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Analysis and Finite Element Simulation of MHD Flows, with an Application to Seawater Drag R

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ANALYSISANDFINITEELEMENTSIMULATIONOFMHDFLOWS,WITHANAPPLICATIONTOSEAWATERDRAGREDUCTION1

A.J.MeirDepartmentofMathematicsAuburnUniversity,AL36849ajm@math.auburn.eduP.G.SchmidtDepartmentofMathematicsAuburnUniversity,AL36849pgs@math.auburn.edu

Abstract.Muchresearchefforthasrecentlybeendevotedtotheelectromagneticcontrolofsaltwaterflows,exploitingthemacroscopicinteractionofsaltwaterwithelectriccurrentsandmagneticfields.ThisinteractionisgovernedbytheequationsofviscousincompressibleMHD,essentially,theNavier-StokesequationscoupledtoMaxwell’sequations.AmajorproblemintheanalysisandnumericalsolutionoftheseequationsisthefactthatwhiletheNavier-Stokesequationsareposedinthefluiddomain,Maxwell’sequationsaregenerallyposedonallofspace.Consequently,electricandmagneticfieldsdonotsatisfystandardboundaryconditions,butjumporcontinuityrelationsonthesurfaceofthefluiddomain(andotherinterfaces).Frequentlytheresultingdifficultiesarecircumventedbyprescribingmoreorlessartificialboundaryconditions.InthispaperwepresentanovelformulationoftheMHDequationsthatavoidssomeinherentdifficultiesofmoretraditionalapproachesbyemployingtheelectriccurrentdensityratherthanthemagneticfieldastheprimaryelectromagneticvariable.Thisformulationleadstoinitial-boundaryvalueproblemsforasystemofintegro-differentialequationsinthefluiddomainandlendsitselfnaturallytotheuseoffinite-elementbaseddiscretizationtechniques.Asafirstapplicationwedescribeamixedfinite-elementmethodforthenumericalsolutionofaclassofstationaryMHDflowproblemsandreportonthecomputationalsimulationofasimpledragreductionexperiment.

