Smoothed Finite Element Method

Transcription

1 Smooted Fte Elemet Metod K.Y. DA ad G.R. LU, 2 Sgapore-MT Allace (SMA), E4-04-0, 4 Egeerg Drve, Sgapore, eter for Advaced omputatos Egeerg Scece (AES), Departmet of Mecacal Egeerg, Natoal Uversty of Sgapore, 0 Ket Rdge rescet, Sgapore, 9260 Abstract ts paper, te smooted fte elemet metod (SFEM) s proposed for 2D elastc problems by corporato of te cell-wse stra smootg operato to te covetoal fte elemets. We a costat smootg fucto s cose, area tegrato becomes le tegrato alog cell boudares ad o dervatve of sape fuctos s eeded computg te feld gradets. Bot statc ad dyamc umercal eamples are aalyzed te paper. ompared wt te covetoal FEM, te SFEM aceves more accurate results ad geerally ger covergece rate eergy wtout creasg computatoal cost. addto, as o mappg or coordate trasformato s performed te SFEM, te elemet s allowed to be of arbtrary sape. Hece te well-kow ssue of te sape dstorto of soparametrc elemets ca be resolved. de Terms fte elemet metod (FEM), Gauss quadrature, soparametrc elemet, smooted fte elemet metod (SFEM), stra smootg. A. NTRODUTON fter more ta alf a cetury of developmet, fte elemet metod (FEM) as become a very powerful tecque for umercal smulatos egeerg ad scece. Mapped elemets, suc as te well-kow soparametrc elemets, play a very mportat role FEM. We usg a mapped elemet, a basc requremet s tat te elemet as to be cove ad a volet dstorto s ot permtted so tat a oe-to-oe coordate correspodece betwee te pyscal ad atural coordates assocated wt a elemet ca be guarateed. umercal mplemetato, te determat of te Jacoba matr sould be always cecked for ts postvty to avod severely dstorted elemets [, 2]. Recetly, a stablzed coformg odal tegrato as bee proposed usg a stra smootg tecque for a Galerk mes-free metod wc sows ger effcecy, desred accuracy ad coverget propertes []. addto, a lear eactess ca be guarateed te soluto of Galerk weak-form based mes-free metods. ts paper, we mplemet te stra proecto dea to formulate ad code a ovel metod, smooted fte elemet metod (SFEM) [4, 5], wc combes te estg FEM tecology wt te stra smootg tecque. We wll demostrate troug tesve case studes te sgfcat beefts arsg from ts ovel combato.. STRAN SMOOTHNG A 2D statc elastcty problem ca be descrbed by equlbrum equato te doma bouded by Γ σ + b = 0 (), wc subect to te boudary codtos: σ = t o Γ t ad u = u o Γ u, were σ s te compoet of stress tesor ad b te compoet of body force; s te ut outward ormal. ts varatoal weak form s derved as δ s ( u) Dkl s ( u) kl d δutdγ = 0 (2) Γt te SFEM, elemets are used as te FEM. Galerk weak form gve Eq. (2) s appled ad tegrato s performed o te bass of elemet. Depedg o te requremet of stablty, a elemet may be furter subdvded to several smootg cells (S) [4]. A smootg operato s performed for eac smootg cell wt a elemet, as epressed by u ( ) = u ( ) Φ( ) d () were Φ s a smootg fucto. For smplcty, a pecewse costat fucto s appled ere, as gve by Φ( ) = / A ( ) ad Φ( ) = 0 ( ), were te area of te cell A = d ad s te doma occuped by te smootg cell (see Fg. ). Substtutg Φ to Eq. (), oe ca get te smooted gradet of dsplacemet u ( ) = u ( ) ( ) Φ( ) dγ = A Γ Γ u ( ) ( ) dγ Te dsplacemet feld a elemet ca be appromated as te FEM by u ( ) = N u (5) NP (4) Mauscrpt receved November 7, 2006.

