Quality Improvement Algorithm for Tetrahedral Mesh Based on Optimal Delaunay Triangulation

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1 Intellgent Informaton Management, 013, 5, Publshed Onlne November 013 ( Qualty Improvement Algorthm for Tetrahedral Mesh Based on Optmal Delaunay Trangulaton Shul Sun, Haoran Bao, Mnghu Lu, Yuan Yuan State Key Laboratory for Turbulence and Complex System, Department of Mechancs and Engneerng Scence, College of Engneerng, Pekng Unversty, Bejng, Chna Emal: Receved September 0, 013; revsed October 1, 013; accepted November 15, 013 Copyrght 013 Shul Sun et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. ABSTRACT The concept of optmal Delaunay trangulaton (ODT) and the correspondng error-based qualty metrc are frst ntroduced. Then one knd of mesh smoothng algorthm for tetrahedral mesh based on the concept of ODT s examned. Wth regard to ts problem of possble producng llegal elements, ths paper proposes a modfed smoothng scheme wth a constraned optmzaton model for tetrahedral mesh qualty mprovement. The constraned optmzaton model s converted to an unconstraned one and then solved by ntegratng chaos search and BFGS (Broyden-Fletcher-Goldfarb- Shanno) algorthm effcently. Qualty mprovement for tetrahedral mesh s fnally acheved by alternately applyng the presented smoothng scheme and re-trangulaton. Some testng examples are gven to demonstrate the effectveness of the proposed approach. Keywords: Tetrahedral Mesh; Mesh Qualty Improvement; Smoothng; Topologcal Optmzaton; Optmal Delaunay Trangulaton 1. Introducton The mesh mprovement procedure can be bascally dvded nto two categores, topologcal optmzaton and geometrcal optmzaton (also called smoothng). Topologcal optmzaton changes the topology of a mesh,.e. node-element connectvty relatonshp, whle the geometrcal optmzaton or smoothng mproves mesh qualty by smply movng or adjustng the poston of nodes wthout changng the topology of a mesh. Usually, the optmzaton-based smoothng procedure would lead to better results. However, due to the complexty of models and algorthms, the optmzaton-based smoothng s dffcult to mplement, and wth the ncrease of the number of desgn varables, the computatonal cost sgnfcantly ncreases. Therefore, n many works, smoothng was commonly performed by Laplacan smoothng technque,.e. shftng each nteror free node to the center of the surroundng polygon or polyhedron. Although computatonally nexpensve, ths method, may produce nvald or llegal elements occasonally that are unacceptable n numercal analyss. The topologcal optmzaton mproves local and whole mesh qualty by changng the topology of a mesh,.e. node-element connectvty relatonshp. The most frequently used operatons of topologcal optmzaton are so-called basc or elementary flps. Delaunay trangulaton can also be proved to be a topologcal optmzaton tool to mprove the qualty of a mesh. Chen et al. [1] presented the concept of ODT (Optmal Delaunay Trangulaton) n 00 and developed a knd of qualty smoothng algorthm [] for trangular mesh by mnmzng the lnear nterpolaton error-based mesh qualty metrc. The algorthm can calculate the optmal locaton of each nteror node drectly accordng to the locaton of ts neghbor nodes wthout solvng the optmzaton model. The computatonal cost of such algorthm s as low as that of Laplacan smoothng. Combnng ths smoothng algorthm and the concept of ODT, Allez et al. [3] proposed a Delaunay-based varatonal approach for tetrahedral mesh by mnmzng a smple mesh-dependent energy through global and updated both vertex postons and connectvty, whch s very effectve to mprove the qualty of bad tetrahedrons. However, through numercal nvestgaton and analyss we fnd that the smoothng formula used [3] for updatng vertex postons also suffers the problem of producng llegal elements. To ths end, ths paper proposes

