Stress analysis by local integral equations

Transcription

1 Boundary Elements and Oter Mes Reduton Metods XXIX 3 Stress analyss by loal ntegral euatons V. Sladek 1, J. Sladek 1 & C. Zang 2 1 Insttute of Construton and Arteture, Slovak Aademy of Senes, Bratslava, Slovaka 2 Department of Cvl Engneerng, Unversty of Segen, Germany Abstrat Ts paper s a omparatve study for varous numeral mplementatons of loal ntegral euatons developed for stress analyss n plane elastty of solds wt funtonally graded materal oeffents. Besdes two mesless mplementatons by te pont nterpolaton metod and te movng least suares approxmaton, te element based approxmaton s also utlzed. Te numeral stablty, auray, onvergene of auray and ost effeny (assessed by CPU-tmes) are nvestgated n numerous test examples wt exat benmark solutons. Keywords: elastty, funtonally graded materals, boundary value problems, fore eulbrum, mesless mplementatons. 1 Introduton A rapd progress an be observed n te development of varous mesless tenues espeally n flud problems. Smultaneously, a onsderable expanson of su tenues an be found also n varous applatons to engneerng and sene problems. Ts an be explaned by te fat tat tere are known ertan lmtatons of standard dsretzaton tenues espeally wen appled to some lasses of problems (e.g. problems n separable meda, problems wt free or movng boundares; rak problems; problems wt large dstortons, et.). Altoug te standard dsretzaton tenues are applable to te numeral soluton of boundary value problems n ontnuously nonomogeneous elast meda, te formulatons developed for omogeneous meda are not applable dretly, sne te governng euatons are now gven by partal dfferental euatons wt varable oeffents. Tere as not been a unue lassfaton of mesless tenues up to now. Mostly tey are lassfed aordng to te employed approxmaton. Some of WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne) do: /be070011

2 4 Boundary Elements and Oter Mes Reduton Metods XXIX te tenues utlze mesless approxmaton of feld varables but a bakground mes s stll reured for numeral ntegraton espeally n approaes based on global formulatons. On te oter and, te loal formulatons brng a possblty to avod te mes ompletely wt usng nodes alone for approxmaton. Ten, te pysal prnples an be formulated n ntegral forms on loal sub-domans. A large group of mesless tenues are denoted as mesless loal Petrov Galerkn metods [1, 2] wt bearng n mnd tat te Petrov Galerkn weak form dea s appled n a loal sense wt seletng te tral and test funtons ndependently and approxmatng te feld varables n a mesless way. Some omparatve studes mgt be desred n ts stage of rapd nrease of lterature devoted to varous mesless tenues as well as ter applatons to varous problems. 2 Loal ntegral euatons Under assumpton of stat loadng ondtons, te demand of te fore eulbrum n an arbtrary but small part of te elast body results n te strong formulaton of te governng euatons gven by te partal dfferental euatons σ j, j ( x) + X ( x ) = 0 n Ω, (1) supplemented by te generalzed Hooke s law σ j ( x) = jkl ( x) u k, l ( x ). (2) In te ase of sotrop FGM, te spatal varaton of te tensor of materal oeffents s usually gven va te varable Young s modulus as jkl ( x) = E( x ) jkl o 1 2, o ν jkl = δ 2(1 ) k δ jl + δ l δ jk + δ 1 2 j δ + ν ν kl, (3) wt te materal parameter ν beng expressed n terms of te onstant Posson rato ν by ν /(1 + ν), for plane stress ondtons ν =. ν, oterwse Insertng (3) nto (1), one obtans te governng PDE for dsplaements o o E( x) u ( x) + E ( x) u ( x) = X ( x ). (4) jkl k, lj, j jkl k, l Te standard boundary ondtons presrbe a alf of te boundary uanttes { u ( η), t ( η)} for ( = 1,..., d ) at ea boundary pont η Ω, wt te traton vetor beng gven by t ( η) = n j ( η) jkl ( η) u k, l ( η ), (5) WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

