Code_Aster. Finite element method isoparametric

Transcription

1 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 1/18 Finit lmnt mthod isoparamtric Abstract : This documnt prsnts th bass of th finit lmnts isoparamtric introducd into for th modlization of th continuums 2D and 3D. On first of all rcalls th transition of a strong formulation to a variational formulation, thn on dtails th discrtization by finit lmnts: us of an lmnt of rfrnc, computation of th shap functions and valuating of th lmntary trms. On also brifly dscribs th principl of th assmbly of ths trms and th imposition of th boundary conditions, and on voks th mthods of matric rsolution usd. Finally ar xposd th main stps of a computation by finit lmnts such as it ar concivd and stablishd in.

2 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 2/18 Contnts 1 Introduction Obtaining a formulation variationnll Modlization of th physical problm Principls and notation Equations of th systèm Mthod of th balancd rsidus intgral Formulation fort intgral Formulation faibl Mthod of th lmnts finis Principls généraux Approximation of th géométri Princip Elmnts of référnc Intrpolation functions géométriqu Jacobian matrix of th transformation Rprsntation of th inconnus Approximation nodal Bas polynomial Functions of form Corrspondnc btwn polynomial bas and functions of form Rsults of xistnc and unicity Mthod of Ritz Construction of th systm matricil Nw notation (notation of Voigt) Systm discrétisé Computation of th élémntairs13 trms Transformation of th dérivés Chang of fild of intégration Intgration numériqu Rsolution of th systm matricil Imposition of th boundary conditions cinématiqus Résolution Organization of a computation by finit lmnts in Notion of finit lmnt in Initializations of th élémnts Computation of th trms élémntairs Rsolution global Bibliographi Dscription of th vrsions of th documnt17...

3 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 3/18 1 Introduction th finit lmnt mthod is mployd in many scintific disciplins to solv partial drivativ quations. It maks it possibl to build a simpl approximation of th unknowns to transform ths continuous quations into a systm of quations of finishd siz, which on can schmatically writ in th following form: [ A]. {U }= {L } (1) whr {U } is th vctor of th unknowns, [ A] a matrix and { L } a vctor. Initially, on transforms th partial drivativ quations into an intgral formulation (or strong formulation of th problm), oftn this first intgral form is modifid (waknd) by mans of th formula of Grn (on obtains a wak formulation thn). Th approximat solution is sought lik linar combination of functions givn. Ths functions must b simpl but nough gnral to b abl wll to approach th solution. Thy must in particular mak it possibl to gnrat a spac of finishd siz which is as clos as on wants spac of functions in which th solution is. From this old ida (mthod of th balancd rsidus), th various ways of choosing ths functions caus various numrical mthods (collocation, mthods spctral, finit lmnts, tc). Th originality of th finit lmnt mthod is to tak as functions of approximation of th polynomials which ar null on almost all th fild, and thus tak part in computation only in th vicinity of a particular point. Thus, th matrix [ A] is vry hollow, containing only th trms of intraction btwn clos points, which rducs th computing tim and th cor mmory ncssary to storag. Morovr, th matrix [ A] and th vctor { L } can b built by assmbly of matrixs and lmntary vctors, calculatd locally. 2 Obtaining a variational formulation On can obtain th variational formulation of a problm starting from th partial drivativ quations, by multiplying thos by functions tsts and whil intgrating by parts. In mchanics of solids, th wak formulation thn obtaind is idntical to that givn by th Principl of th Virtual wors and in th consrvativ cas, th minimization of th total potntial nrgy of structur. Lt us not howvr that for crtain problms, th quations of th modl ar asir to stablish in th variational fram (cas of th plats and th shlls for xampl). 2.1 Modlization of th physical problm Principls and notation a physical systm is gnrally modlld by partial drivativ quations which act on unknowns u who can b: A scalar lik th tmpratur in th problms of thrmal; A vctor lik displacmnts in th problms of mchanics; A tnsor lik th strsss in th problms of mchanics; On can also us svral filds of unknowns simultanously, connctd by partial drivativ quations. Thy ar coupld problms. In, on can quot as xampl th problms of thrmo-hydro-mchanics which coupl displacmnts, prssur and tmpratur. Th filds of unknowns ar paramtrizd by: Th spac, which can b dscribd by a coordinat systm Cartsian or any othr typ of paramtrization. In th continuation of th documnt, on will not it x ; Tim, notd t ; 2.2 Equations of th systm a continuous physical systm can b rprsntd by a systm of quations with th partial drivativs which on will writ in th fild : L uf =0 dans (2 This systm is associatd with th boundary conditions on th bordr of th fild : Cu=h sur = (3)

