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1、JournalofComputationalPhysics198(2004)450–468www.elsevier.com/locate/jcpAphasefieldapproachinthenumericalstudyoftheqelasticbendingenergyforvesiclemembranesQiangDu*,ChunLiu,XiaoqiangWangDepartmentofMathematics,PennsylvaniaStateUniversity,218McAllisterBuilding,Universi
2、tyPark,PA16802,USAReceived29July2003;receivedinrevisedform13January2004;accepted20January2004Availableonline26February2004AbstractInthispaper,wecomputetheequilibriumconfigurationsofavesiclemembraneunderelasticbendingenergy,withprescribedvolumeandsurfacearea.Avariatio
3、nalphasefieldmethodisdevelopedforsuchaproblem.Discretefinitedifferenceapproximationsandnumericalsimulationsarecarriedoutintheaxialsymmetricalcases.Differentenergeticbifurcationphenomenaarediscussed.Ó2004ElsevierInc.Allrightsreserved.Keywords:Phasefield;Elasticbendingener
4、gy;Bio-membrane;Finitedifferencemethods;Numericalsimulation;Axial-symmetry;Bifurcationdiagram1.IntroductionRecently,therehavebeenmanyexperimentalandanalyticstudiesontheconfigurationsanddeforma-tionsofelasticbio-membranes[7,17,32,36,38,39].Inthispaper,westudytheshapetr
5、ansformationofasinglecomponentvesicle,usingaphasefieldapproach.Whileourgoalistostudythedeformationandinteractionofsuchvesiclesundervariousexternalfields,suchastheflowfieldsorelectricfields,herewefirstfocusonthefundamentalsofthephasefieldformulationanditsnumericalapproximat
6、ionsintheabsenceoftheappliedfields.Theusualsinglecomponentvesiclesareformedbycertainamphiphilicmoleculesassembledinwatertobuildbilayers.Theyarethesimplestmodelsforthebiologicalcellsandmolecules.Theequilibriumshapeofsuchanmembraneisdeterminedbytheshapeenergy.Intheisot
7、ropiccase,itusuallyconsiststhefollowingbendingenergy[10,11]:qTheresearchofQ.DuandX.WangissupportedinpartbyNSF-DMS0196522andNSF-ITR0205232.TheresearchofC.LiuissupportedinpartbyAmericanChemicalSociety.*Correspondingauthor.Tel.:+814-8653674;fax:+814-8653735.E-mailaddre
8、sses:qdu@leibniz.math.psu.edu,qdu@math.psu.edu(Q.Du),liu@math.psu.edu(C.Liu),wang@math.psu.edu(X.Wang).0021-9991/$-seefrontmatterÓ2004Else
