESH: A Self-Organizing Mesh Generation Program, SLA-73-1088, Sandia National Laboratories, 1979. [6] W. H. Frey and D.A. Field, Mesh Relaxation: a New Tehnique for Improving Triangulations, International Journal for Numerial Methods in Engineering, Vol. 31, pp. 1121-1133 (1991). [7] D.A. Field and W. H. Frey, Strutural Improvement of Planar Triangulations: Some Constraints and Pratial Issues, Communiations in Numerial Methods in Engineering, Vol. 11, pp. 191-198 (199).