Qurlterl et Heerl seuo-conform Fnte Elements E. DUBACH R. LUCE J.M. THOMAS Lbortore e Mtémtques Applquées UMR 5 u Frnce GDR MoMs Métoes Numérques pour les Flues. rs écembre 6
Wt s te problem? Loss of conergence for te RT BDM BDFM wen we use qurlterl or eerl elements. { } { } L p A M q q q M L p q L q M RT H L Ω Ω Ω Ω Ω Ω u u u f soluton of fn ; ; : Stnrt moel u oes not conerge n H wen te mes s bse on qurlterl or eerl elements.
D 3D uu Ω >.996 pp Ω >.98933 uu Ω >.76 logerreur 3 5 6.5.5 3 3.5 logps.5 5 5.5 Loss of conergence on u Loss of conergence on u n u
Were oes te problem come from? ˆ : element of te trngulton : reference element F s blner J F etdf Q F s trlner J F etdf Q
ol trnsform: u DFˆ ûˆ p J F ˆ pˆˆ u.n ps ˆ û.nˆ ps ˆ ˆ u p ˆ û p ˆ û ˆ u ûˆ Mˆ u J D F ˆ π u C u Ω Ω u u Non lner reltons 3D π u Cu Ω Ω u π u C u Ω Ω u u π u C Ω Ω
Solutons - Increse te spce of scretston to control te non-lner prt of te trnsformton F. Arnol-Boff-Flk D D rllelogrm RT Qurlterl ABF Degrees of freeom: Degrees of freeom: 6 By usng te sme wy we obtn 3D rlleleppe RT Heeron Degrees of freeom: 6 Degrees of freeom: 36!!
Solutons - «Remes» te qurlterls or eerls Yu. uznetso n S. Repn J. Numer. Mt 5 Obously t s possblty - Bul pseuo-conform fnte elements wtout ng egrees of freeom We e cosen ts lst wy
Cs D Let us emne te Q conform ppromton on qurlterls. Wy? No loss of conergence wt te Q fnte element but te bckgroun s fferent of te fnte element on trngles Work on Tˆ s equlent to work on T pˆ T p ˆ T pˆ ˆ Q p Q p s not polynoml
Consequence: All te ntegrls must be clculte on ^ p p DF. pˆ ˆ DF pˆ. DF ˆ J F ˆ Gol: Construct new fnte element stsfyng: p s polynoml on Degrees of freeom re te sme On prllelogrm recoer te clsscl Q fnte element rce to py: Te egree of te polynomls must be ncrese. Te ppromton s pseuo-conform
Geometry n nottons : cone qurlterl R [ ]... eges : : ertces : of center 3 gen by te bss of 3 3 e e R e e
F : ffne prt off. ByF s eformton of te unt squre ˆ s trnsforme nto F cone qurlterl <
Construct on of new fnte element Σ We wnt Σ { w w } Coce of? Q We obtn non conformng ppromton tt oes not stsfy te ptc-test. In te error estmton we on t control te followng term: Soluton : fn u n [ u I u] σ [ u ] suc tt σ
We roposton: must e For ny cone qurlterl tere est polynoml ω suc tt for te coce: We e Vect ω w Σ s fnte element. wσ { w w }... Remrk : ω s not unque
ω Moreoer t s of nterest to obtn: epen contnously on te storson prmeter t te lmte te Lgrnge reference ω σ... te fnte element Te smplest coce s to cooseω ω... Σ ˆ Q fnte element on te unt squre n 3 Q ω cn be clculte eplctely ccorng to stsfyng : or numerclly Remrk: We cn compute te fnte element bss n 3 Q wtout clcultng ω Σˆ
...6.8...3..5.6.7.8.9...6.8...3..5.6.7.8.9 Mes n ceron Mes n léole wt 3 3 7 3 en en ω
Q conform seuo-conform meto
new fnte element Constructon of Σ Σ. We wnt σ wn w ect ψ Let us return to te ntl problem roposton: For ny cone qurlterl tere est polynoml ector ψ suc tt for te coce: We e..... s fnte element. Σ s s s σ σ σ n w wn w
At te lmte te fnte element Σ ˆ te RT reference fnte element on te unt squre Σ ˆ Te smplest coce s to cooseψ De Rm grm ψ rot Anoter coce s to coose BDM H y y ect y y y n ψ. nσ sψ. nσ H ψ BDM rot ω stsfyng :... Remrk: We cn compute te fnte element bss n BDM wtout clculte ψ
uu Ω >.9966 pp Ω >.9936 uu Ω >.993 logerreur 3 5 6.5.5 3 3.5 logps.5 5 5.5 Te conergence orers re goo
Concluson If te qurlterls re prllelogrms we e ˆ ˆ Σ Q Σ ˆ ˆ Σ Σ n te ppromtons re conform. If te qurlterls re not prllelogrms te ppromtons re not conform. Te use of Σ llows us to e te contnuty of p t te noes n only te contnuty of te men lues troug ec ege of te mes. Te use of ˆ ˆ Σ llows us to e te contnuty of te men lue of u.n n te contnuty of te frst momentum of u.n troug ec ege. Te mplementton s smple but te bss functons epen on of te spe of.
3D cse We pply te sme pproc wt te 3D cse. roblems wt te eerons Descrpton of eeron eformton of te unt cube lne fces n non-plne fces? Is t rel problem? rmetrston of te fces
3D cse epens on 6 prmeters f te fces re plne n 9 prmeters f te fces re not plne. σ ˆ ˆ non plne fce : M ˆ nˆ ˆ σ plne fce : Mˆ nˆ M ˆ nˆ s not polynoml
Construct on of new fnte element Σ We wnt Σ { w w 8} How to bult? cse of plne fces Let bev m V V Σ polynomlspce suc tt { } w w...8; w σ...6 s unsolnt onv wen ˆ
... cobn t ponts te te cofcteur mtr of epen only on te lues of te fce te noe of numbers of re serl... were...6 l l l ω ω σ Bulng of ntegrton formuls...6 ; s V l ω σ
ossble coces of polynoml spce V { } 3 Q z z y y z y yz yz z y z y V y z y z z y z y z y z y Q V
roposton: For ny qurlterl not too muc eforme compre to unt cube we e...6 s fnte element. Σ l ω σ Remrk: Te mn fference wen te fces re not plne concerns te ntegrton formuls: plne cse must be numerclly clculte n te non - re eplctly known n te plne cse ω ω
Interor fces re not plne.8.6...5...6.8 Non plne fces Destructure mes
lne fces Q conform Non-plne fces seuo-conform meto>
Wt s bout te ppromton n H Constructon of We wnt new fnte element w wn. σ 6 Σ Σ How to bult? Let bev m V Σ polynomlspce suc tt V 8 { } u un. σ...6; w su. nσ...6 s unsolnt onv wen ˆ
. cobn on te te cofcteur mtr of epen only on lues of...6 ;.. s ω σ ω σ n n Bulng of ntegrton formuls 6... ;.. ; s V ω σ σ n n ossble coce of polynoml spce BDM V yz y yz z z y y y z z y z y y z y z z y ect BDM
roposton: For ny qurlterl not too muc eforme compre to unt cube we e...6 ;.. s fnte element. Σ s ω σ σ n n roblem wt te ntegrton formul wen te fces re not plne.
seuo-conform meto plne fces
Conclusons D cse: Te numercl results re gree wt te teoretcl results. ossblty to generlse te results to FEM of ger egrees. 3D cse: Necessty to work on te geometry of te eerons. Clrfy te problem of plne fces n non-plne fces. Use te De Rm grm to connect te H cse to te H cse De Rm grm : H H curl rot H