4 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 4/18 th diffrntial oprator can xprss himslf on svral partial drivativ quations. On could writ: L 1 u f 1 =0 L 2 u f 2 =0 (4) L i u is a diffrntial oprator acting on th vctor of th unknowns u. In a mor gnral way, th diffrntial oprator L i u is writtn according to th unknowns and thir partial drivativs: L i u, u 2 u,,,, m u x 1 x 1. x 2 x,t, u m t,, p u t, p (5) Such an oprator is known as of ordr m in spac and ordr p in tim. If it dos not dpnd on tim (and its drivativs), it is said that th problm is stady. In th continuation of th documnt on will considr only th stady problms. 2.3 Mthod of th balancd rsidus strong intgral Formulation On will dfin th rsidu R u as bing th quantity canclling itslf whn u is th solution of th physical problm: R u=lu f=0 dans (6) th mthod of th balancd rsidus consists: 1 A to build a solution approachd u by th linar combination of judiciously slctd functions N u x= c i. i x (7) Whr i x ar th shap functions of th approximation and c i th cofficints to b idntifid. 2 A to solv th systm in intgral form: Trouvr u E u tl qu P E P Avc W = Ru.Pu.d [ Cu h ].P u. d =0 (8) W usd th sam wight functions for th principal systm and th limiting conditions, but it is not compulsory. P u ar th wight functions blonging to a st of functions E P. Th solution u blongs to th spac E u of th rgular functions sufficintly (diffrntiabl until th ordr m). Th choic of th wight functions P u maks it possibl to crat svral mthods: If th function P u is a distribution of Dirac, on obtains th collocation mthod by points. If th function P u is constant on subdomains, on obtains th collocation mthod by subdomains. If th wight functions P u us th sam shap functions i x as th approximation of th solution (7), on obtains th mthod of Galrkin. Th strong intgral form thus is obtaind. 2.4 Wak intgral formulation th intgral formulation (8) rquirs diffrntiabl spacs of function to th ordr m for E u. Th wak formulation consists in carrying out an intgration by parts (by application of th formula of Grn) of th systm (8). On th othr hand on incrass th rquirmnts for rgularity on th wight functions P u. Th formula of Grn is statd as follows: u.. P. d = whr n is th outgoing norm at th bordr of th fild. 3 Finit lmnt mthod P..u. d u. P. n. d (9)

5 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 5/ gnral Principls th sarch of a suitabl approximat function on all th fild bcoms difficult in th gnral cas of a gomtry of an unspcifid form. Th ida of th finit lmnt mthod is thus to build this approximation in two tims: To idntify subdomains gomtrically simpl which pav th fild; To dfin a function approachd on ach subdomain; A crtain numbr of charactristics of this construction ar thus had a prsntimnt of: Th paving of th fild by th subdomains must b as prcis as possibl; Th function approachd on th subdomain must obsrv conditions of continuity btwn th various subdomains; Th function approachd on th subdomain must hav cohrnt proprtis with th conditions of drivability and in kping with th physical dscription of th solution (what can imply to us a waknd formulation for xampl). 3.2 Approximation of th gomtry Principl On idntifis N th subdomains (or lmnts ) which pav th spac of solid: N = (10) =1 Lt us not x =1,3 th punctual coordinats x in th absolut coordinat systm. Th gomtry of th subdomain is built with a nodal approximation, that is to say for an lmnt with nods: x = x i. N i or x = x,i This paving (msh) is an opration bing abl to b complx, spcially in 3D. Thr xist gnral algorithms to nt. On uss triangls or quadrangls in 2D and ttrahdrons or hxahdrons in 3D (mor som lmnts bing usd as connctions). Th triangls and ttrahdrons what is calld giv fr mshs, th quadrangls and th hxahdrons form structurd mshs. Th fr mshs ar rlativly asy to build thanks to largly tstd tchniqus: clls of Voronoï building a triangulation of Dlaunay or mthods of propagation (mthods known as frontal), th structurd mshs ar much mor dlicat to gnrat. Th msh ncssarily inducd a gomtrical rror of discrtization For xampl, on th figur ( 1 ), on ss that a curvd bordr only is imprfctly approachd by linar lmnts.. N i (11) Illustration 1: Gomtrical rror of discrtization In th sam way th msh must b in conformity: no hols or of covring (s figur (2).

6 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 6/18 Covring of both maillstrou btwn th two maillsillustration 2: Nonconformity of th msh to obsrv this condition of conformity, it is nough to two ruls: 1.Each lmnt must b dfind in a singl way starting from th coordinats of its gomtrical nods (and not thos of its nighbors!); 2.Th bordr of an lmnt must b dfind in a singl way starting from th nods of ths bordrs, ths nods bing common btwn th lmnts dividing this bordr. Ths conditions of conformity ar an important diffrnc compard to finishd volums which do not hav ths rquirmnts. Th paving of th fild maks it possibl to apply th finit lmnt mthod to complx gomtris, contrary to th mthods by finit diffrncs. Th gomtrical paving of th fild inducs a first rror: it is not possibl, in th gnral cas, to rprsnt a ral gomtry by a msh by rgular polygons, in particular on th bordr of th fild Elmnts of rfrnc A bautiful msh is a good msh Th computation of th shap functions for an unspcifid lmnt can b rathr complicatd. This is why on oftn prfrs to bring back onslf to an lmnt known as of rfrnc, from which on can gnrat all th lmnts of th sam family by a gomtrical transformation. Th shap functions ar thn calculatd on this notd gnric lmnt r, and th transport of th quantitis on th ral lmnt is accomplishd thanks to th knowldg of th gomtrical transformation. (0,1) 2x1x2 (0,0) (1,0) 1 Illustration 3: Transition of th spac of rfrnc to ral spac th points of th lmnt of rfrnc will b dscribd in paramtric trms of coordinats =1,3. Th transformation must b bijctiv and transform th tops and sids of th lmnt of rfrnc into tops and sids of th ral lmnt: Intrpolation functions gomtrical x (12)

