T-Base: A Triangle-Based Iterative Algorithm for Smoothing Quadrilateral Meshes

Transcription

1 T-Base: Trangle-Based Iteratve lgorthm for Smoothng Quadrlateral Meshes Gang Me, John.Tpper and Nengxong Xu 2 bstract We present a novel approach named T-Base for smoothng planar and surface quadrlateral meshes. Our motvaton s that the best shape of quadrlateral element square can be vrtually dvded nto a par of equlateral rght trangles by any of ts dagonals. When move a node to smooth a quadrlateral, t s optmal to make a par of trangles dvded by a dagonal be equlateral rght trangles separately. The fnally smoothed poston s obtaned by weghtng all ndvdual optmal postons. Three varants are produced accordng to the determnaton of weghts. Tests by the T-Base are gven and compared wth Laplacan smoothng: The Var. of T-Base s effectvely dentcal to Laplacan smoothng for planar quad meshes, whle Var.2 s the best. For the quad mesh on underlyng parametrc surface and nterpolaton surface, Var.2 and Var. are best, respectvely. Keywords Mesh smoothng Iteratve smoothng Quad meshes Laplacan smoothng Length-weghted Introducton The qualty of meshes s crtcal to obtan relable smulaton results n fnte element analyses. Usually after generatng computatonal meshes, t s necessary to mprove the qualty of meshes n further. There are two mportant categores of qualty mprovement methods. One s called clear-up technques, whch alters the connectvty between elements. The other s called mesh smoothng, whch only relocates the nodes. There are numerous publcatons on the topc of mesh smoothng. nd we just refer some popular and representatve ones. Gang Me, John.Tpper () Insttut für Geowssenschaften Geologe, lbert-ludwgs-unverstät Freburg, lbertstr. 23B, -7904, Freburg m Bresgau, Germany e-mal: {gang.me, john.tpper}@geologe.un-freburg.de Nengxong Xu () School of Engneerng and Technology, hna Unversty of Geoscences, Bejng,00083, hna e-mal: xunengxong@yahoo.com.cn

2 2 G. Me, J..Tpper and N. Xu The most popular smoothng methods s Laplacan smoothng [5, 7], whch repostons each node at the centrod of ts neghbourng nodes n one teraton. The popularty of ths method comes from ts effcency and effectveness. smpler but more effectve method s angle-based approach [6], n whch new locatons are calculated by conformng specfc angle ratos n surroundng polygons. geometrc element transformaton method [4], whch s based on a smple geometrc transformaton, s proposed and appled to polygons. Shmada et al [2] proposed a method whch treats nodes as the centre of bubbles and nodal locatons are obtaned by deformng bubbles wth each other. projectng/smoothng method s proposed for smoothng surface meshes [4], where the new poston of each free node s obtaned by mnmzng the mean rato of all trangles sharng the free node. Based on quadrc surface fttng and by combnng vertex projectng, curvature estmatng and mesh labellng, Wang and Yu [5] proposed a novel method and appled t n bomedcal modellng. varatonal method for smoothng surface and volume trangulatons s proposed by Jao [9], where the dscrepances between actual and target elements s reduced by mnmzng two energy functons. lso, a general-purpose algorthm called the target-matrx paradgm s ntroduced n [0], and can be appled to a wde varety of mesh and element types. To smooth meshes better, two or more basc methods can be combned nto a hybrd approach [,2,6],.e., an analytcal framework for mesh qualty metrcs and optmzaton drecton computaton n physcal and parametrc space are proposed for smoothng surface quad meshes n [3]. In ths paper we ntroduce a novel teratve method named T-Base to smooth planar and surface quad meshes. The best shape of a quadrlateral element s square, whch can be vrtually dvded nto a par of equlateral rght trangles by any of ts dagonals. Hence, when move a node to smooth a quad element, t s optmal to make the two trangles dvded the dagonal conssted by the node and ts opposte one be equlateral rght trangles separately. The fnal smoothed poston s obtaned by weghtng all the separate optmal postons. When smooth surface quad meshes, we frstly compute the local coordnates system for each vrtual trangle and then calculate the optmal poston, and fnally obtan the smoothed node by transformng t from local coordnates to the global coordnates and weghtng all ndvdual optmal postons. fter generatng the optmal smoothed postons, they should be moved agan n order to preserve the features of ntal surfaces. For quad mesh on parametrc surfaces, we project the smoothed node onto the orgnal parametrc surface along the normal. For quad mesh on nterpolaton surfaces, we re-nterpolate the smoothed nodes to ft them wth the ntal surfaces. The rest of ths paper s organzed as follows. Sect.2 descrbes the detals of the T-Base ncludng ts three varants for smoothng planar quad meshes. In Sect.3, we smply extend T-Base to smooth surface quad meshes. Then we gve several examples n Sect.4 to test the performance of the T-Base and compare t wth Laplacan smoothng. Fnally, Sect.5 concludes ths work.

