Smoothing and Adaptive Stretching of Discontinuous Fields in Finite Element Applications

Transcription

1 Smoothng and Adaptve Stretchng of Dscontnuous Felds n Fnte Element Applcatons INVYDAS RAGULSKIS Department of athematcal Research n Systems Kaunas Unversty of echnology Studentu 50-, Kaunas L-5368 LIHUANIA ALGIANAS FEDARAVICIUS Department of heoretcal echancs Kaunas Unversty of echnology Kestuco 7, Kaunas L-4405 LIHUANIA JURAE RAGULSKIENE Department of echncal Scences Kaunas College Pramones 0, Kaunas L LIHUANIA Abstract: - A technque for adaptve conjugate smoothng of dscontnuous felds s presented n the paper. he descrbed technque s applcable n varous engneerng problems and s especally effectve when hybrd numercal epermental methodologes are used. Adaptve mage smoothng and stretchng strategy s llustrated for a dscontnuous plan stress feld problem when photoelastc frnges representng the varaton of stress are constructed n vrtual projecton plane. Key-Words: - Dscontnuous feld, Smoothng, Fnte element method, Interference frnges Introducton Vsualsaton technques of the results from fnte element analyss procedures are mportant due to several reasons. Frst s the meanngful and accurate representaton of processes takng place n the analysed structures. Second, and perhaps even more mportant, s buldng the ground for hybrd numercal - epermental technques. A typcal eample of fnte element analyss n developng a hybrd technque s presented n []. Unfortunately, conventonal FE analyss technques are based on the appromaton of nodal dsplacements (not stresses) va the shape functons. Ramesh et al [] have correctly noted that photoelastc sochromatcs can be effectvely used for the detecton of FE meshng problems. Conventonal FE would requre unacceptably dense meshng for producng suffcently smooth photoelastc patterns. ultscale meshng s not affordable ether - the whole doman of the structure must be analysed wth the same accuracy. herefore there ests a need for the development of a technque for smoothng the generated photoelastc frnge patterns representng the stress dstrbuton and calculated from the dsplacement dstrbuton. he proposed smoothng technque s based on conjugate appromaton used for the calculaton of nodal values of stresses and provdes the dgtal mages of acceptable qualty on relatvely rather coarse meshes. he purpose of the paper s the development of technques for smoothng of dscontnuous felds applcable n fnte element analyss enablng effectve nterpretaton of epermental results what provdes nsght nto the physcal processes takng place n the analysed objects.

2 Smoothng of Dscontnuous Felds he components of stresses n the doman of the analysed fnte element can be calculated n the usual way [3]: σ σ y τ y [ D][ B]{ } δ 0, () where {δ 0 } s the vector of nodal dsplacements of the egenmode; [B] s the matr relatng the strans wth the dsplacements; [D] s the matr relatng the stresses wth the strans; σ, σ y, τ y are the components of the stresses n the problem of plane stress. It can be noted that the dsplacements are contnuous at nterelement boundares, but the calculated stresses are dscontnuous due to the operaton of dfferentaton. he most natural way for the calculaton of the nodal values of stresses s the mnmsaton of the squared dfference between the dscontnuous stran feld defned by eq.() and the nterpolated stress feld through the nodal values of the stress components. hat dfference s ntegrated n the domans of the approprate elements: ([ N ]{} σ ), δ ddy () e where σ denotes correspondng component of stress; {δ} s the vector of nodal values of σ; [N] s the row of the shape functons of the fnte element; e stands for the doman of the -th fnte element; summaton denotes the drect stffness procedure n the global doman [3]. Unfortunately, the soluton of unknown nodal values {δ} from eq.() s unsatsfactory for generaton of dgtal photoelastc mages as the dervatves of the nterpolated stress fleds are stll dscontnuous. hat s llustrated n the numercal results. herefore, addtonal penalty terms for fast varaton of the stress felds are ntroduced: σ σ λ +, (3) y where λ s a parameter of smoothng. Addton of the penalty term defned n eq.(3) to the resdual n eq.() after elementary transformatons leads to: e ([ N]{ δ} σ ) + + λ{}[ δ C] [ C]{} δ ddy, (4) where [C] s the matr of the dervatves of the shape functons. he mnmsaton of resduals defned by eq.(4) leads to the followng systems of lnear algebrac equatons for the determnaton of each of the component of the stresses: ([ N ] [ N] + [ C] λ[ C] ) ddy {} δ [ N ] e e σddy (5) he selecton of the smoothng parameter λ can be performed usng the error norms of the fnte elements [3] and the components of the stresses are nterpolated from ther nodal values by usng the shape functons of the fnte elements. 3 One-Dmensonal Eample he formulaton of the smoothng procedure s llustrated by one-dmensonal model comprsng three fnte elements each consstng from 3 nodes (Fg. ). he co-ordnates of the nodes are k k ; k,,..., 7. Fg.. One-dmensonal eample. hen, the shape functons of the -th fnte element are: N + + ; ; (6) () N N () he dstrbuton of stran and stress felds n the doman of the -th element s nterpolated as: ε ( ) () N ε () + N ε () N ε ; + 3 +