7 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 7/18 th gomtry of th lmnt is thus approximat by th mans of intrpolation functions gomtrical. Ths notd functions N ar dfind on th lmnt of rfrnc; thy mak it possibl to know th unspcifid x punctual coordinats of th ral lmnt from its coordinats of its antcdnt in th lmnt of i rfrnc and of th coordinats x of th nods (of local numbr I ) of th ral lmnt: x = x i. N i or x = x, i. N i (13) Jacobian matrix of th transformation th jacobinn of th transformation is th matrix of drivativs partial of th ral coordinats x compard to th coordinats in th lmnt of rfrnc: J = x (14) By taking account of th dfinition of th coordinats x according to th coordinats x,i of th nods, on obtains an quivalnt statmnt of th jacobian matrix: Whr N i J = i =1 N i. x,i (15) ar th trms of th matrix [ N ] T, of which th numbr of lins is th numbr of dirctions of spac, and th numbr of columns th numbr of nods of th lmnt. Lt us not that th matrix [ N ] T dpnds only on th dfinition of th lmnt of rfrnc and not of that of th ral lmnt. Th dtrminant of th jacobian matrix, usful in computations which will follow, is calld th jacobian of th gomtrical transformation. It is non-zro whn th transformation which maks pass from th lmnt of rfrnc to th ral lmnt is bijctiv, and positiv whn rspcts th dirctional sns of spac. 3.3 Rprsntation of th unknowns J =dt [ N 0 (16) ] to solv th problm, on considrs an approximation by finit lmnts of an unknown fild. Spacs E P and E u ar rprsntd by spacs discrt E h. Thr ar two quivalnt ways to rprsnt th unknowns in an lmnt: by th cofficints of thir polynomial approximation, or by thir nodal valus. Ths two possibilitis corrspond to th two ways complmntary to dfin an lmnt: by th data of a bas of studnts' rag procssions, or by th data of th shap functions associatd with th nods. In a gnral way, on builds th function approachd by writing th following linar rlation on ach lmnt: u = a i. i (17) Whr ar i to thm indpndnt linar functions. Thy constitut th bas of th approximation, th gnral paramtrs of th approximation bing th cofficints a i Nodal approximation th first ida of th finit lmnt mthod is to build approximation of a nodal typ for which th cofficints u i =a i corrspond to th solution in ths nods:

8 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 8/18 u = u i. N i (18) On thn obtains a nodal approximation with N i th intrpolation functions on th lmnt of rfrnc. On ach on of ths subdomains on builds an approximat function diffrnt from on subdomain to anothr. Th approximation finit lmnts is lmntary bcaus th function dpnds only on th nodal valus constituting th lmnt: u x = u i. N i x (19) an lmnt is isoparamtric whn it is basd on idntical intrpolations for its gomtry and its unknowns: N =N. To nsur th continuity of th solution on th lmnt and, possibly, th continuity of its drivativs, on nds that th functions N i ar continuous and, possibly, with continuous drivativs. In th sam way if on wants to nsur th continuity of th solution and of its drivativs at th bordrs of th lmnts (conformity of th approximation), it is ncssary that th solution and its drivativs dpnd in a singl way of th nodal variabls on th nods of th bordr Bas polynomial th way simplst to dfin an lmnt is to choos a polynomial bas mad up of a crtain numbr of indpndnt studnts' rag procssions. For a givn unknown, th numbr of studnts' rag procssions usd must b qual to th numbr of nodal variabls, i.. with th numbr of nods usd to rprsnt th unknown. On gnrally dfins th polynomial bas on th lmnt of rfrnc; it contains studnts' rag procssions of th form , whr, and ar positiv or null whol xhibitors. Th dgr of such a studnts' rag procssion is th intgr. Th bas is known as complt of dgr n whn all th studnts' rag procssions of dgr n ar prsnt. In crtain cass, incomplt bass ar mployd. On nots P p p ièm th studnts' rag procssion of th bas (which undrstands som m ). Th componnts of th vctor displacmnt u in th lmnt ar thn givn by th formula: m u = a, p.p p (20) p =1 On will not th matrix giving th valus takn by th studnts' rag procssions of th polynomial bas on th nods of th lmnt of rfrnc: Ip =P p I (21) whr p is th squnc numbr of th studnts' rag procssion in th bas, I numbr of th nod locally to th lmnt and th I coordinatd of th nod I in th lmnt of rfrnc. This matrix is squar, its dimnsion is th squar amongst nods of th lmnt. I With th nod I displacmnt u is worth: u I, =a, p. Ip (22) On distinguishs thr larg lmnt typs finishd frquntly usd: finit lmnts of Lagrang which rst on bass polynomial complt and diffrnt standard from gomtris (symplctic for th triangls and th ttrahdrons, with tnsorial structur for th quadrangls and th hxahdrons or of prismatic typ); th finit lmnts of th Srndip typ, which ar of th finit lmnts of Lagrang with incomplt bass; th finit lmnts of Hrmit, of utmost prcision, which us th nodal unknowns and thir drivativs; Finit lmnts of Lagrang symplctic to dtrmin if a polynomial bas is complt with th lmnts symplctic, it is nough to us th triangl of Pascal: Linar