3 T-Base: Trangle-Based Iteratve lgorthm for Smoothng Quadrlateral Meshes 3 2 T-Base for Planar Quad Meshes The best shape of a quadrlateral element s square, whch can be vrtually dvded nto a par of equlateral rght trangles by any of ts dagonals. When move a node n order to smooth a quadrlateral, t s optmal to make the two trangles dvded by the dagonal conssted wth the node and ts opposte one be equlateral rght trangles separately (Fg.). * * B B + Fg. Smoothng of quad element B based on a par of vrtual trangles onsder a sngle quadrlateral element B shown n Fg.. It s vrtually dvded nto a par of trangles B and. * s are the postons to whch node would have to be moved to make B and be equlateral rght trangles, assumng that nodes B, and were fxed. The coordnates of * n trangles B and are: * X = X B + YB Y * Y = X B + YB + X * X = X, * Y = X Y + Y + Y X () Now assume B s part of a quadrlateral mesh. Each node of B for nstance node s then shared wth several other elements, and * can be calculated for each of these. The fnal poston of ts optmal smoothed poston s obtaned by consderng all the separately calculated * s (Fg.2). The tradtonal Laplacan smoothng assumes that the new poston of a node should be an average of the postons calculated for t for each of the elements of whch t s part (Fg.2). Ths assumpton s not essental, however, and t can be relaxed by makng that new poston a weghted average of those postons, wth the weghts beng proportonal to the lengths of the opposte edges. Then, for node : X = ( w X ), Y = ( w Y ) (2)

4 4 G. Me, J..Tpper and N. Xu,where n s the number of elements of whch node s part, and w s the weght of the th separate poston * whch can be calculated accordng to the length l of relevant th edge * Fg. 2 Node belongs to 5 quad elements. * can be calculated for each trangle separately (black crcles). Optmal smoothed poston for s produced from all * s ccordng to the determnaton of the weghts w, we produce three varants: Varant. Ths varant s termed as the average verson of the T-Base: 0 w = = l 0 l (3) The averagng process s effectvely dentcal to that used n the tradtonal Laplacan smoothng, whch explans why test results obtaned usng Laplacan smoothng are dentcal to that of T-Base for planar quad meshes (see Fgs 3, 4). But the above concluson s no longer true for surface quadrlateral meshes. Varant 2. Ths varant s termed as the ( /2) nverse-length verson: w = l l = l / 2 / 2 l (4) Varant 3. Ths varant s termed as the ( ) nverse-length verson: w = l = l l l (5) The ntroducton of nverse-length weghtng s n many respects advantageous, because hgh qualty elements such as equlateral quadrlateral element or even square generally have nearly or exactly same-length edges. In order to transform

