A Static Geometric Medial Axis Domain Decomposition in 2D Euclidean Space

Transcription

1 A Sttic Geometric Meil Axis Domin Decomposition in D Euclien Spce LEONIDAS LINARDAKIS n NIKOS CHRISOCHOIDES College of Willim n Mry We present geometric omin ecomposition metho n its implementtion, which prouces goo omin ecompositions in terms of three bsic criteri: () The bounry of the subomins crete goo ngles, i.e., ngles no smller thn given tolernce Φ, where the vlue of Φ is etermine by the ppliction which will use the omin ecomposition. () The size of the seprtor shoul be reltively smll compre to the re of the subomins. (3) The mximum re of the subomins shoul be close to the verge subomin re. The omin ecomposition metho uses n pproximtion of Meil Axis s n uxiliry structure for constructing the bounry of the subomins (seprtors). The N-wy ecomposition is bse on the ivie n conquer lgorithmic prigm n on smoothing proceure tht elimintes the cretion of ny new rtifcts in the subomins. This pproch prouces well shpe uniform n gre omin ecompositions, which re suitble for prllel mesh genertion. Ctegories n Subject Descriptors: I.3.5 [Computer Grphics]: Computtionl Geometry n Object Moeling; J.6 [Computer-Aie Engineering]: Generl Terms: Algorithms Aitionl Key Wors n Phrses: omin ecomposition, prllel mesh genertion, Deluny tringultion. INTRODUCTION Although the Domin Decomposition (DD) problem hs been stuie for more thn yers in the context of prllel computing, there re mny spects of this problem which re unsolve. DD methos hve been use for solving numericlly prtil ifferentil equtions using prllel computing (cf. [Smith et l. 996]. Here we exmine the Geometric Domin Decomposition problem (GDD) in the context of prllel mesh genertion. We focus on the formultion, solution n implementtion of the GDD problem for continuous -imensionl (D) omin Ω into non-overlpping subomins Ω i, so tht the subomins Ω i crete no new rtifcts, such s smll ngles between the seprtors Ω i, n the seprtors n This work ws prtilly sponsore by the Ntionl Science Fountion uner Grnt No. 986, Funing support ws lso provie through the Virgini Institute of Mrine Science n the Southestern Universities Reserch Assocition uner the grnts ONR N n NOAA NANOS735. Any opinions, finings, n conclusions or recommentions expresse in this mteril re those of the uthors n o not necessrily reflect the views of the forementione institutes. Permission to mke igitl/hr copy of ll or prt of this mteril without fee for personl or clssroom use provie tht the copies re not me or istribute for profit or commercil vntge, the ACM copyright/server notice, the title of the publiction, n its te pper, n notice is given tht copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to reistribute to lists requires prior specific permission n/or fee. c 6 ACM 98-35/6/- $5. ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6, Pges 6.

2 Fig.. Left: Prt of the Chespeke by geometry ecompose uniformly by the MADD metho. The ngles crete by the seprtors re greter thn 7. Right: Detil of the Deluny mesh of the ecompose geometry. The ecompositions prouce by the MADD re suitble for stble prllel gre mesh genertion. the externl bounry Ω. These ecompositions re suitble for stble prllel gre mesh genertion proceures (see Fig. ), where the termintion of these proceures n the qulity of the resulting elements epen on the fetures of the subomins. Furthermore, the sme ecompositions cn be use for the next step, by the prllel FEM or FD solver. However, the geometric omin ecomposition we escribe oes not epen on how the mesh is use, or wht is the PDE solving metho. Prllel mesh genertion methos ecompose the originl meshing problem into smller subproblems tht cn be solve (i.e., meshe) in prllel. There re two pproches tht cn be employe in orer to ecompose the problem: mesh t ecomposition n geometric omin ecomposition techniques. Mesh t ecomposition techniques compute t-subsets of the mesh tht cn be processe in prllel [Chrisochoies n Nve 3; Kow n Wlkington 3; Chernikov n Chrisochoies ]. The ecomposition for these pproches is n esier problem thn the geometric omin ecomposition problem, but communiction n locl synchroniztion re unvoible uring the prllel mesh genertion in orer to mintin the conformity n qulity of the istribute mesh. Geometric omin ecomposition techniques prtition the omin into subomins; the subomins re crete by inserting internl bounries (seprtors) into the omin. Prllel mesh genertion proceures tht follow this pproch require low communiction [Chew et l. 997], or no communiction t ll [Gltier n George 996; Si et l. 999; Linrkis n Chrisochoies 6], n thus re very efficient. When communiction is require by the prllel mesh genertion proceure, this will be nlogous to the lengths of the seprtors. Hence, one of the ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