3 Boundary Elements and Oter Mes Reduton Metods XXIX 5 were n j ( η ) denotes te Cartesan omponents of te unt outward normal vetor on te boundary Ω. In numeral formulatons for soluton of b.v.p., weak formulatons are freuently utlzed nstead of te strong formulaton. Te governng euaton s satsfed n a weak sense f te wegted ntegral of te governng euaton s fulflled (, ) σ j j ( x) + X ( x) w k ( x) dω ( x ) = 0. (6) Ω Sne te test (or wegt) funtons an be arbtrary, te weak formulaton mgt ave no pysal nterpretaton. In order to apply te formulaton wt lear pysal nterpretaton, we sall use te test funtons gven by te Heavsde funtons wt support on loal sub-domans Ω of te wole analysed doman Ω δ, x Ω w ( x) = k k. 0, x Ω Ten, te weak formulaton (6) after usng te Gauss dvergene teorem yelds te well known fore eulbrum on loal sub-domans Ω η η η η x x, (7) nj( ) jkl ( ) uk, l( ) d Γ ( ) = X ( ) d Ω ( ) Ω Ω w s te weak formulaton wt te lear pysal nterpretaton. Reall tat te loal ntegral euatons (7) are non-sngular, sne tere are no sngular fundamental solutons nvolved n ontrast to te sngular ntegral euatons employed n te boundary ntegral euaton metod. Moreover, te ntegraton of unknown (approxmated) feld varables s onstrant to te boundary of loal sub-domans even n te ase of non-omogeneous meda. Ts an be effetvely utlzed by dereasng te amount of ntegraton ponts as ompared wt te formulatons nvolvng te doman ntegrals. 3 Numeral mplementatons In numeral solvng, n general, te amount of degrees of freedom s dereased from nfnty to a fnte number by approxmatng te feld varable n terms of ertan sape funtons and nodal unknowns. Te nodal unknowns are determned by te set of euatons obtaned by olloatng te presrbed boundary ondtons at boundary nodes and fore eulbrum euatons at nteror nodal ponts. We sall onsder doman-type approxmatons ( x) for te dsplaements u x wtn a sub-doman Ω ( Ω Ω ). Ten, t s possble to get also te ( ) s approxmatons for dsplaement gradents by dfferentatng te approxmated dsplaements. Tus, te dsretzed euatons take te form u WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

4 6 Boundary Elements and Oter Mes Reduton Metods XXIX u () ζ = u () ζ at ζ Ω were u () ζ s presrbed, (8a) n j () ζ jkl () ζ ukl, () ζ = t () ζ at ζ Ω were t () ζ s presrbed, (8b) ( ) ( ) n j η jkl η u k, l ( η) dγ ( η) = X ( x) dω( x ) (8) Ω Ω on sub-domans Ω around nteror nodes y. 3.1 Quadrlateral uadrat elements A 2-d plane doman Ω s assumed to be subdvded nto m onformng uadrlateral serendpty elements S [3] wt uadrat polynomal nterpolaton e for te approxmaton of bot te geometry and dsplaements. Ten, m 8 ae a 8 ae a Ω= Se, x = ( 1, 2) S x N ξ ξ, u ( x) = u ( x ) N ( ξ, ξ ), (9) e 1 2 e= 1 a= 1 S e a = 1 ae were x are te Cartesan oordnates of te a -t nodal pont on S e and N a represent te sape funtons. Sne te nterpolaton polynomals are expressed as funtons of ntrns oordnates, te expressons for dsplaement gradents are not trval [4] and ntegratons are to be arred out n te transformed ntrns spae. Te loal sub-doman Ω s spefed as unon of elements adjaent to te nteror node y. 3.2 Pont nterpolaton metod (PIM) As n all mesless approxmaton tenues, te sape funtons derved for te approxmaton of te feld varable u (x) wtn a sub-doman Ω s utlze only nodes sattered arbtrarly n te analyzed doman wtout any predefned mes to provde a onnetvty of te nodes. Wtout gong nto detals [5, 6], we present te nterpolaton formula for dsplaements n surroundngs of te nodal pont x n terms of te sape funtons and nodal values as N n (, α) (, α) u ( x) = u ( ) ( ) x ϕ x, (10) Ω α = 1 were nα (, ) stands for te global number of nodes from te nterpolaton doman nvolved n Ω. If Ω s defned as a rle wt te radus N t Ω s gven as N = H( r a ) x x, a= 1 r te number of nodes WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