3 σ () () B ε () + B ε () B ε, (7) where () B s the dervatve of shape functon j () N, j,..., 3. hen, elementary transformatons j lead to: [ N ] [ N ] ( ) [ C] ( ) [ N ] ( ) d ; [ C] d ; (8) ε + ε ε () σ d ε + ε ε ε + ε Summaton over the global doman and soluton of algebrac system of equatons n eq. (5) produces a set of dscrete nodal components of stress {δ} whch can be nterpolated n the domans of approprate fnte elements: Fg.. Smoothed felds of stresses. Eplctly, I m m m a a 0 * I a +, (0) a ma a ma where I a s the smoothed feld; I* s the stretched feld; m nf mn S,0 ; ( ) ( ( S ) (, )) 0 0 sup ma 0 ; m a a ( ( mn S ) (, R) ) nf λ ; () ( ( ma S ) (, R) ) sup λ. S ( ) (, λ) () () () δ ( λ) N + δ ( λ) N δ ( λ) N (9) Partcularly, for ε 0.4; ε 0.35; ε 3 0.; ε 4 0.4; ε ; ε and ε the theoretcal dscontnuous, nterpolated contnuous and smoothed contnuous felds of stresses are presented n Fg.. Array of thn sold lnes correspond to ncreasng values of the smoothng parameter λ. It can be noted, that the smoothed felds of stresses s much more compressed comparng wth the theoretcal dscontnuous feld of stresses. herefore the number of reconstructed nterference frnges would be too small and would not represent the physcal effects takng place n the analysed system. hs problem can be avoded by stretchng the smoothed feld of stresses to the lmts of the non-smoothed feld. Fg. 3. Smoothed and stretched felds of stresses. Fg. 3 presents the effect of stretchng when the feld of stress s not only smoothed but also stretched up to range of numercal values whch guarantees correct representaton of stress dstrbuton n the global doman. 3 Fnte Element odel he problem of smoothng of dscontnuous felds s llustrated by fnte element analyss of bendng

4 vbratons of mcromechancal components comprsng photoelastc coatngs. Bendng vbratons are common n dfferent engneerng and physcal applcatons. Bendng vbratons of centrally clamped rotatng crcular dsks play crucal role n the functonalty of hard dsk drves. Lots of efforts are spent for dynamc stablsaton, control and measurement of bendng vbratons n such mcro-mechancal systems [4]. easurement of mcroscopc deflectons from the state of equlbrum s a challengng epermental problem. Dfferent optcal measurement technques are developed for epermental nvestgaton of bendng vbratons [5]. Unfortunately, nterpretaton of epermental measurement results s a nontrval nverse engneerng problem often havng non-unque solutons. herefore there ests a defnte need for hybrd numercal epermental technques that could help to nterpret the measurement results (Fg. 3). Fg. 4. Fnte element mesh of dsk wth clamped nternal radus. Plate bendng element wth the ndependent nterpolaton of the dsplacement w and the rotatons about the approprate aes θ and θ y s used [3]. he schematc representaton of the plate and the coatng s shown n Fg.., y and z are the aes of the orthogonal Cartesan system of coordnates; w s the dsplacement of the plate; θ and θ y are the rotatons h of the plate about the approprate aes; u θ y h and v θ are the dsplacements on the surface of the plate; h s the thckness of the plate; d s the thckness of the photo-elastc coatng. Fg. 3. Schematc dagram of hybrd numercalepermental technques. Such technques usually comprse a numercal model of the system coupled wth optcal and geometrcal parameters of the measurement set-up. hen the predcted response of the epermental optcal measurement system can be mmcked n vrtual numercal envronment when the dynamcal parameters of the analysed object are pre-defned. Fnte element mesh of a vbratng mcromechancal centrally clamped dsk s shown n Fg. 4. he mesh n the status of equlbrum s grey and deflected accordng to the egenmode s black. Fnte element technques are used to construct the numercal model of a centrally clamped crcular dsk. Fg. 4. Schematc model of the fnte element. he nodal varables of the plate bendng element are the deflecton of the plate w, the rotaton of the plate about the as θ and the rotaton of the plate about the y as θ y. he prncpal stresses σ, σ at each node are calculated as the egenvalues of the matr: σ τ y τ y, () σ y and the normalsed egenvectors of ths matr {V }, {V } are the drectons of the prncpal stresses. he vector of polarsaton s gven as: cosα P, (3) snα { }

