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EFFICIENTIMPLEMENTATIONOFADAPTIVEP1-FEMINMATLAB

S.FUNKEN,D.PRAETORIUS,ANDP.WISSGOTT

Abstract.WeprovideaMatlabpackagep1afemforanadaptiveP1-finiteelementmethod(AFEM).Thisincludesfunctionsfortheassemblyofthedata,differenterrorestimators,andanindicator-basedadaptivemesh-refiningalgorithm.Throughout,thefocusisonanefficientrealizationbyuseofMatlabbuilt-infunctionsandvectorization.Numericalexperimentsunderlinetheefficiencyofthecodewhichisobservedtobeofalmostlinearcomplexitywithrespecttotheruntime.AlthoughthescopeofthispaperisonAFEM,thegeneralideascanbeunderstoodasaguidelineforwritingefficientMatlabcode.

1.Introduction

Inrecentyears,Matlabhasbecomeadefactostandardforthedevelopmentandprototyping

ofvariouskindsofalgorithmsfornumericalsimulations.Inparticular,ithasproventobean

excellenttoolforacademiceducation,e.g.,inthefieldofpartialdifferentialequations,cf.[20,21].

In[1],aMatlabcodefortheP1-GalerkinFEMisproposedwhichwasdesignedforshortness

andclarity.Whereasthegivencodeseemstobeoflinearcomplexitywithrespecttothenumber

ofelements,themeasurementofthecomputationaltimeprovesquadraticdependenceinstead.

SincethisismainlyduetotheinternaldatastructureofMatlab,weshowhowtomodifythe

existingMatlabcodesothatthetheoreticallypredictedcomplexitycanevenbemeasuredin

computations.

Moreoverandinadditionto[1],weprovideacompleteandeasy-to-modifypackagecalled

p1afemforadaptiveP1-FEMcomputations,includingthreedifferentaposteriorierrorestima-

torsaswellasanadaptivemesh-refinementbasedonared-green-bluestrategy(RGB)ornewest

vertexbisection(NVB).Forthelatter,weadditionallyprovideanefficientimplementationof

thecoarseningstrategyfromChenandZhang[10,15].Allpartsofp1afemareimplemented

inaway,weexpecttobeoptimalinMatlabasacompromisebetweenclarity,shortness,and

useofMatlabbuilt-infunctions.Inparticular,weusefullvectorizationinthesensethat

for-loopsareeliminatedbyuseofMatlabvectoroperations.

ThecompleteMatlabcodeofp1afemcanbedownloadfromtheweb[18],andthetechnical

report[17]providesadetaileddocumentationoftheunderlyingideas.

Theremainingcontentofthispaperisorganizedasfollows:Section2introducesthemodel

problemandtheGalerkinscheme.InSection3,wefirstrecallthedatastructuresof[1]as

wellastheirMatlabimplementation.Wediscussthereasonswhythiscodeleadstoquadratic

complexityinpractice.Evensimplemodificationsyieldanimprovedcodewhichbehavesalmost

linearly.Weshowhowtheoccurringfor-loopscanbeeliminatedbyuseofMatlab’svector

arithmeticswhichleadstoafurtherimprovementoftheruntime.Section4givesashort

overviewonthefunctionalityprovidedbyp1afem,andselectedfunctionsarefurtherdiscussedin

theremainderofthepaper:Section5isfocusedonlocalmesh-refinementandmesh-coarsening

basedonNVB.Section6providesarealizationofastandardadaptivemesh-refiningalgorithm

steeredbytheresidual-basederrorestimatorduetoBabuˇskaandMiller[4].Section7

concludesthepaperwithsomenumericalexperimentsand,inparticular,comparisonswith

otherMatlabFEMpackageslikeAFEM[13,14]oriFEM[11,12].2.ModelExampleandP1-GalerkinFEM

2.1.ContinuousProblem.Asmodelproblem,weconsidertheLaplaceequationwithmixed

Dirichlet-Neumannboundaryconditions.Givenf∈L2(Ω),uD∈H1/2(ΓD),andg∈L2(ΓN),

weaimtocomputeanapproximationofthesolutionu∈H1(Ω)of

−∆u=finΩ,

u=uDonΓD,

∂nu=gonΓN.(2.1)

Here,ΩisaboundedLipschitzdomaininR2whosepolygonalboundaryΓ:=∂Ωissplitinto

aclosedDirichletboundaryΓDwithpositivelengthandaNeumannboundaryΓN:=Γ\ΓD.