2 Smlarly te smooted stra dscrete form ca be obtaed as follows ε ( ) = B ( ) d (6) NP were B s te smooted stra matr. For a 2D case b ( ) 0 B ( ) = 0 b 2 ( ) b ( ) ( ) 2 b were bk ( ) = N ( ) k ( ) dγ, ( k =, 2). Te A Γ smooted elemet stffess matr ca be obtaed by assembly of all te smootg cells assocated wt te elemet,.e., T K e = BDB (7) 8(0.5, 0, 0, 0.5) (.0, 0, 0, 0) 8 S=4 4 4(0, 0, 0,.0) S= 5 (a) 7 9(0.25, 0.25, 0.25, 0.25) 5(0.5, 0.5,0, 0) (b) 7(0, 0, 0.5, 0.5). SFEM SHAPE FUNTONS AND STABLTY ONDTON te SFEM, as oly te sape fucto tself s volved calculatg te gradet matr, very smple sape fuctos ca be utlzed at Gauss pots o te edges of a 6 2 (0, 0,.0, 0) 6(0, 0.5, 0.5, 0) 2(0,.0, 0, 0) Fg. Dvso of S ad ostructo of sape fuctos cell. Te SFEM sape fuctos sould possess te followg crtera: () Delta fucto: N ( ) = δ ; (2) Partto of uty: N = N = () = ; () Lear compatblty: ( ) = ; (4) N ( ) 0. Ay sape fuctos satsfyg te four codtos ca be used te SFEM. Let s take a quadrlateral elemet for eample. For ay pot o ts sde, e.g., te mdpots #5, #6, #7 ad #8 sow Fg. (b), te values of te sape fuctos are calculated learly usg sape fuctos of two related odes o te sde. Te values of te sape fuctos at pot #9, te tersecto of two bmedas, are te average of tose at te four mdpots. f ts pot appes to cocde wt oe feld ode, ts sape fuctos sould adopt te values of ts ode accordgly. Sape fuctos for oter teror pots eeded le tegratos ca be easly obtaed a smlar way [4]. egevalue aalyss of a free elemet usg SFEM, f a etre elemet s used as oe smootg cell (S=) as sow Fg. (a), fve spurous zero-eergy modes are foud, wc s smlar to te case of FEM usg oe Gauss pot. Later o t s foud tat oe S s equvalet to tree depedet relatos ( σ = ). Te dfferece betwee te dsplacemet freedoms u ad te costrats s te umber of spurous modes, ad tus u σ = 5. Te oe cell smooted tegrato caot suppress te well-kow ourglass modes. Ts meas tat te use of smooted tegrato ca stll gve rse to stabltes. Te we subdvde te elemet to four cells (S=4) ad ow u σ < 0. omparg te results wt tose of FEM usg 2 2 quadrature, Oce aga te modes of te two metods cocde wt eac oter. t s also foud tat, ecept tree rgd-body-movemet modes, o zero-eergy modes est tem, wc demostrates te stable tegrato. Terefore, S=4 s recommeded 4-ode SFEM ts paper. Furter study of more geeral polygoal elemets, we ca coclude tat to esure stffess stablty, te umber of costrats arsg from S sould ot be less ta tat of te free dsplacemet freedoms [6],.e, u σ. A dodecagoal elemet s vestgated to verfy ts pot. Te elemet s dvded to 2, 4, 6 ad 2 cells. For te frst tree te rak of te stffess matr s te same as σ, wc s smaller ta te u ad accordgly stadard patc test fals. For te last case (S=2), te patc test s well passed. V. DYNAM ANALYSS We erta ad dampg effects are cosdered, te dscrete goverg equatos ca be obtaed as FEM [7] et M d& + d& t + f = f (8) were d s te vector of geeral odal dsplacemets ad (8)

