Code_Aster. Finite element method isoparametric

Transcription

1 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 1/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : Finit lmnt mthod isoparamtric Summary: This documnt prsnts th bass of th isoparamtric finit lmnts introducd into Cod_Astr for th modling of th continuous mdiums 2D and 3D. On first of all rcalls th passag of a strong formulation to a variational formulation, thn on dtails th discrtization by finit lmnts: us of an lmnt of rfrnc, calculation of th functions of form and valuation 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 principal stags of a calculation by finit lmnts such as it ar concivd and stablishd in Cod_Astr. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

2 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 2/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : Contnts 1 Introduction Obtaining a variational formulation Modling of th physical problm Principls and notation Equations of th systm Mthod of th balancd rsidus strong intgral Formulation Wak intgral formulation Finit lmnt mthod Principls gnrals Approximation of th gomtry Principl Elmnts of rfrnc Functions of gomtrical intrpolation Matrix jacobinn of th transformation Rprsntation of th unknown factors Nodal approximation Bas polynomial Functions of form Corrspondnc btwn polynomial bas and functions of form Rsults of xistnc and unicity Mthod of Ritz Construction of th matric systm Nw notation (notation of Voigt) Discrtizd systm Calculation of th lmntary trms Transformation of th drivativ Chang of fild of intgration Digital intgration Rsolution of th matric systm Imposition of th boundary conditions kinmatics Rsolution Organization of a calculation by finit lmnts in Cod_Astr Concpt of finit lmnt in Cod_Astr Initializations of th lmnts Calculation of th lmntary trms Total rsolution Bibliography Dscription of th vrsions of th documnt...19 Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

3 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 3/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : 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 unknown factors 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 unknown factors, [ A] a matrix and { L } a vctor. Initially, on transforms th partial drivativ quations into an intgral formulation (or formulation strong problm), oftn this first intgral form is modifid (waknd) by using th formula of Grn (on obtains a formulation thn wak). 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 digital 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 worthlss on almost all th fild, and thus tak part in calculation 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 plac mmory ncssary to storag. Morovr, th matrix [ A] and th vctor { L } can b built by assmbly of matrics 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 th solids, th wak formulation thn obtaind is idntical to that givn by th Principl of Virtual Work and in th consrvativ cas, th minimization of th total potntial nrgy of th structur. Lt us not howvr that for crtain problms, th quations of th modl ar asir to stablish within th variational framwork (cas of th plats and th hulls for xampl). 2.1 Modling of th physical problm Principls and notation A physical systm is gnrally modlld by partial drivativ quations which act on unknown factors u who can b: A scalar lik th tmpratur in th problms of thrmics; A vctor lik displacmnts in th problms of mchanics; A tnsor lik th constraints in th problms of mchanics; On can also us svral filds of unknown factors simultanously, connctd by partial drivativ quations. Thy ar problms coupld. In Cod_Astr, on can quot as xampl th problms of thrmo-hydromchanics which coupl displacmnts, prssur and tmpratur. Th filds of unknown factors ar paramtrizd by: Th spac, which can b dscribd by a Cartsian fram of rfrnc or any othr typ of paramtrization. In th continuation of th documnt, on it will not x ; Tim, notd t ; 2.2 Equations of th systm A continuous physical systm can b rprsntd by on systm partial drivativ quations which on will writ in th fild : L uf =0 dans (2) This systm is associatd with th boundary conditions on th bordr fild : Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