5 T-Base: Trangle-Based Iteratve lgorthm for Smoothng Quadrlateral Meshes 5 quadrlateral elements to be equlateral as more as possble, we let longer edges of an element have less mportance n the smoothng than shorter ones. The dsadvantage of nverse-length versons (Var.2 and 3) over the average verson (Var.) s of course that t brngs a tme penalty, as the weghts have to be calculated afresh at each teraton step. Ths s also the reason why nverse-length versons need more teraton steps to converge than that of the average verson. The mplementaton of T-Base for planar quadrlateral meshes s very smple: () search all ncdent elements for each node; (2) calculate of smoothed postons of each node by makng relevant vrtual trangles be equlateral rght; (3) terate prevous step untl a tolerance dstance s reached. 3 T-Base for Surface Quad Meshes Eq. computes the optmal poston for a vrtual trangle n 2. For surface quad meshes, we frstly compute the local coordnates system for each vrtual trangle and then calculate the optmal poston va Eq., and fnally obtan the smoothed nodal poston by recoverng t to global coordnates and weghtng all * s. fter obtanng the optmal smoothed postons, updatng should be done for dfferent type of dscrete surfaces n order to preserve the shape of ntal surfaces. For quad mesh on parametrc surfaces, we compute the normal at each vertex and then project the smoothed node onto the orgnal parametrc surface along the normal to obtan fnal poston (Fg.5). For quad mesh on nterpolaton surfaces, we re-nterpolate the smoothed nodes to ft them wth the ntal surfaces (Fg.6). Flow of the T-Base for surface quad meshes s lsted n lgorthm.. lgorthm T-Base for Smoothng Surface Quad Meshes Input: n orgnal surface quad mesh Output: smoothed surface quad mesh : Search the ncdent elements for each node v. 2: whle teratons not termnate do 3: for each node v do 4: for each ncdent element Q j (0 j < n) of v do 5: vde Q j nto two trangles and calculate local coordnates separately. 6: Obtan a par of * s locally and transform them back to global. 7: alculate a par of weghts w s n Q j. 8: end for 9: Obtan optmal smoothed poston of v : X = ( w X ), Y = ( w Y ), Z = ( w Z 0: Update v by projectng t to ntal parametrc surface or re-nterpolatng. : end for 2: end whle )

6 6 G. Me, J..Tpper and N. Xu 4 pplcatons and scusson 4. Mesh Qualty The smplest way to measure mesh qualty s to calculate the dstorton values for each of the mesh elements separately, and then to compare the dstrbutons, ncludng mean qualty (MQ) and mean square error (MSE), of those values. For a quad B, we use the measure λ [8], shown n Eq.6. The value of λ les between 0 and ; λ = 0 when any three nodes are collnear; λ = when B s square. B B B B λ = 24 (6) ( B + )( B + B )( + B )( + ) For a quad element n 3, t s warped generally. In ths paper, we propose a new measurement γ n whch shape and warpage are taken nto account. quad B can be dvded nto four trangles: B, B, and B. We frstly calculate the local coordnates system of these trangles and project the orgnal quad element B onto each local coordnates system to obtan four planar quads B P, B P, B P and B P, respectvely. Let λ, λ 2, λ 3 and λ 4 denote the λ values of the four planar quads, then γ = (λ + λ 2 + λ 3 + λ 4 ) / 4. The value of γ also les between 0 and ; we specally set λ = 0 when any three nodes are collnear; λ = when B s coplanar square. Thus, we have: n 2 MQ = γ, MSE = ( γ MQ) (7) n n n, where n s the number of elements n a quad mesh. 4.2 Tests of Smoothng Planar Quad Meshes (a) Orgnal (b) By Var./LS (c) By Var.2 (d) By Var.3 Fg. 3 Test of smoothng planar quad mesh by T-Base and Laplacan smoothng (LS)