3 3 gols of the omin ecomposition is to prouce smll seprtors. On the other hn, the seprtors will be prt of the geometry, n consequently prt of the finl mesh, so the ecomposition hs to meet certin qulity criteri (like the size of the ngles tht it imposes), so tht the qulity of the mesh will not be istorte. Geometric omin ecomposition methos cn be chrcterize s topology bse or geometry bse. Typiclly topology bse techniques prtition mesh of the omin, or the ul grph of bckgroun mesh, giving ecomposition of the omin. This pproch is followe by the Metis librry [Krypis n Kumr 995b]. On the other hn, geometry bse techniques tke into ccount the geometric chrcteristics of the omin. For exmple, the Recursive Coorinte Bisection pproch [Berger n Bokhri 985] recursively bisects the omin long the xes, while the Inertil metho [Nour-Omi et l. 986] uses the inerti xis of the omin to prouce ecomposition. Finlly, librries like Chco [Henrickson n Leln 995] provie both topology n geometry-bse pproches.. THE GEOMETRIC DOMAIN DECOMPOSITION PROBLEM We exmine the GDD problem in the context of prllel mesh genertion. In the rest of this pper we efine s omin Ω the closure of n open connecte boune set in R. The bounry Ω is efine by plnr stright line grph (PSLG), which is forme by set of line segments, intersecting only t their en points. Formlly -wy omin ecomposition is efine s follows. We ssume the omin Ω is the closure of n open connecte boune set n the bounry Ω is PSLG tht forme set of liner segments which o not intersect. A complete seprtor H Ω is finite set of simple pths ( continuous - mp h : [, ] Ω), which we cll prtil seprtors, tht o not intersect n efine ecomposition Ω, Ω of Ω, such tht: Ω n Ω re connecte sets, with Ω Ω = Ω, n for every pth P Ω, which connects point of Ω to point of Ω, we hve P H. Gurntee qulity mesh genertion lgorithms [Chew 989; 993; Ruppert 995] prouce elements with goo spect rtio n goo ngles. These lgorithms require tht the initil bounry ngles re within certin goo bouns. For exmple, Ruppert s lgorithm [Ruppert 995] requires bounry ngles (the ngles forme by the bounry eges) no less thn 6, in orer to gurntee the termintion. When these lgorithms re use in prllel, omin ecomposition bse, mesh genertion proceures, the seprtors re trete s externl bounry of ech subomin. So, the omin ecomposition shoul crete seprtors tht meet the requirements of the mesh genertion lgorithm. Therefore the constructe seprtor shoul form ngles no less thn given boun Φ, which is etermine by the sequentil mesh genertion proceure tht will be use to mesh the iniviul subomins. Even in cses like [Tringle ], where mesh genertor cn hnle smll input ngles, these ngles re rtifcts n will be permnent, istorting the qulity of the finl mesh prouce by the prllel mesh genertor. The performnce of the prllel mesh genertion is ffecte by the require communiction n the work-lo blnce mong the processors. If there is communiction, this is usully proportionl to the size of the seprtor, therefore, This efinition oes not llow internl bounries. The lgorithm we present cn be extene to hnle internl bounries, if neee. ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

4 Fig.. Left: The Pipe geometry. The ngles prouce by grph-bse prtitioners, like Metis, epen on the bckgroun mesh, n cn be s smll s the smller ngle of the mesh. The ngles mrke with ots re smll (less thn 6 ). Right: Prt of the Chespeke by geometry. When the geometry is complicte, methos like the Recursive Coorinte Bisection n the Inertil Metho cn prouce rbitrry smll ngles between the seprtors n the omin bounry, n lso cn plce seprtors rbitrry close to the bounry. one of our objectives in the omin ecomposition step is to minimize the size of the seprtors. On the other hn, the lo blncing problem is best resse by over-ecomposing the omin [Chrisochoies 996]. Over-ecomposition llows both sttic n ynmic lo blncing methos to istribute eqully the worklo mong the processors more effectively [Brker et l. ; Linrkis n Chrisochoies 6]. These methos though will be less effective, if some of the subomins represent much lrger work-lo thn the verge. Therefore, we shoul keep the mximum re of the subomins close to the verge subomin re 3. In conclusion, geometric omin ecomposition is suitble for stble prllel mesh genertion, if it stisfies the following criteri. C. Crete goo ngles, i.e., ngles no smller thn given tolernce Φ < π/. The vlue of Φ is etermine by the sequentil, gurntee qulity, mesh genertion lgorithm (for Ruppert s lgorithm we use the vlue Φ = 6 ). C. The length of the seprtor shoul be reltively smll. C3. The mximum re of the subomins shoul be close to the verge subomin re. Previous DD pproches re very successful for tritionl prllel PDE solvers, but they were not evelope for prllel mesh genertion proceures, n thus o Smll work los o not crete lo-blncing problems, on the contrry, the resulting grnulrity cn be use to improve the lo blnce, especilly on heterogenous environments. 3 The re of the subomins oes not lwys reflect to work-lo of the mesh genertion proceure. However, for well shpe subomins, s the ones prouce by MADD, n Deluny mesh genertors, the work-lo is nlogous to the re of the subomin [Linrkis n Chrisochoies 6] ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

5 5 Fig. 3. Left: The Deluny tringultion of the Pipe omin. The circumcenters of the tringles pproximte the meil xis. Right: The circumcenters re the Voronoi noes. The seprtor is forme by selecting subset of the Voronoi noes n connecting them with the bounry. not ress the problem of the forme ngles. For exmple, grph bse prtitioning lgorithms, like Metis, give well-blnce ecompositions with smll seprtors, but the ngles forme by the seprtors epen on the bckgroun mesh, n they cn be s smll s the smllest ngle of the mesh (see Figs. Left, n 9). On the other hn, methos like the Recursive Coorinte Bisection n the Inertil Metho cn crete rbitrry smll ngles, n lso plce the seprtors rbitrry close to the bounry (see Fig. Right), so they re unsuitble for prllel mesh genertion proceures. The geometric omin ecomposition pproch we present resses ll of the three bove criteri, n is suitble for stble n efficient prllel mesh genertion proceures. 3. MEDIAL AXIS DOMAIN DECOMPOSITION METHOD The Meil Axis Domin Decomposition (MADD) metho ws first introuce in [Linrkis n Chrisochoies 6] in the context of the Deluny Decoupling metho n it is bse on n pproximtion of the meil xis (MA) of the omin. The MA ws introuce in [Blum 967], n hs been stuie n utilize extensively (cf. [Attli et l. ]. The pproximtion of the MA in the MADD metho is use s n uxiliry structure to etermine seprtors tht form goo ngles. In this pper we present n expne n improve version of the MADD metho which inclues gre N-wy omin ecomposition proceure, n smoothing proceure for improving the qulity of the seprtors. A circle C Ω is si to be mximl in omin Ω, if there is no other circle C Ω such tht C C. The closure of the locus of the circumcenters of ll mximl circles in Ω is clle the meil xis Ω n will be enote by MA(Ω). The intersection of the bounry of Ω n mximl circle C is not empty. The points C Ω, where mximl circle C intersects the bounry, re clle contct points of c, where c is the center of C. If b is contct point of c, then the ngles forme by the segment cb n the bounry re t lest π/ [Linrkis n Chrisochoies 6]. The meil xis of the omin cn be pproximte by Voronoi noes of is- ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

