[PDF] A Finite Element Methods 2. Piecewise linear global basis





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PE281 Finite Element Method Course Notes - Stanford University

May 23 2006 · any set of linearly independent functions will work to solve the ODE Now we are ?nally going to talk about what kind of functions we will want to use as basis functions The ?nite element method is a general and systematic technique for constructing basis functions for Galerkin approximations In 5

A

FiniteElementMethods

A.1FiniteElementSpaces

arejustcalledfaces.? bedenotedbydimP(K)=NK.? meshcellKisdenotedbyP1(K): P

1(K)=?

a 0+d? i=1a ixi:x=(x1,...,xd)T?K? N

K=d+1follows.?

whicharelinearlyindependent.Therearedifferent typesoffunctionalswhich canbeutilizedinfiniteelementmethods: •pointvalues:Φ(v)=v(x),x?K, n

80AFiniteElementMethods

•integralmeanvaluesonK:Φ(v)=1 |K|?

Kv(x)dx,

|E|?

Ev(s)ds.

Thesmoothnessparametershastobe chosensuchthatthefunctionals p?P(K)with i=1,...,NK.

K,i(φK,j)=δij.

basis.? termsoftheknownbasis K,j=N K? k=1c jkpk,cjk?R,j=1,...,NK, tothelinearsystemofequations

K,i(φK,j)=N

K? k=1c coefficientscjkaredetermineduniquely.?

A.1FiniteElementSpaces81

notvanishatthevertices. thefollowingsetofequations (0 0 11 0 10 1 1) (a b c) (1 0 0) •Itholds

Ω=?K?ThK.

m?{0,...,d-1}.

Ω)→Rcon-

willbedenotedbyωi.? avertexofK,thenωiistheunionofallmeshcellswhich possessthisvertex, seeFig.A.1.? functionalΦi:Ω→Rif i(v|K1)=Φi(v|K2)

82AFiniteElementMethods

Fig.A.1.Subdomainsωi.

forallK1,K2?ωi.

Thespace

S=? i,i=1,...,N? iscalledfiniteelementspace.

Theglobalbasis{φj}N

j=1ofSisdefinedbythefollowingcondition: j?S,Φi(φj)=δij,i,j=1,...,N. suchafunctioniscalledhatfunction.?

A.1FiniteElementSpaces83

theglobalbasisfunctions. to{Φi}N thiscase,one canreallyspeakofvaluesoffiniteelementfunctionsonm-faces withmP(K)=? p:p=ˆp◦F-1

K,ˆp?ˆP(ˆK)?

.(A.1) •Thelocalfunctionalsaredefinedby

K,i(v(x))=ˆΦi(v(FK(ˆx))),(A.2)

itholdsx=FK(ˆx). A.12offiniteelementsspacesisverygeneral.Forinstance,different types finiteelementsareconsidered.?

A.1.1FiniteElementsonSimplices

A=( (a

11a12...a1,d+1

a

21a22...a2,d+1.

a d1ad2...ad,d+1

1 1...1)

whereai=(a1i,a2i,...,adi)T,i=1,...,d+1. tetrahedrons.?

84AFiniteElementMethods

points{ai}d+1 i=1,a possibleparametrizationofKreadsasfollows K=? x?Rd:x=d+1? i=1λ i=1λ i=1? Thecoefficientsλ1,...,λd+1inthisparametrizationare calledbarycentric coordinatesofx?K. d+1? i=1a i=1λ i=1. coordinatesaredetermineduniquely. vanishesinallotherverticesajwithj?=i.

Thebarycenterofthesimplexisgivenby

S K=1 d+1d+1? i=1a i=d+1? i=11d+1ai. K=? x?Rd:d? mappings F plicesK?Rd.? affinefamilyofsimplicialfiniteelements.

A.1FiniteElementSpaces85

P k=span? d? i=1x

αii=xα:αi≥0fori=1,...,d,d?

i=0α

ItholdsdimP0(K)=1.?

Fig.A.4.ThefiniteelementP0(K).

ementspaceisasubspaceofC(

Ω).Thelinearfunctionalsarethevaluesof

86AFiniteElementMethods

dimP1(K)=d+1.

Fig.A.5.ThefiniteelementP1(K).

{λi}d+1 v elementspaceisalso asubspaceofC(

Ω).Itconsistsofpiecewisequadratic

Thelocalbasiswhich belongstothefunctionals{Φi(v)=v(ai),i=

1,...,d+1},isgivenby

{φi(λ)=λi(2λi-1)}. correspondinglocalbasisisgivenby {φij=4λiλj,i,j=1,...,d+1,iA.1FiniteElementSpaces87

Fig.A.6.ThefiniteelementP2(K).

spaceofC( (2?d givenby dimP3(K)=(d+1)+d(d+1)+(d-1)d(d+1)

6=(d+1)(d+2)(d+3)6.

