FEM II - Lctur Pag of 4 FINITE ELEMENT METHOD II Autumn 05 Lcturs (5h):. Accuracy, rror stmaton and adaptv rmshng. Hat flow and thrmal strsss n FEM 3. Introducton to structural dynamcs, fr vbratons 4. Nonlnar problms n mchancs of structurs - basc numrcal tchnqus 5. Orthotropc matrals and compost structurs 6. Paramtrc modlng and dsgn optmzaton Computr lab (5h): Modlng smpl problms of: thrmal strsss, contact mchancs, plastcty and rsdual strsss, fr vbratons, buclng, paramtrc modlng Rfrncs: [] Lctur nots from th wb st: http://ml.pw.du.pl/zwm/zwmk/dla-studntow/fnt-elmnt-mthod-ii [] Moavn S.: Fnt lmnt analyss. Thory and applcatons wth ANSYS. Parson Educaton, 05. [3] Klbr M. (rd.): Komputrow mtody mchan cał stałych, sra Mchana Tchnczna XI, Warszawa PWN 995. [4] Xaoln Chn, Yjun Luv: Fnt Elmnt Modlng and Smulaton wth ANSYS. Worbnch, CRC Prss 04 [5] Hubnr K. H., Dwhrst D. L., Smth D.E., Byrom T. G.: Th fnt lmnt mthod for ngnrs, J. Wly & Sons, Inc., 00. [6] Znwcz O.C., Taylor R.: Th Fnt Elmnt Mthod - dffrnt publshrs and dtons [7] Krzsńs G., Zagraj T., Mar P., Borows P.: MES w mchanc matrałów onstrucj. Rozwązywan wybranych zagadnń za pomocą programu ANSYS, Of. Wyd.PW 05 [8] Bja-Żochows M., Jawors A., Krzsńs G., Zagraj T.: Mchana Matrałów Konstrucj, Tom, Warszawa, Of. Wyd. PW, 04 Assssmnt basd on th fnal tst and th rsults of computr lab wor
FEM II - Lctur Pag of 4. ACCURACY OF FE ANALYSIS. ERROR ESTIMATION AND ADAPTIVE REMESHING Ral objct gomtry, boundary (ntal) condtons matral proprts, laws of physcs Mthmatcal modl (contnuous) dscrtzaton approxmaton Ral rsult w r Exact soluton of th mathmatcal modl w s ε = w w modlng rror, s s r ε = w w dscrtzaton rror d d s ε = w w numrcal rror n n d Total rror: ε = ε + ε + ε = w w c s d n n r Th most ffctv FEM analyss: ε s ~ ε d ~ ε n Dscrt modl Exact soluton of th dscrt modl w d numrcal calculatons Numrcal rsult - w n Approxmat mthods flow chart (vald also for othr mthods.g. Boundary Elmnt Mthod - BEM and Fnt Dffrncs Mthod - FDM)
FEM II - Lctur Pag 3 of 4 Modlng rror ε s Dpnds on th accuracy of th avalabl nformaton about th problm and nowldg of th analyst (D D 3D modls, lnar, nonlnar, assumd smplfcatons, rlabl nformaton concrnng matral proprts, loads.) Dffrnt modls for th problm Th xampl woodn board on dmnsonal modl - bam a) modl bl b) two- modl dmnsonal płyty modl - plat q 0 N m N p 0 m Mathmatcal modl:,, 3 dmnsonal proprts of wood: sotropc-orthotropc, mostdry, homognous-nhomognous, rat dpndntrat-ndpndnt) contact ntracton modl (frcton) loadngs c) thr modl dmnsonal trójwymarowy modl sold bryły volum N 0 m 3
FEM II - Lctur Pag 4 of 4 Dscrtzaton rror ε d Dpnds on th msh dnsty, typs of th lmnts shap functons, th shaps of th fnt lmnts.86398 5 50 00 50 00 50 300 400 FE msh and von Mss strss dstrbuton Dscrt soluton vrsus xact soluton of th contnuous problm Thr ar som mathmatcal convrgnc rqurmnts n FEA concrnng th msh, shap functons, and ruls of FE modl buldng.