OnΓN,weprescribethenormalderivative∂nuofu,i.e.theflux.Fortheoreticalreasons,we

identifyuD∈H1/2(ΓD)withsomearbitraryextensionuD∈H1(Ω).With

u0=u−uD∈H1D(Ω):={v∈H1(Ω):v=0onΓD},(2.2)

theweakformreads:Findu0∈H1D(Ω)suchthat󰀋

Ω∇u0·∇vdx=󰀋

Ωfvdx+󰀋

ΓNgvds−󰀋

Ω∇uD·∇vdxforallv∈H1D(Ω).(2.3)

Functionalanalysisprovidestheuniqueexistenceofu0intheHilbertspaceH1D(Ω),whence

theuniqueexistenceofaweaksolutionu:=u0+uD∈H1(Ω)of(2.1).Notethatudoesonly

dependonuD|ΓDsothatonemayconsidertheeasiestpossibleextensionuDoftheDirichlet

traceuD|ΓDfromΓDtoΩ.

2.2.P1-GalerkinFEM.LetTbearegulartriangulationofΩintotriangles,i.e.•TisafinitesetofcompacttrianglesT=conv{z1,z2,z3}withpositivearea|T|>0,•theunionofalltrianglesinTcoverstheclosure

−1

−0.500.51−1−0.8−0.6−0.4−0.200.20.40.60.81

1123

2253

3543

4413

5457

65107

71097

8947

9568

106118

1111108

121058112

225

356

4611

11110

2109

394

441wherethenodesaregivenincounterclockwiseorder,i.e.,theparametrizationoftheboundary

∂Tℓismathematicallypositive.

TheDirichletboundaryΓDissplitintoKaffineboundarypieces,whichareedgesoftriangles

T∈T.ItisrepresentedbyaK×2integerarraydirichlet.Theℓ-thedgeEℓ=conv{zi,zj}

ontheDirichletboundaryisstoredintheform

dirichlet(ℓ,:)=[ij].

Itisassumedthatzj−zigivesthemathematicallypositiveorientationofΓ,i.e.

nℓ=1

1function[x,energy]=solveLaplace(coordinates,elements,dirichlet,neumann,f,g,uD)

2nC=size(coordinates,1);

合集下载

midas有限元理论-5 实体单元

midas有限元理论-5 实体单元

Solid Elements

The Finite Element Library of the MIDAS Family Programs includes the following Solid Elements: 4-

node tetrahedron, 6-node pentahedron, and 8-node hexahedron shown in Fig. 1. The finite element

formulation of all element types is based on the isoparametric procedure (the element geometry and

displacements are interpolated in the same way).

Z

Y

Xζ

34

ξη

21u

4w

4v

436

ξη

2ζ

14

5u

6wv

66

u

4wv

44

u

2w

v2

2

u

1wv

11

u

3wv

33u

5w

v5

5

8

5

17

ξη

23ζ

6

4u

7wv

77

u

1w

v1

1

u

2w

v2

2u

3w

v3

3

u

1w

v1

1u

2w

v2

2u

3w

v3

3u

5w

v5

5u

6w

v6

6u

8wv

88

u

1wv

44

Figure 1 3D Solid Finite Elements

The nodal degrees of freedom (DOF’s) are illustrated in Fig. 1. The element geometry and displacement

field are defined in terms of nodal coordinated and DOF’s by the following functions:

where

iu

,, =

displacements in direction of global

iv

iwX,Y,Z−

axes, respectively, at th node −i

ix

Finite element studies of flow and temperature evolution in single screw extruders

Finite element studies of flow and temperature evolution in single screw extruders

Published by Maney Publishing (c) IOM Communications LtdFiniteelementstudiesofflowandtemperature

evolutioninsinglescrewextruders

C.Rauwendaal*

AfiniteelementapproachisusedtomodelHDPEmeltflowsandmelttemperaturedistributionsina63.5mmsinglescrewextruder.Theeffectoflengthscaleisexaminedusinga38mmextruder,includingtheeffectofflightclearance.Transientmelttemperaturefieldsarealsopresented.Keywords:Finiteelementanalysis,Non-isothermalflow,Non-Newtonianflow,Melttemperature,Melttemperaturedevelopment,Screwextruders,Flightclearance