4 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 4/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : Cu=h sur = (3) 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 unknown factors u. In a mor gnral way, th diffrntial oprator L i u is writtn according to th partial unknown factors and of thir drivativ: 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 of ordr p in tim. If it dos not dpnd on tim (and its drivativ), it is said that th problm is stationary. In th continuation of th documnt on will considr only th stationary problms. 2.3 Mthod of th balancd rsidus strong intgral Formulation On will dfin it rsidu R u as bing quantity canclling itslf whn u is th solution of th physical problm: R u=lu f=0 dans (6) mthod of th balancd rsidus consists: 1/ To build an approximat solution u by th linar combination of judiciously slctd functions N u x= c i. i x (7) Whr i x ar th functions of form of th approximation and c i cofficints to b idntifid. 2/ 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 obligatory. P u ar th wight functions blonging to a st of functions E P. Th solution u blongs to spac E u rgular functions sufficintly (drivabl until th ordr m). Th choic of th wight functions P u allows to crat svral mthods: If th function P u is a distribution of Dirac, on obtains th mthod of collocation by points. If th function P u is constant on undr-filds, on obtains th mthod of collocation by undrfilds. If wight functions P u us th sam functions of form i x that th approximation of th solution (7), on obtains th mthod of Galrkin. On obtains thus strong intgral form. 2.4 Wak intgral formulation Th intgral formulation (8) rquirs drivabl 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. Lformula of Grn has is statd as follows: u.. P. d = whr n is th outgoing normal at th bordr fild. P..u. d u. P. n. d (9) Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

5 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 5/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : 3 Finit lmnt mthod 3.1 Principls gnrals Th sarch for a suitabl approximat function on all th fild bcoms difficult in th cas gnral 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 undr-filds gomtrically simpl which pavs th fild; To dfin a function approachd on ach undr-fild; A crtain numbr of charactristics of this construction ar thus had a prsntimnt of: Th paving of th fild by th undr-filds must b as prcis as possibl; Th function approachd on th undr-fild must obsrv conditions of continuity btwn th various undr-filds; Th function approachd on th undr-fild 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 Thy ar idntifid N undr-filds (or lmnts ) who pav spac solid: N = (10) =1 Lt us not x =1,3 punctual coordinats x in th absolut rfrnc mark. Th gomtry of th undr-fild is built with a nodal approximation, that is to say for on lmnt with nods: x = x i. N i or x = x,i This paving (grid) is an opration bing abl to b complx, spcially in 3D. Thr xist algorithms gnrals to nt. Triangls or quadrangls in 2D and ttrahdrons or hxahdrons in 3D ar usd (mor som lmnts bing usd as connctions). Th triangls and ttrahdrons giv grids what is calld fr, th quadrangls and th hxahdrons form grids rgulatd. Th fr grids 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 rgulatd grids ar much mor dlicat to gnrat. Th grid inducs ncssarily a gomtrical rror of discrtization For xampl, on th figur ( 1 ), it is sn that a curvd bordr only is imprfctly approachd by linar lmnts.. N i (11) Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

6 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 6/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : Illustration 1: Gomtrical rror of discrtization In th sam way th grid must b in conformity: no hols or of covring (s figur (2)). Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

7 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 7/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : Covring of th two mshs Hol ntrs th two mshs Illustration 2: Nonconformity of th grid 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 finishd diffrncs. Th gomtrical paving of th fild inducs a first rror: it is not possibl, in th cas gnral, to rprsnt a ral gomtry by a grid by rgular polygons, in particular on th bordr of th fild Elmnts of rfrnc A bautiful grid is a good grid Th calculation of th functions of form 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 functions of form ar thn calculatd on this notd gnric lmnt r, and th transport of th sizs on th ral lmnt is carrid out thanks to th knowldg of th gomtrical transformation. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

8 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 8/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : 2 X 2 (0.1) (0.0) (1.0) 1 Illustration 3: Passag 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: Functions of gomtrical intrpolation x (12) Th gomtry of th lmnt is thus approximat by th mans of functions of gomtrical intrpolation. Ths notd functions N ar dfind on th lmnt of rfrnc; thy mak it possibl to know th coordinats x of an unspcifid point of th ral lmnt starting from its coordinats of its antcdnt in th i lmnt of rfrnc and th coordinats x nods (of local numbr I ) ral lmnt: X 1 x = x i. N i or x = x, i. N i (13) Matrix jacobinn of th transformation Th jacobinn of th transformation is th matrix of th drivativ 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 nods, on obtains an quivalnt xprssion of th matrix jacobinn: 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. L E dtrmining of th matrix jacobinn, usful in calculations which will follow, is calld th jacobin of th gomtrical transformation. It is nonnull whn th transformation who maks pass from th lmnt of rfrnc to th ral lmnt is bijctiv, and positiv whn rspct th orintation of spac. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