5 Boundary Elements and Oter Mes Reduton Metods XXIX 7 were H ( z) s te Heavsde unt step funton and Nt s te total number of nodes. Te numerally stable development of te sape funtons an be aeved by ombnng te polynomals and RBFs as bass funtons n a PIM(P+RBF) approa [5, 6]. Te explt expresson for te sape funtons beng gven elsewere [6]. Reall tat te Kroneker-delta property s satsfed (, α) n(, β) ϕ ( x ) = δ αβ. Fnally, te dsplaement gradents are approxmated as gradents of approxmated dsplaements u N n(, α ) (, α ), j = u ϕ, j Ω α = 1 ( x) ( x ) ( x ), (11).e., n terms of te nodal values of dsplaements and te dervatves of te sape funtons. Note tat te sape funtons and ter dervatves are not avalable n losed form. Tus, ertan omputatonal algortm s to be repeated at ea evaluaton pont. Neverteless, n te present formulaton, some of te nverse matres an be pre-omputed and stored n te memory for ea nodal pont n order to save CPU-tme. 3.3 Movng least suares (MLS) approxmaton Te MLS-approxmaton s wdely used n mesless metods. Te dsplaements are approxmated n terms of ertan sape funtons and nodal unknowns as N t a a a= 1 u ( x) = φ ( x ) uˆ. (12) Te sape funtons are expressed n terms monomal bass funtons and wegts assoated wt ea nodal pont. Tey ave to be omputed aordng to ertan algortm at ea evaluaton pont. Sne te sape funtons do not a b possess te Kroneker delta property, φ ( x ) δ ab, n general, te nodal unknowns are expanson oeffents (fttous nodal dsplaements) w are dfferent from te atual nodal values of dsplaements. Sne te number of nodal ponts w ontrbute to te sum n E. (12) s ontrolled by te wegts, one as to onsder all te nodes n te summaton. To derease te amount of te onsdered nodes, te Central Approxmaton Node (CAN) onept an be used. Ten, te number of onsdered nodes n ea evaluaton at x s redued from N t to N, were N < N t s te number of nodes supportng te approxmaton at te entral approxmaton node x. Te nodes supportng te approxmaton wt te CAN loated at x le n te Ω CAN spefed by te radus r. Ten, nstead of te approxmaton gven by E. (12), one an use WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