IntroductionTheanalysisofflowinsinglescrewextrudershasbeenthesubjectofmanystudies,due,atleastinpart,tothesubstantialimportanceofscrewextrusioninavarietyofmanufacturingoperations.Earlyanalyses1–3dealtwithtwo-dimensionalflowofNewtonianfluidswithtem-peratureindependentviscosity.Thisproblemallowsanalyticalsolution;thissolutionisfrequentlyusedintheanalysisofpolymerextrusionoperations.Mostpolymers,however,arestronglynon-NewtonianandtheassumptionofNewtonianbehaviourwilloftenresultinsubstantialerrors.Asaresult,flowanalyseswereextendedtodealwithnon-Newtonian,temperatureindependentfluids.4–6Thisextensiontonon-Newtonianbehaviourcomplicatestheproblemtothepointthatanexactanalyticalsolutionforthetwo-orthree-dimen-sionalvelocityprofilesisnolongerpossible.Variousapproximatesolutionshavebeenpro-posed;7–10however,thesesolutionshavealimitedrangeofvalidityandlimitedaccuracy.Ontheotherhand,theseapproximatesolutionsaretobepreferredovertheNewtoniansolutionswhentheactualfluidisstronglynon-Newtonian.Theonlyanalyticalsolutiontothecombineddragandpressureflowofanon-Newtonianfluidisthesimplecaseofone-dimensionalflowofaPrandtl-Eyringfluid.10Theanalysisofflowattainsanadditionaldegreeofcomplexitywhenthefluidisconsideredbothnon-Newtonianandtemperaturedependent.Thisisbecauseforthisproblem,theequationofmotioniscoupledwiththeenergyequation;theextentofcouplingcanbeassessedbyexaminationoftheNahmenumber.Onlythesimplestcaseoffullydeveloped,puredragflowbetweenparallelplateshasbeensolvedanalytically.11Extensiontoa2Dor3Danalysisinvariablyleadstonumericaltechniques.Anumberofworkershavestudiednon-isothermalflowinextruders.AgoodreviewonthissubjectwasgivenbyFenner.12AnextensivereviewofcomputeraidedanalysisofvariouspolymerprocessingoperationswascompiledbyPittman.13Becauseofthesereviews,earlierworkonnon-isothermalflowinextruderswillnotbediscussedindetail.SomeofthesignificantcontributionsarelistedinTable1.ExaminationofTable1showsthatessentiallyallearliernon-isothermalanalysesneglectleakageflowbetweenthescrewflightandthebarrel.Ithasbeenshown,however,thatleakageflowcanhaveamajoreffectonthetem-peraturedistributioninthescrewchannel.23,26Arealisticanalysis,therefore,shouldconsiderallthreevelocitycomponentsinthescrewchannelaswellasintheflightclearance.Inthefollowing,afiniteelementanalysiswillbedescribedoffullydevelopedthree-dimensionalflowofanon-Newtonianfluidwithtemperaturedependentviscosityinascrewchannelincludingleakageflow.FiniteelementformulationThestartingpointoftheanalysisistheweakformsforthelinearmomentum,massandenergybalances.Becauseofthedistinctrolethez-directionplaysinthisanalysis,thenotationthatGreekindicesstandforcomponents1and2willbeemployed,whilethez-directionwillcontainanexplicitzsubscript(super-script).Thelinearmomentumbalancereadsð

LS-prepost finite-element

LS-prepost finite-element

Mesh Overview

zShapeM-Simple shell/solid meshing interface

zAutoMesh-Surface mesh interface.

zSolidMesh-Solid mesh interface.

zBlockM-3D block mesher interface.

zNlMesh-2, 3, and 4 line mesh interface.

zTetMesh-Tetrahedron mesher interface.

zBMesh-Blank mesher interface.

zElemGen-Create beam, shell, and solid elements.

zNodeEdit-Show, create, delete, and modify nodes.

zElemEdit-Show, create, delete, and modify elements.

zMassTrim-Mass trim interface.

zSpot Weld-Generate spotweld elements from a spotweld file.

zSphGen-Sph particle generation interface.

ShapeM

Purpose: This interface, often used for generating solid meshes for ALE simulations, can be used to generate solid and shell meshes.

Function:Box_Solid : Define a box and generate solid mesh of it.Box_Shell : Define a box and generate shell mesh of it. 4N_Shell : Define four node surface and generate shell mesh of it. Sphere_Solid : Define a sphere and generate solid mesh of it. Sphere_Shell : Define a sphere and generate shell mesh of it. Cylinder_Solid : Define a cylinder and generate solid mesh of it.Cylinder_Shell : Define a cylinder and generate shell mesh of it. Circle_Shell : Define a circle and generate shell mesh of it.

有限元分析英文文献

有限元分析英文文献

The Basics of FEA Procedure 有限元分析程序的基本

知识

2.1 Introduction

This chapter discusses the spring element, especially for the purpose of introducing various

concepts involved in use of the FEA technique.

本章讨论了弹簧元件，特别是用于引入使用的有限元分析技术的各种概念的目的

A spring element is not very useful in the analysis of real engineering structures; however, it

represents a structure in an ideal form for an FEA analysis. Spring element doesn’t require

discretization (division into smaller elements) and follows the basic equation F = ku.

在分析实际工程结构时弹簧元件不是很有用的;然而,它代表了一个有限元分析结构在一

个理想的形式分析。弹簧元件不需要离散化(分裂成更小的元素)只遵循的基本方程 F = ku

We will use it solely for the purpose of developing an understanding of FEA concepts and

procedure.

我们将使用它的目的仅仅是为了对开发有限元分析的概念和过程的理解。 2.2 Overview 概述

Finite Element Analysis (FEA), also known as finite element method (FEM) is based on the