From the Finite Element Method toward the Isogeome

CzechTechnicalUniversityinPragueFacultyofCivilEngineeringDepartmentofMechanicsDanielRypl,BoˇrekPatz´akFromtheFiniteElementMethodtowardtheIsogeometricAnalysisinanObjectOrientedComputingEnvironmentPragueSIGA2011February2011FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akPresentationOutlineMotivationB-splinebasisT-splines=NURBS+PB-splinesPrinciplesofOOdesignOOFEMOOdesignofIGAmoduleNumericalexampleSummarySIGA20112/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akMotivationIsogeometricAnalysisrecentlyintroducedalternativetotheFEMemploysthesamefunctionsforthedescriptionofgeometryandfortheapproximationofthesolutiononthatgeometryeliminatescostlyFEmeshgenerationgeometricpreprocessingstillrequiredoutperformsclassicalFEMinvariousaspectsstillmanyopenissues(trimmedgeometry,boundaryconditions,integration,efficiencyissues,implementation,performance:::SIGA20113/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akMotivationIGAoriginallydevelopedforNURBSconvenientforfree-formsurfacemodellingexactrepresentationofquadricsurfacesstableandefficientalgorithmsavailablepresentinmostCADsystemsgapsandoverlapscannotbeavoidedtrimmedNURBSnothandledbyIGAgenerallyonlyC0continuityacrosspatchboundariestensorproductstructureofNURBSnotefficientforrepresentationoflocalfeaturesandforconnectionofadjacentsurfacesmostshapescannotberepresentedasasinglewatter-tightNURBSSIGA20114/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akMotivationT-splineBasedIsogeometricAnalysisgeneralizationofNURBStechnologyinheritsgeometricalflexibilityofNURBSallowsefficientlocalrefinementallowswatter-tightmergingofadjacentNURBST-splinesareforwardandbackwardcompatiblewithNURBStrimmedNURBScanberepresentedasT-splinenon-straightforwardrefinementaroundextraordinarypointsnon-trivialrepresentationofsolidspreservingexactlyboundarysurfacegeometrylimitedavailabilityincommercialCAD(Maya,Rhino,SolidWorks)SIGA20115/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akMotivationImplementationmanysimilarfeaturesbetweenFEMandIGAnoneedtostartimplementationfromscratchmostoftheFEcodescanbereusedobjectorienteddesignrecognizedasveryappropriateprovedtobeaviableconceptsignificantlyenhancingmodularity,extensibility,maintainability,androbustnessofthecodewithoutsacrificingitsperformancesupportsteamwork,allowsfurtherdevelopmentswithoutparticipationoforiginalauthorsSIGA20116/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akUnivariateB-splinebasisfunctionsNi;p(t)=ttiti+ptiNi;p1(t)+ti+p+1tti+p+1ti+1Ni+1;p1(t)forp0Ni;0(t)=8:1iftitti+10otherwisetttttti+2ii+1i+3i+41i,3Ni+4tNi+1,2i+3ttNi,2−t−tt−t−ti+4i+1ii1Ni−2,0Ni−1,0Ni,0Ni+1,0i+2,0Nttttttti−2ii+1i+2i+3i−11Ni,1Ni+1,1Ni−2,1Ni−1,1t1i−2,2NNi−1,2Ni,2tSIGA20117/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akBivariateB-splinebasisfunctionsNuvij;pq(u;v)=Nui;p(u)Nvj;q(v)=Nk(u;v)iui+4uvj+3jviui+4ujvvj+3vvj+1j+2i+1i+2i+3uuuN(u)i,3j,2uvN(v)N(,)uvi3j2,uv=)RationalbivariateB-splinebasisfunctionsRk(t)=Nk(u;v)wkPnm=1Nm(u;v)wmk=1;2;:::;nwk0SIGA20118/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akQuadraticNURBScurvet=f0;0;0;1;3;3;4;4;4gP1P2P3P4P5P6w1=1w2=1w3=4w4=1w5=2w6=10,0,013,34,4,401123456NRr(t)=6Xj=1Ri(t)PiRi(t)=Ni(t)wiP6j=1Nj(t)wjSIGA20119/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akNURBS–NonuniformRationalB-splinesaNURBSpatchisdefinedby2setofcontrolpoints(coordinatesandweights)topologicallyformingregulargrid2globaldegreesofB-splinebasisfunctionsforeachparametricdirectionofthepatch2globalknotvectorsforeachparametricdirectionofthepatch=)NURBSisfullystructuredv(2−quadratic)u(3−cubic)1.00.80.00.90.10.40.01.00.20.30.60.70.80.90.50.3SIGA201110/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akNURBS–NonuniformRationalB-splinesaNURBSpatchisdefinedby2setofcontrolpoints(coordinatesandweights)topologicallyformingregulargrid2globaldegreesofB-splinebasisfunctionsforeachparametricdirectionofthepatch2globalknotvectorsforeachparametricdirectionofthepatch=)NURBSisfullystructuredv(2−quadratic)u(3−cubic)1.00.80.00.90.10.40.01.00.20.30.60.70.80.90.50.3SIGA201110/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akPB-splines–Point-basedB-splinesaPB-splinepatchisdefinedby2setofcontrolpoints(coordinatesandweights)topologicallyirregular2localdegreesofB-splinebasisfunctionsforeachparametricdirectionofeachcontrolpoint2localknotvectorsforeachparametricdirectionofeachcontrolpoint=)PB-splineisfullyunstructuredv(2)u(3)v(3)u(1)0.00.10.20.40.60.21.00.60.00.20.30.50.81.00.90.60.4SIGA201111/26FromtheFEMtowardtheIGAinanOOComputingEnvironmentDanielRypl,BoˇrekPatz´akPB-splines–Point-basedB-splinesaPB-splinepatchisdefinedby2setofcontrolpoints(coordinatesandweights)topologicallyirregular2localdegreesofB-splinebasisfunctionsforeachparametricdirectionofeachcontrolpoint2localknotvectorsforeachparam