5 where α s the angle of the vector of polarsaton. hen the ntensty n the photoelastc mage of the plane polarscope (soclncs and sochromatcs ntertwned) s calculated as stresses up to the orgnal range of values (Fg. 5.C) provdes correct number of frnges wth acceptable smoothness. (({ V } { P} ){ ( V } { P} ) ( σ )) I C, (4) sn σ where C s the constant dependent on the thckness of the analysed structure n the state of plane stress and on the materal from whch t s produced. he ntensty of the photoelastc mage for the crcular polarscope (sochromatcs) s calculated as: ( sn ( σ )) I C σ. (5) he relatonshps presented above form the bass for the generaton of dgtal photoelastc mages. he procedure of constructon of dgtal mages n projecton planes from soparametrc fnte elements s descrbed n detal n [6]. Fg. 6. Isoclncs and sochromatcs ntertwned. Fnally, soclncs and sochromatcs ntertwned are presented n Fg. 6 (polarzaton angle α π/8). Such numercal reconstructon provdes hgh qualty dgtal mages well acceptable for hybrd numercal epermental technques. Fg. 5. Photoelastc mages of a clamped dsk: A unsmoothed mage; B smoothed mage; C smoothed and stretched mage Photoelastc frnges (crcular sochromatcs) of the vbratng dsk are presented n Fg. 5. hree mages are overlapped due to the space lmtatons. It can be clearly seen that the frnges are broken n the unsmoothed mage A, what s unacceptable for stress feld analyss. he smoothng procedure produces better pattern of frnges (Fg. 5.B). Nevertheless, t can be notes that the number of frnges n the smoothed mage s less that n the orgnal unsmoothed one. Stretchng of the feld of 4 Concludng Remarks he dsplacement based FE formulatons are coupled wth stress based photoelastcty analyss. As the stress feld s dscontnuous n the nterelement boundares, the ntroduced smoothng and stretchng procedure enables the generaton of hgh qualty dgtal mages acceptable for hybrd numercal epermental technques. It can be noted that the generaton of the dgtal photoelastc mages s not a straghtforward procedure. It nvolved such steps as the constructon of the numercal model of the analysed object; fnte element calculatons based on the loadng scheme, boundary condtons; determnaton of the nodal vales of stress components and ther smoothng; generaton of approprate dgtal mages. References: [] A. Holsten, L. Salbut,. Kujawnska, W. Juptner, Hybrd epermental-numercal concept of resdual stress analyss n laser

6 weldments, Epermental echancs, Vol.4, No.4, 00, pp [] K. Ramesh, P.. Pathak, Role of photoelastcty n evolvng dscretzaton schemes for FE analyss, Epermental echnques, Vol.3, No.4, 999, pp [3] K.H. Huebner, D.L. Dewhrst, D.E. Smth,.G. Byrom, Fnte element method, Wley, 00. [4] P. anzone, A.H. Nayfeh, Instablty mechansms of a centrally clamped rotatng crcular dsk under a space-fed sprngmass-dashpot system, Journal of Vbraton and Control, Vol.7, No.7, 00, pp [5] A.S. Kobayash, Handbook on epermental mechancs, nd ed. SE, 993. [6]. Ragulsks, A. Palevcus, L. Ragulsks, Plottng holographc nterferograms for vsualsaton of dynamc results from fnte element calculatons, Internatonal Journal for Numercal ethods n Engneerng, Vol.56, No., 003, pp