I.INTRODUCTIONIthaslongbeenknownthattheflowofanelectricallyconductingfluid,suchasseawater,isaffectedbyLorentzforces,inducedbytheinteractionofelectriccurrentsandmagneticfieldsinthefluid.OnlyrecentlyhasitbeendemonstratedthatsuchLorentzforcescanbeusedtocontroltheflowandtoattainspecificengineeringdesigngoalssuchasflowstabilization,suppressionordelayofflowseparation,reductionofnear-wallturbulenceandskinfriction,dragreductionandthrustgeneration(see,forexample,[4,9,10]andthereferencescitedtherein).Thetheorythatdescribesthemacroscopicinteractionofanelectricallyconductingfluidwithelectriccurrentsandmagneticfieldsismagnetohydro-dynamics(orMHD).Assumingthefluidtobeviscous,incompressible,andfinitelyconducting,thegoverningequationsaretheNavier-Stokesandpre-Maxwellequations,coupledviatheLorentzforceandOhm’slaw.WhiletheNavier-Stokesequationsareposedinthefluiddomain,Maxwell’sequationsaregenerallyposedonallofspace,andtypicallybothinteriorandexteriorfieldsmustbedetermined.Onlyunderspecialcircumstances,mostnotablyinthepresenceofperfectlyconductingwalls,isitlegitimatetoconfineattentiontothebodyofconductingfluidandtoneglectitselectromagneticinteractionwiththeoutsideworld.Ingeneralthisinteractionisofcriticalimportance;infact,itconstituteswhatmostlydistinguishesMHDfromordinaryhydrodynamicsandisasourceofchallengingmathematicalandcomputationalproblems.Traditionally,theMHDequationsareformulatedasasystemofevo-lutionequationsforthefluidvelocityandthemagneticfield,alongwithanauxiliaryequationfortheelectricfieldoutsidethefluidregion.Thefactthatthemagneticfieldextendstoallofspaceandmayexhibitjumpdiscon-tinuitiesacrossinterfacesseparatingmediawithdifferentelectromagneticpropertiescausesanalyticalaswellascomputationaldifficulties,whicharefrequentlycircumventedbyprescribingmoreorlessartificialboundaryconditions.In[5–8]and[12]wedevelopedanovelapproachtoviscousincompressibleMHDthatavoidssomeintrinsicdifficultiesofthetraditionalmethodbyemployingfluidvelocityandelectriccurrentdensity(ratherthanfluidvelocityandmagneticfield)astheprimaryvariables.This“velocity-currentformulation”exploitsthefactthatwhilemagneticfieldsmayextendthroughoutspace,theunknowncurrentsinducingthosefieldsaretypicallycarriedbyconductorsoffiniteextent.Ifweconsider,forexample,asinglebodyofconductingfluidandassumeallexternalfieldsourcestobeknown,theonlyunknowncurrentflowsinthefluidregionitself.Inthiscase,thevelocity-currentformulationallowsustoperformallcomputationsonthefluiddomainwhilestillaccountingexactlyfortheeffectsoftheuniversalelectromagneticfield.Ingeneral,thevelocity-currentformulationleadstoasystemofevolutionequationsforthefluidvelocityandtheunknowncurrentdensityinthefluidsandadjacentsolidconductors,alongwithanauxiliarylineardiv-curlsystem,whichcanusuallybesolvedanalyticallyintermsofsingularintegrals.Thevelocity-currentformulationlendsitselfnaturallytotheuseoffinite-elementbaseddiscretizationtechniquesandprovidesatheoreticalframeworkforthedevelopmentofefficientcomputationaltoolsforthesimulationofawidevarietyofMHDflowproblems,includingtheelec-tromagneticcontrolofseawaterflow.Whilethemethodhasnotyetbeenappliedonanindustrialscale,ithasbeenshowntobeeffectiveintheanaly-sisandnumericalsolutionofaclassofstationaryMHDflowproblems(see[8]).Inthefollowingwedescribethegeneralapproach(SectionII),deriveamixedvariationalformulationforthestationarycase(SectionIII),discussafinite-elementmethodbasedonthisformulation(SectionIV),andre-portonthecomputationalsimulationofasimpledragreductionexperiment(SectionV).Despitetheacademicnatureofthissimulation,itillustratesthepotentialusefulnessofourapproachinsolvingavarietyofMHDflowcontrolanddesignproblems.II.THEVELOCITY-CURRENTFORMULATIONWeareconcernedwiththeflowofaviscous,incompressible,electri-callyconductingfluid,confinedtoaboundedregionofspaceandinteractingwithvariousbodyforces,electriccurrents,andelectromagneticfields.Un-dertheassumptionsoftheMHDapproximation,theflowisgovernedbytheNavier-Stokesequations,posedinthefluiddomain,andthepre-Maxwellequations,posedonallofspace;botharecoupledviatheLorentzforceandOhm’slaw.Asdiscussedintheintroduction,weseektoformulatetheproblemasasystemofevolutionequationsforthefluidvelocityuandtheelectriccurrentdensityJinthefluid;botharesolenoidalvectorfields,dependingontimeandposition.TheevolutionofthevelocityfieldisgovernedbytheNavier-Stokesequations,thatis,themomentumbalanceu∆u+(u)u+JB=Fext(1)alongwiththecontinuityequationu=0(2)reflectingtheincompressibilityofthefluid.Hereanddenotethe(con-stant)densityandviscosityofthefluid;Fextisagivenexternalbodyforce;andisthescalarpressure,anauxiliaryunknownthatplaystheroleofaLagrangemultiplierassociatedwiththedivergenceconstraint(2).Equa-tions(1)and(2)arecoupledtoMaxwell’sequationsthroughtheLorentzforce,JB,andOhm’slaw,J=(E+uB)(3)whereEandBdenotethe(unknown)electricandmagneticfields;isthe(constant)electricconductivityofthefluid.AdditionalcurrentsJextmay

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Anand本构方程在焊点可靠性研究中的应用

Anand本构方程在焊点可靠性研究中的应用

Anand本构方程在焊点可靠性研究中的应用王旭艳;徐仁春;刘刚【摘要】焊点可靠性直接决定了电子产品的使用寿命.因此,在微电子领域,对焊点可靠性提出了更高的要求.有限元模拟技术是研究焊点可靠性的重要手段.综合评述了一种统一了蠕变和塑性变形的非线性本构方程-Anand本构模型；概述了其发展演变过程及研究现状；介绍了该本构方程中9个参数的计算规则,并进一步分析了目前国内外对于本构方程参数的确定以及进一步的改进情况.在焊点可靠性研究方面,评论了该模型在无铅QFP、BGA焊点应力-应变分析及焊点疲劳寿命预测方面的应用,为焊点可靠性的研究提供了理论指导.同时,为了更好的研究无铅焊点的可靠性,对该模型的构建及修正提出了新的需求.%Service life of the electronics is decided by the reliability of solder joint .therefore,higher reliability of the solder joint is required in the field of micro-electronics.Finite Element Modeling technology is an important method to study the reliability of solder joint.Anand constitutive model which is a nonlinear equation unified with creep and plastic deformation was reviewed systematically in the text.Evolution,study status and the calculation of nine constants of this constitutive equation were introduced,and the determination of parameters and modification of the constitutive equation at home and abroad were analysed.In aspect of study of solder joint reliability .applying of the constitutive model to stress-strain analysis and fatigue life of QFP and BCA solder joint was discussed,which can provide the theory guide for practical applications.Meanwhile,new requirement for building andamending the constitutive model was proposed to study the reliability of solder joint better.【期刊名称】《电焊机》【年(卷),期】2012(042)012【总页数】4页(P66-69)【关键词】Anand本构模型;蠕变;微电子焊接;焊点可靠性【作者】王旭艳;徐仁春;刘刚【作者单位】南京电子技术研究所,江苏南京210039;南京汽车集团有限公司,江苏南京210061;南京电子技术研究所,江苏南京210039【正文语种】中文【中图分类】TG453+.90 前言在微电子工业中，随着手机、笔记本电脑等便携式电子产品的微型化需求，促进了电子元器件也向微型化、高密度方向转化，导致焊点的尺寸也越来越小。

