Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 1/12 Eemen of Fouie fo xiymmeic ucue Abc he eemen of Fouie e inended o ccue he ucue epone fo xiymmeic geomey oicied by nonxiymmeic oding boken up ino Fouie eie. One expoe in hi documen gene heoy of Anyi of Fouie wih couping of he ymmeic nd kewymmeic mode in he nioopic ce. The ce of he ioopic, o ohoopic mei of xi Oz, whee he mode e uncouped, i udied excep fo. The eemen of Fouie e ube in fom modeizion AXIS_FOURIER. Mehe he uppo of hee eemen e inge nd qudnge of degee 1 nd 2.
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 2/12 Conen 1Inoducion3... 2Anyze of Fouie nioope3... 2.1Théoie génée3... 2.2Coupge nd decouping of he ymmeic mode nd niyméique5... 2.3Ccu of he conine6... 3Ccu of he mix of igidié7... 3.1C géné7... 3.2Ccu of in he ce ioope8... 4Chgemen11... 5Concuion nd Pepecive12... 6Bibiogphie12... 7Decipion of he veion of he documen12...
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 3/12 1 Inoducion he nyi of Fouie e inended o ccue he ucue epone fo xiymmeic geomeie ubjeced o nonxiymmeic oding. In hi ce, i i necey o deveop he oding in Fouie eie. Geney convegence i eched fo 4 o 5 hmonic, bu he peed of hi convegence depend on he nue of he oding: he moe egu he oding i nd he moe quicky he coeponding eie convege. The mo unfvoube ce i h of concened foce fo which he pcice how h i i necey o go o beyond ( e 7 hmonic). In he, he decompoiion of he oding in Fouie eie i uppoed o be mde peiminy by he ue. The mke i poibe o ccue he epone wih hi oding, hmonic by hmonic (modeizion AXIS_FOURIER), nd ove fe ecombinion of he hmonic beween hem (opeo COMB_FOURIER). One wi expoe in fi chpe he gene fme of he nioopy, whie iniing on he decouping of he mode in he ohoopic ce. The econd chpe cifie he compuion of he iffne mix in he ioopic ce. Fo he ue of he eemen of Fouie in, one eun o he noe of ue of he modeizion Fouie [U2.1.7]. 2 Anyze of nioopic Fouie 2.1 gene Theoy A he fied conideed (foce, dipcemen, in, ee) e expeed in cyindic coodine wih foowing convenion on he ode of he componen: 1 di componen ccoding o 2 xi componen ccoding o z 3 ngeni componen ccoding o Exmpe: u,u z,u,f,f z,f z θ M u z u θ u The meh i ocied in he pne, z, he ymmey of evouion being done ound he xi Oz. The ihedon, z, i dieced in he diec mening.
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 4/12 z ez C θ θ e Dipcemen i boken up u (o he oding f ) ccoding o u=u u whee u (ep. u ) indice he ymmeic p (ep. kew-ymmeic) of he deveopmen in Fouie eie fom u io o he vibe. One obin: u = u, z co u z = v,z co u = w,z in }pie yméique u u = u z = u = u,z in v, z in niyméique u w, z co}pie A o noe he choice of he ign fo u, which mke i poibe o impify e compuion. If one noe U = u, v,w ep. U ième he ymmeic componen (ep. kew-ymmeic) fom he deveopmen in Fouie eie of u, one obin: in [ co u= co U in in U co ] éq 2.1-1 If one indice by he veco in ineized, one eize h cn be boken up ino foowing Fouie eie: = co I 4 4,2 = 2,4 in I 2 in I 4 4,2 2,4 co I 2 éq 2.1-2
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 5/12 (ee [bib1]): =[ = B U 1 B = B U 1 ] wih ={, z, q, z, q, zq } One h B =B (hi i due o he choice of he ymmeic deveopmen of u in co, co, in ined of co,co,in ). One wi omi fom now he indice nd one wi noe B he opeo owing o ccue he in coeponding o he hmonic. 2.2 Couping nd decouping of he mode ymmeic nd kewymmeic By king gin he peceding noion, one : u= co I 2 2,1 1,2 in u wh i wien, by inoducing mixe M e M : M U M U u= u = M U M U in I 2 2,1 1,2 co u One fom of deduced h: =M ' M '
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 6/12 = vec M ' co I 4 2,4 4,2 in I 2 = M ' in I 4 2,4 4,2 co I 2 Compuion of in enegy 2 2 W = = 2 D dd d d Puique M ' DM ' = 2 nd h vec d=ddz 2 M ' DM ' d 2 M ' DM ' d in I 4 M ' DM ' = co I 2 d M ' DM ' d d M ' DM ' d D 1 D 3 D 3 D 2 D 1 in co D 3 in 2 D 3 co 2 D 2 in co co I 4 in I 2 in co d =, i D 3 = hee i hu no em, ou, in W. Thee i hen no couping U,U ou U,U. 2.3 Compuion of he ee u, cn be boken up ino foowing Fouie eie: = Hooke' w =D, one deduce: = co D 1 in D 3 co D 3 in D 2 M ' M ' in D 1 co D 3 in D 3 co D 2 Mybe, whie eveing he mixe M ' e M ' : = M ' [ M ' [ D 1 2,4 4,2 D 2 4,4 D 3 D 3 2,2 ] 4,4 D 3 D 3 2,2 D 1 4,2 2,4 D 2 ]