6 8 Boundary Elements and Oter Mes Reduton Metods XXIX N ˆ na (, ) u ( ) n(, a) x = u φ ( x ). (13) a = 1 Te gradents of dsplaements are approxmated as gradents of approxmated dsplaements gven by Es. (12) and (13). In ontrast to te mplementaton based on fnte elements, te ntegraton n mesless approaes s arred out n te global Cartesan oordnate system. Te oe of smple sape for sub-domans yelds smple ntegraton. Te Gaussan uadrature proved to be more onvenent tan te trapezodal rule for ntegraton over te boundary of rular sub-domans, sne te later exbts very slow mprovement of auray wt fnng te subdvson of te ntegraton nterval, wat results n enormous nreasng te omputatonal tme needed for evaluaton of sape funtons. 4 Numeral tests In order to test te proposed numeral metods, we onsder examples for w analytal solutons are avalable. Te body fores are vansng n Ω, te Posson rato s onstant ν = 0.25, plane stress ondtons are assumed and for onseness, we present only te numeral results for exponental gradaton E( x ) = Eo exp(2 δ x 2 / L) wt δ = 3. Te onsdered doman s a suare L L wt appled tenson load on te top, fxed bottom n vertal dreton and tratons on te lateral sdes are gven by te analytal soluton [7]. In te study of te onvergene and auray of te numeral results wt respet to te nreasng densty of nodal ponts, we use te dsplaement norm % error defned as 1/2 1/2 N N t a a t ex a ex a dspl. norm error (%) = 100 u u / u ( ) u ( ) x x a = 1 a = 1 a num a ex a u = u ( x ) u ( x ), (14) were N t s te total number of nodes on te losed doman Ω Ω. In most of te presented omputatons, we sall use a omogeneous a a b dstrbuton of nodes wt = mn{ x x } = onst =. b In te PIM, we ave used ombnaton of polynomal funtons (gven by sx monomal bass) wt RBFs for w we onsdered multuadrs, Gaussan RBFs, and te 8-order splne. Smlar n te ase of MLS-approxmaton, we ave used tree dfferent knds of wegts gven by Gaussan, exponental, and 8-order splne wegts. Altoug te sape and sze of sub-domans an be osen arbtrarly, te results of numeral omputatons may depend on tese aspets and smlarly on te sape parameters nvolved n bot te RBFs and WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

7 Boundary Elements and Oter Mes Reduton Metods XXIX 9 wegts of te MLS-approxmaton. Terefore, frstly we ave nvestgated te stablty of numeral omputatons wt respet to tose tree ndators. Te use of suare sape for sub-domans yelds better results as ompared wt te rular sape. In te next omputatons, we used optmal values for te sape parameter and te sze of sub-domans w guarantee te numeral stablty. Te numeral nstablty wt respet to aeptable auray was observed n ase of exponental wegts used n MLS-approxmaton. Te CAN onept wt te nearest node to te evaluaton pont proved to gve te best results n bot te mesless tenues. Te radus of te nterpolaton doman s taken as a r = Fg. 1 sows te onvergene of te numeral solutons by varous PIM(P+RBF) approaes. Te nreasng densty of nodes s represented by te dereasng parameter / L. Fg.2 llustrates te varaton of te dsplaement feld u2( L/2, x2) along te vertal lne ( L/2, x 2 ) wt x 2 [0, L/ 2]. It an be seen tat exellent auray s aeved even n te ase of strong gradaton of Young s modulus wen te varaton of dsplaements dffers dramatally from te ase of omogeneous medum. Te auray of numerally omputed nteror stresses s also reasonable (te results wll be summarzed n Tab.1). Tus, te PIM based on te ombnaton of polynomals and te multuadrs wt m = 5/2seems to be approprate even for strong non-omogenety δ = 3, wen te Young modulus on te top of te suare doman s 403 tmes ger tan on te bottom. a Fgure 1: Convergene study. Fgure 2: Dsplaement results. It s nterestng to ompare te results by two varants of te MLSapproxmaton: standard formulaton vs CAN-nearest node onept. Fg. 3 sows te omparson of te onvergene of numeral results by usng tese two dfferent approaes wt Gaussan wegts. It an be seen tat te auray s almost nvarant wt respet to te predefnton of supportng nodes. On te oter and, te nfluene on te CPU-tmes s mu more sgnfant (Fg. 4). Altoug we an see neglgble dfferenes between te CPU-tmes resultng WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