7 T-Base: Trangle-Based Iteratve lgorthm for Smoothng Quadrlateral Meshes 7 These two orgnal quad meshes are generated by Q-Morph []. Fg.3 and Fg.4 dsplay the results of the two planar quad meshes by Laplacan smoothng and T-Base. From the comparson of mesh qualty lsted n Table., we can learn that Var. s effectvely dentcal to Laplace smoothng, Var.3 and Var.2 are better than Var., but Var.2 s best. onvergence We do not gve the algebrac proof of convergence for T-Base n theory. But our tests show that T-Base does converge for planar meshes, and the numbers of teraton steps of Var., 2 and 3 always ncreases when t converges. (a) Orgnal. (b) By Var./LS (c) By Var.2 (d) By Var.3 Fg. 4 Test 2 of smoothng planar quad mesh by T-Base and Laplacan smoothng (LS) 4.3 Tests of Smoothng Surface Quad Meshes We frstly generate the planar meshes n a crcular area, and then project t onto the parametrc surface z = (x 2 + y 2 ) to obtan the mesh n Fg.5a. The orgnal mesh n Fg.6a s for heght nterpolaton by Krgng method [3]; only z - value/heght s nterpolated whle coordnates x and y are fxed. Fg.5 shows the results of quad meshes on a underlyng parametrc surface. Notceably, only Laplacan smoothng converges after 49 teratons. We just let T- Base terate 49 tmes as that of Laplacan smoothng. From the comparson of mesh qualty n Table., we can learn that T-Base s better than Laplacan smoothng. In further, Var.2 s the best, and the Var. s better than Var.3 snce the dstrbuton of element qualtes s better than that of Var.3. In Fg.6, all optmal smoothed by only Laplacan smoothng or T-Base s generated frstly, and then z -value s re-nterpolated by Krgng method. ue to the expensve cost of re-nterpolaton, we only terate 0 tmes. Smlar to the quad mesh on parametrc surface, T-Base s better than Laplacan smoothng. But Var. s the best, and then the Var.2, whle Var.3 s the worst.

8 8 G. Me, J..Tpper and N. Xu (a) Orgnal (b) By LS (c) By Var. (d) By Var.2 (e) By Var.3 Fg. 5 Smoothng results of surface quad mesh on underlyng parametrc surface (a) Orgnal (b) By LS (c) By Var. (d) By Var.2 (e) By Var.3 Fg. 6 Smoothng results of surface quad mesh on nterpolaton surface

9 T-Base: Trangle-Based Iteratve lgorthm for Smoothng Quadrlateral Meshes 9 onvergence Only Laplacan smoothng converges for the quad meshes on parametrc underlyng surface. ccordng to the convergence analyss for planar quad meshes, T-Base needs to terate more tmes than Laplacan smoothng, hence, we can frstly record the teraton number of Laplacan smoothng for surface quad meshes, and then set the number from Laplacan smoothng to be the maxmum teratons n T-Base. Ths s the reason we only terate 49 tmes n T- Base. When smooth quad meshes on nterpolaton surface, snce re-nterpolaton s too expensve, we just test the results after a specfed-number of teratons. Ths soluton of endng teratons s acceptable and valuable n practcal applcatons. Table Mesh qualty results of smoothng planar and surface quad meshes Mesh Fg.3 Fg.4 Fg.5 Fg.6 lgorthm Element qualty (0.0~.0) 0.0~ ~ ~ ~ ~.0 MQ MSE Orgnal 0.00% 0.00%.3% 6.53% 92.34% Var./LS 0.00% 0.00% 0.00% 2.93% 97.07% Var % 0.00% 0.00% 3.5% 96.85% Var % 0.00% 0.00% 3.60% 96.40% Orgnal 0.00% 0.00% 0.45% 6.47% 93.08% Var./LS 0.00% 0.00% 0.30% 6.47% 93.23% Var % 0.00% 0.45% 5.86% 93.69% Var % 0.00% 0.90% 4.8% 94.29% Orgnal 0.00% 0.99% 67.26% 8.8% 2.94% LS 0.00% 0.00% 62.45% 26.03%.53% Var. 0.00% 0.00% 0.64% 75.74% 23.62% Var % 0.00% 3.8% 6.0% 35.7% Var.3 0.4%.3% 0.04% 48.30% 40.38% Orgnal 0.00% 0.00% 2.40% 9.47% 88.2% LS 0.00% 0.00%.98% 8.0% 90.0% Var. 0.00% 0.00% 0.4% 5.70% 94.6% Var % 0.00% 0.99% 5.04% 93.97% Var % 0.28%.37% 4.90% 93.45% oncluson We present a novel teratve smoothng algorthm called T-Base for planar and surface quad meshes based on vrtually dvdng a quad element nto a par of trangles by ts dagonal. We relocate a node by makng all of the ncdent vrtual trangles be equlateral rght trangles separately, and then weghtng all separate smoothed postons. ccordng to the determnaton of weghts based on length of