3 et T T f = N bd + N tdγ (9) Γt M = N T ρnd (0) = N T cnd () Now SFEM te teral odal force s epressed as t T f = B σ d (2) f o dampg or forcg terms ests Eq. (8), after troducg te smooted versos of stra ad stress, for lear problems t reduces to M d& + Kd = 0 () were te smooted stffess matr s gve as T K J = B DB J d (4) A geeral soluto of suc a equato ca be wrtte as d = d ep( ωt). Te o ts substtuto to Eq. (), te frequecy ω ca be foud from 2 ( ω M + K) d = 0 (5) Total Lagrage formulato s used we geometrcal olear beavor s cosdered. Te tal posto of a materal pot a body s gve by a fed referece cofgurato ad te total dsplacemet at tme t s deoted as u te te curret deformed cofgurato s descrbed by = + u (6) Te curret deformato s measured by te deformato gradet matr relatve to gve by F = (7) a smlar way as used for stra, deformato gradet Eq. (7) eeds to be smooted SFEM as u u F ( ) = + δ d = d + δ A A (8) = u ( N ) dγ + δ = e ( ) + δ A were Γ A = d s te tal area of te smootg cell L study ad e ( ) = ( u ) dγ, or A e ( ) = b d (9) were b Γ = ( ) A Γ N dγ. V. NUMERAL EAMPLES A. Stadard Patc Test te stadard patc test, lear dsplacemets are mposed alog te boudares of a square patc wt at least oe teror ode. Satsfacto of te patc test requres tat te dsplacemets of all te teror odes follow eactly (to mace precso) te same fucto of te prescrbed dsplacemets. Two types of dscretzato are used, as sow Fg. 2: oe wt 0 0 regular elemets ad te oter wt rregular teror odes. t s foud tat te SFEM ca pass te patc test wt mace precso regardless of te umber of S used ad te sape of elemets. Fg. 2. Meses for stadard patc test. B. fte Plate wt a rcular Hole A plate wt a cetral crcular ole (as sow Fg. ) s vestgated tat subected to a udrectoal tesle load of.0 N/m at fty te -drecto [8]. Plae stra 2 codto s cosdered ad E=.0 0 N / m, v = 0.. Eac elemet s dvded to four smootg cells. Te computed dsplacemet ad stress are selectvely demostrated Fg. 4 ad compared wt te eact solutos. We calculatg te eergy tree scemes are adopted. S44, S/GP=4 s used for calculato of bot dsplacemet ad stress (or eergy). Lkewse S, S/GP= s employed all te tme. stead S4, S/GP=4 s used oly for dsplacemet wle reduced tegrato S/GP= s used for post-processg of stress ad eergy. From our results t s observed tat te

4 computed dsplacemets ad stresses are good agreemet wt te aalytcal solutos. Te covergece rates dsplacemet ad eergy are demostrated Fg. 5. t s observed tat a comparable covergece speeds dsplacemet ad eergy ave bee obtaed but tose of SFEM are oce aga more accurate ta of FEM. Dsplacemet u SFEM (Regular mes) Aalytcal solu. SFEM (rregular mes) y r σ 0 a O θ σ 0 Stress σ Aalytcal solu. SFEM (Regular mes) SFEM (rregular mes) y (=0) Fg. 4. Te eact ad computed dsplacemets ad stresses. Fg.. fte plate wt a crcular ole ad ts meses. Fg. 5. Te covergece rates dsplacemet ad eergy.

