A comparative Study of Contact Problems Solution Based on the Penalty and Lagrange Multiplier Approaches

Transcription

1 Journal of he Serbian Soiey for ompuaional Mehanis / Vol. / o., 2007 / pp A omparaive Sudy of ona Problems Soluion Based on he Penaly and Lagrange Muliplier Approahes S. Vulovi, M. Zivovi,. Grujovi, R. Slavovi Fauly of Mehanial Engineering Universiy of Kragujeva, S. Janji 6, Kragujeva, Serbia zile@g.a.yu Absra umerial models based on he penaly and Lagrange muliplier mehod for ona problems wih friion are ompared in his paper. he presened approahes, wih use of oulomb s friional law, elaso-plasi angenial slip deomposiion, and onsisen linearizaion, resul in quadrai raes of onvergene wih he ewon-raphson ieraion. A sandard ona searh algorihm independen of he formulaion is used for he deeion of ona beween previously separae meshes and for he appliaion of displaemen onsrains where ona was idenified. he models have been implemened ino a version of he ompuaional finie elemen program PAK [3]. umerial examples ha illusrae performane of he desribed proedures are given. Key words: ona problem, penaly mehod, Lagrange mehod, oulomb s law. Inroduion ona mehanis has is appliaion in many engineering problems. he ineraion beween soil and foundaions in ivil engineering, general bearing problems as well as bol and srew joins in mehanial engineering, are examples of small deformaion ona problems. On onrary, he impa of ars, ar ire-road ineraion and meal forming are large deformaion ona problems. Here, nonlinear maerial laws, damage, dynami faigue, friion, wear, e. mus be aen ino aoun o design opimal omponens and assemblies. Effeive appliaion of finie elemen ona solvers demands a high degree of experiene sine he general robusness and sabiliy anno be guaraneed. For his reason he developmen of more effiien, fas and sabile finie elemen ona disreizaions is sill a ho opi, espeially due o he fa ha engineering appliaions beome more and more omplex. he aim of his paper is o provide a framewor for ona problems wih friion, based on he penaly [4-8] and he Lagrange muliplier mehod [,2]. he Lagrange muliplier mehod provides exa soluions bu have addiional degrees of freedom. he penaly formulaion is purely geomerially based and herefore no addiional degrees of freedom need be aivaed or inaivaed, bu he soluion is dependen on he inrodued penaly faor. umerial examples

2 Journal of he Serbian Soiey for ompuaional Mehanis / Vol. / o., are shown o demonsrae omparison of he presened algorihms when applied o ona problems. 2. Formulaion of he muli-body friional ona problem A ona beween wo deformable bodies is onsidered. As he onfiguraion of wo bodies oming ino he ona is no a priori nown, he ona represens a nonlinear problem even when he oninuum behaves as a linear elasi maerial. 2. ona inemais wo bodies are onsidered: () B and B, Fig.. We will denoe he ona surfae Γ as he par of he body B (i) suh ha all maerial poins where ona may our a any ime are inluded. Using a sandard noaion in ona mehanis we will assign o eah pair of ona () surfaes involved in he problem as slave and maser surfaes. In pariular, le Γ is aen o be he slave surfae and Γ is he maser surfae. he ondiion whih mus be saisfied is ha any slave parile anno penerae he maser surfae. Le x be he projeion poin of he urren posiion of he slave node x ono urren posiion of he maser surfae Γ, defined as () i x x x x 2 ( ξ, ξ ) 2 ( ξ, ξ ) a = 2 ( ξ, ξ ) 0 () where =, 2 and 2 a (, ) are he angen ovarian base veors a he poin. ξ ξ he definiion of he projeion poin allows us o define he disane beween any slave node and he maser surfae. he normal gap or he peneraion g for slave node is defined as he disane beween urren posiions of his node wih respe o he maser surfae Γ g = ( x x) n where n refers o he normal o he maser fae Γ a poin x (Fig. ). ormal o be defined using angen veors a he poin x is a a2 n = (3) a a 2

