精品文档17中英文双语外文文献翻译成品高阶时间行进式局部水平集函数的重新初始化.docx

精品文档17中英文双语外文文献翻译成品高阶时间行进式局部水平集函数的重新初始化.docx

《精品文档17中英文双语外文文献翻译成品高阶时间行进式局部水平集函数的重新初始化.docx》由会员分享，可在线阅读，更多相关《精品文档17中英文双语外文文献翻译成品高阶时间行进式局部水平集函数的重新初始化.docx（24页珍藏版）》请在冰豆网上搜索。

精品文档17中英文双语外文文献翻译成品高阶时间行进式局部水平集函数的重新初始化

此文档是毕业设计外文翻译成品（含英文原文+中文翻译），无需调整复杂的格式！

下载之后直接可用，方便快捷！

本文价格不贵,也就几十块钱！

一辈子也就一次的事！

外文标题：

High-ordertime-marchingreinitializationforregionallevel-setfunctions

外文作者：

ShuchengPan,XiuxiuLyu,XiangyuY.Hu∗,NikolausA.Adams

文献出处：

《JournalofComputationalPhysics》,2017,354（如觉得年份太老，可改为近2年，毕竟很多毕业生都这样做）

英文4879单词，23887字符（字符就是印刷符），中文7482汉字。

High-ordertime-marchingreinitializationforregionallevel-setfunctions

ShuchengPan,XiuxiuLyu,XiangyuY.Hu∗,NikolausA.Adams

Abstract

Inthiswork,thetime-marchingreinitializationmethodisextendedtocomputetheunsigneddistancefunctioninmulti-regionsystemsinvolvingarbitrarynumberofregions.Highorderandinterfacepreservationareachievedbyapplyingasimplemappingthattransformstheregionallevel-setfunctiontothelevel-setfunctionandahigh-ordertwo-stepreinitializationmethodwhichisacombinationoftheclosestpointfindingprocedureandtheHJ-WENOscheme.Theconvergencefailureoftheclosestpointfindingprocedureinthreedimensionsisaddressedbyemployingaproposedmultiplejunctiontreatmentandadirectionaloptimizationalgorithm.Simpletestcasesshowthatourmethodexhibits4th-orderaccuracyforreinitializingtheregionallevel-setfunctionsandstrictlysatisfiestheinterface-preservingproperty.ThereinitializationresultsformorecomplexcaseswithrandomlygenerateddiagramsshowthecapabilityourmethodforarbitrarynumberofregionsN,withacomputationaleffortindependentofN.Theproposedmethodhasbeenappliedtodynamicinterfaceswithdifferenttypesofflows,andtheresultsdemonstratehighaccuracyandrobustness.

Keywords:

Reinitialization,Regionallevel-setfunction,High-orderaccuracy,Interfacepreserving

Introduction

Thelevel-setmethod[15]isawell-establishedinterface-capturingmethodandisbeingwidelyusedformultiphaseflowcomputation,imageprocessingandcomputervision[14].Areinitializationprocessisemployedtoreplacethedistortedlevel-setfunctionφ0:

Rd→Rduringadvectionbythesigneddistancefunctionφ:

Rd→RwhichisthesolutionoftheEikonalequation,

Successfulnumericalmethodsfordirectlysolvingthisstationaryboundaryvalueproblemincludethefastmarchingmethod[21]andthefastsweepingmethod[27].OnecanalsotransformEq.

（1）toatime-marchingproblem[25,24],whichisaHamilton–Jacobi（HJ）equationwithadiscontinuouscoefficientacrosstheinterface.

NumericalapproximationsofEq.

（2）mayexhibitoscillationsorinterfaceshifting[18],asthediscretizationofthederivativesacrosstheinterfacemayemployerroneouslevel-setinformationfromtheothersideoftheinterface.High-orderschemesspeciallydevelopedforHJequations[11]maysufferfromorderdegenerationandlargetruncationerrors[18,5].Modifications[24,18,13,5,9,3]havebeenproposedtocopewithspuriousdisplacementoftheinterfaceandsuccessfullyreducethemassloss[7].

Thelevel-setmethodhasbeenextendedtosimulatetheinterfaceevolutionofamulti-regionsysteminvolvingarbitrarynumberofregionsbyusingmultiplelevel-setfunctions[26,23]orasingleregionallevel-setfunction[28,20,17],i.e.,acombinationoftheunsigneddistancefunctionϕ（x）≥0andtheintegerregionindicatorχ（x）,wherex∈isapointinthecomputationaldomain.Formultiplelevel-setfunctions,reinitializationisappliedtoeachlevel-setfunction[23].

Reinitializationfortheregionallevel-setfunctioncanbeaccomplishedbythefastmarchingmethod[20].Althoughthetime-marchingmethodisconsideredtobemorecostly,itismoreflexibleandeasiertoparallelize[3].Whenappliedtotheregionallevel-setmethod,high-orderaccuracy,toourknowledge,hasnotbeendemonstratedintheliterature.Forexample,thetime-marchingreinitializationmethodusedinRef.[17]toregularizetheregionallevel-setfunctionsislimitedto1storderwithoutpreservingtheinterfacelocation.Itscomputationalcostdependsonthenumberofregionsduetotheregion-by-regionupdating.Thesedrawbacksmotivatethispaperwhichisdedicatedtodemonstratinghowtoachievehighorderandstrictinterfacepreservationforreinitializingregionallevel-setfunctions.Weemployasimplemapping,whichpreviouslyhasbeenusedtoconstructmultiplelocallevel-setsforsolvingregionallevel-setadvectioninRef.[17],totransformtheregionallevel-setfunctiontothelevel-setfunctionandapplyahigh-ordertwo-stepreinitializationmethodwhichisacombinationoftheclosestpointfindingprocedure[4]andtheHJ-WENOmethod[11].Comparedtothetime-marchingregionallevel-setreinitializationmethodinRef.[17],thepresentmethodcan（i）achievehigh-orderaccuracy,（ii）simplifytheupdatingoftheregionallevel-setfunctionbyimposingtheinterface-preservingproperty,and（iii）reducethecomputationalcosttobeapproximatelythesameastheoriginallevel-setreinitializationproblemfortwophases.