5 . Free Vbrato Aalyss of a Varable ross-secto Beam A catlever beam wt varable cross-secto s eamed as sow Fg. 6. Te followg parameters are used: 7 L=0; H(0)=5, H(L)=, t=.0, E =.0 0, v=0., ad ρ =.0. Te frst four atural frequeces are computed usg FEM/SFEM as gve Table. t ca be observed tat te SFEM gves better results ta FEM we usg te same mes. y O L () D. Eplct forced vbrato of a spercal sell To clearly demostrate te effect of geometrcal olearty f compared wt te lear elastc aalyss, we use te followg materal propertes as well as te geometrc parameters of te spercal sell ts eample: o R = 2. cm, t = 0.04 cm, φ = 0.9, P = 445 N, E = 68.9 GPa, v = 0.. ρ = kg/m. A cetral dfferece procedure s used to tegrate te kematcs eplctly troug tme. As te metod s codtoally stable a very small tme step s permtted tat solely depeds o te computatoal model. ts eample, Δt = 0 8 sec s used. Fg. 7(a) sows te comparso of dyamc resposes betwee lear ad olear elastc solutos. t s observed tat bot te perod ad ampltude of olear respose are about two tmes of tose of lear case. Fg. 7(b) dyamc 4 relaato s troduced wt α = 0 ad β = 0. We otce tat te respose s dampg out gradually ad statc deflecto ca be appromately retreved. Te statc lear deflecto s located rougly at te mddle part of te curves wereas te statc olear deflecto s very close to te peak ampltude, wc s agreeable wt tose usg statc olear aalyss of Newto-Rapso procedure [9]. Fg. 6. A catlever beam wt varable cross-secto ad ts meses. TABLE Frst four atural frequeces ( 0 rad/s) of a varable cross-secto catlever beam usg FEM/SFEM No. of No. of FEM FEM SFEM elemets odes (4-ode) (8-ode) (4-ode (8-odes (4-ode 66 (8-odes (4-ode Fg. 7. Lear ad olear elastc dyamc resposes of a spercal sell uder cocetrated loadg.

6 V. ONLUSON ts work, te smooted fte elemet metod (SFEM) s preseted based o te framework of FEM by corporatg a stra-smootg tecque. SFEM, feld gradets are computed drectly oly usg sape fuctos tself. As o coordate trasformato or mappg s performed SFEM, restrcto placed o te sape of elemets FEM ca be removed. ts covergece rates bot dsplacemet ad eergy of SFEM are comparable as compared wt ts couterpart of 4-ode soparametrc fte elemets but te umercal results of SFEM are geerally more accurate ta FEM solutos. Te cocluso s also verfed by te free vbrato aalyss. Te eergy rate obtaed usg S4 s two tmes ger as compared wt tat usg S44. Numercal epermets also sow tat te SFEM s geerally more effcet ta FEM especally for a mes dvded wt very large umber of elemets. REFERENES [] O.. Zekewcz ad R. L. Taylor, Te fte elemet metod (Fft edto). Oford: Butterwort Heema, [2] G. R. Lu ad S. S. Quek, Te fte elemet metod: a practcal course. Oford: Butterwort Heema, 200. [] J. S. e,. T. Wu, S. Yoo ad Y. You, A stablzed coformg odal tegrato for Galerk mesfree metod, t. J. Numer. Met. Egg.,; vol. 50. pp , 200. [4] G. R. Lu, K. Y. Da, T. T. Nguye, A smooted fte elemet metod for mecacs problems, omput. Mec., ( press, appear ole), [5] G. R. Lu, T. T. Nguye, K. Y. Da ad K. Y. Lam, Teoretcal aspects of te smooted fte elemet metod (SFEM), t. J. Numer. Metod Egg. ( press), [6] K. Y. Da, G. R. Lu ad. Ha, A -sded polygoal smooted fte elemet metod (SFEM) for sold mecacs, Fte Elemets Aal. Desg (Revsed), [7] K. Y. Da ad G. R. Lu, Free ad forced vbrato aalyss usg te smooted fte elemet metod (SFEM), J. Soud Vb. ( press), [8] S. P. Tmoseko, J. N. Gooder, Teory of Elastcty, rd Edto. New York: McGraw-Hll, 970. [9] K. Kleber ad. Wozak, Nolear Mecacs of Structures, Dordrect: Kluwer Academc Publsers, 99.