Finite Element Solution Algorithm for Nonlinear Elasticity Problems by Domain Decomposition Method

Transcription

1 Fnte Element Soluton Algorthm for Nonlnear Elastcty Problems by Doman Decomposton Method Bedřch Sousedík Department of Mathematcs, Faculty of Cvl Engneerng, Czech Techncal Unversty n Prague, Thákurova 7, Prague 6 sousedk@matfsvcvutcz Abstract The purpose of the present work s to gve a bref descrpton of the fnte elastcty and of ts approxmaton va fnte element method We formulate the problem for the case of compressble elastcty Weak formulaton allows to use any sotropc hyperelastc materal model that satsfes polyconvexty assumptons Dscretzaton usng FEM leads to systems of non-lnear equatons Such systems of equatons can be solved by Newton s method and ts modfcatons In each teraton of Newton s method a lnearzed system of equatons has to be solved We propose to use BDDC precondtonng teratve method for the soluton of lnear system of equatons 1 Introducton The man object of the fnte three-dmensonal elastcty s to predct changes n the geometry of sold bodes The startng pont of the classcal theory of lnear elastcty s the concept of small strans: the deformaton of structures under workng loads are not detectable by human eye In contrast, many modern stuatons nvolve large deformatons The nonlnear behavor of polymers and synthetc rubbers are such examples Applcatons n bomechancs are even more crtcal because the most of vtal organs such as eye, heart trachea or vocal apparatus fulfll ther functon only because of ther large deformatons In ths framework the concept of fnte elastcty covers the smplest case where nternal forces (stresses) depend only on the present deformaton of the body and not on the hstory In the paper we show the fnte element approxmaton and strategy of soluton of ths nonlnear and vsble stress-stran relatonshp 1

2 2 Formulaton of elastcty problem Let us descrbe the deformaton of a body by ϕ : ϕ, and the occuped doman we decompose n an nteror and boundary as ϕ = ϕ Γ ϕ, (e Γ ϕ = ϕ ) The classc formulaton of equlbrum equaton we wrte as 21 Weak formulaton T ϕ j x ϕ j + f ϕ = 0 (1) Multplcaton by test functon v ϕ, ntegraton over the whole doman ϕ and applcaton of Green s theorem gves T ϕ j nϕ j vϕ daϕ T ϕ v ϕ j dx ϕ + f ϕ vϕ dx ϕ = 0 (2) ϕ ϕ ϕ x ϕ j On the parts of boundary Γ ϕ = Γ ϕ v Γ ϕ τ we prescrbe followng boundary condtons T ϕ j nϕ j = g ϕ on Γ ϕ τ (3) v ϕ = 0 on Γ ϕ v (4) After some rearrangements we obtan the weak formulaton of equlbrum equatons of a body after deformaton T ϕ v ϕ j ϕ x ϕ j dx ϕ = f ϕ vϕ dxϕ + ϕ Γ ϕ τ g ϕ vϕ daϕ (5) Unfortunatelly, n fnte elastcty, ϕ s unknown and may be very dfferent from the known reference confguraton Therefore, t s more convenent to rewrte the equlbrum equatons on, usng the formula for changes of varables n multple ntegrals Dong ths, usng Pola transform and consderng that x = ϕ 1 (x ϕ ) we fnally obtan the followng weak form of equlbrum equaton n the reference confguraton v T k dx = f v dx + g v da v V, (6) x k Γ τ 2

