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Chapter5
FiniteElementMethods
5.1FiniteElementSpaces
dimP(K)=NK.? onameshcellKisdenotedbyP1(K): P1(K)=?
a 0+d? i=1a ixi:x=(x1,...,xd)T?K? d+1.? whicharelinearlyindependent.Therearedifferent typesoffunctionalswhichcan beutilizedinfiniteelementmethods: •pointvalues:Φ(v)=v(x),x?K, istheoutwardpointingunitnormalvectoronE, •integralmeanvaluesonK:Φ(v)=1 |K|?Kv(x)dx,
|E|?Ev(s)ds.
Thesmoothnessparametershastobe choseninsuchawaythatthefunctionals 60p?P(K)with thei-thfunctional,i=1,...,NK.
K,i(φK,j)=δij,i,j=1,...,NK.
basis.? knownbasis K,j=N K? k=1c jkpk,cjk?R,j=1,...,NK, thelinearsystemofequationsK,i(φK,j)=N
K? k=1c c jkaredetermineduniquely.? setofequations( (0 0 11 0 10 1 1) (a b c) (1 0 0) Atriangulationiscalledregular,seethedefinitioninCiarlet Ciarlet(1978),if: •ItholdsΩ=?K?ThK.
61{0,...,d-1}.
Ω)→Rcontin-
cellsKj,forwhichthereisap?P(Kj)withΦi(p)?=0,willbedenoted byωi.? iistheunionofallmeshcellswhich possessthisvertex,seeFigure5.1.?Figure5.1:Subdomainsωi.
functionalΦi:Ω→Rif i(v|K1)=Φi(v|K2),?K1,K2?ωi.Thespace
S=? i,i=1,...,N? iscalledfiniteelementspace.Theglobalbasis{φj}N
j=1ofSisdefinedbythecondition j?S,Φi(φj)=δij,i,j=1,...,N. iscalledhatfunction.? 62globalbasisfunctions. {Φi}N case,one canspeakofvaluesoffiniteelementfunctionsonm-faceswithm
K,ˆp?ˆP(ˆK)?
.(5.1) •ThelocalfunctionalsaredefinedbyK,i(v(x))=ˆΦi(v(FK(ˆx))),(5.2)
where holdsx=FK(ˆx). finiteelementsspacesisverygeneral.Forinstance,different typesofmeshcells considered.?5.2FiniteElementsonSimplices
63tothenon-singularityofthematrix A=( (a
11a12...a1,d+1
a21a22...a2,d+1.
a d1ad2...ad,d+11 1...1)
whereai=(a1i,a2i,...,adi)T,i=1,...,d+1. points{ai}d+1 readsasfollows K=? x?Rd:x=d+1? i=1λ i=1λ i=1? d+1? i=1a i=1λ i=1. natesaredetermineduniquely. vanishesinallotherverticesajwithj?=i.Thebarycenterofthesimplexisgivenby
S K=1 d+1d+1? i=1a i=d+1? i=11d+1ai. K=? x?Rd:d? mappings F {K}?Rd.? 64each non-degeneratedsimplex.Thesetoftheselocalspacesiscalledaffinefamily ofsimplicialfiniteelements.? P k=span? d? i=1x
αii=xα:αi?N?{0}fori=1,...,d,d?
i=1αLagrangianfiniteelements.?
dimP0(K)=1.?Figure5.4:ThefiniteelementP0(K).
spaceisasubspaceofC( 65Figure5.5:ThefiniteelementP1(K).
functionsfromP1arecontinuous.? mentspaceisalsoasubspaceofC( i=1i=d(d+1)/2edges.Figure5.6:ThefiniteelementP2(K).
Thepartofthelocalbasiswhich belongstothefunctionals{Φi(v)=v(ai), i=1,...,d+1},isgivenby 66spondingpartofthelocalbasisisgivenby {φij=4λiλj,i,j=1,...,d+1,i
Figure5.7:ThefiniteelementP3(K).
Forthefunctionals
i(v)=v(ai),i=1,...,d+1,(vertex), thelocalbasisisgivenby i(λ)=12λi(3λi-1)(3λi-2),
iij(λ)=92λiλj(3λi-1),
ijk(λ)=27λiλjλk? 67C(
Figure5.8:ThecubicHermiteelement.
intheimplementationofthisfiniteelement. Becauseofthisproperty,one canusethederivativesinthedirectionoftheedges asfunctionals i(v)=v(ai),(vertices) ijk(v)=v(aijk),iFigure5.9:ThefiniteelementPnc1.
i(λ)=1-dλi,i=1,...,d+1.5.3FiniteElementsonParallelepipeds
69affinemappingsoftheform F