9 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 9/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : 3.3 Rprsntation of th unknown factors 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 unknown factors in an lmnt: by th cofficints of thir polynomial approximation, or by thir nodal valus. Ths two possibilitis corrspond to th two mannrs complmntary to dfin an lmnt: by th data of a bas of studnts' rag procssions, or by th data of th functions of form 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 thm i ar indpndnt linar functions. Thy constitut bas approximation, th paramtrs gnrals of th approximation bing cofficints a i Nodal approximation Th first ida of th finit lmnt mthod is to build approximation of a nodal typ for which cofficints u i =a i corrspond to th solution in ths nods: u = u i. N i (18) On obtains a nodal approximation thn with N i functions of intrpolation on th lmnt of rfrnc. On ach on of ths undr-filds on builds an approximat function diffrnt from on undr-fild 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 unknown factors: N =N. To nsur th continuity of th solution on th lmnt and, possibly, th continuity of its drivativ, it is ncssary that th functions N i ar continuous and, possibly, with drivativ continuous. In th sam way if on wants to nsur th continuity of th solution and of its drivativ at th bordrs of th lmnts (conformity of th approximation), it is ncssary that th solution and its drivativ 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 factor, 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 factor. On gnrally dfins th polynomial bas on th lmnt of rfrnc; it contains studnts' rag procssions of th form , whr, and ar positiv or worthlss whol xhibitors. Th dgr of such a studnts' rag procssion is th ntirty. Th bas is known as complt of dgr n whn all studnts' rag procssions of dgr n ar prsnt. In crtain cass, incomplt bass ar mployd. On nots P p p ièm studnts' rag procssion of th bas (which undrstands som m ). Componnts of th vctor displacmnt u in th lmnt ar thn givn by th formula: Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

10 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 10/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : 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 th numbr of th nod locally to th lmnt and I coordinats 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 grat typs of finit lmnts frquntly usd: th 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 Srndip typ, which ar finit lmnts of Lagrang with incomplt bass; th finit lmnts of Hrmit, of utmost prcision, which us th nodal unknown factors and thir drivativ; 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: Ordr Constant 1 Linar 1 2 Quadratic Cubic 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 finit lmnts quadrangular (or hxahdral), it is nough to tak complt polynomials of th ordr givn and to mak th product of it. Ordr Constant Linar Quadratic 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 Elmnts 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. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

11 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 11/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : Functions of form An quivalnt way to dfin a finit lmnt is to giv, for ach unknown factor, th xprssion of th functions of form of th lmnt. For a givn scalar unknown factor (componnt of displacmnt according to thr for xampl), thr is as much as nods whr th unknown factor must b calculatd. In much of cas, on uss th sam functions of form for all th componnts of an unknown vctor, but it is not obligatory. In what follows, it will b supposd howvr to simplify th writings that it is th cas. Th functions of form 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 function of form associatd with a givn nod thr th valu on taks, whras it is canclld in all th othr nods of th lmnt. Th unknown factors ar xprssd thn lik linar combination of th functions of form, th cofficints u,i combination bing calld nodal variabls: u = u. N, i i (23) By using th transformation ntr th lmnt of rfrnc and th ral lmnt: On a: x (24) u = u. N, i i -1 x (25) Corrspondnc btwn polynomial bas and functions of form Thr 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, i i (27) 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, th following polynomial approximation was writtn; By injcting th quation (29) in th nodal xprssion (27), on obtains: u I, =a, p. Ip (29) u = a..n (30), p Ip i By comparison with th polynomial approximation (26), on from of dducd th following rlation btwn th polynomial bas and th functions from form: Ip. N i =P p (31) In practic, on will find in th litratur th writings of th nodal functions of form for th most currnt lmnts, according to th choic of th polynomial bas. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