基于静态加载和CEL的纯水射流冲击压力有限元分析

基于静态加载和CEL的纯水射流冲击压力有限元分析


运用 ＡＢＡＱＵＳ 有限元分析软件分别建立其纯水射流冲击强化模型， 获取能量历程变化、 冲击压力作用的动态过程规律。 将
冲击压力仿真结果与理论结果、 实验结果进行对比， 结果表明： ２ 种仿真模型的结果均与理论结果相吻合， 而 ＣＥＬ 法与实
验结果吻合程度更高。 所得结论可以为探究纯水射流冲击强化提供一定的参考。
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ｗａｓ ｕｓｅｄ ｔｏ ｅｓｔａｂｌｉｓｈ ｔｈｅ ｐｕｒｅ ｗａｔｅｒ ｊｅｔ ｐｅｅｎｉｎｇ ｍｏｄｅｌｓ ｒｅｓｐｅｃｔｉｖｅｌｙ ｔｏ ｏｂｔａｉｎ ｔｈｅ ｄｙｎａｍｉｃ ｐｒｏｃｅｓｓ ｌａｗ ｏｆ ｔｈｅ ｃｈａｎｇｅ ｏｆ ｅｎｅｒｇｙ ｈｉｓｔｏｒｙ
由于纯水射流冲击压力作用区域较小， 为保证计算结
果准确， 取 几 何 尺 寸 为 长 ３ ５ ｍｍ、 高 １ ２ ｍｍ、 宽
３ ５ ｍｍ 的立方实体。
基于流体力学方程的二维烟雾模拟

基于流体力学方程的二维烟雾模拟


+ %流体力学方程
烟的流态属于气固两相流" 即包括气体和固体颗
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* " \ W& ’ R 方程求解过程被简化为四个步骤& 烟雾速 度场从当前状态转换到下一个状态需要受到外力) 对流) 粘度和投射等四个因素影响" 各因素独立发生 $ 作用&速度场的求解过程如图 ! & 投射算子 \能保持流体速度 " K 44 K + $ cK + ,k 场的涡流特性& 图 #所示为本算法生成的烟雾速度场&用户定 义的烟源位于图片正下方" 烟源处有周期性向上吹 < M # 拂的风力" 图中所示为速度场流线表示" ! # ) ! 是 我们在模拟过程中任意截取的两幅图&从图中可以 看到" 系统生成的烟雾运动轨迹呈现比较自然的流
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c " ! # # 446 ! # # 式! # ) ! 分别称为不可压缩流体的动量方程 J为压力" > 和连续方程&其中 6为速度" 为流体运 C N + 动粘度系数" 为流体的密度" 为外力&* 为矢量 , 点积" * 为微分算子& #+ 从上式可以看到" ’ R 方程的数值求解算法十分 复杂" 通常不能实时地在计算机上产生图像&因此" 8 0领域中的烟雾模拟算法必须建立在简化的 ’ R
$ + D 4 0A ? " # ,4 8 J # % 0# AF ,) L %F 4 ,D + # / 4 ) "
波动方程的高阶间断有限元方法

波动方程的高阶间断有限元方法


Copyright © 2021 by author(s) and Hans Publishers Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). /licenses/by/4.0/
−1
xrk xlk
n
⋅
(−c) Ehk − (−c) E*
ϕ k ( x) dx.
其中：
Jk
为线性映射的
Jacobi， r,(i, j)
=
dϕ j dr
ri
。 r
和 ij 之间存在如下的计算关系：
( ) ∑ ∑ ∫ ( ) ( ) ∫ ( ) ∑ ( ) ∫ ( ) ( ) = = = r (i, j) nN p1= inr,(n, j) nN= p1 −11ϕi r ϕn r ddϕrj rn d= r = −11ϕi r nN p1 ddϕrj rn ϕn r= dr −11ϕi r dϕdjr r dr
Abstract
In recent years, discontinuous finite element method has been widely used in solving hyperbolic equations. Compared with traditional finite element method and difference method, discontinuous finite element method has many advantages. In this paper, the discontinuous finite element analysis is applied to the hyperbolic wave equation. By constructing the strong form of the discontinuous finite element and selecting the appropriate numerical flux, the smaller error and higher convergence order are obtained compared with the exact solution, and the theoretical convergence order of the discontinuous finite element is reached.