3 Consderng the followng consttutve relaton for compressble materal we get T j (u) = Ŵ F j (x,f(u)), (7) Ŵ (x,f(u)) v dx = f v dx + g v da v V (8) F j x j Γ τ 3 Consttutve relatons The smplest law uses a quadratc sotropc functon of the Green stran tensor E = 1 2 (C I) So called St Venant materal unfortunately can reach nfnte compresson rates wth fnte energy and do not satsfy the polyconvexty assumptons used n the exstence theory For these reason we do not use ths materal model here and suggest to use polyconvex functons gven n terms of nvarants The smplest example of such materals s the neo-hookean materal W = C 10 (I 1 3) (9) If we add a lnear term, we get well-known Mooney-Rvln materal W = C 10 (I 1 3) + C 01 (I 2 3) (10) Ths energy functon was further generalzed nto the thrd order polynomal n nvarants I 1,I 2 whch fts well to numerous expermental data W = C 10 (I 1 3) + C 01 (I 2 3) + C 20 (I 1 3) 2 + C 02 (I 2 3) 2 + C 11 (I 1 3)(I 2 3) + C 30 (I 1 3) 3 (11) Theorem (Rvln-Erksen representaton theorem): For any sotropc hyperelastc materal, the elastc potental W satsfes: W(x,F) = W(x,I 1 (E),I 2 (E),I 3 (E)) (12) Invarants and ther dervatves are gven n the followng table: I 1 = tre I 1 E j = δ j I 2 = 1 2 tre2 = 1 2 E je j I 2 E j = E j (13) I 3 = 1 3 tre3 = 1 3 E je jk E k I 3 E j = E k E kj 3

4 4 Numercal soluton technque Equlbrum equatons are generally nonlnear n the dsplacements u We would lke to fnd the feld of dsplacements u so that F(u) = 0 usng Newton s method n the followng way: n the (k+1)-th teraton of Newton s method we are lookng for the feld of dsplacements u k+1 The feld of dsplacements from k-th teraton u k s known and we must fnd ncrement h R n satsfyng the equaton DF(u k ) h = F(u k ), (14) then we add the ncrement h to the prevous teraton so that Naturally, n our case, F(u) = u k+1 = u k + h (15) Ŵ (x,f(u)) w dx fw dx gw da (16) F j x j Γ τ and the Fréchet dervatve of F(u) s D v F(u) = ( 2 Ŵ(x,F(u)) v k F j F kl x l 5 Fnte Element Formulaton ) w x j dx, (17) Frst we use the equvalence relaton W = Ŵ(F) = W(C) followng from defnton C j = F k F kj = δ j + 2E j to transform F(u) and DF(u) n order to use nternal energy functons n terms of nvarants and stran tensors E j F(u) = [( W (δ k + u )] k wk dx fw dx gw da (18) E j x x j Γ τ and usng the second Pola-Krchhoff stress tensor S j = 2 W C j we get D v F(u) = { 2 W E j E kl [( δ r + u r x 4 ) vr x j ][( δ sk + u s x k ) vs x l ] } v k w k + S j dx(19) x x j

5 Takng w = (0,N t,0) and v = h = (h x,h y,h z ) where h x = N h u=1 h xun u h y = N h u=1 h yun u h z = N h u=1 h zun u (20) we get the components of the stffness matrx and rght hand sde vectors nto Newton s method Fnally, the (k+1)-th teraton of Newton s method conssts of two steps: () Frst, we solve the followng system of equatons A k h = b k, (21) where the components of the matrx A and vectors h, b are defned as A = a xx a xy a xz a yx a yy a yz a zx a zy a zz (22) b = b x b y b z h = h x h y h z (23) () Next, we update the feld of dsplacements u k+1 = u k + h (24) 6 Formulaton of BDDC method Clearly, n each step of Newton s teratons we solve a lnear system of equaton Here, we descrbe the BDDC precondtonng method Ths method s closely related to FETI-DP method that s also brefly descrbed Let us decompose the doman nto the nonoverlappng set of substructures (subdomans), where = 1,N and N means the number of substructures Let K be the stffness matrx and v the vector of degrees of freedom (dofs) for substructure We want to solve the problem n decomposed form 1 2 vt Kv v T f mn (25) 5

