ACCURACY OF DIRECT TREFFTZ FE FORMULATIONS

Transcription

1 COMPUTATIONAL MECHANICS Nw Trnds and Applications S. Idlsohn, E. Oñat and E. Dvorkin (Eds.) c CIMNE, Barclona, Spain 1998 ACCURACY OF DIRECT TREFFTZ FE FORMULATIONS Vladimír Kompiš,L ubor Fraštia, Michal Kaukič and Pavol Novák Faculty of Mchanical Enginring Univrsity of Žilina Vlký Dil, Sk Žilina, Slovakia -mail: kompis@fstroj.utc.sk Faculty of Managmnt Scinc and Informatics Univrsity of Žilina Vlký Dil, Sk Žilina, Slovakia -mail: mik@frcatl.utc.sk Ky words: Btti s rciprocity, Trfftz functions, larg lmnts, local filds Abstract. This papr prsnts th FE formulation for which th only incompatibility is in th intr-lmnt tractions. Th boundary tractions and displacmnts ar dfind using th Btti s rciprocity, similarly as that usd by BEM. As th intgration for th stiffnss matrix formulation is takn ovr th lmnt boundaris only, th shap of matrix can b mor gnral than it is in classical FEM. Thr is always th possibility of including local filds in th lmnts, so w can dscrib local ffcts with rlativly larg lmnts. Arbitrary Trfftz functions can b usd for th auxiliary stats and so nonsingular formulation of th Btti s rciprocity is achivd. 1

2 1 INTRODUCTION Th ffort to prdict mor ffctivly th rspons of th complx problms of continuum mchanics motivats th authors to sk for th formulations which idntically satisfy th govrning quations insid th approximatd domain. Such analytically drivd functions ar calld Trfftz functions (or simply T-functions) which can b in th form of polynomials, Lgndr, harmonic, Bssl, Hankl, Kupradz functions, tc. 1. Corrsponding domain formulations can b basd on th hybrid-fem using variational formulation 2,3, or in th form of BEM basd upon th gnralizd Btti s 4.Inourapproach, th FE formulation is basd on th Btti s rciprocity conncting th boundary displacmnt and traction filds and th variational (wak intr-lmnt quilibrium) principls. Th formulation will b applid to th lasticity, although th xtnsion to othr problms is straightforward. 2 BASIC (A) FE FORMULATION Lt us considr th lasticity problm without body forcs for simplicity. Th boundary displacmnts ũ i ( x) and tractions t i ( x) of ach approximatd subdomain (lmnt) will b rlatd by th Btti s rciprocity thorm T i ( x)ũ i ( x)dγ( x) = U i ( x) t i ( x)dγ( x) (1) Γ Γ whr x dnots a fild variabl, U i (x) ar displacmnt filds satisfying th quilibrium conditions insid th lmnt (Trfftz functions) and T i (x) ar corrsponding tractions on th lmnt boundaris Γ. Tild dnots corrsponding fild variabl on th lmnt boundary. Th Trfftz functions for displacmnts can b in th polynomial, Lgndr, harmonic, Bssl, Hankl, Kupradz s form 1, or th fundamntal solutions. In th last cas th problm (1) has to b writtn in th form of singular intgral quations 4 cũ i (ỹ)+ T ij (ỹ, x)ũ j ( x)dγ( x) = U ij (ỹ, x) t j ( x)dγ( x) (2) Γ Γ whr ỹ dnots a sourc point, T ij and U ij ar tractions and displacmnts of th fundamntal solution, as thy ar known from BEM formulations 4. Summation on rpatd indics (dnoting th componnts of th fild variabls) is considrd. Th boundary displacmnts can b xprssd by thir nodal valus d (j) (th uppr indx corrsponds to th nodal point) and shap functions N ũ i (ξ) =N u (j) (ξ)d (j) i (3) ξ is a local coordinat of a point on th lmnt boundary. Th quation (3) can b writtn in th matrix form {ũ} =[N u ]{d } (4). 2