3 76 S. Vulovi e al.: A omparaive Sudy of ona Problems Soluion Fig.. Geomery of a 3D node-o-segmen ona elemen. his gap gives he non-peneraion ondiions as follows: g = 0 perfe ona; g > 0 no ona; g < 0 peneraion (4) If he analyzed problem is friionless, he funion (4) ompleely defines he ona inemais. However, if he friion is modeled, angenial relaive displaemen mus be inrodued. In his ase he sliding pah of he node x over he ona surfae Γ is desribed by oal angenial relaive displaemen as wihin a ime inerval from 0 o. g = & d = d = a d (5) β g & ξ a & ξ & ξ β he ime derivaives of parameer ξ in equaion (6) an be ompued from he relaion (), [8]. In he geomerially linear ase we obain he following resul: & ξ = x & x& a = & (6) β aβ g where a β = a a β is he meri ensor a poin x of he maser surfae Γ. From he equaions (5) and (6) we an dedue he relaive angenial veloiy a he ona poin 2.2 onsiuive equaions for ona inerfae g& = & ξ a = g& a (7) For mahemaial and ompuaional modeling he surfae haraerisis have o be pu ino he onsiuive inerfae onsrain. A ona sress veor wih respe o he urren ona inerfae a normal and angenial pars, Γ an be spli ino = + = a (8)

4 Journal of he Serbian Soiey for ompuaional Mehanis / Vol. / o., where a is he onravarian base veor. he sress as on boh surfaes obeying he aionreaion priniple: ( ξ, ξ ) = a he ona poin x. he angenial sress is equal o 2 zero in he ase of friionless ona. When he ona ours, one has he ondiion < 0. If here is no peneraion beween he bodies, hen he relaions g > 0 and = 0 hold. his leads o he saemens g 0, 0, g = 0 (9) whih are nown as Kuhn-uer ondiions. In he angenial direion a disinion is made beween si and slip. As long as no sliding beween o bodies ours, he angenial relaive veloiy is equal o zero. If he veloiy is zero, also he angenial relaive displaemen (5) is zero. his sae is alled he si ase wih he following resriion: g& = 0 g = 0 (0) A relaive movemen beween wo bodies ours if he sai friion resisane is overome and he loading is large enough suh ha he sliding proess an be ep. herefore, he relaive sliding veloiy, wih respe o he sliding displaemen, is in he opposie direion o he friion fore. Wih his, he angenial sress veor is resried as follows: sl sl = μ sl g& where μ is he friion oeffiien. In he simples form of oulomb s law (), μ is onsan and no disinion is made beween sai and sliding friion. Afer he inroduion of he si and slip onsrains, one needs an indiaor o deide wheher si or slip aually aes plae. herefore, an indiaor funion g& () f = -μ is evaluaed, whih respe o he oulomb s model for friional inerfae law. In equaion β he firs erm is = a. hen he following ona saes an be disinguished: β f -μ 0 Si = - μ >0 Slip (3) Using he penaly mehod for normal sress, he onsiuive equaion an be formulaed as = ε g (4) where ε is he normal penaly parameer. he angenial par is differen for he si and for he slip ases. For he si, a simple linear onsiuive model an be used o desribe he angenial sress = ε g (5) si where ε is he angenial penaly parameer. For he slip, he angenial sress is given by he onsiuive law for friional sliding (). A baward Euler inegraion sheme and reurn mapping sraegy are employed o inegrae he friion equaions. If a sae of si is

5 78 S. Vulovi e al.: A omparaive Sudy of ona Problems Soluion assumed, he rial values of he angenial ona pressure veor, and he indiaor funion f a load sep an be expressed in erms of heir values a load sep n, as follows he reurn mapping is ompleed by = + ε Δ g = + ε a Δ ξ (6) rial β n n β f = μ (7) rial rial rial = rial μ n if f 0 if f > 0 (8) wih = (9) rial rial rial n he penaly mehod an be illusraed as a group of linear elasi springs ha fore he body ba o he ona surfae when overlapping or sliding ours. 2.3 Equilibrium equaion for bodies in ona When wo bodies a ime are in ona, he priniple of virual wor an be wrien as (for a deailed legend of he symbols see [8]) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) σ : gradδu dv ρ b u&& δu dv σ n δu da = 0 (20) σ 2 ( ) = ( ) ( ) ( ) V V S where is ona onribuion. For he Lagrange muliplier mehod for ona wih friion, he ona onribuion are formulaed for si as ( λδ δ ) = g + λ g da S and for ase of sliding i is ( λδ δ ) = g + g da (22) S where δ g and δ g are he variaions of gap and angenial displaemen; λ and λ are normal and angenial Lagrange mulipliers and is angenial sress veor whih is deermined from he onsiuive law for friional slip. oe ha he Lagrange muliplier λ an be idenified as he ona sress. ona onribuion for he penaly mehod are formulaed as follow ( ε δ δ ) = g g + g da (23) S