9 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 9/18 Constant 1 ordr 1 2 Quadratic formula Ordr a complt polynomial bas of ordr two compriss six studnts' rag procssions: {1; 1 ; 2 ; 1 2 ; 2 2 ; 1. 2 } and thus th gomtric standard support will b a triangl with six nods. Finit lmnts of Lagrang with tnsorial structur to dscrib of th finit lmnts quadrangular (or hxahdral), it is nough to tak complt polynomials of th ordr givn and to mak th product of it. Quadratic Constant ordr Linar Cubic Constant Linar Quadratic Cubic a polynomial bas on - complt of ordr two for a quadrangular lmnt compriss nin studnts' rag procssions: {1; 1 ; 1 2 ; 2 ; 2 2 ; 1. 2 ; ; ; }, which mans nin nods. Such an lmnt compriss trms of ordr 3 and 4. Finit lmnts of Srndip th lmnts of Srndip, for a polynomial of ordr s, xclud th cross trms from dgr highr than s1 not to hav nods insid th lmnts. For xampl, for an lmnt of Srndip of ordr two, th studnts' rag procssions will b {1; 1 ; 1 2 ; 2 ; 2 2 ; 1. 2 ; ; }, that is to say ight nods Shap functions an quivalnt way to dfin a finit lmnt is to giv, for ach unknown, th statmnt of th shap functions of th lmnt. For a givn scalar unknown (componnt of displacmnt according to thr for xampl), thr is as much as nods whr th unknown must b calculatd. In much of cas, on uss th sam shap functions for all th componnts of an unknown vctor, but it is not compulsory. In what follows, it will b supposd howvr to simplify th writings that it is th cas. Th shap functions can b dfind on th ral lmnt : thy thn ar notd N x, thy dpnd on th gomtry of th ral lmnt, and ar thus diffrnt from on lmnt to anothr. It is simplr to xprss thm on th lmnt of rfrnc, which givs th functions N indpndnt of th gomtry of th ral lmnt. Lt us rcall that ths functions ar polynomial on th lmnt, and that th shap function associatd with a nod givn thr taks th valu on, whras it is canclld in all th othr nods of th lmnt. Th unknowns ar dscribd thn lik linar combination of th shap functions, th cofficints u,i of th combination bing calld th nodal variabls: u = u. N, i i (23) By mans of th transformation btwn th lmnt of rfrnc and th ral lmnt: Thr a: x (24)

10 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 10/18 u = u. N, i i -1 x (25) Corrspondnc btwn polynomial bas and shap functions On ar two rlations. Th first coms from th approximation of th solution by a polynomial bas: th scond is th nodal approximation: m u = a, p.p p (26) p =1 u = u. N, (27) i i th matrix giving th valus takn by th studnts' rag procssions of th polynomial bas on th nods of th lmnt of rfrnc: Ip =P p I (28) In a nod I, on wrot th following polynomial approximation; By injcting th quation (29) in th nodal statmnt (27), on obtains: u I, =a, p. Ip (29) u = a..n, p Ip i (30) By comparison with th polynomial approximation (26), on from of dducd th following rlation btwn th polynomial bas and th shap functions: Ip. N i =P p (31) In practic, on will find in th litratur th writings of th nodal shap functions for th most currnt lmnts, according to th choic of th polynomial bas. 3.4 Rsults of xistnc and unicity On can writ th problm in a mor abstract way: Trouvr u E u tl qu v E v (32) au, v= f v E u and E v ar vctor spacs of functions dfind on. Thy ar spacs of Hilbrt. au,v is a bilinar form on E u E v (it was supposd that L u rprsnts a linar physical problm compard to u ). f v is a linar and continuous form on E v. To stablish th conditions of xistnc and unicity, on applis th thorm of Lax-Milgram. Initially, it is supposd that th solution blongs to th sam spac as th functions tst E u = E v If th form au,v is corciv i..: Thn problm: u E u au,u c. u Eu 2 avc c0 (33) admits on and only on solution. Trouvr u E u tl qu v E u a u, v= f v (34) 4 Mthod of Ritz