3 Similarly, tractions can b givn by thir valus q (j) in th nodal points and by corrsponding shap functions, N (j) t,as t i (ξ) =N (j) t (ξ)q (j) i or t =[N t ]{q } (5) which lads to matrix form of th Equations (1) and (2) [T ]{d } =[U]{q } (6) q is a vctor of nodal tractions. Elmnts of matrics [T ]and[u] inequation(6)canbdtrmindasfollows T kl = T (k) ( x(ξ))n u (l) (ξ)dγ = Γ j U kl = U (k) ( x(ξ))n (l) Γ t (ξ)dγ = j T (k) ( x(ξ (j) ))N (l) u (ξ (j) )J(ξ (j) )w (j) U (k) ( x(ξ (j) ))N (l) t (ξ (j) )J(ξ (j) )w (j) (7) whr ξ (j) and w (j) ar co-ordinats and wights of Gauss quadratur formulas J is th Jacobian. T (k) and U (k) ar tractions and displacmnts of th arbitrary stats of th lmnt satisfying all th govrning quations (indpndnt stats according to Eq.(1) 5, or thos corrsponding to th sourc point ỹ in Eq.(2)). Not that th intgration and thus also th summation in Equations(7) is mad on th total lmnt boundary. W will assum that th whol domain will b dcomposd into subdomains (lmnts) and th displacmnts btwn th subdomains will b compatibl, i.. th displacmnts on th lmnt boundaris ar common to th nighbour lmnts. Th tractions, howvr, will b incompatibl btwn th lmnts and so th intr-lmnt quilibrium can b satisfid only in a wak (intgral) sns. W us th variational formulation δũ T ( t t)dγ + Γ t δũ T ( t A t B )dγ = Γ i δũ T t dγ Γ i Γ t δũ T t dγ =0 Γ t (8) In this quation th suprscript T dnots transposition, Γ i Γ t ar th intr-lmnt boundaris and th domain boundary with prscribd tractions, t s, t t and t n rspctivly. Th uppr indics A and B dnot th nighbouring lmnts to th common boundary. With a bar w dnot th prscribd valus. For th purpos of numrical implmntation w can writ Equation (8) in th form j N (k) u (ξ (j) )N (l) t (ξ (j) )J(ξ (j) )w (j) q (l) = i N (k) u (ξ (i) ) t(ξ (i) )J(ξ (i) )w (i) q (i) (9) 3

4 or in th quivalnt matrix form [M ]{q } = {p } (10) Not that th Equation (8) contains intgrals on all intr-lmnt boundaris and on th domain boundary with prscribd boundary tractions and thus, also th summation in Eq.(9) and matrics in Eq.(10) corrspond to th whol rgion. From Equation(6) w can xprss th nodal tractions in ach lmnt by its nodal displacmnts as {q } =[U] 1 [T ]{d } (11) and substitut this into Eq.(10), which yilds [M ][U] 1 [T ]{d } = {p } (12) or [K]{d} = {p} (13) which is th rsulting systm of quations in th discrtizd form and introducs, whr K is th global stiffnss matrix. W did not tak into account prscribd boundary displacmnts in th formulas abov for th sak of simplicity. This is a simpl problm, whn w raliz, that th givn (non-zro) boundary displacmnt ar includd in Eq. (11) and can b sparatd from thos, which ar to b computd. Solving th Eq. (13) w obtain th unknown part of th nodal displacmnts. In th solution of Equation (11), th numbr of Trfftz functions (auxiliary stats of th lmnt) hav to b qual to th numbr of d.o.f. in displacmnts in ordr to rciv a uniqu solution. If it is largr, thn th problm of Btti s rciprocity has to b solvd in th last squars (LS) sns. Similarly, for th uniqu solution of th intr-lmnt quilibrium (in th wak sns) th numbr of componnts of th lmnt displacmnt vctor should b qual to th numbr of componnts of its nodal tractions. If it is largr, thn th quilibrium can b satisfid in th (LS) sns only. 3 ACCURACY ASSESSMENT AND COMPUTATION OF TRACTIONS AND STRESS Having obtaind th nodal displacmnts from Eq.(13) th tractions in ach lmnt can b computd from th Equation (11). Th tractions will b an xclnt masur of th local and global accuracy of th dirct Trfftz FE solution, bcaus th incompatibility btwn th tractions of th nighbouring lmnts (i.. thir sum) and also 4

5 that btwn th calculatd and givn valus on th domain boundary ar th only incompatibility of th solution. Th local rror of th solution valuatd in this way is influncd mainly by th rror of approximation of th local displacmnt fild dpndnt upon th ordr and finnss of th boundary displacmnts. Th strsss on th lmnt boundaris can b calculatd bst in th local coordinats. Two componnts for 2D (normal and shar componnt) and thr componnts in 3D ar idntical with th tractions (transformd into th corrsponding dirctions). Th othr componnts can b found from strains obtaind from th boundary displacmnt filds 6. Dnoting, t s, t t and t n, th traction componnts in two orthogonal dirctions s and t and in th normal dirction, rspctivly, w can writ σ nt = t t σ nn = t n σ tt = 1 1 ν (νt n +2Gε tt ) (14) for plan strain (ν = ν/(1 + ν) for plan strss) stat and σ sn = t s σ tn = t t σ nn = t n σ st = Gε st σ ss = 1 1 ν [νt t +2G(ε ss + νε tt )] σ tt = 1 1 ν [νt n +2G(ε tt + νε ss )] (15) for 3D problms. Th strains ε ss, ε tt, ε st hav to b computd from th displacmnts in th lmnt nodal points and corrsponding shap functions by standard mthod known from FEM formulations. Continuous strss filds can b obtaind using th moving last squar (MLS) tchniqus from displacmnts and tractions in th nodal points. Similar idas has bn usd in 7,8. W assum th displacmnt fild (in a fild point x), {u(x)}, givn in th form {u(x)} =[U(x)]{c} (16) whr [U(x)] is a matrix of displacmnt-trfftz functions and {c} is th vctor of unknown cofficints. If Trfftz polynomials ar usd for th Trfftz functions, w can calculat strain and strss filds from displacmnts (16). Th strss fild can b thn formally writtn as {σ(x)} =[S(x)]{c} (17) whr th matrix of strss-trfftz functions [S(x)] is drivd from th matrix [U(x)]. Similarly, w can xprss traction-trfftz function as {t(x)} =[T (x)]{c} (18). 5