FEM II - Lctur Pag 5 of 4 Numrcal rror ε n [ K ]{ q} = { F}, [ K + δ K ]{ q + δ q} = { F + δ F}. [ K ] [ K ] { δ F} { } [ δ ] [ ] δ q K +, q F K condton numbr of th matrx K ([ ]) [ ] [ ] cond K = K K. Norm of a matrx (vctor) a masur of magntud L - Eucldan norms, = Vctor norm { q} ( q ) Matrx norm [ K ] ( j ) (matrx norm nducd by th vctor norm) j = Max norms (L ~ norms) { q} = max q, [ K ] max j =. j A problm wth a lsmall condton numbr s sad to b wll-condtond, whl a problm wth a hgh condton numbr s sad to b ll-condtond. cond(k) cond(k ) - problm wll-condtond cond(k )» - problm ll-condtond Rasons of ll-condtonng of th problms n FE strss analyss grat dffrncs btwn stffnss of FE lmnts, unstabl boundary condtons
FEM II - Lctur Pag 6 of 4 Th xampl (ll condtond problm): q A q B q 3 [ ] q = 0 = Lt s assum: A Th soluton: Th rsult: A A q F + q = F A A B B B B q 3 F 3 A + B B q F = q F B B 3 3 = = 000 B F = F3 F q A A F = q3 F3 + A A B q 0 = q3 0.00 F 3 Th xampl of th vry small prturbaton For changd forc vctor : { δ F} δ F 0.00 δ F3 0 = = F 0.999 = F3.0 q 0.00 = q3 0.00 δ F F { δ q} δ q 0.00 δ q3 0.00 = = δ q = = q 3 0.707 0.44 3 cond( K) 4 0
FEM II - Lctur Pag 7 of 4 Frst quaton Scond quaton q q + F = q A B 3 B = q + 3 F 3 B B q 3 Systm ll-condtond f A + B B A 0 Snstvty to th slop chang K A <<K B systm ll condtond B + A B tgα = B α α tgα = Round off rror As a gnral rul, f th condton numbr cond(k) = 0, thn you may los up to maxmum dgts of accuracy durng th soluton of th systm of lnar quatons. Howvr, th condton numbr dos not gv th xact valu of th maxmum naccuracy that may occur n th algorthm. ( ([ ])) r p log 0 cond K p numbr of sgnfcant dgts n th computr rprsntaton of numbrs r numbr of sgnfcant dgts of th rsult In FE modls cond(k) rachs 0 8 q
FEM II - Lctur Pag 8 of 4 A postror rror approxmaton tchnqus Elmnt and nodal soluton n FE program postprocssors (PLESOL, PLNSOL n ANSYS) FE soluton provds th contnuous dsplacmnt fld ( from lmnt to lmnt), and th dscontnuous strss fld. To obtan smooth strss dstrbuton, th avragng of th strsss n th nods s prformd ( nodal strsss). MN PowrGraphcs EFACET= D =.086853 SMN =-4.333 S =379. -4.333 9.383 73.099 6.85 60.53 04.47 47.963 9.679 335.395 379. MN EFACET= AVRES=Mat D =.086853 SMN =-0.37 S =370.5-0.37 3.99 74. 6.50 58.793 0.085 43.376 85.667 37.959 370.5 Y Y Z X Z X JUN 008 00:3:3 PLOT NO. 5 ELEMENT SOLUTION STEP= SUB = TIME= SY (NOAVG) RSYS=0 PowrGraphcs EFACET= D =.05 SMN =6.884 S =379. 6.884 46.0 75.57 04.93 33.49 6.566 9.70 30.838 349.975 379. JUN 008 00:3:37 PLOT NO. 6 NODAL SOLUTION STEP= SUB = TIME= SY (AVG) RSYS=0 PowrGraphcs EFACET= AVRES=Mat D =.05 SMN =6.884 S =370.5 6.884 45.036 73.88 0.339 9.49 57.643 85.795 33.946 34.098 370.5 Rctangular plat wth a hol undr tnson. Th modl of th quartr of th structur. Th strss componnt σ y Dscontnuous lmnt soluton (lft) and avragd contnuous nodal soluton (rght). Sx-nod trangular plan lmnts