6 c 6 c externl bounry Deluny tringultion prtil seprtors rejecte seprtors Fig.. Left is prt of the Deluny tringultion n right re the prtil seprtors. Tringle 3 5 is junction tringle, while the other tringles re not. cretiztion of the omin [Brnt n Algzi 99]. The Voronoi noes re the circumcenters of the Deluny tringultion of the iscretize omin (see Fig. 3 right). As the Voronoi noes pproximte the MA, the segments tht connect them to the bounry ten to crete ngles close to π/. The pproximtion of MA(Ω) is chieve in two steps: () iscretiztion of the bounry, n () computtion of bounry conforming Deluny tringultion using the points from step (). The circumcenters of the Deluny tringles re the Voronoi noes of the bounry points. The seprtors will be forme by either connecting these circumcenters to two of the vertices of the Deluny tringles, giving two segments, or from eges of the Deluny tringles, giving one segment. In both cses these segments re chosen so tht they form goo ngles, with ech other n the externl bounry, n they re clle prtil seprtors. A complete seprtor will be forme by set of one or more prtil seprtors. Fig. 3 epicts the bounry conforming mesh of the cross section of rocket n the meil xis pproximtion (left), n -wy seprtor for the sme geometry (right). Our gol is to crete ecompositions tht form ngles no less thn tolernce Φ. The prtil seprtors we choose re of two types (see Fig. ): () non-bounry eges of the Deluny tringultion tht form ngles Φ with the bounry, n (b) segments tht connect tringle circumcenter with the tringle vertices. The first type of prtil seprtor is esy to ientify. We only hve to scn the non-bounry eges of the Deluny tringultion n select the ones tht crete ngles t lest equl to our tolernce boun Φ. In orer to ientify the secon type of prtil seprtor we efine specil type of tringles. Let D be Deluny tringultion of iscretiztion D of the bounry Ω. We cll tringle t D junction tringle if: () it inclues its circumcenter c, () t lest two of its eges re not in D, (3) t lest two of the segments efine by the circumcenter n the vertices of t form ngles Φ, both with the bounry n ech other. The secon type of prtil seprtors re inclue in junction tringles. In Fig., ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

7 7 tringle 3 5 stisfies ll the bove criteri n is junction tringle. The other tringles re not junction tringles. 3 n 5 6 o not inclue their circumcenter n violte property (); 3 5 hs two eges on the bounry, violting property (); 6 7 oes not inclue prtil seprtor tht hs cceptble ngles (both ngles t n 7 re less thn the tolernce Φ, where Φ = 6 ), so it violtes property (3). The prtil seprtors re either internl Deluny eges, like, 3 n 6 7, or re forme by connecting the circumcenter of junction tringle to its vertices. In our exmple c 3, c 5 n 3 c 5 re the three possible prtil seprtors insie the junction tringle 3. The prtil seprtors lwys connect two points of the bounry, since D is bounry conforming tringultion. The complete seprtor is forme by choosing subset of prtil seprtors tht will gurntee the ecomposition of the geometry into two connecte subomins. The existence n the qulity of complete seprtor epens on the number n qulity of the prtil seprtors, which in turn epens on the level of the iscretiztion of the bounry segments. It is ifficult problem to pre-etermine the level of the refinement tht woul give n optiml ecomposition. Incresing the bounry refinement results better pproximtion of the meil xis, n more n better in terms of the C-C3 criteri prtil seprtors. However, over-refinement cretes number of problems. First, it increses the time for ecomposing the geometry, since the time for creting the Deluny tringultion epens on the number of input points. Furthermore, it will tke more time to ientify the prtil seprtors n form complete seprtor. Secon, it coul result into rithmetic rouning errors when clculting geometric entities, like circumcenters n ngles. There re three prmeters tht effect the level of require refinement of the bounry: () the number of subomins we wnt to crete, () the chrcteristics of the initil geometry, n (3) the ngle lower boun Φ. In our implementtion we compute refining size bse on the verge length of the initil bounry eges n on the squre root of the number of subomins (see Section 6). The ngle lower boun Φ cn be s lrge s 8, epening on the geometry n the refining fctor; for vlues lrger thn this the lgorithm my not fin seprtor tht stisfies the ngle lower boun conition.. THE MADD ALGORITHM The MADD lgorithm uses s strting point the pproximtion of the meil xis by the Deluny tringultion D, s escribe in the previous section. Any lgorithm tht gives Deluny bounry conforming tringultion cn be use to crete it. For our implementtion we hve use Tringle [Shewchuk 996], which is consiere to be stte of rt Deluny mesher for plnr geometries. The MADD lgorithm uses the Deluny tringultion to ientify set of cnite prtil seprtors. Then it will form complete seprtor by set of prtil seprtors, tht will gurntee the ecomposition of the omin into two subomins. The selection of prtil seprtors is bse on minimizing the size of the seprtors, while mintining the blnce of the res. The MADD lgorithm mps the Deluny tringultion D into grph G D, ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