11 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 11/18 th mthod of Galrkin, in crtain cass, is quivalnt making stady a functional calculus. It is th cas if th bilinar form au,v is symmtric and positiv: u, v E u au, v=a v,u t au,u 0 (35) In this cas th problm (34) admits on and only on solution u minimizs on E u th following functional calculus: u= 1. a u, u f u (36) 2 From th mchanical point of viw, that mans that th principl of virtual powr can b also writtn lik th minimization of a scalar quantity: th total nrgy of structur. This way writ th quilibrium is vry frquntly mployd. W hr will hav som rsults of thm. W point out initially that a functional calculus is a function of a st of functions (and of its drivativs). This functional calculus will b writtn. On will limit onslf to th formulations in displacmnt, knowing that thr is th diffrnt on. In this cas, th functional calculus will b writtn: u= u, u x (37) For th consrvativ problms, on can show that to writ that th first variation of is null (condition of stationarity of th functional calculus) is quivalnt applying th principl of th virtual wors, or to us th mthod of Galrkin by taking virtual displacmnts lik wight function. On calls that th mthod of Galrkin consists starting from th problm with drivativs partial stablishing th quilibrium of structur, that is to say: L u f =0 dans avc. n= g sur N t u=u D sur D (38) On thn sks to solv th problm in intgral form by mans of wight functions which ar of th sam natur as th approximat solution: W= [ L u f ]. u.d =0 (39) Avc.n= g sur N t u=u D sur D If on chooss lik wight function th variation of th unknowns =u and aftr having intgratd by parts onc, on obtains: u=w u=0 avc u=u D sur D (40) To find th form xact of th functional calculus is not immdiat in th gnral cas. In mchanics, for th consrvativ cass, it is that this functional calculus is quivalnt to th total potntial nrgy of th systm. Aftr discrtization of th functional calculus (by an approximation finit lmnts), on finds onslf with a matric systm strictly quivalnt to that of th mthod of Galrkin (or its mchanical principl ar quivalnt, th mthod of th virtual powrs). Intuitivly, it is undrstood that a wak variation u of th solution is a fild which can b kinmatically admissibl and which thus corrsponds wll to th assumptions of th mthod of th virtual powrs. 5 Construction of th matric systm W now will prsnt th various ingrdints lading to th construction of th matric systm which will mak it possibl to solv th problm. 5.1 Nw notation (notation of Voigt) In ordr to undrstand wll th construction of th discrt trms in th finit lmnt mthod, w will us a mor compact notation: V is a vctor lin

12 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 12/18 {V } is a vctor column [ A ] is a matrix Thus th gomtrical intrpolation is writtn according to thr dimnsions of spac: x = x 1 = x 1,i. {N i }= N i. { x 1,i } } } y = y 1 = y 1,i.{N i }= N i. { y 1,i z = z 1 = z 1,i. { N i }= N i. {z 1,i Or in a mor compact way in vctorial form: { x }=[ N i ]. {x i }= x i. [ N i ] T (42) With th matrix N of th shap functions. By considring an lmnt with two nods, on obtains in dvlopd form: {x }=[ y N N } y z 1 0 N N 2 0 (43) ].{x1 x z 0 0 N N 2 2 y 2 z 2 (41) 5.2 discrtizd Systm On is placd in th cas hypr lastic in small strains, th problm of mchanics to b solvd on writs in a mor compact way: To find u E h such as u E ĥ with au, ul u=0 With au, u a bilinar, symmtric form which rprsnts th potntial nrgy of structur and l u th potntial 1 voluminal and surfac forcs: au, u= l u= h h u : u. d h f. u.d h N h th discrtization consists in choosing a bas of spac h (44) g. u.d h (45) and with calculating th trms of th matrix numrically A and vctor L. For that, on xprsss th bilinar form a.,. and th linar form l. lik a sum on lmnts, dfind by basic fild division:, u j = {aui l u i = élémnts kl u i. kl u j.d élémnts f i. u i.d N g i. u i.d N th trms A ij, which rprsnt th intraction btwn two dgrs of frdom i and j ar built by assmbling (th notd opration... ) th contributions coming from ach lmnt which contains th élémnts corrsponding nods; on procds in th sam way to build th scond mmbr vctor L i. Ths (46) 1 potntil1l of th xtrnal forcs dos not dpnd on th displacmnt of structur, it is what is calld a dad loading or NON-followr. In th cass of th larg dformations, th loadings of typ prssur cannot rspct this assumption.