6 Journal of he Serbian Soiey for ompuaional Mehanis / Vol. / o., Finie elemen formulaion 3. Finie elemen formulaion of friional ona he virual wor of boundary nodes whih are in ona is formulaed for a slave node : = F δg + F δg = Aδg + Aδg = Aδg + Aδξ = δu F (24) Here, he quaniies are: F = A he normal fore; F = A he angenial fore [8]; A he area of he ona elemen; F he ona fore veor. For he penaly mehod we define a displaemen veor for he five-node ona elemens (,, 2, 3, 4) and he veors n H n = H 2n H 3n H 4n { 2 3 4} δu = δu δu δu δu δu (25) aβ H a β H β = 2aβ H3a β H 4aβ β D = a (26) hus, he ona fore veor an be expressed by (26) for he slave node whih is in ona, by F = F + F D (27) he ona fores F and F in (27) an be obained by muliplying he onsiuive inerfaes laws (5), (6) and (8) by he area of he ona elemen A. β 3.3 Algorihm for friional ona In order o apply ewon s mehod for he soluion of nonlinear sysem of he equilibrium equaion (20), a linearizaion of he ona onribuions is neessary. he linearizaion of he equaion (25), for he infiniesimal heory, gives Δ δg +Δ δξ = δukδu (28) where K is he ona siffness marix of ona elemen. I is assumed ha he ona area A is no hanging signifianly so he area A is onained wihin he penaly parameers. he angen siffness marix for he normal ona is K = ε (29) Analogous o (29) we obain he symmeri angen siffness marix for si ondiion, K = ε D D (30) si a β β For he slip ondiion, he angen siffness marix is

7 80 S. Vulovi e al.: A omparaive Sudy of ona Problems Soluion K με g = με n D + ε a δ n n D D slip rial β rial rial β γ rial βγ (3) he seond erm in he angen marix is non-symmeri. his is beause he oulomb s of friion an be viewed as a non-assoiaive onsiuive equaion. Friional ona algorihm using penaly mehod is shown in able. LOOP over all ona segmen (he for ona ) IF g 0 HE (he firs ieraion) IF i= HE se all aive nodes o sae si, si (5), ompue marix K (30) ELSE rial rial ompue rial sae: n + (6) and f (7) rial IF f 0 HE rial si =, ompue marix K (30) GO O (a) ELSE rial slip = μ n, ompue marix K (3) EDIF EDIF EDIF (a) ED LOOP able. Friional ona algorihm using he penaly mehod he linearizaion of he equaions and (22) gives he siffness marix for Lagrange muliplier mehod Δ δg +Δ δξ = δukδu (32) λ Deailed desripion of he Lagrange muliplier mehod ona siffness marix is given in referene []. Finally, we obain he global nonlinear finie elemen equaion for he penaly mehod as and for Lagrange muliplier mehod, [ ] () MU&& + K + K U = F F (33) M 0 K 0 0 K ( ) λ ΔU F F λ + + = Δ 0 Kλ λ gn (34) where: M is mass marix; K is siffness marix and veor F () orresponds o an exernal fore. he ona fore veor for he 3D ona elemens for he Lagrange muliplier mehod is = λ [ H H H H ] F λ λ λ λ λ (35) 2 3 4

8 Journal of he Serbian Soiey for ompuaional Mehanis / Vol. / o., Examples 4. ompression of a ylinder ompression of a ylinder beween wo parallel plaes is onsidered. Iniial dimensions of he ylinder are: radius r = 6.35mm, heigh h= 2r. Elaso-plasi maerial model wih following yield urve is used δ e ( )( e ) σ = σ + σ σ + Hδe (36) p y y0 y y0 p Maerial onsans are: E = GPa, ν = 0.38, K = GPa, G = GPa, σ y0 = 0.45 GPa, σ y = 0.75 GPa, δ = 6.96 GPa, H = GPa. Due o symmery, one-eigh of he ylinder is modeled, wih symmery ondiions for nodes lying in oordinae planes. I is assumed ha here is no friion, and he soluions are obained using ona elemen based on he Lagrange muliplier (see []) and penaly mehods. Fore [] Lagrange 0E 00E 000E Displaemen [mm] Fig. 2. Fore displaemen relaionship. Deformaion of he ylinder is inreased by presribed displaemen a he plae. Soluion is obained by 25 seps of displaemen inremens equal o 0.2 mm; and by full-ewon ieraion mehod wih line searh. he hree differen values for normal penaly parameer are onsidered: a) ε = 0 E ; b) ε = 00 E and ) ε = 000 E. he fore - displaemen diagram is shown in Fig. 2. In his example, a penaly number whih is hosen have o be a leas 00 imes larger hen E, for good approximaion of he normal fore. I is obvious ha he value of he penaly parameer has he effe on auray of he resuls in ona problems. Iniial and deformed onfiguraions a he final sep are shown in Fig. 3. Fig. 3. Iniial and final deformed onfiguraions.