12 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 12/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : 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 rprsnt 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 th 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 Th mthod of Galrkin, in crtain cass, is quivalnt making stationary a functional calculus. It is th cas if th bilinar form au,v is symmtrical 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 minimiz 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 th virtual powrs can b also writtn lik th minimization of a scalar siz: th total nrgy of th structur. This mannr of writing balanc 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 drivativ). On will writ this functional calculus. 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 worthlss (condition of stationnarity of th functional calculus) is quivalnt applying th principl of virtual work, or using th mthod of Galrkin by taking virtual displacmnts lik wight function. On calls that th mthod of Galrkin consists starting from th problm with th drivativ partial stablishing balanc of th 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 using wight functions which ar of th sam natur as th approximat solution: Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

13 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 13/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : 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 unknown factors = 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 cas gnral. 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 solution is a fild which can b kinmatically accptabl 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 {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 functions of form. 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 placs onslf in th cas hyprlastic in small dformations, th problm of mchanics to b solvd on writs in a mor compact way: Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

14 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 14/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : To find u E h such as u E ĥ with au, ul u=0 With au, u a bilinar, symmtrical form which rprsnts th potntial nrgy of th structur and l u potntial 1 voluminal and surfac fforts: au, u= l u= h h u : u. d h f. u.d h N h (44) g. u.d h (45) Th discrtization consists in choosing a bas of spac h and to calculat th trms of th matrix numrically A and of th vctor L. For that, th bilinar form is xprssd 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 (46) Trms A ij, which rprsnts th intraction btwn two dgrs of frdom i and j ar built in assmbling (th notd opration... ) contributions coming from ach lmnt which contains th corrsponding élémnts nods; on procds in th sam way to build th vctor scond mmbr L i. Ths contributions, calld lmntary trms, ar calculatd at th tim of 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 constraints of Cauchy and displacmnts u is givn by th rlation of bhavior, 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 tnsor of lasticity 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 th modifid shap of th componnts of dformation to mak it possibl to xprss th contractd product, is: = xx yy zz 2. xy 2. xz 2. yz (51) (47) Notic important: In th intgration of th laws of bhavior, componnts of sharing of th constraints and dformations usd by Cod_Astr ar: = xx yy zz 2. xy 2. xz 2. yz 1 Th potntial of th fforts xtrnal dos not dpnd on displacmnt of th structur, it is what is calld a loading did or not-followr. In th cass of th grat dformations, th loadings of typ prssur cannot rspct this assumption. Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

15 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 15/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : = xx yy zz 2. xy 2. xz 2. yz Th product of ths two vctors givs th sam rsult wll as th doubl contractd product ( 49 ). With this nw notation, w hav in lasticity: { }=[ A]. {} (52) W st out again of writing EF of th fild of displacmnts: {u }=[ N i ]. {u i }= u i.[ N i And, in similar mannr, th fild of virtual displacmnts: { u }=[ N i ]. {u i }= u i.[ N i ] T (53) ] 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 dformations (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) Matrics [ 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 grat dformations or th grat transformations (grat rotations and/or grat displacmnts). 2. [ ] is th matrix of bhavior. It bcoms dpndnt on displacmnts in th cas of (and othr variabls) th non-linar and/or inlastic bhaviors. In ths two cass, th procss of rsolution of th quations will imply a spcific tratmnt (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 Calculation of th lmntary trms Th lmntary trms to calculat ar form: u x f ux, x. d x (57) Thr typs of oprations ar to b carrid out: 1. th transformation of th drivativ compard to x in drivd compard to ; 2. th passag of an intgration on th ral lmnt with an intgration on th lmnt of rfrnc, 3. th digital ralization of this intgration which is gnrally mad by a formula of squaring Transformation of th drivativ Th transformation of th drivativ is carrid out thanks to th matrix jacobinn J, according to th rul of drivation in chain: whr u nod u x = x. u = J -1.[ N T nod ].u is th vctor of th nodal valus of th componnt displacmnt Chang of fild of intgration Th passag to intgration on th lmnt of rfrnc is carrid out by multiplying th intégrand by th dtrminant of th matrix jacobinn, calld jacobin: (58) Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