6 where K 1 0 K = 0 K N, v = v 1 v N, f = subject to contnuty dofs between substructures Parttonng the dofs n each subdoman nto nternal and nterface (boundary) K = [ K K b T K b K bb ] [ v, v = v b ], f = and elmnatng the nteror dofs we obtan the problem reduced to nterfaces 1 2 wt Sw w T g mn, S = dag(s ), S = K bb [ f f b f 1 f N ], K b T K 1 K b, agan subject to contnuty of dofs between substructures In BDD type methods, the contnuty of dofs between substructures s enforced by mposng common values on substructures nterfaces: w = Ru for some u, where R = and R s the operator of restrcton of global dofs on the nterfaces to substructure In FETI type methods, contnuty of dofs between substructures s enforced by the constrant Bw = 0, where the entres of B are typcally 0, ±1 By constructon, we have R 1 R N R R T = I, range R = nullb A BDDC or FETI-DP method s specfed by the choce of coarse dofs and the choce of weghts for ntersubdoman averagng To defne the coarse problem for BDDC, choose a matrx Q T P that selects coarse dofs u c from global nterface dofs u, eg as values at corners or averages on sdes: u c = Q T P u We defne W as the space of all vectors of substructure nterface dofs that are contnuous between substructures, W = {w W : u c : Cw = R c u c } = {w W : Cw range R c } (26) 6

7 where C 1 0 C = R c Q T P RT, C = 0 C N, R c = R c1 R cn, (27) and the matrx R c restrcts the vector all coarse dof values nto a vector of coarse dof values that can be nonzero on substructure The dual approach n FETI-DP s to construct Q D such that W = { w W : Q T D Bw = 0} (28) In BDDC, the ntersubdoman averagng s defned by the matrces that form a decomposton of unty, D P = dag (D P ) R T D P R = I The correspondng dual matrces n FETI-DP are are B D = [D D1 B 1, D DN B N ], where the dual weghts D D are defned so that B T D B + RRT D P = I (29) The BDDC method s then the method of conjugate gradents for the assembled system Au = R T g wth the system matrx and the precondtoner P defned by A = R T SR Pr = R T D P (Ψu c + z), where u c s the soluton of the coarse problem and z s the soluton of Ψ T SΨu c = Ψ T D T P Rr Sz + C T µ = DP TRr, Cz = 0 whch s a collecton of ndependent substructure problems The coarse bass functons Ψ are defned by energy mnmzaton, [ ][ ] [ ] S C T Ψ 0 = C 0 Λ R c 7

8 7 Concluson In the paper we showed fnte element approxmaton of the non-lnear threedmensonal fnte elastcty problem for compressble materal model We also descrbed the strategy of soluton of the non-lnear system of equatons arsed The lnearzed system of equatons wll be solved by an teratve solver of conjugate gradent type wth BDDC precondtonng proposed and studed n [1], [3] Currently we are developng and testng ths fnte element code n programmng language FORTRAN We wll also contnue n developng of adaptve strateges for the selecton of constrants studed n the paper [4] The whole code wll be fnally parallelzed usng MPI lbrary Acknowledgement The work was supported by grant GA ČR 106/05/2731 and by the Program Informaton Socety under project IET References [1] Dohrmann, CR: A precondtoner for substructurng based on constraned energy mnmzaton, SIAM J Sc Comput, 25(1): , 2003 [2] Le Tallec, P: Numercal Methods for Nonlnear Three-dmensonal Elastcty, n Handbook of Numercal Analyss, Vol III, (P G Carlet and J L Lons eds) North-Holland, 1994 [3] Mandel, J, Dohrmann, CR: Convergence of balancng doman decomposton by constrants and energy mnmzaton, Numer Lnear Algebra Appl 10: , 2003 [4] Mandel, J, Sousedík, B: Adaptve Coarse Space Selecton n BDDC and FETI-DP Iteratve Substructurng Methods: Optmal Face Degrees of Freedom, 16th Internatonal Doman Decomposton Conference, New York, January 2005, Lecture Notes n Computatonal Scence, submtted [5] Sousedík, B, Burda, P: Fnte Element Formulaton of the Threedmensonal Non-lnear Elastcty Problem, Semnar n Appled Mathematcs, Czech Techncal Unversty n Prague, Prague, Aprl