8 c externl bounry Deluny tringultion prtil seprtors grph eges noe contrction Fig. 5. An exmple of creting the MADD grph. Left is prt of the Deluny tringultion n the cretion of the corresponing initil grph G D. Center, the proceure of contrcting the grph by combining the vertices of G D. The vertices connecte by ouble lines re combine. Right is the finl grph G D tht correspons to this prt. which is moifie element ul grph. The informtion encpsulte in this grph inclues: () the topology of D, (b) the length of the prtil seprtors, n (c) the re of the subomins tht will be crete. This informtion will be use to : () gurntee tht the inserte prtil seprtors form complete seprtor, () minimize the length of seprtors, n (3) keep the subomin res blnce. After G D is constructe, the grph is contrcte, so tht only the prtil seprtors of D re represente s grph eges (see Section.). Then the contrcte grph is prtitione in wy tht minimizes the cut cost n gives blnce subomin weights. In our implementtion we hve use Metis [Krypis n Kumr 995b], which is consiere to be stte of rt grph prtitioner. Finlly the grph prtition is trnslte bck into insertions of prtil seprtors, which results -wy ecomposition (see Section.3). The mjor steps of the lgorithm re: () Crete moifie element ul grph G D from the Deluny tringultion D. () Contrct G D into the grph G D, so tht only the cnite prtil seprtors re represente s eges of G D. (3) Prtition the grph G D, optimizing the cut-cost to subgrph weight rtio. () Trnslte the cuts of the previous prtition into the corresponing prtil seprtors n insert them into the geometry.. Construction of the Grph G D In this step the junction tringles of the Deluny tringultion D re ivie into three tringles, n the finl tringultion is represente s weighte ul grph. Ech of the three tringles inclue into junction tringle re represente by three grph vertices. Non-junction tringles re represente by single grph vertex. Vertices tht represent jcent tringles re connecte by grph ege. The weight of ech vertex is set equl to the re of the corresponing tringle, while the weight of grph ege connecting two vertices is set equl to the length of the common tringle ege tht is shre by the two corresponing tringles. Fig. 5 (left) epicts the step for constructing the grph G D. Tringles 3, ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

9 9 5 6, 3 5 n 6 7 re not junctions, n ech is represente by one vertex,, 6, 5, n 7 respectively. Tringle 3 5 is junction tringle n is ivie in three tringles: c 3, c 5 n 3 c 5, where c is the circumcenter of 3 5. These tringles re represente by the vertices,, n 3 respectively. The weight of ech vertex is equl to the re of the corresponing tringle. For exmple, the vertex hs weight equl to the re c 3. Vertices tht represent jcent tringles re connecte by grph ege, with weight equl to the length of their common tringle ege. For exmple, the vertices n re connecte by grph ege with weight equl to the length 3, while the vertices for n 3 re connecte by grph ege with weight equl to c 3. The bove proceure is escribe by Algorithm. Algorithm.. for ll the tringles i j k in D o. if i j k is junction tringle then 3. let c be the circumcenter of i j k ;. crete three vertices corresponing to tringles i c j, i c k, j c k with weight equl to their res; 5. else 6. crete one vertex with weight equl to i j k ; 7. enif 8. enfor 9. for ll vertices G D o. fin the jcent tringles n connect the corresponing vertices by grph ege with weight equl to the length of their common tringle ege;. enfor. Grph Contrction In this step the grph G D prouce from the previous step is contrcte into new grph G D, so tht only the cceptble prtil seprtors re represente s eges in G D. In orer to contrct the grph G D we iterte through ll the grph eges n eliminte those tht correspon to not cceptble tringle eges. A tringle ege is not cceptble if t lest one of the ngles tht it cretes is less thn Φ. The grph ege tht correspons to non-cceptble tringle eges is elete, n the two grph vertices tht were connecte by the eliminte ege re combine into one vertex; the new vertex represents the totl re of the tringles represente by the contrcte vertices. Fig. 5 (center) illustrtes the proceure of contrcting the grph. The tringle ege 3 5 forms smll ngles with the bounry n is not cceptble. The corresponing grph ege 3 5 is eliminte, while the vertices 3 n 5 re combine into new vertex. The new vertex represents the polygon 3 c 5 n its weight is equl to the polygon re, which is the sum of the two previous res. The new vertex lso inherits ll the externl grph eges of the two previous vertices, which ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

10 c c externl bounry Deluny tringultion prtil seprtors grph eges noe contrction Fig. 6. An exmple of contrction of the vertices insie of junction tringle. Left is prt of the Deluny tringultion n the cretion of the corresponing initil grph G D. Center, the proceure of contrcting the grph, in this cse the two vertices of the junction tringle 3 5 re combine. Right is the finl grph G D n the corresponing cnite prtil seprtors. in this cse re the two eges 3 n 3. The sme proceure is followe for eliminting the eges 6 n 6 7. In Fig. 5 right the finl grph G D is epicte with the corresponing res n prtil seprtors. In Fig. 6 we hve slightly ifferent geometry, which epicts the elimintion of n internl ege of junction tringle. The tringle ege c 3 forms smll ngle with the bounry, so it is not cceptble n it is eliminte. The two vertices n 3 in the junction tringle 3 5, which re seprte by this ege, re combine into new vertex. The new vertex inherits two grph eges connecting it to the sme vertex. These two eges hve totl weight equl to the length of the prtil seprtor c 5. The bove proceure is escribe by Algorithm. Algorithm.. for ll eges i j G D o. if the corresponing tringle ege forms n ngle < Φ then 3. elete the ege i j ;. crete new vertex with weight equl to the sum of the weights of the vertices i, j ; 5. trnsfer ll the externl grph eges of i n j to the new vertex ; 6. enif 7. enfor.3 The Construction of the Seprtor The result of the previous step is grph G D, whose eges represent the prtil seprtors tht cn be use to ecompose the omin. The next step is to prtition the grph in two connecte subgrphs n trnslte this prtition into geometric omin ecomposition. ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