2 19 S. L. SUN ET AL. a modfed smoothng scheme for tetrahedral mesh whch avods the possblty of producng llegal elements whle mantanng hgh effcency of the orgnal scheme. The correspondng optmzaton model s then solved by ntegratng chaos search and BFGS algorthm effcently. Qualty mprovement for tetrahedral mesh s fnally acheved by alternately applyng the presented smoothng scheme and re-trangulaton. An outlne of ths paper s as follows. Secton ntroduces the concept of ODT and the correspondng errorbased qualty metrc. Then n Secton 3, smoothng scheme through mnmzng the nterpolaton error among all trangulatons wth the same number of vertces n the local patch s dscussed and tested. A modfed smoothng scheme whch avods the possblty of producng llegal elements s proposed n Secton. Secton 5 gves the qualty mprovement cycle by alternately applyng the presented smoothng scheme and re-trangulaton. Several examples are gven n Secton 6 to llustrate performance of the proposed method.. ODT and Error-Based Qualty Metrc Chen et al. [1] presented the concept of ODT and proved the followng theorem: For a gven fnte pont set V n R n, let Ω be a convex hull of V, T a trangulaton of Ω, and f I,T the pecewse lnear fnte element nterpolaton of a gven contnuous functon f defned on Ω. Defne,, IT, L p QT f p f f p f x fit, x dx Then for the Delaunay trangulaton of V, DT, 1 p (1) Q DT, x, p mn Q T, x, p,1 p () TT V where T V denotes all possble trangulatons of Ω by usng the ponts n V. Ths theorem shows that for all the possble trangulatons, Delaunay trangulaton makes the lnear nterpola- ton error QT, x, p take the mnmum. Based on the lnear nterpolaton error defned n above theorem,,, QT f p, Chen et al. [] derved an error-based mesh qualty metrc,.e. the energy metrc named as E ODT n [3], E Q T N 1 ODT, x,1 x x d n x (3) 1 1 where n s dmenson (for tetrahedral mesh n = 3) and Ω denotes the frst rng of vertex x. Chen et al. [] presented that E ODT acheves the mnmum f all edge lengths of the trangulaton are equal; If reducng the value of E ODT, the dfferences between edge length and volume of mesh can also be reduced. Therefore, reducng the value of E ODT by topologcal or geometrcal optmzaton approach can lead to mprovement of the mesh qualty. 3. Mesh Smoothng Based on Interpolaton Error 3.1. Smoothng Scheme When performng mesh qualty mprovement, the errorbased mesh qualty metrc defned n the local patch Ω for free vertex x s used. The local qualty metrc for tetrahedral patch Ω, can be defned accordng to [] and derved as 1 E T x x () j k Tj Tj x where T j s the volume of tetrahedron T j wth respect to vertex x, Ω denotes the volume of Ω. If reducng the value of ths local qualty metrc, the qualty of whole mesh wll be mproved. Then the smoothng scheme can be establshed by mnmzng the local qualty metrc n Equaton () by the manner of pont by pont. Accordng to [,3], ths optmzaton-based smoothng model could be solved explctly, so that the computatonal cost s as low as that of Laplacan smoothng. In [3], the optmal poston of x s derved and gven by x x T x x * 1 x j k (5) Tj Tj x where x T j s the gradent of the volume of the tetrahedron T j wth respect to x. Formula (5) can be used to drectly update the locaton of free vertex x n tetrahedral mesh through the postons of ts neghbors and thus mproves the mesh qualty. Combnng ths smoothng algorthm and the concept of ODT, Allez et al. [3] proposed a Delaunay-based varatonal approach for tetrahedral meshng by mnmzng a smple mesh-dependent energy through global updates of both vertex postons and connectvty, whch s very effectve to mprove the qualty of poorly tetrahedrons. However, through numercal nvestgaton and analyss we fnd Formula (5) for updatng poston of x also suffers the defcency of producng llegal elements. 3. Algorthm Analyss and Testng Formula (5) can be derved by makng the dervatve of E zero. However, the feasble doman of x * should be restrcted n a certan regon nsde Ω. The poston of x