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 7/12 Whie poing D D 1 4,2 2,4 D 2 nd D = 4,4 D 3 D 3 2,2 nd kew-ymmeic of he e eing o he hmonic : Noe::, one fom of deduced he p ymmeic { = D D =D B u D B u = D D = D B u D éq 2.3-1 B u In he ce of he ohoopy comped o Oz, one h D = nd [éq 2.1-1] i educed o: { = D B u =D B u I.e. if dipcemen e ymmeic (o kew-ymmeic), he ee e i oo. 3 Compuion of he iffne mix 3.1 gene Ce Ae u nd wo unpecified kinemicy dmiibe fied. By ppying he pincipe of he viu wo o he voume eemen v, one cn wie: v. dv= u. f dv v Afe decompoiion in Fouie eie nd inegion comped o, one obin, fo unpecified,,u, u fied A.C. nd ny hmonic :.. d = u. f u. f d Mybe, by men of [éq 2.3-1] nd whie poing: K = B D B d K = B D B d =K =K K = B D B d The foowing yem of equion i obined: whee K = K { K u K u = f K u K u éq 3-1 = f i i een h if D, he decouping of he mode in ymmeic nd kewymmeic hmonic i no poibe ny moe. On he ohe hnd, if D = (ohoopy comped o Oz ) hen K = nd [éq 3-1] i educed o: { K u = f K u = f
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 8/12 Whie king fo veco dipcemen (ep. foce) coeponding o he hmonic he veco: u ={u,u z,u,u,u z,u } f ={ f, f z, f, f, f z, f } One h hen: = K g u =f vec K g K K K K 3.2 Compuion of K g in he ioopic ce In hi ce one hu h K =. One dei in he coninuion he compuion of K = B D B d =[ In he ioopic ce, one : D1 D2 D2 D2 D1 D2 D=D D2 D2 D1 D3 D3 D3 ] vec D1= E 1 1 1 2 D2= One cn wie: E 1 1 2 u D3= E 2 1 { z z z}=b {u u z u }=B ' { u, u z, u, u, u z, u, u, u z, u } =[ vec B ' fc of fom deived fom he fc of fom 1 1 1 1 1 1] 1 1
u Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 9/12 =[ Whie indicing by {W } =1 à 4n he hpe funcion of he eemen conideed, one : noeud W ]=[ u z W u W u ] { W u u U z W u z u W u u W u z W u W boc P } One noe P = P 1,, P N whee N i he numbe of node of he eemen. Then K = P B ' DB ' P d K ymmeic nd i fomed by bock K 3 3 : K = P I B ' DB ' P d The compuion bock K i cified beow:
=[ D11 B ' DB ' Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 1/12 P I B ' DB ' = {K 11 = K 22 = K 33 2 D3 D1D3 D2 D 3 D2 2 D3 D 3 D1D3 2 D1D3 D 2 D3 D 2 D2 D 2 D1 D2 D3 D3 ] D 3 D3 D3 D3 D3 D2 D 2 D2 D1 D 3 D3 K 11 K 12 K 13 K 21 K 22 K 23 vec K 31 K 32 K 33 =[ P = K ij 3 3 D1 2 D3 2 2 D3 2 2 D1D3 2 K 12 = D2 W I W K 21 = D3 W I W W D1 W I I W W D3 W I I W W D3 I W I ] W D3 W I W D2 W W I W W I W D1 W I W W W I D3 W I W D2 W W I W D2 W I W D2 W W I K 13 = D1 D3 W W 2 I D2W W I D3W I W D3 W W I W I W W K 31 = D1 D3 W W 2 I D2W W I D3W W I K 23 = D2 W I W D3W W I K 32 = D2W W I D3 W I W The bock K e no ymmeic excep fo I = (on he digon of K ). One noice in fc h he bock K cn be wien fo ny hmonic ( incuding).
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 11/12 {K 11 = K 11 2 D3 W W 2 I K 22 = K 22 2 D3 2 K 33 = K 33 2 D1 K 12 = K 12 K 21 = K 21 K 13 = K 13 K 31 = K 31 K 23 = K 23 K 32 = K 32 2 W I W W I W whee he bock K e independen of he hmonic. 4 Loding I i uppoed h he oding w boken up ccoding o he me be which dipcemen, h i o y: f = co [co F in in F in co Thee i no couping fo he me hmonic beween he p ymmeic nd kew-ymmeic of he oding becue of ohogoniy of he goniomeic funcion in nd co, hi fo he ype of oding. Thi wn o y in picu h he equiven nod foce e he me one fo he hmonic ymmeic nd kew-ymmeic if he mpiude F e F e he me one. Fo he nue of he ccepbe oding wih he modeizion Fouie, one eun o he noe of ue [U2.1.7]. ]
Tie : Éémen de Fouie pou e ucue xiyméi[...] De : 13/2/213 Pge : 12/12 5 Concuion nd Popec Cueny, i i uppoed h he decompoiion of he oding w mde peiminy by he ue, i.e. { F, F } i known. Thi decompoiion coud be eized by n opeo of which woud pojec he oding on he mode of Fouie. Fo ime, ony he nonnioopic ce i ebihed, i.e. hee i neve couping of he mode. The exenion o he nioopy cn coniue e deveopmen. 6 Bibiogphy 1 DUVAUT G.: Mechnic of he coninuum p282 2 ASKA HS.: Axiymmeic Sucue in Fouie eie, My 1982, ISD 7 Decipion of he veion of he documen Ae Auho (S) Ognizion (S) 5 X.Deoche EDF- R&D/AMA Decipion of he modificion inii Tex