16 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 16/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : f u x ux, x. d x= u f r u,. dt J. d (59) Th passag 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) Digital intgration In crtain typical cass, on can calculat th intgrals analytically. For xampl, for a triangl in two dimnsions, Jacobin is constant on th triangl, and th intégrands ar brought back to studnts' rag procssions which on can intgrat xactly thanks to th formula of digital 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 (60) g. d g. g g (61) g=1 Scalars g th wights of intgration, and th coordinats ar calld g ar th coordinats of r points of intgration in th lmnt of rfrnc. In th mthods of intgration 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 Cod_Astr, th points of intgration ar calld thn points of Gauss. Th numbr of points of slctd Gauss 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 spac 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 rigidity: [a]= { }.. d (62) Digital intgration implis that on valuats th constraints and th dformations at th points of intgration: [a]= {}.. d g=1 r g.{ g }. g (63) What mans that th constraints and th dformations ar most xact (or th last fals) at th points of intgration (filds known as ELGA in Cod_Astr). Th simpl fact of xtrapolating ths valus with th nods for posting introducs an rror. It is bsids about a mthod valuation of th rror, calld indicator of rror of Zhu-Zinkiwicz. In lasticity 2D, a triangl displaying a jacobin constant, only on point of Gauss is sufficint to intgrat xactly th trms of th matrix and th scond mmbr (if it is constant). Th cost calculation incrass with th numbr of points of intgration, particularly for th non-linar laws of bhavior. For xampl, a hxahdron with 27 nods nds 27 points of Gauss 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 grid. Bsids this systmatic 2 By abus languag, on frquntly calls th digital diagrams of intgration diagrams of Gauss although thr ar svral kinds (Hammr for th triangl, Gauss-Radau, Nwton-Dimnsions, tc). Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

17 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 17/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : rror, this undr-intgration must b mad with prcaution bcaus it can produc dfcts 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 boundary conditions kinmatics Tratmnt of th boundary conditions kinmatics of th typ u=u D is don in two diffrnt ways: 1. Kinmatic mthod (AFFE_CHAR_CINE in Cod_Astr) 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. Mthod by dualisation (AFFE_CHAR_MECA in Cod_Astr) consists in introducing a vctor of multiplirs (or paramtrs) of Lagrang, which incrass th numbr of unknown factors but maks it possibl to trat all th cass. {[ A]. {u}[q] T { }={L } [Q]. {u}= {u D (66) } 6.2 Rsolution Th linar systm can b solvd by a crtain numbr of digital mthods. Mthods usd in Cod_Astr ar a factorization LDL T by blocks, a multi-frontal mthod (or its quivalnt with swivlling, MUMPS), a combind gradint prpackd as wll as th collction of itrativ solvurs PETSC. Th mthods of rsolution ar dividd into two catgoris: Th dirct mthods which solv xactly (with th digital rrors nar) Th itrativ mthods which build a vctor sris convrging towards th solution Th matrics rsulting from th finit lmnt mthod ar vry hollow (thy compris a majority of worthlss trms). In practic, on systms of standard siz (a fw tns of thousands of quations), th dnsity of nonworthlss trms sldom xcds th 0.01%. Thy ar thus stord in form digs (or spars ) and tak littl plac in mmory. A contrario, th matrics ar not built to b usd ffctivly with th mathmatical libraris of programs optimizd ddicatd to th full matrics (booksllrs BLAS for xampl). Solvurs ar thus dvlopd spcifically for ths problms. A dirct solvor has as a principl of braking up th matrix into a product of particular matrics 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 symmtrical matrics. 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 matrics. It is th opration of factorization. Ths rmarkabl matrics mak it possibl to solv th vry fast problm of mannr. It is th phas of dscnt-incras. 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 th 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 Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

