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1、Delaunay-basedmeshingAlgorithmicissuesAnintroductiontomeshgenerationPartII:Delaunay-basedmeshgenerationtechniquesJean-FrançoisRemacleDepartmentofCivilEngineering,UniversitécatholiquedeLouvain,BelgiumJean-FrançoisRemacleMeshGenerationDelaunay-basedmeshingAlgorithmicissu
2、esMediatorLetS1andS2betwopointsofR2.WenoteS1S2thelinegoingfromS1toSS2S.ThemediatorM(S1,S2)isthelocusofallthepointswhichareR2SSS1212equidistanttoS1andS2.S2M(S1,S2)S1S2M(S,S)={P∈R2,d(P,S)=d(P,S)}12212M(S1,S2)={P∈R
3、d(P,S1)=d(P,S2)},d(·,·)whered(.,.)istheeuclidiandistanceb
4、etweentwopointsofR2.Geometrically,itistheorthogonalbissectorofthesegmentoflinebetweenthetwopoints.Pd(P,S1)S1d(P,S2)S2MJean-FrançoisRemacleMeshGenerationS1S1S2"Delaunay-basedmeshingAlgorithmicissuesPartitionoftheplaneThemediatorseparatestheplaneintotworegions.Thefirstreg
5、ioncontainsallthepointsthatareclosertoS1,thesecondonecontainstheonesthatareclosertoS2.AnypointofR2Marethereforeassociatedtooneofthosetwopoints.Notethatthisrelationdependsonthewaydistancesarecomputed,thisrelationcanbedefinedinRd.Jean-FrançoisRemacleMeshGenerationDelaunay
6、-basedmeshingAlgorithmicissuesTheVoronoïdiagramLetusconsiderasetofS={Si}i=1,...,NNpointsS=N{S1,...,SN}.S2S11S10S6S1S3SS87S5S12S9S4Jean-FrançoisRemacleMeshGenerationS2S11S10S6S1S3SS87S5S12S9S4!Delaunay-basedmeshingAlgorithmicissuesS9TheVoronoïdiagramS4Weassumethattheree
7、xistsnotripletofpointsinSthatarecolinearS2S11S10S6S1S3SS87S5S12S9S4!Jean-FrançoisRemacleMeshGenerationDelaunay-basedmeshingAlgorithmicissuesTheVoronoïdiagramS={Si}i=1,...,NNWeassumethatthereexistsnoquadrupletofpointsinSthatarecocircularS2S2S11S11S10S10S6S6SS3S1S31SS8S7
8、S87S5S5S12S12S9S9S4S4C(Si)SiJean-FrançoisRemacleSMeshGenerationiS2C(S)={P∈R2
9、d(P,S)≤d(P,S),∀j%=i}.S11iijS10S6S1S3SS87S5S12S9S4!!Delaunay-basedmeshingAlgorithmicissuesTheVoronoïdiagramC(Si)SiSiTheVoronoïcellC(Si)associatedtopointSiisthelocusofpointsofR2S2thatarecloserto
10、C(Si)S=ithananyotherpoint{P∈R2
11、d(P,SSi)j,≤j=d1,...,(P,SNj,),i∀6=jj.%=i}.S11S10S6S1S3SS87S5S12S9S4Jean-FrançoisRemacle