13 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 13/18 contributions, calld lmntary trms, ar calculatd during a loop on th lmnts and dpnd only on th only variabls of th lmnt : {a l = = kl. kl.d f i. w i. d N g i.w i. d N th rlation btwn th tnsor of th strsss of Cauchy and th displacmnts u is givn by th bhavior modl, and is indpndnt of th writing of th variational formulation. In th lastic cas, on a: ij w i = ijkl. ku w i (48) ijkl is th lasticity tnsor of Hook. This tnsorial form is not vry practical, on prfrntially uss th notation of Voigt, which maks it possibl to writ: : =. {} (49) In Cartsian coordinats, on a: = xx yy zz xy xz yz (50) And a form modifid of th componnts of strain to allow to xprss th contractd product, ar: = xx yy zz 2. xy 2. xz 2. yz (51) (47) important Rmark: In th intgration of th constitutiv laws, th componnts of shars of th strsss and strains usd by ar: = xx yy zz 2. xy 2. xz 2. yz = xx yy zz 2. xy 2. xz 2. yz th product of ths two vctors givs th sam on wll rsult as th doubl contractd product ( 49 ). With this nw notation, w hav in lasticity: { }=[ A]. {} (52) W st out again of th writing EF of th fild of displacmnts: {u }=[ N i ]. {u i }= u i.[ N i ] T (53) And, of similar way, th fild of virtual displacmnts: { u }=[ N i ]. {u i }= u i.[ N i ] T (54) By proccupation with a simplification of th notations, on will omit th rfrnc to th lmnt. It is ncssary of xprimr th tnsor of th strains (virtual or ral): { }= [B ]. {u}= u.[b ] T and { }= [B ]. { u}= u.[b ] T (55) On obtains thn for th matrix rlating to th bilinar form: [a] = u. [ B ] T.[ ]. [B ].d. {u} (56) th matrixs [ B ] and [] contain th possibl non-linarity of th bhavior and will dpnd on displacmnts: 1. [ B ] is a function of displacmnts if on is in th situation of th larg dformations or th grat transformations (larg rotations and/or larg displacmnts). 2. [] is th matrix of bhavior. It bcoms dpndant on displacmnts in th cas of (and othr variabls) th nonlinar and/or inlastic bhaviors.

14 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 14/18 In ths two cass, th procss of rsolution of th quations will imply a spcific procssing (gnrally, a linarization of th Nwton-Raphson typ). In a similar way, on will asily obtain th lmntary form for th scond mmbr. 5.3 Computation of th lmntary trms th lmntary trms with calculating ar form: u x f ux,.d x (57) x Thr typs of oprations ar to b carrid out: 1.th transformation of drivativs compard to x in drivd compard to ; 2.th transition of an intgration on th ral lmnt with an intgration on th lmnt of rfrnc, 3.th numrical ralization of this intgration which is gnrally mad by a formula of squaring Transformation of drivativs th transformation of drivativs is carrid out thanks to th jacobian matrix J, according to th drivativ rul in charactr string: whr u nod u x = x. u = J -1.[ N T nod.u ] is th vctor of th nodal valus of th componnt of displacmnt Chang of fild of intgration th transition to intgration on th lmnt of rfrnc is carrid out by multiplying th intégrand by th dtrminant of th jacobian matrix, calld jacobian: u x u f ux, f x u,. dt J. d (59). d x= r th transition of th lmnt of rfrnc to th ral lmnt implis th bijctivity of th transformation. It is thus ncssary dt J 0, which implis that th lmnt should not b turnd ovr or dgnrat (for xampl it is not ncssary that th quadrangl dgnrats into triangl) Numrical intgration In crtain typical cass, on can calculat th intgrals analytically. For xampl, for a triangl in two dimnsions, th Jacobian ar constant on th triangl, and th intégrands ar brought back to studnts' rag procssions which on can intgrat xactly thanks to th formula of numrical intgration known as of Gauss 2 : !! 2. d 1. d 2 = 2! Howvr, ths typical cass ar rar, and on prfrs to valuat th intgrals numrically by calling on formulas of squaring. Thos giv an approximation of th intgral in th form of a balancd sum of th valus of th intégrand in a crtain numbr of points of th lmnt calld points of intgration: r r (58) (60) g. d g.g g (61) g=1 2 abus languag, on frquntly calls th numrical diagrams of intgration diagrams of Gauss although thr ar svral kinds (Hammr for th triangl, Gauss-Radau, Nwton-Dimnsions, tc).