Fig.A.7.ThefiniteelementP3(K).

88AFiniteElementMethods

Forthefunctionals

{Φi(v)=v(ai),i=1,...,d+1,(vertex), thelocalbasisisgivenby {φi(λ)=1

2λi(3λi-1)(3λi-2),

iij(λ)=9

2λiλj(3λi-1),

ijk(λ)=27λiλjλk}.

ExampleA.29.Pbubble1.?

Fig.A.8.ThefiniteelementPnc1.

i(λ)=1-dλi,i=1,...,d+1.

ExampleA.31.Pdisc1.?

ExampleA.32.Pbubble2.?

A.1FiniteElementSpaces89

A.1.2FiniteElementsonParallelepipeds

ofbijectiveaffinemappingsoftheform F

Kˆx=Bˆx+b,B?Rd×d,b?Rd.

enlarged,seeSectionA.1.3.? Q k=span? d? i=1x dimensionalbasisfunctions.? anditholdsdimQ0(K)=1.? elementspaceisasubspaceofC(

Ω).Thefunctionalsarethevaluesofthe

2 d. product,aregiven by

φ1(ˆx)=1

2(1-ˆx),ˆφ2(ˆx)=12(1+ˆx).

90AFiniteElementMethods

Fig.A.9.ThefiniteelementQ1.

thatQ2?C(

Fig.A.10.ThefiniteelementQ2.

by

φ1(ˆx)=-1

Thebasisfunction?d

A.1FiniteElementSpaces91

niteelementspaceisasubspaceofC(

Ω).Thefunctionalsonthereference

dimQ3(K)=4d.

Fig.A.11.ThefiniteelementQ3.

φ1(ˆx)=-1

16(3ˆx+1)(3ˆx-1)(ˆx-1),

φ2(ˆx)=9

16(ˆx+1)(3ˆx-1)(ˆx-1),

φ3(ˆx)=-9

16(ˆx+1)(3ˆx+1)(ˆx-1),

φ4(ˆx)=1

16(3ˆx+1)(3ˆx-1)(ˆx+1).

ExampleA.40.Qrot

P

1(K)=2d.The

spaceonthereferencemeshcellisdefinedby Q rot

1?ˆK?

Q rot

1?ˆK?

Notethatthetransformedspace

Q rot

1(K)={p=ˆp◦F-1

K,ˆp?Qrot

1(ˆK)}

92AFiniteElementMethods

Fig.A.12.ThefiniteelementQrot

1.

1(ˆx,ˆy)=-3

8(ˆx2-ˆy2)-12ˆy+14,

2(ˆx,ˆy)=3

8(ˆx2-ˆy2)+12ˆx+14,

3(ˆx,ˆy)=-3

8(ˆx2-ˆy2)+12ˆy+14,

4(ˆx,ˆy)=3

8(ˆx2-ˆy2)-12ˆx+14.

ExampleA.41.Pdisc1.?

missiblereferencemapshastobeenlarged. F

K(ˆx)=?F1K(ˆx)

F

2K(ˆx)?

=?a11+a12ˆx+a13ˆy+a14ˆxˆy a

21+a22ˆx+a23ˆy+a24ˆxˆy?

,FiK?Q1,i=1,2, lateralKiscalledadmissibleif

A.1FiniteElementSpaces93

K.Theseareingeneral

F Kon anedgeofˆKisanaffinemap.E.g.,inthecaseoftheQ1finiteelement, continuous,seeExampleA.26.?

A.1.4TransformofIntegrals

F isdenotedbyF-1 (Jacobians)ofFKandF-1

Kareneeded

DFK(ξ)ij=∂xi

∂ξj,DF-1

K(x)ij=∂ξi∂xj,i,j=1,...,d.

K v(x)dx=?

Kˆv(ξ)|detDFK(ξ)|dξ,(A.3)

whereˆv(ξ)=v(FK(ξ)).? ∂v ∂xi(x)=d?

K(x)?T?

i =?ξˆv(ξ)·??DF-1

K(FK(ξ))?T?

i ,(A.4) ∂ˆv ∂ξ(ξ)=d? (DFK(ξ))T? i =?v(x)·??DFK(F-1

K(x))?T?

i .(A.5)quotesdbs_dbs14.pdfusesText_20
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