8 10 Boundary Elements and Oter Mes Reduton Metods XXIX from bot te formulatons provded tat te denstes of nodes are low, te rates for te CPU-tme nrease are sgnfantly dfferent for ea of te employed approaes. Te dfferene between te CPU-tmes by te CAN-nearest node approa and standard MLS approa s nreasng remarkably wt nreasng te densty of nodes. Fnally, we present some omparsons of te best formulatons based on te use of tree dfferent knds of doman-type approxmatons. Te best mesless formulatons utlze te CAN-nearest node onept and are araterzed by seleton of suare sapes for sub-domans, and optmal values of te sape parameter (nvolved n RBFs and/or wegts) as well as te sub-doman sze parameter. Te QE-approa exbts relable onvergene of auray wt nreasng te densty of nodes, but lower auray s aeved n te FGM sample wt strong gradaton of te Young modulus ( δ = 3 ) as ompared wt te mesless PIM and/or MLS results (Fg. 5). Fgure 3: Auraes by two MLS onepts. Fgure 4: CPU by two MLS onepts. Fgure 5: Comparson of auraes by varous numeral tenues. WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

9 Boundary Elements and Oter Mes Reduton Metods XXIX 11 A omparson of te CPU-tmes by te QE-approa wt varous mesless approaes s gven n Fg. 6. Te CPU-tmes by mesless approaes onverge to ea oter by nreasng te densty of nodes and te dfferenes between te QE and mesless approaes are dmnsed. Ts an be explaned by te fat tat wt nreasng te amount of nodes te tme needed for soluton of te system of dsretzed euatons s beomng domnant n omparson wt te tme needed for evaluaton of te system matrx. Fgure 6: Table 1: Comparson of CPU-tmes by varous numeral tenues. Maxmal % errors for dsplaements and stresses omputed at nteror ponts along te vertal lne (L/2. x 2 ) n suare sample. omputatonal metod LIE-QE (400 elem.) 1281 nodes LIE- PIM(P+MQ) (441 nodes) LIE-MLS (441 nodes) grade parameter δ = max % error u σ σ δ = δ = δ = δ = δ = δ = δ = δ = CPU [se] WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)

10 12 Boundary Elements and Oter Mes Reduton Metods XXIX Te slgtly ger relatve error for te stresses σ11 s due to small value of ts omponent n te onsdered b.v.p. Smlar results ave been obtaned also for stress analyss n bot te transversal and axal ross-seton of te tk-wall tube. 5 Conlusons Bot te mesless tenues proved to be useful for numeral mplementaton of te LIEs appled to stress analyss problems even n te ase of strong gradaton of te Young modulus. Aeptable auray, onvergene of auray and numeral stablty are guaranteed by usng te proposed tenues. Great savngs n te CPU-tme are aeved by usng te CAN-nearest node onept. Te auray by te QE-approa s slgtly worse tan by mesless approaes, but relable onvergene s aeved wt nreasng te densty of nodes. Aknowledgements Te resear as been supported by te Slovak Grant Agenes VEGA, APVV and German Resear Foundaton (DFG), w are gratefully aknowledged. Referenes [1] Atlur S.N., Sen S., Te mesless loal Petrov-Galerkn (MLPG) metod, Te Sene Press: Enno, [2] Atlur S.N., Te mesless metod (MLPG) for doman & BIE dsretzatons, Te Sene Press: Forsyt, [3] Huges T.J.R., Te Fnte Element Metod. Lnear Stat and Dynam Fnte Element Analyss. Prente-Hall, In.: Englewood Clffs, [4] Sladek V., Sladek J., Zang C., Loal ntegro-dfferental euatons wt doman elements for numeral soluton of PDE wt varable oeffents. J. Eng. Matemats 51, pp , [5] Lu G.R., Mes Free Metods, Movng beyond te Fnte Element Metod. CRC Press: Boa Raton, [6] Sladek V., Sladek J., Tanaka M., Loal ntegral euatons and two mesless polynomal nterpolatons wt applaton to potental problems n nonomogeneous meda. Computer Modelng n Engneerng & Senes 7, pp , [7] Sladek V., Sladek J., Zang C., A mesless Pont Interpolaton Metod for Loal Integral Euatons n elastty of non-omogeneous meda. Advanes n te Mesless Metod, eds. J. Sladek, V. Sladek, Te Sene Press: Forsyt, WIT Transatons on Modellng and Smulaton, Vol 44, ISSN X (on-lne)