15 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 15/18 th scalars g ar calld th wights of intgration, and th coordinatd g ar th coordinats of r th points of intgration in th lmnt of rfrnc. In th intgration mthods of Gauss, th points and wights of intgration ar givn so as to intgrat xactly polynomials of a natur givn. It is this kind of mthod which on uss in, th points of intgration ar calld thn Gauss points. Th numbr of Gauss points slctd maks it possibl to intgrat xactly in th lmnt of rfrnc. In fact, bcaus of th possibl non-linarity of th gomtrical transformation or th spatial dpndnc of th cofficints (for xampl for lmnts dformd or of scond ordr), intgration is not xact in th ral lmnt. For ach lmnt, on knw to calculat th trms known as lmntary: lmntary matrix A and lmntary vctor L. Th matrix A and th vctor L ar obtaind by a procdur that on calls th assmbly of th lmntary trms. If on rgains th lmntary shap of stiffnss: [a]= { }..d (62) numrical intgration implis that on valuats th strsss and th strains at th points of intgration: [a]= {}.. d g=1 r g.{ g }. g (63) What mans that th strsss and th strains ar most xact (or th last fals) at th points of intgration (filds known as ELGA in ). Th simpl fact of xtrapolating ths valus with th nods for th display introducs an rror. It is bsids about a mthod valuation of th rror, calld rror indicator of Zhu- Zinkiwicz. In lasticity 2D, a triangl xhibant a jacobian constant, only on Gauss point is sufficint to intgrat xactly th trms of th matrix and th scond mmbr (if it is constant). Th cost computation incrass with th numbr of points of intgration, particularly for th nonlinar constitutiv laws. For xampl, a hxahdron with 27 nods nds 27 Gauss points to intgrat th quantitis. It thus arrivs frquntly that on undr-just, i.. that on uss lss points of intgration than th rquird minimum, thus making a mistak that on will possibly compnsat by a finr msh. Bsids this systmatic rror, this undr-intgration must b mad with prcaution bcaus it can produc s of row of th matrix and thus mak th systm linar noninvrtibl. 6 Rsolution of th matric systm On thus obtains a linar systm to solv: u.[ A]. {u} u. {L }=0 (64) Whatvr th fild of virtual displacmnts, thrfor: [ A]. {u }= {L } (65) 6.1 Imposition of th kinmatical boundary conditions th procssing of th kinmatical boundary conditions of th typ u=u D is don in two diffrnt ways: 1.Th kinmatical mthod (AFFE_CHAR_CINE in ) consists in modifying th matrix and th scond mmbr. This mthod is fast and dos not introduc additional variabls. On th othr hand, it is not gnral and dos not allow to apply complx limiting conditions of th styl u i. a i =u D. 2.Th mthod by dualisation (AFFE_CHAR_MECA in ) consists in introducing a vctor of multiplirs (or paramtrs) of Lagrang, which incrass th numbr of unknowns but maks it possibl to trat all th cass. {[ A]. {u}[q] T { }={L } [Q]. {u}= {u D (66) }

16 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 16/ Rsolution th linar systm can b solvd by a crtain numbr of numrical mthods. Th mthods usd in ar factorization LDL T pr blocks, th mulitfrontal mthod (or its quivalnt with swivlling, MUMPS), and th prconditiond conjugat gradint. Th mthods of rsolution ar dividd into thr catgoris: Th dirct mthods which solv xactly (with th numrical rrors nar) th itrativ mthods th hybrid mthods, vry much usd in th mthods of dcomposition of filds. (s mthod FETI [R ]). Th matrixs rsulting from th finit lmnt mthod ar vry hollow (thy compris a majority of null trms). In practic, on systms of standard siz (a fw tns of thousands of quations), th dnsity of non-zro trms sldom xcds th 0.01%. Thy ar thus stord in form digs (or spars ) and tak littl cor in mmory. A contrario, th matrixs ar not built to b usd ffctivly with th mathmatical libraris of programs optimizd ddicatd to th full matrixs (booksllrs BLAS for xampl). Solvrs ar thus dvlopd spcifically for ths problms. A dirct solvr has as a principl braking up th matrix into a product of particular matrixs of form. For xampl, dcomposition LDL T : [ A]=[ L].[ D].[L ] T (67) Whr th matrix D is diagonal and th matrix L is triangular lowr. This dcomposition is valid only for th symmtric matrixs. If it is not th cas, othr dcompositions should b usd. Th principl is th following: From th initial matrix (vry hollow), on builds a product of rmarkabl matrixs. It is th opration of factorization. Ths rmarkabl matrixs mak it possibl to solv th vry fast problm of way. It is th phas of dscntincras. Th phas of factorization is most xpnsiv. For th most sprad dcompositions, th cost machin is in n 3 whr n is th numbr of quations. Th cost rport will dpnd on th profil of th matrix (of classification of th finit lmnts). Automatic procsss sk to optimiz this classification to hav a structur as compact as possibl. Evn with this optimization, it is frqunt that th factorizd matrix tak svral hundrds of tims, vn svral thousands of tims mor mmory than th initial matrix. Th dirct solvrs thus consum much mmory and that bcoms crippling about it from svral hundrds of thousands of dgrs of frdom, vn on th most powrful machins. On th othr hand, ths dirct mthods ar particularly robust. Th problms in structural mchanics and of solids vry oftn lad to matrixs with a bad conditioning (it is particularly th cas of all th last numrical innovations which us mixd mthods with many Lagrang multiplirs). Whn it is possibl, itrativ mthods whos principl consists in finding an approximation of th rvrs of th matrix and to procd thn to an itrativ rsolution, not by stp, which uss only products matrix-vctors, vry ffctiv and inxpnsiv in mmory ar prfrntially usd. Howvr, ths itrativ mthods hav svral s: Thy ar lss robust than dirct mthods, particularly whn conditioning is bad th mthods of prconditionning ar vry numrous and thr ar som as much as diffrnt problms (vn svral possibl by problm). What obligs th usr to juggl with th various mthods, without nvr bing assurd to obtain rsult at th nd. Thy ar itrativ mthods, which implis stopping critria of th procss, and thus a paramtr to b managd but also problms of offic plurality of round-offs. Finally th hybrid mthods try to rconcil th advantags of th two approachs. Gnrally, on uss thm in th mthods of dcomposition of filds, whr ach fild is tratd by a dirct solvr whil th problm of intrfac is solvd by an itrativ solvr. On can quot th mthod FETI (and its altrnativs, to s [R ]) or th variations of th mthod LATIN. It is about an xtrmly activ fild of sarch. 7 Organization of a computation by finit lmnts in th On vry brifly dscribs how and at which plac th aspcts vokd in this documnt ar stablishd in.

