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ID:34527014
大小:768.58 KB
页数:18页
时间:2019-03-07
《on stabilized finite element methods in relaxed micromagnetism》由会员上传分享,免费在线阅读,更多相关内容在教育资源-天天文库。
1、ONSTABILIZEDFINITEELEMENTMETHODSINRELAXEDMICROMAGNETISMSTEFANA.FUNKENANDANDREASPROHLAbstract.Themagnetizationstateofaferromagneticbodyisgivenasthesolutionofanon-convexvariationalproblem.Arelaxationofthismodelbyconvexifyingtheenergydensityresolvesessentialm
2、acroscopicinformationthatappliedphysicistsandengineersareafter.Thenumericalanalysisoftherelaxedmodelfacesanonlinearlyconstrainedconvexbutdegeneratedenergyfunctionalinmixedformulationformagneticpotentialudandmagnetizationm.In[CPr],theconformingP1?(P0)-eleme
3、ntind=2;3spatialdimensionsisshowntoleadtoanill-poseddiscreteprobleminrelaxedmicromagnetismandsuboptimalconvergencebehavior,eveniftheoriginalproblemiswell-posed.Thisobservationmotivatedtointroduceanon-conformingmethodtotheproblemthatimpliesawell-poseddiscre
4、teproblemwithsolutionsconvergingatoptimalrate.Inthiswork,weintroduceanddiscusstwostabilizedconformingmethodsthatemployinter-elementjumpsofthemagnetizationinnormaldirectionandtotalmagnetizationjumps,respectively.Bothstabilizedschemesconvergeatoptimalrates,w
5、hereasonlythelatteroneleadstoadiscreteproblemwithuniquesolution.1.IntroductiondThemathematicaldescriptionofmagnetizationstatesm:!!Rinrigidferromagneticdbodiescoveringadomain!R,d=2;3reliesontheclassicalmodelofWeiss,LandauandLifshitz[B],wherethemagnetizatio
6、nminimizesthemagnetostaticenergyZZZZ122(1.1)E(m)=jrmjdx+(m)dx?fmdx+jrujdx2!!!subjecttothefollowingconsequenceofMaxwell'sequations,?1(1.2)?u+div(m)=0inH():!Here,isthecharacteristicfunctionoftheset!,i.e.,(x)=1ifx2!and(x)=0if!!!not.Further,wechooseabou
7、ndedsetc!forpracticalsimplicationsinthesubsequentdanalysisinsteadofthewholeRtoconsidernon-localenergycontributions,seealso[CPr]onthismatter.Foradierentapproachusingthecouplingofpenalizedniteelementmethodsandboundaryelementmethodswereferto[CF].Themagneti
8、cenergyisassembledfromdierentcontributionsattributingtodierentpropertiesofminimizers:therstcontribution(surfaceenergy")isscaledby>0andfavorsnon-oscillatorystructures.Thenon-convexanisotropicenergydensity0
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