18 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 18/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : svral hundrds of tims, vn svral thousands of tims mor mmory than th initial matrix. Th dirct solvurs thus consum much mmory and that bcoms crippling about it starting 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 mchanics of th structurs and th solids vry oftn lad to matrics with a bad conditioning (it is particularly th cas of all th last digital innovations which us mixd mthods with many multiplirs of Lagrang). 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 dfcts: Thy ar lss robust than th dirct mthods, particularly whn conditioning is bad Th mthods of prpacking 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 gt a rsult at th nd. Thy ar itrativ mthods, which implis a critrion of stop of th procss, and thus a paramtr to b managd but also problms of offic plurality of rounding rrors. 7 Organization of a calculation by finit lmnts in Cod_Astr On vry brifly dscribs how and at which plac th aspcts vokd in this documnt ar stablishd in Cod_Astr. 7.1 Concpt of finit lmnt in Cod_Astr A kind of finit lmnt is dfind by: a kind of msh a list of nods functions of form options of calculation An lmnt in th grid is dfind by a kind of msh, a gomtry (coordinatd nods) and a topology (ordrd list of th nods). It is th typ of modling chosn in th command fil which maks it possibl to assign to ach msh of th grid a kind of finit lmnt. Th ordr AFFE_MODELE [U ] assigns to ach msh a kind of finit lmnt corrsponding to th modling spcifid for this msh. Notic important: On should not forgt to assign finit lmnts to th mshs of dg which on nds to impos th boundary conditions and loadings, and that on will hav takn car to crat during th manufacturing of th grid. Th oprator AFFE_CHAR_MECA [U ], which affcts boundary conditions and loadings, also will crat 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 ]. Th oprator AFFE_CARA_ELEM [U ] allows to dfin additional charactristics for crtain typs of lmnts: for xampl, th thicknss of th hulls, orintation of th bams, matrics of mass and rigidity of th discrt lmnts. An option of calculation indicats th lmntary typ of calculation that th lmnt is abl to calculat. For xampl RIGI_MECA rlat to th calculation of th lmntary matrix of mchanical rigidity: 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 finit lmnts of dg individuals, and not th bordrs of th finit lmnts of volum (3D) or surfac (2D). Not: Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

19 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 19/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : A dvlopr can somtims hav th choic btwn crating a nw finit lmnt or adding an option of calculation 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 to onc and for all carry out a crtain numbr of calculations 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 points of GAUSS; Th numbr of points of GAUSS; Wights of intgration g ; Valus of th functions of form at th points of Gauss N i g ; Valus of th drivativ of th functions of form at th points of Gauss N i g. For a givn lmnt, on invitably dos not intgrat all th lmntary trms with th sam numbr of points of Gauss: for xampl, on in gnral uss mor points of Gauss for th matrix of mass than for th matrix of rigidity, bcaus th products of functions of form ar of dgr highr than th products of thir drivativ. Anothr xampl is th undr-intgration usd in crtain cass. On calls family of points of Gauss ach whol of points of Gauss likly to b usd. 7.3 Calculation of th lmntary trms During th calculation of th lmntary trms (in th routins TE.), on carris out for ach point of Gauss th following oprations: Calculation of th drivativ of th functions of form on th ral lmnt starting from th coordinats of th nods of th lmnt and th drivativ of th functions of form on th lmnt of rfrnc; Calculation of th matrix jacobinn; Rcovry of th wight of intgration multiplid by Jacobin at th point of GAUSS considrd; Evaluation of th intégrand (according to th calculatd option). Th lmntary trm is calculatd by nap on th points of Gauss whil balancing by th wights of intgration. 7.4 Total rsolution Th total rsolution taks plac in th routins OP. high lvl corrsponding to th ordrs 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 Vrsion 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 mchanics Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (

20 Cod_Astr Vrsion dfault Titr : La méthod ds élémnts finis isoparamétriqus Dat : 09/10/2013 Pag : 20/20 Rsponsabl : ABBAS Mickaël Clé : R Révision : Warning : Th translation procss usd on this wbsit is a "Machin Translation". It may b imprcis and inaccurat in whol or in part and is providd as a convninc. Copyright 2017 EDF R&D - Licnsd undr th trms of th GNU FDL (