17 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 17/ Notion of finit lmnt in A kind of finit lmnt is dfind by: a kind of msh nods list of th shap functions of th computation options an lmnt in th msh is dfind by a kind of msh, a gomtry (coordinatd nods) and a topology (ordrd list of th nods). It is th typ of modlization chosn in th command fil which maks it possibl to assign to ach msh of th msh a kind of finit lmnt. Th command AFFE_MODELE [U ] assigns to ach msh a kind of finit lmnt corrsponding to th modlization spcifid for this msh. Notic important: On should not forgt to assign of th finit lmnts to mshs dg which on nds to impos th boundary conditions and loadings, and that on will hav takn car to crat during th fabrication of th msh. Oprator AFFE_CHAR_MECA [U ], which affcts boundary conditions and loadings, also will crat of th finit lmnts, for xampl th finit lmnts which will carry th dgrs of frdom of LAGRANGE usd in th dualisation of th boundary conditions [R ]. Oprator AFFE_CARA_ELEM [U ] allows to dfin additional charactristics for som lmnt typs: for xampl, th thicknss of th shlls, dirctional sns of th bams, mass matrixs and of stiffnss of th discrt lmnts. A computation option indicats th lmntary typ of computation that th lmnt is abl to calculat. For xampl RIGI_MECA rlats to th computation of th lmntary matrix of mchanical stiffnss: A = ijkl. ij N x. kl N x.d (68) th data of this option ar th gomtry and th matrial, supplmntd by th tmpratur if th matrial dpnds on it. Lt us rcall that to apply th loadings of bordr, on uss dg individuals of th finit lmnts, and not th bordrs of th finit lmnts of volum (3D) or surfac (2D). Not: A dvlopr can somtims hav th choic btwn crating a nw finit lmnt or adding a computation option to an xisting lmnt; th choic btwn ths two solutions in gnral taks account of critria of data-procssing facility (.g. lmnts undr - intgratd). 7.2 Initializations of th lmnts th us of lmnts of rfrnc maks it possibl onc and for all to carry out a crtain numbr of computations at th bginning of th xcution. On dfins, for ach typ of lmnt of rfrnc: Th numbr of nods and thir coordinats; Th numbr of familis of Gauss points; Th numbr of Gauss points; Wights of intgration g ; Valus of th shap functions to Gauss points N i g ; Valus of drivativs of th shap functions to Gauss points N i g. For a givn lmnt, on invitably dos not intgrat all th lmntary trms with th sam numbr of Gauss points: for xampl, on uss in gnral of Gauss points for th mass matrix than for th stiffnss matrix, bcaus th products of shap functions ar of dgr highr than th products of thir drivativs. Anothr xampl is th undr-intgration usd in crtain cass. On calls of Gauss points family ach whol of Gauss points likly to b usd.

18 Titr : La méthod ds élémnts finis isoparamétriqus Dat : 10/01/2011 Pag : 18/ Computation of th lmntary trms During th computation of th lmntary trms (in th routins YOU.), on carris out for ach Gauss point th following oprations: Computation of drivativs of th shap functions on th ral lmnt starting from th coordinats of th nods of th lmnt and drivativs of th shap functions on th lmnt of rfrnc; Computation of th jacobian matrix; Rcovry of th wight of intgration multiplid by th Jacobian at th Gauss point considrd; Evaluating of th intégrand (according to th calculatd option). Th lmntary trm is calculatd by sum on Gauss points whil balancing by th wights of intgration. 7.4 Total rsolution th total rsolution taks plac in routins OP. high lvl corrsponding to th commands usr (MECA_STATIQUE [U ], STAT_NON_LINE [U ], THER_LINEAIRE [U ], tc). 8 Bibliography 1) P.G. Ciarlt, Th finit lmnt mthod for lliptic problms, Studis in Applid Mathmatics, North Holland, ) A. Ern, J. - L. Gurmond, Finit lmnts: thory, applications, put in work, Springr, ) G. Dhatt, G. Touzot, E. Lfrançois, Finit lmnt mthod: a prsntation, Hrms, Dscription of th vrsions of th documnt Astr Author (S) Organization (S) Dscription of th modifications 3 I.VAUTIER initial Txt 10.2 M.Abbas partial Rwriting, anonymization of th concpts compard to th mchanical