6 W us low (typically scond) ordr polynomials for Trfftz functions. For a point of intrst, in which w want to calculat strsss, w choos a domain of influnc in which w will hav nough discrt nodal points whr th displacmnts and tractions wr calculatd as mntiond abov. Th unknown cofficints {c} ar thn computd by LS mthod as wi d ([U(x i)]{c} {d i }) 2 + wi t ([T (x i)]{c} {t i }) 2 = min (19) i i whr {d i } and {t i } ar th displacmnts and tractions in th nodal points and w d i, wt i ar xponntial wighting functions chosn in th form w i = k d c(r/r m) (20) whr c is a constant, r is th distanc btwn th nod and th point of intrst, r m is th radius dfining th domain of influnc and k d is a constant to insur th sam dimnsion of both trms of th lft sid of Eq. (19). Th co-ordinats in Eq.(16) (20) hav th origin in th point of intrst. Whn w find th unknown constant vctor {c} by solving th LS problm (19), w can find th strss in th point of intrst. Not, that th man of th tractions of th two nighbouring lmnts must b calculatd with rgard to th dirction of corrsponding normal. 4 AN ALTERNATIVE (B) FE FORMULATION Th Equation (1) can b also writtn in th form T i ũ i dγ = U i t i dγ + U i t i dγ (21) Γ Γ i Γ u Γ t Th scond trm of th r.h.s. contains known tractions. Th Equation (6) will b now in th form [T ]{d } =[U]{q } + {f } (22) Th componnts of th vctor {f } ar dfind by f k = U i t dγ = U k ( x(ξ (j) )) t(ξ (j) )J(ξ (j) )w (j) (23) Γ t j Thn w find {q } =[U] 1 [T ]{d } [U] 1 {f } (24) W hav now th only incompatibility in intr-lmnt tractions, which will b satisfid in th wak sns as δũ T t dγ = 0 (25) Γ i 6

7 or, in th matrix form [M ]{q } = 0 (26) From Eqs. (24) and (26) w hav [M ][U] 1 [T ]{d } = [M ][U] 1 {f } (27) which introducs th systm of linar algbraic quations (13). W can idntify that both, A and B, formulations ar idntical if th lmnt contains th intr-lmnt boundaris only. Th rror assssmnt in th part of th boundary with prscribd tractions is mor complicatd in this formulation than in th A vrsion. Th suprconvrgnt xtraction of strsss and displacmnts dscribd in 9 11 can b usd on this part of boundary. Th tchniqus is similar to th on commonly practisd in th dirct BEM 4,12. Anothr possibility is to us th tchniqus dscribd in th Chaptr 3 in this papr, i.. by comparing computd tractions with th smoothd valus obtaind by our approach. 5 SOME NUMERICAL EXPERIMENTS Th numrical bhaviour can b shown on a simpl xampl. Th squar rgion x, y 0, 2 was loadd with th tractions dfind from th polynomials satisfying th govrning quations insid th whol rgion. Th rgion was modld with a cours msh 2x2 and th displacmnts on th lmnt boundaris wr approximatd by quadratic polynomials and th tractions with linar polynomials only. Th xtrnal loads wr chosn from th filds highr ordr than thos on th lmnt boundaris and th rsults approximatd th filds of 3rd and 7th ordr ar givn in th Tabls 1 to 3 as follows: Tab. 1 computd xact σ σ σ Tab. 2 computd xact σ σ σ