FEM II - Lctur Pag 9 of 4 Basc rlatons btwn dsplacmnts, strans, strsss and stran nrgy wthn fnt lmnts (th rlatons dscussd durng FEM I lcturs) Dsplacmnt fld ovr th lmnt s ntrpolatd from th nodal dsplacmnts: { u} = [ N( x, y, z) ] { q}, whr { } q - nodal dsplacmnts vctor, [ ] For xampl for th smplst trangular lmnt wth 3 nods and 6 DOF th rlaton s N - shap functons matrx. u u υ u( x, y) N( x, y) 0 N( x, y) 0 N3( x, y) 0 u =, υ( x, y) 0 N( x, y) 0 N( x, y) 0 N3( x, y) υ u 3 υ3 whr N ar th lnar functons Shap functons N j ar usually polynomals dfnd n local (lmnt) coordnat systms. Dsplacmnts, strans and strsss wthn ach lmnt ar dfnd as th functons of th nodal dsplacmnts { u} = [ N ]{ q}, { ε} = [ R]{ u} = [ R][ N ]{ q} = [ B]{ q} { σ } = [ D]{ ε} = [ D][ B]{ q}. Th stran nrgy of th lmnt Ω s: U = Ω, [B] stran-dsplacmnt matrx, [ R ] -gradnt matrx T ε { σ} d, [ ] [ ] [ ] { } Ω T whr [ ] [ ] [ ] [ ] U = q B D B q d Ω, U = [ ] { } Ω q q. s th stffnss matrx of th lmnt (symmtrcal, sngular, sm-postv dfnd) Ω = B D B dω LWE=3 LWE=6 LWE=4 LWE=8 lmnty trójwymarow w u v
FEM II - Lctur Pag 0 of 4 {σ} {σ} Avragd strss vctor at nod n {σ} n av = Σ {σ} n / ( =6) Nod n {σ} dscontnuous contnous strss vctor at nod n of lmnt n σ x σ y σ 3 =... τ xy τ yz τ xz z { σ} { σ} { σ} { σ} n n n n n avragd strss vctor at nod n: { } av = σ = n { σ } n
FEM II - Lctur Pag of 4 Strss rror vctor at nod n of lmnt { σ } = { σ} { σ} av n n n Th strss rror vctor { σ} wthn th lmnt may b dtrmnd by standard approxmaton usng th strss rror vctors at nods of lmnt { σ }. Stran nrgy of th lmnt n U = σ { ε} dω Thn for ach lmnt so calld nrgy rror can b stmatd = σ [ D] { σ} dω (ETABLE-SERR) Ω [ Ω { ε} [ D] { σ } =, D] strss-stran matrx U = σ [ D] { σ } dω Ω JUN 008 0:49:0 PLOT NO. ELEMENT SOLUTION STEP= SUB = TIME= SERR (NOAVG) D =.086853 SMN =.30E-08 S =.96E-03.30E-08.07E-03.4E-03.3E-03.47E-03.534E-03.64E-03.748E-03.855E-03.96E-03 JUN 008 0:49:6 PLOT NO. ELEMENT SOLUTION STEP= SUB = TIME= SDSG (NOAVG) D =.086853 SMN =.00959 S =.3.00959.367.74 4.08 5.438 6.795 8.5 9.509 0.866.3 Y Y Z X Z X Intal msh. SERR and SDSG rror dstrbuton SDSG - σ = maxmum absolut valu of any componnt of { } { } { } av σ = σ σ for all nods connctd to lmnt n n n
FEM II - Lctur Pag of 4 Th nrgy rror ovr th modl l. l. = = Th nrgy rror can b normalzd aganst th stran nrgy SEPC = 00 U + U total stran nrgy ovr th ntr modl SEPC prcntag rror n nrgy norm Th valus can b usd for adaptv msh rfnmnt. It has bn shown that f s qual for all lmnts, thn th modl usng th gvn numbr of lmnts s th most ffcnt on. Ths concpt s also rfrrd to as rror qulbraton ( = const, SEPC< S 0 ).
FEM II - Lctur Pag 3 of 4 Adaptv Mshng Tchnqus - Automatc rfnmnt of FE mshs untl convrgd rsults ar obtand - Usr s rsponsblty rducd to gnraton a good ntal msh START Intal msh IF ( /N )/(/N )<ps SEPC<ps END YES NO Msh rfnmnt (dffrnt msh gnraton tchnqus) Y Z X Y Z MN X TIME= SY (AVG) RSYS=0 D =.08685 SMN =-.74 SMNB=-4.773 S =374.04 SB=375.98 -.74 3.68 74.49 7.3 60.033 0.834 45.636 88.437 33.39 374.04 Fnal FE msh and th rsults - Sy dstrbuton (prcntag rror n nrgy norm SEPC=0.8%, unform rror dstrbuton = const)
FEM II - Lctur Pag 4 of 4 Slctv adaptv mshng If msh dscrtzaton rror (masurd as a prcntag) s rlatvly unmportant n som rgons of th modl, th procdur may b spd up by xcludng such rgons from th adaptv mshng opratons. Also - nar sngularts causd by concntratd loads and at boundars btwn dffrnt matrals. Slctv adaptv mshng n ANSYS Typs of rfnmnt n adaptv mshng : h-rfnmnt: rducton of th sz of th lmnt ( h rfrs to th typcal sz of th lmnts) p-rfnmnt: ncras of th ordr of th polynomals on an lmnt (shap functons from lnar to quadratc, tc.) r-rfnmnt: r-arrangmnt of th nods n th msh hp-rfnmnt: combnaton of th h- and p-rfnmnts.