11 c externl bounry prtil seprtors grph eges Fig. 7. Left is epicte the grph G D with the corresponing res. Center The grph is prtitione by eleting two grph eges. Right The corresponing prtil seprtor is inserte to the geometry. The weights of the vertices of G D represent the size of the corresponing res, while the weights of the eges represent the length of the corresponing prtil seprtors. The objective of the grph prtitioner is to minimize the rtio of the cutcost to the subgrph weight. The grph prtitioning problem is chllenging n hs been the source of goo lgorithms n softwre [Kernighn n Lin 97; Brnr n Simon 99; Henrickson n Leln 995b; 995c; Krypis n Kumr 995; Wlshw et l. 997], The grph contrction step, escribe in the previous section, hs the itionl merit of reucing significntly the size of the grph, resulting smller prtitioning problem. In our implementtion we hve use Metis librry of grph prtitioning lgorithms [Krypis n Kumr 995b]. After prtitioning the grph G D into two connecte subgrphs, the finl step is to construct the seprtor of the geometry, by trnslting the grph ege cuts to insertions of prtil seprtors. The prtil seprtors, tht correspon to eges cut by the grph prtitioner, re inserte into the geometry. In Fig. 7 (left) the grph G D is epicte, the grph prtition cuts of the two eges 3 n (mile), n the corresponing prtil seprtor c 5 is inserte to the geometry (right). The construction of the seprtor is escribe in Algorithm 3. Algorithm 3.. for ll the eges i j G D o. if i n j belong to ifferent subgrphs then 3. insert the prtil seprtor, corresponing to i j, into the geometry;. enif 5. enfor If the grph G D hs t lest two vertices, then -wy prtition exists n it will give ecomposition of the omin into two subomins (for the proof see [Linrkis n Chrisochoies 6]). Provie tht the grph prtitioner gives smll cut cost n blnce subgrph weights, the length of the seprtor will ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

12 Fig. 8. N-wy prtitions, where N =,, 8, 6, by the MADD ivie n conquer metho. be reltively smll n the res of the subomins will be pproximtely equl. Moreover, since ll the prtil seprtors, by the construction of G D, form goo ngles, the constructe seprtor will lso form goo ngles. There re cses though where the grph prtition will result the insertion of two prtil seprtors tht meet in the sme bounry point. The ngle forme between these two seprtors might be less thn the boun Φ, giving non-cceptble ecomposition. We hve e routine tht checks for these cses, moifies n reprtitions the grph, so tht only ngles Φ re crete uring the insertion of seprtors. In generl these cses correspon to high cut costs, ue to the length of the two intersecting seprtors, n in our experiments they rrely occure. In summry, the constructe seprtor meets the ecomposition criteri C - C3 escribe in Section. 5. N-WAY DECOMPOSITION The proceure escribe in the previous section ecomposes the omin into two subomins, our gol though is to obtin much lrger number of subomins. N- wy ecompositions cn be obtine by pplying the MADD proceure recursively, in ivie n conquer wy (see Fig. 8). In the first step the omin is ecompose into two subomins. Next, the lrgest subomin is chosen n is ecompose gin into two subomins. The proceure is repete until we hve crete the require number of subomins. The N-wy ecomposition is escribe by Algorithm. Algorithm.. Re the efinition of the omin Ω;. Initilize n mintin list of the subomins; 3. while the current number of subomins is less thn N o. ecompose the lrgest subomin in two subomins using the MADD lgorithm; 5. upte the list of the subomins; 6. enwhile The recursive pproch hs both vntges n isvntges. The MADD lgorithm is pplie from scrtch for every subomin tht is ecompose, so the ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

13 3 b b b b b Fig. 9. The Pipe omin ecompose in 6 subomins using the MADD lgorithm. On the left no smoothing is use. Most of the seprtors on t meet t their en-points, n they crete smll segments on their common bounries. On the mile the smoothing proceure is use, giving conforming seprtors. Right, the points of the first type () n secon type (b) re epicte. whole proceure, from the cretion of the Deluny tringultion to the grph prtition n the insertion of seprtors, is repete, iscring ny previous informtion. Another isvntge rises from the fct tht ech subomin is ecompose inepenently from the neighboring subomin. This might cuse the insertion of seprtors tht crete unnecessry smll segments on their common bounry of two neighboring subomins (see Fig. 9, left). In orer to solve this problem we introuce smoothing proceure, which is escribe in Section 5.. The mjor vntge of the ivie n conquer pproch is tht it pts to the current geometry. The Deluny tringultion of the subomins re re-clculte, fter we insert the seprtors, incorporting the informtion of the so fr ecomposition. The new seprtors re forme tking into ccount the current shpe of the geometry. Fig. 8 emonstrtes this fct. In contrst, if we use n N-wy grph prtitioning pproch, this informtion woul not be vlible, resulting poor ecompositions. 5. Smoothing the Seprtors One of the the isvntges of inepenently ecomposing ech subomin is the possible cretion of smll fetures. The inepenent computtion of the seprtors might crete smll segments long their common bounry (see Fig. 9, left). The size of these segments epens on the level of the bounry refinement. As we increse the number of segments, we lso increse the probbility of creting these smll segments. On the other hn, the grph prtitioner hs informtion only bout the size of the seprtors, n not bout their qulity, i.e., the ngles tht they form. Although ll the permissible seprtors form ngles greter thn preefine lower boun Φ, we woul like to choose the ones tht re not only smll, but lso form the best possible ngles (tht is ner π/). In orer to el with these two issues we introuce smoothing proceure tht improves the qulity of the ecomposition. The smoothing proceure tkes plce in two steps. The first step tkes plce uring the construction of the grph G D. In this step we incorporte into the weight of the grph eges two types of itionl informtion: () the qulity of ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