3 S. L. SUN ET AL. 193 that makes the dervatve of E zero n ths regon not always exsts. Under some crcumstances, especally for poorly mesh, some vertexes after updatng by Formula (5) wll locate outsde of Ω. Some numercal tests also demonstrate Formula (5) has possblty to lead to llegal elements. For example, for the Delaunay trangulaton n a cubc regon shown n Fgure 1, perform smoothng approach for the nteror nodes by Formula (5). As we see, some new postons of vertexes locate far outsde the cubc regon. (a) (b) Fgure 1. Mesh n cubc regon before and after smoothng. (a) Intal mesh; (b) New mesh after smoothng.. Modfed Mesh Smoothng Scheme.1. Modfed Smoothng Scheme and Correspondng Optmzaton Model It s unacceptable for producng llegal elements n smoothng process. There are some strateges to handle ths problem. The smplest strategy s to dscard the new poston of x f llegal element s detected. We test ths strategy wth the mesh n Fgure 1, and no llegal element s found to be produced. However, for such a modfcaton, the effect of qualty mprovement s not satsfactory despte of ts hgh effcency. The better way may be to drectly solve the optmzaton-based model by mnmzng the local qualty metrc n Equaton () wth constrant of feasble doman, whch needs to calculate feasble doman F frst, and then solve a constraned optmzaton model: 1 mn E x T x x x F j k T j Tj x To skp the extra calculaton of the feasble doman, we prefer to realze the equvalent constrant by ntroducng the condton of postve volume,.e. T j > 0. Thus we get a modfed smoothng scheme and correspondng optmzaton model as follows: Step 1) Calculate the new poston of x * by Formula (5); Step ) Check f x * causes llegal element. A contaner of Ω can be frst constructed to reduce the checkng cost. If x * locates outsde of ths contaner, there must exst llegal element; otherwse, further volume calculaton for tetrahedral elements usng the new poston of x * s needed. Step 3) If no llegal element s found to be produced, accept the new poston; otherwse, solve the optmzaton problem mn E wth volume constrants Tj 0 to obtan the new poston of x *. In order to obtan better numercal stablty, the local qualty metrc n Equaton () can be expressed as 1 E x T x x (6) j k (7) Tj Tj x by coordnate translaton. Then the constraned optmzaton model n Step 3) can be defned by 1 mn E x T x x Tj Tj x s. t. T 0 j j k (8)

4 19 S. L. SUN ET AL. In order to solve ths knd of constraned optmzaton model effcently, we can convert t to an unconstraned one wth dfferentable objectve functon accordng to []. Frst, the volume constrant T j > 0 can be converted to the followng maxmum constrant: 1 max 0 Tj x x (9) Tj x Then construct the L penalty functon 1 x, E x max 0, T x x x Tj x j k (10) and replace the non-dfferentable term n Equaton (10) by ts aggregate functon. Thus the dfferentable objectve functon s obtaned as: x, E x q ln 1 exp T x x x q j Tj x j k (11) When the penalty factor α s greater than some threshold value α 0 and the parameter q approaches to nfnty, the soluton of the unconstraned model for mnmzng the dfferentable objectve functon n Equaton (11) approaches to the soluton of the orgnal constraned model (8)... Soluton Algorthm For consderaton of effcency and precson, the scheme that ntegrates chaos search and BFGS s employed to solve the optmzaton model. Chaos search has good global search ablty, but ts convergence rate s low. BFGS method s one of the most representatve algorthms n solvng unconstraned optmzaton problem wth good numercal stablty. The combnaton of chaos search and BFGS can make full use of both global search ablty of chaos search and fast convergence rate near the optmal soluton of BFGS. Based on the above consderaton, the soluton of optmzaton problem n Step 3) can be dvded nto two steps: frst obtan the approxmated soluton by chaos search algorthm as the ntal nput values; then do optmzaton by BFGS algorthm. The procedure of chaos search algorthm s smply descrbed as follows: frst map the desgn varable x to chaotc varable, then to the objectve functon E n Equaton (8) drectly search the best soluton that satsfes the constrants. The volume of each tetrahedron s calculated before evaluatng the objectve functon; therefore, f the current poston does not satsfy the con- strants, drectly go to the next searchng. Then usng the approxmated soluton by chaos search algorthm as the ntal nput values, do optmzaton to the dfferentable objectve functon n Equaton (11) by the BFGS algorthm for optmal soluton. Due to stochastc property of chaos search algorthm, the approxmated solutons may be dfferent. In practce calculaton, four or fve approxmated solutons obtaned by chaos search are taken as the ntal nputs for BFGS algorthm. Then the best soluton by BFGS algorthm s chosen as the fnal optmal soluton. 