10 0 G. Me, J..Tpper and N. Xu relevant edges, three varants of T-Base are produced. The T-Base s appled on planar and surface quad meshes, and compared wth Laplacan smoothng. The Var. of T-Base s effectvely dentcal to Laplacan smoothng for planar quad meshes, whle Var.2 and 3 are better. For the quad mesh on a underlyng parametrc surface, Var.2 s the best; and Var. s the best for the quad mesh on a nterpolaton surface. Tests also show that T-Base always converges for planar meshes. cknowledgments. Ths research was supported by the Natural Scence Foundaton of hna (Grant Numbers and ) and the Fundamental Research Funds for the entral Unverstes of hna. References. anann, S.., Trstano, J.R., Staten, M.L. (998). n approach to combned Laplacan and optmzaton-based smoothng for trangular, quadrlateral and quad-domnant meshes. In: Proceedngs of 7th nternatonal meshng roundtable, pp hen, Z., Trstano, J.R., Kwok, W. (2003). ombned Laplacan and optmzaton-based smoothng for quadratc mxed surface meshes. In: Proceedngs of 2th nternatonal meshng roundtable, pp hles, J.P., elfner, P. (202). Geostatstcs: Modelng Spatal Uncertanty (d edton). Wley-Interscence. 4. Escobar, J.M., Montenegro, R., Rodrguez, E., Montero, G. (20). Smultaneous algnng and smoothng of surface trangulatons. Eng. omput. 27, Feld,.. (988). Laplacan smoothng and elaunay trangulaton. ommun. ppl. Numer. Methods. 4, Fretag, L.. (997). On combnng Laplacan and optmzaton-based mesh smoothng technques. M trends n unstructured mesh generaton. SME. 220, Herrman, L.R. (976). Laplacan-soparametrc grd generaton scheme. J. Eng. Mech. EM Hua, L. (995). utomatc generaton of quadrlateral mesh for arbtrary planar domans. Ph.. thess, alan Unversty of Technology, hna (In hnese). 9. Jao, X., Wang,., Zha, H. (20). Smple and effectve varatonal optmzaton of surface and volume trangulatons. Eng. omput. 27, Knupp, P.M. (200). Introducng the target-matrx paradgm for mesh optmzaton va nodemovement. In: Proceedngs of 9th nternatonal meshng roundtable, pp Owen, S.J., Staten, M.L., anann, S.., Sagal, S. (999). Q-Morph: an ndrect approach to advancng front quad meshng. Int. J. Numer. Meth. Eng. 44(9), Shmada, K. (997). nsotropc trangular meshng of parametrc surfaces va close packng of ellpsodal bubbles. In: Proceedngs of 6th nternatonal meshng roundtable, pp Shvanna, K., Grosland, N., Magnotta, V. (200). n analytcal framework for quadrlateral surface mesh mprovement wth an underlyng trangulated surface defnton. In: Proceedngs of 9th nternatonal meshng roundtable, pp Vartzots,., Wpper, J. (2009). The geometrc element transformaton method for mxed mesh smoothng. Eng. omput. 25, Wang, J., Yu, Z. (2009). novel method for surface mesh smoothng: applcatons n bomedcal modelng. In: Proceedngs of 8th nternatonal meshng roundtable, pp Zhou, T., Shmada, K. (2000). n angle-based approach to two-dmensonal mesh smoothng. In: Proceedngs of 9th nternatonal meshng roundtable, pp