8 Tab. 3 computd xact σ σ σ In Tab.1 and Tab.3, thr ar strsss in th intrnal point (1.5, 1.5) for loads corrsponding to th polynomial filds of th 3rd and 7th ordr, rspctivly, whil in Tab.2 thy ar in th point (1.5, 2), i.. on th boundary. Th computd valus givn in th tabls corrspond to th minimal and maximal valus. W can s from th rsults, that alrady a simpl man valus in th nodal points from th nighbour lmnts can improv th accuracy of th computd strsss. Th accuracy of th mthod is ncouraging vn for a low ordr lmnts also in th rgions with high gradints in displacmnt and strss filds. 6 CONCLUSIONS In our contribution a formulation of th FE calld dirct Trfftz FE is shown. It is basd, similarly as th dirct boundary lmnt formulation on th Btti s rciprocity principl, howvr, also non-singular functions, which can b polynomials, harmonic, Bssl, Hankl, Kupradz functions, tc. may b usd as tst (Trfftz) functions. Th displacmnts ar considrd to b continuous btwn th lmnts (this diffrs prsnt formulation from th hybrid-trfftz FE formulation) and thus, only compatibility of tractions (global quilibrium) has to b satisfid in a wak (intgral) sns. In our formulations, th Trfftz polynomials ar usd as th tst functions. Th intgral quations rsultd from th Btti s formulation ar in th polynomial form and so, th numrical implmntation is vry simpl unlik thos in th common BEM formulations. Th accuracy of th formulation can b assssd in a simpl way from th discontinuity of th tractions in th nighbouring lmnts. An altrnativ formulation includs th known static boundary conditions dirctly into th Btti s formulation, so that th traction incompatibility is prsnt in th intrlmnt boundaris only. Such formulation rducs th problm, howvr, th accuracy assssmnt has to b don with diffrnt (lss dirct) approach. Th advantags of prsnt formulation is th possibility to us lmnts of mor complicatd form as by th classical FEM. Th lmnts nabl to modl local filds in vry ffctiv way and th assssmnt of th local accuracy of th numrical modl is vry simpl. Th vry larg suprlmnts can b dfind in th sam way as a subdomain by th common BEM formulation, for som massiv parts of th domain (a similar aim was intndd in 13 for th hybrid coupld FEM/BEM formulation). On th othr sid th mthod nabls to connct ffctivly th lmnts (subdomains) and so 8

9 th rsulting systm of quations is spars, symmtrical form (but with non-symmtric lmnts). Nonlinar and dynamical problms can b formulatd in a similar way as it is in th common BEM REFERENCES [1] A.P. Ziliński, On trial functions applid in th gnralizd Trfftz mthod, Advancs in Eng. Softwar, 24, (1995). [2] J. Jirousk and A. Wróblwski, T-lmnts: Stat of th art and futur trnds, Archivs of Comput. Mch., 3, (1996). [3] P. Ladvèz, E.A.W. Maundr, A gnral mthod for rcovring quilibrating lmnt tractions, Comp. Mth. Appl. Mch. Eng., 137, (1996). [4] J. Balaš, J. Sládk and V. Sládk, Strss analysis by boundary lmnt mthods, Elsvir, (1989). [5] V. Kompiš, M. Kaukič andm.žmindák, Modlling of local ffcts by hybrid- displacmnt FE, J. Comput. Appl. Math., 63, (1995). [6] R. Bausingr, G. Kuhn, Th boundary lmnt mthod (in Grman), Exprt Vrlag, Grmany, (1987). [7] V. Kompiš andl. Fraštia, Polynomial rprsntation of hybrid finit lmnts, Computr Assis. Mch. in Eng. Sci., 4, (1997). [8] T. Blackr and T. Blytschko, Suprconvrgnt patch rcovry with quilibrium and conjoint intrpolant nhancmnt, Int. J. Num. Mth. Eng., 37, (1994). [9] Q. Niu and M.S. Shphard, Suprconvrgnt xtraction tchniqus for finit lmnt analysis, Int. J. Num. Mth. Eng., 36, (1993). [10] N.E. Wibrg, Suprconvrgnt patch rcovry: A ky to quality assssd FE solutions, Advan. Eng. Softwar, 28, (1997). [11] A.C.A. Ramsay and E.A.W. Maundr, Effctiv rror stimation from continuous, boundary admissibl stimatd strss filds, Comp. & Struct., 61, (1996). [12] C.A. Brbbia, Th boundary lmnt mthod for nginrs, J. Wily, (1978). [13] G.C. Hsiao, E. Schnack and W.L. Wndland, Hybrid coupld finit-boundary lmnt mthods for lliptic systms of scond ordr, to b publishd. [14] A. Forstr and G. Kuhn, A fild boundary lmnt formulation for matrial nonlinar problms at finit strain, Int. J. Solids Struc., 31, (1994). [15] S. Mukhrj and A. Chandra, Nonlinar solid mchanics, in D.E. Bskos, d., Boundary lmnt mthod in mchanics, Elsvir, (1987). [16] L. Gaul and M. Schanz, Dynamics of viscolastic solids tratd by boundary lmnt approachs in tim domain, Eur. J. Mch., A/Solids, 13, (1994). [17] L. Gaul and M. Schanz, Boundary lmnt calculation of transint rspons of viscolastic solids basd on invrs transformation, Mccanica, 32, (1997). 9