14 the ngles tht the corresponing seprtors form, n (b) the conformity with existing seprtors (i.e. if the seprtor s en-points meet t the en-points of n existing seprtor). The weight of ech grph ege is multiplie by coefficient f, which reflects the qulity of the minimum ngle φ tht the corresponing seprtor forms. This coefficient is compute s f = φ Φ +, for φ π/, n for φ π/. The coefficient f tkes vlues from π/ Φ + π/ Φ +,, when φ π/, up to, when the minimum ngle is equl to the minimum cceptble boun φ = Φ. So, the weight of the grph ege is ecrese proportionlly to the qulity of the minimum ngle. We woul lso like to encourge the grph prtitioner to choose seprtors tht conform with existing seprtors, i.e., tht meet on the common bounry with the existing prtil seprtors of the jcent subomins. To this en we ientify two types of bounry points (see Fig. 9, right). Points of the first type re either initil points prt of omin bounry, or re en-points of n existing seprtor. In orer to encourge the grph prtitioner to choose conforming seprtors, we ecrese the weight of the grph eges when these correspon to seprtors efine from points of the first type. These re en-points of existing seprtors (or of the initil bounry), n new seprtors tht meet t these points re conforming with the existing seprtors. The secon type of points re the mile points of segments efine by the first type points. We lso reuce the weight of the grph eges corresponing to seprtors efine from secon type points. In this wy we increse the probbility tht seprtor will be chosen tht hs en-points either on existing en-points (first type points), or wy from them (secon type points). The previous step wrs conforming seprtors, n the ones tht form better ngles, but it oes not gurntee tht these will be chosen by the grph prtitioner. In orer to improve further the qulity of the seprtor we introuce secon smoothing step, n hoc heuristic, fter the grph prtitioning proceure. Inste of inserting the prtil seprtors chosen by the grph prtitioner, we exmine ll the possible seprtors tht re close to the initil ones, n insert the optiml, ccoring to n optimlity function. The neighboring seprtors re efine by the neighboring points to the en-points of the initil seprtor. The optimlity function computes the egree of qulity bse on : () the size of the seprtor, (b) the minimum ngle tht it forms, n (c) the type of its en-points. The computtion of this function is similr s in the previous smoothing step. The smoothing proceure, lmost lwys, gives conforming seprtors tht form goo ngles. This epens though on the initil prtition of the grph, the blnce of the ecomposition, n of course, the geometric chrcteristics of the omin. 5. N-wy Gre Decomposition The proceure tht we hve escribe so fr for N-wy ecompositions prouces uniform omin ecompositions, i.e. the res of the subomins re pproximtely equl. This pproch is well suite for uniform mesh genertion, but in mny cses we woul like to hve gre, loclly refine, mesh. Certin prts of the omin, where the moel inictes higher ctivity, require smller size of mesh elements n thus enser mesh. These prts coul be etermine in vnce, bse on the properties of the geometry n the moel, or s result of n error estimtion ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

15 5 Fig.. Gre MADD bse on bounry weights. Left, moel of the Chespeke by is ecompose in 5 subomins, with weights on ll the bounry points n interpoltion fctor set to zero. Right, etil of the ecomposition, the irregulr inner polygons represent islns n re prt of the initil omin. function from previous FEM proceure. The locl mesh refinement proceure will result isproportionl mesh sizes for the subomins tht inclue these criticl res. In orer to mintin blnce memory requirements, n consequently worklo, uring the mesh genertion proceure, we hve to follow gre pproch in creting the omin ecomposition. The res of the crete subomins shoul be proportionl to the expecte mesh size, n the subomins tht require higher refinement shoul be ecompose into smller subomins. The problem of etermining the element size, n thus the grtion of mesh, hs been stuie extensively in the mesh genertion n refinement literture (cf. [Borouchki et l. 997; Löhner 997; Owen n Sigl 997; Deister et l. ; Zhu et l. ]). Usully the size of the elements is compute s function of: () the geometry of the omin (curvture), (b) the istnce from sources of ctivity in the moel (like het sources), (c) grtion control boun, n () error estimtors, typiclly compute from previous solution over corse mesh. In most cses bckgroun mesh n n interpoltion proceure is employe to efine the esire element size in ech position of the omin. In this section we escribe proceure tht prouces gre omin ecompositions. using the MADD metho. There re two wys to efine the grtion of the subomins. The first is to efine the require re for ech subomin. The secon is to ssign reltive ensity weight for ech subomin, n use it s grtion criterion. While the first pproch is nturl extension of the existing pproches for efining the grtion of the mesh, it oes not llow the user to preefine the number of subomins she wnts to crete. The number of subomins epens not only on the expecte size of the mesh, which cn be estimte through n re criterion, but lso by the number of processors tht we wnt to utilize n the vilble memory. Using ensity weights llows us to prouce gre ecompositions n t the sme time to preefine the number of subomins tht will be crete. N-wy gre omin ecompositions cn be prouce in similr wy s the non-gre ones, by recursively pplying the MADD proceure. The only step tht ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

16 6 nees to be moifie is the wy we choose the subomin to be ecompose in line of Algorithm. In this step the subomin with the greter re to require re rtio, or with the greter ensity weight, is chosen to be ecompose, inste of the subomin with the lrger re. In the cse of using the re rtio, no subomin with re rtio greter thn user-efine boun will be in the finl ecomposition. In the cse of using ensity weight criterion the prts of the geometry tht hve been ssigne greter ensity weights will be ecompose more intensively, n the number of the crete subomins is preefine. In our implementtion the grtion of the ecomposition cn be controlle in the following three wys: () Using ensity weights on the bounry points. () Using ensity-weight or require-re vlues over n unstructure bckgroun mesh. (3) Using ensity-weight function or require-re function over structure bckgroun gri. Cse (). The use of ensity weights on the bounry points is the simplest cse, n cn be viewe sub-cse of the cse (). We escribe it seprtely becuse it is simple to efine, n in some cses (like crck propgtion) we nee better refinement ner the bounry. The weights ssigne to the bounry re efine in the PSLG file tht escribes the geometry. Ech point, in ition to its coorintes, is ssigne n integer ensity weight vlue. A vlue of zero mens tht the point will not contribute to the ensity. Ech subomin is ssigne ensity weight vlue, which is the sum of its bounry weights. An interpoltion fctor llows the user to efine the weights of the crete internl bounries; we use liner interpoltion proceure. An interpoltion fctor of zero will ssign zero weights to the interfces. Exmples of this pproch re epicte in Fig.. Cse (). In this cse we use ensity-weight or require-re bckgroun mesh. A set of points in the interior, or on the bounry, of the geometry is ssigne either with ensity weights, which inicte the require level of refinement t the neighborhoo of these points, or with require re vlues, which inicte the re of the subomin incluing this point. The points typiclly woul be vertices of previous mesh (see Fig, left). The ensity weight of ech subomin is compute s the sum of the weights of the points inclue in the subomin. An exmple of this pproch is epicte in Fig., which is moel use to stuy the incompressible turbulent flow pst circulr cyliner [Dong n Krnikis 5], n in Fig.. The size of the bckgroun mesh shoul be proportionl to the number subomins we wnt to crete. Creting lrge number of subomins using few bckgroun points will result poor qulity of the subomin grtion, with much lrger subomins jcent to smll ones. This will increse the subomin connectivity n the cost for the strt-up in the communiction of the FEM solver. On the other hn, too mny bckgroun points will unnecessrily slow the proceure, without improving the qulity of the grtion. Cse (3). In this cse we use ensity weight function, or require re function, to control the grtion of the ecomposition. These functions re evlute ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