Reinitializationofaregionallevel-setfunction

LetX={r∈N|1≤r≤N}betheindexsetforallregions.Regionallevel-setreinitializationcorrespondstofindingthesolutionofoneachregiondomainχwhichisanopensubsetof,suchthatEq.

（1）isdecomposedintoNsub-problems.

Agoodreinitializationmethodforregionallevel-setfunctionsshouldpreservetheinterfaceandachievehigh-orderaccuracywithacomputationaleffortweaklydependingonthenumberoftheregionsN.Thedisplacementoftheinterfacemayleadtoasignchangeofϕ,correspondingtoaregion-indicatorchange,whichisdifficulttohandleforN>2.Thus,werequirethatthedevelopedmethoddoesnotchangethesignofϕsothattheindicatorχremainsinvariantduringthereinitialization.Althoughmanymethods,especiallythesubcellfixscheme[18,5,13],workwellforreinitializingoriginallevel-setorsigneddistancefunction[7],thesignofϕisnotensuedtobeinvariantwhenthosemethodsareappliedtosolvetheregionallevel-setreinitializationproblem.

Directimplementationofthetime-marchingreinitializationmethodonaregionallevel-setfunctiongives

whichisincorrectneartheinterfaceforreinitializationasϕgivesthewrongcharacteristicdirectionacrosstheinterface.Toaddressthisissue,weemployamappingCr:

R×N→Rdefinedas

Ittransformstheregionallevel-setfunctiontoalevel-setfunctionforeachregionχr.Eq.（4）isreformulatedaswhichisastandardHJequation.

Now,reinitializationcanbeperformedregionbyregion[17],orbysolvingNscalarevolutionequations[3].BothhavecomputationalcostscalinglinearlywithNduetoglobalmappingfromϕandχtoφ.Thesamecostcanbeachievedwithtime-marchingreinitializationmethods[25,13,5,9].

Toreducethecomputationalcosttobeapproximatelythesameasfortheoriginallevel-setreinitializationmethod,wecanlocallyapplythemappingoneverystencilofthespatialdiscretizationschemes.Consideringa2Dmulti-regionsystemdefinedbyϕi,jandχi,jonauniformCartesiangrid,withiandjbeingtheindicesofthecoordinatedirections,weapplyamappingforeachgridpointonthestencilofthediscretizationscheme,where−tkt,−tlt,and2t+1istherequiredstencilwidth.

NotethatEq.（7）isalocaloperationandthelocallevel-setfunctions,φr+k,jandφr,j+l,aretemporaryvariables,unliketheregionallevel-set（ϕ,χ）whichisdefinedglobally.Thus（ϕ,χ）atothergridpointsremainsinvariant.Nowtheinformationispropagatedfromtheregionboundary∂χi,jtotheinterioroftheregiondomainχi,j.ThemappingCrservesthesamepurposeasthesignumterminEq.

（2）.Thesemi-discreteformofEq.（6）iswhereHGistheGodunovnumericalHamiltonian[15,11],HG（a,b,c,d）=max（|a−|2,|b+|2）+max（|c−|2,|d+|2）,withf+=max（f,0）andf−=min（f,0）.

IfN=2,thisisidenticaltotraditionaltime-marchinglevel-setformulationwhichcanbesolvedbyexistinghigh-orderschemes[16,11].Howeverdirectlyapplyingthesehigh-orderschemesrequiresadditionaloperationswhenN>2.WhenthesolutionofEq.（8）changesitssignafteronetime-step,updatingofχisrequired,whichisnotencounteredwiththeoriginallevel-setproblem.Weemphasizethatunliketraditionallevel-setreinitializationwhereinterface-preservationservestoachievegoodmassconservationandhighaccuracyoftheinterfacelocation,inourproblemwerequiretheinterface-preservationpropertyasitimpliesthatthesignchangeofthesolutioninEq.（8）isavoidedandthusthecorrespondingχupdating.Notethatonlystrictinterface-preservationallowstoomitupdatingofχ,althoughhigh-ordermaybeachievedwithoutsatisfyingthiscondition[13,5].

High-ordertwo-stepreinitializationmethod

Toensureinterface-preservationandhigh-orderaccuracy,weperform2operations.First,aswiththeinitializationstepofthefastmarchingmethod[21],wetagthegridpoint（i,j）adjacenttotheinterfaceasAlive,othersasFar.Acell[i−1,i]×[j−1,j]isacut-cellifitcontainsanAlivegridpoint.Insideeverycut-cell,thelevel-setfunctionisapproximatedbyabicubicpolynomial[4]whosecoefficientsaredeterminedbyinterpolationusingall16grid-pointvaluesaroundthiscelltoachieve4th-orderaccuracy.Ifthecut-cell[i−1,i]×[j−1,j]issharedbymorethan2regions,say{χa,χb,χc},weneedtoreconstructmultiplepolynomialswherecrξηisdeterminedbyinterpolatingφr+i,j+j=Cr（ϕi+i,j+j）,withχr∈{χa,χb,χc}andi,j∈[−2,1].

AmodifiedNewtonmethod[4]isusedtofindtheclosestpointxthatsatisfiespr（x）=0andpr（x）·（x−x）=0simultaneous