5. Mesh Qualty Improvement To mprove mesh qualty, smoothng or topologcal optmzaton alone may not acheve deal results. In general, alternately applyng smoothng and topologcal optmzaton technque can mprove mesh qualty sgnfcantly [5]. Smoothng process changes the locaton and dstrbuton of nodes, and may provde further space of mprovng mesh qualty for topologcal process. Therefore, ths paper combnes these two approaches, alternately applyng smoothng and topologcal optmzaton to acheve better results. The most frequently used operatons of topologcal optmzaton are so-called basc or elementary flps. Recently, Lu et al. [6] proposed a new topologcal optmzaton operaton named small polyhedron reconnecton (SPR), to search the optmal topologcal confguraton n an enlarged regon whch usually ncludes 30 ~ 0 tetrahedrons. Delaunay trangulaton can also be proved to be a topologcal optmzaton tool to mprove the qualty of mesh. Ths paper adopts re-trangulaton as topologcal approach, makes a new Delaunay trangulaton after nodes updatng by the proposed smoothng scheme. Such an approach can reduce the value of qualty metrc based on nterpolaton error further [3] (Among all the trangulatons, Delaunay trangulaton makes the value take mnmum []). Meanwhle, reconnecton of nodes also provdes optmzaton space for next smoothng step. 6. Examples and Conclusons The proposed approach s tested by a number of examples to llustrate ts effectveness. Qualty mprovement s performed by applyng the presented smoothng scheme and re-trangulaton for topologcal optmzaton alternatvely. Only nteror nodes are repostoned durng smoothng, whle the nodes on boundares keep fxed. Three testng meshes are shown n Fgure. The frst testng mesh conssts of 10,96 nodes and 58,615 tetrahedrons ntally. The second testng mesh conssts of 19,19 nodes and 76,188 tetrahedrons. The thrd mesh ncludes 653 nodes and 5,177 tetrahedrons ntally. Table 1 shows the statstcs of ntal qualty and qualty after 10

5 S. L. SUN ET AL. 195 (a) (b) (c) Fgure. Intal meshes for the three testng examples. (a) The frst example; (b) The second example; (c) The thrd example. Table 1. Statstcs of ntal mesh qualty and the qualty after optmzaton. Intal mesh After 10 optmzaton cycles mn Average Number of elements wth Number of elements wth mn Average qualty below 0.01 qualty below 0.01 Example 1 5.7E Example.9E Example optmzaton cycles. The radus rato ρ [5] s adopted as the qualty measurement for tetrahedral element, whch takes the value of 1 for regular tetrahedron and 0 for degenerated element. From the optmzaton results, t can be seen that the qualty optmzaton approach presented s able to mprove sgnfcantly the average qualty of the mesh, even the boundary nodes are fxed. More mportant, the number of poorly elements decreases remarkably and the worst mesh qualty also be mproved. The presented smoothng scheme s effcent and robust. No llegal element s produced. Testng results show that the proposed smoothng approach s sutable for combnng wth the topologcal optmzaton procedure, and effectve to elmnate poor elements as well as mprove mesh qualty. 7. Acknowledgements The authors would lke to thank the supports of the Natonal Natural Scence Foundaton of Chna (111700, ) and Bejng Muncpal Natural Scence Foundaton (11000). Journal of Computatonal Mathematcs, Vol., No., 00, pp [] L. Chen, Mesh Smoothng Schemes Based on Optmal Delaunay Trangulatons, Proceedngs of 13th Internatonal Meshng Roundtable, Wllamsburg 19- September 00, pp [3] P. Allez, D. Cohen-Stener, M. Yvnec and M. Desbrun, Varatonal Tetrahedral Meshng, ACM Transactons on Graphcs, Vol., No. 3, 005, pp [] X. S. L, An Effcent Approach to a Class of Non- Smooth Optmzaton Problems, Scence n Chna Seres A, Vol., No., 199, pp [5] S. L. Sun and J. F. Lu, An Effcent Optmzaton Procedure for Tetrahedral Meshes by Chaos Search Algorthm, Journal of Computer Scence and Technology, Vol. 18, No. 6, 003, pp [6] J. F. Lu, S. L. Sun and D. C.Wang, Optmal Tetrahedralzaton for Small Polyhedron: A New Local Transformaton Strategy for 3-d Mesh Generaton and Mesh Improvement, CMES: Computer Modelng n Engneerng & Scences, Vol. 1, No. 1, 006, pp REFERENCES [1] L. Chen and J. Xu, Optmal Delaunay Trangulatons,

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