17 7 Fig.. Gre MADD bse on weighte bckgroun mesh. Top is the weight bckgroun mesh vertices of the Cyliner omin, n bottom is the corresponing ecomposition in 8 subomins. Fig.. Gre MADD bse on weighte bckgroun mesh. Left is the weight bckgroun mesh vertices of the Pipe omin, n right is the corresponing ecomposition into 5 subomins. ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

18 8 Fig. 3. Left, the Key is ecompose in 5 subomins using liner weight function, proportionl to x coorinte. Right, the Pipe ecompose in 35 subomins using n re function proportionl to ρ, where ρ is the istnce from the center of the inner circle. over structure gir crete on the fly uring the ecomposition proceure. The ensity-weight function ssigns weight to ech point of the crete bckgroun mesh, n, s in cse (), the ensity weight of ech subomin is compute s the sum of these weights. An exmple of this pproch is epicte in Fig. 3 (left). The require-re function ssigns to ech point the mximum subomin re tht is expecte for the subomin tht inclues this point. The require-re for subomin is compute s the minimum of the require-re function vlues of ll the mesh points contine in the subomin. In ech step the subomin with the highest rtio of re over require re is chosen to be ecompose. The proceure is repete, until no rtio is greter thn user-efine boun (efult is ), or until mximum number of subomins is reche. An exmple of this pproch is epicte in Fig. 3 (right). 6. IMPLEMENTATION The progrmming lnguge for our implementtion is ANSI C++, n the whole librry is encpsulte into the m clss. The [Tringle ] librry ([Shewchuk 996]) ws use for the cretion of the Deluny tringultion uring the MADD proceure. Also, the [Metis ] librry ([Krypis n Kumr 995b]) ws use for the grph prtitioning step in the MADD proceure. The MADD metho requires grph prtition into two connecte subgrphs; we implemente routine tht restores the connectivity in the cses where Metis returns non-connecte components. All the librries where use without moifictions. The bsic methos re escribe in Tble I, etile escription is provie in the source progrm. The librries for the re n weight function re loe ynmiclly through the l ynmic linker. The file formt (.poly) of the omins is the sme s in [Tringle ], n escribes the omin in three sections: () set of bounry points in x y coorintes, (b) set of segments, in enpoint enpoint form, n (c) set of holes in x y coorintes. The file formt of the ecompositions is similr. It escribes the ecomposition in four sections: () set of points, () set of segments, (3) set of subomins efine by segments of section (), n () set of holes for ech subomin. Exmples of files re provie with the softwre. ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

19 9 repolyfile(chr* nme) Res poly file (the initil omin). ecompose(int subdomins) The min ecomposition routine. rebckgrounweights(chr* nme) Res weighte bckgroun mesh noes from file. rebckgrounares(chr* nme) Res re bckgroun mesh noes from file. writepolyfileall(chr* filenme) Writes ll the subomins to ifferent poly files. writesubominfile(chr* nme) Writes the ecomposition in points, segments, n subomins - segments. setweightfunction(chr *librrynme, Sets the weight function. chr *functionnme) setarefunction(chr *librrynme, Sets the re function. chr *functionnme) setphi(ouble phivlue) Sets minimum cceptble ngle forme uring the ecomposition. setdecarertio(ouble rtiovlue) Sets the min re rtio for ecomposing subomin. setuniformrefinelevel(ouble uniformvlue) Sets the uniform refining n ecomposition fctor. setaptiverefinelevel(ouble ptivevlue) Sets the gre refining n ecomposition fctor. setmximblncelevel(ouble imblncevlue) Sets the mximum cceptble imblnce uring the m prtition. setmxsmoothlevel(int smoothlevel) Defines the number of the smoothing itertions. setinterpoltionweightlevel(ouble weightvlue) Sets the interpoltion coefficient for the weights of new points. Tble I. The bsic methos of the m clss. A line commn user interfce is provie with the librry. This interfce progrm (mi) cn receive n execute number of simple commns; the bsic commns re re escribe in Tble II. An exmple of set of commns is: re pipe.poly set f=.35 ec write pipe.poly writesubomins pipe.t exit The user cn set number of optionl prmeters, through the set commn, tht controls vrious functionl spects, like the level of refinement, the level of smoothing, blncing, etc. The bsic prmeters re escribe in Tble III. The level of refinement is bse on user-efine uniform refinement fctor r, n the N, where N re the number of subomins. The squre root function of the number of subomins ws chosen in heuristic wy, bse on the fct tht the squre of the lengths of the seprtors is nlogous to the res of the subomins. The verge re of subomin is A/N, where A is the totl re n N the number of subomins. So, the seprtor lengths will be proportionl to /N, n consequently the level of refinement shoul be nlogous to N. The refinement level, n the ecomposition times, for the Pipe n the Chespeke by ten to reflect this squre root behvior. This is not the cse for the Key, which hs few initil segments, n requires more intense refinement in orer to get goo ecompositions. The refinement fctor r efines uniform refinement level, ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