9 82 S. Vulovi e al.: A omparaive Sudy of ona Problems Soluion 4.2 he ona beween an elasi ring and a foundaion An elasi ring onsiss of an ouer and inner rings of he same hiness = 5 UL wih differen maerials. he geomery and maerial parameers are given in Fig. 4. A oal downward displaemen of u = 40 UL is applied o he ring a is op end in 80 seps. he 5 ompuaion is performed for boh Lagrange and penaly formulaions ( ε = 0 UL ). Fig. 4. he elasi ring wih he foundaion. Inner ring: E=0 5 UF/UL 2 ν=0.3 Ouer ring: E=0 3 UF/UL 2 ν=0.3 Foundaion: E=0 8 UF/UL 2 ν=0.0 Fig. 5. Verial sress field and deformaion onfiguraion, lef panel: he Lagrange muliplier formulaion; righ panel: he penaly formulaion (upper figures sep 44; lower figures sep 80).

10 Journal of he Serbian Soiey for ompuaional Mehanis / Vol. / o., he foundaion onsiss of 52x0 four-node elemens. he ring is disreized wih 64x2 elemens. In he final sae, he ring has a lif off in he middle. he Lagrange muliplier and he penaly formulaions show good agreemen of resuls. Verial sress field for boh formulaions are shown in Fig. 5, for sep 44 sep onlusions In he paper a model for hree-dimensional ona problem wih friion based on he penaly and Lagrange muliplier mehod was desribed. Due o he inrinsi similariy beween friion and he lassial elaso-plasiiy [9,0], he onsiuive model for friion an be onsrued following he same formalism as in lassial elaso-plasiiy. Using he penaly mehod, he ompuaion ime is smaller bu he resuls are srongly dependen on he value of he penaly faor. he Lagrange muliplier mehod leads o exa soluion bu wih more ieraions and signifian inrease of he number of degrees of freedom, i.e. equaions, and hus redues ompuaional effiieny. he numerial examples indiae a possibiliy of easy omparaive simulaneous use of boh proedures in he analysis of finie deformaion problems wihin he same ompuer ode. Referenes [] Grujovi., umerial soluion of ona problems, Monograph, Fauly of Meh. Eng. Univ. of Kragujeva, Kragujeva, [2] Slavovi R., M. Zivovi, M. Koji,. Grujovi, Large srain elasoplasi analysis using inompaible displaemens, in: XXII Yugoslavian ongress of he heoreial and applied mehanis, June 2-7, Vrnjaa banja, 997. [3] Koji M., R. Slavovi, M. Zivovi,. Grujovi, he sofware paages PAK, Fauly of Mehanial Engineering of Kragujeva, Serbia and Monenegro. [4] Fisher K.A., Moral ype mehods applied o nonlinear ona mehanis, Ph.D. hesis, Insiu für Bumehani und umerishe Mehani Univ. of Hannover, Hannover, [5] Laursen.A., J.. Simo, A oninuum-based finie elemen formulaion for he implii soluion of mulibody, large deformaion friional ona problems, Iner. J. um. Meh. Eng , 993. [6] Peri Đ., R.J. Owen, ompuaional model for 3-D ona problems wih friion based on he penaly mehod, Iner. J. um. Meh. Eng , 992. [7] Wriggers P.,.V. Van, E. Sein, Finie elemen formulaion of large deformaion impa-ona problems wih friion, ompuers and Sruures , 990. [8] Wriggers P., ompuaional ona Mehanis, J. Wiley & Sons Ld, Wes Sussex, England, [9] Koji M., K. J. Bahe, Inelasi Analysis of Solids and Sruures, Springer, Berlin- Heidelberg, [0] Zivovi M., onlinear sruural analysis, Monograph, Fauly of Meh. Eng. Univ. of Kragujeva, Kragujeva, 2006.