20 re <filenme> Res poly file. ec [<noofsubomins>] Decomposes the omin. write <filenme> [ll] Writes the ecompose poly file. re_weights <filenme> Res weighte bckgroun mesh noes from file. re_res <filenme> Res re bckgroun mesh noes from file. writesubomins <filenme> Writes the ecomposition into file. setweightfunction <librrynme> <functionnme> Sets the weight function. setarefunction <librrynme> <functionnme> Sets the re function. set <prmeter>=<vlue> Sets the <vlue> to the <prmeter>. exit Exits. Tble II. The bsic commns for the mi interfce. for exmple, if bounry segment is to be ivie to four subsegments, fctor r = will cuse it to be ivie into 8 subsegments. In ll our experiments the vlues for r were between n. The grtion fctor controls the level of grtion, in ssocition with the uniform refinement fctor r. The totl ensity weight of the subomin is compute s r subomin re + subomin weight. In this wy the reltion between n b controls the intensity of the grtion. 7. EXPERIMENTAL RESULTS For our experiments we use three moel omins. The Pipe moel is n pproximtion of cross section of regenertive coole pipe geometry. It consists of 576 bounry segments n 9 holes. The Key is omin provie with Tringle [Shewchuk 996], n hs 5 bounry segments n hole. The Chespeke by (Cby) moel efine from 3,5 points n it hs 6 islns. We rn three sets of experiments. In the first set of experiments we prouce uniform ecompositions for the three test omins. In the secon set of experiments we prouce gre ecompositions, using the three pproches escribe in Section 5.: () we use weights on the bounries to prouce gre ecompositions, (b) we experimente using weight n re bckgroun meshes, n (c) we use weight n re functions over structure gris. The bove experiments were performe on Pentium IV 3GHz processor, n we use lower ngle boun of f =.3333 rs ( 6 ). A thir set of experiments ws performe in orer to sses the qulity n efficiency of the MADD n compre it to Metis, which is stte of the rt grph prtitioner. For these experiments we ecompose the Key geometry, using Dul Pentium 3.GHz processor. Our results show tht the time to ecompose omin is irectly relte to the size of the omin (mesure in number of segments), n the level of the refinement we pply on it (see Figs. -5). The problem size for ll the mjor routines (Deluny tringultion, grph cretion n prtition) is proportionl to the number of the input segments, n thus we shoul expect this behvior. The level of refinement is nlogous to N, where N re the number of subomins. The refinement level, n the ecomposition times, for the Pipe n the Chespeke by ten to reflect this squre root behvior. This is not the cse for the Key, ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.

21 f Defines the minimum ngle boun crete by the seprtors (in rs). r Defines the uniform refinement level. The expecte length of the segments, fter refining, is multiplie by /r. Equivlently, the number of the segments fter refining is multiplie by r. The grtion fctor for ensity weights. The ensity weight of the subomins is multiplie by, while the re is multiplie by the uniform refinement level r. The totl sum gives the ensity weight of the subomin. p The interpoltion fctor for ensity weights. The weights of the new points of the seprtors re compute by liner interpoltion of the weights of the seprtor en points, multiplie by p. A vlue p = elimintes the weights on the seprtors. i The level of cceptble imblnce uring the smoothing. The vlues shoul be between.6 n.9 (efult is.75). Tble III. The bsic prmeters use in the set commn. which hs few initil segments, n requires more intense refinement in orer to get goo ecompositions. For the first group of gre ecomposition experiments we use bounry weights on the three omins. The user cn control the grtion level, by setting grtion fctor, n the weight interpoltion, p, tht will be pplie on the interfces. The prmeters for the Pipe n the Key were r = 3, = 3, p =.5, while for the Cby they were r =, = 3, p =. One of the ifficulties for ecomposing geometry bse on bounry weights (n lso weight or re functions) is tht the bounry refinement will hve to be ptive on the locl weights. In our implementtion, segments tht hve greter weight will be refine more. These ifficulties cn be solve by computing on the fly, loclly n inepenently, the level of refinement require for ech subomin. The secon group of experiments ws performe on the Pipe omin using bckgroun mesh of, points. Both re n weight vlues over the bckgroun mesh were use, n they prouce similr ecompositions for the sme number of subomins (see Fig. ). The qulity of the grtion epens on the rtio of the number of mesh points to the number of subomins, s well s the grtion of the bckgroun mesh. Domin ecomposition into lrge number of subomins, while using smll number of bckgroun mesh points, will result poor grtion. We lso teste the Pipe n the Key omins using weight n re functions, evlute over structure gri. This gri is crete on the fly, when ech subomin is crete; it inclues totl of,68 points for the Pipe omin n 8,5 points for the Key. This high number of the points results in goo pproximtion of the ensity for ech subomin (the ecompositions re epicte in Fig. 3), while the cost to crete them is smll (see Fig. 7). Of course, efining the functions nlyticlly hs the vntge of voiing the interpoltion proceure, which cn hve significnt cost. The weight n re functions re efine by the user n re linke ynmiclly, uring the execution of the progrm. For the thir set of experiments we prtitione the Key geometry up to, subomins uniformly, n we compre the results obtine by MADD to those obtine by Metis. For the Metis ecompositions we crete bckgroun Deluny meshes of size pproximtely tringles per subomin, The Deluny mesh genertion proceure is the only one tht provies qulity gurntees, creting ACM Trnsctions on Mthemticl Softwre, Vol. V, No. N, October 6.